Sharp complexity asymptotics and topological trivialization for the (p, k) spiked tensor model

TL;DR

This paper analyzes the asymptotic behavior of deep minima in the (p,k) spiked tensor model, providing explicit formulas for ground state energy and showing finite expected minima at high signal-to-noise ratios.

Contribution

It introduces precise asymptotic formulas for deep minima and ground state energy in the (p,k) spiked tensor model, extending understanding of its energy landscape.

Findings

Explicit asymptotics for deep minima count

Formula for limiting ground state energy

Finite expected deep minima at high SNR

Abstract

We provide O(1) asymptotics for the average number of deep minima of the (p,k) spiked tensor model. We also derive an explicit formula for the limiting ground state energy on the N-dimensional sphere, similar to the work of Jagannath-Lopatto-Miolane. Moreover, when the signal to noise ratio is large enough, the expected number of deep minima is asymptotically finite as N tends to infinity and we determine its limit as the signal-to-noise ratio diverges.