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Found 76 papers, 32 papers with code
Supervised Quantile Normalization for Low Rank Matrix Factorization
Low rank matrix factorization is a fundamental building block in machine learning, used for instance to summarize gene expression profile data or word-document counts.
Unsupervised Hyper-alignment for Multilingual Word Embeddings
This paper extends this line of work to the problem of aligning multiple languages to a common space.
On a Neural Implementation of Brenier's Polar Factorization
In 1991, Brenier proved a theorem that generalizes the $QR$ decomposition for square matrices -- factored as PSD $\times$ unitary -- to any vector field $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$.
Careful with that Scalpel: Improving Gradient Surgery with an EMA
Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e. g. performance on another dataset, robustness, agreement with a prior).
Structured Transforms Across Spaces with Cost-Regularized Optimal Transport
Matching a source to a target probability measure is often solved by instantiating a linear optimal transport (OT) problem, parameterized by a ground cost function that quantifies discrepancy between points.
A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport
Kernel-based optimal transport (OT) estimators offer an alternative, functional estimation procedure to address OT problems from samples.
Entropic (Gromov) Wasserstein Flow Matching with GENOT
Optimal transport (OT) theory has reshaped the field of generative modeling: Combined with neural networks, recent \textit{Neural OT} (N-OT) solvers use OT as an inductive bias, to focus on ``thrifty'' mappings that minimize average displacement costs.
Simulation-based Inference for Cardiovascular Models
Over the past decades, hemodynamics simulators have steadily evolved and have become tools of choice for studying cardiovascular systems in-silico.
Learning Costs for Structured Monge Displacements
Because of such difficulties, existing approaches rarely depart from the default choice of estimating such maps with the simple squared-Euclidean distance as the ground cost, $c(x, y)=\|x-y\|^2_2$.
The Monge Gap: A Regularizer to Learn All Transport Maps
That gap quantifies how far a map $T$ deviates from the ideal properties we expect from a $c$-OT map.
Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps
Optimal transport (OT) theory focuses, among all maps $T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measure onto another, on those that are the ``thriftiest'', i. e. such that the averaged cost $c(x, T(x))$ between $x$ and its image $T(x)$ be as small as possible.
Supervised Training of Conditional Monge Maps
To account for that context in OT estimation, we introduce CondOT, a multi-task approach to estimate a family of OT maps conditioned on a context variable, using several pairs of measures $\left(\mu_i, \nu_i\right)$ tagged with a context label $c_i$.
Rethinking Initialization of the Sinkhorn Algorithm
While the optimal transport (OT) problem was originally formulated as a linear program, the addition of entropic regularization has proven beneficial both computationally and statistically, for many applications.
Low-rank Optimal Transport: Approximation, Statistics and Debiasing
The matching principles behind optimal transport (OT) play an increasingly important role in machine learning, a trend which can be observed when OT is used to disambiguate datasets in applications (e. g. single-cell genomics) or used to improve more complex methods (e. g. balanced attention in transformers or self-supervised learning).
Simultaneous Multiple-Prompt Guided Generation Using Differentiable Optimal Transport
Recent advances in deep learning, such as powerful generative models and joint text-image embeddings, have provided the computational creativity community with new tools, opening new perspectives for artistic pursuits.
Averaging Spatio-temporal Signals using Optimal Transport and Soft Alignments
These complex datasets, describing dynamics with both time and spatial components, pose new challenges for data analysis.
The Schrödinger Bridge between Gaussian Measures has a Closed Form
The static optimal transport $(\mathrm{OT})$ problem between Gaussians seeks to recover an optimal map, or more generally a coupling, to morph a Gaussian into another.
Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein
Optimal transport tools (OTT-JAX) is a Python toolbox that can solve optimal transport problems between point clouds and histograms.
Randomized Stochastic Gradient Descent Ascent
RSGDA can be parameterized using optimal loop sizes that guarantee the best convergence rates known to hold for SGDA.
Proximal Optimal Transport Modeling of Population Dynamics
We propose to model these trajectories as collective realizations of a causal Jordan-Kinderlehrer-Otto (JKO) flow of measures: The JKO scheme posits that the new configuration taken by a population at time $t+1$ is one that trades off an improvement, in the sense that it decreases an energy, while remaining close (in Wasserstein distance) to the previous configuration observed at $t$.
Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs
The ability to align points across two related yet incomparable point clouds (e. g. living in different spaces) plays an important role in machine learning.
Efficient and Modular Implicit Differentiation
In this paper, we propose automatic implicit differentiation, an efficient and modular approach for implicit differentiation of optimization problems.
Low-Rank Sinkhorn Factorization
Because matrix-vector products are pervasive in the Sinkhorn algorithm, several works have proposed to \textit{approximate} kernel matrices appearing in its iterations using low-rank factors.
Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form
Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e. g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry.
Learning with Differentiable Pertubed Optimizers
Machine learning pipelines often rely on optimizers procedures to make discrete decisions (e. g., sorting, picking closest neighbors, or shortest paths).
On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification
Yet, the behavior of minimum Wasserstein estimators is poorly understood, notably in high-dimensional regimes or under model misspecification.
Projection Robust Wasserstein Distance and Riemannian Optimization
Projection robust Wasserstein (PRW) distance, or Wasserstein projection pursuit (WPP), is a robust variant of the Wasserstein distance.
Linear Time Sinkhorn Divergences using Positive Features
Although Sinkhorn divergences are now routinely used in data sciences to compare probability distributions, the computational effort required to compute them remains expensive, growing in general quadratically in the size $n$ of the support of these distributions.
Equitable and Optimal Transport with Multiple Agents
When there is only one agent, we recover the Optimal Transport problem.
Entropic Optimal Transport between (Unbalanced) Gaussian Measures has a Closed Form
Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e. g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry.
Statistics Theory Statistics Theory
Debiased Sinkhorn barycenters
However, entropy brings some inherent smoothing bias, resulting for example in blurred barycenters.
Noisy Adaptive Group Testing using Bayesian Sequential Experimental Design
Our goal in this paper is to propose new group testing algorithms that can operate in a noisy setting (tests can be mistaken) to decide adaptively (looking at past results) which groups to test next, with the goal to converge to a good detection, as quickly, and with as few tests as possible.
Learning with Differentiable Perturbed Optimizers
Machine learning pipelines often rely on optimization procedures to make discrete decisions (e. g., sorting, picking closest neighbors, or shortest paths).
Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm
We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of $m$ discrete probability measures supported on a finite metric space of size $n$.
Missing Data Imputation using Optimal Transport
Missing data is a crucial issue when applying machine learning algorithms to real-world datasets.
Regularized Optimal Transport is Ground Cost Adversarial
In this work we depart from this practical perspective and propose a new interpretation of regularization as a robust mechanism, and show using Fenchel duality that any convex regularization of OT can be interpreted as ground cost adversarial.
Supervised Quantile Normalization for Low-rank Matrix Approximation
Low rank matrix factorization is a fundamental building block in machine learning, used for instance to summarize gene expression profile data or word-document counts.
Fast and Robust Comparison of Probability Measures in Heterogeneous Spaces
Building on \cite{memoli-2011}, who proposed to represent each point in each distribution as the 1D distribution of its distances to all other points, we introduce in this paper the Anchor Energy (AE) and Anchor Wasserstein (AW) distances, which are respectively the energy and Wasserstein distances instantiated on such representations.
Differentiable Ranking and Sorting using Optimal Transport
From this observation, we propose extended rank and sort operators by considering optimal transport (OT) problems (the natural relaxation for assignments) where the auxiliary measure can be any weighted measure supported on $m$ increasing values, where $m \ne n$.
Ground Metric Learning on Graphs
Optimal transport (OT) distances between probability distributions are parameterized by the ground metric they use between observations.
Spatio-Temporal Alignments: Optimal transport through space and time
In this paper, we propose Spatio-Temporal Alignments (STA), a new differentiable formulation of DTW, in which spatial differences between time samples are accounted for using regularized optimal transport (OT).
Multi-subject MEG/EEG source imaging with sparse multi-task regression
Magnetoencephalography and electroencephalography (M/EEG) are non-invasive modalities that measure the weak electromagnetic fields generated by neural activity.
On the Complexity of Approximating Multimarginal Optimal Transport
This provides a first \textit{near-linear time} complexity bound guarantee for approximating the MOT problem and matches the best known complexity bound for the Sinkhorn algorithm in the classical OT setting when $m = 2$.
Deep multi-class learning from label proportions
We propose a learning algorithm capable of learning from label proportions instead of direct data labels.
Differentiable Ranks and Sorting using Optimal Transport
Sorting an array is a fundamental routine in machine learning, one that is used to compute rank-based statistics, cumulative distribution functions (CDFs), quantiles, or to select closest neighbors and labels.
Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport
On the other hand, one of the greatest achievements of the OT literature in recent years lies in regularity theory: Caffarelli showed that the OT map between two well behaved measures is Lipschitz, or equivalently when considering 2-Wasserstein distances, that Brenier convex potentials (whose gradient yields an optimal map) are smooth.
Precision-Recall Curves Using Information Divergence Frontiers
Despite the tremendous progress in the estimation of generative models, the development of tools for diagnosing their failures and assessing their performance has advanced at a much slower pace.
Subspace Detours: Building Transport Plans that are Optimal on Subspace Projections
A popular approach to avoid this curse is to project input measures on lower-dimensional subspaces (1D lines in the case of sliced Wasserstein distances), solve the OT problem between these reduced measures, and settle for the Wasserstein distance between these reductions, rather than that between the original measures.
Group level MEG/EEG source imaging via optimal transport: minimum Wasserstein estimates
Inferring the location of the current sources that generated these magnetic fields is an ill-posed inverse problem known as source imaging.
Tree-Sliced Variants of Wasserstein Distances
Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions.
Subspace Robust Wasserstein Distances
Making sense of Wasserstein distances between discrete measures in high-dimensional settings remains a challenge.
Stochastic Deep Networks
This allows to design discriminative networks (to classify or reduce the dimensionality of input measures), generative architectures (to synthesize measures) and recurrent pipelines (to predict measure dynamics).
Semi-dual Regularized Optimal Transport
Variational problems that involve Wasserstein distances and more generally optimal transport (OT) theory are playing an increasingly important role in data sciences.
Unsupervised Hyperalignment for Multilingual Word Embeddings
This paper extends this line of work to the problem of aligning multiple languages to a common space.
Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport
Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data.
Wasserstein regularization for sparse multi-task regression
We argue in this paper that these techniques fail to leverage the spatial information associated to regressors.
Generalizing Point Embeddings using the Wasserstein Space of Elliptical Distributions
We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets.
Computational Optimal Transport
Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site.
Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning
Wasserstein barycenters) between dictionary atoms; such atoms are themselves synthetic histograms in the probability simplex.
Sliced Wasserstein Kernel for Persistence Diagrams
To incorporate PDs in a learning pipeline, several kernels have been proposed for PDs with a strong emphasis on the stability of the RKHS distance w. r. t.
Ranked #2 on
Graph Classification
on NEURON-MULTI
GAN and VAE from an Optimal Transport Point of View
This short article revisits some of the ideas introduced in arXiv:1701. 07875 and arXiv:1705. 07642 in a simple setup.
Learning Generative Models with Sinkhorn Divergences
The ability to compare two degenerate probability distributions (i. e. two probability distributions supported on two distinct low-dimensional manifolds living in a much higher-dimensional space) is a crucial problem arising in the estimation of generative models for high-dimensional observations such as those arising in computer vision or natural language.
Soft-DTW: a Differentiable Loss Function for Time-Series
We propose in this paper a differentiable learning loss between time series, building upon the celebrated dynamic time warping (DTW) discrepancy.
Wasserstein Training of Restricted Boltzmann Machines
This metric between observations can then be used to define the Wasserstein distance between the distribution induced by the Boltzmann machine on the one hand, and that given by the training sample on the other hand.
Wasserstein Discriminant Analysis
Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace.
Stochastic Optimization for Large-scale Optimal Transport
We instantiate these ideas in three different setups: (i) when comparing a discrete distribution to another, we show that incremental stochastic optimization schemes can beat Sinkhorn's algorithm, the current state-of-the-art finite dimensional OT solver; (ii) when comparing a discrete distribution to a continuous density, a semi-discrete reformulation of the dual program is amenable to averaged stochastic gradient descent, leading to better performance than approximately solving the problem by discretization ; (iii) when dealing with two continuous densities, we propose a stochastic gradient descent over a reproducing kernel Hilbert space (RKHS).
On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests
In this work, our central object is the Wasserstein distance, as we form a chain of connections from univariate methods like the Kolmogorov-Smirnov test, PP/QQ plots and ROC/ODC curves, to multivariate tests involving energy statistics and kernel based maximum mean discrepancy.
Wasserstein Training of Boltzmann Machines
The Boltzmann machine provides a useful framework to learn highly complex, multimodal and multiscale data distributions that occur in the real world.
Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric
Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family.
Fast Optimal Transport Averaging of Neuroimaging Data
Data are large, the geometry of the brain is complex and the between subjects variability leads to spatially or temporally non-overlapping effects of interest.
A Smoothed Dual Approach for Variational Wasserstein Problems
Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures.
Iterative Bregman Projections for Regularized Transportation Problems
This article details a general numerical framework to approximate so-lutions to linear programs related to optimal transport.
Numerical Analysis Analysis of PDEs
Sinkhorn Distances: Lightspeed Computation of Optimal Transport
Optimal transportation distances are a fundamental family of parameterized distances for histograms in the probability simplex.
Fast Computation of Wasserstein Barycenters
We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric.
Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances
Optimal transportation distances are a fundamental family of parameterized distances for histograms.
White Functionals for Anomaly Detection in Dynamical Systems
The candidate functionals are estimated in a subset of a reproducing kernel Hilbert space associated with the set where the process takes values.
