5,402 research outputs found
Zero-Delay Rate Distortion via Filtering for Vector-Valued Gaussian Sources
We deal with zero-delay source coding of a vector-valued Gauss-Markov source
subject to a mean-squared error (MSE) fidelity criterion characterized by the
operational zero-delay vector-valued Gaussian rate distortion function (RDF).
We address this problem by considering the nonanticipative RDF (NRDF) which is
a lower bound to the causal optimal performance theoretically attainable (OPTA)
function and operational zero-delay RDF. We recall the realization that
corresponds to the optimal "test-channel" of the Gaussian NRDF, when
considering a vector Gauss-Markov source subject to a MSE distortion in the
finite time horizon. Then, we introduce sufficient conditions to show existence
of solution for this problem in the infinite time horizon. For the asymptotic
regime, we use the asymptotic characterization of the Gaussian NRDF to provide
a new equivalent realization scheme with feedback which is characterized by a
resource allocation (reverse-waterfilling) problem across the dimension of the
vector source. We leverage the new realization to derive a predictive coding
scheme via lattice quantization with subtractive dither and joint memoryless
entropy coding. This coding scheme offers an upper bound to the operational
zero-delay vector-valued Gaussian RDF. When we use scalar quantization, then
for "r" active dimensions of the vector Gauss-Markov source the gap between the
obtained lower and theoretical upper bounds is less than or equal to 0.254r + 1
bits/vector. We further show that it is possible when we use vector
quantization, and assume infinite dimensional Gauss-Markov sources to make the
previous gap to be negligible, i.e., Gaussian NRDF approximates the operational
zero-delay Gaussian RDF. We also extend our results to vector-valued Gaussian
sources of any finite memory under mild conditions. Our theoretical framework
is demonstrated with illustrative numerical experiments.Comment: 32 pages, 9 figures, published in IEEE Journal of Selected Topics in
Signal Processin
A domain-specific language and matrix-free stencil code for investigating electronic properties of Dirac and topological materials
We introduce PVSC-DTM (Parallel Vectorized Stencil Code for Dirac and
Topological Materials), a library and code generator based on a domain-specific
language tailored to implement the specific stencil-like algorithms that can
describe Dirac and topological materials such as graphene and topological
insulators in a matrix-free way. The generated hybrid-parallel (MPI+OpenMP)
code is fully vectorized using Single Instruction Multiple Data (SIMD)
extensions. It is significantly faster than matrix-based approaches on the node
level and performs in accordance with the roofline model. We demonstrate the
chip-level performance and distributed-memory scalability of basic building
blocks such as sparse matrix-(multiple-) vector multiplication on modern
multicore CPUs. As an application example, we use the PVSC-DTM scheme to (i)
explore the scattering of a Dirac wave on an array of gate-defined quantum
dots, to (ii) calculate a bunch of interior eigenvalues for strong topological
insulators, and to (iii) discuss the photoemission spectra of a disordered Weyl
semimetal.Comment: 16 pages, 2 tables, 11 figure
Fast space-variant elliptical filtering using box splines
The efficient realization of linear space-variant (non-convolution) filters
is a challenging computational problem in image processing. In this paper, we
demonstrate that it is possible to filter an image with a Gaussian-like
elliptic window of varying size, elongation and orientation using a fixed
number of computations per pixel. The associated algorithm, which is based on a
family of smooth compactly supported piecewise polynomials, the
radially-uniform box splines, is realized using pre-integration and local
finite-differences. The radially-uniform box splines are constructed through
the repeated convolution of a fixed number of box distributions, which have
been suitably scaled and distributed radially in an uniform fashion. The
attractive features of these box splines are their asymptotic behavior, their
simple covariance structure, and their quasi-separability. They converge to
Gaussians with the increase of their order, and are used to approximate
anisotropic Gaussians of varying covariance simply by controlling the scales of
the constituent box distributions. Based on the second feature, we develop a
technique for continuously controlling the size, elongation and orientation of
these Gaussian-like functions. Finally, the quasi-separable structure, along
with a certain scaling property of box distributions, is used to efficiently
realize the associated space-variant elliptical filtering, which requires O(1)
computations per pixel irrespective of the shape and size of the filter.Comment: 12 figures; IEEE Transactions on Image Processing, vol. 19, 201
Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)
Although the ``scale-free'' literature is large and growing, it gives neither
a precise definition of scale-free graphs nor rigorous proofs of many of their
claimed properties. In fact, it is easily shown that the existing theory has
many inherent contradictions and verifiably false claims. In this paper, we
propose a new, mathematically precise, and structural definition of the extent
to which a graph is scale-free, and prove a series of results that recover many
of the claimed properties while suggesting the potential for a rich and
interesting theory. With this definition, scale-free (or its opposite,
scale-rich) is closely related to other structural graph properties such as
various notions of self-similarity (or respectively, self-dissimilarity).
Scale-free graphs are also shown to be the likely outcome of random
construction processes, consistent with the heuristic definitions implicit in
existing random graph approaches. Our approach clarifies much of the confusion
surrounding the sensational qualitative claims in the scale-free literature,
and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet
Mathematics (2005
Anyonic entanglement renormalization
We introduce a family of variational ansatz states for chains of anyons which
optimally exploits the structure of the anyonic Hilbert space. This ansatz is
the natural analog of the multi-scale entanglement renormalization ansatz for
spin chains. In particular, it has the same interpretation as a coarse-graining
procedure and is expected to accurately describe critical systems with
algebraically decaying correlations. We numerically investigate the validity of
this ansatz using the anyonic golden chain and its relatives as a testbed. This
demonstrates the power of entanglement renormalization in a setting with
non-abelian exchange statistics, extending previous work on qudits, bosons and
fermions in two dimensions.Comment: 19 pages, 10 figures, v2: extended, updated to match published
versio
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