On linear-combinatorial problems associated with subspaces spanned by $\{\pm 1\}$-vectors

TL;DR

This paper provides a complete probabilistic analysis of subspaces spanned by random $\

Contribution

It offers a precise asymptotic probability estimate for the existence of additional $\\pm 1\\

Findings

Probability that the span intersects with additional $\\pm 1\\

Main term is related to triples of vectors with specific linear combinations.

Asymptotic probability formula as dimension grows.

Abstract

A complete answer to the question about subspaces generated by $\{\pm 1\}$-vectors, which arose in the work of I.Kanter and H.Sompolinsky on associative memories, is given. More precisely, let vectors $v_1, \ldots , v_p,$ $p\leq n-1,$ be chosen at random uniformly and independently from $\{\pm 1\}^n \subset {\bf R}^n.$ Then the probability ${\mathbb P}(p, n)$ that $$span \ \langle v_1, \ldots , v_p \rangle \cap \left\{ \{\pm 1\}^n \setminus \{\pm v_1, \ldots , \pm v_p\}\right\} \ne \emptyset \ $$ is shown to be $$4{p \choose 3}\left(\frac{3}{4}\right)^n + O\left(\left(\frac{5}{8} + o_n(1)\right)^n\right) \quad \mbox{as} \quad n\to \infty,$$ where the constant implied by the $O$-notation does not depend on $p$. The main term in this estimate is the probability that some 3 vectors $v_{j_1}, v_{j_2}, v_{j_3}$ of $v_j$, $j= 1, \ldots , p,$ have a linear combination that is a $\{\pm 1\}$-vector different from $\pm v_{j_1}, \pm v_{j_2}, \pm v_{j_3}. $