A Unifled Learning Framework: Multisets Modeling Learning 1

A unifled learning framework is proposed. Its difierent special cases will automatically lead us to current existing major types of neural network learnings, e.g, data clustering, various PCA- type self-organizations and their localized extensions, self-organizing topological map, as well as supervised learning for feedforward network and modular architecture. Not only this new framework is useful for a deep understanding of the existing learnings, but also it provides new insights that can guide the extensions of the existing learning methods. With the new insights, we have obtained several new intersting unsupervised learnings, and introduced regularization or generalization to unsupervised learnings, particularly proposed approaches for solving a classical hard problem|how to decide the number of clusters in clustering analysis or competitive learning. (1) where p(x) means that x satisfles the properties specifled by a general predicate proposition p. For examples, p(x) may mean that x is a root of an equation F(x) = 0, or that x = (»;·) satisfles an explicit function · = f(»), as well as that x makes F(x) > 0 or F(x) < 0. Moreover, p(x) can also be a combined proposition that consists of a number of such simple propositions p1(x);¢¢¢, pr(x) via logic connectives ^;_;:. As a result, a model M can be a point, a curve, a surface, an area or volume with any shape or any of their combinations and it is either able or unable to be described by mathematical equations. Moreover, it even can be described by languages. In summary, this model can be any set in R d . To use such M to model a given data set, we deflne that M represents xi with an error given by " q (xi;M) = min y2M\R jT(xi i y)j