Using Special-Purpose Computing to Examine Chaotic Behavior in Nonlinear Mappings

Studying chaotic behavior in nonlinear systems requires numerous computations in order to simulate the behavior of such systems. The Standard Map Machine was designed and implemented as a special computer for performing these intensive computations with high-speed and high-precision. Its impressive performance is due to its simple architecture specialized to the numerical computations required of nonlinear systems. This report discusses the design and implementation of the Standard Map Machine and its use in the study of nonlinear mappings; in particular, the study of the standard map.

A Theory of Networks for Appxoimation and Learning

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nolinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.

A Theory of How the Brain Might Work

I wish to propose a quite speculative new version of the grandmother cell theory to explain how the brain, or parts of it, may work. In particular, I discuss how the visual system may learn to recognize 3D objects. The model would apply directly to the cortical cells involved in visual face recognition. I will also outline the relation of our theory to existing models of the cerebellum and of motor control. Specific biophysical mechanisms can be readily suggested as part of a basic type of neural circuitry that can learn to approximate multidimensional input-output mappings from sets of examples and that is expected to be replicated in different regions of the brain and across modalities. The main points of the theory are: -the brain uses modules for multivariate function approximation as basic components of several of its information processing subsystems. -these modules are realized as HyperBF networks (Poggio and Girosi, 1990a,b). -HyperBF networks can be implemented in terms of biologically plausible mechanisms and circuitry. The theory predicts a specific type of population coding that represents an extension of schemes such as look-up tables. I will conclude with some speculations about the trade-off between memory and computation and the evolution of intelligence.

Sequence-Seeking and Counter Streams: A Model for Information Processing in the Cortex

This paper presents a model for the general flow in the neocortex. The basic process, called "sequence-seeking," is a search for a sequence of mappings or transformations, linking source and target representations. The search is bi-directional, "bottom-up" as well as "top-down," and it explores in parallel a large numbe rof alternative sequences. This operation is implemented in a structure termed "counter streams," in which multiple sequences are explored along two separate, complementary pathways which seeking to meet. The first part of the paper discusses the general sequence-seeking scheme and a number of related processes, such as the learning of successful sequences, context effects, and the use of "express lines" and partial matches. The second part discusses biological implications of the model in terms of connections within and between cortical areas. The model is compared with existing data, and a number of new predictions are proposed.

On Interpreting Stereo Disparity

The problems under consideration center around the interpretation of binocular stereo disparity. In particular, the goal is to establish a set of mappings from stereo disparity to corresponding three-dimensional scene geometry. An analysis has been developed that shows how disparity information can be interpreted in terms of three-dimensional scene properties, such as surface depth, discontinuities, and orientation. These theoretical developments have been embodied in a set of computer algorithms for the recovery of scene geometry from input stereo disparity. The results of applying these algorithms to several disparity maps are presented. Comparisons are made to the interpretation of stereo disparity by biological systems.

Using Recurrent Networks for Dimensionality Reduction

This report explores how recurrent neural networks can be exploited for learning high-dimensional mappings. Since recurrent networks are as powerful as Turing machines, an interesting question is how recurrent networks can be used to simplify the problem of learning from examples. The main problem with learning high-dimensional functions is the curse of dimensionality which roughly states that the number of examples needed to learn a function increases exponentially with input dimension. This thesis proposes a way of avoiding this problem by using a recurrent network to decompose a high-dimensional function into many lower dimensional functions connected in a feedback loop.

Keyword Join: Realizing Keyword Search in P2P-based Database Systems

In this paper, we present a P2P-based database sharing system that provides information sharing capabilities through keyword-based search techniques. Our system requires neither a global schema nor schema mappings between different databases, and our keyword-based search algorithms are robust in the presence of frequent changes in the content and membership of peers. To facilitate data integration, we introduce keyword join operator to combine partial answers containing different keywords into complete answers. We also present an efficient algorithm that optimize the keyword join operations for partial answer integration. Our experimental study on both real and synthetic datasets demonstrates the effectiveness of our algorithms, and the efficiency of the proposed query processing strategies.