Screening Sinkhorn Algorithm for Regularized Optimal Transport

International audienceWe introduce in this paper a novel strategy for efficiently approximating the Sinkhorn distance between two discrete measures. After identifying neglectable components of the dual solution of the regularized Sinkhorn problem, we propose to screen those components by directly setting them at that value before entering the Sinkhorn problem. This allows us to solve a smaller Sinkhorn problem while ensuring approximation with provable guarantees. More formally, the approach is based on a new formulation of dual of Sinkhorn divergence problem and on the KKT optimality conditions of this problem, which enable identification of dual components to be screened. This new analysis leads to the Screenkhorn algorithm. We illustrate the efficiency of Screenkhorn on complex tasks such as dimensionality reduction and domain adaptation involving regularized optimal transport

Theoretical Guarantees for Bridging Metric Measure Embedding and Optimal Transport

We propose a novel approach for comparing distributions whose supports do not necessarily lie on the same metric space. Unlike Gromov-Wasserstein (GW) distance which compares pairwise distances of elements from each distribution, we consider a method allowing to embed the metric measure spaces in a common Euclidean space and compute an optimal transport (OT) on the embedded distributions. This leads to what we call a sub-embedding robust Wasserstein (SERW) distance. Under some conditions, SERW is a distance that considers an OT distance of the (low-distorted) embedded distributions using a common metric. In addition to this novel proposal that generalizes several recent OT works, our contributions stand on several theoretical analyses: (i) we characterize the embedding spaces to define SERW distance for distribution alignment; (ii) we prove that SERW mimics almost the same properties of GW distance, and we give a cost relation between GW and SERW. The paper also provides some numerical illustrations of how SERW behaves on matching problems

Metric and Representation Learning

All data has some inherent mathematical structure. I am interested in understanding the intrinsic geometric and probabilistic structure of data to design effective algorithms and tools that can be applied to machine learning and across all branches of science. The focus of this thesis is to increase the effectiveness of machine learning techniques by developing a mathematical and algorithmic framework using which, given any type of data, we can learn an optimal representation. Representation learning is done for many reasons. It could be done to fix the corruption given corrupted data or to learn a low dimensional or simpler representation, given high dimensional data or a very complex representation of the data. It could also be that the current representation of the data does not capture the important geometric features of the data. One of the many challenges in representation learning is determining ways to judge the quality of the representation learned. In many cases, the consensus is that if d is the natural metric on the representation, then this metric should provide meaningful information about the data. Many examples of this can be seen in areas such as metric learning, manifold learning, and graph embedding. However, most algorithms that solve these problems learn a representation in a metric space first and then extract a metric. A large part of my research is exploring what happens if the order is switched, that is, learn the appropriate metric first and the embedding later. The philosophy behind this approach is that understanding the inherent geometry of the data is the most crucial part of representation learning. Often, studying the properties of the appropriate metric on the input data sets indicates the type of space, we should be seeking for the representation. Hence giving us more robust representations. Optimizing for the appropriate metric can also help overcome issues such as missing and noisy data. My projects fall into three different areas of representation learning. 1) Geometric and probabilistic analysis of representation learning methods. 2) Developing methods to learn optimal metrics on large datasets. 3) Applications. For the category of geometric and probabilistic analysis of representation learning methods, we have three projects. First, designing optimal training data for denoising autoencoders. Second, formulating a new optimal transport problem and understanding the geometric structure. Third, analyzing the robustness to perturbations of the solutions obtained from the classical multidimensional scaling algorithm versus that of the true solutions to the multidimensional scaling problem. For learning optimal metric, we are given a dissimilarity matrix $hat{D}$, some function $f$ and some a subset $S$ of the space of all metrics and we want to find $D in S$ that minimizes $f(D,hat{D})$. In this thesis, we consider the version of the problem when $S$ is the space of metrics defined on a fixed graph. That is, given a graph $G$, we let $S$, be the space of all metrics defined via $G$. For this $S$, we consider the sparse objective function as well as convex objective functions. We also looked at the problem where we want to learn a tree. We also show how the ideas behind learning the optimal metric can be applied to dimensionality reduction in the presence of missing data. Finally, we look at an application to real world data. Specifically trying to reconstruct ancient Greek text.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169738/1/rsonthal_1.pd