Information geometry is a new powerful method of information sciences. Information geometry is applied to manifolds of neural networks of various architectures. Here is proposed a new theoretical approach to the manifold consisting of feedforward neural networks, the manifold of Boltzmann machines and the manifold of neural networks of recurrent connections. This opens a new direction of studies on a family of neural networks, not a study of behaviors of single neural networks.

Topographic maps begin to be recognized as one of the major computational structures underlying neural computation in the brain. They provide dimension-reducing projections between feature spaces that seem to be established and maintained under the participation of selforganizing, adaptive processes. In this contribution, we investigate how well the structure of such maps can be replicated by simple adaptive processes of the kind proposed by Kohonen. We will particularly address the important issue, how the dimensionality of the input space affects the spatial organization of the resulting map.

Applications of neural networks are prevailing in speech recognition research. In this paper, first, suitable role of neural network (mainly back-propagation based multi-layer type) in speech recognition, is discussed. Considering that speech is a long, variable length, structured pattern, a direction, in which neural network is used in cooperation with existing structural analysis frameworks, is recommended. Activities are surveyed, including those intended to cooperatively merge neural networks into dynamic programming based structural analysis framework. It is observed that considerable efforts have been paid to suppress the high nonlinearity of network output. As far as surveyed, no experiment in real field has been reported.

This paper attempts to account for intelligibility of practices-based learning (so-called 'learning control') for skill refinement from the viewpoint of Newtonian mechanics. It is shown from an axiomatic approach that an extended notion of passivity for the residual error dynamics of robots plays a crucial role in their ability of learning. More precisely, it is shown that the exponentially weighted passivity with respect to residual velocity vector and torque vector leads the robot system to the convergence of trajectory tracking errors to zero with repeating practices. For a class of tasks when the endpoint is constrained geometrically on a surface, the problem of convergence of residual tracking errors and residual contact-force errors is also discussed on the basis of passivity analysis.

This paper reviews the capability of the three layer neural network (TLNN) with one output neuron. The input set is restricted to a finite subset S of Eⁿ, and the TLNN implements a function F such as F : S I={1, -1}, i,e., F is a dichotomy of S. How many functions (dichotomies) can it compute by appropriately adjusting parameters in the TLNN? Brief historical review, some theorems on the subject obtained so far, and related topics are presented. Several open problems are also included.

This paper deals with an adaptive credit assignment algorithm to select strategies having higher capabilities in the learning classifier system (LCS) based upon the genetic algorithm (GA). We emulate a kind of prizes and incentives employed in the economies with imperfect information. The compensation scheme provides an automatic adjustment in response to the changes in the environment, and a comfortable guideline to incorporate the constraints. The learning process in the LCS based on the GA is realized by combining a pair of most capable strategies (called classifiers) represented as the production rules to replace another less capable strategy in the similar manner to the genetic operation on chromosomes in organisms. In the conventional scheme of the learning classifier system, the capability s(k, t) (called strength) of a strategy k at time t is measured by only the suitableness to sense and recognize the environment. But, we also define and utilize the prizes and incentives obtained by employing the strategy, so as to increase s(k, t) if the classifier provide good rules, and some amount is subtracted if the classifier k violate the constraints. The new algorithm is applied to the portfolio management. As the simulation result shows, the net return of the portfolio management system surpasses the average return obtained in the American securities market. The result of the illustrative example is compared to the same system composed of the neural networks, and related problems are discussed.

This paper introduces a modeling of the human rotation invariant recognition mechanism at the neural level. In the model, mechanisms of memory search and mental rotation are realized in the process of minimizing the energy of a bi-directional connection network. The thrust of the paper is to explain temporal mental activities such as successive memory retrievals and continuous mental rotation in terms of state transitions of collective neurons based on nonequilibrium dynamics. We conclude that regularities emerging in the dynamics of intermittent chaos lead the recognition process in a structural and meaningful way.

Threre are two types of nonlinear associative silicon retinas. One is a sparse Hopfield type neural network which is called a H-type retina and the other is its dual network which is called a DH-type retina. The input information sequences of H-type and HD-type retinas are given by nodes and links as voltages and currents respectively. The error correcting capacity (minimum basin of attraction) of H-type and DH-type retinas is decided by the minimum numbers of links of cutset and loop respectively. The operation principle of the regeneration is based on the voltage or current distribution of the neural field. The most important nonlinear operation in the retinas is a dynamic quantization to decide the binary value of each neuron output from the neighbor value. Also, the edge is emphasized by a line-process. The rates of compression of H-type and DH-type retinas used in the simulation are 1/8 and (2/3) (1/8) respectively, where 2/3 and 1/8 mean rates of the structural and binarizational compression respectively. We could have interesting and significant simulation results enough to make a chip.

The purpose of the present paper is to review a state of the art of nonlinear analysis with the self-validating numerical method. The self-validating numerics based method provides a tool for performing computer assisted proofs of nonlinear problems by taking the effect of rounding errors in numerical computations rigorously into account. First, Kantorovich's approach of a posteriori error estimation method is surveyed, which is based on his convergence theorem of Newton's method. Then, Urabe's approach for computer assisted existence proofs is likewise discussed. Based on his convergence theorem of the simplified Newton method, he treated practical nonlinear differential equations such as the Van der Pol equation ahd the Duffing equation, and proved the existence of their periodic and quasi-periodic solutions by the self-validating numerics. An approach of the author for generalization and abstraction of Urabe's method are also discribed to more general funcional equations. Furthermore, methods for rigorous estimation of rounding errors are surveyed. Interval analytic methods are discussed. Then an approach of the author which uses rational arithmetic is reviewed. Finally, approaches for computer assisted proofs of nonlinear problems are surveyed, which are based on the self-validating numerics.

A type of infinite dimensional homotopy method is considered for numerically calculating a solution curve of a nonlinear functional equation being a Fredholm operator with index 1 and an A-proper operator. In this method, a property of so-called A-proper homotopy plays an important role.

As the values of parameters in periodic systems vary, a nodal point appearing on a locus of period doubling bifurcation points crosses over a locus of turning points. We consider the nodal point lying just on the locus of turning points and consider its accurate location. To compute it, we consider an extended system which consists of an original equation and an additional equation. We present a result assuring that this extended system has an isolated solution containing the nodal point.

Efficient probabilistic decision trees are required in various application areas such as character recognition. This paper presents a polynomial-time approximate algorithm for designing a probabilistic decision tree. The obtained tree is near-optimal for the cost, defined as the weighted sum of the expected test execution time and expected loss. The algorithm is advantageous over other reported heuristics from the viewpoint that the goodness of the solution is theoretically guaranteed. That is, the relative deviation of the obtained tree cost from the exact optimum is not more than a positive constant ε, which can be set arbitrarily small. When the given loss function is Hamming metric, the time efficiency is further improved by using the information theoretical lower bound on the tree cost. The time efficiency of the algorithm and the accuracy of the solutions were evaluated through computational experiments. The results show that the computing time increases very slowly with an increase in problem size and the relative error of the obtained solution is much less than the upper bound ε for most problems.

Quantization of the impulse response coefficients due to finite word length causes the moments to deviate from their ideal values. This deviation is found to have a linear variation with the output roundoff noise of the filter realized in direct form. Since the zeros and poles of a given filter also move away from their designed locations due to quantization, we show a relation between the zeros and poles and the moments of the impulse response.