A note on quantitative stability in Hilbert spaces

TL;DR

This paper investigates the quantitative stability of the inner product in Hilbert spaces, establishing exponential growth bounds and analyzing stability under nonlinear transformations.

Contribution

It provides explicit bounds on the stability of the inner product and its nonlinear variants, revealing exponential growth patterns in stability measures.

Findings

Inner product stability is exponential in 1/ε for all k≥exp(π/ε).

Stability bounds for power-type predicates depend on the exponent β, with different growth rates.

Stability under nonlinear connectives varies, with exponential bounds for β<1 and bilinear scale for β>1.

Abstract

We study stability theory in Hilbert spaces quantitatively. We prove that the inner product on the unit ball is $(k,\epsilon)$-stable for all $k\ge \exp(\pi/\epsilon)$, and it is not $(k,\epsilon)$-stable for $k\le \exp(\log 2/\epsilon)$, showing that the growth is necessarily exponential in $1/\epsilon$. We then analyze how stability scales under nonlinear connectives applied to the inner product. In particular, for power-type predicates $f(x,y)=\langle x,y\rangle_+^\beta$ with $\beta<1$ we obtain upper and lower bounds of the form $\exp(C\epsilon^{-1/\beta})$, and for $\beta>1$ and integer powers $\langle x,y\rangle^d$ we retain the bilinear scale $\exp(C/\epsilon)$.