Testing Proximity to Subspaces: Approximate $$\ell _\infty $$ Minimization in Constant Time

Abstract

We consider the subspace proximity problem: Given a vector $${\varvec{x}} \in {\mathbb {R}}^n$$ and a basis matrix $$V \in {\mathbb {R}}^{n \times m}$$, the objective is to determine whether $${\varvec{x}}$$ is close to the subspace spanned by V. Although the problem is solvable by linear programming, it is time consuming especially when n is large. In this paper, we propose a quick tester that solves the problem correctly with high probability. Our tester runs in time independent of n and can be used as a sieve before computing the exact distance between $${\varvec{x}}$$ and the subspace. The number of coordinates of $${\varvec{x}}$$ queried by our tester is $$O(\frac{m}{\epsilon }\log \frac{m}{\epsilon })$$, where $$\epsilon $$ is an error parameter, and we show almost matching lower bounds. By experiments, we demonstrate the scalability and applicability of our tester using synthetic and real data sets.