Proximal Optimal Transport Modeling of Population Dynamics

Bunne, Charlotte; Meng-Papaxanthos, Laetitia; Krause, Andreas; Cuturi, Marco

doi:10.48550/arxiv.2106.06345

Cited by 6 publications

(9 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Pushing forward a probability distribution by the proximal operator corresponds to one step of the JKO scheme for Wasserstein gradient flow of a linear functional in the space of distributions [Jordan et al, 1998, Benamou et al, 2016. Compared to recent works on neural Wasserstein gradient flow [Mokrov et al, 2021, Hwang et al, 2021, Bunne et al, 2021, where a separate network is needed to parameterize the pushforward map for every JKO step, our linear functional yields a pushforward map that is identical for each step; this property allows us to use a single neural network as a parameterization.…”

Section: Related Workmentioning

confidence: 99%

Learning Proximal Operators to Discover Multiple Optima

Li¹,

Aigerman²,

Kim³

et al. 2022

Preprint

View full text Add to dashboard Cite

Finding multiple solutions of non-convex optimization problems is a ubiquitous yet challenging task. Typical existing solutions either apply single-solution optimization methods from multiple random initial guesses or search in the vicinity of found solutions using ad hoc heuristics. We present an end-to-end method to learn the proximal operator across a family of non-convex problems, which can then be used to recover multiple solutions for unseen problems at test time. Our method only requires access to the objectives without needing the supervision of ground truth solutions. Notably, the added proximal regularization term elevates the convexity of our formulation: by applying recent theoretical results, we show that for weakly-convex objectives and under mild regularity conditions, training of the proximal operator converges globally in the over-parameterized setting. We further present a benchmark for multi-solution optimization including a wide range of applications and evaluate our method to demonstrate its effectiveness.

show abstract

Section: Related Workmentioning

confidence: 99%

Learning Proximal Operators to Discover Multiple Optima

Li¹,

Aigerman²,

Kim³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Their work does not, however, consider the invertible mapping as a multivariate quantile function and-like other work on normalizing flows trained using maximum likelihooduses a parametrization in the "reverse" direction (from the target to the Gaussian reference distribution). The same idea of using the gradient of an input-convex neural network model to define functions that are solutions to optimal transport problems under quadratic cost has also been proposed in Bunne et al (2021), albeit embedded in a larger architecture for modeling population dynamics.…”

Section: Related Workmentioning

confidence: 99%

“…Input convex neural network (ICNN) (Amos et al, 2017) is a neural network with special constraints on its architecture such that it is convex with respect to (a part of) its input. ICNN has demonstrated successful applications in various optimal transport and optimal control problems (Bunne et al, 2021;Chen et al, 2019;Huang et al, 2021;Makkuva et al, 2020). Moreover, it has been proved that, under mild assumptions, ICNN and its gradient can universally approximate convex functions (Chen et al, 2019) and their gradients (Huang et al, 2021), respectively.…”

Section: Partially Input Convex Neural Networkmentioning

confidence: 99%

Multivariate Quantile Function Forecaster

Kan¹,

Aubet²,

Januschowski³

et al. 2022

Preprint

View full text Add to dashboard Cite

We propose Multivariate Quantile Function Forecaster (MQF 2 ), a global probabilistic forecasting method constructed using a multivariate quantile function and investigate its application to multi-horizon forecasting. Prior approaches are either autoregressive, implicitly capturing the dependency structure across time but exhibiting error accumulation with increasing forecast horizons, or multi-horizon sequence-to-sequence models, which do not exhibit error accumulation, but also do typically not model the dependency structure across time steps. MQF 2 combines the benefits of both approaches, by directly making predictions in the form of a multivariate quantile function, defined as the gradient of a convex function which we parametrize using inputconvex neural networks. By design, the quantile function is monotone with respect to the input quantile levels and hence avoids quantile crossing. We provide two options to train MQF 2 : with energy score or with maximum likelihood. Experimental results on real-world and synthetic datasets show that our model has comparable performance with state-ofthe-art methods in terms of single time step metrics while capturing the time dependency structure.

show abstract

“…The time and spatial discretizations of Wasserstein gradient flows are extensively studied in literature (Jordan et al, 1998;Junge et al, 2017;Carrillo et al, 2021a,b;Bonet et al, 2021;Liutkus et al, 2019;Frogner & Poggio, 2020). Recently, neural networks have been applied in solving or approximating Wasserstein gradient flows (Mokrov et al, 2021;Lin et al, 2021b,a;Alvarez-Melis et al, 2021;Bunne et al, 2021;Hwang et al, 2021;Fan et al, 2021). For sampling algorithms, di Langosco et al (2021) learns the transportation function by solving an unregularized variational problem in the family of vector-output deep neural networks.…”

Section: Introductionmentioning

confidence: 99%

Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimization

Wang¹,

Chen²,

Pilancı³

et al. 2022

Preprint

View full text Add to dashboard Cite

The computation of Wasserstein gradient direction is essential for posterior sampling problems and scientific computing. The approximation of the Wasserstein gradient with finite samples requires solving a variational problem. We study the variational problem in the family of two-layer networks with squared-ReLU activations, towards which we derive a semi-definite programming (SDP) relaxation. This SDP can be viewed as an approximation of the Wasserstein gradient in a broader function family including two-layer networks. By solving the convex SDP, we obtain the optimal approximation of the Wasserstein gradient direction in this class of functions. Numerical experiments including PDE-constrained Bayesian inference and parameter estimation in COVID-19 modeling demonstrate the effectiveness of the proposed method.

show abstract

Proximal Optimal Transport Modeling of Population Dynamics

Cited by 6 publications

References 0 publications

Learning Proximal Operators to Discover Multiple Optima

Learning Proximal Operators to Discover Multiple Optima

Multivariate Quantile Function Forecaster

Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimization

Contact Info

Product

Resources

About