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arXiv · 2412.06063

On Socially Fair Low-Rank Approximation and Column Subset Selection

Zhao Song·Ali Vakilian·David P. Woodruff·Samson Zhou
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Abstract

Low-rank approximation and column subset selection are two fundamental and related problems that are applied across a wealth of machine learning applications. In this paper, we study the question of socially fair low-rank approximation and socially fair column subset selection, where the goal is to minimize the loss over all sub-populations of the data. We show that surprisingly, even constant-factor approximation to fair low-rank approximation requires exponential time under certain standard complexity hypotheses. On the positive side, we give an algorithm for fair low-rank approximation that, for a constant number of groups and constant-factor accuracy, runs in $2^{\text{poly}(k)}$ time rather than the na\"{i}ve $n^{\text{poly}(k)}$, which is a substantial improvement when the dataset has a large number $n$ of observations. We then show that there exist bicriteria approximation algorithms for fair low-rank approximation and fair column subset selection that run in polynomial time.

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Zhao Song, Ali Vakilian, David P. Woodruff, Samson Zhou. 2024-12-08. On Socially Fair Low-Rank Approximation and Column Subset Selection. https://arxiv.org/abs/2412.06063

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