Nathan Ross

Johannes Bäumler, Miklós Z. Rácz, Nathan Ross et al.

We introduce and study a new model of correlated uniform attachment (UA) trees, where correlation is sprinkled throughout the time evolution of the process. In this model, two UA trees are grown in parallel, and at each time step a new node is added to each tree, with an edge between it and a uniformly chosen existing vertex in the respective tree. The two choices of attachment are correlated: with probability $α$, the edges attach to nodes with the same time label in both trees, and with probability $1-α$, the choices are made independently. We study fundamental detection and estimation questions for this model, given two \emph{unlabeled} trees. In our main result, we construct a consistent estimator of the correlation parameter $α$, as the size of the trees goes to infinity. The construction of our statistic relies on two key ideas. First, we use Jordan centrality to identify subsets of vertices of each tree whose intersection has a sufficient number of common early vertices. The second idea is that, across multiple time scales, it is possible to approximately determine the labels of vertices that have attached to these early vertices, using the sizes of fringe subtrees. Our analysis includes novel quantitative bounds on the fraction of early vertices that remain central, which are of independent interest in the network archaeology literature.

Krishnakumar Balasubramanian, Larry Goldstein, Nathan Ross et al.

We derive upper bounds on the Wasserstein distance ($W_1$), with respect to $\sup$-norm, between any continuous $\mathbb{R}^d$ valued random field indexed by the $n$-sphere and the Gaussian, based on Stein's method. We develop a novel Gaussian smoothing technique that allows us to transfer a bound in a smoother metric to the $W_1$ distance. The smoothing is based on covariance functions constructed using powers of Laplacian operators, designed so that the associated Gaussian process has a tractable Cameron-Martin or Reproducing Kernel Hilbert Space. This feature enables us to move beyond one dimensional interval-based index sets that were previously considered in the literature. Specializing our general result, we obtain the first bounds on the Gaussian random field approximation of wide random neural networks of any depth and Lipschitz activation functions at the random field level. Our bounds are explicitly expressed in terms of the widths of the network and moments of the random weights. We also obtain tighter bounds when the activation function has three bounded derivatives.

Krishnakumar Balasubramanian, Nathan Ross

We study the Finite-Dimensional Distributions (FDDs) of deep neural networks with randomly initialized weights that have finite-order moments. Specifically, we establish Gaussian approximation bounds in the Wasserstein-$1$ norm between the FDDs and their Gaussian limit assuming a Lipschitz activation function and allowing the layer widths to grow to infinity at arbitrary relative rates. In the special case where all widths are proportional to a common scale parameter $n$ and there are $L-1$ hidden layers, we obtain convergence rates of order $n^{-({1}/{6})^{L-1} + ε}$, for any $ε> 0$.