# Robust Mahalanobis Metric Learning via Geometric Approximation Algorithms

## TL;DR

This paper introduces a fast, parallelizable algorithm for robust Mahalanobis metric learning that effectively handles adversarial label noise, with theoretical guarantees and practical improvements demonstrated on various datasets.

## Contribution

It presents a fully polynomial-time approximation scheme for robust Mahalanobis metric learning, leveraging linear programming tools and ensuring near-optimal performance despite adversarial label corruption.

## Key findings

- Algorithm is nearly-linear time and fully parallelizable.
- Effective in recovering metrics with adversarial label noise.
- Experimental results show robustness on real, synthetic, and poisoned data.

## Abstract

Learning Mahalanobis metric spaces is an important problem that has found numerous applications. Several algorithms have been designed for this problem, including Information Theoretic Metric Learning (ITML) [Davis et al. 2007] and Large Margin Nearest Neighbor (LMNN) classification [Weinberger and Saul 2009]. We study the problem of learning a Mahalanobis metric space in the presence of adversarial label noise. To that end, we consider a formulation of Mahalanobis metric learning as an optimization problem, where the objective is to minimize the number of violated similarity/dissimilarity constraints. We show that for any fixed ambient dimension, there exists a fully polynomial-time approximation scheme (FPTAS) with nearly-linear running time. This result is obtained using tools from the theory of linear programming in low dimensions. As a consequence, we obtain a fully-parallelizable algorithm that recovers a nearly-optimal metric space, even when a small fraction of the labels is corrupted adversarially. We also discuss improvements of the algorithm in practice, and present experimental results on real-world, synthetic, and poisoned data sets.