Geometry and Expressive Power of Conditional Restricted Boltzmann Machines

Conditional restricted Boltzmann machines are undirected stochastic neural networks with a layer of input and output units connected bipartitely to a layer of hidden units. These networks define models of conditional probability distributions on the states of the output units given the states of the input units, parametrized by interaction weights and biases. We address the representational power of these models, proving results their ability to represent conditional Markov random fields and conditional distributions with restricted supports, the minimal size of universal approximators, the maximal model approximation errors, and on the dimension of the set of representable conditional distributions. We contribute new tools for investigating conditional probability models, which allow us to improve the results that can be derived from existing work on restricted Boltzmann machine probability models.Comment: 30 pages, 5 figures, 1 algorith

Invariant Generation for Multi-Path Loops with Polynomial Assignments

Program analysis requires the generation of program properties expressing conditions to hold at intermediate program locations. When it comes to programs with loops, these properties are typically expressed as loop invariants. In this paper we study a class of multi-path program loops with numeric variables, in particular nested loops with conditionals, where assignments to program variables are polynomial expressions over program variables. We call this class of loops extended P-solvable and introduce an algorithm for generating all polynomial invariants of such loops. By an iterative procedure employing Gr\"obner basis computation, our approach computes the polynomial ideal of the polynomial invariants of each program path and combines these ideals sequentially until a fixed point is reached. This fixed point represents the polynomial ideal of all polynomial invariants of the given extended P-solvable loop. We prove termination of our method and show that the maximal number of iterations for reaching the fixed point depends linearly on the number of program variables and the number of inner loops. In particular, for a loop with m program variables and r conditional branches we prove an upper bound of m*r iterations. We implemented our approach in the Aligator software package. Furthermore, we evaluated it on 18 programs with polynomial arithmetic and compared it to existing methods in invariant generation. The results show the efficiency of our approach

Evaluability: an alternative approach to polarity sensitivity

Based on Brandtler (2012), this paper argues that polarity items are sensitive to evaluability, a concept that refers to the possibility of accepting or rejecting an utterance as true in a communicative exchange. The main distinction is made between evaluable and non-evaluable utterances. The evaluable category comprises any clause that asserts, presupposes or entails the truth of an affirmative or a negative proposition. In contrast, the non-evaluable category contains clauses that do not assert, presuppose or entail the truth of an affirmative or a negative proposition. According to the Evaluability Hypothesis, non-evaluable environments are natural hosts for both NPIs and PPIs. Hence, the occurrence of polarity items in non-evaluable clauses does not require formal licensing, and this is the reason we find both PPIs and weak NPIs in yes/no-questions and conditionals. Evaluable clauses, in contrast, are restricted environments and may only host polarity items that are formally (i.e. syntactically) licensed. Hence, NPIs require the presence of a licensing element, and PPIs require the absence of such elements. This analysis leads to an important change of perspective, as the occurrence of polarity items in negative and affirmative sentences becomes the marked, or exceptional, case

Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory

We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the invertible case and is, to our knowledge, one among not very many instances in which a natural invariant measure for a non-invertible dynamical system is well-understood.Comment: v3. Exposition improved. Final version, to appear in Ann. Scient. de l'EN