A PTAS for $\ell_0$-Low Rank Approximation: Solving Dense CSPs over Reals

We consider the Low Rank Approximation problem, where the input consists of a matrix $A \in \mathbb{R}^{n_R \times n_C}$ and an integer $k$, and the goal is to find a matrix $B$ of rank at most $k$ that minimizes $\| A - B \|_0$, which is the number of entries where $A$ and $B$ differ. For any constant $k$ and $\varepsilon > 0$, we present a polynomial time $(1 + \varepsilon)$-approximation time for this problem, which significantly improves the previous best $poly(k)$-approximation. Our algorithm is obtained by viewing the problem as a Constraint Satisfaction Problem (CSP) where each row and column becomes a variable that can have a value from $\mathbb{R}^k$. In this view, we have a constraint between each row and column, which results in a {\em dense} CSP, a well-studied topic in approximation algorithms. While most of previous algorithms focus on finite-size (or constant-size) domains and involve an exhaustive enumeration over the entire domain, we present a new framework that bypasses such an enumeration in $\mathbb{R}^k$. We also use tools from the rich literature of Low Rank Approximation in different objectives (e.g., $\ell_p$ with $p \in (0, \infty)$) or domains (e.g., finite fields/generalized Boolean). We believe that our techniques might be useful to study other real-valued CSPs and matrix optimization problems. On the hardness side, when $k$ is part of the input, we prove that Low Rank Approximation is NP-hard to approximate within a factor of $\Omega(\log n)$. This is the first superconstant NP-hardness of approximation for any $p \in [0, \infty]$ that does not rely on stronger conjectures (e.g., the Small Set Expansion Hypothesis).