About subspaces the most deviating from the coordinate ones

TL;DR

The paper constructs specific subspaces in b^n that deviate significantly from coordinate subspaces, supporting a hypothesis about the maximal deviation value using graph-based constructions.

Contribution

It introduces a novel construction of subspaces based on series-parallel graphs that achieve the hypothesized maximal deviation from coordinate subspaces.

Findings

Constructed subspaces deviate by at least cos(1/ n) from all coordinate subspaces.

The construction is based on scaled star spaces of 2-connected series-parallel graphs.

For fixed graphs, the weighting used is uniquely optimal for the deviation property.

Abstract

Using the largest principal angle as a distance between same-dimensional linear subspaces of $\mathbb{R}^n$, we construct $k$-dimensional subspaces which deviate from every coordinate $k$-subspace by at least $\arccos(1/\sqrt n)$. The construction is motivated by the hypothesis of Goreinov, Tyrtyshnikov and Zamarashkin that this value is the largest possible one for all $n > k > 0$. The subspaces are scaled star spaces of $2$-connected series-parallel graphs with $k+1$ vertices and $n$ edges, equipped with a particular positive edge weighting, while the largest principal angles take two values -- $\arccos(1 / \sqrt{n})$ and $\pi/2$, depending on whether a $k$-edge subgraph corresponding to a coordinate $k$-subspace is a spanning tree or not. For a fixed series-parallel graph, we also prove that the constructed weighting is the unique positive one, up to scaling, for which the corresponding $k$-subspace deviates from all coordinate $k$-subspaces by at least $\arccos(1 / \sqrt{n})$.