7,864 research outputs found
A Homological Approach to Belief Propagation and Bethe Approximations
We introduce a differential complex of local observables given a
decomposition of a global set of random variables into subsets. Its boundary
operator allows us to define a transport equation equivalent to Belief
Propagation. This definition reveals a set of conserved quantities under Belief
Propagation and gives new insight on the relationship of its equilibria with
the critical points of Bethe free energy.Comment: 14 pages, submitted for the 2019 Geometric Science of Information
colloquiu
Weighted projective spaces and iterated Thom spaces
For any (n+1)-dimensional weight vector {\chi} of positive integers, the
weighted projective space P(\chi) is a projective toric variety, and has
orbifold singularities in every case other than CP^n. We study the algebraic
topology of P(\chi), paying particular attention to its localisation at
individual primes p. We identify certain p-primary weight vectors {\pi} for
which P(\pi) is homeomorphic to an iterated Thom space over S^2, and discuss
how any P(\chi) may be reconstructed from its p-primary factors. We express
Kawasaki's computations of the integral cohomology ring H^*(P(\chi);Z) in terms
of iterated Thom isomorphisms, and recover Al Amrani's extension to complex
K-theory. Our methods generalise to arbitrary complex oriented cohomology
algebras E^*(P(\chi)) and their dual homology coalgebras E_*(P(\chi)), as we
demonstrate for complex cobordism theory (the universal example). In
particular, we describe a fundamental class in \Omega^U_{2n}(P(\chi)), which
may be interpreted as a resolution of singularities.Comment: 26 page
The Hatsopoulos-Gyftopoulos resolution of the Schroedinger-Park paradox about the concept of "state" in quantum statistical mechanics
A seldom recognized fundamental difficulty undermines the concept of
individual ``state'' in the present formulations of quantum statistical
mechanics (and in its quantum information theory interpretation as well). The
difficulty is an unavoidable consequence of an almost forgotten corollary
proved by E. Schroedinger in 1936 and perused by J.L. Park, Am. J. Phys., Vol.
36, 211 (1968). To resolve it, we must either reject as unsound the concept of
state, or else undertake a serious reformulation of quantum theory and the role
of statistics. We restate the difficulty and discuss a possible resolution
proposed in 1976 by G.N. Hatsopoulos and E.P. Gyftopoulos, Found. Phys., Vol.
6, 15, 127, 439, 561 (1976).Comment: RevTeX4, 7 pages, corrected a paragraph and added an example at page
3, to appear in Mod. Phys. Lett.
Why Does a Kronecker Model Result in Misleading Capacity Estimates?
Many recent works that study the performance of multi-input multi-output
(MIMO) systems in practice assume a Kronecker model where the variances of the
channel entries, upon decomposition on to the transmit and the receive
eigen-bases, admit a separable form. Measurement campaigns, however, show that
the Kronecker model results in poor estimates for capacity. Motivated by these
observations, a channel model that does not impose a separable structure has
been recently proposed and shown to fit the capacity of measured channels
better. In this work, we show that this recently proposed modeling framework
can be viewed as a natural consequence of channel decomposition on to its
canonical coordinates, the transmit and/or the receive eigen-bases. Using tools
from random matrix theory, we then establish the theoretical basis behind the
Kronecker mismatch at the low- and the high-SNR extremes: 1) Sparsity of the
dominant statistical degrees of freedom (DoF) in the true channel at the
low-SNR extreme, and 2) Non-regularity of the sparsity structure (disparities
in the distribution of the DoF across the rows and the columns) at the high-SNR
extreme.Comment: 39 pages, 5 figures, under review with IEEE Trans. Inform. Theor
Mapping constrained optimization problems to quantum annealing with application to fault diagnosis
Current quantum annealing (QA) hardware suffers from practical limitations
such as finite temperature, sparse connectivity, small qubit numbers, and
control error. We propose new algorithms for mapping boolean constraint
satisfaction problems (CSPs) onto QA hardware mitigating these limitations. In
particular we develop a new embedding algorithm for mapping a CSP onto a
hardware Ising model with a fixed sparse set of interactions, and propose two
new decomposition algorithms for solving problems too large to map directly
into hardware.
The mapping technique is locally-structured, as hardware compatible Ising
models are generated for each problem constraint, and variables appearing in
different constraints are chained together using ferromagnetic couplings. In
contrast, global embedding techniques generate a hardware independent Ising
model for all the constraints, and then use a minor-embedding algorithm to
generate a hardware compatible Ising model. We give an example of a class of
CSPs for which the scaling performance of D-Wave's QA hardware using the local
mapping technique is significantly better than global embedding.
We validate the approach by applying D-Wave's hardware to circuit-based
fault-diagnosis. For circuits that embed directly, we find that the hardware is
typically able to find all solutions from a min-fault diagnosis set of size N
using 1000N samples, using an annealing rate that is 25 times faster than a
leading SAT-based sampling method. Further, we apply decomposition algorithms
to find min-cardinality faults for circuits that are up to 5 times larger than
can be solved directly on current hardware.Comment: 22 pages, 4 figure
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