On optimal nonlinear associative recall
The paper [see also the author, ``On optimal discrete estimation.'' in: Symposium on Nonlinear Systems, Caltech, G. D. McCann (ed.)and P. Z. Marmareli (ed.) (1975)] considers the problem of determining the nonlinear mapping which optimally associates, in the least square sense, two sets of discrete, finite, column vectors, forming two matrices \(X\) (''input'') and \(Y\) (''output''), with the same number of columns and an arbitrary number of rows. The optimum mapping is seeked in the class of polynomial mappings of degree \(k\). The mapping is defined by \[ Y = L_0 + L_1(X) + L_2(X, X) + \ldots + L_k(X, \ldots,X) \tag{1} \] where \(L_i\) \((i = 1, \ldots, k)\) is an \(i\)-linear, symmetric mapping \(V\times V\times\cdots\times V\rightarrow W\). \(V\) and \(W\) are vector spaces over the real field. With \((L_m)_{i,\alpha_1\cdots\alpha_m}\) defined as the \(m\)-way matrix associated to the mapping \(L_n\), equation (1) can be rewritten as \[ Y_{ij} = (L_0)_{ij} + \sum_{\alpha_i}(L_1)_{i,\alpha_1}X_{\alpha_1j}+ \sum_{\alpha_1\alpha_2}(L_1)_{i,\alpha_1\alpha_2}X_{\alpha_1j} X_{\alpha_2j} + \ldots + \sum_{\alpha_1\cdots\alpha_k} (L_k)_{i,\alpha_1\cdots\alpha_k} X_{\alpha_1j}\cdots X_{\alpha_kj}. \tag{2} \] Since equation (2) is a linear mapping on appropriate cross products of the elements of \(X\), the optimum polynomial of degree \(k\) can be found in terms of the pseudo-inverse of the corresponding matrix containing all the cross products of \(X\) up to the degree \(k\). It is shown in the paper that an alternative iteration scheme, based on cycles of successive optimal corrections until the order \(k\), converges to the optimal polynomial estimator of order \(k\). Various properties of polynomial estimators are further characterized. Conditions on the input matrix \(X\) under which the optimal estimate is linear are derived. ``Stationarity'' of the random processes of which \(X\) and \(Y\) are assumed to represent a set of sample sequences implies that the optimal linear estimate is given by a discrete counterpart of the Wiener-Hopf equation. If the input signals are ``noise-like'', the ``classical'' holographic scheme of associative memory is obtained as the optimal estimator. Nonlinear encoding of the ``input'' \(X\) either before or after the polynomial estimator affects, and in some cases simplifies, the polynomial estimator problem. For instance, a suitable input (output) coding followed (preceded) by a linear mapping can be equivalent to equations (1), (2). The problem of identifying nonlinear mappings from ``input-output experiments'' is also easily treated in terms of this discrete approach and provides a method essentially equivalent to the ``white-noise'' identification method for functionals. Finally, the paper suggests some implications on models of associative distributed memory in brain sciences.
scientific article; zbMATH DE number 3503117