Generalized Holographic Principle, Gauge Invariance and the Emergence of Gravity a la Wilczek

We show that a generalized version of the holographic principle can be derived from the Hamiltonian description of information flow within a quantum system that maintains a separable state. We then show that this generalized holographic principle entails a general principle of gauge invariance. When this is realized in an ambient Lorentzian space-time, gauge invariance under the Poincare group is immediately achieved. We apply this pathway to retrieve the action of gravity. The latter is cast a la Wilczek through a similar formulation derived by MacDowell and Mansouri, which involves the representation theory of the Lie groups SO(3,2) and SO(4,1).Comment: 26 pages, 1 figur

Distributed Robust Set-Invariance for Interconnected Linear Systems

We introduce a class of distributed control policies for networks of discrete-time linear systems with polytopic additive disturbances. The objective is to restrict the network-level state and controls to user-specified polyhedral sets for all times. This problem arises in many safety-critical applications. We consider two problems. First, given a communication graph characterizing the structure of the information flow in the network, we find the optimal distributed control policy by solving a single linear program. Second, we find the sparsest communication graph required for the existence of a distributed invariance-inducing control policy. Illustrative examples, including one on platooning, are presented.Comment: 8 Pages. Submitted to American Control Conference (ACC), 201

Multiple solutions for semilinear Robin problems with superlinear reaction and no symmetries

We study a semilinear Robin problem driven by the Laplacian with a parametric superlinear reaction. Using variational tools from the critical point theory with truncation and comparison techniques, critical groups and flow invariance arguments, we show the existence of seven nontrivial smooth solutions, all with sign information and ordered

Information-Geometric Optimization Algorithms: A Unifying Picture via Invariance Principles

We present a canonical way to turn any smooth parametric family of probability distributions on an arbitrary search space $X$ into a continuous-time black-box optimization method on $X$, the \emph{information-geometric optimization} (IGO) method. Invariance as a design principle minimizes the number of arbitrary choices. The resulting \emph{IGO flow} conducts the natural gradient ascent of an adaptive, time-dependent, quantile-based transformation of the objective function. It makes no assumptions on the objective function to be optimized. The IGO method produces explicit IGO algorithms through time discretization. It naturally recovers versions of known algorithms and offers a systematic way to derive new ones. The cross-entropy method is recovered in a particular case, and can be extended into a smoothed, parametrization-independent maximum likelihood update (IGO-ML). For Gaussian distributions on $\mathbb{R}^d$, IGO is related to natural evolution strategies (NES) and recovers a version of the CMA-ES algorithm. For Bernoulli distributions on $\{0,1\}^d$, we recover the PBIL algorithm. From restricted Boltzmann machines, we obtain a novel algorithm for optimization on $\{0,1\}^d$. All these algorithms are unified under a single information-geometric optimization framework. Thanks to its intrinsic formulation, the IGO method achieves invariance under reparametrization of the search space $X$, under a change of parameters of the probability distributions, and under increasing transformations of the objective function. Theory strongly suggests that IGO algorithms have minimal loss in diversity during optimization, provided the initial diversity is high. First experiments using restricted Boltzmann machines confirm this insight. Thus IGO seems to provide, from information theory, an elegant way to spontaneously explore several valleys of a fitness landscape in a single run.Comment: Final published versio

On the Gibbs-Liouville theorem in classical mechanics

In this article, it is argued that the Gibbs-Liouville theorem is a mathematical representation of the statement that closed classical systems evolve deterministically. From the perspective of an observer of the system, whose knowledge about the degrees of freedom of the system is complete, the statement of deterministic evolution is equivalent to the notion that the physical distinctions between the possible states of the system, or, in other words, the information possessed by the observer about the system, is never lost. Furthermore, it is shown that the Hamilton equations and the Hamilton principle on phase space follow directly from the differential representation of the Gibbs-Liouville theorem, i.e. that the divergence of the Hamiltonian phase flow velocity vanish. Finally, it is argued that the statements of invariance of the Poisson algebra and unitary evolution are equivalent representations of the Gibbs-Liouville theorem

Digraphs, Knowledge Hypernets, and Neurons

A current flow network of switches, with input node I and output node O, are represented by a directed graph G. In G we define a model of a neuron, and introduce another model in which neurons are theoretically linked. In this second model, we cover invariance, information flow and noise. We show how this model arises from G, how it can be taught, and how it can be declaratively interpreted. The system is made dynamic due to the closing, from O to I, through the environment of the combined models, of a feedback circuit

Abstraction and Invariance for Algebraically Indexed Types

Reynolds’ relational parametricity provides a powerful way to reason about programs in terms of invariance under changes of data representation. A dazzling array of applications of Reynolds’ theory exists, exploiting invariance to yield “free theorems”, non-inhabitation results, and encodings of algebraic datatypes. Outside computer science, invariance is a common theme running through many areas of mathematics and physics. For example, the area of a triangle is unaltered by rotation or flipping. If we scale a triangle, then we scale its area, maintaining an invariant relationship be-tween the two. The transformations under which properties are in-variant are often organised into groups, with the algebraic structure reflecting the composability and invertibility of transformations. In this paper, we investigate programming languages whose types are indexed by algebraic structures such as groups of geometric transformations. Other examples include types indexed by principals–for information flow security–and types indexed by distances–for analysis of analytic uniform continuity properties. Following Reynolds, we prove a general Abstraction Theorem that covers all these instances. Consequences of our Abstraction Theorem include free theorems expressing invariance properties of programs, type isomorphisms based on invariance properties, and non-definability results indicating when certain algebraically indexed types are uninhabited or only inhabited by trivial programs. We have fully formalized our framework and most examples in Coq