42 research outputs found
Clustering in Block Markov Chains
This paper considers cluster detection in Block Markov Chains (BMCs). These
Markov chains are characterized by a block structure in their transition
matrix. More precisely, the possible states are divided into a finite
number of groups or clusters, such that states in the same cluster exhibit
the same transition rates to other states. One observes a trajectory of the
Markov chain, and the objective is to recover, from this observation only, the
(initially unknown) clusters. In this paper we devise a clustering procedure
that accurately, efficiently, and provably detects the clusters. We first
derive a fundamental information-theoretical lower bound on the detection error
rate satisfied under any clustering algorithm. This bound identifies the
parameters of the BMC, and trajectory lengths, for which it is possible to
accurately detect the clusters. We next develop two clustering algorithms that
can together accurately recover the cluster structure from the shortest
possible trajectories, whenever the parameters allow detection. These
algorithms thus reach the fundamental detectability limit, and are optimal in
that sense.Comment: 73 pages, 18 plots, second revisio
Wireless network control of interacting Rydberg atoms
We identify a relation between the dynamics of ultracold Rydberg gases in
which atoms experience a strong dipole blockade and spontaneous emission, and a
stochastic process that models certain wireless random-access networks. We then
transfer insights and techniques initially developed for these wireless
networks to the realm of Rydberg gases, and explain how the Rydberg gas can be
driven into crystal formations using our understanding of wireless networks.
Finally, we propose a method to determine Rabi frequencies (laser intensities)
such that particles in the Rydberg gas are excited with specified target
excitation probabilities, providing control over mixed-state populations.Comment: 6 pages, 7 figures; includes corrections and improvements from the
peer-review proces
Almost Sure Convergence of Dropout Algorithms for Neural Networks
We investigate the convergence and convergence rate of stochastic training
algorithms for Neural Networks (NNs) that, over the years, have spawned from
Dropout (Hinton et al., 2012). Modeling that neurons in the brain may not fire,
dropout algorithms consist in practice of multiplying the weight matrices of a
NN component-wise by independently drawn random matrices with -valued
entries during each iteration of the Feedforward-Backpropagation algorithm.
This paper presents a probability theoretical proof that for any NN topology
and differentiable polynomially bounded activation functions, if we project the
NN's weights into a compact set and use a dropout algorithm, then the weights
converge to a unique stationary set of a projected system of Ordinary
Differential Equations (ODEs). We also establish an upper bound on the rate of
convergence of Gradient Descent (GD) on the limiting ODEs of dropout algorithms
for arborescences (a class of trees) of arbitrary depth and with linear
activation functions.Comment: 20 pages, 2 figure
Matrix concentration inequalities with dependent summands and sharp leading-order terms
We establish sharp concentration inequalities for sums of dependent random
matrices. Our results concern two models. First, a model where summands are
generated by a -mixing Markov chain. Second, a model where summands are
expressed as deterministic matrices multiplied by scalar random variables. In
both models, the leading-order term is provided by free probability theory.
This leading-order term is often asymptotically sharp and, in particular, does
not suffer from the logarithmic dimensional dependence which is present in
previous results such as the matrix Khintchine inequality.
A key challenge in the proof is that techniques based on classical cumulants,
which can be used in a setting with independent summands, fail to produce
efficient estimates in the Markovian model. Our approach is instead based on
Boolean cumulants and a change-of-measure argument.
We discuss applications concerning community detection in Markov chains,
random matrices with heavy-tailed entries, and the analysis of random graphs
with dependent edges.Comment: 69 pages, 4 figure
Achievable Performance in Product-Form Networks
We characterize the achievable range of performance measures in product-form
networks where one or more system parameters can be freely set by a network
operator. Given a product-form network and a set of configurable parameters, we
identify which performance measures can be controlled and which target values
can be attained. We also discuss an online optimization algorithm, which allows
a network operator to set the system parameters so as to achieve target
performance metrics. In some cases, the algorithm can be implemented in a
distributed fashion, of which we give several examples. Finally, we give
conditions that guarantee convergence of the algorithm, under the assumption
that the target performance metrics are within the achievable range.Comment: 50th Annual Allerton Conference on Communication, Control and
Computing - 201