A Review of Learning with Deep Generative Models from Perspective of Graphical Modeling

This document aims to provide a review on learning with deep generative models (DGMs), which is an highly-active area in machine learning and more generally, artificial intelligence. This review is not meant to be a tutorial, but when necessary, we provide self-contained derivations for completeness. This review has two features. First, though there are different perspectives to classify DGMs, we choose to organize this review from the perspective of graphical modeling, because the learning methods for directed DGMs and undirected DGMs are fundamentally different. Second, we differentiate model definitions from model learning algorithms, since different learning algorithms can be applied to solve the learning problem on the same model, and an algorithm can be applied to learn different models. We thus separate model definition and model learning, with more emphasis on reviewing, differentiating and connecting different learning algorithms. We also discuss promising future research directions.Comment: add SN-GANs, SA-GANs, conditional generation (cGANs, AC-GANs). arXiv admin note: text overlap with arXiv:1606.00709, arXiv:1801.03558 by other author

Social Spiral Pattern in Experimental 2x2 Games

With evolutionary game theory, mathematicians, physicists and theoretical biologists usually show us beautiful figures of population dynamic patterns. 2x2 game (matching pennies game) is one of the classical cases. In this letter, we report our finding that, there exists a dynamical pattern, called as social spiral, in human subjects 2x2 experiment data. In a flow/velocity vector field method, we explore the data in the discrete lattices of the macro-level social strategy space in the games, and then above spiral pattern emergent. This finding hints that, there exists a macro-level order beyond the stochastic process in micro-level. We notice that, the vector pattern provides an interesting way to conceal evolutionary game theory models and experimental economics data. This lattice vector field method provides a novel way for models evaluating and experiment designing.Comment: in Chinese, 4 Figures, Keyword: evolutionary game theory, experimental economics, spiral pattern, mixed equilibrium, lattice vector field, JEL: C91, C7

Beurling's Theorem And Invariant Subspaces For The Shift On Hardy Spaces

Let $G$ be a bounded open subset in the complex plane and let $H^{2}(G)$ denote the Hardy space on $G$. We call a bounded simply connected domain $W$ perfectly connected if the boundary value function of the inverse of the Riemann map from $W$ onto the unit disk $D$ is almost 1-1 rwith respect to the Lebesgure on $\partial D$ and if the Riemann map belongs to the weak-star closure of the polynomials in $H^{\infty}(W)$. Our main theorem states: In order that for each $M\in Lat(M_{z})$, there exist $u\in H^{\infty}(G)$ such that $M = \vee\{u H^{2}(G)\}$, it is necessary and sufficient that the following hold: 1) Each component of $G$ is a perfectly connected domain. 2) The harmonic measures of the components of $G$ are mutually singular. 3) % $P^{\infty}(\omega) The set of polynomials is weak-star dense in$ H^{\infty}(G). \noindent Moreover, if G$satisfies these conditions, then every$M\in Lat(M_{z})$is of the form$u H^{2}(G)$, where$u\in H^{\infty}(G)$and the restriction of$u$to each of the components of$G$ is either an inner function or zero

The Riemann Mapping Problem

In this article we investigate the century-old continuous extension problem of the Riemann map. Let $G$ be a simply connected domain. We call $\lambda$ in $\partial G$ a multiple point if there are simply connected subdomains $U$ and $V$ such that $\lambda \in\partial U \cap\partial V$ and $dist (\partial U\cap G , \partial V\cap G )>0$. We show that the Riemann map of $G$ has a continuous extension to $\overline G$ if and only if $\partial G$has no multiple points. All of the results in this paper, together with the Riemann mapping theorem, give a complete and desirable solution to the mapping problem that was originally raised by Riemann in 1851 and intensively investigated by many famous mathematicians throughout history

Approximate Flavor Symmetry in Supersymmetric Model

We investigate the maximal approximate flavor symmetry in the framework of generic minimal supersymmetric standard model. We consider the low energy effective theory of the flavor physics with all the possible operators included. Spontaneous flavor symmetry breaking leads to the approximate flavor symmetry in Yukawa sector and the supersymmetry breaking sector. Fermion mass and mixing hierachies are the results of the hierachy of the flavor symmetry breaking. It is found that in this theory it is possible to solve the flavor changing problems. Furthermore baryogenesis of the universe can be well described and neutron electric dipole moment is closely below it experimental bound by assuming approximate CP violating phase $\sim 10^{-2}$ and superpartner mass around 100 GeV.Comment: 14 pages, latex file, no figur

On the error rate of conditional quasi-Monte Carlo for discontinuous functions

This paper studies the rate of convergence for conditional quasi-Monte Carlo (QMC), which is a counterpart of conditional Monte Carlo. We focus on discontinuous integrands defined on the whole of $R^d$, which can be unbounded. Under suitable conditions, we show that conditional QMC not only has the smoothing effect (up to infinitely times differentiable), but also can bring orders of magnitude reduction in integration error compared to plain QMC. Particularly, for some typical problems in options pricing and Greeks estimation, conditional randomized QMC that uses $n$ samples yields a mean error of $O(n^{-1+\epsilon})$ for arbitrarily small $\epsilon>0$. As a by-product, we find that this rate also applies to randomized QMC integration with all terms of the ANOVA decomposition of the discontinuous integrand, except the one of highest order

Consequence of doping in spatiotemporal rock-paper-scissors games

What determines species diversity is dramatic concern in science. Here we report the effect of doping on diversity in spatiotemporal rock-paper-scissors (RPS) games, which can be observed directly in ecological, biological and social systems in nature. Doping means that there exists some buffer patches which do not involve the main procession of the conflicts but occupied the game space. Quantitative lattices simulation finds that (1) decrease of extinction possibility is exponential dependent on the increase of doping rate, (2) the possibility of the conflict is independent of doping rate at well mix evolution beginning, and is buffered by doping in long time coexistence procession. Practical meaning of doping are discussed. To demonstrate the importance of doping, we present one practical example for microbial laboratory efficient operation and one theoretical example for human-environment co-existence system better understanding. It suggests that, for diversity, doping can not be neglected.Comment: 4 Pages, 4 Figur

Cyclic motions in Dekel-Scotchmer Game Experiments

TASP (Time Average Shapley Polygon, Bena{\=\i}m, Hofbauer and Hopkins, \emph{Journal of Economic Theory}, 2009), as a novel evolutionary dynamics model, predicts that a game could converge to cycles instead of fix points (Nash equilibria). To verify TASP theory, using the four strategy Dekel-Scotchmer games (Dekel and Scotchmer, \emph{Journal of Economic Theory}, 1992), four experiments were conducted (Cason, Friedman and Hopkins, \emph{Journal of Economic Theory}, 2010), in which, however, reported no evidence of cycles (Cason, Friedman and Hopkins, \emph{The Review of Economic Studies}, 2013). We reanalysis the four experiment data by testing the stochastic averaging of angular momentum in period-by-period transitions of the social state. We find, the existence of persistent cycles in Dekel-Scotchmer game can be confirmed. On the cycles, the predictions from evolutionary models had been supported by the four experiments.Comment: 7 Page, 3 Figure; keywords: experimental economics; angular momentum; period by period transition; social motion; stochastic averaging method; tumbling cycl

The Structure of the Closure of the Rational Functions in LqL^{q}(μ\mu)$

Let $K$ be a compact subset in the complex plane and let $A(K)$ be the uniform closure of the functions continuous on $K$ and analytic on $K^{\circ}$. Let $\mu$ be a positive finite measure with its support contained in $K$. For $1 \leq q < \infty$, let $A^{q}(K,\mu)$ denote the closure of $A(K)$ in $L^{q}(\mu)$. The aim of this work is to study the structure of the space $A^{q}(K,\mu)$. We seek a necessary and sufficient condition on $K$ so that a Thomson-type structure theorem for $A^{q}(K,\mu)$ can be established. Our results essentially give perfect solutions to the major open problem in the research filed of theory of subnormal operators and aproximation by analytic functions in the mean

Continuous extension of conformal maps

For a simply connected domain $G$, let $\partial_{a}G$ be the set of accessible points in $\partial G$ and let $\partial_{n} G=\partial G-\partial_{a}G$. A point $a\in\partial G$ is called semi-unreachable if there is a crosscut $J$ of $G$ and domains $U$ and $V$ such that $G-J=U\cup V$ and $a\in(\partial_{n} U\cup\partial_{n} V)-J$. We use $\partial_{sn}G$ to denote the set of semi-unreachable points. In this article we show that a univalent analytic function $\psi$ from the unit disk $D$ onto $G$ extends continuously to $\overline D$ if and only if $\partial_{sn}G=\emptyset$. As a consequence, we provide a very short and elementary proof for the Osgood conjecture: if $G$ is a Jordan domain, then $\psi^{-1}$, the Riemann map, extends to be a homeomorphism from $\overline G$ to $\overline D$.Comment: arXiv admin note: substantial text overlap with arXiv:1307.2740. This is to supercede the arXiv:1307.2740 since I am unable to replace the content in that pape