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A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regression
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Low-rank tensors are an established framework for high-dimensional least-squares problems. We propose to extend this framework by including the concept of block-sparsity. In the context of polynomial regression each sparsity pattern corresponds to some subspace of homogeneous multivariate polynomials. This allows us to adapt the ansatz space to align better with known sample complexity results. The resulting method is tested in numerical experiments and demonstrates improved computational resource utilization and sample efficiency.
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(MPO)$^2$: Multivariate Polynomial Optimization based on Matrix Product Operators
Writing both feature maps and polynomial coefficient tensors as MPOs yields feature-order-independent polynomial models that beat CPD/TT polynomial baselines on most benchmarks.
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