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Sparse Linear Regression and Lattice Problems
Furthermore, for well-conditioned (essentially) isotropic Gaussian design matrices, where Lasso is known to behave well in the identifiable regime, we show hardness of outputting any good solution in the unidentifiable regime where there are many solutions, assuming the worst-case hardness of standard and well-studied lattice problems.
Characterizing the Implicit Bias of Regularized SGD in Rank Minimization
We study the bias of Stochastic Gradient Descent (SGD) to learn low-rank weight matrices when training deep neural networks.
Continuous LWE is as Hard as LWE & Applications to Learning Gaussian Mixtures
Under the (conservative) polynomial hardness of LWE, we show hardness of density estimation for $n^{\epsilon}$ Gaussians for any constant $\epsilon > 0$, which improves on Bruna, Regev, Song and Tang (STOC 2021), who show hardness for at least $\sqrt{n}$ Gaussians under polynomial (quantum) hardness assumptions.
The Fine-Grained Hardness of Sparse Linear Regression
Sparse linear regression is the well-studied inference problem where one is given a design matrix $\mathbf{A} \in \mathbb{R}^{M\times N}$ and a response vector $\mathbf{b} \in \mathbb{R}^M$, and the goal is to find a solution $\mathbf{x} \in \mathbb{R}^{N}$ which is $k$-sparse (that is, it has at most $k$ non-zero coordinates) and minimizes the prediction error $\|\mathbf{A} \mathbf{x} - \mathbf{b}\|_2$.
