95 research outputs found
On the second eigenvalue of random bipartite biregular graphs
We consider the spectral gap of a uniformly chosen random
-biregular bipartite graph with , where
could possibly grow with and . Let be the adjacency matrix
of . Under the assumption that and we show
that with high probability. As a corollary,
combining the results from Tikhomirov and Youssef (2019), we confirm a
conjecture in Cook (2017) that the second singular value of a uniform random
-regular digraph is for with high
probability. This also implies that the second eigenvalue of a uniform random
-regular digraph is for with high
probability. Assuming and , we further prove that for a
random -biregular bipartite graph,
for all with
high probability. The proofs of the two results are based on the size biased
coupling method introduced in Cook, Goldstein, and Johnson (2018) for random
-regular graphs and several new switching operations we defined for random
bipartite biregular graphs.Comment: 37 pages, 3 figures. Corollary 1.4 added, a few typo fixe
A non-backtracking method for long matrix and tensor completion
We consider the problem of low-rank rectangular matrix completion in the
regime where the matrix of size is ``long", i.e., the aspect
ratio diverges to infinity. Such matrices are of particular interest in
the study of tensor completion, where they arise from the unfolding of a
low-rank tensor. In the case where the sampling probability is
, we propose a new spectral algorithm for recovering the
singular values and left singular vectors of the original matrix based on a
variant of the standard non-backtracking operator of a suitably defined
bipartite weighted random graph, which we call a \textit{non-backtracking wedge
operator}. When is above a Kesten-Stigum-type sampling threshold, our
algorithm recovers a correlated version of the singular value decomposition of
with quantifiable error bounds. This is the first result in the regime of
bounded for weak recovery and the first for weak consistency when
arbitrarily slowly without any polylog factors. As an application,
for low-rank orthogonal -tensor completion, we efficiently achieve weak
recovery with sample size , and weak consistency with sample size
Overparameterized random feature regression with nearly orthogonal data
We investigate the properties of random feature ridge regression (RFRR) given
by a two-layer neural network with random Gaussian initialization. We study the
non-asymptotic behaviors of the RFRR with nearly orthogonal deterministic
unit-length input data vectors in the overparameterized regime, where the width
of the first layer is much larger than the sample size. Our analysis shows
high-probability non-asymptotic concentration results for the training errors,
cross-validations, and generalization errors of RFRR centered around their
respective values for a kernel ridge regression (KRR). This KRR is derived from
an expected kernel generated by a nonlinear random feature map. We then
approximate the performance of the KRR by a polynomial kernel matrix obtained
from the Hermite polynomial expansion of the activation function, whose degree
only depends on the orthogonality among different data points. This polynomial
kernel determines the asymptotic behavior of the RFRR and the KRR. Our results
hold for a wide variety of activation functions and input data sets that
exhibit nearly orthogonal properties. Based on these approximations, we obtain
a lower bound for the generalization error of the RFRR for a nonlinear
student-teacher model.Comment: 39 pages. A condition on the activation function is added in
Assumption 2.
Global eigenvalue fluctuations of random biregular bipartite graphs
We compute the eigenvalue fluctuations of uniformly distributed random
biregular bipartite graphs with fixed and growing degrees for a large class of
analytic functions. As a key step in the proof, we obtain a total variation
distance bound for the Poisson approximation of the number of cycles and
cyclically non-backtracking walks in random biregular bipartite graphs, which
might be of independent interest. As an application, we translate the results
to adjacency matrices of uniformly distributed random regular hypergraphs.Comment: 45 pages, 5 figure
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