Latent Stochastic Interpolants
Authors: Saurabh Singh
Dmitry Lagun
Venue: ICLR 2026 Poster (2026)
Keywords: Generative Models
Stochastic Interpolants
Flow Models
Primary area: generative models
TL;DR: Novel formulation for joint training with stochastic interpolants in a latent space enabling simultaneous learning of encoder, decoder and latent space generative model.

Abstract: Stochastic Interpolants (SI) is a powerful framework for generative modeling,
capable of flexibly transforming between two probability distributions.
However, its use in jointly optimized latent variable models remains unexplored
as it requires direct access to the samples from the two distributions.
This work presents Latent Stochastic Interpolants (LSI) enabling joint learning
in a latent space with end-to-end optimized encoder, decoder and latent SI
models. We achieve this by developing a principled Evidence Lower Bound (ELBO)
objective derived directly in continuous time.
The joint optimization allows LSI to learn effective latent representations
along with a generative process that transforms an arbitrary prior distribution
into the encoder-defined aggregated posterior.
LSI sidesteps the simple priors of the normal diffusion models and mitigates the
computational demands of applying SI directly in high-dimensional observation
spaces, while preserving the generative flexibility of the SI framework.
We demonstrate the efficacy of LSI through comprehensive experiments on the
standard large scale ImageNet generation benchmark.

Published as a conference paper at ICLR 2026
LATENT STOCHASTIC INTERPOLANTS
Saurabh Singh*
Dmitry Lagun Google DeepMind d1 dllagnn@google.com
Poetiq AI iq.raabbepoetiq..a.
ABSTRACT
Stochastic Interpolants (SI) is a powerful framework for generative modeling, capable of flexibly transforming between two probability distributions. However, its use in jointly optimized latent variable models remains unexplored as it requires direct access to the samples from the two distributions. This work presents Latent Stochastic Interpolants (LSI) enabling joint learning in latent space with end- to-end optimized encoder, decoder and latent SI models. We achieve this by developing principled Evidence Lower Bound (ELBO) objective derived directly in continuous time. The joint optimization allows LSI to learn effective latent representations along with a generative process that transforms an arbitrary prior distribution into the encoder-defined aggregated posterior. LSI sidesteps the simple priors of the normal diffusion models and mitigates the computational demands of applying SI directly in high-dimensional observation spaces, while preserving the generative flexibility of the SI framework. We demonstrate the efficacy LSI through comprehensive experiments on the standard large scale ImageNet generation benchmark.
INTRODUCTION
Diffusion models have achieved remarkable success in modeling complex, high-dimensional data distributions across various domains. These models learn transform simple "prior" distribution Po. such as a standard Gaussian, into a complex data distribution p1. While early formulations were constrained to use specific prior distributions that are Lévy Stable, recent advancements, particularly Stochastic Interpolants (SI) (Albergo et al., 2023) offer powerful, unifying framework capable of bridging arbitrary probability distributions. However, assumes that both the prior Po and the target Pı distributions are fixed and the samples from both are directly observed. This requirement limits their use jointly learned latent variable models where the generative model learned, along with encoder and decoder, in a latent unobserved space. Further, the latent space, often lower dimensional, evolves as the encoder and decoder are jointly optimized. Lack of support for joint optimization implies that arbitrary fixed latent representations may not be optimally aligned with the generative process resulting in inefficiencies.
To address this, present Latent Stochastic Interpolants (LSI), novel framework for end-to-end learning of a generative model in an unobserved latent space. Our key innovation lies in deriving a principled, flexible and scalable training objective continuous time. This objective, like SI, provides data log-likelihood control, while enabling scalable end-to-end training of the three components: an encoder mapping high-dimensional observations to a latent space, a decoder reconstructing observations from latent representations, and latent SI model operating entirely within the learned latent space. Our approach allows transforming arbitrary prior distributions into the encoder- aggregated posterior, simultaneously aligning data representations with a high-fideligg generative process using that representation.
Evidence Lower Bound (ELBO) directly in
LSI's single ELBO objective provides a unified, scalable framework that avoids the need for simple priors of the normal diffusion models, mitigates the computational demands of applying directly in high- -ish-iimensioal observation spaces and offers an alternative to ad-hoc multi-stage training. Our formulation admits simulation-Fis training analogous to observation-spar diffusion and SI models, while preserving the flexibility of SI framework. We empirically validate LSI's strengths through
Work done while DeepMind.

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Published as a conference paper at ICLR 2026
comprehensive experiments the challenging ImageNet generation benchmark, demonstrating competitive generative performance and highlighting its advantages efficiency.
Our key contributions are: Latent stochastic interpolants (LSI): novel and flexible framework for scalable training of a latent variable generative model with continuous time dynamic latent variables, where the encoder, decoder and latent generative model are jointly trained, 2) Unifying perspective: a novel perspective on integrating flexible continuous-tine formulation of SI within latent variable models, leveraging insights from continuous time stochastic processes, 3) Principled ELBO objective: a new ELBO as principled training objective that retains strengths of SI - simple simulation free training and flexible prior choice while enabling the benefits of joint training in latent space.
2 BACKGROUND
Notation. We use small letters 2, y. etc. to represent scalar and vector variables, f. etc. to represent functions, Greek letters (3, etc. to represent Jlyper-)parametent Lower case letters are used to represent both the random variable and particular value ~ p(æ). Dependence on an argument & indicated as a subscript u, or argument u(t) interchangeably.
Our work builds upon two key results briefly reviewed below. The first result (Li et al. 2020 Theodorou] 2015) states an Evidence Lower Bound (ELBO) for models using continuous time dynamic latent variables. We state a more general form than the original aid the discussion of the prior distributions. The second result is well known method for constructing a stochastic mapping between two distributions. We exploit it to construct variational approximation in the latent space.
2.1 VARIATIONAL. LOWER BOUND USING DYNAMIC LATENT VARIABLES
Consider two SDEs. starting with the same starting point 30 = 20 at t = 0, sharing the same dispersion coefficient (24, t) but potentially different initial distributions - 20 Po(2)) ) for the model, with path measure Po, and ~ 40(20) for the variational posterior, with path measure Q
dē, = ho( t)di + o ( ()dur, dæ, ha(2,, t)dt + o (z,, t)dwr,
(model, path measure Po)
(variational posterior, path measure
(2)
Where W; and wi Wiener processes under corresponding path measures. The first equation be viewed as the latent dynamics under the model hg we are interested in learning and the second the latent dynamics under some variational approximation produce samples 2, Further, let Ht, be observations at time & that are assumed to only depend on the corresponding unobserved latent state 24,, then the ELBO be written
the posterior that be used to
4o (20)
/ [lu(zr)))] 1)1 dl
In ( Ba
In po (xre, (2c,) In
(3)
In po (Ze, (2r.) (LL(Q()P))
Where satisfies
a(2, (e,,,))(20) ho(2,t) ho(2,t)
(5)
We provide the proof of the above general form in Section A Similar to the ELBO for the VAEs (Kingma et al. 2013), the first second term penalizes the mismatch between the variational and model path distributions. In the following, we focus on the case of go Po and draw attention the general case when needed.
in eq. (4) explains observations given the latent path and the
2.2 DIFFUSION BRIDGE
Given two arbitrary points 20 and 21, diffusion bridge between the two is random process constrained to start and end at the two given end points. A diffusion bridge can used to specify the stochastic dynamics of particle that starts at 2g at and is constrained to land at 21 at 1.
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Consider a stochastic process starting at 2g with the dynamics specified by eq. (2). Using Doob's 's h-transform, the SDE for the end point conditioned diffusion bridge, constrained to end at 21 at time t= 1 can be written
(6) where p(= the conditional density for 2 under the original dynamies in eq. (2) and depends on has Note that Brownian bridge is special case of Diffusion bridge where the dynamics are
d - [ 1) + (ozz, (ifiz,)) V,, (=))(==)) + ( L)dw,
specified by the standard Brownian motion. Diffusion bridges can be used to construct a stochastic
mapping between two distributions by considering the end points 20 po(zo) and 21 p1 (1(2)) to be sampled from the two distributions of interest.
LATENT STOCHASTIC INTERPOLANTS
Stochastic Interpolants (SI) and their limitation: SI (Albergo al. 2023) is powerful frame- work for generative modeling, capable of learning a model that can flexibly transform between two probability distributions. Let p(x11 be observation from the data distribution p(æ1) that we want to sample from. In SI framework, another distribution po (o(z)) is chosen as a prior with samples To ~ (xa).. Typically, Po easy to sample from, e.g. a Gaussian distribution. A stochastic inter- polant is then constructed with the requirement that the marginal distribution (x(22) of xı equals Po at = 0 and P1 at / 1. For example, the interpolant x = (1 - (1-1)ag + + + V(1 - -(j,,,,, satisfies this requirement. The velocity field and the score function for the generative model are then estimated as solutions to particular least squares problems. The learned velocity field and the score function can then be used to transform a sample from po to produce a sample from p1. SI requires that the samples 2o and Ti are observed, though Ii could be an output of a fixed model, hence still observed. We use the term observation space SI to emphasize this.
N(, T)
However, we interested in jointly learning generative model in a latent space to leverage efficiency of low dimensional representations while also aligning the latents with the generative process. Therefore, we want to jointly optimize an encoder pø (21 (x1 that represents high dimensional observations in the latent space and a decoder pe (21[21) that maps given latent representation to the observation space, along with the generative model in latent space. To use SI, we need to interpolate between fixed prior Po(2o) in the latent space and the true marginal posterior Pi (21) = p(2 Jn dz). However, we only have access to the posterior model po (21) that is optimized concurrently and is approximation to the true intractable posterior. Consequently, we can not directly construct an interpolant in the latent space that satisfies the requirements of SI. the following, we address this issue by deriving Latent Stochastic Interpolants (LSI), though from an entirely different perspective than is considered by SI.
Since we want to jointly learn the generative model in latent space, we propose a latent variable model where the unobserved latent variables assumed to evolve in continuous time according to the dynamies specified by an SDE of the form in eq. (I. Let pa [2) (x1) be a parameterized stochastic decoder and he parameterized drift for eq. [1. Then, the generation process using our model is as following first sample zn Po(20) is produced from a prior (2(2), then 20 evolves according to the dynamics specified by eq. (1) using fi from using the decoder pe (z1 (21). In theory, we can now utilize the ELBO presented in section 2. to
Generative model with dynamic latent variables:
0 to f = to yield 21, and finally an observation space sample is produced
train this model. Note that, although the ELBO in eq. 3 supports arbitrary number of observations Ie, at arbitrary times f,. in this paper we focus on single observation X2 at f = 1. The ELBO in
eq. (3) needs a variational approximation to the posterior (2 (x)) which can be used to sample 21. This approximation is constructed as another dynamical model specified by the SDE in eq. (2).
Unfortunately, for a general variational approximation specified by an arbitrary hig simulating eq. (2) would lead to significant computational burden for large problems during each training iteration and open the door to additional issues resulting from approximations needed for simulation of the SDE.
Instead, explicitly construct the drift he in eq. (2) such that 2; can be sampled directly without simulation for any time t. Our scheme provides a scalable alternative that allows simulation free efficient training, is common in the observation space diffusion models.
Variational posterior with simulation free samples: Next we construct a variational posterior approximation, that enables easy sampling of 2 without requiring the simulation of the SDE in
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eq. 2 Let 21 (2 (21) be stochastic encoding of the observation #1 providing direct access to 21 at f = 1. Next, using the Diffusion Bridge specified by eq. (6) we construct a stochastic mapping between the prior Po())) and the aggregated approximate posterior (2 (ay )dx, at t 1. The diffusion bridge, coupled with the encoder (=(21())) yields our approximate posterior po(= However, p(21|2)) is unknown in general. If we additionally assume that ha(z,,1) t) h42 and
(zz, t) = or, then the original SDE in eq. (2) becomes linear with additive noise
dæp h,24dt + ordue
It is well known that for linear SDEs of the above form, the transition density p(zelza), > gaussian (z; Oat bacl) ) (see section G) for some functions asts bat that depend on hes dr. Consequently, compute V., In pm2 24) for given 2 as
v. Imp(112))) 4.1 dx1
(8)
b
The transformed SDE in terms of the simplified drift and dispersion coefficients be expressed as
dz, = [h, 22 + of In (22112=)]dt + ordw;
(9)
Further, if we condition on the starting point zo, then the conditional density p(=x|=1, 20) can be expressed as following using the Bayes rule
p(=+=11 21, 20) p(z1l=r. [-t, 2o =o)p(z))2))
(10)
p(21 =20)
p(=1|20)
where p(z1/zz, 20) p(21 because of the Markov independence assumption inherent in eq. (2) Note that all the factors on the right are gaussian. It can be shown that the conditional density (122221, 20) is also gaussian if the transition densities are gaussian and takes the following form
p(2,221,20 = (
b01 2T borbri
cxp (-1 bo1
2x bor &1 + br ao
I')
(11)
2 bn.bar
bp
Where a(-), b(.) are constant or time dependent scalars and d is the dimensionality of 2,- Their specific forms depends on the choice of h4, d, Refer to section for details. 2, can now be directly sampled without simulating the SDE, given sample za and the encoded observation ž1. Note that the assumptions made for eq. while restrictive, do not limit the empirical performance.
Latent stochastic interpolants: We can now define latent stochastic interpolants using reparame- terization trick in conjuction with eq. (IT) to parameterize 2n
e + #5 + U:
N(O,
(12)
For some functions ne, Re. 24 that depend on a()). b(g- Note that 17u = 171 0, Ku 11 0, K1 Vo = 1 since is sampled from a diffusion bridge with the two end points fixed at 21. Equa- tion (2) specifies a general stochastic interpolant, akin to the proposal in (Albergo al. 2023). but now in the latent space. If we choose the encoder and decoder to be identity functions, then above be viewed as an alternative way to construct stochastic interpolants in the observation space. Instead instead choose K4, Vi and infer the corresponding h4, ot. For example, leads to oe = O, constant, and we arrive at the following
of choosing h4, oe first,
choosing K4 = f, 1/-
((1 l)e 121 + (1 1)=o,
(O,
(13)
See section T for a detailed derivation. We use the above form for all the experiments in the paper. Further, if Po(2)) is chosen to be standard gaussian then the interpolant simplifies to t)zo (section M). With the above interpolants, we can now define the ELBO and optimize it efficiently with simulation free samples 2, We also derive the expressions for variance preserving choices of Kr = Vt, nz + vg = in section K however we do not explore this interpolant empirically.
2, |21 v(L (1-t) ()
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Constructing training objective using ELBO (eq. (3): We first define u(2,. using eq.
(9
u(2e,) 1) = of [1+2 2: + of V., In [=+) hz: 1)
(14)
For the general latent stochastic interpolant 2 The + K,21 2t20 (eq. (12). we show that u(2,, t) takes the following form
[(dr
of deg
du
u u t) =0;
20 ho (z(zz,))
(15)
dt
2np
dt
dt
See section H for the proof. This u(z,,t) t) can be substituted into the ELBO eq. training objective. For example, with the choices L, = 1 1, we get ha 2e,
3 to construct
t
u( o
(16)
See sectionDfor for details. We write generalized loss based on the ELBO as
8t t 2
Bptthp(er,:oopo0a in (=))p==/))
In (x +
o
+ 21 20 h2 t)
(17)
Where 3r (discussed further in section 4 is relative weighting term, similar in spirit to 8 VAE (Higgins et al.] 2017 Alemi et al.) (2118). allowing empirical re- babaalnncigg for metrics of interest, c.g. FID. Above loss is reminiscent of the SI training objective, but with an additional reconstruction term and the interpolants &n arising from the variational posterior. We use this training objective for all the experiments, and optimize it using stochastic gradient descent to jointly train all three components encoder pa (=(zi [æ1).. decoder pe i
(z)) ) and latent SI model ha (ze, t). Note that we choose pe [21) be conditional gaussian in all experiments, resulting in simple L2 decoder loss.
Observation-space stochastic interpolants: To elucidate the connection with observation-space SI (Albergo al. 2023) we derive the corresponding training objective in our framework, yielding:
Bz 2
(pp())(a1.220pp) (ar .m0)
c + x1 - To - ho (2 t)
(18)
where 3r has the same interpretation as in eq. 17). with Br oー- "correpponding to exact ELBO. See Section[] detailed proof. Comparing with the LSI loss (eq. (17)) the observation-space ELBO
precisely the LSI objective with the reconstruction term - In po [=1 _ removed and replaced by LSI recovers observation-spe space stochastic interpolants when the encoder and decoder are identity functions. All parameterizations (Section 4 and sampling procedures (Section (5) apply directly with
z replaced by x. Lastly, the likelihood control property of the above objective is trivially established the objective corresponds to (L[Q)[(p) for 3, -2 and KL(P) llpo) s KL(QQ[(P)) (eq. (41p). where p1 is the true data distribution and pa is the data likelihood under the model.
Learnable priors: When the prior Po is parameterized (e.g., po(2)) N(uo, 2,,), the default construction above uses the same learnable prior for both processes (qo = Po), so KL (u [Pp)) = 0 and the ELBO retains the same form. The prior parameters distribution of 2n in the path integral EgI ||u|| df), and gradients flow through 20 ~ po(2)) via the reparameterization trick. Alternatively, if the variational process uses fixed reference go * po. the kT (Ipe) term appears as an additional regularizer penalizing deviation from the reference. Same carries over the observation space stochastic interpolants as well.
still learned: they affect the
PARAMETERIZATION
Directly using the loss in eq. 17 leads to high variance in gradients and unreliable training due to the -I-t t in the denominator of the second term. Consequently, we consider several alternative parameterizations for the second term, including denoising and noise prediction (see section C for details). Among the alternatives considered, we found the following parameterization, referred to as it in all our experiments.
InterpFlow, to reliably lead to better results and
3
ovie + vI ((21 zo) + vlza ho
(19)
2
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Where ho(ze, t)
vtz, + vI -i--thhlle t2,(2,, t) and B, 3/(1 -) is time t dependent weighting term, with 3 a constant. Instead of explicitly using the weights 5i. due to 1 - -t in the denominator, we consider a change of variable for t with the parametric family t(s) = 1 (1 s) with 24(0, 1] uniformly sampled. It can be shown that p(t) x (1 - the reweighting and we simply set B3 = 1, constant. Empirically, we found that value of c = 1 (i.e. a uniform schedule) works the best for all parameterizations during training and sampling, except for NoisePred and Denoising, which preferred R during sampling. < 1 led to degradation FID. Figure in appendix visualizes ((8) for various values of c. While the ELBO suggests using 1/o² we compute the two terms in eq. (17) as averages and experiment with different weightings. When used with optimizers like Adam or AdamW, can be interpreted as the relative weighting of the gradients from the two terms for the encoder po [21)) A lower value of 3 leads the encoder to focus purely on the reconstruction and is akin to using a pre-trained encoder-decoder pair as 3 + 0. A higher value of 3 forces the encoder adapt its representation for the second term as well. We empirically study the effect of 3 in the experiments.
+) (1-t)}-a
therefore the change of variable provides
5 SAMPLING
For the InterpFlow parameterization, the learned drift hg (21. [z) t) is related to the original drift ho (2e, t) as tz) = (h(2, t) vtz)/vI-z (see section F22 We can sample from the model by discretizing the SDE eq. ( where o, = for the choices of Kg = t. = 1 1- However, to derive flexible family of samplers where we can independently tune the dispersion without retraining, we exploit Corollary 1 from Singh Fischer (2024) to introduce a family of SDEs with the same marginal distributions as that for eq. (1)
dz = ha 1) (1 - 7o002 V. Impelze) dt + nadw,
(20)
2
Where 7t 2 0 can be chosen control the amount of stochasticity introduced into sampling. For example, setting = 0 yields the probability flow ODE for deterministic sampling. In general, to use eq. 20p for Yt * 1, the score function Vz, In (z(2)) is needed well. For the interpolant
2, o / f(1 t) + t2 + (1 t) the score can be estimated using
E [[c|2e]
V Inp)
(21)
o / - t)
See section for the proof. However, for Gaussian 20, score can be computed from the drift ho t) (Singh & Fischer 2024) as following (see section D for details)
(exr) = -2 + the (2t, t)
(22)
Section F provides detailed derivation of samplers for various parameterizations. For classifier free
guided sampling (Ho Salimans| [202 Xie et al. [024] Dao et al.) 2023 Zheng et al. 2023 Singh & Fischer 2024), we define the guided drift a linear combination of the conditional drift ho (2, c)
and the unconditional drift ho( f, c = 2) as
(x,, (,) = (1 + A)ho(z1,4,)) - Xho(z,,,c
(23)
where À is the relative weight of the guidance, c is the conditioning information and = Ø denotes no conditioning. Note that À = -1 corresponds to unconditional sampling, À = corresponds to
conditional sampling and À > 0 further biases towards the modes of the conditional distribution.
6 EXPERIMENTS
We evaluate LSI on the standard ImageNet (2012) dataset (Deng et al.) 2009 Russakovsky et al., 2015). We train models at various image resolutions and compare their sample quality using the Frechet Inception Distance (FID) metric (Heusel et al. 2017) for class conditional samples. All models were trained for 1000 epochs, except for the comparison in table I which reports FID at 2000 epochs. All results use deterministic sampler, using % = 0, unless otherwise specified. A key implementation detail to note is that the encoder uses normalization and tanh to bound the scale of the latents. See sections and P for additional details.

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Table LSI enables joint learning for SI and cheaper sampling: The latent space models achieve FID similar to observation space models of comparable size. However, the latent space model L has fewer parameters (reported millions (M) and FLOPs (reported in Giga (G)). as part of the parameters live in the encoder E and the decoder D. During sampling, encoder is not used, decoder is used only once, while the latent model is run repeatedly, once for each sampling step. Therefore, FLOP savings from computationally cheaper latent model accumulate with sampling steps.
FID @ 2K epochs
Params (M)
Flops (G)
Resolution
Latent
Observ.
Latent (E/D/L)
Observ.
Latent (E/D/L)
Observ.
64 x 64
2.62
2.57
392 (5/5/382)
398
15/15/161
201
128 x 128
3.12
3.46
392 (5/5/382)
400
59/59/327
466
256 x 256
3.91
3.87
393 (5/5/383)
405
240/240/450
1288
42
18
6
17
Fixed c
41
BSNR
Learned
40
4
4
10-6
10-5
10 10-4
10
2
4
Encoder Scale
-10
Figure 1: Effect of loss trade-off and encoder noise scale e: In the left panel, we evaluate the effect of loss trade-off weight /3 for 128 x 128 models and observe that FID improves with 3, until the degradation in reconstruction quality (PSNR) starts degrading FID. In the right panel, we evaluate the effect of encoder noise scale on FID. We also plot the FID for model with learned scale as dashed line. A deterministic encoder performs the worst (c = 0), with FID improving with until degrades again. Encoder with learned (dashed line) is outperformed by fixed in our experiments.
LSI enables joint learning for SI While SI doesn't t allow latent variables, LSI enables joint learning of Encoder (E), Decoder (D), and Latent SI models (L). In table [w compare FID across various resolutions for LSI models against SI models trained directly in observation (pixel) space. LSI models achieve FIDs similar to the observation space models indicating on par performance in terms of the final FID. Models for both were chosen with similar architecture and number of parameters and trained for 2000 epochs. Reference comparison with other methods provided in section R
LSI enables computationally cheaper sampling: In table we also report the parameter counts (in millions) as well as FLOPs (in Giga) for the observation space SI model as well as E, and L models for the LSI. For the latent L. model, FLOPs are reported for single forward pass. First note that the parameters LSI are partitioned across the encoder E, the decoder and the latent models. At sampling time, encoder run multiple times, once for each step of sampling. Therefore, while the overall FLOP count for LSI and Observation space SI models similar for single forward pass, sampling with multiple steps becomes significantly cheaper. For example, sampling with 100 steps leads to 73.6% reduction in FLOPSs for sampling 128 128 images and 48.6% for 256 x 256 images.
used, decoder is used only once, while the latent model is
Joint learning is beneficial: In fig. Ileft panel) we plot the FID the weighting term 8 is varied (eq. (19). ). A higher 3 forces the encoder to adapt the latents more for the second term of the loss. We observe that FID improves 3 increases, going from 4.53 (for 8 0) 3.75 (+s 17% improvement) for 3 = 0.0001, indicating that this adaptation is beneficial for the overall performance. Eventually, FID worsens 3 is increased further. We also plot the reconstruction PSNR for each of these models
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Table Joint training helps mitigate capacity shift:: We evaluate the effect of moving first is and last & convolutional blocks from the latent model I. to encoder and decoder respectively, for 128 x 128 resolution models. This results in the overall parameter count staying roughly the same, but the number of FLOPs required for sampling changing significantly. We observe that the model trained with 3 > 0 perform better and maintains FID well, in comparison to the independently trained model (3 + 0), even when capacity is shifted away from the latent model L,, resulting in 8.5% reduction in
FLOPs for sampling from =
0 to
k
FID (3 > 0)
FID (3 + 0) #aarass. (E/D/L)
FLOPs (E/D/L)
0
3.76
4.31
392 (5/5/382)
59/59/327
3
3.91
4.55
389 (9/8/372)
68/66/313
6
3.96
4.87
387 (13/12/362)
75/73/299
9
4.61
4.98
383 (16/16/351)
82/80/284
in orange and observe that increasing 33 essentially trades-off reconstruction quality with generative performance. For too large B, poor reconstruction quality leads to worsening FID. The dashed line indicates the performance when the encoder-decoder are trained independently of the latent model, limit of B -> 0. We implement it stop gradient operation in implementation, where the gradients from the second term of the loss are not backpropagated into To further assess the benefits of joint training, in table we compare the FIDs between jointly trained model (B >> 0) and independently trained model (B 0) as parameters are shifted from the latent model L to the encoder E and decoder models, by moving first i and last & convolutional blocks from the latent model to the encoder and the decoder respectively. While this keeps the total parameter count roughly the same, the number of FI.OPs required for sampling changes significantly. The jointly trained model performs better and maintains FID well even when capacity shifts away from the latent model, resulting in 8.5% reduction in FLOPs required for sampling from it t k 6.
Encoder noise scale affects performance: The stochasticity of the encoder (a (z) has significant impact on the performance. We parameterize the encoder as a conditional Gaussian (o (1), Ex(r)) where E(xx) is assumed to be diagonal. We experimented with purely deter- ministic encoder (Eo(x) = O), learned Eg (xr) and constant noise Eg(x) = cT. In fig. right panel) we plot FID as the encoder output stochasticity c is varied. Dashed line indicates performance with learned Eg(r) (x). A deterministic encoder (c 0) performs poorly. FID improves as the noise scale c increased, until eventually it degrades again. While learned Dig (z) (dashed line) performs well, fixed models achieved higher FID.
InterpFlow parameterization performs better than alternatives: In table 3 we compare dif- ferent parameterizations discussed in section and section C The InterpFlow parameterization consistently led to better FID. Both OrigFlow and NoisePred parameterizations exhibited higher variance gradients and noisy optimization. While Denoising parameterization resulted in less noisy training, InterpFlow parameterization led to fastest improvement in FID.
LSI supports diverse Pn: In table we report FID achieved by LSI using different prior Po(2)) distributions. While Gaussian Po performs the best, other choices for Po yield competitive results indicating that LSI retains one of the key strengths of SI support for diverse Po distributions. See section [ for additional details. To allow flexible sampling using eq. 220). we modified latent SI model output extra output channels and augmented the loss with another term to estimate E[e]æ:]. Equation 21 was used to compute the score and sample with the deterministic sampler using 7 0.
LSI supports flexible sampling: In fig. 2 and fig. we qualitatively demonstrate flexible sampling with LSI model for popular use cases. Figure 2 demonstrates compatibility of classifier free guidance (CFG) with LSI, using eq. (22). Increasing guidance weight À results in more typical samples. First 20 is sampled from po))o). Gaussian in this example, following which eq. (00) is simulated forward in time, using class conditional drift with different guidance weights X. In fig. 3| given 'Original' image (shown leftmost) is first encoded to yield it's representation 21, which is then inverted by simulating probability flow ODE (setting %/ = in eq. (20) backward in time from t = 1 to 0.
yielding 2n (similar DDIM inversion (Song et al. 2020al). Using this 2g as starting point, eq. 20
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Table 3: Effect of parameterization: We com- pare various parameterization schemes at 128 x 128 resolution. InterpFlow parameterization performs better against the alternatives.
Table 4: LSI supports diverse po: LSI retains one of the key strengths of SI support for arbi- trary po distribution. Different po achieve com- petetive FID for 128 x 128 resolution model.
Parameterization
FID @ K epochs
Po
FID @ IK epochs
OrigFlow
4.56
Uniform
4.81
NoisePred
4.73
Laplacian
4.45
Denoising
4.28
Gaussian
3.76
InterpFlow
3.76
Gaussian Mixture
4.26
==
==1.
=
A=5.
A
==0
=
==3.
==5.
Figure 2: LSI supports CFG sampling. Class conditional samples are visualized with increasing guidance weight leading more typical samples for the class. See text for details.
is simulated forward is time using 71 = >1 t) for different values of y. We show three samples for each value of 7 and observe increasing diversity with increasing 7. See section Q for additional details and results.
7 RELATED WORK
I Latent Stochastic Interpolants (LSII) draw from insights in diffusion models, latent variable models, and continuous-time generative processes. We discuss key works from these areas in the following.
Diffusion Models: Diffusion models, originating from foundational work on score matching (Vincent) 2011 Song Ermon 2019) and early variational formulation (Sohl-Dickstein et al. 2015),, gained prominence with Denoising Diffusion Probabilistic Models (DDPMs) (Ho et al. 2020). Subsequent improvements focused on architectural choices and learned variances (Nichol Dhariwal (021), faster sampling via Denoising Diffusion Implicit Models (DDIMs) (Song al. (220aa), progressive distillation (Salimans & Ho| 2022), and powerful conditional generation through techniques like classifier-free guidance (Ho& & Salimans, 2022). Further exploration of the design space (Karras et al.) 2022] 2024) has lead to highly performant models. More recently, diffusion inspired consistency models (Song et 2023) have emerged, offering efficient generation. LSI complements these with a flexible method for jointly learning in a latent space using richer prior distributions.
Latent Variable Models and Expressive Priors: Variational Autoencoders (VAEs) (Kingma et al.. 2013 Rezende et al., 2014) learn a compressed representation of data x. but are limited by the expressiveness of the prior p(z) (NVAE (Vahdat Kautz 2020). LSGM (Vahdat al. 2021 as they typically use simple priors (e.g., isotropic Gaussian). LSI addresses this by jointly learning flexible generative process the latent space, enabling powerful transformations of the simple prior. Early work (Sohl-Dickstein et al.) 2015) derived ELBBO for discrete time diffusion models, while Variational Diffusion Models (VDM) (Kingma et al. 2021) interpret diffusion models as a specific type of VAE with Gaussian noising process. In contrast, while LSI also optimizes ELBO, it allows for broader choice of the prior p(2g) and the transforms mapping the prior to the learned aggregated posterior. Our work is similar in spirit to models like NVAE, which employed deep hierarchical latent representations, and LSGM, which proposed training score- score-based models in the latent space of a VAE, but offers flexible framework similar SI allowing rich family of priors and latent space

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Original
?
0.25
Y
0.5 0.5
7= 1.0
Figure 3: LSI supports flexible sampling. We demonstrate inversion of an 'Original' image, using reverse probability flow ODE (similar to DDIM inversion), followed by forward stochastic sampling to yield samples similar to it, with diversity increasing with y (eq. (20)). See text for details.
dynamics. Note that LDM (Rombach et al. 2022 train a diffusion generative model in the latent space of a fixed encoder-decoder pair - making their latents actually observed from the point of view of generative modeling.
Continuous- Time Generative Processes: While diffusion models have been formulated and studied
using continuous time dynamics (Song al. 2020b Kingma et al. [021 Vahdat et al. 22221), their
relation to Continuous Normalizing Flows (CNFs) (Chen et al. [018|Graahwwbl
Grathwohl et al.] 2019) offers
another perspective continuous-tine transformations. Early training challenges with the CNFs have been addressed by newer methods like Flow Matching (FM) (Lipman et al.] [022] Xu et al.)
2022), Conditional Flow Matching (CFM) (Neklyudov al. 2023 Tong et al. (023)) and Rectified Flow (Liu et al. 2022). These approaches propose simulation-free training by regressing vector fields of fixed conditional probability paths. However, likelihood control is typically not possible (Albergo et al.) (023), consequently extension to jointly learning in latent space is ill-specified. In contrast, LSI optimizes an ELBO, offering likelihood control along with joint learning latent space. Stochastic Interpolants (SI) (Albergo et al. 2023) provides a unifying perspective on generative modeling, capable of bridging any two probability lity distributions via continuous-time stochastic process, encompassing aspects of both flow-based and diffusion--ased methods. While SI formulates learning the velocity field and score function directly in the observation space using pre-specified stochastic interpolants, LSI arrives at similar objective in the latent space, as part of the ELBO, from the specific choices of the approximate variational posterior. LSI reduces to SI when encoder and decoder are chosen be Identity functions. SI is related to the Optimal Transport and the Schrödinger Bridge problem (SBP) which have been explored as a basis for generative modeling (De Bortoli et al.) 2021: Wang et al., 202TT Shi et al. 2023). While LSI learns transport, its primary objective is data log- ligliioel maximization via the ELBO, rather than solving a specific OT or SBP.
8 CONCLUSION
In this paper, we introduced Latent Stochastic Interpolants (LSI), generalizing Stochastic Interpolants to enable joint end-to-end training of an encoder, a decoder, and a generative model operating entirely within the learned latent space. LSI overcomes the limitation of simple priors of the normal diffusion models and mitigates the computational demands of applying SI directly in high- observation spaces, while preserving the generative flexibility of the SI framework. LSI leverage SDE-bbaed Evidence Lower Bound to offer a principled approach for optimizing the entire model. We validate the proposed approach with comprehensive experimental studies on standard ImageNet benchmark. Our method offers scalability along with a unifying perspective on continuous -ime generative models with dynamic latent variables. However, to achieve scalable training, our approach makes simplifying assumptions for the variational posterior approximation. While restrictive, and common with other methods, these assumptions do not seem to limit the empirical performance.
ACKNOWLEDGMENTS
We would like to thank Kevin J. Shih and Ian Fischer for proofreading early drafts of this manuscript and providing valuable feedback.
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REPRODUCIBILITY STATEMENT
We have included detailed proofs of all the key theoretical results in the appendix. Sections and provide key training and evaluation setup details. Section P provides the necessary architecture details to reproduce the models used in the experiments. Section provides additional sampling setup details.
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APPENDIX
A PROOF OF ELBO FOR DYNAMIC LATENT VARIABLES (EQ.
We provide self-contained proof of the variational lower bound stated in eq. (3) The proof is based on the approach of Li et al. (2020), but in more general form. We first establish the path-space KL divergence between two diffusion processes via Girsanov's 's theorem (Theorem []. then use it to
derive the ELBO (Theorem 2 Let Pø be the path measure under the model in eq. !I and let Q be
the path measure under the variational posterior process approximation as in eq. (2).
Theorem (Path-space KL via Girsanov's theorem): Consider two SDEs, starting at the same starting point 2o = 20. sharing the same dispersion coefficient o=z, () but potentially different initial distributions - 20 ~ (2)) for Q and 20 ~ Po(zo) for Pp:
dd ho(2,, t) dt + o (2,, t) dwi,
(model, path measure P3)
(eq.
dæ, ho(21, t) dt (z 1) due,
(variational, path measure
(eq.
Define u(z,,t) () by (z, u/ ho( 1) ho(= 1) (eq. ( Then:
S. [1((,,)) dt
( (L((ol)Po) EQ
(24)
When 40 - po (both processes share the same initial distribution, potentially learnable), the initial KL vanishes and the path-space KL reduces to the dynamics mismatch alone.
Proof. The full path measure factorizes as the initial distribution times the conditional path measure given the initial state, The Radon- Radon-Nikodym derivative therefore decomposes as
dQ
go(2))
d
(25)
0
(=))
dlp
where T and 1 denote the conditional path measures given the initial state 20. Next we use Girsanov's theorem to evaluate the second factor. Under Q, the process satisfies dz, he dt + a dw, where w is standard Brownian motion. Define the Po-Brownian motion via dw!"" o (d2) hg dt). Substituting the Q-dynamics for dæ4:
duf ?u, u(22, t) dt
(26)
That is, under Q the process
Py acquires drift u4. By Girsanov's theorem:
dQ exp (f( u du 2// ||el| dt
(27)
dF
Substituting dwp d + or dl and combining with the initial density ratio:
dQ In go(=0) Po(20)
+ S'" uh du + 2 {'
=//// ||uell dt
(28)
dr 0
Taking the expectation under Q:
(L(Q))P)) = Ego In (20) Po(20 (2))
+ 1 R2
L/ 1
+ EE
T uy dw
(29)
KL(g0llpo)
The Itō integral fo of dup is martingale under Q (under the standard integrability condition Ea I|u dt) oo), and the expectation of martingale starting at zero is zero.
Remark 1: Theorem applies to any two diffusion processes sharing the same dispersion, regardless of whether the state space represents latent variables or observations. In particular, it applies directly in observation space (with z replaced by 2:).
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Remark 2 (Learnable prior); When the prior Pu is parameterized (e.g., po(2u) Nu Eg)). the natural construction uses the same learnable prior for both processes (% = po). so (L(golpP)) = 0 and the ELBO retains the same form. The prior parameters still learned: they affect the distribution of 2g in the path integral [ ||u]] dt), and gradients flow through 2g po(=o) via the fixed reference To 7 po, the (L(gollpe) term appears as an additional regularizer penalizing deviation from the reference.
reparameterization trick. Alternatively, if the variational process
Theorem 2 (ELBO for dynamic latent variables, eqs. (3) and (5:: Under the setup of Theorem with potentially learnable initial distributions Po and qu. let It, be observations at times t, E [0,
i == I,...,,7 n, assumed to depend on the latent state only through Ze,, i..., Po ( (2 21,
Then:
-nn (0( Po(2o)
{
In )t,.) > Eg
In po (22 (2),
2 ||u(2e, t)]l
(30)
Eq
In po (2re, (zx)) <I(QIIP))
(31)
When Po. the second term on the right vanishes and we recover the special case of above as stated in Li al. (2020).
Proof. Under the model, the latent path evolves according to Po (eq. 1 and observations are generated conditionally at each time t, The marginal likelihood is obtained by integrating over all latent paths:
/Ill (o z4,) dP)(22)
(32)
Since both Ps and Q share the same dispersion and initial distribution, they are mutually absolutely continuous (by Girsanov's theorem, under standard regularity). We therefore re-express the integral using the variational path measure Q (eq. (2)
as a proposal:
dlP 8
po (Zt, ................................ Iz,.) Eq
II po (2 l2))
(2)
(33)
dQ
Taking the logarithm both sides and using Jensen's incquality:
d.P
In po Lt..)
E2
Σ po 12,) + In
(z)
(34)
dQ
From the proof of Theorem D (with to = Po, so the initial density ratio cancels), the log Radon- Nikodym derivative expressed in terms of the Q- (Q-Brownian motion is:
In
e
2
(0(2)) po(z))
S u dw,
1 - L.
|| dt
(35)
2
Substituting the above get:
(0(20)
1-" u/ du,
2 f
1
In (I tt,)) 2 Eg
In pa (It, (2.,) - po(2o)
l|ue 112 dt
(36)
1
(0(20)
2 %.
Eq
In po (2 (2),,2))
||ue||2 dt
(37)
(o(2))
T
Eg
In (2 (2))
KL Po)
(38)
Where the Itô integral So u. dw? vanishes under Eq (as established in Theorem
Remark 3: The bound has a natural interpretation: the first term reconstruction likelihood (how well the model explains observations given the latent path) and the second term penalizes the mismatch between the variational and model path distributions. 1 [ ||ue|| can also be seen as the control cost required steer the model process Pp match the variational process Q (Theodoroul 2015). The bound is tight when Q = Po(. Dt, ..., Lt.,) ), i.c., when the variational process equals the true posterior process.
2
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B OBSERVATION-SPACE STOCHASTIC INTERPOLANTS
The LSI framework (Section 3 jointly trains an encoder, decoder, and latent generative model. Here consider the special case where the generative process operates directly in observation space, without an encoder or decoder. This corresponds to setting 2 = 2t for all i, making the latent process identical the observation process.
Setup: The generative model an SDE directly in observation space,
dē, = h 1) di + o, düe, 3o ~ Po(
(39)
i.c., eq. ! with 2 - T. The prior po( may be fixed or learnable (c.g., pe(zo) = NN Eg)). The marginal at t defines the model distribution pe(z1).
The variational process Q is constructed using the same diffusion bridge machinery as in Section 2.2 now bridging Po (m)) and P1 (z1) = Pauma (11) directly observation space. As in the previous section, both Q and Po can use potentially different initial distributions qu and Pu- Starting from the linear SDE dæ. = he di + ar duy, (cf. eq. (7)) with = + x) and applying Doob's 's h-transform condition on ending at Pr. the drift is (cf. eq. 9):
( 1) hy V ar Inp(21 | xx)
(40)
By construction, Q has marginal Pdata P1 at t 1, while Py has marginal pe (x1).
ELBO:
Since Pdata = P1 and 2j a deterministic function of the path X' ~ Q, the data processing
inequality gives
KL lipo)
(P)
(41)
Expanding the left side and rearranging yields the evidence lower bound:
Ep, (x))) 2 E,, (m KL(((Pp))
(42)
= -Hffp - (L(Q))F)
(43)
Where H[pı] is the entropy of the data distribution Pl. Using Theorem the path-space KL can be evaluated via Girsanov's 's theorem (with x)):
KL(Q = (T(gol)eo) + Ee L. [|u(2,, 1)1 au
(44)
where or u(z,,t) t) ha( t) he t), in eq. (5) If go Po. the first term the right vanishes, leaving only the dynamics cost. As main text, we assume qu = Po in the following.
Simulation-fiee training: All the simulation-free machinery from section 3 carries over with
2 + σ. Using the observation-speee interpolant (cf. eq. (12):
St = 1) + K4 31 + 14 Ro: 6~ N0,
(45)
u(.,, 1) takes the following form similar to the result in section 3 (cf. eq. 15
[ dm: dt
of
drg du xo ho (o2,
u(z,, t) -,
21 dt
(46)
2mp dt
Final objective:
Substituting 44 and 46)) into (43) the observation-space ELBO is:
-H(pi)
F 14 x1 =I zo Pe. e ((0. [1lu( [)
(47)
where the entropy H[p,] of the data distribution constant and independent of the model parameters. For the linear choice Ky = 1, minimized (cf. eq. (17):
I, this specializes to the following loss to be
27
3 2
E
OV - c in No ho t)
(48)
where 9r has the same interpretation as in eq. (17 . of generalized weighting term, and the constant term has been dropped.
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Remark 4: Comparing with the LSI loss (eq. [77) the observation-space ELBO is precisely the LSI objective with the reconstruction term Ia 21) removed and - x throughout. This confirms the consistency of the framework: LSI reduces to observation-space stochastic interpolants when the encoder and decoder are identity functions. All parameterizations (Section 4 and sampling procedures (Section 5) apply directly with
Remark 5 (Learnable prior): The ELBO in eq. (47 supports learnable prior pa (zo) without modification. If both Q and Py start from the same (Iro), the initial KL vanishes regardless of the prior's 's form (Theorem I The prior parameters are still learned through the interpolant des It T 2 + MtTo and the target dt 2o in eq. (46), providing gradients via the reparameterization trick. Note that the interpolant coefficients T), Nt, Vt depend only on the base SDE parameters hG dz. not on Po, so changing the prior affects only the sampling distribution of zo not the interpolant structure.
C PARAMETERIZAATIONS
For the linear choice of t, (section J used for experiments in this paper, the loss term with u(2,, t) is
E,
Bptz 21 zo) 20
1-t E
ho(ze, t)
(49)
Where (). If 20 is also Gaussian, 20 (0, 1).
we can combine e, 20 to yield žt
(21 v (1-1)( + +1 l)zn and rewrite the above as
2t 1-t
E Ep (1 (p(z)(a
21
2u - ho(zzz
(50)
Directly using above forms leads to high variance in gradients and unreliable training with frequent - f in the denominator. Consequently, we consider alternative parameterizations
NaNs due to the v1- discussed in the following. Two of the parameterizations OrigFlow and InterpFlow applicable for arbitrary Po. while the remaining two Demoising and NoisePred are applicable when 2p is Gaussian. For each of these parameterizations, we also derive the corresponding sampler in section
C.1 OrigFlow
With straightforward manipulation of the term inside the expectation we arrive
1
VT- ((21 - 20) -ovle ho(=z, o||
(51)
2x
where t2 v1 tho(2e,t). t). We rewrite above in terms of time dependent weighting
B. = σ-(1-1] as following.
Be
V1 V1- 11-t(22 t(21 20) - oV=te -
(52)
2
When 2g Gaussian, we can rewrite as
VI V tzo ho (Ze, ( 0
(53)
This objective can be viewed as estimating ho(2,, t) = E[VI 12 -o t20/20 with time dependent weighting 3r.
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C.2 InterpFlow
Again, starting with the loss term with u(z, () and straightforward manipulations we arrive at the parameterization
0
e
+ 21 - 50 - ho( t)
(54)
202
2
1
1
1-t c 21 20 + V 1-t V 1- 2 - (o
(55)
20²
3t
-oVie + vI-t(2, - 20) + vta- - ho(ze t)
(56)
2
Where ha(z). () = Vlzi + vT- the (ze, 1) and 3. = ल
To gain insights into this parameteri-
zation, let's consider the term inside the norm and substitute 21
-Vlle VI- - z0) + vtz
(57)
= -ovlle + V1- - 20) + vi(lz + (1 - z)zo + + 11110111101111111111111 1)e)
(58)
(VI-+++ tvt)2 + +VE([1- -) 11-1)2a0 o Vte
(59)
Leading to
B1 2
(VI-++tVBp)
(Vt(1-t)- - VI- aovlooovl-t--- Vtyc ho 1)
(60)
The term
(vi(1 (V = 0 and 21 Oee at t = 1. Since this term appears to interpolate between the two, we refer to this parameterization as InterpFlow. When 2u is also Gaussian, we can combine c, 20 and rewrite as
t vf) reduces to 21 2g at
8. [ (V1- 1 +
(Vt(1 t) 1) +1 tzo - ho ho |||
(61)
2
Observe that, with o= = 1, the term (VI- + (vt(1 - t) - 1)20 reduces to 21 - 20 both at t = 0 and t 1.
C.3 Denoising
This parameterization is applicable only when 20 is Gaussian. Starting with the loss term with u(z,, t) and using the fact that 121 + - (1-1))(02 + l)z0, we can manipulate the objective as following
a2t + t - ho((),))
41
(62)
2
2
1
2t 4-t21 121 ((1-
31
ho(2e, t)
(63)
2
1-t
1
4 tzı ho( t)
(64)
2
1-t
||21 - = - (1 - -1)ho(z),
(65)
2 (1 2)2
1
1
|2 - -ho(zen)) t)|?
(66)
2 (1 t)2
3
Φ1 hh((2,,)) 1)
(67)
where ha (2) = 4 + (1 - t)ho t) and 3, = 1/(1 - t)?. In this form, can be viewed as a
denoiser.
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C.4 NoisePred
This parameterization is applicable only when 20 is Gaussian. Similar to the previous sec- tion, we can construct the noise prediction parameterization by substituting 21 using 4
tz
v(1 ( +
t)
02t + 1
21
1 1 20 - (2(
(68)
2
1
At v (1 () + 1)20
8 0 + -
1-1 20 ho (2ez 1)
(69)
2
1
V1 to - (1 1)=at + t2o - -tVo +1 t2o (2(22,
(70)
2
tv--t
1
VI bz Vort + 1 tzo hop(z, 1)
(71)
2
1/1 -
1
1
V1- - t2- -22 + t20 tv1 - -thoozolle tho( 1
(72)
2 -t)
1 1o*++1-t
-1- - -tv1 - -tv1- t2,
2 (2 ())
20
vo't+1-t
(73)
Be
2 20 - ho(zz; 1
(74)
where (2),) (V1 1a WVT-thol (1 L and = 1/((((1 1))
D LATENT SCORE FUNCTION WITH GAUSSIAN Po
When p0(2)) is gaussian. 20 (0. I). compute the score function estimate V.. In (x(e)) from the learned drift he (Singh & Fischer 2024). When 20 is gaussian, the transition density p(z)(21)
is Gaussian. With 2N The + KEž1 + Pr we can reparameterize as 2N K₂ž1 + Vv² + 1/2 z0 20 N I).
+ p 21) = (26 Ke21, (V²+ )²
(75)
From Singh & Fischer 2024 (eq. 41, Appendix B) we have
2t + ((2,,t) (21 t)²
v., s. (x)) = (p((=)))))
(76)
Substituting
- 2x + Ke
VE, Inpp(z)) = Ep((es)e.)
(77)
vt + nk
-t + kr 3/21 [2|]
(78)
22 + n
Since the interpolation relates 20, 21, as 21 = k221 + V2z + m/ zo. we can rewrite the above expression in terms of 2g as following
E 2
Vz, Impe((t)
(79)
VvZ + m/
E LATENT SCORE FUNCTION WITH GENERAL Po
For a general distribution (0())). it may not be possible to estimate the score function V., In (x,) from the learned drift (1 1) alone. Here derive the expression for estimating the score function
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for general distribution Pu (z)). Recall from eq. (10) that p(zr|20, 21) is Gaussian. From Denoising
Score Matching |Vincent 2011), we can write
O ponpo(zelz0, 21)
V (np) E =1
(80)
dz
where we have conditioned on both variables zu. 11. Since p(21/20 21) is Gaussian, as in the previous section, we can write
21 + 11 21, 1) ( 21, 1)2
V. Imp)(()) = (p
(81)
Now, for 24 = T/E + 1121 + 2e20, we have p(ze/20, 21)
= .V (2e: R421 + 1120, 1) Substituting
+ K421 1420 n
V (np((()) E (p((20)11
(82)
T/e
(83)
mi
E E(c
(84)
V/+
Note that this result mirrors the one for SI(Theorem 2.8, AAlerggo et al. 2023 ), though our derivation straightforward and follows directly from Denoising Score Matching (Vincent 2011).
DETAILED DERIVATION OF SAMPLING
For an SDE of the form
dz, = ho t)dt + o,dwe
(85)
Singh & Fischer (0024) (Corollary derives a flexible family of samplers as following
ho(ze, (1 )0? V. Inpe(zi) dt no
dæ,
(86)
where "Y. is a time dependent weighting that can be chosen to control the amount of stochasticity injected into the sampling. Note that choosing y 0 yields the probability flow ODE (Song et al.) 2020b) and results in deterministic sampler. This general form of sampler requires both the drift he(2,, f) and the score function Vz, In (z.). In general, the score function needs be separately estimated. See section E for an estimator. We can also set 7t = 1, leading to direct discretization of the original SDE in eq. (85). However, for the special case of Gaussian 20, we can infer the score function from the learned drift he (section D.. For this special case, use the general form above to derive a family of samplers for various parameterizations discussed section C Recall that for the choice of Kg I, Mt = I used in this paper, the loss term is specified by eq. (50p. Without any reparameterization, we have
E[
he t)
(87)
1-t
E E////10] = 21 + (1 (2,,
(88)
We can use the above to determine the expression for the score function
-2 t2,
V Impe(E))
(89)
o-1+1-t
Above expressions for the score Va Inpx(z)) can then be plugged into eq. (86 to derive sampler for the original formulation
(1 - 7- 0² -2 t2)
dz,
ho(ze,t) t)
dt + nadw:
(90)
2²1 +
For each the following parameterizations, we calculate the expression for the drift hg and the score function V, In (zx). These expressions can then be plugged into eq. (86) derive the sampler.
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F.1 SAMPLER FOR OrigFlow
For the FrigFlow parameterization, we have
ha t) he (2x. t)
(91)
VI-(
For Gaussian Zp, we can now substitute into the expression for the score function
V (mp)(=)) = -2t + tho(ze,t) r²t + -
(92)
-VI 1 t2r + t2o(22.t) v1 - t(o2t +
(93)
The drift hg and the score function V. (mp) can now be plugged into eq. 86 sampler.
derive the
F.2
SAMPLER FOR InterpFlow
For the InterpFlow parameterization, we have
ho(=,, h(2,, t) vtz
(94)
v1 /1-t
For Gaussian 2o, we can now substitute into the expression for the score function
-2 (ze(
V (np(()))
(95)
02t + 1--
VI 12, + thgt (ze Vl
(96)
V1 - t(o2t + +1
-VI - 1112 + (=)(= 1)
(97)
V1 t(o + 1)
The drift ha and the score function V (mpp(()) can now be plugged into eq. (66 to derive the sampler.
F.3 SAMPLER FOR Denoising
For the Denoising parameterization, we have
he (2 1) = ha ba( 1) 2t
(98)
1-t
For Gaussian 2o, substituting into the expression for the score function
In - (ho(2e, 1)
(99)
o²t +
t)z, the (2,, t2
(100)
(1 - to²t + t) 24 + t2)
(101)
(1-t)(o't+12)
The drift hg and the score function V, pn can now be plugged into eq. sampler.
86 derive the
F.4
SAMPLER FOR NoisePred
Again, we have
-Vo2+ tho + v1- 1-1 lz
ho(2,, 1) =
(102)
tv1
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For Gaussian 20, substituting into the expression for the score function
V Inpo(()) -t + the the((2,)) (2r,
(103)
G21+1
-VII- t2l
t2, + vI t2
(104)
V1- + 1
+
ho(z,.t) t)
(105)
v(1 ()2t + 1-11)
The drift ha and the score function 2x (z)) can now be plugged into eq. (86 to derive the sampler.
G GAUSSIANITY OF CONDITIONAL. DENSITY
We have
p [1, 20) p021| 2, p((=(=o)
(106)
p(2) (20)
Further, for the SDE in eq. (7). using results from section L we have that the transition density px,,(a,) normal with
p(x((=)) N(v Met: Est)
(107)
Mst us exp ([ntt)ar) = prasst
(108)
Est =1 { [(T)) exp (2 M(a)du)) dr) = =bbat
(109)
Then, the conditional density p(ze 21, 20) also normal (z6i pl20, 21, 21, E(so. 21 t)) with
borae1 21 + bei 2000
m(z0, =1,
(110)
b0g
borbel I E(2,, 21,t) bot
(111)
Proof: First note that
001 agr
(112)
doc
0.1
agt ons
(113)
bet
(lv)2 2
(114)
Next
J'' (v)
1n
(115)
( ouu)aa + J. o(u)a2,do
(116)
o(v)
au do bri
(117)
aj l' [o())22 bn
(118)
a bue + +
(119)
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Now
bo1 2n berbor
1
0:1
12
zol
[2]
do1 Zal
?
p(2 zo)
exp
-
be1
Uoe
b0i
(120)
Using the identities do1 age0e1; buz
dus
+ be1 and completing the squares get
1 bu1
$
bu1
21 + bei 20
12)
p(ze(21,20)
exp
2t
(121)
2T bor
borb11
601
We can therefore parameterize as following using the reparameterization trick.
borbes
bordea
b Gor
2t
20:
(),
(122)
bu1
bu1
b01
we can succinctly rewrite the above as
24 = THE + K₂ + 1/20,
(0,1)
(123)
Where Th, Ke, are appropriate scalar functions of time t.
H GENERAL TRAINING OBJECTIVE
Here we derive the form of the general training objective. The first term in the objective is the reconstruction term and remains as is. The second term of the training objective uses u(2,, t), let's recall it's expression
m(2 t) = or [4, + oiV Inp(21l2)) - ho (z+ t)]
(124)
The first two terms in the above serve as the target for hg. Next, we rewrite them in terms of existing variables. Let E(t) denote these two terms and substitute eq. (8) following
E(l) h4 + oo² In p(p(21
(125)
h,2 aaael
(21 a.1
(126)
b11
ofue
h1
2t +
(127)
bi
b
Next, recall the stochastic interpolant and the expressions for aat and bet from section G
2t = T7 + Kig21 + U²=0:
6~ NO,
(128)
7)c V borbel
borael bon
Vi bei
bun
Kr
bo
(129)
0,t exp (s. M())tr).
[" (
but
(130)
(131)
Intuitively, we expect the drift hg to be related to the velocity field. Therefore, we compute the time derivatives of Ki, Mg and 7/ next
dxt
da
dbot
but
+
am dt
(132)
dt
bu1
dt
dv;
bot b dap:
dbeı aut
+
(133)
dt
dt
dt
dn
dbb1
dboo bei
bot
+ dt
(134)
dt
2mb
dt
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From the expression for ast, using differentiation under the integral sign, we have
da
dae -arhe
aothe:
(135)
dt
dt
Similarly, from the expression for bat dbut
0f ai + S/"
alv) = oî + 2borh
(136)
dt
db11 -0
(137)
dt
Since du 1. Substituting back into the equations for the derivatives of and is
disg dt
(-bocarihy + (o) 2b0che)an1) boi (ofacs boc
(138)
b0t
ofam bo1
+ Kehe
(139)
du; 1
(be1aueh4 - 077 aor) = vihe of Aoz
(140)
dt
bo1
bo1
odui1
ut
14
(141)
bi
dnp
(-o, of
a22 Dut + (o + 2boche)ber
(142)
dt
20
- a boc)o + 2borbeshe)
(143)
21,b01
(en dur
(b11 - an buc)o? + 2b
+ of of
(144)
2mp
is di
(br1 + a² bub bo + 2boober due
(145)
2n:b01
dt
bo1o? + b0 du
(146)
211.b1
14 dl
of
+ Ne du
(147)
2n
dt
Where we have used the identity bo1 = bb1 + an bue from eq. du, by eliminating h4 following
119). Further, we can relate de
dxg
and
dx:
oian1
du 11 dt
ofa
Ki du
ofanl
ofaî bei abn bocof1)
Kit
+ be
+ nit
(148)
dt
bn
Di
boi
K₁ dv:
+ oian boran
ofa; at
R4 du
(149)
Vy. boi bot bi1
Vi dt
botbii
du: + afaetoo
(150)
borba vi dt
K, du: + bu
ofan
(151)
Me dt
We substitute into the expression for E(t) in eq. (127
1 du
+ J7an ba
E(t)
(152)
vi di
du
dri
Ke du:
Vi dt (ne + 1421 + m120) +
21
(153)
dt
dt
Te. du
drit
du
21 +
20
(154)
Vt. dt
dt
( (dip.
of c
dre
dut
21 dt
20
(155)
dt 27,
dt
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dt. Published as a conference paper at ICLR 2026
Substituting back into the expression for u(2i,
write the general form following
u u(z;, 1) = _, [(
dni of
drs
du
21
+
20
ho (21,,1))
(156)
dt
277
dt
dt
With the u(2t, 1) above, the ELBO can written using eq.
I DRIFT h,, DISPERSION or AND STOCHASTICITY Th. FROM Kt, 1t
Often, specifying the interpolant coefficients He, u intuitively easier than specifying h4 oe directly. Here we derive expressions for h2 and dy given and V4. We have
dee kell 07
(157)
dt
bo1
dur
of
dt h44
(158)
bu
Multiplying first equation by i and second by He and then subtracting the second from the first
ds: dt dt bu1
Rtt dut oran
of bi1
+ kst
14
(159)
0faml
it
+ me
11
(160)
bo1
bi1
Substituting in the definitions of Ne and vs in RHS and simplifying
dsr
Ktt du
bonogaiin 401
V
dt
dt
+
bn
(161)
aoio?
ao10? buz
(be1
(162)
1/
1
ao10?
(163)
bu1
where we have used doi agraet and b0t
br bocaz..
Therefore
bo1
disc dt
dv: dt
V:
Kt
(164)
a01
Where do1 > 0, a01 are time independent constants that can't be determined by ne 2 alone. In this paper, assume aoi 2 and bos api 02. where or is hyper-parameter. Next, derive the expression for h4, we eliminate of from eqs. (157) and (158).
du, + hu bui duc
buz
kehe -
(165)
dt
ae
dt
+ bei
drg
h
bo
kit
bot
(166)
Oe1
an dt dt
ho1 + bei
bon drg + bi bot dus
(167)
a+
dt
an braage dt
aorbel
drs + dt.
duc
tt
+
age dt
(168)
bon
dr: + dt
du.
he (Ag1 K2 + ut) = aol dt
(169)
dec dus
Gu1
h Op1 K4 + Vi
(170)
As before, agi > 0 is time independent constant that can't 't be determined from the choice of Kgz Vt alone. Finally, to express in terms given K4, 24, note that
boebe1
bo1
borax1
bo1
Kg
(171)
boi
ao14
bp
boi
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where have used the identity 401 dgr0e1- In the following, derive the formulation for the linear Ke, 2/2 schedule used in experiments in this paper. This schedule also corresponds to the choice used in Stochastic Interpolants(Albergg
al. 2023). Note that similar choice is made by the Rectified Flow (Liu et al. 2022), however the missing 1 term implies that they do not have a bound on the likelihood, also observed by [Albergo et al. (2023). We also provide the derivation for the variance preserving schedule as it is quite commonly used for diffusion models. However, we do not empirically explore
J FORMULATION FOR LINEAR Kt, 1/
For linear choice K; i, Vt Further, we assume dol
2, boi
02, Therefore,
drgt dt
dv.
= 1,
(172)
dt
We can write the expressions for h2 and of directly, using eqs.
164) and 170). as
he 1
-²
(173)
1
To express the latent stochastic interpolant, can calculate the coefficient n) for
a0l 1/- Kill = a ((1-1)
(174)
We can now write the expression for the latent stochastic interpolant
2t a a (1- 12) + (1 -1)z0, ~~ N(O, 7).
(175)
Finally, to express u(ze: t) first we calculate
dnc
of a(1 a
o² 0² - 2t) 0²
t
(176)
dt 2T 2 201(1-1 20 211 11)
20 20
1(1 -t)
1-t
leading to
| - 1-t 1- - 20 - ho(
u(z,,t)=0-1
(177)
K FORMULATION FOR VARIANCE PRESERVING Kit, 1/
For the variance preserving formulation, we set Ki = Vi and r/ + oz 1-t. Note that 2g (0, I) Gaussian, this setting leads to the latent stochastic interpolant 2; vTzi + v1 fāg. Here and have been combined since they both are Gaussian. Let bot /uot C, then
t CVin 1 1-t- v² 44 = -CV++ + (C2
(178)
(179)
2
Using above, the expressions for h; and of can be derived as
Ga Vi 2V +
C²-4
h4 2001 Vt - CVE+ 4Ct
(180)
C
0 Vi -)t+
(181)
Choosing ap1 = 1 and C = 2 yields
h 0, of Vi
4 =1 Vi
(182)
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The coefficient d7 for can be calculated as
TJ Kil4
bo1
22v((1-
- vi)
(183)
dol
We can now write the expression for the latent stochastic interpolant
2t = =2 vtje+ + viai + (1 Vt)zo, e~ ().
(184)
Finally, to express u(z,.t) t) first we calculate
dnp
of
1
(185)
dt 2T7
22v1(1-V4
with
dk₃
1
dv
(186)
2v
dt
2VF
we arrive at
u(z () 0-
+ 2 Vt)
2vi
2 20
ho(=z.
(187)
Note that above expression for particular choice of api and the ratio b01/an //// 2, which chose for relative simplicity of the final expression above. Other choices can be made, leading to different expressions.
L GAUSSIAN TRANSITION DENSITIES
Let's consider a linear SDE of the form
dz, 2, + w.d + aidw;
(188)
When the SDE is linear with additive noise, we know that the transition densities are gaussian and therefore fully specified by their mean and covariance. From [arrkaä & Solin (2019) (Eq 6.2)
these are specified by the following differential equations
djp
hef4 + Ut
(189)
dt
dE.
Ei + -0.
(190)
dt
The solution to these is given by (eq. 6.3, 6.4, Särkkä & Solin (011
#i = to + / (tt T))(T)d
(191)
Σ, (t, (t, L: ((T)) d dr
(192)
Where (s(ss) 1) the transition matrix. For our specific case of linear SDEs, we have
(s, exp (J." h(r)dr)
(193)
Substituting, get
M: Mro exp ( h(T))rr)
+ S. eep (J. h(s)ds) u(T)dr
(194)
Σ; = Ei, exp I' h(T)dT)
+1 I' o(7)2 exp 2 h(s)ds) dr
(195)
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M GAUSSIAN 20
For the interpolant (section)
2t = -ovt(1-1ee t21 + t)zo, e~ 1),
(196)
if 20 is gaussian, we can replace the linear combination of two normal random variables e, 20 with single random variable &u ~ N(2 Assuming zu _N(0,)) I), the mean f 0 and covariance Σ can be computed as
= () + (1--))) = -1- + (1) (1-)))
(197)
(198)
Using the reparameterization trick, we can express &n in terms of zu and write
20 ~ N(O, 2t = tz + v(1 t) + (1 - 2))
(199)
Note that
21 = 1,2 + -1- 1z0:
if of =
(200)
24 tzj + (1 t)zo,
if o²
(201)
Similarly, recall the expression for uu u(2z, t) from section
t
u( o
e
(2()))))
(202)
If =0 ~ I) is also gaussian, we can combine e, 2n and write
(o
1)1 20 ho(2 8)
u(z,, t) = 0- 21
(203)
we choose o² then the expression simplities
u( 1) = = V1 - ho ha 6)
(204)
Finally, we would like to reiterate that we arrive at the above by assuming 2o is gaussian. The general form derived in other sections make no assumptions about the distribution of zo.
N CHOICE OF PRIOR Po
The Gaussian distribution, along with small set of other distributions, enjoys the special privilege of being Lévy stable. That is, a linear combination of two Gaussian random variables is still a Gaussian random variable. Lévy stability is the main property behind the original formulation of the simulation free training of the Gaussian diffusion models, e.g. as in DDPM. In contrast, Laplacian, Uniform and Gaussian Mixture are not Lévy stable, and thus our experiment with those provides strong evidence for the general nature of the proposed method. The Gaussian mixture used in our experiment was constructed by having a component for each training image. Consequently, it is a mixture with a very large number of components. The current estimate of the encoder being learned was used to encode the training images, yielding the means of the corresponding components. Standard deviation for each dimension
fixed to 0.1. In practice, we simply shuffled the encoding of the training images, added noise, and used a stop_gr hurp__rrdientt operation to prevent the flow of gradient through the prior. Since the encoder is also evolving during training, this experiment required 3x more steps to yield the reported FID. Without stop_gradiens ent, the experiment became unstable.
O IMAGENET TRAINING AND EVALUATION DETAILS
We trained our models using the entire ImageNet training dataset, consisting of approximately 1.2 million images. Models are trained with Stochastic Gradient Descent (SGD) with the AdamW
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t(s) (1 s)
0.8
0.6
0.1
0.2
0.4
0.5
= 1
0.2
2
= 5
10
0.2
0.4
0.6
0.8
Figure 4: Schedule for t. A visualization of the schedule for t(s) with E [0, 1] as is varied. As c increases, larger values are favored, thereby sampling interpolants closer to f more frequently.
optimizer (Kingma & Ba 2014 Loshchilov & Hutter 2017), using 31 = 0.9, B2 0.99, 10 All models are trained for 1000 epochs using a batch size of 2048, except for the ones reported in table [ where they were trained for 2000 epochs. Only center crops were used after resizing the images to the have the smaller side match the target resolution. For data augmentation, only horizontal (left-right) flips were used. Pixel values for image I were scaled to the range 1-1, by computing 2(1/2(())))) /255) - 1 before feeding to the model. For evaluation, a exponential moving average of the model's parameters was used using decay rate of 0.9999. The FIDs were computed over the training dataset, with reference statistics derived from center-cropped images, without any further augmentation. All FIDs are reported with class conditioned samples. To compute PSNR, sampled image pixel values were scaled back to the range [0, 255) and quantized to integer values. Figure visualizes the change of variables discussed in section 4 All reported results use 1, resulting in uniform schedule, for both training and sampling, except for NoiscPred and Denoising both of which resulted in slightly better FID values for c = 2 during sampling.
Each model was trained on Google Cloud TPU v3 with 8 x 8 configuration. For 2000 epochs, the 64 x 64 model took days to train, 128 x 128 took days to train and 256 x 256 took 7 days to train. For 1000 epochs, the training times were roughly the half of that for 2000 epochs. The training times for the models reported in table are roughly similar for similarly sized models. Note that our training setup is not maximally optimized for training throughput.
P ARCHITECTURE DETAILS
The base architecture of our model is adapted from the work described by Hoogeboom et (2023) and modified to separate out Encoder, Decoder and Latent SI models. In the adapted base architecture feature maps are processed using groups of convolution blocks and downsampled spatially after each group, to yield the lowest feature map resolution at 16 x 16. A sequence of Delf-Attention Transformer blocks then operates on the 16 x 16 feature map. Note that the transformer blocks in our adapted architecture operate only 16 x 16 resolution. Consequently, for 64 64 resolution input image, two downsamplings are performed, for 128 x 128 resolution, three downsamplings performed and for 256 x 256 four downsamplings are performed. All convolutional groups have the same number convolutional blocks. The observation space SI models used in this paper constructed using this adapted base architecture. To construct Encoder, Decoder and Latent SI
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(b) Decoder architecture
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(c) Latent stochastic interpolant model architecture. The blocks shown with dashed boundaries optional across different resolutions.
Figure 5: An overview of the architecture of various components for 128 x 128 resolution model. The architecture for 64 x 64 and 256 x 256 resolutions is similar, except for the difference in the spatial feature map sizes. See section P for details.
models, simply partition the base model into three parts. The first part contains groups of convolutional blocks, each followed by downsampling, and forms the encoder. An extra dense layer added reduce the number of channels. Further, the output is normalized and unit standard deviation followed by tanh activation to limit the range to [-1, 1]. Similarly, the last part contains two groups of convolutional blocks, each followed by upsampling, and forms the decoder. An extra dense layer is added at the beginning to increase the number of channels. The remaining middle portion forms the Latent SI model, where two extra dense layers are added, one at beginning and one at end to increase and decrease the feature map sizes respectively. We show an overview the architecture for various components the fig. 5
have zero mean
Note that the tanh activation or other forms of scale control, such as normalization, play a crucial role preventing the encoder from learning arbitrarily large embeddings and allowing it to achieve better FID. Without this constraint, the model makes the encoder outputs have large scale to make denoising easier at later timesteps. This is an important implementation detail that ensures stable training. Empirically, encoder output normalization yielded more stable training and better FID, than without anything, at the same number of steps. Addition of tanh further improved the FID.
For different resolutions, the Encoder and Decoder models fully convolutional and have the architecture. The architecture of Latent SI models differs in the presence/absene abbeenc//eeeeeeeeeee the optional downsampling and upsampling blocks (shown blocks with dashed boundaries). The 64 64 Latent SI model does not contain any nownsanpling/ussappiin 16 16. The 128 x 128 model does not contain "Downsampling 1" and "Upsampling2" blocks. The 256 x 256 model contains all blocks. All models contain 16 Self-Attention Transformer blocks. To increase/decrease number of parameters to match model capacities, only the number of convolutional blocks in groups immediately before and after the Self-Attention Transformer blocks is changed.
blocks as the encoder output is already
All models operate with 3x smaller latent dimensionality that the observations. We focused on this dimensionality ratio to ensure fair comparison with observation baselines while maintaining reasonable latent dimensionality for effective modeling. earlier experiments we tried other com- pression ratios including 2 and 4x, before settling on 3x. The primary effect of the dimensionality ratio is on the reconstruction performance. Higher the dimensionality ratio, the harder it is for the decoder to achieve high PSNR at the same number of training steps, resulting in worse sample quality (FID) and longer training times. Lower the dimensionality ratio, less the computational advantage.
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Table Comparison with state-of-the-ari FID results on ImageNet x28x 128. Note that these models have differing sizes, FLOPs and NFEs. The comparison is provided purely for reference.
Method
FID
Ours
3.12
SiD2 (Hoogeboom al. 2024
1.26
PaGoDA (Kim et al. 2024g
1.48
DisCo-Diff (Xu
2024
1.73
VDM++ Kingma & Gao
2023
1.75
SiD (Hoggeboom et al. 20233)
1.94
RIN Jabri et al. 2022)
2.75
CDM
Ho & Sal Salimans
2022)
3.52
ADM
Dhariwal & Nichol 2021
5.91
ADDITIONAL SAMPLING DETAILS AND RESULTS
All the results reported in the paper the deterministic sampler with 300 steps, setting "rr in
eq. (86). except when otherwise stated. fig. 3 and fig. 6 use stochastic sampling with 7/ = 7 1). where y specified constant. We use Euler (for probability flow ODE) and Euler-Maruyama (for
SDE) discretization for all results, except for qualitative inversion results in fig. 3 and fig. 6 For
the inversion results we experimented with two reversible samplers: 1) Reversible Heun (Kidger et al. 2021) and, 2) Asynchronous Leapfrog Integrator (Zhuang et al. (021). While both exhibited instability and failed invert some the images, we found Asynchronous Leapfrog Integrator to be more stable in experiments and used it for results in fig. 3 and fig. 6 Figure provides additional
samples for qualitative assessment, complementing fig. 2 in the main paper.
Sampling speed (with 100 steps) for pixel space models is roughly 2.2 images/sec/core for 64x64, 0.95 images/sec/core for 128x 128 and 0.21 images/scc/core for 256x256. LSI achieves 2.65 im- ages/sec/core for 64x64, 1.30 images/scc/core, and 0.53 images/sec/core for 256x256. We would like to emphasize that these numbers exhibit high variance, are highly hardware dependent and can be significantly impacted by hardware specific optimizations that are not the focus of this paper.
R COMPARISON WITH OTHER METHODS
While the primary focus of this paper is the theoretical results and their empirical validation, in table 5 we present comparison with other image generation methods for completeness. We provide this table purely for reference as these methods model sizes, FLOPs and NFEs. While our best result is comparable, techniques in these works are complementary our method. We leave future work to explore this direction.
directly comparable due to differing
USE OF LLM
LLMs were used help create some of the figures in the paper.
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Original
? 0.25
== 0.5
7 1.0
Figure 6: LSI supports flexible sampling.
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Published as a conference paper at ICLR 2026
== 0. ==1.
=
A=5.
=
==1.
==3.
1=5.
Figure 7: LSI supports CFG sampling.
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Reviews:
[{"review_id": "ViOBHq0boW", "reviewer": "Reviewer_TLxQ", "summary": "The authors propose a latent stochastic interpolant method, where they aim to learn a generative model with a low-dimensional feature. To ensure the condition of the stochastic interpolant, they come up with a novel parameterization of the stochastic interpolant, conditional on the sample $z_0$ and the encoded output $z_1$. They further optimize the training objectives for reducing the variance in the training and improving the performance. The experiments were conducted on ImageNet and show noticeable improvements in the computational cost.", "questions": "Could you show more results regarding the encoder's performance? How does it perform when you train the interpolant model using varying initial conditions and training objectives?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The goal of jointly optimizing the encoder and the generative model is challenging. And the authors' proposal seems to help address the problem.\n\nThe experiments on varying initial distributions show that the method might help to be used in the case where the Gaussian might not be ideal.\n\nThe overall presentation is clear.", "weaknesses": "The authors should consider giving a more detailed benchmark, including more models with a pre-trained encoder, and discuss the advantages over the comparison methods.\n\nIt would be good if the encoder is evaluated for its linear probing accuracy, since it's very useful to see if the encoder is meaningful or not.", "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": "t2jOzqGfz9", "reviewer": "Reviewer_UcdG", "summary": "This paper proposes Latent Stochastic Interpolants (LSI), which extend stochastic interpolants into a jointly trained latent-variable model by learning the encoder, decoder, and latent dynamics together through a continuous-time ELBO. The method gives a nice unifying view of SI inside latent models, keeps the flexibility of arbitrary priors, and preserves simulation-free training, while avoiding heavy computation in pixel space. I haven’t checked every technical detail, but the approach feels clean, scalable, and grounded in solid continuous-time modeling ideas. It seems to address a hard and meaningful problem in a practical way, and the flexibility to apply this direction beyond ImageNet is the kind of capability I’d like to see more of at ICLR.", "questions": "* How sensitive is performance to design choices in the latent SDE or encoder noise scale?\n\n* Any results (even preliminary) on other modalities or conditional settings?\n\n* Please provide some failure examples, and discuss the limitations of the proposed method.", "limitations": "", "rating": 10, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "* This paper tackles a non-trivial and meaningful problem in generative modeling.\n\n* The approach of using joint training of encoder + latent dynamics + decoder feels principled and elegant.\n\n* Flexible priors and continuous-time formulation are nice advantages.\n\n* Experiments on ImageNet show competitive performance, and this general framework could extend to many domains.\n\n* The paper is generally well written and easy to follow.", "weaknesses": "* Evaluation is mostly on ImageNet, so the broader impact still needs to be validated.\n\n* It’s not fully clear how robust the training is across architectures and hyperparameters.\n\n* The paper is heavily reliant on the supplementary materials and mathematical details.\nThis makes the paper less accessible to the readers who are not interested in going through the theoretical details.\nThe authors could provide the key insights in a more intuitive way, possibly using graphical illustrations.", "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": "LnfKs93cPu", "reviewer": "Reviewer_xrNR", "summary": "The key idea of this paper is to jointly train an encoder, decoder, and latent stochastic interpolant model using a continuous-time ELBO. Unlike traditional SI, which requires direct access to samples from both prior and target distributions, LSI constructs interpolants in the latent space, allowing end-to-end optimization.", "questions": "This paper is well motivated and well presented. I do not have many questions, but I am curious about whether the method could be extended to a learnable prior, also formulated within the joint learning scheme? The proposed method mentioned \"arbitrary prior\", which, however, are mainly simple, known distributions (e.g., Gaussian, Laplace). Would you also consider a learnable prior (e.g., an EBM prior or another more sophisticated prior)?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper provides generalization of stochastic interpolants into latent spaces, enabling joint encoder–decoder–generator learning.\n\n2. The derivation of a continuous-time ELBO in latent space seems theoretically grounded.\n\n3. Demonstrates comparable or better FIDs across multiple resolutions on ImageNet", "weaknesses": "The assumptions of linear SDEs and Gaussian posteriors may limit its expressivity. It’s unclear how much these approximations affect performance or generalization.", "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 (Poster)