Iván Medri

Yikun Bai, Ivan Medri, Rocio Diaz Martin et al.

Optimal transport (OT) has gained popularity due to its various applications in fields such as machine learning, statistics, and signal processing. However, the balanced mass requirement limits its performance in practical problems. To address these limitations, variants of the OT problem, including unbalanced OT, Optimal partial transport (OPT), and Hellinger Kantorovich (HK), have been proposed. In this paper, we propose the Linear optimal partial transport (LOPT) embedding, which extends the (local) linearization technique on OT and HK to the OPT problem. The proposed embedding allows for faster computation of OPT distance between pairs of positive measures. Besides our theoretical contributions, we demonstrate the LOPT embedding technique in point-cloud interpolation and PCA analysis.

Sumati Thareja, Gustavo Rohde, Rocio Diaz Martin et al.

We describe a method for signal parameter estimation using the signed cumulative distribution transform (SCDT), a recently introduced signal representation tool based on optimal transport theory. The method builds upon signal estimation using the cumulative distribution transform (CDT) originally introduced for positive distributions. Specifically, we show that Wasserstein-type distance minimization can be performed simply using linear least squares techniques in SCDT space for arbitrary signal classes, thus providing a global minimizer for the estimation problem even when the underlying signal is a nonlinear function of the unknown parameters. Comparisons to current signal estimation methods using $L_p$ minimization shows the advantage of the method.

Rocio Diaz Martin, Ivan Medri, Yikun Bai et al.

The optimal transport problem for measures supported on non-Euclidean spaces has recently gained ample interest in diverse applications involving representation learning. In this paper, we focus on circular probability measures, i.e., probability measures supported on the unit circle, and introduce a new computationally efficient metric for these measures, denoted as Linear Circular Optimal Transport (LCOT). The proposed metric comes with an explicit linear embedding that allows one to apply Machine Learning (ML) algorithms to the embedded measures and seamlessly modify the underlying metric for the ML algorithm to LCOT. We show that the proposed metric is rooted in the Circular Optimal Transport (COT) and can be considered the linearization of the COT metric with respect to a fixed reference measure. We provide a theoretical analysis of the proposed metric and derive the computational complexities for pairwise comparison of circular probability measures. Lastly, through a set of numerical experiments, we demonstrate the benefits of LCOT in learning representations of circular measures.

Rocío Díaz Martín, Ivan V. Medri, James M. Murphy

This paper considers the problem of estimating a matrix that encodes pairwise distances in a finite metric space (or, more generally, the edge weight matrix of a network) under the barycentric coding model (BCM) with respect to the Gromov-Wasserstein (GW) distance function. We frame this task as estimating the unknown barycentric coordinates with respect to the GW distance, assuming that the target matrix (or kernel) belongs to the set of GW barycenters of a finite collection of known templates. In the language of harmonic analysis, if computing GW barycenters can be viewed as a synthesis problem, this paper aims to solve the corresponding analysis problem. We propose two methods: one utilizing fixed-point iteration for computing GW barycenters, and another employing a differentiation-based approach to the GW structure using a blow-up technique. Finally, we demonstrate the application of the proposed GW analysis approach in a series of numerical experiments and applications to machine learning.

Shourav B. Rabbani, Ivan V. Medri, Manar D. Samad

Despite groundbreaking success in image and text learning, deep learning has not achieved significant improvements against traditional machine learning (ML) when it comes to tabular data. This performance gap underscores the need for data-centric treatment and benchmarking of learning algorithms. Recently, attention and contrastive learning breakthroughs have shifted computer vision and natural language processing paradigms. However, the effectiveness of these advanced deep models on tabular data is sparsely studied using a few data sets with very large sample sizes, reporting mixed findings after benchmarking against a limited number of baselines. We argue that the heterogeneity of tabular data sets and selective baselines in the literature can bias the benchmarking outcomes. This article extensively evaluates state-of-the-art attention and contrastive learning methods on a wide selection of 28 tabular data sets (14 easy and 14 hard-to-classify) against traditional deep and machine learning. Our data-centric benchmarking demonstrates when traditional ML is preferred over deep learning and vice versa because no best learning method exists for all tabular data sets. Combining between-sample and between-feature attentions conquers the invincible traditional ML on tabular data sets by a significant margin but fails on high dimensional data, where contrastive learning takes a robust lead. While a hybrid attention-contrastive learning strategy mostly wins on hard-to-classify data sets, traditional methods are frequently superior on easy-to-classify data sets with presumably simpler decision boundaries. To the best of our knowledge, this is the first benchmarking paper with statistical analyses of attention and contrastive learning performances on a diverse selection of tabular data sets against traditional deep and machine learning baselines to facilitate further advances in this field.

Akram Aldroubi, Rocio Diaz Martin, Ivan Medri et al.

This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [ACHA 45 (2018), no. 3, 616-641] to arbitrary (signed) signals on $\overline{\mathbb{R}}$. We present both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity, linear separability and others. Finally, we describe a metric in transform space, and demonstrate the application of the transform in classifying (detecting) signals under random displacements.