Sparse Approximation in Lattices and Semigroups

TL;DR

This work analyzes how well a fixed integer matrix A can sparsely approximate image vectors b ∈ A·X by vectors Ay with at most k nonzeros in x, focusing on X being the lattice Z^n or the semigroup Z^n_≥0. It develops exponential-in-k bounds for the best approximation, first in the lattice case via δ(A) and a lattice-covering lemma, then in semigroups under simplicial-cone and parallelepiped-norm settings with a parameter μ capturing column sizes. Notably, it provides tight results for m=1 (including a sharp k=2 bound via Sylvester’s sequence) and general bounds for m≥2 using a norm ||·||_{P(B)} and det(B), along with lower-bound constructions that demonstrate limits of these techniques. The results connect to Carathéodory-type representations and sparse-recovery literature, offering both exact-sparsity thresholds and quantitative approximation guarantees, with implications for interpretable sparse integral solutions in integer programs.

Abstract

This paper deals with the following question: Suppose that there exist an integer or a non-negative integer solution $x$ to a system $Ax = b$, where the number of non-zero components of $x$ is $n$. The target is, for a given natural number $k < n$, to approximate $b$ with $Ay$ where $y$ is an integer or non-negative integer solution with at most $k$ non-zero components. We establish upper bounds for this question in general. In specific cases, these bounds are tight. If we view the approximation quality as a function of the parameter $k$, then the paper explains why the quality of the approximation increases exponentially as $k$ goes to $n$. This paper is a complete version of an extended abstract that appeared at the 26th International Conference on Integer Programming and Combinatorial Optimization (IPCO).

Theorems & Definitions (32)

• Theorem 1
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thm_lattice_main']}
• Example 1
• Example 2
• Corollary 1
• proof
• Theorem 2
• Theorem 3