Proportional Volume Sampling and Approximation Algorithms for A-Optimal Design

We study the $A$-optimal design problem where we are given vectors $v_1,\ldots,v_n\in\mathbb{R}^d$, an integer $k\geq d$, and the goal is to select a set $S$ of $k$ vectors that minimizes the trace of $(\sum_{i\in S}v_iv_i^\top)^{-1}$. Traditionally, the problem is an instance of optimal design of experiments in statistics where each vector corresponds to a linear measurement of an unknown vector and the goal is to pick $k$ of them that minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placement in wireless networks, sparse least squares regression, feature selection for $k$-means clustering, and matrix approximation. In this paper, we introduce proportional volume sampling to obtain improved approximation algorithms for $A$-optimal design. Given a matrix, proportional volume sampling picks a set of columns $S$ of size $k$ with probability proportional to $\mu(S)$ times $\det(\sum_{i\in S}v_iv_i^\top)$ for some measure $\mu$. Our main result is to show the approximability of the $A$-optimal design problem can be reduced to approximate independence properties of the measure $\mu$. We appeal to hard-core distributions as candidate distributions $\mu$ that allow us to obtain improved approximation algorithms for the $A$-optimal design. Our results include a $d$-approximation when $k=d$, an $(1+\epsilon)$-approximation when $k=\Omega\left(\frac{d}{\epsilon}+\frac{1}{\epsilon^2}\log\frac{1}{\epsilon}\right)$ and $\frac{k}{k-d+1}$-approximation when repetitions of vectors are allowed in the solution. We consider generalization of the problem for $k\leq d$ and obtain a $k$-approximation. The last result implies a restricted invertibility principle for the harmonic mean of singular values. We also show that the problem is $\mathsf{NP}$-hard to approximate within a fixed constant when $k=d$.