arXiv subjects

Rishi Sonthalia

Publications and source records attributed to Rishi Sonthalia.

23 records · Page 2Linked to original sources

arXiv2020

Dual Regularized Optimal Transport

In this paper, we present a new formulation of unbalanced optimal transport called Dual Regularized Optimal Transport (DROT). We argue that regularizing the dual formulation of optimal transport results in a version of unbalanced optimal transport that leads to sparse solutions and that gives us control over mass creation and destruction. We build intuition behind such control and present theoretical properties of the solutions to DROT. We demonstrate that due to recent advances in optimization techniques, we can feasibly solve such a formulation at large scales and present extensive experimental evidence for this formulation and its solution.

cs.CG↗

arXiv2020

Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding

Given data, finding a faithful low-dimensional hyperbolic embedding of the data is a key method by which we can extract hierarchical information or learn representative geometric features of the data. In this paper, we explore a new method for learning hyperbolic representations by taking a metric-first approach. Rather than determining the low-dimensional hyperbolic embedding directly, we learn a tree structure on the data. This tree structure can then be used directly to extract hierarchical information, embedded into a hyperbolic manifold using Sarkar's construction \cite{sarkar}, or used as a tree approximation of the original metric. To this end, we present a novel fast algorithm \textsc{TreeRep} such that, given a $δ$-hyperbolic metric (for any $δ\geq 0$), the algorithm learns a tree structure that approximates the original metric. In the case when $δ= 0$, we show analytically that \textsc{TreeRep} exactly recovers the original tree structure. We show empirically that \textsc{TreeRep} is not only many orders of magnitude faster than previously known algorithms, but also produces metrics with lower average distortion and higher mean average precision than most previous algorithms for learning hyperbolic embeddings, extracting hierarchical information, and approximating metrics via tree metrics.

cs.LG↗

arXiv2019

Generalized Metric Repair on Graphs

Many modern data analysis algorithms either assume or are considerably more efficient if the distances between the data points satisfy a metric. These algorithms include metric learning, clustering, and dimension reduction. As real data sets are noisy, distances often fail to satisfy a metric. For this reason, Gilbert and Jain and Fan et al. introduced the closely related sparse metric repair and metric violation distance problems. The goal of these problems is to repair as few distances as possible to ensure they satisfy a metric. Three variants were considered, one admitting a polynomial time algorithm. The other variants were shown to be APX-hard, and an $O(OPT^{1/3})$-approximation was given, where $OPT$ is the optimal solution size. In this paper, we generalize these problems to no longer consider all distances between the data points. That is, we consider a weighted graph $G$ with corrupted weights $w$, and our goal is to find the smallest number of weight modifications so that the resulting weighted graph distances satisfy a metric. This is a natural generalization and is more flexible as it takes into account different relationships among the data points. As in previous work, we distinguish among the types of repairs permitted and focus on the increase only and general versions. We demonstrate the inherent combinatorial structure of the problem, and give an approximation-preserving reduction from MULTICUT. Conversely, we show that for any fixed constant $ς$, for the large class of $ς$-chordal graphs, the problems are fixed parameter tractable. Call a cycle broken if it contains an edge whose weight is larger than the sum of all its other edges, and call the amount of this difference its deficit. We present approximation algorithms, one which depends on the maximum number of edges in a broken cycle, and one which depends on the number of distinct deficit values.

cs.DS↗

arXiv2019

Unsupervised Metric Learning in Presence of Missing Data

For many machine learning tasks, the input data lie on a low-dimensional manifold embedded in a high dimensional space and, because of this high-dimensional structure, most algorithms are inefficient. The typical solution is to reduce the dimension of the input data using standard dimension reduction algorithms such as ISOMAP, LAPLACIAN EIGENMAPS or LLES. This approach, however, does not always work in practice as these algorithms require that we have somewhat ideal data. Unfortunately, most data sets either have missing entries or unacceptably noisy values. That is, real data are far from ideal and we cannot use these algorithms directly. In this paper, we focus on the case when we have missing data. Some techniques, such as matrix completion, can be used to fill in missing data but these methods do not capture the non-linear structure of the manifold. Here, we present a new algorithm MR-MISSING that extends these previous algorithms and can be used to compute low dimensional representation on data sets with missing entries. We demonstrate the effectiveness of our algorithm by running three different experiments. We visually verify the effectiveness of our algorithm on synthetic manifolds, we numerically compare our projections against those computed by first filling in data using nlPCA and mDRUR on the MNIST data set, and we also show that we can do classification on MNIST with missing data. We also provide a theoretical guarantee for MR-MISSING under some simplifying assumptions.

cs.LG↗

arXiv2018

Generalized Metric Repair on Graphs

Many modern data analysis algorithms either assume that or are considerably more efficient if the distances between the data points satisfy a metric. These algorithms include metric learning, clustering, and dimensionality reduction. Because real data sets are noisy, the similarity measures often fail to satisfy a metric. For this reason, Gilbert and Jain [11] and Fan, et al. [8] introduce the closely related problems of $\textit{sparse metric repair}$ and $\textit{metric violation distance}$. The goal of each problem is to repair as few distances as possible to ensure that the distances between the data points satisfy a metric. We generalize these problems so as to no longer require all the distances between the data points. That is, we consider a weighted graph $G$ with corrupted weights w and our goal is to find the smallest number of modifications to the weights so that the resulting weighted graph distances satisfy a metric. This problem is a natural generalization of the sparse metric repair problem and is more flexible as it takes into account different relationships amongst the input data points. As in previous work, we distinguish amongst the types of repairs permitted (decrease, increase, and general repairs). We focus on the increase and general versions and establish hardness results and show the inherent combinatorial structure of the problem. We then show that if we restrict to the case when $G$ is a chordal graph, then the problem is fixed parameter tractable. We also present several classes of approximation algorithms. These include and improve upon previous metric repair algorithms for the special case when $G = K_n$

cs.DS↗