2022/01/11 by Adrian Fan, Fan, Adrian, Jack Montemurro +11 · 1 citation
Computer Science · Mathematics · #Matrix Theory and Algorithms #Mathematical Approximation and Integration #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2201.04238
Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions A:ℝ→ Mn× n and lower ℓ2-norm bounds associated with their restriction to certain subspaces. We prove that for any such A with unit-length columns, there exists a continuous choice of subspaces t↦ U(t)⊂ ℝn such that for v∈ U(t), ‖A(t)v‖≥ c‖v‖ where c is some universal constant. Furthermore, the U(t) are chosen so that their dimension satisfies a lower bound with optimal asymptotic dependence on n and supt∈ ℝ‖A(t)‖. We provide two methods. The first relies on an orthogonality argument, while the second is probabilistic and combinatorial in nature. The latter does not yield the optimal bound for dim(U(t)) but the U(t) obtained in this way are guaranteed to have a canonical representation as joined-together spaces spanned by subsets of the unit vector basis.