Elchanan Mossel

Han Huang, Pakawut Jiradilok, Elchanan Mossel

Recent work studied the problem of finding clusters and denoising pairwise distances from noisy distances of points sampled on a manifold. We study the same problems in more general metric measure spaces under \lowerphiregularity{}. We give an algorithm that extracts large localized clusters around every sampled point and uses them to denoise distances to any fixed accuracy, with near-linear running time in the dense fixed-accuracy regime. We also show how to achieve much higher accuracy with a non-efficient algorithm. This suggests that unlike the Riemannian case, denoising to higher accuracy in more general metric spaces has a statistical-computational gap.

Neil Krishnan, Elchanan Mossel

We study the role of depth in ReLU networks with discrete (Boolean) inputs and real-valued outputs, complementing two established lines of work. For Boolean inputs, striking depth separation results were proven for $\mathsf{AC}^0$ but with threshold ($\mathsf{TC}^0$) or ReLU gates depth separation is only established for depth two vs. three. On the other hand, for {\em real-valued} functions and ReLU networks, Telgarsky's (2016) constructed a simple one variable class of functions which establishes separation at higher depths. In this paper we are interested to establish an all-depths depth separation for ReLU networks on $\{0,1\}^n$. We do so by exhibiting an explicit family of functions computable exactly by a ReLU network of depth $n+1$ and constant width, such that any ReLU network of depth $d$ and width $w$ computing the function exactly must satisfy $w^d = Ω(2^n)$; in particular, no network of depth $d = o(n/\log n)$ can compute it with width polynomial in $n$. We note that our lower bound relies on \emph{exact, infinite-accuracy} computation as an exponential precision truncation of the output is computable by a polynomial-size $\mathsf{TC}^0$ circuit.