357,120 research outputs found
Information geometric methods for complexity
Research on the use of information geometry (IG) in modern physics has
witnessed significant advances recently. In this review article, we report on
the utilization of IG methods to define measures of complexity in both
classical and, whenever available, quantum physical settings. A paradigmatic
example of a dramatic change in complexity is given by phase transitions (PTs).
Hence we review both global and local aspects of PTs described in terms of the
scalar curvature of the parameter manifold and the components of the metric
tensor, respectively. We also report on the behavior of geodesic paths on the
parameter manifold used to gain insight into the dynamics of PTs. Going
further, we survey measures of complexity arising in the geometric framework.
In particular, we quantify complexity of networks in terms of the Riemannian
volume of the parameter space of a statistical manifold associated with a given
network. We are also concerned with complexity measures that account for the
interactions of a given number of parts of a system that cannot be described in
terms of a smaller number of parts of the system. Finally, we investigate
complexity measures of entropic motion on curved statistical manifolds that
arise from a probabilistic description of physical systems in the presence of
limited information. The Kullback-Leibler divergence, the distance to an
exponential family and volumes of curved parameter manifolds, are examples of
essential IG notions exploited in our discussion of complexity. We conclude by
discussing strengths, limits, and possible future applications of IG methods to
the physics of complexity.Comment: review article, 60 pages, no figure
Curve Diagrams, Laminations, and the Geometric Complexity of Braids
Braids can be represented geometrically as curve diagrams. The geometric
complexity of a braid is the minimal complexity of a curve diagram representing
it. We introduce and study the corresponding notion of geometric generating
function. We compute explicitly the geometric generating function for the group
of braids on three strands and prove that it is neither rational nor algebraic,
nor even holonomic. This result may appear as counterintuitive. Indeed, the
standard complexity (due to the Artin presentation of braid groups) is
algorithmically harder to compute than the geometric complexity, yet the
associated generating function for the group of braids on three strands is
rational.Comment: 33 pages, 19 figure
No occurrence obstructions in geometric complexity theory
The permanent versus determinant conjecture is a major problem in complexity
theory that is equivalent to the separation of the complexity classes VP_{ws}
and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a
strengthened version of this conjecture over the complex numbers that amounts
to separating the orbit closures of the determinant and padded permanent
polynomials. In that paper it was also proposed to separate these orbit
closures by exhibiting occurrence obstructions, which are irreducible
representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit
closure, but not in the other. We prove that this approach is impossible.
However, we do not rule out the general approach to the permanent versus
determinant problem via multiplicity obstructions as proposed by Mulmuley and
Sohoni.Comment: Substantial revision. This version contains an overview of the proof
of the main result. Added material on the model of power sums. Theorem 4.14
in the old version, which had a complicated proof, became the easy Theorem
5.4. To appear in the Journal of the AM
Riemannian-geometric entropy for measuring network complexity
A central issue of the science of complex systems is the quantitative
characterization of complexity. In the present work we address this issue by
resorting to information geometry. Actually we propose a constructive way to
associate to a - in principle any - network a differentiable object (a
Riemannian manifold) whose volume is used to define an entropy. The
effectiveness of the latter to measure networks complexity is successfully
proved through its capability of detecting a classical phase transition
occurring in both random graphs and scale--free networks, as well as of
characterizing small Exponential random graphs, Configuration Models and real
networks.Comment: 15 pages, 3 figure
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
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