Higher degree sum-of-squares relaxations robust against oblivious outliers

TL;DR

This paper introduces a family of sum-of-squares algorithms that are robust against symmetric and heavy-tailed noise, extending previous results and providing near-optimal guarantees for tensor PCA and quasipolynomial guarantees for sparse PCA.

Contribution

It shows that sum-of-squares algorithms designed for Gaussian noise can be adapted to robustly handle symmetric and adversarial noise, including heavy-tailed distributions.

Findings

Algorithms recover signals in heavy-tailed noise with guarantees matching Gaussian noise cases.

Tensor PCA is solvable in polynomial time under certain SNR conditions.

Sparse PCA admits quasipolynomial time algorithms with guarantees matching the best known for Gaussian noise.

Abstract

We consider estimation models of the form $Y=X^*+N$, where $X^*$ is some $m$-dimensional signal we wish to recover, and $N$ is symmetrically distributed noise that may be unbounded in all but a small $\alpha$ fraction of the entries. We introduce a family of algorithms that under mild assumptions recover the signal $X^*$ in all estimation problems for which there exists a sum-of-squares algorithm that succeeds in recovering the signal $X^*$ when the noise $N$ is Gaussian. This essentially shows that it is enough to design a sum-of-squares algorithm for an estimation problem with Gaussian noise in order to get the algorithm that works with the symmetric noise model. Our framework extends far beyond previous results on symmetric noise models and is even robust to adversarial perturbations. As concrete examples, we investigate two problems for which no efficient algorithms were known to work for heavy-tailed noise: tensor PCA and sparse PCA. For the former, our algorithm recovers the principal component in polynomial time when the signal-to-noise ratio is at least $\tilde{O}(n^{p/4}/\alpha)$, that matches (up to logarithmic factors) current best known algorithmic guarantees for Gaussian noise. For the latter, our algorithm runs in quasipolynomial time and matches the state-of-the-art guarantees for quasipolynomial time algorithms in the case of Gaussian noise. Using a reduction from the planted clique problem, we provide evidence that the quasipolynomial time is likely to be necessary for sparse PCA with symmetric noise. In our proofs we use bounds on the covering numbers of sets of pseudo-expectations, which we obtain by certifying in sum-of-squares upper bounds on the Gaussian complexities of sets of solutions. This approach for bounding the covering numbers of sets of pseudo-expectations may be interesting in its own right and may find other application in future works.