Formalizing Triggers: A Learning Model for Finite Spaces

In a recent seminal paper, Gibson and Wexler (1993) take important steps to formalizing the notion of language learning in a (finite) space whose grammars are characterized by a finite number of parameters. They introduce the Triggering Learning Algorithm (TLA) and show that even in finite space convergence may be a problem due to local maxima. In this paper we explicitly formalize learning in finite parameter space as a Markov structure whose states are parameter settings. We show that this captures the dynamics of TLA completely and allows us to explicitly compute the rates of convergence for TLA and other variants of TLA e.g. random walk. Also included in the paper are a corrected version of GW's central convergence proof, a list of "problem states" in addition to local maxima, and batch and PAC-style learning bounds for the model.

Recognition of Topological Invariants by Iterative Arrays

A study is made of the recognition and transformation of figures by iterative arrays of finite state automata. A figure is a finite rectangular two-dimensional array of symbols. The iterative arrays considered are also finite, rectangular, and two-dimensional. The automata comprising any given array are called cells and are assumed to be isomorphic and to operate synchronously with the state of a cell at time t+1 being a function of the states of it and its four nearest neighbors at time t. At time t=0 each cell is placed in one of a fixed number of initial states. The pattern of initial states thus introduced represents the figure to be processed. The resulting sequence of array states represents a computation based on the input figure. If one waits for a specially designated cell to indicate acceptance or rejection of the figure, the array is said to be working on a recognition problem. If one waits for the array to come to a stable configuration representing an output figure, the array is said to be working on a transformation problem.

Boundaries and Topological Algorithms

This thesis develops a model for the topological structure of situations. In this model, the topological structure of space is altered by the presence or absence of boundaries, such as those at the edges of objects. This allows the intuitive meaning of topological concepts such as region connectivity, function continuity, and preservation of topological structure to be modeled using the standard mathematical definitions. The thesis shows that these concepts are important in a wide range of artificial intelligence problems, including low-level vision, high-level vision, natural language semantics, and high-level reasoning.

On the V(subscript gamma) Dimension for Regression in Reproducing Kernel Hilbert Spaces

This paper presents a computation of the $V_gamma$ dimension for regression in bounded subspaces of Reproducing Kernel Hilbert Spaces (RKHS) for the Support Vector Machine (SVM) regression $epsilon$-insensitive loss function, and general $L_p$ loss functions. Finiteness of the RV_gamma$ dimension is shown, which also proves uniform convergence in probability for regression machines in RKHS subspaces that use the $L_epsilon$ or general $L_p$ loss functions. This paper presenta a novel proof of this result also for the case that a bias is added to the functions in the RKHS.