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Tensor cumulants for statistical inference on invariant distributions
This basis lets us unify and strengthen previous results on low-degree hardness, giving a combinatorial explanation of the hardness transition and of a continuum of subexponential-time algorithms that work below it, and proving tight lower bounds against low-degree polynomials for recovering rather than just detecting the signal.
Low coordinate degree algorithms I: Universality of computational thresholds for hypothesis testing
These results are the first computational lower bounds against any large class of algorithms for all of these models when the channel is not one of a few special cases, and thereby give the first substantial evidence for the universality of several statistical-to-computational gaps.
The Average-Case Time Complexity of Certifying the Restricted Isometry Property
A matrix has the $(s,\delta)$-$\mathsf{RIP}$ property if behaves as a $\delta$-approximate isometry on $s$-sparse vectors.
Notes on Computational Hardness of Hypothesis Testing: Predictions using the Low-Degree Likelihood Ratio
These notes survey and explore an emerging method, which we call the low-degree method, for predicting and understanding statistical-versus-computational tradeoffs in high-dimensional inference problems.
Subexponential-Time Algorithms for Sparse PCA
Prior work has shown that when the signal-to-noise ratio ($\lambda$ or $\beta\sqrt{N/n}$, respectively) is a small constant and the fraction of nonzero entries in the planted vector is $\|x\|_0 / n = \rho$, it is possible to recover $x$ in polynomial time if $\rho \lesssim 1/\sqrt{n}$.
