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We construct an explicit family of real $m\times n$ restricted isometries with $m\to\infty$, $m/n\to 0$, $δ_k\to 0$, and $k=m^{2/3-o(1)}$. We combine Gershgorin's circle theorem with a flat RIP estimate to reduce the problem to choosing a large vertex set in a strongly regular graph with the property that every small subset spans few edges. We select this vertex set using an incidence geometric result due to Zahl. The proof only uses the entrywise absolute value of the Gram matrix despite the conventional wisdom that such proof techniques exhibit a barrier of $m^{1/2}$. We prove that this barrier is actually $m^{2/3}$, and we show that our particular construction exhibits the same barrier.

We prove that the set $\mathcal{N}_N$ of ReLU networks with one hidden layer and at most $N\in\mathbb N$ neurons is weakly closed in $W^{1,p}(Ω)$, for a bounded connected open set $Ω\subset\mathbb R^d$. Consequently, every coercive, weakly lower semicontinuous energy on this space attains its minimum over $\mathcal{N}_N$. This covers the Deep Ritz method with convex integrands, Sobolev training and Lipschitz learning. Moreover, as the weight decay parameter tends to zero, minimizers of the regularized problem converge, up to subsequences, to minimizers with minimal parameter norm. We also prove that the set of ReLU networks with two hidden layers of bounded widths is not closed in $W^{1,p}(Ω)$ or in $C(\overlineΩ)$. This disproves a conjecture of Petersen, Raslan and Voigtlaender.

In this note we prove that the Langevin Monte Carlo (LMC) algorithm can minimize the population risk of appropriately regularized depth-2 nets with no restrictions on the size of the net and for any choice of training data --- and we give non-asymptotic convergence rates for it. We achieve this via showing that in q-Renyi divergence, the iterates of LMC converge to the Gibbs distribution of certain Frobenius norm regularized losses for these nets --- in both classification and regression settings. The key technicality of the work is two-fold (a) to keep the amount of regularization needed be independent of the size of the net and (b) to obtain convergence rates that scale with just the Poincare constant of the corresponding Gibbs measure. This work synthesizes several recent observations about isoperimetry conditions under which LMC converges and that two-layer neural loss functions can always be regularized by a constant amount such that they satisfy the Villani conditions, and thus their Gibbs measures satisfy a Poincare inequality.