Hardness of Low Rank Approximation of Entrywise Transformed Matrix Products
This work addresses computational hardness in machine learning and numerical linear algebra, providing foundational insights for algorithm design in areas like NLP, but it is incremental as it builds on prior sublinear approximation results.
The paper tackles the problem of low rank approximation for entrywise transformed matrix products, showing that certain conditions are necessary for efficient algorithms and providing matching lower bounds and algorithms. It proves that without these conditions, achieving relative error approximation requires nearly quadratic time, and presents tight algorithms with runtime depending on parameters like polynomial degree.
Inspired by fast algorithms in natural language processing, we study low rank approximation in the entrywise transformed setting where we want to find a good rank $k$ approximation to $f(U \cdot V)$, where $U, V^\top \in \mathbb{R}^{n \times r}$ are given, $r = O(\log(n))$, and $f(x)$ is a general scalar function. Previous work in sublinear low rank approximation has shown that if both (1) $U = V^\top$ and (2) $f(x)$ is a PSD kernel function, then there is an $O(nk^{ω-1})$ time constant relative error approximation algorithm, where $ω\approx 2.376$ is the exponent of matrix multiplication. We give the first conditional time hardness results for this problem, demonstrating that both conditions (1) and (2) are in fact necessary for getting better than $n^{2-o(1)}$ time for a relative error low rank approximation for a wide class of functions. We give novel reductions from the Strong Exponential Time Hypothesis (SETH) that rely on lower bounding the leverage scores of flat sparse vectors and hold even when the rank of the transformed matrix $f(UV)$ and the target rank are $n^{o(1)}$, and when $U = V^\top$. Furthermore, even when $f(x) = x^p$ is a simple polynomial, we give runtime lower bounds in the case when $U \neq V^\top$ of the form $Ω(\min(n^{2-o(1)}, Ω(2^p)))$. Lastly, we demonstrate that our lower bounds are tight by giving an $O(n \cdot \text{poly}(k, 2^p, 1/ε))$ time relative error approximation algorithm and a fast $O(n \cdot \text{poly}(k, p, 1/ε))$ additive error approximation using fast tensor-based sketching. Additionally, since our low rank algorithms rely on matrix-vector product subroutines, our lower bounds extend to show that computing $f(UV)W$, for even a small matrix $W$, requires $Ω(n^{2-o(1)})$ time.