Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)
Zhao Song, David Woodruff, Peilin Zhong
There are a number of approximation algorithms for NP-hard versions of low rank approximation, such as finding a rank-k matrix B minimizing the sum of absolute values of differences to a given n-by-n matrix A, min, or more generally finding a rank-k matrix B which minimizes the sum of p-th powers of absolute values of differences, \min_{\textrm{rank-}k~B}\|A-B\|_p^p. Many of these algorithms are linear time columns subset selection algorithms, returning a subset of \poly(k \log n) columns whose cost is no more than a \poly(k) factor larger than the cost of the best rank-k matrix. The above error measures are special cases of the following general entrywise low rank approximation problem: given an arbitrary function g:\mathbb{R} \rightarrow \mathbb{R}_{\geq 0}, find a rank-k matrix B which minimizes \|A-B\|_g = \sum_{i,j}g(A_{i,j}-B_{i,j}). A natural question is which functions g admit efficient approximation algorithms? Indeed, this is a central question of recent work studying generalized low rank models. In this work we give approximation algorithms for {\it every} function g which is approximately monotone and satisfies an approximate triangle inequality, and we show both of these conditions are necessary. Further, our algorithm is efficient if the function g admits an efficient approximate regression algorithm. Our approximation algorithms handle functions which are not even scale-invariant, such as the Huber loss function, which we show have very different structural properties than \ell_p-norms, e.g., one can show the lack of scale-invariance causes any column subset selection algorithm to provably require a \sqrt{\log n} factor larger number of columns than \ell_p-norms; nevertheless we design the first efficient column subset selection algorithms for such error measures.