2013 Conference Abstracts: Annual Undergraduate Research Conference at the Interface of Biology and Mathematics

URC Schedule and Abstract Book for the Fifth Annual Undergraduate Research Conference at the Interface of Biology and Mathematics Date: November 16-17, 2013Plenary Speaker: Mariel Vazquez, Associate Professor of Mathematics at San Francisco State UniversityFeatured Speaker: Andrew Liebhold, Research Entomologist for the USDA Forest Servic

Decomposition of Gaussian processes, and factorization of positive definite kernels

We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $K$, vs for Gaussian process $V$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $K$ is the exact same as that which yield factorizations for $V$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems

Guarantees for Efficient and Adaptive Online Learning

In this thesis, we study the problem of adaptive online learning in several different settings. We first study the problem of predicting graph labelings online which are assumed to change over time. We develop the machinery of cluster specialists which probabilistically exploit any cluster structure in the graph. We give a mistake-bounded algorithm that surprisingly requires only O(log n) time per trial for an n-vertex graph, an exponential improvement over existing methods. We then consider the model of non-stationary prediction with expert advice with long-term memory guarantees in the sense of Bousquet and Warmuth, in which we learn a small pool of experts. We consider relative entropy projection-based algorithms, giving a linear-time algorithm that improves on the best known regret bound. We show that such projection updates may be advantageous over previous "weight-sharing" approaches when weight updates come with implicit costs such as in portfolio optimization. We give an algorithm to compute the relative entropy projection onto the simplex with non-uniform (lower) box constraints in linear time, which may be of independent interest. We finally extend the model of long-term memory by introducing a new model of adaptive long-term memory. Here the small pool is assumed to change over time, with the trial sequence being partitioned into epochs and a small pool associated with each epoch. We give an efficient linear-time regret-bounded algorithm for this setting and present results in the setting of contextual bandits

Decomposition of Gaussian processes, and factorization of positive definite kernels

We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $K$, vs for Gaussian process $V$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $K$ is the exact same as that which yield factorizations for $V$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems