14,058 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
Chord Diagrams and Gauss Codes for Graphs
Chord diagrams on circles and their intersection graphs (also known as circle
graphs) have been intensively studied, and have many applications to the study
of knots and knot invariants, among others. However, chord diagrams on more
general graphs have not been studied, and are potentially equally valuable in
the study of spatial graphs. We will define chord diagrams for planar
embeddings of planar graphs and their intersection graphs, and prove some basic
results. Then, as an application, we will introduce Gauss codes for immersions
of graphs in the plane and give algorithms to determine whether a particular
crossing sequence is realizable as the Gauss code of an immersed graph.Comment: 20 pages, many figures. This version has been substantially
rewritten, and the results are stronge
Gauss paragraphs of classical links and a characterization of virtual link groups
A classical link in 3-space can be represented by a Gauss paragraph encoding
a link diagram in a combinatorial way. A Gauss paragraph may code not a
classical link diagram, but a diagram with virtual crossings. We present a
criterion and a linear algorithm detecting whether a Gauss paragraph encodes a
classical link. We describe Wirtinger presentations realizable by virtual link
groups.Comment: 12 pages, 12 figures, v2: new results have been adde
An Introduction to Virtual Spatial Graph Theory
Two natural generalizations of knot theory are the study of spatial graphs
and virtual knots. Our goal is to unify these two approaches into the study of
virtual spatial graphs. This paper is a survey, and does not contain any new
results. We state the definitions, provide some examples, and survey the known
results. We hope that this paper will help lead to rapid development of the
area.Comment: 9 pages, 7 figures, presented at the International Workshop on Knot
Theory for Scientific Objects at Osaka City University, March 200
- …