When big data actually are low-rank, or entrywise approximation of certain function-generated matrices

Abstract

The article concerns low-rank approximation of matrices generated by sampling a smooth function of two $m$-dimensional variables. We identify several misconceptions surrounding a claim that, for a specific class of analytic functions, such $n \times n$ matrices admit accurate entrywise approximation of rank that is independent of $m$ and grows as $\log(n)$ -- colloquially known as ''big-data matrices are approximately low-rank''. We provide a theoretical explanation of the numerical results presented in support of this claim, describing three narrower classes of functions for which function-generated matrices can be approximated within an entrywise error of order $\varepsilon$ with rank $\mathcal{O}(\log(n) \varepsilon^{-2} \log(\varepsilon^{-1}))$ that is independent of the dimension $m$: (i) functions of the inner product of the two variables, (ii) functions of the Euclidean distance between the variables, and (iii) shift-invariant positive-definite kernels. We extend our argument to tensor-train approximation of tensors generated with functions of the ''higher-order inner product'' of their multiple variables. We discuss our results in the context of low-rank approximation of (a) growing datasets and (b) attention in transformer neural networks.