Abstract — A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser and Donoho. These measures include the p-norm-like (`(p 1)) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation for the gradient and involves successive relaxation of the Lagrangian necessary condition. This yields algorithms that are intimately related to the Affine Scaling Transformation (AST) based methods commonly employed by the interior point approach to nonlinear optimization. The algorithms minimizing the `(p 1) diversity measures are equivalent to a recently developed class of algorithms called FOCal Underdetermined System Solver (FOCUSS). The general nature of the methodology provides a systematic approach for deriving this class of algorithms and a natural mechanism for extending them. It also facilitates a better understanding of the convergence behavior and a strengthening of the convergence results. The Gaussian entropy minimization algorithm is shown to be equivalent to a well-behaved p =0norm-like optimization algorithm. Computer experiments demonstrate that the p-norm-like and the Gaussian entropy algorithms perform well, converging to sparse solutions. The Shannon entropy algorithm produces solutions that are concentrated but are shown to not converge to a fully sparse solution. I.

We consider the AdaBoost procedure for boosting weak learners. In AdaBoost, a key step is choosing a new distribution on the training examples based on the old distribution and the mistakes made by the present weak hypothesis. We show how AdaBoost 's choice of the new distribution can be seen as an approximate solution to the following problem: Find a new distribution that is closest to the old distribution subject to the constraint that the new distribution is orthogonal to the vector of mistakes of the current weak hypothesis. The distance (or divergence) between distributions is measured by the relative entropy. Alternatively, we could say that AdaBoost approximately projects the distribution vector onto a hyperplane dened by the mistake vector. We show that this new view of AdaBoost as an entropy projection is dual to the usual view of AdaBoost as minimizing the normalization factors of the updated distributions.

This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems whichwe call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use these results to explore the class GLP and show new applications to geometric optimization, and also to prove Helly theorems. In general, a GLP is a set...

We consider the speed scaling problem of minimizing the average response time of a collection of dynamically released jobs subject to a constraint A on energy used. We propose an algorithmic approach in which an energy optimal schedule is computed for a huge A, and then the energy optimal schedule is maintained as A decreases. We show that this approach yields an efficient algorithm for equi-work jobs. We note that the energy optimal schedule has the surprising feature that the job speeds are not monotone functions of the available energy. We then explain why this algorithmic approach is problematic for arbitrary work jobs. Finally, we explain how to use the algorithm for equi-work jobs to obtain an algorithm for arbitrary work jobs that is O(1)-approximate with respect to average response time, given an additional factor of (1 + ffl)energy.

This paper explores a method of manipulating a planar rigid body on a conveyor belt using a robot with just one joint. This approach has the potential of offering a simple and flexible method for feeding parts in industrial automation applications. In this paper we outline our approach, develop some of the theoretical properties, present a planner for the robot, and describe an initial implementation. 1

. This paper explores a method of manipulating a planar rigid part on a conveyor belt using a robot with just one joint. This approach has the potential of offering a simple and flexible method for feeding parts in industrial automation applications. In this paper we develop a model of this system and of a variation which requires no sensing. We have been able to characterize these systems and to prove that they can serve as parts feeding devices for planar polygonal parts. We present the planners for these systems and describe our implementations. Key Words. Robotics, Manipulation, Mechanics, Planning, Minimalism, Automation, Manufacturing, Parts feeding. 1. Introduction. The most straightforward approach to planar manipulation is to use a rigid grasp and a robot with at least three joints, corresponding to the three motion freedoms of a planar rigid part, but three joints are not really necessary to manipulate a part in the plane. In this paper we achieve effective control of all t...

A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, convert the original problem into primal and relaxed dual subproblems that provide valid upper and lower bounds respectively on the global optimum. Theoretical properties are presented which allow for a rigorous solution of the relaxed dual problem. Proofs of ffl-finite convergence and ffl-global optimality are provided. The approach is shown to be particularly suited to (a) quadratic programming problems, (b) quadratically constrained problems, and (c) unconstrained and constrained optimization of polynomial and rational polynomial functions. The theoretical approach is illustrated through a few example problems. Finally, some further developments in the approach are briefly discussed.

The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in high-dimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty. We establish a rate of con-vergence in the Frobenius norm as both data dimension p and sample size n are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlation-based version of the method exhibits better rates in the operator norm. The estimator is required to be positive definite, but we avoid having to use semi-definite programming by re-parameterizing the objective function

A conic linear system is a system of the form P (d) : find x that solves b \Gamma Ax 2 C Y ; x 2 CX ; where CX and C Y are closed convex cones, and the data for the system is d = (A; b). This system is"well-posed" to the extent that (small) changes in the data (A; b) do not alter the status of the system (the system remains solvable or not). Renegar defined the "distance to ill-posedness," ae(d), to be the smallest change in the data \Deltad = (\DeltaA; \Deltab) for which the system P (d + \Deltad) is "ill-posed," i.e., d + \Deltad is in the intersection of the closure of feasible and infeasible instances d 0 = (A 0 ; b 0 ) of P (\Delta). Renegar also defined the "condition measure" of the data instance d as C(d) := kdk=ae(d), and showed that this measure is a natural extension of the familiar condition measure associated with systems of linear equations. This study presents two categories of results related to ae(d), the distance to ill-posedness, and C(d), the condition me...

With technology scaling, the trend for high performance integrated circuits is towards ever higher operating frequency, lower power supply voltages and higher power dissipation.