Multiobjective distribution matching
Authors: Xiaoyuan Zhang
Peijie Li
Yingying Yu
Yichi Zhang
Han Zhao
Qingfu Zhang
Venue: ICML 2025 poster (2025)
Keywords: multiobjective optimization; multitask learning; distribution matching
Primary area: deep_learning
TL;DR: Discuss several issues in generating a distribution which resembles multiple distributions at the same time.

Abstract: Distribution matching is a key technique in machine learning, with applications in generative models, domain adaptation, and algorithmic fairness. A related but less explored challenge is generating a distribution that aligns with multiple underlying distributions, often with conflicting objectives, known as a Pareto optimal distribution.
In this paper, we develop a general theory based on information geometry to construct the Pareto set and front for the entire exponential family under KL and inverse KL divergences. This formulation allows explicit derivation of the Pareto set and front for multivariate normal distributions, enabling applications like  multiobjective variational autoencoders (MOVAEs) to generate interpolated image distributions.
Experimental results on real-world images demonstrate that both algorithms can generate high-quality interpolated images across multiple distributions.

Multiobjective Distribution Matching
Xiaoyuan Zhang'
Peijie Li Yingying Yu?
Yichi Zhang? Han Zhao Qingfu Zhang'
Abstract
can also be expressed in its inverse form: ming D(plipa), since the divergence D-Il-) is not necessarily symmetrie. This classical distribution matching problem has been ex- tensively studied, with various approaches proposed for calculating the divergence such as the Wasserstein distance W(pellp) (Villani et al., 2009) and f-divergence Minimiza- tion (Nowozin et al., 2016). A widely used but underex- plored problem addressed in this paper is aligning a single distribution with multiple distributions simultaneously, for- mulated as:
Distribution matching is core concept in ma- chine learning, with applications in generative models, domain adaptation, and algorithmic fair- ness. A closely related but less explored challenge is generating a distribution that aligns with multi- ple underlying distributions, often with conflict- ing objectives, known as Pareto optimal distri- bution. In this paper, we develop general the- ory based on information geometry to construct the Pareto set and front for the entire exponen- tial family under KL and inverse KL divergences. This formulation allows explicit derivation of the Pareto set and front for multivariate normal distri- butions, enabling applications like multiobjective variational autoencoders (MOVAEs) to generate interpolated image distributions. Experimental re- sults on real-world images demonstrate that both algorithms can generate high-quality interpolated images across multiple distributions.
min f(0) = fm 0
(I)
-D(pollpi),.... D(pollpm))
where each objective can also be presented in its inverted form, D(p:llpa). MODM has broad applications in machine learning, including the controllable generation of interme- diate distributions (e.g., images, drugs, speeches) between multiple underlying distributions, the development of mod- els that balance multiple domain adaptations (Han and Pi- mentel, 2024; Jin et al., 2020; Wu et al., 2021), multi-source domain adaptation (Wen et al., 2020; Zhao al., 2018), and group fairness with multiple sub-populations (Chen et al., 2023; Xian and Zhao, 2024; Xian et al., 2023).
1. Introduction
Distribution Matching (DM) is a fundamental concept in machine learning and has rich applications across multiple applications, including generative modeling (Goodfellow al., 2014; Ho and Ermon, 2016; Li et al., 2015), domain adaptation (Baktashmotlagh et al., 2016; Ganin al., 2016; Gong et al., 2024; Tachet des Combes et al., 2020; Zhao et al., 2018), causal representation learning (Johansson et al., 2016; Shalit et al., 2017), and algorithmic fairness (Zhang et al., 2018; Zhao et al., 2019b; 2022), just to name few. typical DM problem formulated as:
This paper presents two approaches to study the MODM problem. The first approach assumes that the distributions Pe: PI..... Pm belong to a specific distribution family, such as the Exponential family. Under this condition, we first investigate a more general case: how generate Pareto solutions when the decision space is dually flat manifold endowed with Riemannian metric. Building these gen- eral results, we characterize the Pareto set for multivariate normal (MVN) distributions under both Kullback- (KL) divergence and inverse KL divergence as illustrative examples. Furthermore, we highlight a direct application of generating Pareto-optimal MVN distributions the con- text of multiobjective Variational Autoencoder (MOVAE) algorithm. This algorithm learns multiple latent MVN dis- tributions and employs non-linear decoders to map them into complex real-world applications. Experimental results demonstrate the effectiveness of the proposed method in this context. The contributions this paper are summarized as follows:
min (pellp),
where denotes the target distribution, and po is the distribu- tion which needs to be optimized. Alternatively, the problem
'Department of Computer Science, City *Department of Mathematics, HKK. 'Department of Statistics, IU. "Department of Computer Science, UIUC. Correspondence to: Qingfu Zhang Siingfu.zhangeee witta..uunettu..uu...........
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Multiobjective Distribution Matching
1. We investigate the geometric structure of the multiob- jective distribution matching problem with the tools of information geometry. We derive the explicit form of Pareto set on the dually flat manifold under the canonical divergence. Then, we give the shape of PS for the entire exponational family and use multivariate normal (MVN) distributions as a specific example.
2.2. Distribution matching
Distribution matching (DM), or distribution alignment, is a core concept in machine learning with broad applications, including domain adaptation (Ganin et al., 2016; Nguyen et al., 2023; Xiao et al., 2024; Zhang et al., 2022; Zhao et al., 2018: 2019a), algorithmic faimess (Prost et al., 2019; Quadrianto and Sharmanska. 2017; Zhao and Gordon, 2022; Zhao et al., 2019b), and generative models (c.g., GANs, VAEs, diffusion models, imitation learning) (Higgins et al., 2017; Ho and Ermon, 2016; Jin et al., 2020; Yin et al., 2023).
2. Besides exponational family, we derive the form of the Pareto set under the a-divergence (-Il-). Based on our theoretical results, we design an algorithm called multiob- jective variational auto-encoder MOO-VAE) to generate interpolated images between multiple distributions.
For generative models like GANs, the objective is to gener- ate a distribution that closely aligns with the data distribu- tion. Similarly, in generative adversarial imitation learning (GAIL) (Ho and Ermon, 2016), the goal is to learn a pol- icy such that the state-action density function matches that of the expert. DM also help learn a representation that aligns two distributions and has been applied to enhance robustness and enforce constraints in domain generalization, causal discovery, and fair representation learning (Gong et al., 2024).
3.
We evaluate the performance of the proposed multiobjec- tive distribution matching method not only on synthetic distributions but also on real-world image distributions. The proposed method can generate high-quality image distributions among multiple underlying distributions.
2. Related Work
3. Preliminaries
Since this paper is both related with gradient- basedd multiob- jective optimization (MOO) and distribution matching. We briefly discuss those two related topics separately.
3.1. Multiobjective optimization
For a MOP (Equation (1)). is difficult to compare the quality of solutions since vector objectives do not admit total order. To describe the optimality for a MOP, we first introduce the concept of Pareto optimality.
2.1. Gradient-based MOO
tradient-based MOO methods gained growing popularity with the successful application of Multiple Gradient De- scent Algorithm (MGDA) (Sener and Koltun, 2018) in deep multitask learning. Methods to find the Pareto set (PS) using two strategies: identifying a diverse set of Pareto solutions or modeling the entire PS directly. Notable works include Pareto Multi-Task Learning (PMTL) (Lin et al., 2019). find- ing solutions restricted in specific objective regions; Exact Pareto Optimization (EPO) (Mahapatra and Rajan, 2020; 2021), Weighted Chebyshev (WC)-MGDA (Momma et al., 2022), and Preference-based MGDA (PMGDA) (Zhang et al., 2024), which align objectives with preference vec- tors; and Gradient-based Hypervolume maximization (HV-
Definition 1 (Pareto Optimal (PO), Pareto Set (PS), Pareto
Front (PF)). A solution 0 is PO if no other solution 6' E θ dominates it, denoted as f(0)) Sstritt f(0), i.e., fi(0") < (0()) for all i E [m]] with at least one strict inequality hold. The set of all PO solutions is called the Pareto set, and its image is the Pareto front.
In addition to PO solutions, weakly PO solutions are those that cannot be strictly dominated by the other solutions, i.e., no solution O' exists such that Vi E [m]] (O())) < f0()), or f(0') < f(0).. The simplest way to find PO solu- tions is to use aggregation functions (x(+)= RT + R to convert vector optimization problem into a scalar one. Widely-used aggregation functions include linear scalar- ization, (f)) = > (0) and Tchebycheff func- tions, 9> (f) = maxie [m] (.) 2,). where z is a reference point such that x = f(0), for 0 E O. When applying aggregation functions to solve a MOP, we have the following lemma for the optimal solution of an aggregation function.
Grad (Deist et al., 2020; 2021; Emmerich et al., 2007)), which optimizes the hypervolume of a set of solutions to achieve both Pareto optimality and diversity. The most com- mon approach, however, remains optimizing some aggre- gation functions that convert a multiobjective optimization problem (MOP) into a single-objective one. Besides finding single Pareto optimal solution or a set of Pareto solutions, another popular based MOO paradigm is Pareto set learning (Chen and Kwok, 2024; Dimitriadis et al., 2023; Navon et al., 2020; Ruchte and Grabocka, 2021), which trains a single model to predict the entire PS, typically us- ing a hypernetwork or a neural network with a low-rank adaptation structure.
Lemma 2 (Adapted from (Miettinen, 1999), Theorem 2.6.2). If (f)) is decreasing w.r.t, f()), i.e., f((()) Sstrict the optimal solu- tion of (f) is a weakly PO solution. If (x)
((0 (x((((())) S (x(f(((h)),
2

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Multiobjective Distribution Matching
is stricily decreasing w.E., f()), iee, when ((((a) Sstriet f(0()), (0()) < of 9x is PO solution.
Given (local) coordinate system d [0]] on S, denoted @ J the coordinate vector fields, the metric g can be characterized by the smooth functions Pij (0,, 0₃) (the metric components), and the connection V can be char- acterized by the smooth functions iij = (V,0ū, au.) (the Christoffel symbols), A connection V is said to be flat if there exists a coordinate system & such that the correspond- ing Christoffel symbols of V are all zero, and & is said to be an affine coordinate system of the flat connection V.
()))f the optimal solution
by {a ă
Lemma guarantees that, optimizing aggregation functions with specific preference vector can find (weakly) Pareto optimal solution. The remaining issue is whether the en- tire PS can be recovered by using all preference vectors that span the probabiliiyy simplex. For this question we know that (1) when each objective function f0( is con- vex, for any Pareto solution, there exists a preference vec- tor where the optimal solution is the result of the linear aggregation function under this preference vector (Boyd, 2004)]Section 4.7J, (2) for any weakly PO solution, there exists preference vector such that the optimal value of the Tchebycheff scalarization function corresponds to this solution (Choo and Atkins, 1983). Under mild conditions (c.f. (Zhang et al., PProp 2)), optimizing the Tcheby- cheff function yields the "exact" Pareto solution, where the optimal solution 0² of the Tchebycheff function satisfies: 2m).
Statistical models S = ={pa) Jege. that is, parametrized probability distributions (density functions), are considered smooth manifolds in information geometry. Moreover, these statistical manifolds are often endowed with the Fisher in- formation metric gf given by
(0) == E. 10:ta 0,60];
and the a-connections ((a) given by
((a ij.k (0) = FE (
0,0,C0 + 1- 0.lo 0,60 Oxlo
A1 (f) =1) = ... Afm
2
Another popular MOO paradigm is called Pareto Set Learn- ing (PSL) (Lin et al., 2020; 2022), which learns a model (A)= Am + PS that maps preference vector to a PO solution. Different from previous methods, PSL aims to learn the entire PS with a single model rather than to find a finite set of Pareto solutions.
where &o == log pe and Eg]f] = f(x) po (a dæ. Next, we introduce dualistic structures on statistical manifolds, essential for analyzing information geometry.
Definition 5 (Dual connection V*). Lel (5,g) be a Rieman- nian manifold and V, V be two connections on S. If
7 Y) (VzX,Y)
3.2. Information geometry
holds for all X, Y.Z E T(S), then we say that V and 4 are duals of each other with respect to g.
This section introduces some basic concepts of information geometry (Amari, 2016; Ay et al., 2017; Nielsen, 2020), begin with the concept of Riemannian manifold.
Under a coordinate system, the condition for dual connec- tion can be rewritten as
Definition 3 (Riemannian manifold (Lee, 2012)). A Rieman- nian manifold is a pair (S. g). where S is a smooth manifold locally resembling the Euclidean space, and g :: p (:))) a Riemannian metric that equips each tangent space (S) with an inner product (:-:)p : (2(5) xT, + R...
0.gjk Pij.A + Fikk
Clearly, V(a) and (--a) are dual with respect to gf. In particular, the pair (L) and v(-1) are of special interest. We call them exponential connection V(e) = ((1) and
A connection on a Riemannian manifold describes how to differentiate vector fields.
mixture connection ((m) == (-1) respectively.
In this paper, we focus on dually flat manifolds V.V" where the pair of dual connections V and V on (S,9) are both flat. On dually flat manifold, we can always find pair of dual coordinate systems 0 = [0 n [15]. such that & and n are the affine coordinate systems of V and V respectively, and (a,, a²) = 6 for 0 : के.
Definition 4 (Connection of a smooth manifold (Lee, 2012)). A connection v a manifold S is bilinear map the space of smooth vector fields T(S):
V: I(S) X r(S) - I(S).
2 ai := on,
Moreover, we can also find a pair of potential functions v o corresponding to 0, 17 such that 0, = mi 0' = d and v + 4 = Σ; U'n Under these settings, the canonical divergence of (S, g, V, V*), which is the key notion in our discussion, is then defined as
satisfying the following properties:
1. v = /V
2. Vx(fY) = SVXY + (Xf)Y,
X,Y,Ze for vector fields Y, Z E.Y,Ze((S) and smooth snoothfunction f E (S)(S), where Xf is the derivative of f along X.
D(pllg) == (pp) + p(g)) Eo(())
3

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Multiobjective Distribution Matching
Note that the canonical divergence is independent of the choice of dual coordinate systems and potential fune- tions (Amari and Nagaoka, 2000). Dually flat manifolds are closely related to the widely-used Bregman divergences in machine learning.
where H is the Shannon entropy given by
H(p) ( : p(xx) log p(x) dæ.
The canonical divergence is then given by
Definition 6 (Bregman divergence). For strictly convex smooth function f defined on some open E S IR", the corresponding Bregman divergence D, is defined as
"
D(pla)= Σ -g) (D)) (=(9) + xp() xb())
i=1
= / (log((z) q(æ)) log p(x)) ((x) dr Dgl(gl)),
Dy(ølly) = f(x) -(x))-(()) - Σ O.f( (x'
which is indeed the dual of the Kullback-Leibler diver- gence. Many important probabilistic models belong to expo- nential families, such as multivariate normal distributions, chi-squared distributions, gamma distributions, Poisson dis- tributions, multinomial distribution (with fixed trial). dis- tributions on a finite space (probability simplex).. ... In fact, arbitrarily given density functions pu: Pr,... .
for y E E, where f is the i-th partial derivative of f.
Given pair of dual coordinate systems &, 17 and the corre- sponding potential functions v, on a dually flat manifold (S, 9. V, V*), the canonical divergence can be expressed as a Bregman divergence:
Pro then
D(pll)) = Du (((p)) = De(((a)ll(()))
(=( = po( 1¹
DI (2x) pm())
1 po(x))- 0¹ p1(2)) pn(())) dx
Conversely, given a strictly convex smooth function f on c R", the Riemannian metric characterized by
gives an exponential family. Furthermore, (Banerjee et al., 2005) showed that any dually flat manifold can be realized as an exponential family (see also (Amari, 2016)).
955 (x) == fx)),
and the pair of affine connections characterized by
Here present the model of multivariate normal distribu- tions (MVNs), which is widely used in machine learning:
T (x) = 0, Tija 0,0,
po |2m exp { - H) -1 Σ - -x- (x- m1)
form a dually flat structure on E, with canonical divergence given by the Bregman divergence Dy- In the following, we present some examples of dually flat statistical manifolds.
= ((0A)T(( Fa(x) +Tr(00 Tr(O0FF(x))) -b()) (0))
Example 7 (Exponential family). An re-dimensional model
is expressed as an exponential family in terms of
S = (pok heee is called an exponential family, if it can be expressed in terms of functions {C, Fi....,,FF FnF on the base space and a function tb on θ as
C(x) 0, FA(x)= =x, Fg(x)) = xx." T
and
" Σ
1 E "u+ log 2
po(x) = exp C(x) exp
00 F(F)()) v00)
v(0)
2
(0A) (1 1 1
2 log (
with
with respect the natural parameters 1 =10 :881:
/ exp C(x)
a Σ
(0) =og
00 Fx()))
dx.
0- 01=Σ-1 n, 9B --1x- Σー
For an exponential S, (S.g v(e), (m)) is indeed a dually flat munifold. The vie) -ifi-ne natural parume- ters 4- [0'] and the vim))-affine expectation parameters n = [ng] defined by (0) == Es [Fy] give pair of dual coordinate systems. The corresponding potential functions are given by the cumulaatt function vb and
and the expectation parameters 17 = =mn>A .n
7A = pb B Σ + p4p
Example 8 (Mixture family). An n-dimensional model S (po Jeee is called mixture family, it can be expressed in terms of functions {C,, FC,FF,.... Fi,..., Fn) on the base space as
a (0) Σ (n) w00) -H(p)) Es [C],
pa(x) = C(x) + Σ ('F((z).

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Multiobjective Distribution Matching
For a mixture family S, (f.g.. ((m), v(e)) is indeed a dually flat manifold. The ((m) -affine and the V(e)- parameters n [ng] defined by
where O(p) denote the coordinate of p under the affine coondinate system i and the same for n(p).
-affeeffine parameters 0 [0*]
The Pareto set of the MOP
my(0)= : Fa())) log pe (a:) da
min (D(pllm)) D(pllp2),.... D(p))) (()))
PES
give pair of dual coordinate systems. The corresponding potential functions are given by
is given by the convex hull of pi,..., Pm enclosed by 4 -geddesics, that is
(()) = Hp())), ((0) =- -00==/1 (0)==(((o)) log po (x) da.
C(x)
E 8 n(p) = Σ (um(ps), Amaj. )
m
The canonical divergence is then given by
The proof is postponed to Appendix A. The theorem builds that for non-degrade cases (dim(S) and p1.--- Ps linearly independent in corresponding coordinates), the PS is indeed an isomorphism of the m-dimensional simplex.
- D(plla) = Σ -x()))
- (p() + -
= f p(z) (logp(x) logg(z)) dx DKL
Next, we present the direct results for MVNs:
which is exactly the Kullback-Leible divergence.
A frequently used formulation is the mixture of distributions: arbitrarity given density functions po, Pi.....Pr. Pus then
For the MOP:
min (DKL (pl(p)e)) DKL (pllp)) PEMVNS
(1 1
po(x) =
Σ
gi
po(x) +
Σ 0 (x) i=1
where Pae is the MVN of (Ms, Ex), the PS contains MVNs of (n, E) such that
i=1
= =po(x)= +
Σ
8² (p:( po(x))
Σ
4
A k
is expressed as mixture family in terms of
k=1
A E Am
(2)
C(x) p)))), F((x) = px))) pplxx) [n].
Σ
X E
4. Multiobjective distribution matching theories
For the MOP:
min (DKI. (Pill)))))), DKI,(22lP), (1)2.ll))..... DKr.. l!p))
In this section, first introduce the underlying geometry of multiobjective distribution matching (MODM) based dually flat spaces, and later derive the explicit forms of PS and PF of MODM under some specific divergences.
PEMVNS
where ps is the MVN of (Mus Ex), the PS contains MVNs of (u, E) such that,
4.1. Geometric structure of MODM
App
Theorem 9. Let (S.g. V, V*) be dually flat manifold, 0 and n be pair of dual coordinate systems, and (ppll4) be the canonical divergence.
A E A
Σ + mp
Ex
+
(3)
-The Pareto set of the MOP with the
min D(pall), ((pmllp))
The corresponding PFs can then be obtained by computing the KL divergence between the MVNs on the PS and the given MVNs. Note that the KL divergence between the MVN pr of (μ1, En) and the MVN p2 of (μ2 Eg) is given by
PES
is given by the convex hull of Pi.... Pm enclosed by
V -ggeddssics, that is
PES: 0(p)
Σ Apo(pa), X E Am J
DKL(Pill22) = ( ΣΣ + D' (41, (41,
Am
5

---

Multiobjective Distribution Matching
where
Du ( ( log Σ Σ
Parato front Opt cation curve Prolerancc vedee
Bregman divergence between positive-definite matrices corresponding to + - log | and
D P (1,, 12) = (1l - 122 Σσ' ( - f42)
is the (squared) Mahalanobis distance between 14 and 142 with respect to Σσ Note also that (Ej اا is ex- actly the KL. divergence between the MVNs covariant ma-
(a) PF of a 3-objective MODM(h) The optimization trajectory using Tchehycheff aggregation function.
problem.
trices E and Eg with the same mean, and 222 (#1, u2) is exactly the KL. divergence between the MVNs of means 14 and 222 with the same covariant matrix Σ.
Figure 2. Results on a 3-obj problem under KL divergence using
parameters, p [0.0]
u2
[1.1]
44
[2, and
Figures and 2a illustrate the Pareto fronts (PFs) of MVN distributions with two and three objectives under KL diver- gence. In Figure blue dots represent gradient descent solutions with Σ obtained via LU decomposition, while the red curve shows the theoretical PF from Theorem 9. The numerical results from gradient descent align well with the theoretical predictions. For completeness, we provide the PS for MVNs under Wassertein distance.
5
2
0 E2 3
Es
2
4
The PS can be formulated as:
X
km
A E Am
(4)
1//
E 1/2
1.2
Solutions by GD PF by Theoream
1.0
0.8
Equations (2) to (4) provide principled approach for merg- ing multiple distributions, which is also closely related to recently proposed model merging techniques (Yang et al., 2024; Zeng et al., 2025). Our results show that, under certain divergences, the entire Pareto set can be recov- ered through appropriate manipulations in the parametric space- speci-specifically, by manipulating mean vectors and vari- ance matrices.
0.6
0.4
0.0
f
Figure The PF of 2-objective, 3-dimensional multivariate nor- mal (MVN) distribution matching under the KL divergence, where []" E; diag([1, 1, 1)), and
f1
[0,0,0]
42
4.2. MODM on the preference simplex
2.5
-0.5
Ea
-0.5
2.5
Discrete solutions (in blue) are opti-
In this section, we provide a result of knowing the exact from of the PS of the MODM problem. Once we know the of the PS, it is easy to locate a specific Pareto solution under preference vector. The following theorem shows that "optimizing over the entire probability space" is equivalent to optimizing on the preference simplex.
0
mized by gradient descent (GD), and the theoretical PF (in red) curve is predicted by Theorem 9.
Example 10 (PS under 2- Wasserstein distance Wz). The Wasserstein distance between two MVN distributions owns an explicit form:
Theorem 11. For decreasing aggregation function, opti- mizing it on the PS is equivalent to optimizing it over the entire decision space O.
Wa(p.Pa)2 = 11p aiu--pal2- 12 Tr (I -
/
The gradients w.rd ar and E are:
For example, with MVN distributions under KL divergence, optimizing a solution in θ reduces to optimizing within smaller m-dim space Am:
(W.(p.pk) 2114 px)),
Op
(F ({EEee 110,. 1)
(P2(p.P)
2/ /2_ - 1.?
min dA xEA, m
min ga (f()).
051/2
aee
6

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Multiobjective Distribution Matching
Proof. Assuming 0" E (+ PS), but there exists solution 0' PS such that f f(0)) Sstrict f(0"), which contradicts decreasing aggregation function.
Here we use Px. PI...., Pm as shorthand notations for the density functions of the distributions px Pt, and p2 on the underlying space, respectively.
G
These results show that MODM with aggregation functions reduces to single-objective optimization with a simplex constraint, while the original problem involves more param- eters and requires costly semi-definite programming due to the positive definite constraint on Σ. Additionally, gx is convex w.r.t X, and the decision space, an m-dim simplex, convex. The gradient of gA under KI. divergence is:s
Proof. Fix A E Ams consider the convex combination:
m71
f())
Dx
(6)
Then we have the difference between (() and (px) can be formulated as:
aA& ax
f {<<
VSA vf
Σ
0
Ok
f()) f(xx)
Na D (pxlllpp))
(7)
k=1
which convex with respect to A. The proof in Ap- pendix B. Optimizing the modified Tchebycheff function on
1 a-a2
4
/
AEPR
He Px
-
12
dx (8)
the simplex using projective gradient descent (Algorithm 1) yields the "exact" Pareto solution, allowing precise con- trol over its position. The optimization curve is shown in Figure 2b.
If we set pa to be the following form
I Px
m
ApP
(9)
Algorithm MODM on the Preference Simplex.
1=
Initialization: The initial preference vector Am where Am is the preference simplex.
then by replacing the term Px
in Equation (8) we have,
f()) - f(PN)
2: for epoch = 1 to N Nepoch do
3: A - A - 17- VAOa (f (,0,))
1 a ((2) /
4
24a
1+a
4: À < Projam (A)
Px
(Px
dæ
5: end for
= +())) 1-a2 a²
1
P>
1 مت
p
da
4.3. The Pareto set under a-divergence
= (() 1 (px (pp) 2
We first recap that, The a-divergence is defined as:
Thus, Px minimizes f(p), and the minimum given by
4 T- 1 (1- f p(x) - ((x)) dæ
(pllg)
f(x)) =
A DDDD(pllps
k=1
recovers the KL divergence at a + 1 and the reverse KI. divergence at a = 1, making it a special case of f- divergence.
4
1 1-
/
Σ Nm/
AMPk
1-a
P>
140
dæ
a
4
Theorem 12. For the MOP:
(1- (1))
a²
min (D Ipp) D (p2llp) (p)... .... D (pmlll)) Ip))
over all distributions, the PS is given by:
Example 13. Especially for +1, we have
)
When a -1, D-1 DRL Hence for the MOP:
px
AuP
X E Ami
(5)
(b)
1=1
min (pi DKL. (p2ll)) .... DRL (pollp))
P
where
over all distributions, the is given by:
[
2.
((A))
Σ >xpx(x)
1-0
dx
PA
AkPR: A E A

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Multiobjective Distribution Matching
Furthermore,
decoder networks have around 157K parameters. Number of training images is around 12K. The optimizer is Adam with a learning rate of 3e-5.
m
deD = D
(pill -.. 1-Ipmm),
The inference results of Figure 3 demonstrate that MOVAE is able to generate smooth interpolations between the (alarm,clock) distribution and the other circle distribution. We use five uniform preferences from [1, 0] to [0, 1] as ex- amples, though any preference within this range can serve as input to the MOVAE network. By taking the preference from [1, 0] to [0, 1]. the interpolated images gradually resem- bles the second distribution. For an intermediate preference, MOO generate a blending distribution of images under this preference.
which the skseew Jensen-Shannon divergence.
When a = =1 D = DRL. Hence for the MOP:
min (DKL() DKL (plpp)) ..... (pl(p))mm)
over all distributions, the PS is given by:
PA= = exp DB
( (pill ... (l + Σ Xa In
D
(1,
(0.75, 0.25) (0.5, 0.5)
(0.25, 0.75)
(0,1)
D2
(A) (p 1.mm) == In s var is the skew Bhattacharyya divergence. Furthermore,
where D
Alarm
Circle
DOO
m \eDRI(PNlle) D/""(1n (pill --
D (1,0)
(0.75, 0.25)
(0.3, 0.5)
(0.25, 0.75)
(0,1)
D2
4.4. Multiobjective Varational AutoEncoders (MOVAEs)
The MOVAE algorithm has two parts, which is shown in Al- gorithm 2 in Appendix C. During training, a VAE model is trained with all samples from multiple distribution with shared encoder (Φ:) and decoder (1%) parameters. After passing through the encoder network, each image is con- verted into a multivariate normal distribution.
Apple
bired
Figure 3. MOVAE Results under inverse KL divergence.
During inference, for any preference vector, a Pareto- optimal distribution is generated using Equation (2), Equa- tion (3) or Equation (4). This interpolated MVN distribution is then input to the decoder network. MOVAE's core idea is that directly generating PO distributions is challenging, so encoders and decoders transform normal distributions into real-world image distributions. Generating PO distribution and then converting to other distributions approximates the generation of PO distributions. Another advantage of MOVAF over VAES is that it requires only single model, Pareto MVN distributions in the hidden layers are computed using explicit formulations, avoiding the need for neural models. In contrast, mixture models with a controlling ratio A require training separate models for each A. Therefore, the proposed method is more efficient in training and storage compared to VAEs with mixture distributions.
6. Conclusion, further work, and limitations
This paper explores the less-studied MODM by modeling general multiobjective optimization problem the dually flat Riemannian manifold. This approach pro- vides explicit formulations for the entire exponential family with KL and inverse KL divergence. We also discuss the explicit form of the PS under a-divergence. The theoretical results have direct applications, including multiobjective variational autoencoders (MOVAE). We evaluate MOVAE the QuickDraw dataset, demonstrating their ability to generate blending images across different preference vec- tors.
Conelusion.
Future work. Beyond the geometry explored here, distri- bution matching and information geometry share an intricate connection. Future work will address advanced topics, such as projecting distributions outside a manifold onto the mani- fold.
5. Experiments
In this section, we present the results of MOVAE Results for various tradeoff levels are shown in Figure 3 under the inverse KL divergence. For other results, please refer to Ap- pendix D. The size of the image 28. Both the encoder and
This paper mainly focus on the theoretical results on multiobjective distribution matching and do not
Limitations.
8

---

Multiobjective Distribution Matching
discuss much on the applications. In the future, we will discuss the relationship between multiobjective distribution matching LLM, trustworthy machine learning and other topics.
T. M. Deist, S. C. Maree, T. Alderliesten, and P. A. Bosman. Multi-objective optimization by uncrowded hypervolume gradient ascent. 2020.
T. M. Deist, M. Grewal, F. J. Dankers, T. Alderliesten, and
P. A. Bosman. Aulti--bjeective learning to predict pareto fronts using hypervolume maximization. arXiv preprint 2021.
Acknowledgement
This work was supported by the Research Grants Council of Hong Kong, GRF Project No. CityU 11212524.
N. Dimitriadis, P. Frossard, and F. Fleuret. Pareto manifold learning: Tackling multiple tasks via ensembles of single- task models. In International Conference on Machine Learning, pages 8015-8052, 2023.
Impact Statement
This paper presents work whose goal is to advance the field of multiobjective optimization and its applications in distri- bution matching. Given the scope of this research, we do not anticipate immediate ethical concerns or direct societal consequences. Therefore, we believe there are specific ethical considerations or immediate societal impacts to be emphasized in the context of this work.
M. Emmerich, A. Deutz, and N. Beume. Gradient- based/evolutionary relay hybrid for computing Pareto front approximations maximizing the s-metric. In Hybrid Metaheuristics: 4th International Workshop, 2007.
Y. Ganin, E. Ustinova, H. Ajakan, P. Germain, H. Larochelle, F. Laviolette, M. March, and V. Lempitsky. Domain- adversarial training of neural networks. Journal of ma- chine learning research, 17(59):1-35, 2016.
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---

Multiobjective Distribution Matching
A. Proof of Theorem 9
Proof. Due to the following property of the potential functions:
dΦ - 9, 00
-
the gradients of Bi 0g 8m 80
are given by:
0
Ô (ppalp)) 0m(p) (vbpp)) + p(p) - (npp ((p))) -(pp))
@m(p)
04 (p) - ((pk)
dn
(p) ((p))
0 D(pllp))
0 (vbp) (pp)) A(p) [pp)TT(px))
(p)
(p(p)
(p) 17 (pk)
00
npp) (pk)
Hence for any À Am, we have
i (m(p)
m
D (1) = = 0(p) Σ
\ (px)
A (p)
((plppk) =0 > n(v)
{um(pk)
k=
which completes the proof.
B. Proof of Theorem 11
If (x) is convex function, then f m the simplex A = (A E Rm A; 2 0, Ee A = 1}. Given that f(x) is convex, for any points Fı,... F' and any set of weights At,..., Am. the following inequality holds:
E= A,A,)) is also convex with respect to the weights A, provided that À belongs to
m
m
ART
fxx)).)
To examine the convexity of (Eie ARo;)) with respect X, let us consider X and A two distinct points the simplex, and k1, ka 2 such that ki l2 = 1. Then, have:
f k1 M
X x. h2
I kiif
Σ Afoz + k2f
Σ
This inequality demonstrates that f (E \,a;)) convex with respect to the decision variables A. Essentially, the convexity of f over its argument translates into the convexity of the function with respect the weight parameters X; as long as remains within the simplex constraints.
Further since for any fi are convex, take gx is non-decreasing convex function, the overall function 9x (f \stz)) is convex.
C. Algorithms
12

---

Multiobjective Distribution Matching
Algorithm 2 Multiobjective VAE (MOVAE)
1: # Step 1. Offline VAE Training.
2: Input: m datasets 11,...;2,
3: for epoch = Nepoch do
4: Sample batch size of data, Di...., D
5: for mm do
6: f = {BCE + (p(( || 0). I)).
7: end for
8:
l 1 Ei= E.
9: i m do
Update: di t Φi
DE 17 B0;
1/ + 1/: 17 7aa where rj the learning rate.
11: end for
12: end for
13: # Step 2. MOVAE Prediction.
Input: ( JE=...... (0, { jKK obtain m parameters Ei,,..,... (Mmm Em) from the encoder network.
15: Given preference vector X, generate the PO distribution with parameters using Equation (2),Equation (3) or Equa- tion (4).
16: Given a preference vector X, generate the PO distribution with parameters using Equation (2), Equation (3) or Equation (4).
17: Output: Image distribution under the Pareto optimal distribution.
D. Extra experimental results
In this section, we present extra results under KL and Wasserstein distance Figures 4 and 5.
D
(1,
(0.75,0.2k 0.25)
(0.5, 0.5)
(0.25, 0.75)
(0,
D2
Alarm
Circle
P
X
(1,0)
(0.75, 0.25)
(0.5, 0.5)
(0.25, 0.76)
(0,1)
Apple
Circle
Figure 4. MOVAE Results under KL divergence.
13

---

Multiobjective Distribution Matching
(L, 0)
(0.75, 0.25)
(0.5, 0.5)
(0.25, 0.75)
(0,1)
D2
Gircle
(0.75, 0.25)
(0.5,0.5)
(0.25, 0.76)
(0,1)
Apple
Circle
Figure 5. MOVAE Results under Wasserstein distances.
14


Reviews:
[{"review_id": "MasiA4WOu4", "reviewer": "Reviewer_b6z2", "summary": "The paper develops a theoretical framework for multiobjective distribution matching using information geometry, deriving explicit forms for the Pareto set and Pareto front for exponential family distributions. It applies these insights to design multiobjective generative models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thanks for all valuable comments from you. We thanks for that your believe our theoretical results is strong and hope that our following response can address your concerns more or less.\n\n----\n\n**W1. Limited experimental validation.**\n \nTo response with that, we have included a new dataset called ageing of real world images.\n\n**W2. Rely on strong assumption that may not work in practice**\nWe address this issue in two parts. First, we derive the explicit Pareto set formulation for the dually-flat manifold, encompassing the Exponential and mixture families. This applies to a wide range of distributions, including MVN, Poisson, Gamma, Wishart, Beta, and Hypergeometric. Second, we extend our approach using nonlinear models like VAE and GAN to transform MVN into complex real-world distributions, further broadening its applicability.\n\n**Q1. Would relaxing the dually flat manifold assumption significantly affect the applicability of the method in practice?**\n\nIn fact, most of the commonly used statistical models can be covered by the case of dually flat manifold, e.g., MVN model, Possion model, Gamma model, probability simplex ... If more generally, a non dually flat statistical models is considered, similar analysis can still be applied to its canonical divergence function, but there will be an additional error term of 4th order in the result (see Section 3.8 in \"Methods of Information Geometry'' by S. Amari and H. Nagaoka).\n\n**Q2. How scalable are both MOVAE and MOGAN with respect to the dimension of the dataset?**\n\nWe use MOOVAE as an illustrative example. In MOOVAE, the decoder network generates the output image from a latent vector $ z $, which typically has a low-dimensional representation. The Pareto-optimal distribution is constructed within this latent space and subsequently decoded into an image. By keeping the latent vector relatively small, the model maintains efficiency. The scalability for generating larger images primarily depends on the capacity and power of the neural network.  \n\nTo empirically show the scalability power, we also have added a new experiments on the ageing dataset, where the images are realworld $3 \\times 512 \\times 512$ images. \n\n---\nReference\n\n[1] https://github.com/royorel/FFHQ-Aging-Dataset.", "claims_and_evidence": "The derivation of the Pareto set is well supported by theory, but the paper lacks comparisons or ablation studies with conventional methods. Overall, the experimental section is limited and does not provide enough evidence that the proposed approach is more effective than simpler alternatives. For example, a baseline comparison using a vanilla VAE optimized with MSGD would provide a useful reference for readers.", "methods_and_evaluation_criteria": "The paper primarily demonstrates its concepts through theoretical derivations and illustrative experiments (e.g., with multivariate normal distributions and image interpolation). While this approach makes sense for exploring tradeoffs and Pareto optimality, it can be challenging to directly compare these results without quantitative analysis.", "theoretical_claims": "To the best of my knowledge the theoretical claims and proofs in the appendix seems to be correct and extensively derived.", "experimental_designs_or_analyses": "There is a lack of comprehensive ablation studies, error bars, or detailed comparisons with alternative multiobjective optimization approaches.", "relation_to_broader_scientific_literature": "This theoretical contribution builds on and complements prior work in multiobjective optimization and generative modeling.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strength\n- Paper is well written and the theoretical claims and derivations are extensive. \n- Extends Pareto optimality in the setting of generative models.\n\nWeakness\n- Limited experimental validation, specifically a lack of a good baseline model. Hard to judge practical benefit.\n- Rely on strong assumptions that may not work in practice.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- Would relaxing the dually flat manifold assumption significantly affect the applicability of the method in practice?\n- How scalable are both MOVAE and MOGAN with respect to the dimension of the dataset?", "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": "BvqkmbGxyC", "reviewer": "Reviewer_nugw", "summary": "This paper studies matching a distribution to multiple target distributions. The authors use information geometry to find Pareto optimal solutions, particularly for the exponential family.  They apply this to multivariate normal distributions and a MOVAE.  They also propose a multi objective GAN.  Experiments demonstrate the performance of MOVAE and MOGAN.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thanks for your valuable comments and hope that our following response can address some of your concerns.\n\n**W4 The theoretical analysis is restricted to special families of distributions (mainly exponential families like multivariate normals), which limits the generality. Some discussions on this is needed.**\n\nActually, exponential family already covers many statistical models, e.g., normal, exponential, log-normal, gamma, chi-squared, beta, Dirichlet, Bernoulli, Poisson, geometric $\\ldots$ (see Wikipedia artical “Exponential family”). Meanwhile, our discussion works on dually flat manifolds, which cover most commonly used statistical models including not only exponential families, but also mixture families (convex combinations of distributions), and more generally, $\\alpha$-affine manifolds and $\\alpha$-families (see Section 3.6 in \"Methods of Information Geometry'' by S. Amari and H. Nagaoka). We will complement more concrete examples of exponential families and mixture families in the revised version.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "The evaluation criteria is limited. Please see Weaknesses for more details.", "theoretical_claims": "The theoretical claims appear sound.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "The paper extends core ideas from information geometry and multiobjective optimization (e.g., Pareto set learning and MGDA) to generative modeling and distribution matching.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n1. Considers an important problem that has not been explored enough.\n2. Provides novel theoretical framework based on information geometry.\n\nWeaknesses and Questions:\n1. One of my major concerns is regarding the limited experimental verification. Only one (arguably toy) image dataset is considered. It is not clear if the method performs well for more complex real-world datasets, such as natural images. \n2. The considered experiments do not show the usefulness of the proposed method. Some real use case needs to be included to show the significance of the considered problem and efficacy of the proposed solution. \n3. The experimental section lacks quantitative evaluations and baselines. For example, a possible baseline could be using aggregate function to convert multiobjective into single objective optimization. \n4. The theoretical analysis is restricted to special families of distributions (mainly exponential families like multivariate normals), which limits the generality. Some discussions on this is needed.", "other_comments_or_suggestions": "No other comments.", "questions_for_authors": "Please refer to Weaknesses and Questions", "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": "EDg0RFT8ih", "reviewer": "Reviewer_6Grc", "summary": "The paper propose multiobjective distribution matching (MODM) using tools from information geometry. two concrete algorithms are introduced: a multiobjective variational autoencoder (MOVAE) and a multiobjective generative adversarial network (MOGAN). Experiments on the QuickDraw dataset are provided to demonstrate that the proposed methods are capable of generating high-quality interpolated image distributions.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "The claims are supported by mathematical derivations and by experimental results.", "methods_and_evaluation_criteria": "The proposed method are suitable for multi-obejctive distribution macthing.", "theoretical_claims": "I reviewed the proofs provided for theorems such as Theorem 8 and Theorem 10 but didn't check the details.", "experimental_designs_or_analyses": "I reviewed the experimental designs and analyses for all experiments in Section 6. There are some minor issues, which will be discussed in Weaknesses.", "relation_to_broader_scientific_literature": "The keep contribution of the paper realted to multi-objective optimization, information geometry, and generative modeling, which are dissused in the paper.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "**Strengths:**\n\n1. The consideration of multi-objective distribution matching is interesting and novel.  \n2. The proposed method utilizes rigorous mathematical approaches to solve the problem.  \n3. The experiments demonstrate the effectiveness of the proposed method.  \n\n**Weaknesses:**\n\n1. The proposed method is not compared with the original VAE and GAN on real-world datasets.  \n2. It appears that no experiments have been conducted using the proposed PrefGAN.  \n3. The discussion on Multiobjective VAE vs. VAE over Mixture Distributions could be more solid if supported by additional literature and experiments.  \n4. There are some minor errors in the paper, e.g.,  \n   - Page 1: \"MOVAE (multiobjective Variational Autoencoder)\" should be \"Multiobjective Variational Autoencoder (MOVAE).\"  \n   - Page 4, line 215: \"but letting\" should perhaps be \"by letting.\"", "other_comments_or_suggestions": "For all experiments in the paper, more experimental details should be included, such as the number of epochs, learning rate, and other hyperparameters.", "questions_for_authors": "1. What does Pref-conditioned GAN mean in Figure 3? \n2. Why are different preferences used in Figures 3 and 4?  \n3. How does the runtime of the proposed method compare to the baseline methods?", "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": "zSluAaoxoT", "reviewer": "Reviewer_LE4u", "summary": "The paper tries to figure out how to do distribution matching to generate a distribution that aligns with multiple underlying distributions, often with conflicting objectives, known as a Pareto optimal distribution. The paper develops a theory on information geometry to construct the Pareto set. This allows for explicit derivation of the Pareto set and front for multivariate normal distributions. This leads to algorithms like multiobjective variational autoencoders (MOVAEs) to generate interpolated data distributions that can be used in multiple application fields. Based on the theory, the paper proposes the multiobjective generative adversarial network (MOGAN) algorithm, which is shown to be able to interpolate high quality real world images across domains.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "**Claim #1**: \"A related but less explored challenge is generating a distribution that aligns with multiple underlying distributions, often with conflicting objectives, known as a Pareto optimal distribution.\"\n- This is supported by convincing evidence. The issue that using one distribution to align multiple distributions has really applications in machine learning.\n\n**Claim #2**: \"We figure out the difficulty of multiobjective distribution arises from the constrained parameter space and the complex geodesics formulation.\"\n- This is supported by convincing evidence. The paper provides formal theory for optimization and calculate the Pareto set on this while admitting that there are special cases (the exponential family and multivariate normal distribution). After carefully checked the math, I confirm that it's mathematically sound.\n\n**Claim #3**: \"MOVAE, which employs a non-linear decoder to map MVN distributions to real-world distributions, and MOGAN, which learns preference-conditioned generative models.\"\n- This is supported by convincing evidence. The aforementioned theory corroborates the (theoretical) efficacy of the proposed algorithms.", "methods_and_evaluation_criteria": "Based on my understanding, the evaluation metrics are the visualization of the interpolated data. \nStrengths:\n+ The visualized generated distributions fit nicely the theoretical outcome/results. \nWeaknesses:\n- The data this paper is evaluated on is quite simple - 2-D tiny greyscale images that are fairly simple and lacking in details (i.e. the tiny images of a microwave just consist of few boxes). It's unclear how this would work or scale to harder images such as CIFAR-10/100.\n- The paper motivates the theoretical and algorithmic contributions by claiming that disturbing matching can be used for fields such as domain adaptation, yet the paper didn't use any domain adaption dataset (DomainNet, for example), for interpolation.", "theoretical_claims": "+ The paper's strongest part is its theoretical contribution. Unless I missed anything, the proof looks right.", "experimental_designs_or_analyses": "See Methods And Evaluation Criteria. My main complaint with the evaluation is that the data they used are quite simplistic. It's hard to tell if such a complex algorithm will scale to a more complex dataset.", "relation_to_broader_scientific_literature": "This paper could potentially have a very significant impact on broader scientific literature because distribution match is very important in machine learning. Areas such as domain adaptation, generalization, etc. could benefit greatly from it. However, like I have noted in previous comments, there's no evidence that this proposed algorithm will scale to a larger, more difficult website. Esepcially, since the authors are using a variation of the GANs, it might suffer from the issues that the GANs usually face, such as mode collapse.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- Could you provide some evidence on how the proposed approach fares against harder data? For example, Office-31 or DomainNet.", "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": {"ethical_review_concerns": "N/A"}}]

Author rebuttal:

Meta-review: This paper focuses on the problem of "generating a distribution that aligns with multiple underlying distributions". The core contribution of the paper. The core contributions are characterizing the pareto optimal set in case of exponential family. The paper also has empirical section that shows the practical implementation of the proposed schemes. The reviewers appreciated theoretical contribution but find empirical comparison a bit underwhelming. Overall, this is a good paper to be accepted.
Decision: Accept (poster)