One-pass additive-error subset selection for $\ell_{p}$ subspace approximation

TL;DR

This paper introduces a one-pass subset selection algorithm for $\, ext{l}_p$ subspace approximation that provides an additive error guarantee for any $p$ in $[1, \, ext{infty})$, improving efficiency over previous methods.

Contribution

It presents the first one-pass subset selection algorithm with an additive approximation guarantee applicable to all $p \,\in\ [1, \infty)$, extending beyond special cases.

Findings

Works for any $p$ in $[1, \infty)$ with additive guarantee.

Requires only a single pass over data, improving efficiency.

Generalizes previous special-case algorithms to all $p$.

Abstract

We consider the problem of subset selection for $\ell_{p}$ subspace approximation, that is, to efficiently find a \emph{small} subset of data points such that solving the problem optimally for this subset gives a good approximation to solving the problem optimally for the original input. Previously known subset selection algorithms based on volume sampling and adaptive sampling \cite{DeshpandeV07}, for the general case of $p \in [1, \infty)$, require multiple passes over the data. In this paper, we give a one-pass subset selection with an additive approximation guarantee for $\ell_{p}$ subspace approximation, for any $p \in [1, \infty)$. Earlier subset selection algorithms that give a one-pass multiplicative $(1+\epsilon)$ approximation work under the special cases. Cohen \textit{et al.} \cite{CohenMM17} gives a one-pass subset section that offers multiplicative $(1+\epsilon)$ approximation guarantee for the special case of $\ell_{2}$ subspace approximation. Mahabadi \textit{et al.} \cite{MahabadiRWZ20} gives a one-pass \emph{noisy} subset selection with $(1+\epsilon)$ approximation guarantee for $\ell_{p}$ subspace approximation when $p \in \{1, 2\}$. Our subset selection algorithm gives a weaker, additive approximation guarantee, but it works for any $p \in [1, \infty)$.