82 research outputs found
Chromatic roots are dense in the whole complex plane
I show that the zeros of the chromatic polynomials P_G(q) for the generalized
theta graphs \Theta^{(s,p)} are, taken together, dense in the whole complex
plane with the possible exception of the disc |q-1| < 1. The same holds for
their dichromatic polynomials (alias Tutte polynomials, alias Potts-model
partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate
corollary is that the chromatic zeros of not-necessarily-planar graphs are
dense in the whole complex plane. The main technical tool in the proof of these
results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for
certain sequences of analytic functions, for which I give a new and simpler
proof.Comment: LaTeX2e, 53 pages. Version 2 includes a new Appendix B. Version 3
adds a new Theorem 1.4 and a new Section 5, and makes several small
improvements. To appear in Combinatorics, Probability & Computin
Exponential families on resource-constrained systems
This work is about the estimation of exponential family models on resource-constrained
systems. Our main goal is learning probabilistic models on devices with highly restricted
storage, arithmetic, and computational capabilitiesâso called, ultra-low-power
devices. Enhancing the learning capabilities of such devices opens up opportunities for
intelligent ubiquitous systems in all areas of life, from medicine, over robotics, to home
automationâto mention just a few. We investigate the inherent resource consumption of
exponential families, review existing techniques, and devise new methods to reduce the
resource consumption. The resource consumption, however, must not be reduced at all
cost. Exponential families possess several desirable properties that must be preserved:
Any probabilistic model encodes a conditional independence structureâour methods
keep this structure intact. Exponential family models are theoretically well-founded.
Instead of merely finding new algorithms based on intuition, our models are formalized
within the framework of exponential families and derived from first principles. We do
not introduce new assumptions which are incompatible with the formal derivation of the
base model, and our methods do not rely on properties of particular high-level applications.
To reduce the memory consumption, we combine and adapt reparametrization
and regularization in an innovative way that facilitates the sparse parametrization of
high-dimensional non-stationary time-series. The procedure allows us to load models in
memory constrained systems, which would otherwise not fit. We provide new theoretical
insights and prove that the uniform distance between the data generating process
and our reparametrized solution is bounded. To reduce the arithmetic complexity of
the learning problem, we derive the integer exponential family, based on the very definition
of sufficient statistics and maximum entropy estimation. New integer-valued
inference and learning algorithms are proposed, based on variational inference, proximal
optimization, and regularization. The benefit of this technique is larger, the weaker
the underlying system is, e.g., the probabilistic inference on a state-of-the-art ultra-lowpower
microcontroller can be accelerated by a factor of 250. While our integer inference
is fast, the underlying message passing relies on the variational principle, which is inexact
and has unbounded error on general graphs. Since exact inference and other existing
methods with bounded error exhibit exponential computational complexity, we employ
near minimax optimal polynomial approximations to yield new stochastic algorithms
for approximating the partition function and the marginal probabilities. Changing the
polynomial degree allows us to control the complexity and the error of our new stochastic
method. We provide an error bound that is parametrized by the number of samples, the
polynomial degree, and the norm of the modelâs parameter vector. Moreover, important
intermediate quantities can be precomputed and shared with the weak computational device
to reduce the resource requirement of our method even further. All new techniques
are empirically evaluated on synthetic and real-world data, and the results confirm the
properties which are predicted by our theoretical derivation. Our novel techniques allow
a broader range of models to be learned on resource-constrained systems and imply
several new research possibilities
Feynman integrals and hyperlogarithms
We study Feynman integrals in the representation with Schwinger parameters
and derive recursive integral formulas for massless 3- and 4-point functions.
Properties of analytic (including dimensional) regularization are summarized
and we prove that in the Euclidean region, each Feynman integral can be written
as a linear combination of convergent Feynman integrals. This means that one
can choose a basis of convergent master integrals and need not evaluate any
divergent Feynman graph directly.
Secondly we give a self-contained account of hyperlogarithms and explain in
detail the algorithms needed for their application to the evaluation of
multivariate integrals. We define a new method to track singularities of such
integrals and present a computer program that implements the integration
method.
As our main result, we prove the existence of infinite families of massless
3- and 4-point graphs (including the ladder box graphs with arbitrary loop
number and their minors) whose Feynman integrals can be expressed in terms of
multiple polylogarithms, to all orders in the epsilon-expansion. These
integrals can be computed effectively with the presented program.
We include interesting examples of explicit results for Feynman integrals
with up to 6 loops. In particular we present the first exactly computed
counterterm in massless phi^4 theory which is not a multiple zeta value, but a
linear combination of multiple polylogarithms at primitive sixth roots of unity
(and divided by ). To this end we derive a parity result on the
reducibility of the real- and imaginary parts of such numbers into products and
terms of lower depth.Comment: PhD thesis, 220 pages, many figure
Sublinear Computation Paradigm
This open access book gives an overview of cutting-edge work on a new paradigm called the âsublinear computation paradigm,â which was proposed in the large multiyear academic research project âFoundations of Innovative Algorithms for Big Data.â That project ran from October 2014 to March 2020, in Japan. To handle the unprecedented explosion of big data sets in research, industry, and other areas of society, there is an urgent need to develop novel methods and approaches for big data analysis. To meet this need, innovative changes in algorithm theory for big data are being pursued. For example, polynomial-time algorithms have thus far been regarded as âfast,â but if a quadratic-time algorithm is applied to a petabyte-scale or larger big data set, problems are encountered in terms of computational resources or running time. To deal with this critical computational and algorithmic bottleneck, linear, sublinear, and constant time algorithms are required. The sublinear computation paradigm is proposed here in order to support innovation in the big data era. A foundation of innovative algorithms has been created by developing computational procedures, data structures, and modelling techniques for big data. The project is organized into three teams that focus on sublinear algorithms, sublinear data structures, and sublinear modelling. The work has provided high-level academic research results of strong computational and algorithmic interest, which are presented in this book. The book consists of five parts: Part I, which consists of a single chapter on the concept of the sublinear computation paradigm; Parts II, III, and IV review results on sublinear algorithms, sublinear data structures, and sublinear modelling, respectively; Part V presents application results. The information presented here will inspire the researchers who work in the field of modern algorithms
Approximate sampling and counting for spin models in graphs
En aquest treball abordem els problemes de mostreig i comptatge aproximat en models d'espins en grafs, recopilant els resultats mĂŠs significatius de l'Ă rea i introduĂŻnt els coneixements previs necessaris del camp de la fĂsica estadĂstica. En particular, prestem especial atenciĂł als mètodes generals de disseny d'algorismes desenvolupats per Weitz i Barvinok, aixĂ com els avenços recents en matèria de comptatge i mostreig de conjunts independents de mida donada. AixĂ mateix, discutim com es podrien adaptar aquests arguments als problemes de comptatge i mostreig de coloracions amb les mides de cada color fixades, explicant amb detall la lĂnia de recerca actual que estem duent a terme.En este trabajo abordamos los problemas de muestreo y conteo aproximado en modelos de espines en grafos, recopilando los resultados mĂĄs significativos del campo e introduciendo el conocimiento previo necesario del ĂĄrea de la fĂsica estadĂstica. En particular, prestamos especial atenciĂłn a los mĂŠtodos generales de diseĂąo de algorismos desarrollados por Weitz y Barvinok, asĂ como a los avances recientes en cuanto al conteo y muestreo de conjuntos independientes de un tamaĂąo dado. AsĂ mismo, discutimos cĂłmo se podrĂan adaptar estos argumentos al problema de contar y muestrear coloraciones con el tamaĂąo de cada color fijo, explicando en detalle la lĂnea de investigaciĂłn que estamos llevando a cabo actualmente.We approach the problems of approximate sampling and counting in spin models on graphs, surveying the most significant results in the area and introducing the necessary background from statistical physics. We pay particular attention to the general algorithm design frameworks developed by Weitz and Barvinok, as well as to the newer results on counting and sampling independent sets of given size. In addition, we discuss the adaptation of the arguments behind these results to count and sample colorings with fixed color sizes, explaining in detail the current research line we are undertaking.Outgoin
Quantum Approaches to Data Science and Data Analytics
In this thesis are explored different research directions related to both the use of classical data analysis techniques for the study of quantum systems and the employment of quantum computing to speed up hard Machine Learning task
Quantum Apices: Identifying Limits of Entanglement, Nonlocality, & Contextuality
This work develops analytic methods to quantitatively demarcate quantum
reality from its subset of classical phenomenon, as well as from the superset
of general probabilistic theories. Regarding quantum nonlocality, we discuss
how to determine the quantum limit of Bell-type linear inequalities. In
contrast to semidefinite programming approaches, our method allows for the
consideration of inequalities with abstract weights, by means of leveraging the
Hermiticity of quantum states. Recognizing that classical correlations
correspond to measurements made on separable states, we also introduce a
practical method for obtaining sufficient separability criteria. We
specifically vet the candidacy of driven and undriven superradiance as schema
for entanglement generation. We conclude by reviewing current approaches to
quantum contextuality, emphasizing the operational distinction between nonlocal
and contextual quantum statistics. We utilize our abstractly-weighted linear
quantum bounds to explicitly demonstrate a set of conditional probability
distributions which are simultaneously compatible with quantum contextuality
while being incompatible with quantum nonlocality. It is noted that this novel
statistical regime implies an experimentally-testable target for the Consistent
Histories theory of quantum gravity.Comment: Doctoral Thesis for the University of Connecticu
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