José Luis Montaña

Lower bounds for diophantine approximations

Abstract We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero-dimensional polynomial equation system. This result represents a multivariate version of Liouvilles classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their polynomial character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight-line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newtons algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.

Lower bounds for arithmetic networks

We show lower bounds for depth of arithmetic networks over algebraically closed fields, real closed fields and the field of the rationals. The parameters used are either the degree or the number of connected components. These lower bounds allow us to show the inefficiency of arithmetic networks to parallelize several natural problems. For instance, we show a √n lower bound for parallel time of the Knapsack problem over the reals and also that the computation of the “integer part” is not well parallelizable by arithmetic networks. Over the rationals we obtain results of similar order and that the Knapsack has an √n lower bound for the parallel time measured by networks. A simply exponential lower bound for the parallel time of quantifier elimination is also shown. Finally, separations among classesPKandNCKare available for fieldsK in the above cases.

Lower bounds for arithmetic networks II: Sum of Betti numbers

We show lower bounds for the parallel complexity of membership problems in semialgebraic sets. Our lower bounds are obtained from the Euler characteristic and the sum of Betti numbers. We remark that these lower bounds are polynomial (an square root) in the sequential lower bounds obtained by Andrew C.C. Yao.

Model selection in genetic programming

In this paper we discuss the problem of model selection in Genetic Programming. We present empirical comparisons between classical statistical methods (AIC, BIC) adapted to Genetic Programming and the Structural Risk Minimization method (SRM) based on Vapnik-Chervonenkis theory (VC), for symbolic regression problems with added noise. We also introduce a new model complexity measure for the SRM method that tries to measure the non-linearity of the model. The experimentation suggests practical advantages of using VC-based model selection with the new complexity measure, when using genetic training.

Straight Line Programs: A New Linear Genetic Programming Approach

Tree encodings of programs are well known for their representative power and are used very often in Genetic Programming. In this paper we experiment with a new data structure, named straight line program (slp), to represent computer programs. The main features of this structure are described and new recombination operators for GP related to slps are introduced. Experiments have been performed on symbolic regression problems. Results are encouraging and suggest that the GP approach based on slps consistently outperforms conventional GP based on tree structured representations.

A Dynamic-Bayesian Network framework for modeling and evaluating learning from observation

Abstract Learning from observation (LfO), also known as learning from demonstration, studies how computers can learn to perform complex tasks by observing and thereafter imitating the performance of a human actor. Although there has been a significant amount of research in this area, there is no agreement on a unified terminology or evaluation procedure. In this paper, we present a theoretical framework based on Dynamic-Bayesian Networks (DBNs) for the quantitative modeling and evaluation of LfO tasks. Additionally, we provide evidence showing that: (1) the information captured through the observation of agent behaviors occurs as the realization of a stochastic process (and often not just as a sample of a state-to-action map); (2) learning can be simplified by introducing dynamic Bayesian models with hidden states for which the learning and model evaluation tasks can be reduced to minimization and estimation of some stochastic similarity measures such as crossed entropy.

The non-scalar Model of Complexity in Computational Geometry

An outline on the relation between algebraic complexity theories and semialgebraic sets is presented. First we discuss the concepts of total and non-scalar complexities both for polynomials and semialgebraic sets observing that they are ”geometric complexities” verifying the ”semialgebraic” version of the Benedetti-Risler conjecture [4]. Moreover, we remark that total and non-scalar complexities of semialgebraic sets are decidable theories. Finally, using non-scalar complexity and intersection numbers of semialgebraic sets we get new lower bounds for several problems in computational geometry, generalizing the results obtained by M. Ben-Or using total complexity and number of connected components. An expanded version of the ideas sketched here is [10]

Penalty functions for genetic programming algorithms

Very often symbolic regression, as addressed in Genetic Programming (GP), is equivalent to approximate interpolation. This means that, in general, GP algorithms try to fit the sample as better as possible but no notion of generalization error is considered. As a consequence, overfitting, code-bloat and noisy data are problems which are not satisfactorily solved under this approach. Motivated by this situation we review the problem of Symbolic Regression under the perspective of Machine Learning, a well founded mathematical toolbox for predictive learning. We perform empirical comparisons between classical statistical methods (AIC and BIC) and methods based on Vapnik-Chrevonenkis (VC) theory for regression problems under genetic training. Empirical comparisons of the different methods suggest practical advantages of VC-based model selection. We conclude that VC theory provides methodological framework for complexity control in Genetic Programming even when its technical results seems not be directly applicable. As main practical advantage, precise penalty functions founded on the notion of generalization error are proposed for evolving GP-trees.

A NEW LINEAR GENETIC PROGRAMMING APPROACH BASED ON STRAIGHT LINE PROGRAMS: SOME THEORETICAL AND EXPERIMENTAL ASPECTS

Tree encodings of programs are well known for their representative power and are used very often in Genetic Programming. In this paper we experiment with a new data structure, named straight line program (slp), to represent computer programs. The main features of this structure are described, new recombination operators for GP related to slps are introduced and a study of the Vapnik-Chervonenkis dimension of families of slps is done. Experiments have been performed on symbolic regression problems. Results are encouraging and suggest that the GP approach based on slps consistently outperforms conventional GP based on tree structured representations.

An evolutionary strategy for the multidimensional 0-1 knapsack problem based on genetic computation of surrogate multipliers

In this paper we present an evolutionary algorithm for the multidimensional 0–1 knapsack problem. Our algorithm incorporates a heuristic operator which computes problem-specific knowledge. The design of this operator is based on the general technique used to design greedy-like heuristics for this problem, that is, the surrogate multipliers approach of Pirkul (see [7]). The main difference with work previously done is that our heuristic operator is computed following a genetic strategy -suggested by the greedy solution of the one dimensional knapsack problem- instead of the commonly used simplex method. Experimental results show that our evolutionary algorithm is capable of obtaining high quality solutions for large size problems requiring less amount of computational effort than other evolutionary strategies supported by heuristics founded on linear programming calculation of surrogate multipliers.