Generator Matching: Generative modeling with arbitrary Markov processes
Authors: Peter Holderrieth
Marton Havasi
Jason Yim
Neta Shaul
Itai Gat
Tommi Jaakkola
Brian Karrer
Ricky T. Q. Chen
Yaron Lipman
Venue: ICLR 2025 Oral (2025)
Keywords: Flow matching
Markov process
Diffusion model
Generative Modeling
Primary area: generative models
TL;DR: The core principles of flow matching can be vastly generalized to practically all continuous-time Markov processes using Markov generators, unifying all previous methods and opening the door to new generative models agnostic to data modality.

Abstract: We introduce Generator Matching, a modality-agnostic framework for generative modeling using arbitrary Markov processes. Generators characterize the infinitesimal evolution of a Markov process, which we leverage for generative modeling in a similar vein to flow matching: we construct conditional generators which generate single data points, then learn to approximate the marginal generator which generates the full data distribution. We show that Generator Matching unifies various generative modeling methods, including diffusion models, flow matching and discrete diffusion models. Furthermore, it expands the design space to new and unexplored Markov processes such as jump processes. Finally, Generator Matching enables the construction of superpositions of Markov generative models and enables the construction of multimodal models in a rigorous manner. We empirically validate our method on image and multimodal generation, e.g. showing that superposition with a jump process improves performance.

Published as a conference paper ICLR 2025
GENERATOR MATCHING: GENERATIVE MODELING WITH ARBITRARY MARKOV PROCESSES
Peter Holderrieth Marton Havasi", Jason Yim', Neta Shaul2.3, Itai Gat?,
Tommi Jaakkola', Brian Karrer", Ricky T. Q. Chen", Yaron Lipman" MIT CSAIL, FFIRR, Meta, 'Wizzannn Institute of Science Work done during an internship at FAIR, Meta.
ABSTRACT
We introduce Generator Matching, modality framework for generative modeling using arbitrary Markov processes. Generators characterize the infinitesimal evolution of Markov process, which we leverage for generative modeling in similar vein to flow matching: we construct conditional generators which generate single data points, then learn to approximate the marginal generator which generates the full data distribution. We show that Generator Matching unifies various generative modeling methods, including diffusion models, flow matching and discrete diffusion models. Furthermore, it expands the design space to new and unexplored Markov processes such as jump processes. Finally, Generator Matching enables the construction of superpositions of Markov generative models and enables the construction of multimodal models in a rigorous manner. We empirically validate our method on image and multimodal generation, e.g. showing that superposition with a jump process improves performance.
1 INTRODUCTION
Early deep generative models-like VAEs (Kngmma 2013) and GANs (Goodfellow et al. 2014) generated samples in a single forward pass. With denoising diffusion models (DDMs) (Song et al. 2020 Ho et al. 2020), paradigm shift happened were step-wise updates used transform noise into data. Similarly, scalable training of continuous normalizing flows (CNFs; CCen et al. 2018) via flow matching (Lipman al. 2022 Liu et al. 2022] Allergg et al. 2023) allowed for high-quality and fast generative modeling by simulating an ODE Since then, similar constructions based on diffusion and flows have also been applied to other modalities such as discrete data (Campbell et al., 2022: Gat et al. 2024) or data on manifolds (De Bortoli Huang et al.) 2022) Chen & Lipman 2024) leading to variety of models for different data types.
et all 2022]
The single common property of the aforementioned generative models is their iterative step-wise nature: starting with sample Xn ~ Psimple from an casy-to-sample distribution Psimple. they iteratively construct samples X++a of the next time step depending only on the current state X,. Mathematically speaking, this means that they are all Markov processes. In this work, we develop generative modeling framework that relies on that Markov property. At the core of our framework is the concept of generator that describes the infinitesimal change of the distribution of a Markov process. We show that one can easily learn generator through a family of scalable training objectives- a framework we coin Generator Matching (GM).
Generator Matching unifies many existing generative modeling techniques across data modalities such as denoising diffusion models (Song et al. 2020). flow matching (Lipman et al., 2022), stochastic interpolants (Albergo al. 2023), , discrete diffusion models (Camphell et al. [022]] Gat al. 2024 Lou al., 2024a), among many others (see sec. (8). Most importantly, GM gives rise to new, unexplored models, and allows us combine models across different classes of Markov processes. We make the following contributions:

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Space
Markov Pr.
Cond. Probability Path
Generator Matching
Algorithm 1 Generator Matching
Images
Row
Notse
P. (drrja)
Cond. generator
recipe for constructing Markov gen- erative model (theory in black, imple- mentation in brown)
Solve KFE
Falid
Jump
Training target
Step O: Choose prior Primple
Train
Step 1: Choose p: x)) such that
L(C.
Text
Offlaionn
Marginal Prob. Path
marginal po Psimple and p1 Pdata
CGM loss
Step 2: Find solution L; to KFE
Notes
Pr(dt)
Generator
Step 3: Choose Bregman div. loss
Mullimedal
CTMC
Simulate
C
Step 4: Construct neural net L
distib
Data
Neural network
Step 5: Minimize CGM loss using C
uton
Step 6: Sample using Algorithm 2.
Figure I: Overview of the Generator Matching (GM) framework to construct generative models. GM works on any state space (including multi-modal) and Markov processes. Flower image source; vecteezy. com
1. Generator Matching:
We present Generator Matching (GM), framework for generative modeling with Markov processes on arbitrary state spaces. This framework unifies diversity of prior generative modeling methods into a common framework that is modality-
2. Novel models: On discrete and Euclidean spaces, we universally characterize the space of Markovian generative models identifying jump models as an unexplored model class for Ted.
3. Model combinations: We show how GM allows to combine models in 2 ways: (1) Markov superpositions enable to construct ensembles of generative models; and (2) Multimodal generative models can be constructed by combining GM models built for single data modalities.
4. Experiments: On image and multimodal protein generation experiments, we show that jump models and Markov superpositions allow us achieve competitive results.
2 GENERATIVE MODELING VIA PROBABILITY PATHS
Let S be a state space. Important examples are S = Rd (e.g., images, vectors), S discrete (e.g., language), S a Riemannian manifold (e.g., geometric data) or their products for multimodal generation. In generative modeling, we are given samples Fi...., .NN Pasta from a distribution Pdata on S and our goal is to generate novel samples= z ~ Pdata- GM works for arbitrary distributions, in particular those that do not have densities (e.g., with discrete support). For general probability measures p, we use the notation p(dx) where "dr" is a symbolic expression denoting integration with respect to p in a variable x. If for a distribution p a density exists with respect to a reference measure y on S, we write de (x) for its density.
du
A fundamental paradigm of recent state-of-the-art generative models is that they prespecify transformation of simple distribution Psimple (e.g. Gaussian) into Pdata via probability paths. Specifically, conditional probability path is set of time-varying probability distributions (Pt (dx 2))os(g1 depending data point 2 E S. Together with the data distribution Pdata this induces a corresponding marginal probability path via the hierarchical sampling procedure:
2 Peata I ~ Pr (dxl22
2 ~ 2
(I)
Le. first sample a data point 2 ~ Pdata and then sample x ~ Pt 22) from the conditional path. As we will see, this makes training scalable. The conditional probability path is usually chosen such that (oxr)) 22) Psimple and pı (dx|2) = d_ where 6_ is the Dirac delta distribution at x (i.e. the trivial, deterministic distribution returning z every draw). The associated marginal probability path interpolates between Psimple and Pdata leading to the first design principle of GM:
Principle 1: Given data distribution Pista choose prior Psimple and a conditional probability path such that its marginal probability path (Pt)oses1 fulfills Psimple = Po and Pdata
P1-
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Two common constructions are mixtures (for arbitrary S) and geometric averages (for S = RR):: pc(dx)z) = (1 - -Ke) Psimple (dx) + Ki 5x(dx)
with prob
+ Ft ~
mixture
(2)
20
with prob (1 (1-kt)
Pc(dx]2) 2) = EEz [00.20-as2]
Ie 0,30 +
0x22
geometric average (3)
where 2t pe(-12),250 Psimple; 2 Pduaa and 0t ot Kig I2zo are differentiable functions satisfying
Ki = Cgp = 01 = and K1 = 01 = do = and 0 5 Ki I 1.
Remark. Note that the above includes specific ways of constructing probability paths, e.g. via forward process in diffusion models (Song et al., (020) or an interpolant function (Albergo et al. (023) (see app. A.4). GM works for all these cases including if one conditions on more general latent variables (Tong al. 2023
Pooladian et al. 2023). Note also that in the diffusion literature, time is inverted (1 = 0 corresponds to data).
3 MARKOV PROCESSES
We briefly define time-continuous Markov processes, fundamental concept in this work (Ethier & Kurtz 2009). For t E [0, IJ. let X₁ E S be a random variable. We call (Xe)oste1 Markov process if it fulfills the following condition for all 0 S t t2
tn < tn+1 S and A C S (measurable):
(4) Informally, the above condition says that the process has no memory. we know the present, knowing the past will not influence prediction of the future. In table D we give an overview over important classes of Markov processes. Each Markov process has a transition kernel (0+tee)))) that assigns every r S probability distribution (-1z) such that P[X++A~ A|X x] = K (Alz). Due to the Markov assumption, there is 1:1 correspondence between a Markov process and a transition kernel paired with an initial distribution Po- We impose loose regularity assumptions on X, listed app.
PIX E X _t, x P(X 1.4 E x
Markov assumption
Given a marginal path (pe (p.(dr))osesi (see sec. D. we want to train a model that allows to simulate Markov process such that Xo ~ Psimple + Xx ~ Pt for all 0 < & < 1. That is, if initializing state at t = 0 with Xo Po, the marginals of X, will be Pt for all 0 S t S 1. Once we have found such a Markov process, we can simply generate samples from Pi = Pdata by sampling Xo Po and simulating Xeth
In the context of GM, we use a Markov process as follows:
(-)x))(-(Xx,) step-wiss up to time t = 1. The challenge with such an approach is that an arbitrary general kernel ki is hard to parameterize in a neural network. One of the key insights in the development of diffusion models was that for small h 0, the kernel kithle can be closely approximated by a simple parametric distribution like Gaussians (Soht-Dicksteii et al. 2015
Ho et al. 2020). One can extend this idea to Markov processes leading the concept of the generator.
4
GENERATORS
Let us consider the transition kernel bej for small h > O. Specifically, we consider an informal Ist-order Taylor approximation in f with an error term o(h):
"he = Kuje + lL,, + o(h)", L. d.
dh h==0 Retblee kejo(-lar) = 6.z
(5)
We call the 1st-order derivative L, the generator of ketble (Ethier & Kurtz 2009; Rüschendorf et al. 2016). Similar to derivatives, generators are first-order linear approximations and, as we will see, easier to parameterize than kz- Diffusion, flow, and other generative models can all be seen as algorithms to leam the generator of a Markov process (see table Tp. However, as probability measure is not a standard function, equation 5 is not well-defined yet. We will make it rigorous using test functions.
Test functions. Test functions are a way to "probe" a probability distribution. They serve as a theoretical tool to handle distributions if they were real-valued functions. Specifically, use a family T of bounded,
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Continous-time
Name
Diffusion
Jump process
Markov chain
Space 9
= E
S = Rd
8 arbitrary
|||
(a)) Raddd
Q E RS52, 12Q, - Q:(1(:x) 2 0 A.))
Parameters
Me(x) R
Jump measure Q:idy:.)
of pos. semii def.
Xess X, + (Koee 4 N(0.))
Xeik N2 proh. & (dg::)
Sampling
Xeth X1 (X.)
Xets (I (+)
Xy-A
with prob. & (e(ogy
Generator C
Vffuu
[1 - (xy)]Q. x)
~
KFE
Continuity Equation:
Fokker Planck Equuttio:
Junp Comtinuity Equation: d (2)
Mass preservation: 0g. Qup.
(Adjoint)
Oeps = - lusil
0ge. 1 (poof]
[Q(x;)) (x) Qu(2%an) 2 (x)e(dr)
Marginal
Ba Sanoualll (e
CGM
[+H())=)) (n0))
(Example)
(og
Table I: Examples of Markov models that can be learnt with GM. Derivations are in app. we assume zero drift. KFE is listed its adjoint version, i.c. assumes jump kernel Q du
A.5
For diffusion,
x) and density
(r) exists with respect to reference v. For Lebesgue measure v, we write de: dv
du (a) pc()).
integrable functions ʃ S + R that characterize probability distributions fully, i.e., two probability distributions μ1; /22 are equal if and only Expps (xr) = -Er-pa [x)) for all f E Generally speaking, one chooses to be as "nice" (or regular) as possible. For example, if S Rd. the space = C of infinitely differentiable functions with compact support fulfills that property. We define the action of the marginal pt and transition kernels ketaje for all f E T via the linear function defined via
(pc. def [ f(x)pulde)) = Exup, [f(x)]
marginal action
(6)
(kethite f) (x) def
transition action
(7)
where the marginal action maps each test function I to scalar (pt, E R. while the transition action maps a realevalued function x + f(x) to a another real-valued function x ++ (ke+hhe: (z). The tower property implies that (pe, (ketkte, f)) = (pe+h, f). We note that the above is only a "symbolic" dot product but becomes a "proper" dot product if density de. exists, i.c (p:. = f(z) des (r))(dx))
Generator definition. Let revisit equation 5 and define the derivative of ky41/e- With the test function perspective in mind, we can take derivatives of (ke+njes f) (x) per x E S and define
d I (ketnie (x) lim h-0
(ketalee f) (x) f(i) del (c.f((x).
dh I-o
h
We call this action the generator Le (and define it for all f for which the limit exists uniformly in and t, see app. I.. In table there are several examples of generators listed with derivations in app. A.5 With this definition, the Taylor series in equation 5 has the, now well-defined, form as (ka+Aa f) = hLef + o(h).
Under mild regularity assumptions, there is a unique correspondence between the generator and the Markov process (see (Ethier & Kurtz 2009 Pazy 2012 This allows us parameterize Markov process:
Principle 2: Parameterize a Markov process via parameterized generator C².
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Of course, is hard to parameterize a linear operator L, on function spaces directly via a neural network. simple solution is to restrict ourselves to certain subclasses of Markov processes and parameterize it linearly with neural network (see app. A for details and examples). For example, flow matching restricts itself generators of the form L. - Ví(z) T ug (3:) which correspond to flows. However, as we will show now, we can in fact fully characterize generators on specific spaces.
Theorem (Universal characterization of generators). Under regularity assumptions (see app. generators of a Markov processes X, (0 < t S 1) take the form:
A.2 the
1. Discrete |S| < oc: The generator is given by a rate transition matrix Qs and the Markov process corresponds contnuous-time Markov chain (CTMC).
2. Euclidean space S = Rd; The generator has a representation as a sum of components described in table 7 i.e..
L.f(x) ff()" 1 (
/ |15G) f(x) Qx(yy;))
(9)
flow
diffuzion
jump
where u [0, x Rdd + Rdd is a velocity field, a : [0. 1] x Rd + SJ+ the diffusion coefficient (S =positive semi-definite matrices), and Qz(Alzr) is a finite measure called jump measure. describes the Frobenius inner product.
2 f(x) describes the Hessian of f and (2) of
The proof adapts known result in the mathematical literature and can be found in app. This result allows us to not only characterize a wide class of Markov process models but to characterize the design space exhaustively for S IRd or S discrete. In Euclidean space, people have considered learning the flow parts of the generator and for diffusion models, using fixed for diffusion. Learning or jump models on non-discrete spaces have not (or rarely) been considered. A general recipe to sample from Markov process with universal generator is presented in alg. 2 For S RR we can therefore simplify Principle 2:
Principle 2 (S ) Parameterize Markov process (e.g., using a neural network) via a generator L, that is composed of (a subset of) velocity ue, diffusion coefficient of, and jump measure Q..
5 KOLMOGOROV FORWARD EQUATION AND MARGINAL GENERATOR
Beyond parameterizing a Markov process, the generator has a further use-case in the Generator Matching framework: checking if a Markov process generates desired probability path pt. We discuss the latter now using the Kolmogorov Forward Equation (KFE). Specifically, the evolution of the marginal probabilities pe of Markov process X, are governed by the generator Lt, can be seen by computing:
d
( Pc: dh h=o (bethiss
d
O (pr, f) = dh 1==0 (ptth, =
(pc. Lef)
(10)
where we used that the (pt,)) operation is linear to swap the derivative, and the fact that (pt, (ketk)t f)) (pt+ho f). This shows that given a generator L, of a Markov process X, we can recover its marginal probabilities via their infinitesimal change,
o (pr, f) (pr, L.f)
Kolmogorov Forward Equation (KFE)
(11)
Conversely, if a generator L, of a Markov process X, satisfies the above equation, then X, generates the probability path (pt)0 i.e. initializing Xo Po will imply that X, ~ pt for all 0 S l (see app. A.2 (Rogers& & Williams 2000). Therefore, the key challenge of Generator Matching is:
Principle 3*; Given marginal probability path (pr)ose<1, find a generator satisfying the KFE,
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Remark Adjoint KFE. We note that the above version of the KFE determines the evolution of expectations of test functions/ f. Whenever probability density dp: for examples and app. A.3). In this form, the KFE generalizes many equations used to develop generative models such as the Fokker-Planck or the continuity equation (Song al. 2020 Lipman et al. 2022
du (a) exists, one can use the adjoint KFE (see table |
We now show how to find generator that generates a marginal probability path Pt with conditional path Pe(-12). Assume that for every data point E 5, we found generator L that generates p-(-12). We call C conditional generator. This allows to construct generator for the marginal path (marginal generator):
Proposition 1.
The marginal probability path (Pt)oseg1 is generated by a Markov process X; with generator
L.f(2) (z) =Br-r (e) [=) [f] fx))
(12)
where Prie(dz(a) the posterior distribution (i.e. the conditional distribution over data 2 given an observation r). For S = Rd and the representation in eq.
9). we get marginal representation of (xf(x) given by:
V fxx( (o)
V (x)) Expppe (x<[=]]]]]]=]]
lo + -
f(x)] Berpu (ddy)))
2
Generally, an identity as in eq.
12 holds for any linear parameterization of the generator (see app. A.6
The proof relies on the linearity of the KFE (see app. Proposition I Jimmensely simplifies the construction of Markov process that generates desired probability path. To find the right training target, we only need to find a solution for the KFE for the conditional path. This simplifies Principle 3 to:
Principle 3: Derive a conditional generator L satisfying the KFE for the conditional path Pr Pr(-1z).
In denoising diffusion models, the strategy to find solutions to KFE is to construct probability path via a forward noising process and then use a time-reversal of that process as a solution to the KFE (we illustrate this in app. H.2 Here, illustrate two novel solutions for the KFE for common conditional probability paths on
Rd in fig. We discuss them here for d = (in sec. 7.2 it is discussed how to easily extend it to d > 1).
Example 1 Jump solution to geometric average. Current state-of-thee the--rt models use geometric average
probability path given by (+(2 = (tz, (1 -1-10 t)² called CondOT path (Lipman et al. 2022). We ask the question: there other Markov processes that follow the same probability path? As derived in app. E.7 another solution is given by jump model with rate kernel Q2 : R x R R:
(x)) (=k')
Qx(x':a/2) (1 t)3
ke(x) x2 (t 1)22 (1 (1-1) 1)² + tz
(13)
where [x]+ == max 0). In fig. 2 we illustrate how a jump model trained with this conditional rate has the same marginal probability path as common flow models but with significantly different sample paths.
Example 2 Pure diffusion solution to mixture path. GM allows to learn the diffusion coefficient of of an SDE. We illustrate this for the mixture path pe (dx]=) k,0= + (1 Me))Uiifaaaaa) We introduce a solution that we call "pure diffusion" (see app. E.3). The corresponding Markov process is given by an SDE with no drift (i.., vector field) and diffusion coefficient given by
of (x|2) - 2k, 1 K2 2 02 a
a1
a2)
(z - a1)² + 1 21+
(xr 1004-11---000 1(x=)1)²)
(14)
02 01
We add an additional reflection term at the boundaries of the data support (see app. feature of this model: It only specifies how much noise to add the current state. Still, it able to generate data (see fig. D). This is strictly different than "denoising diffusion models" because they corrupt data (as opposed to generating) with diffusion process and their (x) of is indeeinneeenddet and usually fixed.
E33 Note the striking
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[d] Mixture
(b) Geometric Average
Figure 2: Illustration of Markov models trained with different KFE solutions for the same probability path.
The paths for individual samples are plotted across time in one plot. 2d histograms of generated samples plotted per time point. Although the individual sample paths look very different, the marginal probability path (histogram) are the same up approximation error (Geometric average example 1, mixture example 2).
6 GENERATOR MATCHING
We now discuss how to train parameterized generator LJ to approximate the "true" marginal generator C.. practice, C is linearly parameterized by a neural network F. S x [0, - St where f C V is a convex subset of some vector space V with inner product (,,)) (see app. A.6 details). Our goal is approximate the ground truth parameterization F; S x [0. + Ω of L. For example, F; = ue for flows, F, = of for diffusion, F. Q: for jumps (see table D. We train the neural network F approximate F.. As distance function on S2, we consider Bregman divergences defined via convex function Φ S + R as
D(a, <(a) [o(b) + (a b, Vo(b))], a, b Ω
(15)
which are a general class of loss functions including many examples such as MSE or the KL-divergence (see app. C.3.1). 1). We use D to measure how well F approximates F; via the Generator Matching loss defined as
Lgm (0) def
Ee-Unif,
F.
Generator Matching
(16)
Unfortunately, the above training objective is intractable as we do not know the marginal generator L, and also no parameterization F; of the marginal generator. To make training tractable, let us set F7 to be a linear parameterization of the conditional generator & with data point & (see app. A.6). For clarity, we reiterate that by construction, Fe is unknown. By proposition we can assume that F; has the shape (x) = F? (z) (d)))))d))))) This enables us define the conditional Generator Matching loss
know F F7 p=(+)2), D as well
draw data samples Pdata but the shape of
Conditional Generator Matching (17) This objective is tractable and scalable. It turns out that we can use it to minimize the desired objective.
Legm (0) def ptt.
[D(FF()) E (x))
Proposition 2. For any Bregman divergence, the GM loss Lgm has the same gradients as the CGM loss Legms i.e. VoLgm (0) = (o)ogm(O). Therefore, minimizing the CGM loss with Stochastic Gradient Descent will also minimize the GM loss. Further, for this property to hold, D must necessarily be Bregman divergence.
See app. for a proof. Note the significance of proposition 2 can learn La with scalable objective. Further, we can universally characterize the space of loss functions, including unexplored and new loss functions for diffusion models. In [ we list examples of several CGM loss functions. Often it also possible to derive losses that give lower bounds on the model log-likelihood (ELBO bounds, see app. D).
Principle 4: Train C by minimizing the CGM loss with Bregman divergence.
With this, we arrived at the last principle of GM. In alg. I we summarize the Generator Matching recipe for constructing generative models.
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7
APPLICATIONS OF GENERATIVE MATCHING THEORY
GM provides a unifying framework for many existing generative models (see sec. B). as well as gives rise new models. Beyond that, the generality of GM in itself has several use cases that we discuss in this section.
7.1
COMBINING MODELS
The generator is a linear operator and the KFE 0r properties enable us to combine generative models for the same state space S in different ways.
is a linear equation. These two
Proposition 3 (Combining models), Let Pt. be a marginal probability path, then the following generators solve the KFE for pt and consequently define a generative model with pt as marginal:
1. Markov superposition: a. where Lt, C2 are two generators of Markov processes solving the KFE for pt, and oi, of 2 0 satisfy of + a = 1. We call this Markov superposition.
2. Divergence-free components: L.+ B, where Liiv is a generator such that (pt, L1 f) 0 0.for all f E T, and 3t 2 O. We call such Livr divergence-free.
3. Predictor-corrector: a² Ci + af where Ls is generator solving the KFE,fo in forwad--imme and L. is a generator solving the KFF backward time, and o of 2 0 with af of =
proof can be found in app. C.4 Markov superpositions can be used to combine generative models of different classes, e.g., one could combine a flow and a jump model. These can be 2 networks trained separately or we can train two models in one network simultaneously. We illustrate Markov superpositions in fig. To find divergence-free free components, one can use existing Markov-Char Monte-Cart (MCMC) algorithms- such as Hamiltonian Monte Carlo, Langevin dynamics, or approaches based on detailed balance all of these algorithms are general recipes to find divergence- components.
7.2 MULTIMODAL AND HIGH-DIMENSIONE GENERATIVE MODELING
GM allows us to easily combine generative models from two state spaces S1, S2 into the product space S1 x S2 in a rigorous, principled, and simple manner. This has two advantages: (1) we can design a joint multi-modal generative model easily and (2) we can often reduce solving the KFE high dimensions to the one-dimensional case. We state here the construction informally and provide a rigorous treatment in app. C.5
Proposition 4 (Multimodal generative models- Informal version). Let of (-121), o 222)) be two conditional probability paths on state spaces S1, S2. Define the conditional factorized path on S x S2 as p=(-121,22) at (-1z)) ) ( (-122). Let pe(dz) be its marginal path.
1. Conditional generator: To find a solution to the KFE for the conditional factorized path, we only have to find solutions to the KFE for each S1, S2. We can combine them component-wisg
2. Marginal generator:
The marginal generator of pe pr(dx) can be parameterized as follows: (1) parameterize a generator on each S₁ but make values depend on all dimensions; (2) During sampling, update each component independently one would do for each S; in the unimodal case.
3. Loss function: We can simply take the sum of loss functions for each S..
As conerete example, let consider joint image-text generation with joint flow and discrete Markov model with S = IR R S2 = {1...., Ji....,N], To build a multimodal model, we can simply make the vector field ue(x (1xx, 2x) E Rd depend on both modalities at 2 but update the flow part via X X1 + heu, X²). we give another example for jump models.
Similarly, the text updates depend on both (X) X²). In app.
8 RELATED WORK
GM unifies a diversity of previous generative modeling approaches. We discuss here a selection for S Rd and S discrete. App. inccllues extended overview and models for other S (e.g. manifolds, multimodal).
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Denoising Difffusion and Flows in Rd. From the perspective of GM, a "denoising diffusion model" is a flow model that is learnt using the CGM loss with the mean squared error. During sampling, divergence- free component given via Langevin dynamies (Flow SDE) can be be added for stochastic sampling (see proposition 3) If we set the weight of that component to O, we recover the probability flow ODE (Song et al. 2020). To the best of our knowledge, it has not been explored the literature yet whether one could learn a state-dependent diffusion coefficient or (4) as opposed to fixing it. Our framework allows for that we illustrate in fig. 2 Flow matching and rectified flows (Lipman et al. 2022 Liu et al. (022) are immediate instances of Generator Matching leveraging the flow-specific versions of the KFE given by the continuity
equation (see table Stochastic interpolants (Albergo et al. 2023) extend general flow-based models by
learning an additional divergence- freee Langevin dynamics component separately (see proposition (b)) and showcase the advantages of adding it both theoretically and practically.
Discrete models and LLMs. In discrete spaces, Generator Matching recovers generative modeling via continuous-time Markov chains (Campbell et al. 2022 Santos et al., 2023 Gat et al. 2024). often coined "discrete diffusion models". These models use version of proposition using factorized probability paths
make the generator (=rate matrix Q2) update each dimension independently. SEDD (Lou et al. 2024b) use the same Bregman divergence but with different linear parameterization of the generator, namely via the discrete score. Theoretically, by using auto-regressive probability paths and fixing the jump times via high jump intensities, one can also recover common language model training as an edge case of GM.
Markov generative modeling. The most closely-related work to ours is Benton et al. (2024) that focus on recovering existing denoising diffusion models into a common framework. Here, we try to fully characterize the design space of Markov generative models as a whole and identify novel parts e.g., by introducing jump models on Rd, Markov superpositions, universal characterizations the space of generators, novel solutions to the KFE, Bregman divergence losses as the natural loss classes, among others.
9 EXPERIMENTS
The design space of the GM framework is extraordinarily large. At the same time, single classes of models (e.g., diffusion and flows) have already been optimized over many previous works. Therefore, we choose focus on 3 aspects of GM: (i) Jump models as novel class of models (ii) The ability of combining different model classes into a single generative model (iii) The ability to design models for multiple data modalities.
New models- Jump model. We first study jump models as a novel model class in Eu- clidean space. We use the jump model de-
Method
CIFARIO
ImageNet
fined in eq. (13) and extend to multiple dimensions using proposition 4 The jump kernel is parameterized with a U-Net archi- tecture (see app. F for details). We use the loss from table I In app. [ we show that this corresponds an ELBO loss. We apply the model on CIFARIO 10 and the ImageNet32 (blurred faces) datasets. As jump models do not have yet an equivalent of classifier-f frreee guidance, we focus on unconditional gener- ation. A challenge for fair comparison is that flow models can use higher-order ODE samplers, while sampling for jump models in Rd is only done with Euler sampling so far. Hence, we ablate over this choice. As one high quality. In table we show quantitative results. While lacking behind current state-of-the art models, the jump model shows very promising results as first version of an unexplored class of models.
DDPM (Ho et al. 2020
3.17
6.99
VP-SDE (Song et 2020)
3.01
6.84
EDM (Karras et al. 2022)
1.98
Flow model (Euler)
2.94
4.58
Jump model (Euler)
4.23
7.66
Jump + Flow MS (Euler)
2.49
3.47
Flow model (2nd order)
2.48
3.59
Jump + Flow MS (mixed)
2.36
3.33
Table 2:
Experimental results for image generation. FID scores are listed. MS=Markov superposition. Euler: euler sampling. 2nd order: 2nd order ODE sampler. Mixed: Flow uses 2nd order sampler and jump uses Euler sampling.
in fig. the jump model can generate realistic images of
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Combining models- Markov superposition. Next, we train a flow and jump model in the same architecture. We validate that the flow part achieves the state-of-the-ar results as before. We then combine both models via a Markov superposition. As one can see in table Markov superposition of a flow and jump model boosts the performance of each other. For Euler sampling, we see significant improvements. We can also combine 2nd order samplers for flows with Euler sampling for jumps in a "mixed" sampling method (see table 2 leading to improvements of the current SOTA by flow models. We anticipate that with further improvements of the jump model, the increased performance via Markov superposition will be even more pronounced.
Multimodal state spaces Pro- tein experiments. GM allows to design models for arbitrary and complex state spaces. To il- lustrate this, we show that GM allows to easily improve Mulli- Flow, a state-of- of-the-art art model for joint protein structure and amino acid sequence generation (Camp- bell et all, 2024b), without even re-training the model. Specifically, we derive a novel solution L to the KFE on
Multimodal Div.
Unimodal Div.
Method
Nov.
Nov.
RFdiffusion Watson et al.
2023)
N/A
0.4
0.37
FrameFlow Yim et al. 2024)
N/A
0.39
0.39
FoldFlow (Bose et al. Protpardelle (Chu et al. 2024
(023)
N/A
0.24
0.32
0.1
0.4
0.12
0.41
ProteinGenerator (Lisanza el al.
2023)
0.09
0.31
0.19
0.35
MultiFlow (Campbell et al. 2024b) w/ SO(3) jumps (ours) w/ SO(3) jumps flow (ours)
0.38
0.39
0.52
0.39
0.48
0.41
0.63
0.41
0.47
0.4
0.59
0.40
SO(3) with a jump model (app. G.I We then use proposition to make it multi- dimensional and combine it with a flow model on IR.d and a discrete Markov model on
Table 3: Protein generation results. Diversity (Div) is the share of unique proteins passing a quality check called designability, Novelty (Nov) is the average inverse similarity of each protein passing designability.
n2 for n = 20 (# amino acids), i.e. the state space becomes S = Rdd x SO(3) {1...., 20 Using the pre-trained MultiFlow without any fine-tuning, we mareeuuuaaaii the conditional jumps by predicting E (3) and then taking a conditional step with CJ. In fig. we can see examples of generated proteins. We benchmark our results following Yim et all (2023a). 3
our results of incorporating jumps with MultiFlow compared to baselines. In multimodal setting. both sequence and structure state spaces are sampled jointly while in the unimodal setting only the structure is sampled. We see that including a jump model results in state-of-the-ar performance while greatly increases the diversity metric. See App. G.4 for more results and experiment details.
Additional small-scale experiments. We include a systematic study of the design space of GM on toy data in app. In app. [
we show that new Bregman divergences can improve existing flow and diffusion models.
10 DISCUSSION
We introduced Generator Matching, general framework for scalable generative modeling arbitrary state spaces via Markov generators. The generator abstraction offers key insights into the fundamental equations governing Markov generative models: Generators are linear operators, the KFE is linear equation, and Bregman divergences are linear in the training target. Therefore, any minimization we do conditionally on a data point, implicitly minimizes the training target marginalized across a distribution of data points. These principles allowed us to unify a diversity of prior generative modeling methods such as diffusion models, flow matching, or discrete diffusion models. Further, we could universally characterize the space of Markov models and loss functions. GM allows us to combine generative models of different classes on the same state space (Markov superpositions) and to easily build multimodal generative models. Future work could further explore the design space of GM. For example, we showed how one can learn diffusion coefficient o, of diffusion model. In addition, jump models on Euclidean space offer a large class of models that we could only study here in its simple instances. In addition, future work can explore better samplers or distillation. To conclude, Generator Matching provides both a rigorous theoretical foundation and opens up a large practical design space to advance generative modeling across diverse range of applications.
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ACKNOWLEDGMENTS
P.H. would like to thank Yann Ollivier for his insightful comments and for sharing his mathematical expertise throughout the project. Further, P.H. would like to thank Andreas Eberle for mathematical support and advice as well as for sharing exceleent teaching materials for Markov process theory. Further, we would like to thank Maurice Weiler, Ishaan Chandratreya, Alexander Sauer, and Krunoslav Lehman Pavasovie for extensive feedback on early drafts of the work. We also thank Anuroop Sriram and Daniel Severo for feedback and help along the way. Finally, would like thank Heli Ben-Hamu for sharing code to implement Mises-Fisher distributions.
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CIFAR10 Flow
ME
Insagehat522
ME
Figure 4: Examples of generated images on CIFARIO (top) and ImageNet32 (bottom).
Figure 3: Examples of generated proteins with SO(3) jumps model and MultiFlow, Each protein passes the designability filter check and is structurally unique.
Algorithm Generator Matching recipe for constructing Markov gen- crative model (theory in black, imple- mentation in brown)
Algorithm 2 Euler sampling for S = Rdd
Given: U1 Tt Qt. Psimplc step size h >> 0
Init: Xo ~ Psimple
1: for L in linspace 1, 1/h) do
Step 0: Choose prior Psimple
2: Jump intensity (xx)) = Q+ Xt)
Step 1: Choose (rdx 2) such that
3:
X ~ Qe(: (o));(,))))
marginal Po = Psimple and P1 = Pdata
Step 2: Find solution L to KFE
4:
Bernoulli
N(0,1)
5: X X, + hu Vhoi(Xi)a
Step Choose Bregman div. loss
Step 4: Construct neural net L
6: X mX m) X
Step 5: Minimize CGM loss using Li
7: end for
Step 6: Sample using Algorithm 2.
Return: X1
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A OVERVIEW OF MARKOV PROCESSES AND THEIR GENERATORS
A.1 SETUP AND DEFINITIONS
State space. Throughout this work, we assume (S, d) is Polish metric space, i.e., S set and there is a metric d: S x 5 + Rz 220 defined on S such that (S,d) is complete (i.e., any Cauchy sequence converges) and separable (i.e., has a countable dense subset). We endow with its Borel o-algebra (SS) and consider a set A c S measurable if A E (SS). Any function f S + R considered in this work is assumed to be measurable. Throughout this work, we assume that T is set of functions f S - I on S such two Ex [f]]] for all
T. We call elements in T test functions. Throughout this work, denote reference measure on S (e.g. the Lebesgue measure on Rdd or the count measure on discrete spaces). We write de measure p with respect to the reference measure V.
probability measures J1. F2 are equal if and only if 1e [==)
du (x) for a density of probability
Markov process. Let (2, F)) be probability space. A Markov process (Xt)osesi is a collection of integrable random variables X, + S such that
Markov assumption: P( A| ,,,...., Xt,] = P\X E e(,.,
(18)
in < tn+1 S 1, S S measurable We denote by (++1)((Al)) = B|X++A E A|x, = z] the transition kernel Xx.
for all 0 S t 12
(19)
Semigroup. We define the action of marginals and of transition kernels ke+nje on test functions f as in the main paper via:
(pr, (v. -
marginal action
(20)
(ke-h)e, f) (z) - f(y)keth)(dddll
= (Bf x]
transition action
(21)
where the marginal action maps each test function f to scalar (pt, f) R. while transition action maps real-valued function x (xx to another real-valued function + ())++())))))) The tower property implies that Pt (ke+ f) (pe+hs f). Considering kethle as a linear operator as above, we know by the Markov assumption and the tower property that there are two fundamental properties that hold:
kuje ld, S) (Dals (Paje {)), Ou<<i<sl
composition
(22)
|l (pajes f) lb blo, Ost<s $
contraction
(23)
where || Ilso describes the supremum norm.
A.1.1 DEFINITIONS FOR TIME HOMOGENEOUS MARKOV PROCESSES
A complication in the theory we develop here that we (need to) consider time-inhomogencous Markov processes, while most of the mathematical theory and literature resolves around time-homogeneoue Markov processes. Therefore, we first give the definitions for homogeneus Markov processes and then explain how this can be translated to the time-inhomogeneous case.
homogenogges Markov process is a Markov process X, such that ketNje = kyjo for all 4, 2 -.e. the evolution is constant in time. This implies the semigroup property
k-jo Id, kejo kejo kg+4j0
(24)
Let (S) be the space of continuous functions : - IR that vanish at infinity, i.e. for all c > there exists a compact set K C S such that [f()) for all x E S K.
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Feller process. We call X, a Feller process it holds that
1. Operators on continuous functions: For any f E Co(S) and 2 0, also ka E Co(S).
2. Strong continuity: For any f' E Co(S):
lim
|kjoff - flloo = 110
(25)
Generator.
We define the generator L of a Feller process X, follows: for any f E Co(5) such that the
limit
knjf- LJ
lim EJU
(26)
exists uniformly in S (i.e., in 11ss)) and define LJ as the limit above. We define the core D(C) all for which that limit exists. It holds that ((C) is dense subspace of Co(S) (Pazy 2012) and L is a linear operator.
A.1.2 DEFINITIONS FOR TIME-NIIMOGENEOUS MARKOV PROCESSES
For general time-inhomogeneouu Markov process X,, one can extend the definitions to a two-parameter semigroup, see (Rüschendorf et al. 2016) for example. Another approach to simply consider the corre- sponding time-homogeneous Markov process X, (Xe, () on extended state space S x [0, 1]. The transition kernel on extended state space is then given for A [0, 1] via:
Ph (A, (ar, ()) = (1+n)((A++alo))
Ar == (ul(, E A}
(27)
and the action is given via
Pnuf(xr = EIf(X h)]x, =x]
(28)
Feller process. The Feller assumption is equivalent to:
1. Operators on continuous functions: For any f E Co(S 1]) and 2 O, also ky of Co(S x [0, 1)).
2. Strong continuity: For any Co(S 10. 1)):
lim |iajjof - flloo =
(29)
where here the supremum norm || bl is taken across S and time [0, I].
Time-dependent generator. In the above case, we can reshape the generator via Bif(Xe-n, + h) f(Xe, x) x
Cf(x,t) lim
(30)
h100
Ef t h) f(Xe,t t h) f(Xo,t+h) f(X,,t)|XX x]]
lim
(31)
h
( (+ - 1(Xe, 1 11) = x
[(x L + h1) - -f(xx
lim
+ lim
(32)
hー0
h
h-00
h
[Bff + h) - f(X,, + h) = x]]
lim
+0,flz,t)
(33)
h
-:2(5((m)
=xff((x) + 0.f(2xt)
(34)
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where ft : S R, f(x,t) t) describes the restriction of on S and we defined the time-dependent generator L; as an operator acting spatial components, i.e. functions in Co(S), via
Lig(z) == lim ((g( - (Xx,)xx, xx]
(35)
for any g : S R such that the limit exists. Note that however that when we define the time-dependent generator LtJ', we also need to specify the direction operator è (i.e. in what direction of time it goes). This is always implicitly assumed. Further, we remark that in the above derivation, we have assumed that + f is continuous (in supremum norm) in for any function in the domain of Le. We will state this now again.
A.2 REGULARITY ASSUMPTIONS.
Throughout this work, make the following regularity assumptions:
1. Assumption semigroup: The Markov process X₁ is Feller process in the sense defined in app. A.1.2
2. Assumption on sample paths: In every time interval [s, t]. the expected number of discontinuities of - X, is finite.
3. Assumptions test functions:
There exists a subspace of test functions TT that is dense in Co(S) such that T C dom(C,) and the function & + L is continuous for any f E T. Further, two probability distributions pi, #2 on S equal if and only if for all f T it holds that Exzp, [fr) = Ex-pa [f())]
4. Assumptions on probability path:
Any probability path (pr)osigı considered fulfils that the function t r f) is continuous in for all f T.
5. KFE is sufficient to check probability path: Let (pr)ogegı probability path on S. Then:
Xo ~Po
(start with right initial distribution)
(36)
0: (pt, f) (pt, Lef) for all f T
(fulfill KFE)
(37)
+ X, ~p: for S S
(marginals of X, are given by pt)
(38)
i.e. if X, is initialized with the right initial distribution, its marginals will be given by Pr-
Remark regarding assumption 5. We note that assumption 5 is true under relatively weak assumptions and there is a diversity of mathematical literature on showing uniqueness of the solution of the KFE in pt for different settings. However, to the best of our knowledge, there is no known result that states regularity assumptions for general state spaces and Markov processes, which is why simply state it here For the machine learning practitioner, this assumption holds for any state space interest. illustrate this. point the rich sources in the mathematical literature that show uniqueness that list the regularity assumptions for specific spaces and classes of Markov processes:
assumption.
1. Flows Rd and manifolds: (Villani et al. Lions, 1989), (Ambrosio) 2004)
2009 Mass conservation formula, page 15), DiPerna
2. Diffusion in Rd and manifolds: (Villani et al.
2009 Diffusion theorem, page 16)
3. General Ito-SDEs in IRd; Figalli 2008
Theorem 1.3 and 1.4), (Kurtz 2011
Corollary 1.3),
(Bogachev et al. 2022
4. Discrete state spaces: Here, the KFE is a linear ODE, which has a unique solution under the assumption that the coefficients are continuous.
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Remark regarding assumption In order for a generator of Markov process to be defined, property in the definition of a Feller process must hold (see section A.1.2). Property 1 in the definition of Feller process (see section A.1.2), might not be stricly necessary. However, we use because (i) most of the mathematical literature (including the one we cite) uses it and (ii) any function computer is defined on compact set.
Motivation for assumption Our framework would not break necessarily if we allowed for infinite number of discontinuities. We included this assumption because simulating an infinite number of discontinuities with a finite number of simulation steps would necessarily induce uncontrollable simulation error upon sampling and therefore most likely be not interesting for the purposes of generative modeling.
Motivation for assumption As every probability path generated by a Feller process fulfils this assumption, it is reasonable to only consider probability paths that have that property well.
A.3 ADOINT KFE
The version of the KFE in eq. (11) determines the evolution of expectations of test functions f. This necessary if version of the KFE can be used that directly prescribe the change of the probability densities. To present it, we introduce the adjoint generator Lú, which acts on probability densities du (æ) with respect to reference measure v, namely (e) (2) is implicitly defined by the identity
probability distributions that do not have density. If density exists, more familiar
dp
dpt du
=(c;
dpt du
Lif)
(fi
VfET
(39)
1 dei (=)co(())(())
dp: C
(z)
NfET
(40)
du
Now, equation 39 apppled to the KFE (equation
we get
f 0,
dpc (x)f(z)u(dz)
(41)
du
= a2 /
dyt (z)((z)v(d))
(42)
du
= (pe, f
(43)
(pe, Lef)
(44)
dpc du
Les
(45)
C dpt.
(46)
du
[ct dpt
(())
(47)
where "u(dz)" denotes the integration with respect to the reference measure v on S. As this holds for all test functions f, we can conclude that this is equivalent to
a dp: (σ) = [c;
dp: ((x) for all x S
Adjoint KFE
(48)
dv
dv
which is an equivalent version of the KFE that we call adjoint KFE. In this form, the KFE generalizes many famous equations used to develop generative models such the Continuity Equation the Fokker-Planck Equation (Song et al. 2020 Lipman et al.) (202)) (see table []. Whenever a probability density exists, we use
the adjoint KFE. We will derive several examples of adjoint generators and adjoint KFEs in app. A.5
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A.4 TIME-REVERSAL
In diffusion models (Song et al. 2020), the notion of a time-reversal of a diffusion process plays crucial role in the construction of the generative model. We discuss here the idea of time-reversal of a stochastic process and explain why we build our framework based on probability paths and solutions to the Kolmogorov Forward Equation (KFE) and how this includes the diffusion construction.
Probability paths vs noising/forward process. In the context of diffusion models, one usually corrupts (or "noises") data via a "forward process". With our time parameterization (where = 1 corresponds to data), this is represented by a Markov process (Xe)) running backwards in time. Every such Markov process also defines conditional probability path pa (+1=)) = karu (-12) via its transition kernel h and the marginal probability Pr corresponds to the marginals, i.e. X₁ ~ pr. Therefore, diffusion models specify a probability path via the forward process/ataa corruption process. We use probability path opposed to noising process because (1) it is often casier to define a probability path as opposed to a full transition kernel (2) it is the only thing that is used during training (only the marginals of the probability path are used during training, see eq. (7) this includes diffusion model training as a special example). Further, we show in this work how one probability path can have different Markov processes that generate the same probability path (see fig. for examples). However, an explicit data corruption process via a Markov process can be still a useful heuristic find good probability paths.
The notion of the time-reversal by Anderson (9822) is stronger than the one needed for generative modeling. To explain this, briefly define two notions of time-reversals. Let (X2)osegi be Markov process running in forward time and (X, (Xe)usesi a Markov process running in backward time. The Markov process X, is called time-reversal of X, if the joint distributions coincide, i.e. if
P(X,, E A A X,, E A, = IP( E AI,..., X,, E A, for all 0 I ,4, I 1, A,...., A, C.S measurable A famous example is the time-reversal of stochastic differential equations by [Anderson] (1982). Next, we introduce here the notion of weak time-reversal. We call X, weak time-reversal of X, if the marginals coincide, i.c. if
(49)
(50)
PiX, E A] = P(X, E A] for all 0 < 1 S A measurable
(51)
Obviously, any strong time-reversal gives us a weak time-reversal but not vice versa necessarily. For the purposes of generative modeling, generated image) and discard earlier time points. Therefore, whether a Markov process is a "true" time- reversal or weak time-reversal does not matter for most applications. A famous example of weak time reversal that is not a "strong" time-reversal is the probability flow ODE in diffusion models (Song et al, 2020). The probability flow ODE does not constitute a time-reversal of diffusion process in the sense of Anderson (1982) but it constitutes weak time-reversal (i.e. it has the same marginals). As the probability flow ODE is currently the state-of-the-art method for low NFEs, it illustrates that finding "true" " time-reversals is a harder mathematical problem to solve but (often) not necessary to solve for the purposes of generative modeling and might give suboptimal results. Therefore, we restrict ourselves to weak time-reversals in this work.
often only use the final point X₁ of the Markov process (e.g. as
Using the KFE to find weak time-reversals. Let (Xi) Kx)oses1 be a Markov process running in backward time. How can we find or check whether a Markov process (X,) (X,)ogegı running in forward time is a weak time reversal of (Xx)oses1? ? Let (pt)o be the marginals of X,, then pr defines a probability path. Further, by definition (Xe)usts1 is a weak time reversal of (Xx) if and only if X, generates the probability path (pr)osegi- By the KFE (see eq. (11).. this holds if and only if Xo Po and
(pc. Lef) = (pc. f)
L.f)
for all f E T, 1
(52)
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where in the second equation use the KFE reverse-time (we have flip signs as time is running backwards). Therefore, finding time-reversal reduces to finding generator Li such that Lef) for all TT,
(.,)
(53)
This allows us to find weak time-reversal for many Markov processes. We illustrate this app. H.2 on the example of diffusion processes. However, actual time-reversals like the one by Anderson (1982) can still be helpful for our purposes as a way to find a solution to the KFE. If pı is a probability path given by the marginals of Markov process X,, its strong time-reversal is also weak time-reversal find solution to the KFF. Therefore, finding time-reversals can of Markov processes can be tool to find solutions to the KFF. We avoid using time-reversal it imposes additional mathematical complexity that is not needed for the vanilla generative modeling task.
A.5 PROPERTIES OF IMPORTANT MARKOV PROCESSES (DERIVATIONS FOR TABLE
In this section, include the definition of important classes of Markov processes and their properties including their generators.
A.5.1 FLOWS
Definition. Let S Hd and u 12 Rd, (I, ue(x) be a time-dependent vector field. Then flow (x) is the solution to the ODE:
_ (x) = ue (o),, 02,0(x)= =x
(54)
It is clear that &r,a is a deterministic Markov transition kernel Pejar i.e. Peps (=3xr) is a delta distribution for every x E RRd. Fuler sampling of the ODE corresponds to
X1th = X, + hu,((,)) o(h)
(55)
Generator. Let 7 = C (Rd) be the space infinitely differentiable and smooth functions with compact support (test function in R.), Then we can compute the generator via
E [f, + (X,) o(h)) x] f(x)
[Crf](x)
lim h100
(56)
h
hV (V + o(h)
lim h-0
(57)
h
where used first-order Taylor approximation and the limit is uniform by the fact that f is in C (R.).
Adjoint and adjoint KFE. Let's assume that Pe has density d (z) with respect the Lebesgue measure
that is bounded and continuously differentiable. In the following paragraph, we write pr(x) = de (x) for its density. Then can compute the adjoint generator L3 via
(pe, Lzf) =Exop, [Z)((x)
(58)
(uf(z)p(<)ddr
(59)
= / Vf(x) ((())()d)))
(60)
/ [()]-v (u()))
(61)
= / (())C)))))))d
(62)
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by partial integration. Therefore, the adjoint generator is given by Lip: -V -V. (=)(())) Using the adjoint KFE, we get the well-known continuity equation d(po(z) -V [2)
(63)
A.5.2 DIFFUSION
Definition. Let S = R and o; x Rdxd (z.t) 0z(xr be time-dependent function mapping to symmetric positive semi-definite matrices oN continuous fashion. diffusion process with diffusion coefficient oN is defined via the infinitesimal sampling procedure:
X = X, + Vho,(X,)ee
(64)
where 6 (/, I). For more formal definition, see (Oksendal 2013).
Generator. Let C (RR) be the space of infinitely differentiable and smooth functions with compact support (test function in R.). Then we can compute the generator via
[(Xe + (X,)e; + o(h)) = =x]- f(x)
E
(z)
lim h-+0
(65)
E[f(x) + V(oa"" + +
lim
(66)
h-0
h
lim /ta
(67)
h-+0
h
(68)
Trace (o. (,()))**
(69)
Tracefo, (z) f2
(70)
2 (()) v3
(71)
where we used 2nd order Taylor approximation (2nd order because BlllvTTe,)" []] x h) and the symmetry of of. Further, we use A B trace(AB) denote the matrix inner product.
Adjoint and adjoint KFE. Let's assume that pt has a density de (x) with respect the Lebesgue measure that is bounded and continuously differentiable. In the following paragraph, we write (x) density. We can compute the adjoint generator C; via
deu
(x) for its
de
(Pe. Lif) =Buup, [E.f()
(72)
(=(()))(z)dzz
(73)
2 / (f(0) (z)(2))
(74)
/ [x))22 (oi
(75)
= -f (())
(76)
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by partial integration. Using the fact that this holds for all test functions, we can convert the KFE the adjoint KFE recovering the well-known Fokker-Planck equation
(xp = v (z)
(77)
A.5.3 JUMPS
Let assume that consider a jump process defined by time-dependent kernel (dy; i.e. for every 0 S t < and every x E S, Qr(dy; z) is a positive measure over S {x) The idea of a jump process is that the total volume assigned to S
(x - (r (()=))((())))))
(78)
gives the jump intensity, i.. the infinitesimal likelihood of jumping, Further, if Ax(x) > 0, assign jump distribution by normalizing Qe to a probability kernel
Je x) Qr(dy;; a))
(79)
Xx(x)
The infinitesimal sampling procedure is as follows:
Xeth {~ Xe Jeldy; X.)
with probability 1 1 hx((X,) + o(h)
(80)
with probability (xx) + o(h)
We derive the generator here in an informal way. For a rigorous treatment of jump processes, see for example DDviis 1984). Up ((h))))pprxximation error, the generator is then given by
E[f(X(+n) XX xx)
(x)
lim
(81)
h100
h (X((()(+h) (X,,) x. Jump in [t, t + h) Dump in [t, h)]
lim
(82)
h-50
E ( f(X,)|x, No jump in (t, t + P) jump in [t. t + h)]
+ lim
(83)
h +0
h
Py (f(v) f()) (a)(())
lim
(84)
h-10
Ey Je(9ni))))) (f(y) fxx) (x)
(8:5)
= f()))
(86)
where we have used that X, does not jump in [t, + h). then X,th X,.
Adjoint and adjoint KFF. Let's assume that pt has a density des (x) with respect to the Lebesgue measure that is bounded and continuously differentiable. In the following paragraph, we write pı(x) du (x) for its density. Let's assume that the jump measures Q:(yy; is given via a kerne Q S x S RRzo. (v, +
Qe (y; .) such that
/ [()))((d);)))) / (()))(l):)))(()))
(87)
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where v(dy) denotes the integration with respect to the reference measure on S. Then we can derive the adjoint generator as follows:
/ / (f(v) (0000-1
(pe, Lef) (x)
(88)
/ / 1w000atnaaa / sto)R
(89)
/ / [1 / (10))
(90)
= / (()) (Qleroma - Qu ( u(de)
(91)
=:
(92)
where we have seen that L describes the adjoint generator. With this, we get the jump continuity equation as adjoint KFE
depe(a)) / (((::)))a())
Qelu: (dy)
(93)
inflow
couffow
A.5.4 CONTINUOUS-TIME MARKOV CHAIN (CTMC)
Let us consider a continuous-tim Markov chain X, on a discrete space S with |S| < oo. We define this to be a jump process defined in app. A.5.3 However, process. Specifically,
find convenient parameterization for the jump convert integrals into sums and see that there is jump kernel Qe Rx given such that for all æ E S it holds that
f.f(x) ((0))1()))) Σi(e)) f())]Q((xx=))
(94)
yES
A natural convention people follow to set (+(x;))= = Σ 2) This gives us the rate of staying at a uyn (However, note that (1; does not describe kernel of positive measure anymore.) With this constraint, we get
(.f(x) = - /"
(95)
WES
where we consider f f= (f(x)) as a row vector. The adjoint is simply given by the adjoint multiplication Pt r> Qipt and we get that the adjoint KFE is given by
@sp((x) = Ee(x:))
(96)
DES
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To sample the next time step given X, 2, we get
Qe(yer) Q+(wa))
with probability h Σ x) + d(1)
X
(97)
x
with probability 1 Σ Q+(y;=)) + o(h)
Qulyra)
with probability h Σ Q+(;;))
(u(aa))
(98)
x
with probability 1+ hQq + (hh)
y with probability (Q o(h) (for all y E S)
(99)
æ with probability 1 1+ (=(((((())) ohh
(1 + h=))(=
(100)
where we simply sample from the stochastic matrix / + hQ2 matrix whose columns sum to 1).
A.6 LINEAR PARAMETERIZATION OF GENERATORS
We describe here what we understand under a linear parameterization of a generator. This is the basis for parameterizing generator neural network that can be implemented in computer. For simplicity, we fix a E [0, 1]. As before, T be the space of test functions and B(S) be the space of bounded functions on S. Then the space A(T) := {C T + B is linear) of linear operators on T is itself again a vector space. Let W C A(T) be subspace. Then a linear parameterization of W is given by 2 components: (I) a convex closed set Ω C V that is subset of a vector space with an inner product (y and (2) a linear operator K.: T (S;; V) such that every La E W can be written
f,f(x) = (((()))F(())
(101)
for a continuous function F. S + Ω. We list several examples to make this abstract definition more concrete.
1. Flows in S = Rd: T = C (IRd) and = V- Rd. Let W be the space of linear operators given via generators of flows, i.c.
4, VTT Uy : Rd + Rd
(102)
Setting K. V and F. = u, we recover the shape of eq. 101
2. Diffusion in S = R4: T = C (Rd) and Ω = S C Rdxd V. where S denotes the set of all positive semi-definite matrices. Let W the space of linear operators given via generators of diffusion, i.c.
G.f o2 of. or Rd S++
(103)
Setting K.f = f and F; = of we recover the shape of eq. 101
3. Jumps in S 1R T = C (R.)) and ft (a : Rdd T R20 a continuous) C((RR, RR) V. On V, dot product is defined via (a, b) (z) Let W be the space of linear operators given via generators of jumps, i.e.
[()) -
Lxf(x) =
[=)))=(( (DS((),
(104)
where we set K.)(x)) as the function ~ f(y) - f(x). Setting F, = Q: we recover the shape of eq. (TOT).
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4. Continuous-time Markov chains: Let S be discrete and Qr 1858 be rate matrix of continuous-time Markov chain. Then for any f E R
Lif(x) ((((())) = (f(Qi(:z))v
(105)
where V I and Df f and (s-) is the standard Euclidean dot product. With this, we recover the shape of eq. (1OI)
B SAMPLING WITH UNIVERSAL REPRESENTATION (ALG. 2
we summarize a sampling procedure to sample generator for Rd with universal representation described in theorem We briefly want to derive that alg. 2 in fact method sampling procedure that simulates a Markov process with the correct generator up to (h) approximation error. Let us assume that X, is Markov process that is obtained by sampling as in alg. (limit process for h 0). Then its generator given via:
alg. 2
(x)(
(106)
lim (X(J(X++h) ((X))xx a:))
(107)
6-0 h
[[f(X++h) f(x))X I. Jump in [t, PJmmp in [t, h)]
-lim
(108)
h00
Bif(X,+n) f(X,) x. No jump [t, + PN) jump in [t, + h)]
lim
(109)
(y- f(x))AAe(z)
-lim
(110)
4-+0
[X, + hxi( + (hor(Xi)e)) f(X.) No jump in [t, h)]] hAr(w))
lim
400
h
(111)
| -(0 (1(()-f(a)) ((=)))
(112)
+]s(x++hullX + (x)a( -(X.))x x., No jump in [t,t+ h)]
+im
(113)
h-0
f (1(W) - (dd)))
(114)
(INVI(=)) + (X.)e()))e)) + 1 (u + (x (X.)
lim
40
h
(115)
= =/Uw f(=))
(116)
AVS(s)" + lim [Hoo(/be))FPP///cocxxxxx)))
(117)
h
=f (()) (x) + Trace [ox(xx))*
(118)
- -fu - -(=))) (/)) + of (X.) v²f(x)
(119)
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This finishes the proof.
PROOFS
C.I PROOF THEOREM
Let (Xt)ogeg1 be continuous-time time Markov process. For now, we assume that X2 is time-homogeneous. Let be the generator of X₁ defined via
[[J(X++h) [(X))X
Cf lim
(120)
We know that has the following property:
f(z) 0 20
cf(x)20 ("almost positive")
(121)
An operator having the above property is called almost positive. The theory of almost positive operators is well-established. Specifically, we can use universal representations of almost positive operators as established in (von Waldenfels 1965 Satz 1). Related theorems be found in (Courrege 1965) for characterizing differential operators satisfying the absolute maximum principle (that is closely related to the almost positive
principle) and in weaker form for Markov processes (Feller, 1955). Here we use (von Waldenfels) 1965 Satz
1). Specifically, we know that L must have the form
Cf(x) = - +
2 oz) f2)(x)
(122)
/ y/a
Q(dy;:)
(123)
where c(x) 0, u(1) E RRd continuous functions and (1x) a positive semi-definite matrix continuous in x. For every v, Qdyy; x) is a measure on TR {x². The term c(x) corresponds to process that is "dying". As we assume that our process runs from to = 1, we can drop c(x). Further, using the assumption that there only exists a finite number of discontinuities (see app. A.2). we can discard the term V'V 1 (alternatively, one can redefine the drift to ū(z) = u(x) the above formula as:
( Therefore, we can rewrite
V/= 1 - vS((x)
f (1(w) - (dd)) 0Φ
(124)
C.I.1 TIME-NHHMMOGENOU CASE
Let (Xt)osegı be continuous-time Markov process. Now, we allow X, to be time-inhomogenous. To make it time-homogenous, we define X, (Xe,t), t). Let L be the generator of X,. Then for a test function f Rdd R R we get
((, + 1 (x, (2, + / (U(,,>) [(x, ()) ds;
LJ u(x, 1)" V
(125)
where ū, 02, Qi operators over the extended space Rdd x R. iec ū(z,t) = (u. (z (), and
(In t) (x)(2
σ(z,t)
(126)
(x, (2)(2,t)
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However, note that since marginal process in is deterministic and has derivative one, it must necessarily hold (by uniqueness of the representation) that (z, t) and (1, (r,,t),. (2; t),,t o² (x, t)e, = 0 and Q(--;; ;; ;. t) is supported over Rd x {t} i.. is a time-dependend kernel Qc(dy; r). Therefore, we can rewrite the above equation as:
a Čf if= = (0 (( u(t).)))( 20(()) vaf(x,t) + (f(..t) ((a,
(127)
Rewriting as time-dependent generator L, (see app. A.2 we get for a time-independent test function : RR
Lif = 20((0) vif(z)+ / (((G) - ((z))Q((dy;=x
(128)
This finishes the proof.
C.2 PROOF PROPOSITION I
Let Be4Mee(--eeee (-la)) be the conditional transition kernel for the conditional Markov process X with conditional generator & that solves the KFE for the conditional probability path (dz))))) Then compute for the marginal probability path pt and a test function f that:
1
0 (pt. f) lim
(peth: f) (p::
(129)
h-+0 h
1
lim
[ p' [f(('')] Ex-ren (1=)
(130)
h-+0
h
1
lim 2
[f(')) E Bgnpan -- pm (1
(131)
h-50
T
1
lim [p
f(x)]]
(132)
h-50 h
lim Fxppp(()
1
E (dol
B [[(x')] f(x)) [(x)
(133)
4-10 h
= -xx
(dela) (dzlax)
lim h-00
Bla Blars f(x)
(134)
(xxp)())
12 (dela)
[x)((
(135)
Ccf(()
(En (pr, LiS)
(136)
(137)
Therefore, we that the marginal generator L, defined above solves the KFE for the marginal probability path Pt.
Proof that proposition I also holds for any linear parameterization of the generator. Let us assume that the conditional generator has linear parameterization in the sense defined in app. A.6 given via
( (Kf (x
(138)
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for function F7 - V. The proof proceeds exactly as above to get
dc (pr. =Bz (1r) (x [x)]
(139)
= -B (u) [ ( [K](());
(140)
= -E)
K
(141)
=
(pr
(142)
Therefore, we can see that also the marginal generator has linear parameterization given via F; S Q
Note that it holds that Fz( E Ω because fi is convex and closed.
C.3 PROOF OF PROPOSITION
2
C.3.1 EXAMPLE OF BREGMAN DIVERGENCES
We list two motivating examples of an illustration for Bregman divergences (see eq.
1.5
1. Mean squared-error: Setting Φ _ + IR, x ||= leads to
D(x, 9) =6(x) - <(y) - (xr Volu))
(143)
||==||2 ||v|2 fxr y, 2n)
(144)
||a=|| 2 (xx, v) + 11v112
(145)
| y
(146)
2. KL-divergence: Define the probability simplex Ad-1 for E Rd Σ 1). Then define
$ ::AA-1 d-11 + R, +
Wy logr; T;
(147)
D(r,y) ((xx) -d(y) (x
v. vΦ(v))
(148)
d
2i log T
Uo log yi
Σ 1=1
(z;
v1 + 1;)
(149)
Vi log
(150)
C.3.2 BREGMAN DIVERGENCE AS SUFFICIENT CONDITION FOR PROPOSITION
2
First, we show that VoLgm (0) VgLagm (0) for any Bregman divergence. Let V be an arbitrary vector space, Q V be convex subset, and D Ω x Ω R be Bregman divergence on it defined via
D(a, b) = ((a) (b(b) + (a b, vo(b))), a, a
(151)
for convex function $: R. We rewrite D
D(a,b) = A(a) + (a, B(b)) + C(D)
(152)
with functions A(a) d(a), B(b) C(b) = -bb() (b. Vo(b)).
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With this, we get:
Tymm (0) Uni,
(((()
(153)
Uni
(154)
A(F(())) { (FTODDCAA ((P)()))
(155)
A(F((z))
C(("
(156)
( = Uniuuiif. / + (F(()), + ( (dz|z)
const
(157)
const
(158)
Un (DF Fi(r)) + const
(159)
=Legm (0) + const
(160)
Le. both losses only different by constant in 8. Hence, their gradients with respect to are the same. This proves proposition for any Bregman divergence D eq. (62
C.3.3 BREGMAN DIVERGENCE AS NECESSARY CONDITION FOR PROPOSITION 2
We now show that the Bregman divergence property a necessary condition for VgLgm VgLegm (0) hold for arbitrary probability paths and data distributions.
To do so, first prove an equivalent characterization of Bregman divergences. For simplicity and because any representation used in a computer will be dintessimensinnl,, we restrict ourselves here to fine-dimensional vector spaces, i.c. we V F for some N.
Lemma 1. Let Ω C R be a convex closed subset such that its interior Ω2 is convex and dense in Q. Let D: Ω x Q - R>o be an arbitrary cost function, here defined as smooth, function such that D(a, b) = 0 if and only if b. Then the following statements are equivalent:
1. Bregman divergence: The function D) is Bregman divergence, i.e. there exists strictly conver function Φ Ω + IR such that
D(a, b) = =(a) - -
(bbb) + (a b, Vo(())), for all a, E Ω
(161)
2. Target-affine loss function: There exist functions A Ω IR, B: Ω + R² C. IR such that
D(a, b) A(a) (a, B(b) C(b)
(162)
3. Gradient in second argument is linear: For every 01, a2 E Q and Y1, X2 2 0 such that x + it holds that
VaD(A A2uz, b) x V,D(a,, A2 A2
(163)
Proof of (1) (2).
This is trivial. Define A(a) ((a), B(b) Vo(b). = - + (b. vo(b))
Proof of (2) (1). Let us assume have
D)a,b) =(aa) + (a, B(b)) + C(b)
(164)
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The condition that D is valid differentiable cost function implies that arg min D(a, b) =b for all b e Ω
(165)
0 Va. (a)aaa))a==
(166)
=VA(b) + B(b) for all b
(167)
This is equivalent to B = -V A. Therefore, we get that
D(a, b) = A(a) (a, VA(()) + C(b)
(168)
Further, we require that D(a, a) for all a, which implies that
0 = D(a a) = A(a) - (a, VA(a)) + C(a) for all a E Ω (aa) (a, VA(a)) -(()) A(a) for all E ((a,b) b) =A(a) - (a, VA(6)) + (6, VA(6)) A(b) for all a
(169)
(170)
(171)
=A(a) A(b) (a b, A(b))) for all EΩ
(172)
Finally, since D(a, b) 2 0 it must hold that
A(a) S A(b) + (a- b, VA(b)) for all a, E Ω
(173)
The above is equivalent to the fact that A is convex function. Therefore, setting A(a) K(a), we get that is Bregman divergence.
Proof of (2) (3). Let us assume that
D(a, =A(a) + (a, B(b)) + C(b)
(174)
Then:
VoD(A1a + b) = (A101 + A2az, V.B(b)) + V.C(b) =λ1 (01, V,B(b) A2 (02, V,B(b) + + =>iV,D(a,,b b) + V,
(175)
(176)
(177)
Proof of (3) + (2). Let us assume that eq. (163) holds. Fix a Q" We first show that eq. 163) implies that the function + V(, b) is an affine function. To see this, because of the required condition, the function F (Fi...., Fa) ) : + Va, b) fulfills the condition that each is both convex and concave. In particular, v and F are both positive semi-definite. This implies that V 0. In turn, this implics that F.; is constant function. In turn, this implics that the Jacobian FF is constant. This implics that F' is affine, i.c that
VeD(a, b) = B(b)a + C(b)
(178)
for Ō. _2t +PP R and B I + RR
R Further, since D is twice continuously differentiable
V. V.V. = V. B(b)T
(179)
which implies that B(b)" = B(b) for the function B V. (a, b) Ω + R* turn, this implies that C(b) = (b)(b) for the function C(b) = (a, b) (a, (bb). Then we get:
V.D(a, b) Vol(a, B(b) C()) for all a, be Ω
(180)
DDaa, b) = A(a) + (a. B(b) + C(b) for all a, b Ω
(181)
for some function A: Ω R.. This finishes the proof.
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Necessity of Bregman divergence property for proposition 2 to hold. We show the necessity of the Bregman divergence property by giving a counterexample. Fix a S t < 1. Let us assume that the network F is just constant function given by vector E Ω. Now, let us suppose that D not a Bregman divergence. Then by lemma there exist some a 02, b E Ω and A2 2 0 with Å + 22 such that
VDD(1 + X222, b) + Y V.D11, - Va b)
(182)
Then define the data distribution as Pdara = X1 S2, + X ds2 and the conditional distribution at time t simply as (r(+(2) Pdata i.e. distribution that just resamples independently from Puata Then pe pe(dz(z) Pdna((2))
Further, let there be conditional KFE solution parameterized by F2 such that F. (x) = a1, (z) 02 for all (note that we can choose an arbitrary parameterization and we can choose pr (-12) arbitrarily for any
I < 1' < 1). Finally, set 0 = b. Then
Fi(x)= = E Ez-pi (dz)ar) (x) = Eg-pue [F) A1a + A202
(183)
and
V Lym (0) -V VOEID(F(())
(184)
aDD(1 A2
(185)
VolA ((11, 0) X22
(186)
VolEz-pen (FFF(()) 0)]
(187)
V Vo (+1z) (1))
(188)
L
(189)
Therefore, the gradients are not the same and proposition 2 does not hold, This shows that the Bregman divergence property is necessary if we want to desired property to hold for arbitrary data distributions, probability paths, and network parameterizations.
C.4 PROOF OF PROPOSITION 3
Let Lt; L, be two generators of two Markov processes that solve the KFE for probability path pt. Then for at, of E R with ot + at = it holds that:
+ a((())
(190)
=at (pe, L.l) of (pr, fi.
(191)
=ata, f) + a,d, (pc. f)
(192)
=(a at)0, ()
(193)
=o, (p
(194)
i.e., a, afC, is again solution the KFE. A small but important detail is whether a of are positive
or negative and whether L,, C, correspond to actual Markov processes and if yes, in forward or backward time. explained app. A.5. the generator of a tumeiinhomogeneuus Markov process given by a spatial generator L. operating on spatial components plus associated time-derivative Ô operator (as explained, it is usually ignored but relevant here). Therefore, both the spatial components have to sum to and the time-derivative operators. Therefore, we have to add the time-derivative operators up (Markov superposition), set their weight to zero (divergence-fir components), or flip their sign ppredictor-correctoy in order to make the KFE hold. This leads the 3 different use-cases described.
C.5 PROOF OF PROPOSITION 4
First, we state a formal version of proposition] 4 We assume that we have a setup where have a Generator Matching model given for each state space S Sz- Specifically, let g. (-11)) gf (+122) be two conditional
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probability paths on state spaces S Let Li č2 be solutions to the conditional KFE for af (-122), (-122) In the sense defined as in app. A.6 we assume that there are linear parameterizations F. 5 - S2 C
V1, 1 S - 2i C Vi of L given such that
(=)()) (z))) [0s)) ( (K2/2 (12
(195)
for all test functions fi : S1 - R. f2 S2 + E and for two linear operators K1, K2. On the product space V1 V2, we define an inner product via
((a1, b1), (az, b2) b2)) (aI, 02)v + (b1, b2)
(196)
Let x be two marginal Markov processes with corresponding marginal generators Lr, C. with parame- terization F/ F?. We assume that there is simulation kernels T T given that are independent for the Markov process (e.g. Euler sampling in table Dor an ODE solver), i... simulating the Markov process
X T (1x), FX (x)))
(197)
22 T2 CtAlel (1 X2, (x]))
(198)
leads to a valid simulation with the correct generators, i.e. for all test functions f:
1
(x,( = lim - E
E TT, Tam (XX)) - f(x))
(199)
h-00 h
(x)()))) = lim ] Ez, ( (x ([(e))))) - [(x))) h-0 h
1
(200)
Proposition 5 (Multimodal generative models Formal version). Let there be data distribution Paata be given over S1 x S2. Define the conditional factorized path as Pt pe(-1z1; 22) = at (-12))) (-122) and the marginal factorized path via (2) Ip (p((dr) F1, 22)]. Then:
1. Conditional generator: The conditional factorized path pt (-|21, 22) = (112)922 on S1 x S2 admits a KFE solution given by a Markov process X, = (X, X?) where X, X2 are independent Markov processes with marginals qi (- (-Iz1), ( (-122) and generator
facttiion
f) = (2P' (=1) + [2T
== (21,
(201)
where (y) = ((),,)) describes the restriction of a test function f S1 S2 - R on S2. In particular, a linear parameterization of LE is given x = (21, E S1 x S2 via
fif(z1, x2) (x) =(F (11). (x2))
Kff()) (x K2 (x²)
(x2))
(202)
(203)
for all test functions f S x S² + R.
2. Marginal generator: The marginal generator of a multimodal generative model (with conditional generator as in eq. (201) is given by
Lef(x1, = c. p (1)nnaaada /
(204)
(x + (x1,
(205)
In particular, the above generator can be linearly by
(.f(z) (KC((x), F((x)) K.f(x) - (K)
(206)
for a function F, S2 22 f2z
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3. Loss function: For two Bregman divergences D1 22 Un - IR D2 02 S22 - R, the product
D((a1, 02), (b1, b2)) D(a1, b1) D(a2,b2), a1, by E S1, 02 b2 S2
(207)
is again Bregman divergence. train linear parumeterizution represented by neural network F. = (F F1.2) S1 S2 22 f2 we can train it with the of two CGM losses
Uns poi (1(a) [D( (Z (21, + (2 (122) 22)
(208)
4. Sampling: A valid simulation procedure is given by updating each dimension independently. i.e.
X
X+th Sth
X2 (th
(209)
X ~T/t (1X,, [X,, Fi. (x!,x?))
(210)
X ~ Titupel (1xP, F (x! X?))
(211)
where X 144s X are sampled independently. i.e. the above procedure simulates Markov process with generator as given in eq. 2204) for 0 (note that each Ft, depends on both modalities).
Note that there are several striking features about eq. 204 and eq. (06 (1) each summand depends only on the marginal posterior Pijé (1je(dzj/12,,1) 42) per single modality (i 1, 2). The dependency of the posterior across modalities does not influence the marginal generator L. (2) The shape of L, can be easily parameterized into a neural network by taking the parameterizations per modality and making input depend on all modalities jointly (i.e. the dimension of the parameterization scales linearly with the dimension if we parameterized a transition kernels, it would scale exponentially).
Proof for conditional generator. It holds that:
Or (2)()11,22),,111 (at
(212)
(213)
= -lim 1
[E a)) (2), Begcopd
(214)
h-40 h
1
= -im -R
(f - Blg. 22)
(215)
h-+0 h
+ Enj a))
E (1
(216)
lim E
22 P (122) En
E,, ~m
(217)
h-40 h
1
+ Exiope m1)
lim (Eesspan (122) Bes (x2)))
(218)
h h
-Bzzrp(((aa)) (e (C 1 + E Exrppalten) (us (12))
(219)
= =pp(-121,,2) C.f)
(220)
where C, is defined in eq. (001)
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Proof of marginal generator shape.
To show statement we can compute that
()f(21,02) = / 02) (d2), day x2)
(221)
/ Ci (x1) (x1) + 27 (12 (dz) 21 x2)
(222)
= / c.v pm (pde))) + 2 pm (x2) dzg (22)
(223)
= | c. (1nnn + p (122)11
(224)
where used in the last equation that summand i 1,2 only depends 2;.
Proof of Loss shape We use the equivalent characterization of Bregman divergences as target-affine loss functions (see lemma 2)). Let (i,a, D2(c, d) are two Bregman divergence functions on V1, V2 for a, b E V1, c, d E V/2 in shape
Di(a,b) = A1(a) + (a, B(b)),, ((b) D2(c,d) + (c,dd +C(())
(225)
(226)
(227)
Then define:
: (a, + D2(c,d) {a)i(a) + A2(c)] + ((a, c), (B),b), (2(d)) x + +C) C2())
(228)
+
(229)
where the last equation shows that D is again of the shape as outlined in lemma [b). i.e. it is again Bregman E Vi V22 of the generator,
divergence. For parameterization F (#1, = (F², (x2, Pi, (21, w2 we therefore get the loss given by
Legm (0) = =E S Ppp
F(F
(230)
=
(112a)
(P2)) (Di(F (z) + (2(FF((az) (x))
(231)
(232)
where FT F describe the ground truth parameterizations of the conditional generators for each modality. In app. we see concrete example of this loss construction.
Sampling. For readability, we drop the parameter and write F? F2 for two general parameterizations of Markov processes. We assume that eq. (199) and eq. (200) hold. Let X(+h. be Markov process that is simulated as specified in eq. (209). Our goal is to show that X has the desired shape of the generator, i.e. that for a test function S S2 RR it holds
1
(xf(11, = -im
4-50
h
E
T1
(A (f 224m) f(x1
(233)
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We derive
lim
E T (1
-x(1,
h-+
(234)
lim
E m 
f(
x2)]
k-0
(235)
+ lim
lim (1
(236)
h-50 h
(x2+1)
(1-2)
(237)
40 h
lim
E B N Ti (1x
f
(238)
h-10 h
-b 1m (2)) + L (2)
(239)
2x1,
(240)
where we used eq. 199 and eq. 2000) and the uniform continuity in both arguments 2/4h, 224n. The above derivation shows that the sampling procedure defined in eq. (209) leads the correct generator.
D KL-DIVERGENCE (ELBO) LOSSES FOR MARKOV PROCESSES
principle, proposition shows that any Bregman divergence allows us to train a Generator Matching model. However, while the minimum for all Bregman divergences is the same, different Bregman divergences trade-off errors differently. Further, some Bregman divergences can be derived via first principles and have theoretical properties that make seem preferrable. Here, we consider ELBO/KL-divergence loss and derive it for jump models, the model that we implement in sec.
D.I PATH ELBO LOSS GENERAL CASE
For completeness, we give here heuristic derivation the continuous-time ELBO loss. similar derivation below was already used by (Soh-- et al. (2015) derive ELBO loss train diffusion model (for discrete time steps). Let us be given a reference Markov process (Xe)ogrgı with transition kernel ke+ and parameterized Markov process (X," Dosigı with transition kernel 1-8 *++ Then let discretize the process in time and let's define grid points t; = i/n for = 0, 1..., and considered probability measures are absolutely continuous with respect to reference measure v. Let p.p" denote the joint marginals of all grid points, i.c.
Here, we assume that all
( T Ta-l $1 #1, =oo(Fo)
II
Kg k (F441l0)))
(241)
1= (n0 1)/n
p( xy Fo) (P)
II
(242)
1= 110,1//,....d (n-1)/)
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Then we can bound the KL-divergence by using the data processing inequality and the chain rule for KL-divergence to get
Dualpama (ci (x))
(243)
SDKi(p(TI, In 3, (o) (x), ( 2 mo))
(244)
=Dxc(p(zo)112 (20))
E B, upc (heusll le, (-1res 24 (+12%))
(245)
6=
DKL (kes (-12,, M1ke Bs (11204
=0 (1+1 1:) 1e) pp
(246)
1441 t
θ bo
Paiau (F: Peter (3re (x1)
(otto))))))))) in=0
do
(247)
Oh
8 ah
EEsuUilf Si
Dx
(248)
[h=0]
for n + 0o. The above shows that
D DKl (x1n)l)))))))))
(249)
0 (
<Ba-Unif Prr
(250)
0h
|h=0]
D.2 ELBO LOSS FOR CONTINUOUS-TIME MARKOV CHAIN (CTMC)
We work ELBO bound for the case of S discrete using the path ELBO bound given via eq. (249
Let
use assume that there are two CTMC's given with rate matrices Q., Q and transition kernels 4 *+hlt Then, it holds that:
Ree (= = 6g, (1(+h) + (1 + o(h) (e = (x1th) hQt (2++1(00) o(h)
(251)
(252)
Further,
D (=
(253)
(e drith
EE (ess)(Tee log 2 p (fe+a(13))
(254)
Witth
Se, (2e+n) (Q (xc) o(h) dxyth
= (Zets) hQ.. (Ze4hh log 5z, (#++4) + hQ( + o(h)
(255)
hQr (2:t) ohh)
=(1 + (z)()))))))) log
(256)
1 hQ (Ia x5) + o(h)
hQ, ( + o(h)
Σ hQe( log
(257)
hQ (fe+ + ohh)
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Ignoring the -temms, we get:
(258)
(x:
Qe (1+++)()0)) h log Of (2 (r2)
=(1+ (Q(((((r)) log 1 + (x2 (z))
Σ
(259)
It holds that:
0 + hy
a + hæ (a + hr)y (a + hy)a
log
(260)
Oh
+ ha
a + hy
(a + hx)²
oy haya ra hyra
aly 2)
(261)
a + hy
a a hx
(a - hy) + hv)
① a hy ah a ho 1k=0
y
log
(262)
a
And therefore.
0 Oh
Dr D Drch
(263)
-(elledl.)) log
1 (z) (1 + hQe(Z((21)) +1+ (()(()/))))(1 + (2(
Q((z)la)) (2,(z))
(264)
1 (2))
1==
Σ (x( log Qe
(265)
Meth
Σ Qe(Fe-n(xz)) log (e(Fet((21) 09 (Zeen
-Q((x))))) - (x((())) +
(266)
Σ
Qx(Fe+a/z)) log Q (1=+ + C
(267)
Reth
= Σ (Z(HL))))) (e(+4ullz)) log Q. (#14nla7) +C ZEia
(268)
where the last step assumes that Q2 has been normalized to E Qu((lx)) = (x(()()) and C' is constant y74:
in g. The above gives us a simple way of training the Qe-kernel. Plugging this into the Generator Matching loss (see proposition 2) for a conditional rate matrix Qi, this gives us the total loss of:
K (Pama (x)))
(269)
<E tn P
Σ (x(x))) - Q (+)a)) (x) log (ála,) Bypa
+C
(270)
=B)
(271)
The above loss was also found previously in the literature (Opper & Sanguinetti 2007 equation (3)) as an ELBO bound of the KL-divergence of continuous-space jump processes.
Up to constant in 0, we can frame the above conditional Generator Matching loss with a Bregman divergence by defining the convex function Ω
R
via
H: + R,
I
Σ
24 log(x;)
24
(272)
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To see this, it holds that which shows that Φ is convex function. Further, the corresponding Bregman divergence is given by
VΦ(z) = log(z), Vo(()) (1/2) 2 0
(273)
D v) = (xx) - Φ(y) - (xr y, Vo(u))
(274)
-1
Σ
y: log(ys) + x4 log(z:)) (2, yc) (2: y4) log(
(275)
18|-1
Σ
x, log(u)) C
(276)
where C' is constant in This recovers the above loss.
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E SYSTEMATIC STUDY OF PROBABILITY PATHS AND MARKOV MODELS
In this section, we introduce and study several new Markov models for various probability paths. This results in a GM model for every combination of {mixture path, CondOT path} {flow, diffusion, jump, Markov superposition) With this, we hope to provide systematic study of the design space of the Markov models constructed via generator matching (GM), in particular ablating the effect of the probability path and the Markov model used.
E. OVERVIEW AND EMPIRICAL STUDY
Deriving novel Markov models. In table one can see an overview over various Markov models introduced in this section for the two most prominent probability paths (mixtures and the CondOT path). For a given probability path, the approach to find a new Markov model is as follows: (1) Define what class of Markov process used and (2) Find generator for this Markov process that solves the KFE. If density is available, we use the adjoint KFE (see table IT. We illustrate this in several examples in this section. We note that throughout this section, we derive equations in S IR because due to proposition we can easily extend these models to arbitary dimensions. We use standard Bregman divergences to train the model (see proposition 2).
Empirical study of probability paths and Markov models. We implemented the model on a 2d checker- board pattern distribution (Lou al. 2023 Lipman al.) 2022). As one can see, the performance varies for different models depending of the probability path. From this experiments, we can extract several learnings:
1. There is no single optimal probability path or Markov model: In fig. 5 and fig. we plot the marginals generated by the trained Markov models. The flow model performs better on the CondOT path and the jump performs better on the mixture path. This shows that different probability paths are better or worse depending on the Markov model used and that there is no single best probability path. There is an intuitive explanation for our results: the CondOT path geometrically moves probability mass, while the mixture path only up- and down-weighs mass but does not move the support of the distribution. In turn, the flow model "transports" probability mass, while the jump model "teleports" probability mass due to its discontinuous evolution. Therefore, it is intuitive that jump model would perform better on mixture path while a flow model performs better CondOT path. This is also in line with existing result in optimal transport theory connecting the CondOT path with flow (Ambrosio) 2004).
2. Discretization error: In fig. 7 and fig. 8 we plot the final distribution for various number of function evaluations (NFEs) allowing us study the discretization errors. The flow model is highly prone to discretization errors in the mixture model, while showing low discretization error for the CondOT path. For a jump model, the opposite true. This can be explained by the high Lipschitz constant of the learned vector field necessary to learn the transport of mass via a flow in the mixture path. the context of density sampling, this is known problem (see e.g. (Máté & Fleuret 2023). For the jump model, many small changes/movement (like in the CondOT path) lead to discretization error, while larger movements of probability path ("teleportation" ") are easier to handle. This explains the difference in sensitivity in the discretization observed in our experiments.
Therefore, the role of the probability path is determine determine the range of KFE solutions that exist. This can, in fact, influence the performance heavily. However, note that there are many KFE solutions for the same probability paths- even for the same Markov class.
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Name
Mixture
CondOT Pc(a)2) (tz, (1-
Formula
pr(-12) KzO2 (1 Ki Unifia: (a:
Flow
ux(r]z)) Kig 01)
(1-Ke) x=== Fo(x))
Uy (x(2)) = I-f
Diffusion
ox(2)) 2Nt(Co
Golal)]
No KFF solution exists.
(1 No (po(a)
Jump
T-Ne" Je(da'; :2/2) 62
Re..
(x
J+ (x(=) (=k)(rl==)++dk
(1-())
MS
Junp+Diffusion-Foo
Jump+Flow
Table Systematic combinations probability paths and KFE solutions introduced in this section. of these models written as models S R but that be extended to arbitary dimensions using proposition 4 A "pure" diffusion solution to the CondOT path is theoretically impossible to exist meaning a solution with zero drift (we show this in app. E.6). Notation: (x(2]2) x² (f + 1)x: (1 1)² + t
MS: Markov superposition. The derivations for table can be found in the remainder of this section.
t=0.00
t=0.40
t=0.60
t=0.80
t=1.00
reference
flow
diffusion
jump
MS
Figure 5: Illustration of Markov models table trained on a checkerboard pattern data distribution with a mixture probability path. As see, the flow model does perform well on this probability path (due to the high Lipschitz constant), while the jump model performs very well because can "teleport" mass due its discontinuity.

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t=0.00
t=0.40
t=0.60
t=0.80
t=1.00
reference
flow
jump
MS
Figure 6: Illustration of Markov models in table trained checkerboard pattern data distribution with a CondOT probability path. As one can see, the flow model performs better than the jump model on this prohability path suggested by optimal transport theory (Ambrosio) 2004), while the Markov superposition (jump+flow) seems to perform best. We validate these differences in our experiments on CIFAR-10 and ImageNet32 (see sec.
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5 NFEs
10 NFEs
100 NFEs
200 NFEs
500 NFEs
flow
diffusion
jump
MS
Figure 7: Results from sampling with various number of NFEs for mixture path for various models. As can see, the jump model least prone to discretization emror for this probability path.
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5 NFEs
10 NFEs
100 NFEs
200 NFEs
500 NFEs
flow
jump
Figure 8: Results from sampling with a various number of NFEs for CondOT probability path for various models. As one can see, the flow model is least prone to discretization error for this probability path.
E.2 DERIVATION OF FLOW MODEL FOR MIXTURE PATII
Let Po be an arbitrary distribution over = IR. Let Jmin a noise parameter. Let's consider a mixture probability path in 1d of the shape:
Pr (+(+12 = K=N(z, 2iin) + (1 Ke)Po
Let Fo be the cumulative distribution function of Pa and Fo.- be the cumulative distribution function of N (2;; 0, _ Then our goal is find vector field we (x(x)=) E IR that satisfies the continuity equation, i.e.
at (((l=) (N
(2; 2, miin) po(())
θ (Ho(Fo(x) Fs.oo (z)))
Ox
(oo(F(()) F2 (x))
Ox (pr (x)=))
(x(x(2)
(F F= am (x))
Ox (p(())))
NeN 2, o mi + (1 - Ki (x)
0
(=(()))u
dr
Therefore, that (+(+2) generates the above probability path.
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Limit for minn
Note that for minn - O, the conditional vector field becomes:
if x
he
(x(x)a1)= (1 rie (o()))
Fo(Ø) Fo(x)
if 21
if > 21
Uniform prior.
Let po
Unif [n))) Then the vector field would become:
Rt
(a2
01 1xs= -
x + a1
ue (x(2))) = (1 Ktt) ag
02 a1
01
K₁
((a2 a1)l2se x +aa)
Kt
Note that the above vector field is not continuous. In particular, its Lipschitz-constant is infinite. However, it still defines valid flow.
Remark Cross-entropy training. Beyond training flow model, we could also train binary classifier model (a Pf 21 z/. The marginal vector field is then given via:
R, (((x) (1 1- (()o)) (r)
- po()))
Learning (1())) can be done via cross-entropy for every dimension, very different Bregman divergence and generator parameterization (see app. (..)). We do not consider this possibility in this work but mention it an interesting avenue for future work.
E.3 DERIVATION OF DIFFUSION SOLUTION TO MIXTURE PATH
this section, we would like to find a solution based a forward diffusion process that solves for the mixture probability path in R given by:
J~ N(2, ain) with prob Kiz with prob (1 (1)
(r(2))) = =(1
(1- - (6g)) (dxr + Kg (2, miin) + Ig
(277)
Po
where min > is small value (we will later min -> 0). Specifically, we search for KFE solutions given by SDE
dx, ()x, + dL,
(278)
where (Xe, f) is diffuusio coefficient controlling the amount of infinitesimal noise we add and dL, describes x a reflection process. Let Go(x) f } pu(w)dwdy and Gz.omn (x) = -J f N(w; z, of duddy. Then:
-00 -00
-o0 -00
8²
0²
G Gnioni (z) =N 2
Go(x) po(2)
(279)
Pr
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Therefore, for any oe, bt E R we get
= Fi (N(3 min Ja) - po(x))
(280)
0² (Re + bex Gz.mann (x) - Go(z)))
(281)
82
0²
Re(ar + b G2 (x) Go,o (z))
(pe (p(())
(282)
82
(c(x(2) 2)
1 82
28x + ber + G2 (x) Go(x))) NNN 2, min) + (1 Ki (p)
I (p((a)))
(283)
(212)
1 0²
2 8r (1x(=l2)0(((11)))
(284)
Therefore, see that (x(2) satisfies the Bokker-Planck equation. However, 5² can be negative and therefore is not a valid diffusion coefficient in general. We have to pick a, b, such that of is non-negative.
Choice of az bc- Specifically, define a, ct + 1 and b, 0 for value ct that we will define later. Then we get that:
0. [z]2) (Rt + bi + Gz (x) Go(z))
(c fe + [22 x) + Gzoom (x) Go(x))
NxN(2;; o + (1 K₁ Po (z)
NoN(x; 2, min) + (1 ke)po(x)
(285)
The value o is chosen such that (r(z)) non-negative (we will simply define it as the minimum of the residual). For miin 0, it holds that G am (x) - [ z]+ and we have:
(x122) 2hd + 1 + (= - x) + Gz (x) Go(x))
(286)
N,W(1; z, oin) + (1 k2 so))
2hh (ct + f + {/2 - x) + [r 2]+ - Go(x)))
(287)
(1 kx)Po(z))
2ca + 1 + 2/2= 2 - Go(z))))
(288)
(1 -1-ma (x))))
We can define
C = min - (x), g(x) 7"
a 2| Go(
(289)
sER
We know that
g'(z) =1 - Fo(x) < 0,
(x 2 z)
(290)
g(xx) = Fo(x) < 0, (x S 2)
(291)
The above implies that the minimum of g(x) is obtained and holds that:
1
6 = Go(z) 2
(292)
This value be computed numerically. The final function we get is: (x|2) 2kz (Gol2) - _ + + f 2| Go(z))) 2 (Golz) + (x 2]+ (o(r))
(293)
x1=x))p(1z)
(1 ne (x))
the above prior Po has no boundaries (i.e. support R) as for a Gaussian, then we are done here. For a prior with compact support, need to consider reflections.
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Uniform prior. Let's consider uniform prior po Unifja Then the equation becomes
/ = - a1) 2 (=-as)
+ [x 2]+
(z=a,)
2 8g-a1
(294)
(1 - Kie)
The above equation has non-zero boundaries:
(a1 (= - oî - 2ail - ail)
(= + aj 2012)
kz(z 01)²
(295)
(02 -a1)(1 Kit)
(02 - 01 [1 K2)
(02 (1)(1 K&t
(02]2) Kz k=(2² - a3 + 2az l02 - -2])
in(2² + a 2022)
K1 02)2
(296)
(02 a1) Nir)
(a2 ai (1) Ks)
(a2 (1) (1 K+t)
Therefore, we need to consider a reflected SDE of the form
dX, = (X.)ddW + Lt.
where L, describes a reflection on the boundaries [a1, az). We refer to (Pilipenko) 2014) for a rigorous definition reflected SDEs. Here, we simply note that one can imulate the reflected SDE up to o(h)- approximation error with
X1th Rai as (X, Vhoi
where Ras a2 describes the reflection operator along the boundaries [a1 ag]. Beyond satisfying the Fokker- Planck equation, it must then also satisfy the Neumann boundary condition that (Pilipenko) 2014 Lou & Ermon 2023):
(1(()))
(x|2)) 0
(297)
Ox
(x)=)
(298)
Or
Note that here, (x( is constant along the boundary (because the only place where it is not constant around for minn >> 0). Therefore, the above reads as
0
o (1) [=]=))]=== = =0
(299)
Ör
0
lo? (x|z)) =
(300)
dr
That this is fulfilled can be easily seen. Therefore, in total, we have proven that reflected Brownian motion with a uniform initial distribution and o? as defined as in eq. (294) generates the conditional mixture path.
E.4 DERIVATION OF JUMP MODEL FOR MIXTURE PATH
Let us consider again the mixture probability path given by: pr(-Iz) = k₂ + (1 Kx)po
We show here that the KFE is satisfied for jump model with
Qe(de'; x|=) [ >e(x]2) = he
;e(de'; = (d(d'')
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To show this, show that the above jump process fulfils the KFE. Using the form of the generator C₁ derived in app. A.5.3 we derive:
E (a
-Eamp (e) (1) f())]
Re
En (1-p((1>)(((t))
14
Kt (1(x)) Exmpptt))
Re.
1 [f(z) - 1(1 - -K-- Be) xn
[z) ky [x
-=,((=) - K.Bor [xx)
- [x) Kiz Ex (x))
=0,bbarp (x)]f())
a (1)()
i.c. we see that the process fulfils the KFE. Therefore, have established jump model.
E.5 DERIVATION OF FLOW SOLUTION TO CONDOT PATH
Next, we consider the CondOT probability path Pc(-1z) = N(2z, (1 t)9). We consider Markov process defined via a flow determined by the vector field
us (e(c)) 1 t
as already introduced Lipman et al. 2022). For completeness, we show that this generates the probability path (in (Lipman et al. 2022). this is done more generally for Gaussian probability paths). Specifically, one
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has show that it fulfils the continuity equation (the adjoint KFE for flows, see table D. i.c.
0
Pc Pc(x(2)
0t
(- (x - tz)² 2(1 t)2
]
exp
0t
2T [1 t)2
(x - 1z)²
0
[x - 12)²
2(1 - t)3 }
2T (1 1)2 exp
+
exp
0t
2(( t)2
v 2T 1)3 dt
(x (x-tz)²²
0 (x- tz)²
01 [11 - 1) ([(x))
1/2 (122))112200) exp
- -N(x; (1 (1))
2(( t1²
0t [2(1 - t)3
=(1 1 (2T)) 1/2 Nxx; 12, (1 11-1]] (1- 1)2)
(2 1)2)-112 112 -N(xttz, (1 -1- 1)3) á (x - 1=)
0t 2(1 t)²
=(1-t) t) Nr tz, (1- )
0 (z=+z))²
No (1 - )
(1-t) -)-
0t 2 t)2
0 (ar (x- - 1=)²
=(1 1) (2; (1 - -))3) Nax; t2, (1 t)2)
0t 2 t)2
(1 (1-t)= t) 0 (I 2
=N(x;t2, (1 - f)²)
Ōt 2(1 - t)²
(1 - 1) 2(1 2)2 (-) (x t2) (-1-
=M(x;t2, (1 - f)3)
4(1 - +)z
- (1 t) =t2)2 + (x t2)² 2
=N(x;t2, (1 - t)2)
(1 - t) -1
(1 - t)3
(a 1z)
=2 (1 - t)3)
(1 - 1)
(-(1 t)z + (x t2))
(1 t)3
=N(x;t2, (1- - t)2)
(1 - t)
(a t2)
(x - 2)
(1 t)3
(x|=)) + I- 1 (ml2)
2-
x la
(I - t)3
= ( - (1(=)))-((=))))pn(((((
where derive the last identity in app. E.7 This proves that u, (z|2) satisfies the continuity equation. Therefore, the corresponding flow generates the CondOT probability path.
E.6 DERIVATION OF DIFFUSION SOLUTION TO CONDOT PATH
In this section, we show that there is no KFE solution to the CondOT path given by a Markov process that is "purely" defined via diffusion process, i.e. via an SDE dX,, ox(X))WW with zero drift. Using the
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derivation of (zp(())2) in the previous paragraph, we can compute:
(1; (1;;)) - 1)9) (1-1)- (r (z) 12)
(1 (1-1)3 1)3 (x =
pr
(See previous section)
y
02
at b + / / N(eettz, 1789 [11 (1--)--- (z (z-tz) tz)
(2 2) dody
xx
(1 t)3
v-ta
02
at + b + /
[ N(=;0,(1--)))) 1)9) (1 (1-
(1 - ()-))(( + tz 2) dzdy
2
xx
0C
82
[ (1-1))(20011)) (a- [1-1)-
2 (1 tjal2 tz -2) dzdy
Gt b +
8x
y-tz
0
/ N(2;0.1) [1 x² + - 1)z
Q0 + 2 +
dzdy
xx
(1 --tt-- t)²
-00
[I
82
Ot + he + /
(1-1) / N(2;0,1) [
x² + +(1 1)
dzdy
8x x
(1 - t)2
-20
-00
82
I 12
x 1z
1 - 1z
gx
00 + 114t + (1 [1-1G
1-t
- 1-1 1
K
1-t
+ zT.
1 - t
0²
āt + by 4- + +11 []G
z=1z
1-1
- - K
nーtz T-f
+ 2/
T-1
n=tz
(az 11- (1))
8x
N Max; tz, (1-t)²)
8²
N(x; (1 t)2) ār + ((0(((x)= + (1 (()()()))))( - (o()) (6t)
8x
Nax; (1 - t)2) =10
where (xx) = (a - -t2)/(1 - t) and az b,, âz by are arbitrary constants (in x) this characterizes the space of solutions the Pokker-Plande equation fully. However, these constants have be chosen such that is
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non-negative, which is as we will show now impossible. Specifically,
G(z) f
O2; 1)dzy
[(x)
22 0, 1)dady
-00 -00
K(x)= =
N2; 1)dzdy
-00
For very large y, it holds that
G(x)
1)d2= x1
lim
1, lim G(z) 0 24-00
20000
x
x1(2;0,1)d=x00
lim L(x) = 0, +0000
lim L(x) L(x)=0 =0 0 oc
K(x)
2 1(220,1)dd
lim
K(x) = 0
2-000
0c
Therefore, the asymptotic limit of the numerator is
ā + br((z) + (1 - (I - + K (ő(x)) + + =
lim
22000
x
āe bbdz)) + (1 (1- (1- (o(x)) - K (x(x)) + 12 (o(x)) ((x)
lim
2+000
σ(x)
=(b, + (1 (1-
- (1-t) t)
1 1
1-t
1
=b, +
(1 - 1)2
(1 (1- (o(x)) -
ā + bio(z) +
+ ( (o(z))
(o(z))
lim
2-0000
ãe (1-
- 1 K (d(x)) + 17 (6(z)) 6(x)
lim
2-0-00
((x)
-(b. +0+0+00 1-t
b
t
The above poses a problem: we cannot choose by such that the asymptotics are both 2 0 for x: too. The reason for that is that 1/(1 - t)² > for all 0 < t < 1. Hence, the above cannot have a solution with non-negative z. This shows that there is no "pure diffusion" solution to the CondOT path.
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E.7 DERIVATION OF JUMP SOLUTION TO CONDOT PATH
The CondOT probability path in 1d with Gaussian prior is given via:
Pc(-12) ((tz, (1-))?)
(301)
1. With a slight abuse of notation, we write pe (x]22) for its density. The jump continuity equation (see app. A.5.3 for derivation) describes the adjoint KFE in this case given via
for z E 0,
p(( = / V((ee))) (=()dd)
(302)
at
where we omit the dependency of Qi on 2 for conciseness and readability. Setting (z(2)) = Quyy x)) 2 0 and ar) = ( (ar) (similarly, dropping dependency on for readability), we get
0 ot (x)
/ ( ((()))((1)2)
(303)
Time-derivative Pt. Let's compute the left-hand side first:
0
Ot p((2)2)
(304)
0
(x (x-tz)² t2)²
2(1 ((1 )
exp
(305)
0t
v 22x - t)2
0
1
(x - 1z)3
1 θ
(x - 12)2
0t
V 25(1 - 1)2
exp
2(1 - t)2
/ (1 - 1)2 0t
exp
2(1 8)2
(306)
0
(- (x (1x--t2)²
[x tz)
(1 0 1/2 exp
-N(x; (1 1)3)
(307)
12
2(1 t)2
Øt
2((1 - t)²
=(1 - ---1)-2(2))) 2(2)) 1/2 N(x; f2, (1 (1-1)3
θ [(x 12)2
-N (1-t)=)
(308)
(2T(1 - 112)
0t 2(1 - t)2
=(1 - t) 2 Na tz, (1 +)3)
-N tz, (1 (1- (1) 0 [2 tz)²
(309)
(1 t)-
0t 2(1 - t)2
(a - tz)
=(1 - 1)= W(xx; t2, (1 - t)²² - - Noxx; t2, (1 - t)²)
(310)
0t 2(1 +)
(1- à (x tz)
=N(x;tz, - ())
(311)
0t 2(1 - t)2
(1 (1-1)-1 2(1 - 2) - tz)( - (x ()
-N(x, (1 1)3)
(312)
4(1 (111)a
(1- (1 ()(x 12)= + (x 1z)²
=N(x;tz, (1 ))-)
(313)
(1 t33
(x (z)
=N(x,tz, (1-11-10 1)2)
(1 -)-
(-(1 (1- - f) + (x tz))
(314)
(1 f)3
(x - (2)
=N(x;tt, t)9)
(1-1)-1
(1 f)3 (x 2)
(315)
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Let's assume that (z(2)) (z) for state-independent jump distribution Je(x). Further, let's set (x(r)= = (1 - 1). Then jump continuity equation becomes:
(1,,1,(11-0)) || (a (0-tz)(0-2) 2) (x)
-J((z) (1 (1) -
(316)
(1 (1-1)
N(ztzz, (1 - 1)2) [1 (1-1 (x)
=J((x)
(317)
((( (1 t)
To be a valid probability distribution, Je must fulfill:
(x-1z)) 12)(xx (<x-12)(x - 2)
1((x) 20 + )e(x) 2 max
1,0
(318)
(1 - 1)2
1- / Je(x)dr
(319)
0 0- / (1- 1 1 (x 1 t2) 2)
dx
(320)
1)3
00 11 / N(1 (1 - 1)2) (x (z) =)
dx
(321)
(1 t)2
40=1 / N(xttz, (1- 1)2) x²² - (t + 1)2x + 1.22
()
dx
(322)
(1-1) 1)2
#0 =1 (1 - 1)2 + 1222 (1 1)=²t 1 tz
(323)
(1- 1)2
822 (t + 1)2²t to?
(324)
(1-1)
+ +2t+t22
(325)
(1 (1-t)² t)2
00
(326)
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Hence, we can see that (x( is indeed a valid probability distribution. Using that (x(2) (x(x) - t), we get the following result:
(x-tz)(a=2) 1, (1-1)²
max
(x(x)
(327)
1 t
max ((I - tz) (2: -2) - (1 (1))),
(328)
(1 - -133
max (x² (1 + 1)x2 - (1 - 1)2 + 122
(329)
(1 t)
Na; (1 (1-1)3) [1 (z (s-1z](2-5) + (1 (x))
[1-1}²
J((x)
(330)
()+ J \e (i) tz, (1 1 t)
(2-12)(2-2)
N(x; Lz, (1 1)9
(1-1)²
(331)
8 1 (a- (8-12\8-2)
N( t2, (1 - t)
(1-1)²
[(1 - 1)² (x 100-10 (2)(x =)) Max; t2, (1 - 1)3)
(332)
[1 0)3 ( - 2) N (0; f2, (1 - 1)3)dē
(2)
|--² + (1 + (1 -12] N/x; 12, -1)))
(333)
11-22 + (t ( 1)x2 + -1=
M(; (1- - (1
Let's study the 2nd degree polynomial that is used here:
ka(x) = - (t 1)2z (1 - t2
(334)
(1+ 1)= + v (1 1)322 + 4(1 412- 462
112
(335)
2
(1+ 1) + (12 (12 + 21 +
1) + - 81 412 41.2
(336)
2
(++ 1); 1 / (t - 2t +
2 +
8t + 4
(337)
2
(1+ 1)2 +
(1
1)
+ 4(1 - t).
(338)
2
(t 1)2 +
f +4
(339)
2
tz +
22 11
+ |1
+1
(340)
2
4
The above says intuitively that the jump intensity is "most negative" around the area the arithmetic of the currrent mean tz and the final mean 2. Note that for t R 1, it holds that only x with x close to 2 have p(z) < 0, all others must jump.
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As an aside, the above allows to know the support of the marginal J, apriori:
to + + 1 1/1
22
t+1
C
+1 Se
(1
1)
+1
(341)
c2
2
+1
1 +
+
(342)
2
4
4
2
S +1+
(343)
2
where is the upper boundary of the support of the data. A reverse incquality for holds the lower boundary.
Summary. The CondOT probability path is generated by jump process with jump intensity (x(x:) and state-independent jump jump distribution J, given by:
(x [x(x))+
(344)
(1 1)3
(x; = J((x) (( (1-t))
(345)
where kx(x) =x² - 11=z (1- 1)2 122 Q+(y)z)= (()()))((()))))))
(346)
(347)
Intuitively, we jump at z, only (xx;)) has positive value. we jump, we jump to region negative (x(x)) proportional to (x(r) multiplied with the desired density.
F DETAILS FOR EUCLIDEAN JUMP MODEL
We use jump model with Q+(v:x) Ae(z)Je(vxzz) where Xe, Je are described in eqs. 345 and (347
Sampling. For sampling, we can use the fact that A₂(x) = (xx(2) 1)3 factorizes in a part that relatively constant across time and part that is relatively time-dependent. Specifically, set:
[k(()] +
Ar-a (x) RJ
0 058ch h
(348)
(1-1- 8)3
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Which gives:
/ )eto(x))ds)
P PoNo Jump in [t, t hi] = exp(-
(349)
1
exxp(-1k((z)1+
(1 - 55)3 ds)
(350)
exp(
(xx +
(1 - t - 8) 210)
(351)
[xr) +
exp
(352)
(1 h12
(1 1)2
[x))+
1
1
exp
(353)
2
(1-t)) t)2
(1 h)
[k(z))
(1 - f)3
exp
(1 t)3
2
1- t
(1 t h)2
(354)
[1
(1 - t)²
exp
2 (x(1)
(355)
(1 h)2
(>(r))
(356)
Therefore, the above gives us valid scheduler decide whether to jump time-interval [t, + h) or not. For us, this modification made significant difference in the image generation results (e.g. FID 12 vs 4.5 on CIFAR-10).
Extension to multi-dimensional case. We assume our data is multi-dimennional and lies in Rd and we can use proposition 4 to extend the model from 1d to multiple dimensions. Specifically, for x E R2 our model given via
-()) (z(x)= ((..))
(357)
(358)
where (x) 2 0 and (z) is a categorical distribution (using softmax) over fixed set of bins in [-1, 1]
(support of normalized images). On images, we implement this by using a U-Net architecture with + 1 channels where b describes the number of bins. During sampling, for each time update + t + h, updates happen independently per dimension. Specifically.
X =(i X
(359)
mi BBrmoulli(1 Ru,44 (Nr(z)))
(360)
X,
ifm =0 0
X t+h
(361)
((()))
im = 1
Loss function.
As loss function, infinitesimal KL-divergence in 1d via
Q+(1=z) = (v,,,)) Q((rx)) (1(x)
(362)
((Q((w))). Σ ()r))))) Q((x:)) log (y:
(363)
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where the sum of y's is here over regularly spaced bin values in [-1, 1]. We extend the above loss the multi-dimensional case via
"f
(364)
(((1)) D(Q(():)), (((x)))) = Σ 1,
(365)
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G DETAILS FOR PROTEIN JUMP MODEL
G.1 JUMP SOLUTION TO THE KFE
We first derive a general jump solution the jump continuity equation. The jump model described in app. will be special case of the same construction. Let pr( be probability density for every 0 < t S on S where S an arbitrary state space. For a jump intensity (e(x) and jump kernel Je (5; x), the jump continuity equation is given by:
σe P((a)) / (()) - (a
(366)
log (x)) + Xx(x)] = / ((
(3) Ot
(367)
Making J(( = (z( state-independent, we get:
(x) + Ar(xx) -c((x) / / (z) (r(1)] a
(368)
(r(r
log (x + {x() =J((x)
(369)
(F)r dd
We require (z) to be probability density and (x))) 2 0. Therefore, get the two necessary constraints:
Ax(x) 21 a
log (x)]+
(370)
1 = ʃ J((x)xx
(371)
/ /M(Omole)de
/ p((x))
0 (o) + s1(0)
dr
(372)
0
Ø1 Pr (r(dddo
(373)
2
/ pr(x))dx
4 0
(374)
at.
0
/ |dd
8 0
(375)
Øt
0 0
(376)
Therefore, )x(x)) and (z) defined as above are always solution to the jump continuity equation for any
state space. For minimal jump intensity de. get:
\x = ot log]
(377)
| 2 (x)]+
(378)
(r(
(e(=) 0 (z)
(x())
(379)
J[- (o)
a (x))++
(380)
p
(381)
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The above equations are illustrated in app. G.1 for a sphere in R2 The above in f'ac represents general solution to arbitrary state spaces.
1+0.00
t+0.20
t+0.39
t-0.99
t+0.78
t+0.98
Density
Jump
intensity
Jump distribution
Figure 9: Illustration of jump model on manifolds. Left: Illustration of conditional probability p: p.(da) 2) on
sphere with (0, 0, 1)" (North pole). Top: density pe(2(z). Middle: jump intensity >e(x)) = = Q: Bottom: Jump distribution Jr Jdy;; x) Qe(dy; (dy:
G.2 PROBABILITY PATHS AND COMPUTING Je.
We consider the quaternion model for 50(3), namely, each element of 50(3) is represented by a unit vector
I E 8 C R4, where
s = fI E R ||||| =} with ||=||| = 1. We consider a probability path of Fisher-von- distributions given by
(382)
(=( = (x( exp(xx=Tx)
(383)
for a scheduler Ke such that Ko = 0 and Ki >> 0 and normalization constant (N.)). We use custom implementation of the Fisher-von- Mises distribution in way that makes pe (z/221 differentiable with respect to t. We then use automatic diffcrentiation to compute (xp((x) This allows us to compute A and Je (see previous section). To parameterize A on S3)3), simply consider function on SO(3). To parameterize Je on SO(3) we place uniform bins over S0(3) and make each bin represent a rotation. We note that the probability path that FrameFlow path constructed via geodesic interpolation of a uniform to a delta function. Computing Jr on this path was numerically unstable for us (because of sharp boundaries introduced by the uniform distribution). Therefore, we choose a Fisher-von-Mises path as above but selected the scheduler K, to optimally approximate the geodesic path. This ensured numerical stability and an (approximately) faithful recovery of the probability path. The Fisher-von von-Mises path is more numerically stable as its support is all of SO(3) for all 0 < t t 1.
trained is not the probability path above. Rather, it is a probability
G.3 SAMPLING
The FrameFlow model predicts a data point given a state x, at time 0 S i.e.e it predicts the conditional expectation (d [=]. The marginal jump rate kernel is given by
Qe xx) = Ex-p (Q1 x;)
(384)
In order to repurpose the flow model, we make the simplifying assumption that
Q, x.) = (ds) Q): x.)] Rz Q. (f- (y:
(385)
Assuming that the distribution Pe (dz|z) is unimodal, this corresponds to high temperature sampling of 2. We do not employ any temperature sample from Je X, but use plain Euler sampling as described in table]]
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G.4 EXPERIMENT DETAILS
We based our implementation off loaded pre-trained weights from the same repository. We use Euler- Mallyama integrator and 100 discretiza- tion steps for all sampling runs. Each sample uses 100 neural network function evaluations (NFEs). Neural network architecture details can be found in [ampbell et al. (2024b). The size of the model is 17.4 million parameters. All MultiFlow hyperparameters, except those for the jump model, use the default ones provided in the open source code. We found increasing the number of SO(3) jumps bins to improve results and use 2056 bins in our experiments. For our metrics, We deviate slightly from (Yim al. 2023a by only reporting diversity and novelty and exclude designability. As noted in (Yim et al. 2024), designability can be artificially high if the generative model samples the same protein repeatedly. Nevertheles, using designability samples is still important since protein generative models are prone to hallucination and producing proteins that would never be real. A detailed description of designability can be found in (Yim et al., 2023b). Our metrics already take designability into account by first filtering the sampled proteins to only be the designable proteins then clustering the protein structures compute diversity and searching against protein datasets to measure novelty. Following the widely adopted benchmark in Watson et al. (2023), we sample 100 proteins for each length 70, 100, 200, 300 for a total of 400 samples then compute each metric as follows:
and down-
filter bad
1. Diversity (Div): Taking only the samples passing the designability filter, we use MaxCluster (Herbert & Sterbberg, 2008) to compute the number of clusters ne. We report / which is the proportion of designable clusters number of total samples. The higher this value, the more diverse the generated samples are after the designability filter. Diversity is important in protein design where it is ideal to test many diverse protein candidates in the hopes of having many shots make new drug or medicine for instance.
2. Novelty (Nov): Taking only the samples passing the designability filter, we use FoldSeek (Vnn Kem- pen et all 2024) to compute the similarity of each sample to the Protein Data Bank (PDB) (Berman protein structure alignment score called the TM-score (Xu & Zhang 2010) where a value 0.5 or less means the two structures are likely to distinct. The probability of the two structures being similar with the same biological function increases as the (TM-score). We report the average dissimilarity of the outputs of FoldSeck as the novelty since this describes how "novel" the designable samples are on average compared to the known protein structures.
et all 2000). The similarity score is given
score goes to 1.0. Therefore, the dissimilarity
For baselines, we follow the ones described in Campbell et al. (2024b) with the addition of FoldFlow (Bose et all 2023) since it was recently open sourced. Our full results are provided in Table a truncated version presented in the main text Table We remove mention of Des. the main text since this metric can be misleading it can easily be hacked and achieve 100% if the method repeatedly samples the same designable protein. As we see, our jump modifications lead to the best results all categories. The ensemble model of jumps and flows performs similarly the pure jump model.
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Method
Co-design (multi-modal)
PMPNN (unimodal)
Des
Div
Nov
Des
Div
Nov
RFdiffusion
0.87
0.4
0.37
FrameFlow
0.86
0.39
0.39
FoldFlow
0.81
0.24
0.32
Protpardelle
0.63
0.10
0.40
0.90
0.12
0.41
ProteinGenerator
0.37
0.09
0.31
0.89
0.19
0,35
MultiFlow
0.86
0.38
0.39
0.99
0.52
0.39
w/ SO(3) jumps (ours)
0.76
0.48
0.41
0.96
0.63
0.41
w/ (3(3) jumps flow (ours)
0.78
0.47
0.40
0.96
0.59
0.40
Table 5: Full protein generation results.
H EXTENDED DISCUSSION OF RELATED WORKS
H.1 FLOW MATCHING
Flow matching and rectified flows (Lipman al. 2022 Liu al. 2022) are immediate instances of Generator Matching leveraging the flow-specific versions of the KFE given by the continuity equation (see table T and app. for a derivation). We briefly describe here how one can map the propositions from this work to their work. Specifically, flow matching restricts itself to generators of the form f = Vz())² T ug (x) for vector field uf(x) parameterized by a neural network with parameters 0. Given a conditional vector field uz (x]2) and probability path Pe (x]2z, the corresponding marginal vector field in flow matching (see (Lipman et al.
2022 equation (8))) given by
(1)2))aaaa dz
ue (x]lz)
(386)
Pt (c
and corresponds to the marginal generator (see proposition D. The Bregman divergence used is the mean squared error (MSE) obtained by choosing ((x) = ||= in eq. (15). The conditional flow matching loss (Lipman et al. 2022 Theorem is special case of proposition 2 Therefore, Generator Matching can be seen as generalization of the principles of flow matching to the space of Markov process generators for arbitrary state spaces.
H.2 DENOISING DIFFUSION MODELS
From the perspective of Generator Matching, denoising diffusion models (Song al. 2020) flow models with two conceptual differences to flow matching: (1) They allow for stochastic sampling via SDEs by adding divergence-fiee free Langevin component (see proposition and (2) A probability path is defined via forward noising process and a time-reversal of that process serves a solution to the KFE. We explain both differences below.
Teve-reversal to find solutions the KFF. First, let's discuss the idea time-reversal. diffusion models, probability path Pr (dz|z) is constructed via forward diffusion process that noises data (i.e. here,
not only the marginals are specified but a full distribution across all time points). The conceptual idea of
time-reversal allows to find solutions the KFE. We illustrate this here. We derived this already in app. H.2
for general Markov processes and illustrate it here for diffusion models. Specifically, let's consider a Markov noising process X, given by variance-exploding SDE dxX, o,dW, that goes from t 1 to t 0
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backwards in uime" Then we know that the KFE holds in reverse time
O.pif Lif)
KFE in reverse time
(387)
/ P((x)) of
(z)d
generator for diffusions, see table
(388)
/ p1(1)) of V vf(x)dr
definition of Laplacian
(389)
/ (v((t)T v
of
T
((x)dx
partial integration derivative of log
(390)
2
/ / v
ot
v
log
dr
(391)
2
=:Li((w)
= (pr; Lef)
(392)
As see, the operator L.f f V log fulfils the KFE in forward time. Further, Li corresponds
generator of flow with vector z z loggt. Pr. We can see that the corresponding flow must generate the probability path pr- This flow is commonly called the probability flow ODE (Song et al. (020)). The conditional Generator Matching loss with a mean squared error then leads the denoising score matching loss (Vincent, 2011)
Egripr un V px - s0(2, +)
(393)
for neural network Rdd x [0, 1] IR approximating the score vector field. We remark that mathe- matically there is also a stronger notion of time-reversal that also requires the joint distribution (across time points) to the be same opposed just the marginals.
Stochastic sampling by adding a divergence component. Second, let's discuss stochastic sampling, For a general probability path Pt with density (x))), general divergence- free component is given via the Langevin generator corresponding to an SDE with drift of V (x)) and diffusion coefficient v /2,z
4 CLangevia f(x) = of V f(x) V (xg + fx))(x);
(394)
This fact is widely applied in statistical physies and Markov chain Monte Carlo methods in the form of Langevin dynamics (Roberts & Tweedie 1996). To see that this is divergence-free (in the sense as defined in proposition 3), we can simply use partial integration
Pt. C. Langevin
(395)
pi(c)o Vo logd (((=))))A(()ddz
by definition
(396)
V /
p((x)o)
derivative of log
(397)
/ (t)( (((eooAAl())d
partial integration
(398)
(399)
Note that in Song al. 2020 t 0 corresponds data, while we keep the convention here that t corresponds
data
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In (Song et al. 2020), they fix a specific weighting and run several iterations of Langevin sampling iteration at every time step, there called a coddictor-corrctoor scheme (see (Song et al. 2020) A Algorithm 1-3)). However, by proposition [ any positive weighting of the above Langevin component leads a valid sampling procedure. This observation was already made by Karras et al. (2022) section 4) where the optimal weighting of the divergence-free Langevin component is studied experimentally more detail.
Finally, we note that many denoising diffusion models use formulations in discrete time (Sohl-Dickstein et al. 2015| Ho et al., 2020). While these formulations do not incur a time discretization error during sampling. they incur error in the loss formulation. The reason for that is that similar proposition in proposition does not hold for discrete time steps (i.e. the lincarization only holds for the generator and not the transition kemne).. Therefore, formulation with discrete time steps can only us approximate loss via ELBO lower bound using parameterized family of distributions such as a Gaussian (Sohl-Dickstein et al. 2015).
H.3 STOCHASTIC INTERPOLANTS
Stochastic interpolants (Albergo & Vanden-Eijnden 2022 Albergo et al. 2023) is framework for generative
modelling that shares many similarities with diffusion models and flow matching. We explain conceptual differences by placing its findings within the Generator Matching framework.
General flow solution based on interpolant. A difference in the perspective of stochastic interpolants is that they take sample- based perspective, i.c. they specific an initial distribution zg Po and a probability path Fe ~ pr(dr) is constructed implicitly via a simulator function I (see (Albergo et al. 2023 definition 2.)))
7()e where c ~ (O0, Ia) and we impose the condition that 7(0, Ip, =) To. [(1, Fy. 2) = = and (0) = 0, 7(1) =
Zt == (1,, No, 2,e) e) := [(1,
(400)
0. A general flow-based solution can be derived by using the KFE (see (Albergo al. 2023 theorem 2.6)):
① Ot (p. f) O, Zo. 2, e)]] =E ( 2, ) 2o. 2)
(401)
=Ez,.m
(((x.) EE (1, To. [28]]
(402)
-Ba,~m, [xx)(
(403)
The operator L.f is generator of a flow with vector field given by the conditional expectation of the velocity u(x,,t) 0( Fo, 2, E) ar). Hence, we see that the above vector field generates the probability path pr- This can be trained in the same way with a mean-squared error as for denoising diffusion models and flow matching.
Stochastic sampling. Another difference of stochastic interpolants is a generalization of the stochastic sampling procedure for denoising diffusion models (see app. H.2). One advantage of denoising diffusion models is that one gets informally "2 advantages for I": Specifically, both for learning flow-based solution to the KFE and for stochastic sampling, we only need to learn the function log Pt commonly called the score function (see app. H.2) This works because the process is constructed as a diffusion process. However, in the general case, one can still learn the score V log P. separately from the flow and then add a fivergence-free Langevin component during sampling (see eq. (394)). As pointed out in (Albergo et al., 2023 theorem 2.8), the special shape of the stochastic interpolant (see eq. (400p) allows to derive a simple denoising score matching loss. In (Albergo et al. 2023 theorem 2.23), it further shown that adding stochastic sampling leads to the ability to control the KL-divergence between the target distribution and the distribution generated by the model. This highlights KL--diergeece is in general not possible with a pure flow model (although bounds Wasserstein distance exist for flow models, see (Benton et al. 2023)
advantage of adding a noise (SDE) term because bounding the
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H.4 DISCRETE SPACES VIA CONTINUOUS- TIME MARKOV CHAINS ("DISCRETE DIFFUSION")
For discrete state spaces S. the generator of Markov process X, S is given by rate transition matrix Qi E RSxS (see app. A.5.4 | for derivations). Therefore, if we restrict ourselves to generators on discrete state spaces, we recover the framework developed by (Campbell al. 2022) as an instance of Generator Matching. Proposition in Campbell et al. (2022) corresponds to proposition I showing that the marginal generator/rate matrix corresponds a conditional generator weighted by the posterior. Proposition in Campbell et al. (2022) shows that continuous-time ELBO can be derived via a Bregman divergence. We derive a similar, slightly simpler, Bregman divergence loss in app. D.2 that also corresponds to an BLBBO lower bound. Further, they use predictor-correctu scheme (see Campbell al. (2022, Proposition 4)) as outlined here for the general case in proposition 3 In the discrete setting, this leads to significant improvements (Gat et al. 2024).
In many applications such as language modelling, the state space decomposes into dimensions, i.e. is given via S = (1...., No =: [N] where N is the vocabulary size. This state space is usually too large that one cannot store full rate transition matrix Qe (and not even a single row of it) in computer. However, using a factorized probability path, we can use proposition
[ see that we can learn a rates Q, that update each dimension independently (i.e. it has a block structure). This reduces the dimension significantly. This was shown for discrete spaces also in Campbell et al. (2022) Proposition 3) and has since then the de facto standard for discrete diffusion models (Lou et al.J 2024a| Gat et al., 2024). Instead of parameterizing the generator directly, proposition 4 also shows that one can also only learns the marginals pe (z/(z)) for each
2 (zi...., 2n)) E [N] independently. One can then train the marginals of the posterior Pe (2s (z) via the
cross-entropy loss Gat et al., 2024; Campbell et al., 2024b).
Campbell et al. (2022) adapt the idea of time-reversal from diffusion models to find solutions of the KFE for given probability path. Specifically, the time-reversal Qe of process with rate matrix Q: running in backwards-time given via
Pc(z) pe(y)
Qe v) Q, x)
(404)
more recent works (Lou et al., 2024a). it has been shown that parmeterizing the generator by the ratio p(y)ppl /p(x) that is needed to time-reverse the process called the "discrete score" leads to improved results.
As this is linear parameterization of the generator, they can use the same Bregman divergence as we derive in app. D.2 to train the discrete score.
Blackout diffusion. As another example a discrete Markov model, we discuss blackout diffusion (Santos et all, 2023). This model also considers model on discrete state spaces S with rate matrix Qi (there, denoted as Lmm"). ). The KFE in discrete spaces corresponds to (Santos et al. 2023 Equation (2)). A probability path is constructed via a "forward process" or noising process that allows for transitions to neighboring states, in the specific case for the blackout diffusion characterization by a decay process representing by m + m 1 (see app. A.4 for a discussion of how probability paths and time-reversal relate). The loss function in (Santos et al.) 2023 Equation (1 1) and (12)) correspond to the ELBO likelihood loss derived in app. D.2 (up to constants). Further, Santos al. (2023) show that this design of the probability path allows simplify several equations during training (Santos et al. 2023] Algorithm 1) and allows to make Umite-time approximations reducing the discretization error during sampling (Santos et al. 2023 Algorithm 2).
H.5 GEOMETRIC DATA AND MANIFOLDS
We next describe how Generator Matching generalizes models Riemannian manifolds. In the following, let S = MM be a smooth Riemannian manifold with metric g. For x E M, let T.M be the tangent space of x and let TM- = LLeM T.M be the tangent bundle.
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First, we consider flows on M showing how we can recover Riemannian Flow Matching (Chen & Lipman 2024 A flow on M is defined via vector field u M [0, 1] + TM such that ut (x) T for x E M. As every tangent space a vector space, the space of vector fields is a vector space again. Every vector fields defines flow drjs that fulfils
Riemannian Flow Matching (Chen & Lipman 2024).
(e(((x) = for all 0 1 st
(405)
o1 (2) -ue (x)) for all 0 Sast51
(406)
Next, we derive the generator. Let f M + R be smooth function. Then the generator is given via
[()+))(()) f(x) (V](r),
(xf(z) = lim h-+
(407)
h
where (,y-), describes the dot product defining the Riemannian metric g and f describes the gradient of f with respect to g. This coincides with the Lie derivative of function (Jost & Jost 2008), a fundamental concept in differential geometry. Therefore, see that 2n (x) is linear parameterization of the generator. We can then use an arbitrary Bregman divergence on T,M, e.g. the mean-squared error. The CGM loss then recovers the Riemannian Conditional Flow Matching loss (see (Chen & Lipman 2024] equation (8))). This shows that Riemannian Flow Matching a specific instance of Generator Matching with Markov processes manifolds restricted to flows.
Diffusion models on manifolds (De Bortoli et al.) 2022; Huang et al. 2022). Riemannian score-based generative modeling equally be seen a probability path is constructed via a forward noising process and a solution to the KFE is found via a time-reversal of the process (see app. H.2 an example on Euclidean space). Specifically, a forward- SDE that generates data obtained via time-reversa (see (De Bortoli et al. 2022 theorem 3.1.)) and has the shape
instance of GM. Similar to Euclidean diffusion models,
dY, =(- + V Vlogp: + ddB,
(408)
where b M + TM describes a drift, V logg, M + TM describe the score vector field and B:M describes Brownian motion on M. The generator of the above SDE is given via
(.f()= - V(()), + (V Vlogp:())) 1 A
(409)
where f describes the Laplace- Laplace-Beltrami operator on manifolds (Elworthy 1998). Note that b is fixed as a hyperparameter. Therefore. like in the Euclidean case, linear parameterization of the generator again given via a score network $g .M x [0, 1] + TM. Choosing the mean squared error Bregman divergence, one recovers Riemannian denoising score matching (see (De Bortoli et al.) 2022 section 3.2)). However, the fact that the noising process is not analytically tractable does not give us analytically tractable formula for Pr (xr]2)) i.e. GM language the conditional solution to the KFE is not known analytically. Therefore, this approach requires iterative simulation of noising process during training even for geometries with analytic geodesic formulas and requires approximations of score functions as a training target (see (De Bortoli et al. table 2)).
2022
H.6
MULTIMODAL SPACES
Markovian multimodal generative models have been previously described for specific spaces and model classes (Anand & Achim 2022 Campbell et al. 2024b).
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MultiFlow (Campbell al. 2024b). As an example. we illustrate here how the work by Campbell et al. (024bb for multimodal protein generation fits into the GM framework. Campbell et al. (2024b) first construct a generative model on discrete spaces representing amino acids of a protein using the recipe outlined in [Campbell et al. 2022) (see app. H.4 for explanations how this fits into GM framework). Further, they use a Euclidean flow model (Lipman et al. 2022) for the translation components of the protein and a Riemannian Flow model (Yim et al. 2024) is used for frames represented via elements on SO(3). Using a factorized probability, one can use the recipe outlined in proposition 4 to build a multimodal generative model. This corresoonds to Proposition 4.1. and Proposition 4.2. in (Campbell al. 2024b). As outlined in proposition 4 loss functions for individual modalities can be summed up, as done in (Camphell et al.) 2024b) equation (16)). Therefore, the work by Campbell al.] (2024b) is direct example of the power of the recipe outlined in proposition
4 to build Markovian generative model.
H.7 JUMP MODELS ON NON-DISCRETE STATE SPACES
We discuss a selection of models relying on (and including) jump models. In app. H.4 we discussed already continuous-time Markov processes, which are jump models on discrete state spaces. Hence, we focus here on other state spaces.
Piecewise deterministic generative models (Bertazzi et al. 2024). The combination of flow and jump Markov process is commonly called piccewise-determnnsst Markov process (PDMP) (Daiis) 1984
Del Moral & Penev 2017). Recently, PDMPs have been applied by Bertazzi al. 2024) for generative modeling in the form of piecewise-determiiisti generative models". Specifically, they consider models phase space, i.e. where the states is given via a tuple (ar, 2) of location r and velocity v. Similar to diffusion models, they construct a probability path via a forward/noising process and use its time-reversal as solution to the KFE of the corresponding probability path. The jump intensity and jump kernels of the time-reversal can be linearly parameterized via likelihood ratios (see (Bertazzi al. 2024 Equation (4) and (5))) this corresponds to a linear parameterization of the generator (see app. A.6) and is related to the linear parameterization used by (Lou et al. 2024a) discrete state spaces.
Score-based generative models with Levy Processes Yoon et al. 2023). A general class of stochastic processes is given by Lévy processes that are characterized by independent and stationary increments. Lévy processes include certain flows, diffusion, and jump processes and were recently used for generative modeling purposes by Yoon et al.] (202)). Here, Yoon et al. (023) construct SDEs driven by Lévy processes as a forward process implicitly definining probability path (see app. app.[..)) and the time-reversal is giving a solution to the KFE (see app. A.4). There, the time-reversal depends on the fractional score function that determines the flow component of the backwards process (the other parts are hyperparameters and don't ton't have
to be learned). The fractional score serves as a linear parameterization of the generator (see app. A.6). To learn this vector field, we employ the conditional Generator Matching loss with the Bregman divergence given by the MSE - this recovers the fractional denoising score matching loss (i.e. (Yoon et al. 2023 Theorem 4.3.) directly corresponds to proposition 2). While Lévy processes in principle include jump processes (and also their SDEs), we note that the work by YYoon et al., 2023) only consider learning the flow component of the Markov processes (the rest are hyperparameters), while here we show how to also learn the jump kernel or the diffusion coefficient.
Trans- dimenssonlli jump diffusion (Campbell et al., 2024a). Recently, jump processes have recently been leveraged to model generative modeling tasks over spaces that includes various numbers of dimensions (Campbell et al. 2024a). From the perspective of GM, we can consider this to be on state space S = _ {d] R A conditional probability path is constructed via forward noising process and time-reversal serve as a solution to the KFE (Cammpelll et al.) 2024a Proposition 1)). The loss in (Campbell) 2024a Proposition 2) corresponds to combined loss of two Bregman divergences: (1) A mean- equared
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error and (2) The Bregman divergence considered in app. D.2 (the jump measure Q: is denoted there as the product Az 2 (y: second term has an additional time-reversal step similar to (Campbell al. 2022).
2) and uses the fact that (z) (3: = log A (x:) + log( A (v;z))). The
I ADDITIONAL EXPERIMENTS ON LOSS FUNCTIONS
To illustrate the utillity of Bregman divergences for GM models, we ran additional experiments using flow model on image generation. Specifically, we use the following two functions ø R + R:
(xx) exp(ax) + exp(-ax)
cosh(ax))
Cosh
(xx) exp(oa))
Exponential
We construct Bregman divergences with these where we give weight 0.5 to the MSE and weight 0.5 to the above ((x). We then train flow model CIFAR-10 using the resulting Bregman divergences. We ablate over a few options of E R. In fig. 10] one can see the training dynamics are significantly more stable and the final results
better by modifying the Bregman divergence loss function. For example, the MSE model achieves an FID score of 2.62 (existing flow matching or diffusion models), while the Bregman divergence achieves score of 2.54. Note that this achieved by modifying one line in existing codebase. Note that the results slightly differ from fig. because those were obtained with joint flow and jump model trained together. Of course, further exploration in the future is needed to explore the design space of Bregman divergences. Here, this shows the practical utility of the space of loss functions given by Bregman divergences.
MSE
Cosh
Exponential
FID
1000
2000
3000
Epoch
Figure 10: Training evolution of a flow model trained on CIFAR-I0 with different Bregman divergences. As one can see, using Bregman divergences other than the MSE leads to improved training stability and final performance (achieved with changing single line of code). Results are reported using dopri5 sampler.
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Reviews:
[{"review_id": "72C7w0iREj", "reviewer": "Reviewer_oGRr", "summary": "The paper introduces a framework for creating generative models using arbitrary Feller processes. The framework is based on the concept of a generator that describes the infinitesimal change of distribution of a Feller process and on a possibility to learn that generator through minimizing a training objective. Existence of such a framework allows to design novel generative approaches and perform model combinations for building multimodal generative models.", "questions": "1. The authors skip a discussion of Anderson’s result [3], which was an important brick in developing generative diffusion models. Why is it not needed in the general framework? Why does focusing on matching of probability paths remove the need for Anderson's result for generators and adjoint generators? \n2. Why does regularity assumption 2 (finite number of discontinuities) is needed? I see that the authors discard one of the terms in the proof of theorem 1. What does this term break in the framework if we do not discard it?\n3. The authors state several times through the main part of the paper that generator matching works with arbitrary Markov processes while in reality they restrict the family of Markov processes to Feller processes. Do the authors expect that restricting to Feller processes may not be needed and the whole framework can be applied to any process with just Markovian property?  \n4. In proposition 1, the object “conditional generator” appears without any definition. I also believe that $X^z_t$ appears for the first time. Could the authors provide a definition of these objects and notation before using it in the proposition? \n5. Could the authors provide a motivation behind introducing three regularity assumptions on Markov process under consideration? \n\n*Minor:*\n \n6. Page 3, line 120. “Starting with the right initial distribution $X_0 \\sim p_0$.” What is the meaning (or definition) of “right” in the sentence?\n7. Page 3. Line 096-099. The meaning of $\\delta_z(dx)$ is neither defined before nor in the formula. \n8. What does the authors mean under the words “an informal” in lines 129-130? \n9. Page 17, lines 762-763. “... the following two regularity … ” -> “... the following three regularity…”\n\n[3] Anderson, B. D. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313–326, 1982. ISSN 0304-4149.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "**Significance and originality.** The main result of the paper lies in developing a generalization of the existing generative models into one theoretical framework. For example, the universal characterization of generators (provided by Theorem 1) appears for the first time in machine learning literature. Moreover, the result is stated for time-inhomogeneous Markov processes while the vast majority of mathematical literature focuses on studying properties of time-homogeneous ones. The paper also demonstrates immediate benefits of having such a theoretical framework by designing two new generative models based on jump and diffusion processes. \n\n**Clarity and quality.** Overall, the paper is well-executed and well-written. It provides necessary background into Feller processes. The authors usually provide a motivation and an intuition behind introduced mathematical concepts. The paper gives guidance principles to build new generative models in a concise manner. The proposed generator matching framework is both mathematically sound and computationally feasible. The authors provided the empirical results on protein and image generation to demonstrate validity of the approach.", "weaknesses": "The main weakness of the paper is that the authors ignore previous results that have been done for jump processes [1, 2]. For example, [1] introduces a theoretical framework for constructing Diffusion Models in discrete-state spaces for an arbitrary Markov process, which can be either discrete time or continuous-time in nature. [1] uses generators and adjoint generators together with Chapman–Kolmogorov equations. Moreover, [1] constructs a generative model using the pure-death process, which, in its turn, is the example of jump processes. [2] focuses on the Levy process.  I believe the authors should provide a very detailed comparison with these papers in section 8 (Related Work).\n\nI believe that the exposition of the framework can be improved. It still remains unclear from the paper why the generator $L_t$ can be used in forward and reverse processes. It remains unclear why using generators and adjoint KFE leads to “correct” generated samples.\n\n[1] Santos, J. E., Fox, Z. R., Lubbers, N., & Lin, Y. T. (2023, July). Blackout diffusion: generative diffusion models in discrete-state spaces. In International Conference on Machine Learning (pp. 9034-9059). PMLR.   \n[2] Eunbi BI Yoon, Keehun Park, Sungwoong Kim, and Sungbin Lim. Score-based generative\nmodels with Lévy processes. In Advances in Neural Information Processing Systems, volume 36 pages 40694–40707. Curran Associates, Inc., 2023.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0GXsGx1U95", "reviewer": "Reviewer_rGQU", "summary": "This paper introduces the generator-matching (GM) framework for modeling arbitrary Markov processes in both continuous (i.e., $\\mathbb{R}^d$) and discrete (i.e., $\\\\{1,...,N\\\\}^D$) domains. GM generalizes the (continuous and discrete) flow-matching (FM) approach, extending its concept of transforming the complex task of learning *marginal* distributions/scores/flows/generators to a simpler task of specifying *data-conditional* distributions/scores/flows/generators, and approximating the marginal ones through a loss based on conditional expectations. The framework not only unifies a variety of existing methods across continuous and discrete generative modeling, but also holds promise for multimodal applications, where the state space comprises the product of multiple different state spaces.", "questions": "1. In appendix D.1, line 1319 suggests that the solution $\\lambda_ t(x)$ is not unique. The derivation selects the form that achieves equality on line 1319 (see line 1334). Are there alternative choices of $\\lambda_ t(x)$ that also satisfy the constraint? If so, are all choices of $\\lambda_ t(x)$ equivalent in some sense, or do they have different characteristics in practical uses (given that they all correspond to the same conditional path $p_ {t|1}(x|z)$)?\n\n2. I don't fully understand the argument in appendix D.2, starting from line 1473. Specifically, can you have a detailed elaboratation on the reason why the prior has compact support poses a potential issue (i.e., lines 1473-1474)? What implications do the non-zero boundaries of $\\tilde\\sigma_ t^2(\\cdot|z)$ have? Moreover, I hope that the author(s) could have a review of the reflected SDE (the process ${\\rm d}L_ t$) so that the readers can better understand the intuition behind how this pure diffusion solution to mixture path works.\n\n3. Minor comments:\n\n- I personally don't quite appreciate the unconventional notation $p_ t\\odot f$ and $p_ {t+h|t}\\odot f$ in this paper. Typically, $p_ t\\odot f$ would be denoted as $\\mathbb{E}_ {p_ t}f$; in standard textbooks of Markov semigroup (e.g., Bakry et al., 2014), $p_ {t+h|t}\\odot f$ in this paper would appear as $P_ {t,t+h}f$, where the semigroup $(P_ {t,s})_ {s\\in[t,\\infty)}$ is defined as $P_ {t,s}f=\\mathbb{E}(f(X_ s)|X_ t=\\cdot)$. However, I recognize these non-standard notations may be acceptable within the machine learning community.\n- Line 274, *no* diffusion coefficient? Besides, the $x_ 1$ in equation (9) seems to be a typo.\n- Line 667: $\\mathbb{E}_ {x\\sim\\mu_ 1}[f(x)]=\\mathbb{E}_ {x\\sim\\mu_ 2}[f(x)]$.\n- Line 669: $(X_ t)_ {0\\le t\\le1}$.\n- Line 725: $p_ {t+h|t}(A_ {t+h}|x)$.\n- Line 755: the limit operation is missing.\n- Line 863: should mention the definition of matrix inner product $A\\cdot B=\\operatorname{tr}(AB)$ for positive definite $A,B$.\n- Lines 901 and 968-978: should be \"with probability $...+o(h)$ when $h\\to0$\".\n- Lines 917, 1001-1017, 1050, 1060, 1067, 1072: the parentheses are missing: should be $\\int_ {y\\ne x}(f(y)-f(x))Q_ t({\\rm d}y;x)$. \n- Lines 1112-1125: $B(F_ t^\\theta(x))$ is missing.\n- Line 1155: the first generator is $\\bar{\\cal L}_ t$, not ${\\cal L}'_ t$.\n- Appendix C.6, proof for conditional generator: I suggest deriving the equation by directly applying derivative by parts to line 1178, which simplifies the proof and reduces its length.\n- Line 1264, should mention that $\\lambda_ t(x)$, $Q_ t(y;x)$ and $J_ t(y;x)$ may depend on $z$, and here this dependence is omitted for conciseness. Similarly, in line 1426, should mention that $a_ t,b_ t$ may depend on $z$ but are independent of $x$.\n- Appendix D.1, time-derivative $\\frac{\\partial}{\\partial t}p_ t$: I suggest rewriting the proof by directly calculating $\\partial_ t\\log{\\cal N}(x|tz,(1-t)^2)$, which would simplify the presentation.\n- Appendix D.2, I suggest replacing all $p_ {\\rm simple}$ with $p_ 0$.\n- Line 1421: the definitions of $G_ 0(x)$ and $G_ {z,\\sigma_ {\\min}}(x)$ have typos.\n- Line 1476: \"Let's *consider* a ...\"\n\nI will be happy to raise the score if the author(s) could address my concern.\n\n**References**\n\n- Bakry et al. Analysis and Geometry of Markov Diffusion Operators. Springer 2014.\n- Bertazzi et al. Piecewise deterministic generative models. Arxiv 2407.19448, 2024.\n- Bose et al. SE(3)-Stochastic Flow Matching for Protein Backbone Generation. ArXiv 2310.02391, 2023.\n- Campbell et al. Trans-Dimensional Generative Modeling via Jump Diffusion Models. NeurIPS 2023.\n- Campbell et al. Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design. ICML 2024.\n- Del Moral & Penev. Stochastic Processes: From Applications to Theory. Chapman & Hall 2014.\n- Gat et al. Discrete Flow Matching. NeurIPS 2024.\n- Huguet et al. Sequence-Augmented SE (3)-Flow Matching For Conditional Protein Backbone Generation. ArXiv 2405.20313, 2024.\n- Lou et al. Scaling Riemannian Diffusion Models. NeurIPS 2023.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "As discussed above, The GM framework extends the FM approach, building on the well-established relationships between marginal distributions/scores/flows and their data-conditional counterparts. From my persepective, this framework presents several notable innovations:\n\n1. The introduction of jump, a type of stochastic process in the continuous domain with non-continuous trajectories, which largely unexplored within the generative AI community. Particularly compelling is the result showing that the widely-used conditional path $p_ {t|1}(x|z)={\\cal N}(x|tz,(1-t)^2)$ can also be achieved through jumps, highlighting its potential utility for generative tasks.\n\n2. An elegant theorem establishing that the generators of interest in $\\mathbb{R}^d$ for applications can be decomposed to three basic types: flow (ODE), diffusion (scaled Brownian motion), and jump. This decomposition underscores the advantages of using the generator perspective in modeling.\n\nThe paper also introduces methodologies for: (a) combining models that use different generators, and (b) modeling multimodal data distributions. While not entirely novel, these contributions effectively systematize existing techniques through the GM perspective. Prior works have addressed these topics, for example: the equivalence of backward SDE and backward probability-flow ODE, the detailed balance condition for CTMC (Campbell et al., 2024), and the predictor-corrector for CTMC (Gat et al., 2024) are all examples of model combination; similarly, multimodal flows introduced in Campbell et al. (2024) and earlier models such as FrameFlow and FrameDiff have laid groundwork in multimodal data modeling, potentially inspiring aspects of this work.\n\nFinally, the paper is well-written, featuring clear notations, a strong motivation, and a rigorous mathematical foundation.", "weaknesses": "1. The writing style in some part of the paper, especially section 6, feels overly general and abstract. While the GM framework is indeed comprehensive, it primarily applies to only 4 specific examples: flow, diffusion, jump in $\\mathbb{R}^d$, and CTMC in finite state spaces. Although the theoretical concepts like target-affine (TA) loss and Bregman divergences are elegant, they lack practical utility in these contexts. This abstraction, without sufficient illustrative examples, limits the reader's ability to grasp the practical impact of the framework. I suggest exploring these theoretical constructs further to enrich the paper. For instance, theoretically or numerically comparing the performance of the TA-loss and discussing generator choice in more general state spaces would provide valuable insights. Additionally, a discussion on how TA-loss may facilitate the development of better training losses for generative tasks would be appreciated.\n\n2. The literature review for jump-based models is insufficient. Jump is related to piecewise deterministic process (flow + jump) and piecewise diffusion process (flow + diffusion + jump) in the stochastic process literature (see, e.g., Del Moral & Penev, 2014), and there have been several works leveraging such process for generative modeling, e.g., Campbell et al. (2023) and Bertazzi et al. (2024), which the author(s) have not mentioned. Including these citations would better position GM within the broader literature and highlight its contributions.\n\n3. The experiments are not convincing enough and lack the design details necessary for fair comparisons.\n\n- For $\\mathbb{R}^d$, the paper introduces two constructions of the conditional probability path (mixtures and geometric averages), each with realizations based on flow, diffusion, jump, or a mixture of these (Markov superposition). The toy example in figure 2 is worth further explorations: (1) Which conditional probability path is recommended for better empirical performance? Comparing the performance on more complex target distributions, such as $6\\times6$ checkerboard patterns (Lou et al., 2023), would be informative. (2) Although these processes all generate the same marginal probability distribution in theory, practical implementations face approximation and discretization errors. Which realization provides the best sampling quality? (3) One potential advantage of jump over flow and diffusion is the ability of exploring complicated landscape of target distribution due to its non-continuous nature, but this benefit isn’t evident in the experiments. Therefore, it remains unclear whether the new algorithms offer practical advantages over existing methods, such as FM.\n\n- The details of image generation experiment are lacking, such as the model size, neural network structure, and sampling schedule. The baselines (DDPM, VPSDE, and EDM) are outdated, and the datasets (CIFAR10 and ImageNet32) are less challenging than those commonly used today, such as ImageNet256.\n\n- Despite emphasizing the GM framework’s potential for multimodal data, there are no numerical results on joint multimodal generation. For example, instead of considering protein backbone generation or pure image generation (which are unimodal tasks where existing algorithms already achieved amazing performance), evaluating GM on protein structure-sequence co-generation or image-text joint generation (which has been mentioned in section 7.2) would better showcase its advantage in multimodal settings.\n\n- For the protein experiments, critical experimental details are absent, such as sampling NFE, training time, and model size. The target task is restricted to protein with length less than $128$, a relatively unchallenging setting. Designability, as an important metric for evaluating protein backbone, is not presented. Additionally, more up-to-date work like FoldFlow (Bose et al., 2023) and FoldFlow-2 (Huguet et al., 2024) are missing from the benchmarks. The performance gain achieved by the proposed approach seems to be minimal at the cost of training a separate network with more computational resources. Thus, its efficacy for protein generation remains unclear, especially in comparison to existing methods. It would certainly be more compelling to apply GM to multimodal tasks in protein design by incorporating additional sequence information.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ktn4lDXzgq", "reviewer": "Reviewer_MFMu", "summary": "Generator matching is introduced as an ambitious unification of multiple generative modeling methods, both in continuous and discrete spaces. This unified design space encompasses jump processes and in general any Markov process. Experiments on protein modeling and generative image modeling.", "questions": "1. What are the limitations of GM? Is the linearity of the operators a limiting factor in the modeling capacity?\n2. Line 128. What exactly is meant with *\"simple\"*. If I understand correctly, it needs to be linear, why not use that word?\n3. Figure 2. What is *ME* in the legend? This figure could benefit some extra attention. Larger text size, match the naming in the legends with the vertical namings of the rows.\n\n## Minor remarks/typo's:\n\n- The use of $dx$ is somewhat confusing to me. I guess it is not the product of $d$ and $x$, is it supposed to signify $\\mathrm{d}x$?\n- On line 124: (1) and (2) hyperlink to seemingly random equations in the appendix. The same on line 401.\n- Line 429: is *the* probability kernel\n- Line 447: the informal use of *just increase number of channels* can be distracting to readers. In general the Experiments section could benefit from some additional refining.\n- Equation numbering is inconsistent. Absent for large parts of the appendix. Also in the main text numbering is sometimes absent (e.g. lines 223, 251)", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper has the ambitious goal of presenting a unifying framework for many (currently popular) generative modeling paradigms. Even unifying continuous and discrete models, and jump processes. This can be very relevant for the community, and the theory seems adequate and well supported.\n\n2. Experiments are performed on diverse applications: proteins and images", "weaknesses": "1. It is unclear to me, and difficult to gauge, in how far the GM framework has benefits over *simple, naive* combinations of different modeling paradigms. I.e. how will the GM framework help future researchers in practice to build better models, other than only providing a theoretical unificiation?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "e9YxMjjgWf", "reviewer": "Reviewer_RJkg", "summary": "The paper introduces a generalized framework called *generator matching* to train diffusion-based models for continuous-time Markov processes, encompassing various well-known models as special cases. In particular, the proposed framework provides a consistent theory for training generative models of continuous-time Markov processes within the class of Feller processes. Feller processes include various types of continuous-time Markov processes, such as Ito diffusions, continuous-time Markov chains (CTMCs), jump processes, and continuous-time normalizing flows.\n\nSpecifically, this paper first introduces some general properties of the continuous-time Markov processes to motivate the proposed method. Many continuous-time Markov processes of interest exhibit two main characteristics. First, their transition probability follows the Markov property. Second, sample paths created by Markov transitions exhibit some form of continuity (right-continuous or left-continuous). These two characteristics allow the averaged behavior of the transition probability at an infinitesimal time scale to be expressed by a gradient operator on the state space $S$. This operator is known as the (infinitesimal) generator of a Markov process and is uniquely defined for each transition probability. Notably, this generator is not simply a gradient evaluated at a point in the state space $S$ but an operator that yields the gradient when applied to a function $f$ on the process. It is essential to remember that a Markov process is uniquely determined by its transition probability (and initial distribution). Therefore, this averaged gradient operator, or generator, along with the initial distribution, uniquely determines the Markov process. The paper proposes a training method for a model generator based on sample paths of a reference Markov process, in particular, ones to bridge or connect two distributions, as seen in bridge matching or flow matching.\n\nWhile similar works on generative modeling for Feller processes exist [1], this paper proposes a novel parameterization that directly models the generator of Markov processes. This enables direct training of the generator, in contrast to other parameterizations such as score-based training in Euclidean spaces or probability ratio training in discrete state spaces. Recall that the (infinitesimal) generator of a continuous-time Markov process is a type of averaged gradient operator on the stochastic process. When this gradient operator is applied to a function $f$ on the state space, it typically comprises a combination of the gradient, Hessian, divergence, or difference of $f$ and the transition probability-related parameters. The specific form of the generator depends on the chosen Markov process. However, the paper emphasizes that the generator for most Markov processes of interest can be expressed as $\\langle \\mathcal{K}f, F \\rangle$, where $\\mathcal{K}$ is a linear operator independent of the Markov process’s transition probability, making it parameter-independent. Thus, $F_t$ uniquely determines the generator, and the paper proposes to train $F_t$ directly—a technique called *generator matching*. The paper specifically employs a Bregman divergence-based objective for this approach.\n\nThis theory applies to all Feller processes, including continuous-time Markov processes like Ito diffusions, continuous-time Markov chains, jump processes, and continuous-time normalizing flows. Moreover, since generators are linear operators, defining a Markov process with a transition probability that combines various types of transition probabilities becomes straightforward: the generator can simply be represented as the sum of the original generators. This approach enables an intuitive understanding of combined Markov processes by summing their respective generators.\n\nUnfortunately, generator matching faces the same challenges as (explicit) score matching, particularly since the exact $F_t$ of the reference processes is generally unknown. To address this, the authors show that $F_t$ at time $t$ can be reformulated as the expected value of an initial value-conditioned $F_t$ (with the expectation taken with respect to the posterior distribution of $X_0$ given $X_t$). Using this insight, the paper proposes *conditional generator matching*, which serves a similar function to denoising score matching in Euclidean diffusion-based models.\n\nFinally, the paper demonstrates the effectiveness of the proposed method through various experiments.\n\n[1] Joe Benton, Yuyang Shi, Valentin De Bortoli, George Deligiannidis, and Arnaud Doucet. From denoising diffusions to denoising markov models. Journal of the Royal Statistical Society Series B: Statistical Methodology, 86(2):286–301, 2024.\n\n----\nI have updated the overall rating from 6 to 8 and the soundness & presentation from 3 to 4 after the authors' rebuttal.", "questions": "N/A", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "In my understanding, the paper's contributions are clear, and I consider that the results are essential for several reasons.\n\nFirst and foremost, the authors introduce a unified framework that combines a unified theory to explain how distinct Markov processes, defined across various state spaces, can be represented within a consistent language. While this framework may be not be novel itself, the paper exploits this framework to propose a novel method for training these processes in the context of generative modeling. Specifically, the paper’s new generator-based parameterization leverages this unified theory, offering a streamlined and theoretically grounded approach to model various Markov processes in a consistent fashion.\n\nSecond, beyond recovering existing diffusion-based models within the unified framework, the paper uses this framework to establish innovative classes of generative models. For example, one of new class straightforwardly combines different types of Markov processes, paving the way for flexible yet rigorous generative modeling approaches.\n\nI believe this paper provides invaluable insights for comprehensively understanding various diffusion-based models (including flow matching, bridge matching, and others). It offers a cohesive view that can unify these models within a broader framework. This paper could very well become a must-read for those interested in this topic, given the clarity it brings to understanding and advancing diffusion-based generative models.", "weaknesses": "I find the paper original and significant overall. However, the presentation should be improved. In particular, there exists some significant notational issue over the paper.\n\nFirst, the author suggests that $ p_t $ can be interpreted both as a measure function (a set function on a $ \\sigma $-algebra of the state space $ S $) and as a function on $ S $. This dual interpretation creates confusion in the derivations and leads to inconsistency in the notation. Additionally, it introduces an unnecessary operator $ \\odot $, which is intended as an inner product. For the proofs in this submission to be correct, several notational revisions are necessary. Specifically, $ p_t $ should be defined as a density function, represented as a Radon-Nikodym (RN) derivative with respect to a reference measure $ \\mu $ in a general setting.\n\nHere, $ \\mu $ denotes a reference measure on the state space $ S $, such as the Lebesgue measure for $ \\mathbb{R}^n $ or the counting measure for a countable set. Consequently, $ p_t(x) = \\frac{d\\mathbb{P}_t}{d\\mu}(x) $ represents the RN-derivative with respect to $ \\mu $, often referred to as a probability density function in the Euclidean setting or as a probability mass function in a discrete (particularly countable) state space.\n\nAs a result, all equations involving $ \\odot $ can be rewritten in terms of an inner product. For example, Equation (2) can be expressed as\n$$\n\\mathbb{E}[f(X_t)] = \\int f(x) p_t(x) d\\mu(x) = \\langle p_t, f \\rangle.\n$$\nAnother example is Equation (6), which becomes\n$$\n\\partial_t \\langle p_t, f \\rangle = \\langle p_t, \\mathcal{L}_t f \\rangle.\n$$\n\nSecond, the notational issue discussed above introduces further complications, which undermine the motivation for using the $ F $-parameterization rather than directly parameterizing the generator $ \\mathcal{L} $. In my understanding, for the continuous-time Markov processes mentioned, the generator $ \\mathcal{L}f $ can be represented as $ \\langle \\mathcal{K}f, F \\rangle $, where $ \\mathcal{K} $ is a linear operator independent of the transition probability of the Markov process (thus parameter-independent). This formulation, in my opinion, effectively motivates the $ F $-parameterization. For example, in the case of a continuous-time Markov chain (CTMC),\n$$\n\\sum_{y} ( f(y) - f(x) ) Q(y; x) = \\Delta f(x)^\\top Q(x),\n$$\nwhere $ \\Delta f(x) = (f(x_1) - f(x), f(x_2) - f(x), \\dots) $ and $ Q(x) = (Q(x_1; x), Q(x_2; x), \\dots) $. With this notation, transitioning from matching the generator to matching the $ F $-parameterization becomes straightforward. Moreover, it allows the use of well-known objective functions, such as mean-squared error minimization, alongside the Bregman divergence-based loss. This expands the flexibility and applicability of the model training approach.\n\nAs previously mentioned, I consider this paper to be an excellent work that provides valuable insights into unifying various diffusion-based models (including flow matching, bridge matching, etc.). For those interested in this topic, I believe that it could very well become essential reading. However, due to the significance of the notational issues, there is a risk of widespread misunderstanding if these inconsistencies persist. While I believe that reviewers should respect the authors’ narrative, including their notations, I also believe that it is essential to avoid reinventing standard notations with new one, especially when these are well-established in standard textbooks. This is particularly important since once published, changing the notation would be extremely challenging. Therefore, I am inclined to assign a lower score at this stage. I am, however, open to raising the score if the paper is revised to address these issues.\n\nDespite my willingness to increase the score following revisions, I must express a concern based on recent experience: I have frequently reviewed papers with similar notational issues where authors promised corrections but ultimately did not make the necessary changes. As a result, this time, I will not consider adjusting the score unless these notational issues are thoroughly addressed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)