Guessing Efficiently for Constrained Subspace Approximation
This work addresses subspace approximation with constraints, which is incremental but has applications in fair machine learning and clustering.
The paper tackles the constrained subspace approximation problem by introducing a coreset-guess-solve framework, achieving (1+ε)-multiplicative or ε-additive approximations for various constraints and improving known results for problems like fair subspace approximation and projected non-negative matrix factorization.
In this paper we study constrained subspace approximation problem. Given a set of $n$ points $\{a_1,\ldots,a_n\}$ in $\mathbb{R}^d$, the goal of the {\em subspace approximation} problem is to find a $k$ dimensional subspace that best approximates the input points. More precisely, for a given $p\geq 1$, we aim to minimize the $p$th power of the $\ell_p$ norm of the error vector $(\|a_1-\bm{P}a_1\|,\ldots,\|a_n-\bm{P}a_n\|)$, where $\bm{P}$ denotes the projection matrix onto the subspace and the norms are Euclidean. In \emph{constrained} subspace approximation (CSA), we additionally have constraints on the projection matrix $\bm{P}$. In its most general form, we require $\bm{P}$ to belong to a given subset $\mathcal{S}$ that is described explicitly or implicitly. We introduce a general framework for constrained subspace approximation. Our approach, that we term coreset-guess-solve, yields either $(1+\varepsilon)$-multiplicative or $\varepsilon$-additive approximations for a variety of constraints. We show that it provides new algorithms for partition-constrained subspace approximation with applications to {\it fair} subspace approximation, $k$-means clustering, and projected non-negative matrix factorization, among others. Specifically, while we reconstruct the best known bounds for $k$-means clustering in Euclidean spaces, we improve the known results for the remainder of the problems.