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

• URL: http://arxiv.org/abs/2407.03250v2
• Date: Thu, 4 Jul 2024 10:56:45 GMT
• Title: When big data actually are low-rank, or entrywise approximation of certain function-generated matrices
• Authors: Stanislav Budzinskiy,
• Abstract summary: We refute an argument made in the literature that, for a specific class of analytic functions, such matrices admit accurate entrywise approximation of rank that is independent of $m$. We describe three narrower classes of functions for which $n times n$ function-generated matrices can be approximated within an entrywise error of order $varepsilon$ with rank $mathcalO(log(n) varepsilon-2 mathrmpolylog(varepsilon-1)$ that is independent of the dimension $m$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The article concerns low-rank approximation of matrices generated by sampling a smooth function of two $m$-dimensional variables. We refute an argument made in the literature that, for a specific class of analytic functions, such matrices admit accurate entrywise approximation of rank that is independent of $m$. We provide a theoretical explanation of the numerical results presented in support of this argument, describing three narrower classes of functions for which $n \times n$ function-generated matrices can be approximated within an entrywise error of order $\varepsilon$ with rank $\mathcal{O}(\log(n) \varepsilon^{-2} \mathrm{polylog}(\varepsilon^{-1}))$ that is independent of the dimension $m$: (i) functions of the inner product of the two variables, (ii) functions of the squared Euclidean distance between the variables, and (iii) shift-invariant positive-definite kernels. We extend our argument to low-rank tensor-train approximation of tensors generated with functions of the multi-linear product of their $m$-dimensional variables. We discuss our results in the context of low-rank approximation of attention in transformer neural networks.

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