19,768 research outputs found
Divide-and-Conquer Method for Instanton Rate Theory
Ring-polymer instanton theory has been developed to simulate the quantum
dynamics of molecular systems at low temperatures. Chemical reaction rates can
be obtained by locating the dominant tunneling pathway and analyzing
fluctuations around it. In the standard method, calculating the fluctuation
terms involves the diagonalization of a large matrix, which can be unfeasible
for large systems with a high number of ring-polymer beads. Here we present a
method for computing the instanton fluctuations with a large reduction in
computational scaling. This method is applied to three reactions described by
fitted, analytic and on-the-fly ab initio potential-energy surfaces and is
shown to be numerically stable for the calculation of thermal reaction rates
even at very low temperature
Stable Recovery Of Sparse Vectors From Random Sinusoidal Feature Maps
Random sinusoidal features are a popular approach for speeding up
kernel-based inference in large datasets. Prior to the inference stage, the
approach suggests performing dimensionality reduction by first multiplying each
data vector by a random Gaussian matrix, and then computing an element-wise
sinusoid. Theoretical analysis shows that collecting a sufficient number of
such features can be reliably used for subsequent inference in kernel
classification and regression.
In this work, we demonstrate that with a mild increase in the dimension of
the embedding, it is also possible to reconstruct the data vector from such
random sinusoidal features, provided that the underlying data is sparse enough.
In particular, we propose a numerically stable algorithm for reconstructing the
data vector given the nonlinear features, and analyze its sample complexity.
Our algorithm can be extended to other types of structured inverse problems,
such as demixing a pair of sparse (but incoherent) vectors. We support the
efficacy of our approach via numerical experiments
Detectability thresholds and optimal algorithms for community structure in dynamic networks
We study the fundamental limits on learning latent community structure in
dynamic networks. Specifically, we study dynamic stochastic block models where
nodes change their community membership over time, but where edges are
generated independently at each time step. In this setting (which is a special
case of several existing models), we are able to derive the detectability
threshold exactly, as a function of the rate of change and the strength of the
communities. Below this threshold, we claim that no algorithm can identify the
communities better than chance. We then give two algorithms that are optimal in
the sense that they succeed all the way down to this limit. The first uses
belief propagation (BP), which gives asymptotically optimal accuracy, and the
second is a fast spectral clustering algorithm, based on linearizing the BP
equations. We verify our analytic and algorithmic results via numerical
simulation, and close with a brief discussion of extensions and open questions.Comment: 9 pages, 3 figure
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