A Theory of Networks for Appxoimation and Learning

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nolinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.

Using Analogy to Acquire Commonsense Knowledge from Human Contributors

The goal of the work reported here is to capture the commonsense knowledge of non-expert human contributors. Achieving this goal will enable more intelligent human-computer interfaces and pave the way for computers to reason about our world. In the domain of natural language processing, it will provide the world knowledge much needed for semantic processing of natural language. To acquire knowledge from contributors not trained in knowledge engineering, I take the following four steps: (i) develop a knowledge representation (KR) model for simple assertions in natural language, (ii) introduce cumulative analogy, a class of nearest-neighbor based analogical reasoning algorithms over this representation, (iii) argue that cumulative analogy is well suited for knowledge acquisition (KA) based on a theoretical analysis of effectiveness of KA with this approach, and (iv) test the KR model and the effectiveness of the cumulative analogy algorithms empirically. To investigate effectiveness of cumulative analogy for KA empirically, Learner, an open source system for KA by cumulative analogy has been implemented, deployed, and evaluated. (The site "1001 Questions," is available at http://teach-computers.org/learner.html). Learner acquires assertion-level knowledge by constructing shallow semantic analogies between a KA topic and its nearest neighbors and posing these analogies as natural language questions to human contributors. Suppose, for example, that based on the knowledge about "newspapers" already present in the knowledge base, Learner judges "newspaper" to be similar to "book" and "magazine." Further suppose that assertions "books contain information" and "magazines contain information" are also already in the knowledge base. Then Learner will use cumulative analogy from the similar topics to ask humans whether "newspapers contain information." Because similarity between topics is computed based on what is already known about them, Learner exhibits bootstrapping behavior --- the quality of its questions improves as it gathers more knowledge. By summing evidence for and against posing any given question, Learner also exhibits noise tolerance, limiting the effect of incorrect similarities. The KA power of shallow semantic analogy from nearest neighbors is one of the main findings of this thesis. I perform an analysis of commonsense knowledge collected by another research effort that did not rely on analogical reasoning and demonstrate that indeed there is sufficient amount of correlation in the knowledge base to motivate using cumulative analogy from nearest neighbors as a KA method. Empirically, evaluating the percentages of questions answered affirmatively, negatively and judged to be nonsensical in the cumulative analogy case compares favorably with the baseline, no-similarity case that relies on random objects rather than nearest neighbors. Of the questions generated by cumulative analogy, contributors answered 45% affirmatively, 28% negatively and marked 13% as nonsensical; in the control, no-similarity case 8% of questions were answered affirmatively, 60% negatively and 26% were marked as nonsensical.