Conditional Generative Quantile Networks via Optimal Transport and Convex Potentials
Authors: Jesse Sun
Dihong Jiang
Yaoliang Yu
Venue: ICLR 2022 Submitted (2022)
Keywords: Optimal Transport
Generative Models
Quantile Functions
Time-Series Forecasting
Image Generation

Abstract: Quantile regression has a natural extension to generative modelling by leveraging a stronger convergence in pointwise rather than in distribution. While the pinball quantile loss works in the scalar case, it does not have a provable extension to the vector case. In this work, we consider a quantile approach to generative modelling using optimal transport with provable guarantees. We suggest and prove that by optimizing smooth functions with respect to the dual of the correlation maximization problem, the optimum is convex almost surely and hence construct a Brenier map as our generative quantile network. Furthermore, we introduce conditional generative modelling with a Kantorovich dual objective by constructing an affine latent model with respect to the covariates. Through extensive experiments on synthetic and real datasets for conditional generative and probabilistic forecasting tasks, we demonstrate the efficacy and versatility of our theoretically motivated model as a distribution estimator and conditioner.

Under review as a conference paper at ICLR 2022
CONDITIONAL GENERATIVE QUANTILE NETWORKS VIA OPTIMAL TRANSPORT AND CONVEX POTENTIALS
Anonymous authors Paper under double-blind review
ABSTRACT
Quantile regression has natural extension to generative modelling by leverag- ing a stronger convergence in point rather than in distribution. While the pinball quantile loss works in the scalar case, it does not have a provable extension to the vector case. In this work, we consider quantile approach to generative modelling using optimal transport with provable guarantees. We suggest and prove that by optimizing smooth functions with respect to the dual of the correlation maximiza- tion problem, the optimum is convex almost surely and hence construct Brenier map as our generative quantile network. Furthermore, we introduce conditional generative modelling with Kantorovich dual objective by constructing affine latent model with respect to the covariates. Through extensive experiments on synthetic and real datasets for conditional generative and probabilistic forecasting tasks, we demonstrate the efficacy and versatility of our theoretically motivated model as a distribution estimator and conditioner.
INTRODUCTION
A rudimentary result in probability theory states that applying the cumulative distribution function (CDF) to the underlying continuous random variable results in a uniformly random sample from the unit interval. Correspondingly, passing a uniformly random sample from the unit interval through the inverse CDF, a.k.a. the univariate quantile function, results in a random sample from the de- sired CDF, a simple fact that has been exploited in simulating many distributions (Devroye, 1986). Quantile functions also play pivotal role in measuring financial risks (Rüschendorf, 2013) and in building regression functions at various levels (Koenker Gilbert, 1978) and robust statistical esti- mators under model mis-specification (Koenker, 2005). The pin-ball loss in quantile regression has also been used training supervised classifiers (Huang et al., 2014) and more recently building unsupervised generative models (Ostrovski et al., 2018).
The perspective to treat quantile functions as a natural mathematical object for modern generative models is very appealing (Parzen, 1979). Traditional quantile regressors trained at fixed quan- tile levels, and the entire quantile function may be obtained through interpolation or smoothing, which, however, suffers from the so-called quantile crossing problem, namely that the monotonic- ity property of the quantile function is often violated due to training quantile regressors at different levels independently. Despite this nuance, the quantile approach, in its the univariate form, has been successfully applied in many applications (Koenker, 2005) and more recently in probabilistic forecasting (e.g. Gasthaus al., 2019; Gouttes et al., 2021). However, the pinball loss, while ef- fective for univariate datasets, does not extend immediately to multivariate responses, setting of paramount practical importance and our focus in this work. While in principle it is possible to ap- ply quantile regression to each dimension, as in Ostrovski et al. (2018), this naive approach ignores the high-order correlations between components and can lead to incorrectly learned models.
Recently, optimal transport has garnered substantial interest in generative modelling (Arjovsky et al., 2017; Makkuva et al., 2020; Finlay et al., 2020; Huang et al., 2021), domain adaptation (Courty et al., 2017), imaging (Bonneel et al., 2015), semi-supervieed learning (Solomon et al., 2014), and much more (Peyré & Cuturi, 2019). The main goal of optimal transport is to learn a coupling between two distributions such that In fact, under mild assumptions, the optimal coupling between two continuous distributions can be obtained through the gradient of convex function (Brenier, 1991; McCann, 1995). Since gradients
transport mass from one distribution to another at minimal cost.

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of convex function monotonic, and hence coincide with quantile function in the univariate set- ting. This fact motivates attempts, such as here, to construct multivariate quantile functions through the gradient of convex functions. Compared to other approaches for generative modelling, quantile networks have the added advantage of exact estimation of quantile levels. Indeed, a pointwise es- timation of quantile levels removes the necessity for empirical quantile esimation of non-quantile based generative models through say MCMC (Rasul et al., 2021).
In this work, we seek to generalize generative quantile modelling to the multivariate setting and we extend the vector quantile regression of Carlier et al. (2016) to conditional generative quantile modelling. Our main contributions are summarized as follows,
We construct a dual formulation of correlation maximization and extend its usage to generative quantile networks that is highly theoretically motivated.
To the best of our knowledge, we are the first work to propose conditional generative modelling using Kantorovich dual objective.
Previous works that are based the Kantorovich dual restrict their models to input convex neural networks (Amos et al., 2017, ICNNs). Instead, we relax this assumption by parameterizing our model as smooth neural networks and verify the sufficiency and advantage of this relaxation.
We further argue that convex potential gradients natural quantile networks since they automat- ically satisfy the non-crossing property of quantile functions.
In addition to the aforementioned novelties, our proposed conditional generative quantile network is also very flexible and versatile. In particular, we conduct thorough experiments to show that it can serve as a conditioner for generative tasks and can also be used for probabilistic forecasting.
BACKGROUND
In this section, we recall some background on generative quantile functions and optimal transport. In the univariate case, given cumulative distribution function (CDF) F(x) Pr(X I 2) of random variable X, the corresponding quantile function Q is defined as: Vu E 10,
= = == inf F(x) 2 w}.
(1)
While the CDF can be straightforwardly extended to the multivariate case, the quantile function, its inverse, less obvious the loss of the total ordering on the real line made it impossible to "just invert" F. Instead, we can think of the quantile function Q as a mapping that transforms a uniformly random variable U to a sample from the CDF F of interest, be univariate or not. This generative view of the quantile function turns out to be quite fruitful and allows us to learn distributions via quantile regression, through for instance the pinball loss en at a fixed quantile level u E 1);
lu(q; Y) = u(Y a)+ + +u 1) -)
Indeed, for fixed u. minimizing the expected loss B(lu(g;)) yields exactly g Quu) for the quantifule function Q of Y (Koenker& & Gilbert, 1978).
Generative Quantile Networks. Now, estimate the entire quantile function Q, we can simply replace the fixed quantile level u with a univariate random variable U ~ Unif([O, 1]). We can now minimize the expected loss EEl (Q(U);; Y)) w.r.t. Q to learn the entire quantile function. The uni- variate quantile loss has been applied successfully recently in probabilistic time-series forecasting. For instance, Gouttes et al. (2021) proposed stacking an implicit quantile network on top of an re- current neural network (RNN) to model the temporal dynamics of the covariates. By minimizing the quantile loss, exact point-wise quantiles can be inferred when forecasting.
However, when we transition to the multivariate setting where U is uniform over the hypercube [o, 1]d, we can no longer directly minimize the expected loss Elg. A natural alternative is to mini- mize the Lp norm of the vector quantity in eq. (2). For instance, the autoregressive implicit quantile networks (AIQN) of Ostrovski et al. (2018) amounts to adopting the L, norm. Wen & Torkkola (2019) proposed to train Caussian-copule model jointly on top of a generative multivariate quan- tile model with the L norm quantile loss and account for higher-order correlations a two-step procedure. Normalizing flows a popular class of generative models that allow explicit evaluation
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of the likelihood and the inverse of Jacobian. In particular, Wang al. (2019) parameterized their quantile function with an increasing triangular mäp and applied to novelty detection. However, an extension to conditional quantile estimation was not considered.
Optimal Transport. Consider two probability measures μ and V on X' S R and y S R", respectively, and c X' - [0, +oo| (closed) cost function. The Monge problem refers to finding a transport map G : X + y that minimizes the total transport cost:
C( v) = inf G.X+y {/ c(z, G(z))du(x) G#u CwH=vJ.
where G denotes the push-forward measure by G. An ubiquitous transport cost is the squared Euclidean distance c(x, v) | yielding the (squared) 2- 2-Wasserstein distance, i.e. W (u, v). common issue with optimal transport in practice its computation in high dimensions. Kolouri et al. (2019), among many others, propose to employ random projections and to reduce to the uni- variate setting with provable convergence guarantees. However, the number of projections sampled from the unit hypersphere necessarily scales with the dimensions of the distribution. Alternatively, Kantorovich's 's duality offers an objective that can be conveniently approximated by restricting the structure of dual potentials. Formally, the Kantorovich dual problem is given by
/ (ddp(x) + (
+ff (bd s.t. Vo V. E 3 plx) + +b()) S c(a,y) (4)
sup
where o E (2), v E (v) can be chosen to be bounded and continuous (Villani, 2008). fact, should the dual admit solution, Ill 1| yp and ; 311.10 112 + are closed convex functions and are Fenchel conjugates of each other, provided that c is the squared Euclidean distance. A byproduct of this is that the gradient map Id 0ep is an optimal coupling between μ and (Villani, 2008).
Makkuva et al. (2020) proposed parameterizing 11l 11 yp as an Input Convex Neural Network (ICNN) (Amos et al., 2017) and optimize the Kantorovich dual for generative modelling tasks. Huang et al. (2021) utilized Brenier's theorem to introduce Convex-Potentin which they restrict their flow model with strong convexity. Thus, the flow is the convex gradient that is invertible numerically using a convex solver and they use Hutchinson's 's trace estimator to estimate the logarithmic Jacobian. Although similar to ours, both works are conditional and neither are they generative quantile networks since the underlying distribution is not learned through all quantiles jointly. Furthermore, we do not constrain our model to be convex. We show in our work that by optimizing smooth neural networks with our correlation maximization objective below, the dual admits a convex solution.
Flows (CP-Flows) in
3 CONVEX POTENTIALS AS GENERATIVE QUANTILE NETWORKS
We propose novel method for density estimation using quantile approach that generalizes to the multivariate setting and allows for ficxible task-dependent conditioning. Briefly, we first formulate the dual problem of correlation maximization an affine latent model with respect to the covariate embeddings that highly versatile. In particular, any deseddepnddent deep feature extractor can be used to extract latent embeddings of the covariates and can be jointly trained with our quantile objective without any auxiliary regularization. We then provide theoretical guarantees of convergence an optimal convex function.
multivariate quantile loss. Then, construct
3.1 MULTIVARIATE QUANTILES AS CONVEX POTENTIALS
Before we consider the multivariate case, let us first analyze univariate quantile functions to moti- vate the correlation maximization problem. Let Qy((u) j0, 1] + R be defined as an univariate quantile function of Y ~ v == Law(Y') By definition, quantile functions are monotonically in- creasing Qy (u) will be correlated with A quantile function is thus a solution to the correlation maximization problem, as established in the following equality (Galichon & Henry, 2012):
S (Qy(v)d(u) max (E(UY), U Uni 1))
(5)
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Now, we interested in generalizing this equality to the multivariate setting for Y R and U [0, 1)²). For the remainder of this paper, Il and are d-dimensional distributions. Formally, we start with the definition of maximal correlation functionals that motivates goal of using convex potential gradients to parameterize multivariate quantile functions. Recall that 1t denotes the LP space with range space R
Definition (Maximal Correlation Functionals, Galichon & Henry 2012). A functional Pu L3 + R. is called a maximal correlation functional with respect to distribution p if for all Y E L'g
Pa := sup { Y.U YRU ~ nn..
(6)
Should the distribution j be absolutely continuous w.r.t. the Lebesgue measure, then there exists closed convex function Rd - R such that Y f(() μ и-almost surely (Villani, 2008). Indeed, it clear that under these mild assumptions on we can establish a multivariate extension of (5) using the convex potential gradient of f. Furthermore, V achieves an optimal coupling between μ and y (Huang et al., 2021) such that (x( = {BU V](U)] ~ wu). By Brenier's theorem, the existence and uniqueness of the convex potential gradient map between j and can be established.
Theorem (Brenier 1991). Y is a squared-integrable random vector in R2, there is a unique map of the form T = f for some convex function J such that V $
In other words, we say that V f is a Brenier mapping between and u. Now, we are ready to define the generalized quantile function as a convex potential gradient.
Definition 2 (Convex Potential Quantile, Chernozhukov et 2017; Hallin et al. 2021). Let J Rd - R be closed convex function. Assume is absolutely continuous on Rd w.r. the Lebesgue measure. Then, the Convex Potential Quantile Y ~ v is defined as ((U) where ~ p Unif([O, 1 i.e. Vf =
Having adopted the above convex potential quantile function as a natural generalization to the mul- tivariate setting, we now turn
estimation. It turns out that directly addressing the maximal correlation functional is a bit challenging. Instead, we resort to its dual problem to learn the Brenier mapping V.. Revisiting the primal where we have also introduced a covariate X:
(up ~ 14, E(XU) = E(X) = 0}.
Here the mean-independence constraint added to decorrelate with the covariate X. The dual derived as the following:
inf I pdu(U)) + / vodu(Y) s.t. +(U)+ 2 U
(8)
(99))))) 10.11
[XX = E(X) = 0,
(9)
where is smooth. We define vb as the Legendre transformation of s.
sup {UTY y (UU)),
(10)
ue(o.1]"
to reduce the problem down to optimizing only to and b as we expect the optimum &p to satisfy lp* = vb. By complementary slackness, the inequality in eq. (8) becomes equality for optimal U:
(UU) (5( = U.Y
(11)
By differentiating eq. (11) w.r.t U, we arrive Y VD)(). However, this alone sufficient for characterizing the existence and convergence of smooth function y to the optimal convex poten- tial gradient that couples μ and p. In section we provide theoretical justification and necessary conditions for existence and convergence.
3.2 CONDITIONAL GENERATIVE QUANTILE NETWORKS
Suppose an explanatory vector X E RM is given, and we are interested generating samples Y given X. Indeed, the target distribution is now the joint distribution of (X, Law(X,V Y).
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Training
Canvergos
Φx(()
to
= (u)
Smooth
Conver
Inference
u- Unif[O, 1]
z~ N0,=N0001
k
Y= Vepx(z)
Quantic Funccion
Figure I: Training and inference pipeline our proposed conditional generative quantile net- work for univariate component. The quantile level u sampled from the unit interval and pushed into standard normal. During training, our smooth parameterization learns an optimal coupling be- tween the standard normal and the target distribution, and converges convex potential. During inference, the gradients are computed via auto-differentiation to generate the respective quantile Y The convex potential gradient Vix is comonotonic push-forward mapping of μ v.
Carlier al. (2016) have shown that convex function be approximated via affine model. Thus, we parameterize our generative quantile network as an affine latent model,
(12) where pUU): Rd R and b(U) Rd R" are smooth functions and f R R". We
px(U)) = s(U) + b(U) f(X)
parameterize so, and b as neural networks with smooth activation functions while f is any task- specific mapping to yield latent vector fXX.. By learning non-linear transformation of X, high- order moments between components of can be modelled, thereby leading to a more expressive conditional generative quantile model.
Practically, dealing with uniform samples from the unit hyper-cube is less convenient computation- ally, due its bounded domain. Empirically also found the model produce lackluster gener- ated samples. Instead, we pre-process the uniform samples using a bijection $-1 [o, 1] - Rd notably the inverse CDF of standard normal applied component-wise. Now, the support of the model is unrestricted, which is easier to handle computationally.
In the conditional setting parameterized via the affine latent model eq. (12), problem (8) from the previous section now becomes
[ sup Y ((U)- b(U) ((X))dd))))))) UER
inf (p.b) Ra
(d)a(UU)+
(13)
subject zero-mean decorrelation constraints in eq. (9). The constraint can be enforced by applying batch normalization to the f(X) vector without the translation term such that the empirical batch mean fX) )) zero-centered. We also find that enforcing this constraint is necessary for numerical stability and theoretical guarantees. Moreover, the supremum is computed across the batch. As we show in the following section, the dual admits a solution such that (x( is convex almost surely with respect to U.
4 THEORETICAL ANALYSIS
Existing work in generative modeling that optimize the Kantorovich dual problem rely on parame- terizing their network input convex neural network. Such parameterization greatly limits the

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b.
CPQ (Ours) IQN
Splino RNN
Theoretical
Target
0-
-2-
Ours
0.4 0.8
Figure 2: Qualitative results on toy datasets. (a.) Estimated quantile curves of the three-component Gaussian mixture model. (b.) Density estimation Eight Gaussians, Half Moons, and Spirals with 1, 2, and classes respectively.
architecture and expressivity of the model. We suggest that a convex parameterization of the model necessary when optimizing the dual problem of correlation maximization primal. Rather, the model only needs to be smooth which can easily be achieved by using smooth activation func- tion. The following theorems prove that the solution to the dual problem will be convex, and the gradients of the model w.r.t. U is the generative model.
Theorem (Carlier et al. 2017). Assume E TR is random vector with distribution (X, Y) E R. R random vector with joint distribution v. Furthermore, assume B(X B(x) 0. If there exists smooth function
IR + R and smooth function b R - R. such that Pa (u( = (p(u) ) + b(u) TT x is convex for Law(X Lal(oo-almost -almost every value of x such that Vip(U)++ Vb((U)+bb(U)) X, then U solves the correlation maximization problem (7).
The following theorem suggests the existence a solution to the dual problem (13) under mild assumptions
Theorem (Carlier et al. 2017). Let v be an absolutely continuous probability measure over R x Rd with density 9- Assume the support of Ωt where Qi is an open bounded convex subset of R Rd, and is bounded Ω and bounded away from zero compact subsets of Q. Then, the dual problem admits at least one solution.
Theorem 4 (Carlier et al. (2017)). Let Rd be a solution to (7) and let V : R" x Rd + R. p: Rd R, b: Rd + Rm be solutions to the corresponding dual problem (13), Let (t) (t)) + b(t) v(t,z) E [0, 1]d x (mppor([La((X)) Then, ix (U) = xx (U) and E (Y)
almost surely.
By the Cenchel-Morean theorem (Rockafellar, 1970) and Theorem 4, xx (U) is convex almost surely. To summarize, the aforementioned theorems establish the existence of solution to (13), and further indicate that the gradient of the optimal convex function in the dual objective solves the primal correlation maximization problem. Lastly, Brenier's theorem establishes the foundation for parameterizing our generative model as convex potential gradient.
Hence, the transported mass from u E Rd given æ E RT Vi((u= + Vb(u)* æ. Due to convexity of ya w.r.t. Vyp.. is comonotonic thereby automatically satisfies the non-crossing property of quantile networks.
y e Rd is given by y = (.
Proposition If (U) Rd + IR is convex w.E.t. U Rd and Us S Vv, then (V)x(U)) < (V)]a for all k 1, l,....d
Therefore, convex potential gradients serve as a natural parameterization of quantile networks as additional constraints to enforce the non-crossing property of quantile functions are not needed. Furthermore, we prove that any conditional quantile function (u) can be reasonably approximated by the affine parameterization in eq. (12).

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Table 1: Performance of work against previous one-dimensional quantile networks in fitting the three-component Gaussian Mixture Model. Results are averaged over 5 runs of 10 epochs each with same hyperparameters. Lower score is better. We use boldface for the lowest score.
Model
MeanAE
MaxAE
sMAPE
RMSE
QL
Spline Quantile
0.395
1.540
0.378
0.523
0,570
IQN
0.350
1.145
0.463
0.428
0.430
CPQ (Ours)
0.283
0.775
0.309
0.332
0.352
Proposition For any continuous conditional quantile function Qz (z( we can find large n and functions f(x): Rm R", =p()) R - R. and b(u) R + R such that the gradient of (2(()) = (u) + b(u) f(x) approximates Q2 (u) uniformlyy over any compact region of (u, x).
In our experiments we find that setting interesting to explore future work when and how the approximation can be substantially tightened for certain classes of conditional quantiles and network architectures for parameterizing b and f.
suffices to obtain reasonable results. It would be
EXPERIMENTS
In this section, we evaluate our proposed approach to conditional generative modeling variety of experiments. We first evaluate the efficacy of our proposed conditional generative quantile net- work as general conditional generative model synthetic data, MNIST (LeCun et al., 1998), and CelebA (Liu et al., 2015). Then, we evaluate the model as a conditional quantile network for prob- abilistic time-series forecasting UCI Appliancee architecture we use for all experiments for our generative quantile network is smooth feed-forward neural networks parameterize both ip and b. The activation of choice must be smooth, so choose to use CELU activation,
Energy and Google Stocks data. The
CELU(x) = max (0, x) + min(0, a(exx (2) 1))-
(14)
for some a > 0. Unless explicitly stated, has 3 layers while has 2 layers. Furthermore, we define f(X) X unless explicitly stated. For evaluation, compute the max and mean absolute error (MaxAE and MeanAE), the 50th and 90th quantile loss (QL50 and QL90), the root mean squared error (RMSE), and the symmetric mean absolute percentage error (sMAPE). Full details implementation details can be found in Appendix appendix B.2. The code for our method and all experiments will be made publicly available upon acceptance.
5.1
TOY EXAMPLES
The efficacy of our proposed method as a quantile function and conditional generative model are first examined synthetic data. We consider fitting our model to learning quantile curve, and then evaluate density estimation on 2D distributions with class-labels.
5.1.1 FITTING GAUSSIAN MIXTURE MODEL QUANTILE FUNCTION
To evaluate our method as quantile function estimator, we generate 10, 000 time-series sam- ples of 24 i..d. points following Gaussian Mixture Model GMM μ,, σ) with π [0.3, 0.4, 0.3] t = [-3,0, 0, and a [0.4, 0.4, 0.4]. We train the model for 10 epochs and provide evaluation against IQN (Gouttes al., 2021) and Spline Quantile RNN (Gasthaus al., 2019). Figure 2.a shows the learned quantile function the three methods relative to the theoreti- cal quantile curve. Performance metrics displayed on Table 1 in fitting the quantile function of GMM (T,, μ. o). where our CPQ consistenly outperforms the two SOTA baselines across a wide variety of metrics.

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Figure 3: Qualitative results generated images. For the MNIST samples, we conditionally generate the handwritten digits by conditioning digit for each row. (a.) Directly learning mapping onto the image space MNIST. (b.) Using VAE, the quantile network maps to the latent space. (c.) Novel combination of attribute generation on CelebA.
5.1.2 2D DISTRIBUTIONS WITH LABELS
Here, we evaluate the efficacy of our model to generate synthetic labelled 2D distributions condi- tionally and unconditionally. The three densities we use are Eight Gaussians, Half Moons. and Swirls. The Eight Gaussians dataset does not distinguish points based on label, thus we use this dataset to evaluate the efficacy of our method first as an unconditional density estimator. For conditional density estimation, we turn to Half Moons. and Swirls. Each point in the Half Moons dataset assigned one two classes, while each point the Swirls dataset assigned to one of three classes. As such, yp R² + R2 for both datasets, but maps to IR² for Half Moons and R² for Swirls with sbt(b) = R for both. Figure 2.b shows the generated distributions qualitatively.
5.2 CONDITIONAL DIGIT GENERATION ON MNIST
To further examine the ability of our method to learn distributions conditionally, test condi- tional quantile network to generate handwritten digits conditionally. The MNIST Handwritten Digits dataset contains 60, 000 gray-scale images of handwritten digits resolution of 28 x 28px?. We consider two scenarios; (1) use a variational auto-encoder (Kingma & Welling, 2014) for dimension- ality reduction and generate
latent space (Figure 3.b), and (2) directly generate on the image space (Figure 3.a). The first scenario maps E RD to Y E IR where D is the dimension of the latent space while the second scenario involves mapping second case is greater challenge than the first as often we set D << 784 and the encoder and de- coder can be trained to implicitly filter out noise. However, we show that by directly generating on the image space, high quality images of the corresponding digit conditioned can be obtained, albeit slightly noisier than the auto-encoder counterpart. Nevertheless, we show that our proposed method can generate high-quality samples in strongly correlated high-dimensional spaces. Moreover, us- ing a dimensionality reduction algorithm such as VAE can improve visual quality of the generated images by implicitly learning filter out noise.
E 784
1]744 Clearly, the
5.3 MULTI-ATTRIBUTE GENERATION ON CELEBA
Unlike MNIST, CelebA samples have multiple attribute labels rather than a single class label. We consider the model's ability to learn multi-attribute combinations and furthermore generate novel
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Table 2: Performance evaluation the multivariate time-series datasets Energy and Stocks. Re- sults are averaged over runs. Lower score is better. We use boldface for the lowest score.
Dataset
Model
MaxAE
MeanAE
QL50
QL90
RMSE
sMAPE
DeepAR
1.033
0.049
0.024
0.012
0.103
0.203
TimeGAN
0.970
0.129
0.172
0.378
Energy
RealNVP + RNN
1.236
0.052
0.110
0.198
Huber QL
0.980
0.050
0.025
0.015
0.094
0.212
CPQ (Ours)
0.624
0.051
0.026
0.038
0.091
0.210
DeepAR TimeGAN
0.791
0.067
0.033
0.018
0.092
0.408
1.379
0.092
0.181
0.256
Stocks
RealNVP + RNN
0.670
0.040
0.063
0.213
Huber QL
0.740
0.026
0.013
0.007
0.041
0.220
CPQ (Ours)
0.660
0.024
0.012
0.014
0.036
0.220
combinations of attributes. The CelebA dataset contains 202,599 RGB images of celebrity portraits aligned and cropped 64 64px². : Each image corresponds with a 40-dimensional multi-hot vector that summarizes features of the celebrity portrait such as gender, hair-color, and whether the entity wearing eyeglasses to name few. VAE was employed for dimensionality reduction thus our conditional quantile network maps to the latent space. Figure 3.c shows the results of novel attribute combination generation qualitatively.
5.4 PROBABILISTIC FORECASTING ON MULTIVARIATE TIME SERIES
Learning the entire distribution rather than predicting point-wise trajectories of trends valuable and practical in domains like stock market analysis and demand forecasting (Wen & Torkkola, 2019). Thus, we evaluate our method on probabilistic forecasting using the UCI Appliances Energy dataset and Google Stocks data. The UCI Appliances Energy dataset contains 28 temporal continuous valued attributes. The Google Stocks dataset contains 6 temporal features gathered from 2004 to 2019. For each dataset, we generate time-series samples of 23 contiguous time points with a prediction window of 1. We compare our method against other competitive probabilistic time-series prediction methods. We use an 2-layer LSTM the temporal feature extractor, S. that yields latent embeddings the temporal covariates. For consistency, DeepAR (Gregor al., 201 4), RealNVP RNN (Rasul et al., 2021), Huber QL, and CPQ all utilize the same backbone LSTM with same wwyype--prramteen. Comparisons are summarized in Table 2, where our CPQ yields comparable superior performance and indicates more robustness across datasets and metrics than baselines.
TimeGAN (Yoon et al., 2019) and ReaINVP+RNN cannot be probed to extract the 50th and 90th quantile levels without empirical estimation through MCMC which is computationally infeasible. Here, is clear the advantage of a quantile approach to distribution estimation as the exact point- wise quantile level can be extracted in single forward-pass by exploiting comonotonic parama- terization of the density estimator.
6 CONCLUSION
We have proposed a novel method for conditional generative quantile modelling. In particular, we construct dual formulation of the correlation maximization problem subject to the zero mean- independence constraint along with an affine latent parameterization Furthermore, we prove that by optimizing smooth neural networks, the optimum convex almost surely. As such, we construct Brenier map as the gradients of the affine latent model. Through experiments on synthetic data, MNIST, CelebA, and time series forecasting datasets, we demonstrate our method is competitive and versatile as a distribution estimator and conditioner. For future work, we would like to explore higher order parameterizations as well as other optimal transport formulations. Applications to natural language modelling and genomic data would also be of great interest.

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Under review as a conference paper at ICLR 2022
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Reviews:
[{"review_id": "thI4O_dMDGL", "reviewer": "Reviewer_8X3c", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 2, "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": "", "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": "", "main_review": "A large part of the introduction is dedicated to a presentation of classical facts regarding quantile functions and optimal transport, including basic definitions of the pinball loss, of the Kantorovich dual problem, etc. This introduction looks a bit lengthy to me, but I must acknowledge that it is a good read with good references and great clarity.\n\nThe paper really starts its contribution with section 3: \"convex potentials as generative quantile networks\" and I must say that this is precisely where clarity drops in my opinion. Although I could say that I already have some experience in these topics, these parts were particularly unclear to me and definitely do not provide sufficient readability. \n\nConsidering that the majority of the paper is extremely unclear to me, although I have some experience on the topic, I suppose that --up to some nonnegligible extent at least --- it is due to an unclear exposition that combines with my non-expertise. I personally recommend rejection, but I may be willing to change my opinion if some definitely knowledgeable reviewer saw great value in the paper.\n\n\nNow for my detailed review. I apologize in advance if some questions appear as trivial. Here are some questions/remarks:\n* It seems that Q_\\nu would be clearer than Q_Y, since Y is apparently a (vector) realization, while \\nu is its law.\n* in equation (5), over what exactly is expectation taken ? over Y~\\nu AND U~\\mu ?\n* your definition 1 is quite unclear again due to the notations. Y is an element of L_d^2 or is it a vector, like a single realization of \\nu  ? It looks like you are alternating between Y and \\nu in some way I cannot understand. In equation (6), do we really have that \\rho_\\mu is a function of the single Y vector that is appearing on the right-hand side ? or is expectation taken on both U and Y~\\nu with the object you want to present actually being \\rho_\\mu(\\nu) ? This looks unclear to me.\n* Maybe I am mistaken, but it looks like in practice you are using E[max(U^\\top Y)] and not max(E[U^\\top Y)] in your algorithm, when you compute the max per batch. Am I wrong ?\n* in your theorem I, how could a vector in R^d not be square integrable ? Are you rather here talking about \\nu being in L^2_d ?\n* your definition 2 reads meaningless to me. Either you define Y as a realization of \\nu through Y~\\nu, or you define it as Y=\\nabla f(U), but you can't do both. Maybe you mean that \\nabla f # \\mu = \\nu as you write after, it just appears wrong to write it the way you do here. Furthermore, writing Y=\\nabla f(U) doesn't tell me what exactly you call the convex potential quantile.\n* The whole part regarding the covariate X is totally mysterious to me and I don't manage to make sense of it. First, I don't see where this covariate enters the picture in equation (7). It is a fact that U is distributed to \\mu, and then what do you mean exactly with these E(X|U)=0 and E(X)=0 ? Are these further constraints imposed on X ? Then, how do you actually use X anywhere in this equation ? Then, what is it exactly you are trying to achieve with this \"sup\" ? This is all pretty unclear in my opinion.\n* I may be missing something obvious or trivial, but I don't understand how you derive (8) ? I guess it's some special case of (4), but I don't get it. Could you please be more explicit about this, which looks like an important aspect of your paper ?\n* I don't understand again how X enters the picture in (8), and I don't understand what I am seing in (9). Is it an assumption ? Some constraint you want to impose (in which case I guess that U is not distributed iid wrt \\mu, since I understand that X is given)\n* It reads quite strange to say that \"U is pushed into standard normal\" and \"we pre+process the uniform samples.... component-wise\". From all practical purpose, it looks to me that you are basically drawing U as gaussian i.i.d, right ? Why are you presenting things in such a complicated way ? Is there actually some reason for it ? Is theory broken if you take U~N(0,I) directly ?\n* It is not clear what is your contribution when put into perspective of the theorems 2 3 and 4 you are mentioning. Why are you providing these theorems in such a detailed way if they are not a contribution ? Why are these theorems proved in the appendix ? Could you please motivate these and provide some narrative  rather than this enumeration of theorems ? I am sorry to say that this whole section is unclear to me...\n\n* the experiments are nice, although I think it is not clear what you are exactly implementing for the time-series experiment", "summary_of_the_paper": "That paper proposes a multivariate quantile network that may be used for generative purposes. Its contribution is to generalize some scalar characterization of quantile functions (correlation maximization) to the multivariate case and propose a dual formulation of the problem that is advocated as generalizing well.\nExperiments are very limited, although somewhat convincing. ", "summary_of_the_review": "the paper apparently comes with some merit, although I must say that the exposition is so unclear that I couldn't really follow the narrative.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "o1EsL7-haL", "reviewer": "Reviewer_7Kv2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "", "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": "", "main_review": "Pros: The empirical results look encouraging. The generated pictures look realistic on both MNIST and CelebA. It also seems that the proposed method is able to avoid mode collapse and can explore different modes.\n\nCons:\n-\tThe main concern of this paper is its methodological and theoretical novelty over the existing papers. A thorough comparison between the current paper and the existing works is missing. The problem formulation and theoretical results follow somewhat straightforwardly by existing papers, i.e. the cited Carlier et al. (2017). In particular, Eq. (13) is essentially Eq. (4) in Carlier et al. (2017) (with errors in the integration domain). The theorems all resemble those in Carlier et al. (2017). \no\tIt looks that the only novelty is in using neural networks to parameterize the potential function. What is the advantage of this method over that in Carlier et al. (2017), in which an estimation algorithm was also proposed?\no\tCurrently the proofs of the theorems in the supplementary materials look like a direct copy of those in Carlier et al. (2017). It may be worth pointing out if there is any technical difficulty from extending the theorems of Carlier et al. (2017) to the present paper.\no\tFrom the pseudo code in Algorithm 1, I couldn’t find which lines are related to neural networks, although Table 3 does show the results obtained by using LSTM. There seems to be a gap between the claim in the main text and the actual algorithm, and clarification may be needed.\n-\tThe convex potential is learned by minimizing the dual problem. In order to facilitate the use of optimizers like SGD, how to compute the gradient? It is not clear if a typical auto differentiation can work given that the dual problem involves integrals and constraints. As noted in Sect. 4.1 of Carlier et al. (2017), the dual problem can be further formulated as an unconstrained problem, which may be more readily solvable by using SGD type optimizers, and more scalable to a higher dimension. More details may be needed. \n-\tGiven that there are many popular conditional GAN models, e.g. BicycleGAN etc. that can overcome the mode collapse problem, a more comprehensive empirical experiment is necessary to fully showcase the advantage of the proposed method. ", "summary_of_the_paper": "This paper estimates conditional quantile contours of a multivariate random vector. As shown by previous papers, his problem can be reformulated as a correlation maximization problem, and its solution exists and can be represented as the differential of a potential function by Brenier’s theorem. Solution can be found by considering the dual problem, as also shown by previous papers. This paper claims to parameterize the potential function by neural networks, which leads to a new method.", "summary_of_the_review": "Although the empirical results are encouraging, it is unclear the method and theory in this paper is sufficiently novel. Therefore, I am as of now not in favor of recommending to accept this paper.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "CaVSCOjTq0i", "reviewer": "Reviewer_Vf6H", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "", "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": "", "main_review": "#### Novely:\nGoing through the two works of Carlier et al. 2016 and Carlier et al. 2017, I noticed that the paper's exposition (Sections 3.1 and 3.2) is quite similar to the last work of  Carlier et al. 2017. It would be better if the authors highlight their contributions compared to these last works.\n\n#### Weakness:\n- In Eq. (12) the authors parametrize their generative quantile network as an affine latent model using a smoothing function $f$. However, the theoretical analysis (Theorems 2, 3, 4) assumes that $f=Id$. I am wondering if the theoretical guarantees are still valid for general $f$.\n- In Eq. (8), the minimization problem is over $(\\varphi, \\psi, b)$, however in the objective and the constraints this parameter doesn't appear. According to Eq. (3) in Carlier et al. 2017, I think the constraint must be $\\phi(U) + \\psi(Y) + b(U)^\\top X \\geq U^\\top Y.$\n- In Definition 2 and the beginning of Section 3.2, the same notation $\\nu$ is used to denote the distribution of $Y$ variable and the joint distribution of $(X,Y)$.\n- In Figure 1, it is not clear for me if Training and Inference are separate; in other words, is it an end-to-end DNN alternating between Training and Inference?\n\n#### Clarity: \nThe paper is overall clear and easy to follow but contains many typos or grammar mistakes, I have listed some of them below. However, it doesn't impact readability.\n\n#### Typos and inaccuracies:\n\n\n##### Missing: \n- page 4: then there exists a closed convex semi-continuous function f. The word \"semi-continuous\" is missing.\n\n##### Notation:\n- At the beginning of Section 2, it would be better if you use letter \"Y\" to define the CDF, since the pinball loss is defined using \"Y\", and giving the definition of notation \"(.)^+\" and \"(.)^-\"\n- I suggest to write \"Eq.\" when referring to equations instead of \"eq.\"\n\n##### Typos:\n- page 1: \"in its the univariate\" -> \"in its univariate\"\n- page 2: \"esimation\" -> \"estimation\"\n- page 2: \"conduct thorough\" -> \"conduct through\"\n- page 2: \"bet it univariate\" -> \"be univariate\"\n- page 2: \"for the quantifule\" -> \"for the quantile\"\n- page 2: \"on top of an reccurent\" -> \"on top o a reccurent\"\n- page 2: \"level u with a univariate\" -> \"level u with an univariate\"\n- page 3: \"should the dual admit\" -> \"the dual should admit\"\n- page 3: \"neither are they genrative\" -> \"neither are the generative\"\n- page 4: \"Should the distribution mu\" -> \"the distribution mu should be\"\n- page 8: \"Swirls\" -> \"Spirals\" \n- page 8: \"turn to Half Moons.\" -> \"turn to Half Moons\"", "summary_of_the_paper": "Quantile regression is frequently used as an alternative to conventional regression. An important advantage of quantile regression is that provides enough flexibility to capture the whole conditional distribution, rather than the conditional mean, of the response variable for given predictor variables. Standard approaches to tackle the estimation problem of quantile regression are based on the so-called \"pinball loss\" (quantile loss) in the univariate case, or an $L_p$-norm of the vector quantile loss in the multivariate case. This paper builds conditional generative quantile modelling to the multivariate setting. The authors propose convex potential quantile (CPQ), which is an optimal transport approach through a Kantorovih dual objective functional, which maximizes the correlation between the target distribution (dependent variable in the regression case) and the source distribution (uniform distribution over the unit hypercube). Various experiments are conducted on synthetic and real datasets, including generative modelling (MNIST and CelebA) and probabilistic forecasting.", "summary_of_the_review": "The authors propose convex potential quantile (CPQ), which is an optimal transport approach through a Kantorovih dual objective, maximizing the correlation functional between a uniform distribution over the unit hypercube and target distribution. Their approach is heavily based on a recent work of Carlier et al. 2017. In my opinion, the experiment part of the paper highlights the beauty of the conditional vector quantile proposed in the work of Carlier et al. 2016, through neural networks implementation.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "1lwisCZBP8W", "reviewer": "Reviewer_jUXa", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "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": "", "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": "", "main_review": "Strengths:\n* Well-defined model and theoretical guarantee\n* Clear writing \n\nConcerns:\n* The motivation is unclear for me. There seems to be a vacancy model for conditional quantile generative modeling, but why should we consider this way? \n* How do conditioned image generation related with quantiles? On the one hand, no quantitative results are provided for mnist and celeba. On the other hand, the quantiles are not used or specified for mnist and celeba dataset conditions. \n* The results on mnist or celeba is less challenging. See [Ostrovski 2018] for the imagenet experiment.\n* Do time series forecasting experiments assume conditions? Moreover, the CPQ does not have the best performance merics among the time series forecasting results.", "summary_of_the_paper": "This paper proposes a conditional quantile generative model using optimal transport. Theoretical results show that the optimal is convex and a conditional dual objective is further introduced. Existing results shows the efficacy and versatility of this method. ", "summary_of_the_review": "This paper addresses the class conditioned problem which need more elaboration. Moreover, the experimental results are not strong enough to offset the motivation weakness. Thus I am tending to give a reject but still open to see more updates to make the above points more clear.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a conditional quantile generative model using optimal transport. Although the problem addressed in this paper is interesting and important, the proposed convex potential quantile (CPQ) approach is highly relevant to a recent work (Carlier et al. 2017). Due to the lack of clear explanations of the contributions compared to the existing work, none of the reviewers suggested acceptance of this paper.
Decision: Reject