# Computational Hardness of Certifying Bounds on Constrained PCA Problems

## TL;DR

This paper investigates the computational difficulty of certifying upper bounds on quadratic forms over constrained sets in random matrices, showing hardness results linked to the GOE and spin glass models under complexity assumptions.

## Contribution

It establishes a reduction from Wishart detection to certification problems and provides evidence of computational hardness below spectral thresholds using low-degree polynomial methods.

## Key findings

- Certifying bounds is computationally hard for certain constrained sets.
- No polynomial-time algorithm can outperform the largest eigenvalue under assumptions.
- Hardness is demonstrated via indistinguishability from GOE matrices in the low-degree polynomial framework.

## Abstract

Given a random $n \times n$ symmetric matrix $\boldsymbol W$ drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form $\boldsymbol x^\top \boldsymbol W \boldsymbol x$ over all vectors $\boldsymbol x$ in a constraint set $\mathcal{S} \subset \mathbb{R}^n$. For a certain class of normalized constraint sets $\mathcal{S}$ we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of $\boldsymbol W$. A notable special case included in our results is the hypercube $\mathcal{S} = \{ \pm 1 / \sqrt{n}\}^n$, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics. Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for identifying computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is believed to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over $\boldsymbol x \in \mathcal{S}$ is much larger than that of a GOE matrix.