Christino Tamon

On the Boosting Pruning Problem

Boosting is a powerful method for improving the predictive accuracy of classifiers. The ADABOOST algorithm of Freund and Schapire has been successfully applied to many domains [2, 10, 12] and the combination of ADABOOST with the C4.5 decision tree algorithm has been called the best off-the-shelf learning algorithm in practice. Unfortunately, in some applications, the number of decision trees required by ADABOOST to achieve a reasonable accuracy is enormously large and hence is very space consuming. This problem was first studied by Margineantu and Dietterich [7], where they proposed an empirical method called Kappa pruning to prune the boosting ensemble of decision trees. The Kappa method did this without sacrificing too much accuracy. In this work-in-progress we propose a potential improvement to the Kappa pruning method and also study the boosting pruning problem from a theoretical perspective. We point out that the boosting pruning problem is intractable even to approximate. Finally, we suggest a margin-based theoretical heuristic for this problem.

Solving Multilinear Systems via Tensor Inversion

Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree

More efficient PAC-learning of DNF with membership queries under the uniform distribution

n

One-Dimensional Continuous-Time Quantum Walks

. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are satisfied. Moreover, within this group structure framework, multilinear systems are derived and solved for problems of high-dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense. These are cases when there is an odd number of modes or when each mode has distinct dimension. Numerically we solve multilinear systems using iterative techniques, namely, biconjugate gradient and Jacobi methods.

On learning width two branching programs

An efficient algorithm exists for learning disjunctive normal form (DNF) expressions in the uniformdistribution PAC learning model with membership queries [J97], but in practice the algorithm can only be applied to small problems. We present several modifications to the algorithm that substantially improve its asymptotic efficiency. First, we show how to significantly improve the time and sample complexity of a key subprogram, resulting in similar improvements in the bounds on the overall DNF algorithm. We also apply known methods to convert the resulting algorithm to an attribute efficient algorithm. Furthermore, we develop techniques for lower bounding the sample size required for PAC learning with membership queries under a fixed distribution and apply this technique to the uniform-distribution DNF learning problem. Finally, we present a learning algorithm for DNF that is attribute efficient in its use of random bits.

Perfect state transfer, graph products and equitable partitions

AbstractWe survey the equations of continuous-time quantum walks on simple one-dimensional lattices, which include the finite and infinite lines and the finite cycle, and compare them with the classical continuous-time Markov chains. The focus of our expository article is on analyzing these processes using the Laplace transform on the stochastic recurrences. The resulting time evolution equations, classical vs. quantum, are strikingly similar in form, although dissimilar in behavior. We also provide comparisons with analyses performed using spectral methods. PACS: 03.67.Lx

Uniform-distribution attribute noise learnability

Abstract We prove that strict width two branching programs or SW 2 (which are width two branching programs with exactly two sinks, as defined by Borodin et al. (1986)) are properly PAC learnable under any distribution. We also observe that PAC learning monotone width two branching programs (which are width two branching programs with exactly one rejecting sink) is as hard as learning DNF formulae. This work refines both the positive and negative results of the paper by Ergun et al. (1995) and answers one of the open questions in that paper.

On learning branching programs and small depth circuits

We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products (which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If

Oracles and queries that are sufficient for exact learning (extended abstract)

G

QUANTUM PERFECT STATE TRANSFER ON WEIGHTED JOIN GRAPHS

is a graph with perfect state transfer at time