1. A Block-Sparse Tensor Train Format for Sample-Efficient High-Dimensional Polynomial Regression
- Author
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Michael Götte, Reinhold Schneider, and Philipp Trunschke
- Subjects
sample efficiency ,homogeneous polynomials ,sparse tensor networks ,alternating least square ,empirical L2 approximation ,Applied mathematics. Quantitative methods ,T57-57.97 ,Probabilities. Mathematical statistics ,QA273-280 - Abstract
Low-rank tensors are an established framework for the parametrization of multivariate polynomials. We propose to extend this framework by including the concept of block-sparsity to efficiently parametrize homogeneous, multivariate polynomials with low-rank tensors. This provides a representation of general multivariate polynomials as a sum of homogeneous, multivariate polynomials, represented by block-sparse, low-rank tensors. We show that this sum can be concisely represented by a single block-sparse, low-rank tensor.We further prove cases, where low-rank tensors are particularly well suited by showing that for banded symmetric tensors of homogeneous polynomials the block sizes in the block-sparse multivariate polynomial space can be bounded independent of the number of variables.We showcase this format by applying it to high-dimensional least squares regression problems where it demonstrates improved computational resource utilization and sample efficiency.
- Published
- 2021
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