On optimal nonlinear associative recall
type: Item
instance of: scholarly article
title: On optimal nonlinear associative recall
zbMATH Open document ID: 0321.94001
DOI: 10.1007/BF02281970
published in: Biological Cybernetics
publication date: 1975-01-01T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 94A05
Mathematics Subject Classification ID: 92B05
zbMATH DE Number: 3503117
Wikidata QID: Q52113796
author: T. Poggio
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02281970
OpenAlex ID: W2162836072
description: scientific article; zbMATH DE number 3503117