11,684 research outputs found
Fast and exact search for the partition with minimal information loss
In analysis of multi-component complex systems, such as neural systems,
identifying groups of units that share similar functionality will aid
understanding of the underlying structures of the system. To find such a
grouping, it is useful to evaluate to what extent the units of the system are
separable. Separability or inseparability can be evaluated by quantifying how
much information would be lost if the system were partitioned into subsystems,
and the interactions between the subsystems were hypothetically removed. A
system of two independent subsystems are completely separable without any loss
of information while a system of strongly interacted subsystems cannot be
separated without a large loss of information. Among all the possible
partitions of a system, the partition that minimizes the loss of information,
called the Minimum Information Partition (MIP), can be considered as the
optimal partition for characterizing the underlying structures of the system.
Although the MIP would reveal novel characteristics of the neural system, an
exhaustive search for the MIP is numerically intractable due to the
combinatorial explosion of possible partitions. Here, we propose a
computationally efficient search to precisely identify the MIP among all
possible partitions by exploiting the submodularity of the measure of
information loss. Mutual information is one such submodular information loss
functions, and is a natural choice for measuring the degree of statistical
dependence between paired sets of random variables. By using mutual information
as a loss function, we show that the search for MIP can be performed in a
practical order of computational time for a reasonably large system. We also
demonstrate that MIP search allows for the detection of underlying global
structures in a network of nonlinear oscillators
Community detection and stochastic block models: recent developments
The stochastic block model (SBM) is a random graph model with planted
clusters. It is widely employed as a canonical model to study clustering and
community detection, and provides generally a fertile ground to study the
statistical and computational tradeoffs that arise in network and data
sciences.
This note surveys the recent developments that establish the fundamental
limits for community detection in the SBM, both with respect to
information-theoretic and computational thresholds, and for various recovery
requirements such as exact, partial and weak recovery (a.k.a., detection). The
main results discussed are the phase transitions for exact recovery at the
Chernoff-Hellinger threshold, the phase transition for weak recovery at the
Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial
recovery, the learning of the SBM parameters and the gap between
information-theoretic and computational thresholds.
The note also covers some of the algorithms developed in the quest of
achieving the limits, in particular two-round algorithms via graph-splitting,
semi-definite programming, linearized belief propagation, classical and
nonbacktracking spectral methods. A few open problems are also discussed
Uncollapsing the wavefunction by undoing quantum measurements
We review and expand on recent advances in theory and experiments concerning
the problem of wavefunction uncollapse: Given an unknown state that has been
disturbed by a generalized measurement, restore the state to its initial
configuration. We describe how this is probabilistically possible with a
subsequent measurement that involves erasing the information extracted about
the state in the first measurement. The general theory of abstract measurements
is discussed, focusing on quantum information aspects of the problem, in
addition to investigating a variety of specific physical situations and
explicit measurement strategies. Several systems are considered in detail: the
quantum double dot charge qubit measured by a quantum point contact (with and
without Hamiltonian dynamics), the superconducting phase qubit monitored by a
SQUID detector, and an arbitrary number of entangled charge qubits.
Furthermore, uncollapse strategies for the quantum dot electron spin qubit, and
the optical polarization qubit are also reviewed. For each of these systems the
physics of the continuous measurement process, the strategy required to ideally
uncollapse the wavefunction, as well as the statistical features associated
with the measurement is discussed. We also summarize the recent experimental
realization of two of these systems, the phase qubit and the polarization
qubit.Comment: 19 pages, 4 figure
Long time dynamics and coherent states in nonlinear wave equations
We discuss recent progress in finding all coherent states supported by
nonlinear wave equations, their stability and the long time behavior of nearby
solutions.Comment: bases on the authors presentation at 2015 AMMCS-CAIMS Congress, to
appear in Fields Institute Communications: Advances in Applied Mathematics,
Modeling, and Computational Science 201
Rectification effects in coherent transport through single molecules
A minimal model for coherent transport through a donor/acceptor molecular
junction is presented. The two donor and acceptor sites are described by single
levels energetically separated by an intramolecular tunnel barrier. In the
limit of strong coupling to the electrodes a current rectification for
different bias voltage polarities occurs. Contacts with recent experiments of
molecular rectification are also given.Comment: 10 pages, 4 figure
Universal Amplitude Ratios in the Ising Model in Three Dimensions
We use a high-precision Monte Carlo simulation to determine the universal
specific-heat amplitude ratio A+/A- in the three-dimensional Ising model via
the impact angle \phi of complex temperature zeros. We also measure the
correlation-length critical exponent \nu from finite-size scaling, and the
specific-heat exponent \alpha through hyperscaling. Extrapolations to the
thermodynamic limit yield \phi = 59.2(1.0) degrees, A+/A- = 0.56(3), \nu =
0.63048(32) and \alpha = 0.1086(10). These results are compatible with some
previous estimates from a variety of sources and rule out recently conjectured
exact values.Comment: 17 pages, 5 figure
Excited electronic states from a variational approach based on symmetry-projected Hartree--Fock configurations
Recent work from our research group has demonstrated that symmetry-projected
Hartree--Fock (HF) methods provide a compact representation of molecular ground
state wavefunctions based on a superposition of non-orthogonal Slater
determinants. The symmetry-projected ansatz can account for static correlations
in a computationally efficient way. Here we present a variational extension of
this methodology applicable to excited states of the same symmetry as the
ground state. Benchmark calculations on the C dimer with a modest basis
set, which allows comparison with full configuration interaction results,
indicate that this extension provides a high quality description of the
low-lying spectrum for the entire dissociation profile. We apply the same
methodology to obtain the full low-lying vertical excitation spectrum of
formaldehyde, in good agreement with available theoretical and experimental
data, as well as to a challenging model insertion pathway for BeH.
The variational excited state methodology developed in this work has two
remarkable traits: it is fully black-box and will be applicable to fairly large
systems thanks to its mean-field computational cost
Recent Advances in Signal Processing
The signal processing task is a very critical issue in the majority of new technological inventions and challenges in a variety of applications in both science and engineering fields. Classical signal processing techniques have largely worked with mathematical models that are linear, local, stationary, and Gaussian. They have always favored closed-form tractability over real-world accuracy. These constraints were imposed by the lack of powerful computing tools. During the last few decades, signal processing theories, developments, and applications have matured rapidly and now include tools from many areas of mathematics, computer science, physics, and engineering. This book is targeted primarily toward both students and researchers who want to be exposed to a wide variety of signal processing techniques and algorithms. It includes 27 chapters that can be categorized into five different areas depending on the application at hand. These five categories are ordered to address image processing, speech processing, communication systems, time-series analysis, and educational packages respectively. The book has the advantage of providing a collection of applications that are completely independent and self-contained; thus, the interested reader can choose any chapter and skip to another without losing continuity
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