The Informational Complexity of Learning from Examples

This thesis attempts to quantify the amount of information needed to learn certain tasks. The tasks chosen vary from learning functions in a Sobolev space using radial basis function networks to learning grammars in the principles and parameters framework of modern linguistic theory. These problems are analyzed from the perspective of computational learning theory and certain unifying perspectives emerge.

Complexity of Human Language Comprehension

The goal of this article is to reveal the computational structure of modern principle-and-parameter (Chomskian) linguistic theories: what computational problems do these informal theories pose, and what is the underlying structure of those computations? To do this, I analyze the computational complexity of human language comprehension: what linguistic representation is assigned to a given sound? This problem is factored into smaller, interrelated (but independently statable) problems. For example, in order to understand a given sound, the listener must assign a phonetic form to the sound; determine the morphemes that compose the words in the sound; and calculate the linguistic antecedent of every pronoun in the utterance. I prove that these and other subproblems are all NP-hard, and that language comprehension is itself PSPACE-hard.

The Complexity of Safety Stock Placement in General-Network Supply Chains

We consider the optimization problem of safety stock placement in a supply chain, as formulated in [1]. We prove that this problem is NP-Hard for supply chains modeled as general acyclic networks. Thus, we do not expect to find a polynomial-time algorithm for safety stock placement for a general-network supply chain.

Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation

We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased.

Segmentation of Brain Tissue from Magnetic Resonance Images

Segmentation of medical imagery is a challenging problem due to the complexity of the images, as well as to the absence of models of the anatomy that fully capture the possible deformations in each structure. Brain tissue is a particularly complex structure, and its segmentation is an important step for studies in temporal change detection of morphology, as well as for 3D visualization in surgical planning. In this paper, we present a method for segmentation of brain tissue from magnetic resonance images that is a combination of three existing techniques from the Computer Vision literature: EM segmentation, binary morphology, and active contour models. Each of these techniques has been customized for the problem of brain tissue segmentation in a way that the resultant method is more robust than its components. Finally, we present the results of a parallel implementation of this method on IBM's supercomputer Power Visualization System for a database of 20 brain scans each with 256x256x124 voxels and validate those against segmentations generated by neuroanatomy experts.

Financial Information Mediation: A Case Study of Standards Integration for Electronic Bill Presentment and Payment Using the COIN Mediation Technology

Each player in the financial industry, each bank, stock exchange, government agency, or insurance company operates its own financial information system or systems. By its very nature, financial information, like the money that it represents, changes hands. Therefore the interoperation of financial information systems is the cornerstone of the financial services they support. E-services frameworks such as web services are an unprecedented opportunity for the flexible interoperation of financial systems. Naturally the critical economic role and the complexity of financial information led to the development of various standards. Yet standards alone are not the panacea: different groups of players use different standards or different interpretations of the same standard. We believe that the solution lies in the convergence of flexible E-services such as web-services and semantically rich meta-data as promised by the semantic Web; then a mediation architecture can be used for the documentation, identification, and resolution of semantic conflicts arising from the interoperation of heterogeneous financial services. In this paper we illustrate the nature of the problem in the Electronic Bill Presentment and Payment (EBPP) industry and the viability of the solution we propose. We describe and analyze the integration of services using four different formats: the IFX, OFX and SWIFT standards, and an example proprietary format. To accomplish this integration we use the COntext INterchange (COIN) framework. The COIN architecture leverages a model of sources and receivers’ contexts in reference to a rich domain model or ontology for the description and resolution of semantic heterogeneity.