A Block-Sparse Tensor Train Format for Sample-Efficient High-Dimensional Polynomial Regression

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.

Necessary Conditions for Interpolation by Multivariate Polynomials

Let $$\Omega $$ Ω be a smooth, bounded, convex domain in $${\mathbb {R}}^n$$ R n and let $$\Lambda _k$$ Λ k be a finite subset of $$\Omega $$ Ω . We find necessary geometric conditions for $$\Lambda _k$$ Λ k to be interpolating for the space of multivariate polynomials of degree at most k. Our results are asymptotic in k. The density conditions obtained match precisely the necessary geometric conditions that sampling sets are known to satisfy and are expressed in terms of the equilibrium potential of the convex set. Moreover we prove that in the particular case of the unit ball, for k large enough, there are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most k.

A representation theoretic explanation of the Borcea–Brändén characterization

In 2009, Borcea and Brändén characterized all linear operators on multivariate polynomials which preserve the property of being non-vanishing (stable) on products of prescribed open circular regions. We give a representation theoretic interpretation of their findings, which generalizes and simplifies their result and leads to a conceptual unification of many related results in polynomial stability theory. At the heart of this unification is a generalized Grace’s theorem which addresses polynomials whose roots are all contained in some real interval or ray. This generalization allows us to extend the Borcea–Brändén result to characterize a certain subclass of the linear operators which preserve such polynomials.

Analyzing the Dual Space of the Saturated Ideal of a Regular Set and the Local Multiplicities of its Zeros

In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set’s zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition of multivariate polynomials, and then propose an algorithm for decomposing a regular set into a finite number of simple sets. From the output of this algorithm, the multiplicities of zeros could be directly read out, and the real solution isolation with multiplicity can also be easily produced. As a main theoretical result of this paper, we analyze the structure of dual space of the saturated ideal generated by a simple set as well as a regular set. Experiments with a preliminary implementation show the efficiency of our method.