Sharp Restricted Isometry Thresholds for Global Minima of Rank-Restricted Matrix LASSO

We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $δ<δ_{\mathrm{sharp}}(k/r_{\star})$, where $δ_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $δ_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}λ$ for all $λ\gtrsim\|\mathcal{A}^{*}(ξ)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $δ<δ_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.

Towards a mathematical theory of superposition

We develop a mathematical theory of superposition in neural networks using tools from frame theory and compressed sensing. In our model, a sparse binary vector \(x\) of active features is encoded through an overcomplete dictionary \(W\), and feature recovery is performed by applying \(\operatorname{ReLU}(W^\top W x+b)\) with an appropriate bias vector \(b\). We prove several recovery theorems for this model. In the random-support setting, we establish high-probability support recovery for nearly tight, low-coherence dictionaries, with guarantees when the expected sparsity is up to order \(d/\log n\). In the worst-case support setting, we give a sharp and computable criterion for which sparsity levels permit support recovery. We apply this criterion to Gaussian random matrices and equiangular tight frames. For real equiangular tight frames with \(n>d+1\), we determine the exact recovery threshold in terms of the coherence. The proof of this result for real equiangular tight frames relies on a novel characterization---which should be of independent interest to frame theorists---of the distribution of signs in the Gram matrix.