Neurodynamics and nonlinear integrable systems of Lax type

Abstract

It is shown that an averaged learning equation in neurodynamics is an integrable gradient system having a Lax pair representation.In this decade, applied analysis based upon the theory of nonlinear integrable systems has been gradually developed. Though the theory has its origin mostly in classical mechanics, this methodology supplies a new mathematical tool which can be widely applied to:i) linear prediction problems [13], (ii) the geometry of linear systems [10, 14], (iii) maximum likelihood estimation [4, 15], (iv) matrix eigenvalue problems [7, 16, 24], (v) the information geometry [17, 18], (vi) linear programming problems [5, 19],and so on. Here (i), (ii) and (vi) are essentially nonlinear problems being due to certain quotient structures and boundary conditions. The motivation of applied analysis of integrable systems is explained in a review article [20]. Although it looks accidental, the dynamical theoretic approaches to the variety of interesting problems turn out to have properties in common. Namely, the dynamical systems which have emerged are integrable systems of Lax type. Rich information about integrable systems successfully helps us to solve the problems being considered as well as the dynamical systems themselves. Solvability of the original nonlinear problems may be related to integrability of the resulting dynamical systems at a deep level.In the following sections, we consider (vii) neurodynamics as a new topic of applied analysis of nonlinear integrable systems. We here take the position that we regard the problem of learning as a challenging problem in nonlinear dynamical systems. First we discuss a stochastic learning equation of Hebb type with a nonlinear *

All the trajectories of an extended averaged Hebbian learning equation on the quantum state space are the e-geodesics

“…of degree n, whose diagonal entries stand for the eigenvalues of the autocorrelation matrix of the stationary stochastic process governing the Hebbian learning process [8,11,14]. The reason for referring to Eq.…”

Geometry and Dynamics of a Quantum Search Algorithm for an Ordered Tuple of Multi-Qubits

“…Due to the account given below, however, geometric studies were not made in [19] either on the search sequence in the ESOT generated by the Grover-type search algorithm or on the reduced search sequence in the QIS: Instead of geometric studies on the search sequences, it is the gradient dynamical system associated with the negative von Neumann entropy as the potential that is discussed in [19] on inspired by a series works of Nakamura [20][21][22] on complete integrablity of algorithms arising in applied mathematics. The result on the gradient system in [19] has drawn the author's interest to publish [23,24] on the gradient systems on the QIS realizing the Karmarkar flow for linear programming and a Hebb-type learning equation for multivariate analysis, while geometric studies on the search sequences were left undone.…”

“…Similarly, the quotient spaceṀ 1 (2 n , ℓ)/ ∼ is realized asṖ ℓ defined by (22). The projection is given by π (n,l) restricted toṀ 1 (2 n , ℓ).…”

“…The tangent space T ρṖℓ at ρ ∈Ṗ ℓ can be identified with the set of ℓ × ℓ traceless Hermitian matrices (see (23)), which can be dealt with in a smilar way to the Euclidean parallel transport setting-up. Indeed, in view of the definition (22),Ṗ ℓ is understood to be a fragment of an affine subspace of M(ℓ, ℓ). Hence the tangent space T ΦṖℓ at everyṖ ℓ admits the structure (23), which is looked upon as a linear subspace of M(ℓ, ℓ).…”

Jacobi algorithm for symmetric eigenvalue problem and integrable gradient system of Lax form

“…A class of dynamical systems similar to (14) also emerges widely in neural networks (cf. [12], [17] and the references therein). Secondly, we consideran objective function…”