Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, October 06, 2022, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Gleb Novikov

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

We consider estimation models of the form Y = X + N, where X is some m-dimensional structured signal we wish to recover, and N is symmetrically distributed noise that may be unbounded in all but a small α 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 additive noise in order to get the algorithm that works with the symmetric noise model. 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 Õ(n^p/4/α), 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. This is joint work with Tommaso D’Orsi, Rajai Nasser and David Steurer.

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