An Explicit Convergence Rate for Nesterov's Method from SDP

The framework of Integral Quadratic Constraints (IQC) introduced by Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov's accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the convergence rate of NAM, and for arbitrary strongly convex functions it was shown numerically that IQC can improve bounds from Nesterov (2004). Unfortunately, an explicit analytic solution to the SDP was not provided. In this paper, we provide such an analytical solution, obtaining a new general and explicit upper bound on the convergence rate of NAM, which we further optimize over its parameters. To the best of our knowledge, this is the best, and explicit, upper bound on the convergence rate of NAM for strongly convex functions

Occlusion reasoning for multiple object visual tracking

Thesis (Ph.D.)--Boston UniversityOcclusion reasoning for visual object tracking in uncontrolled environments is a challenging problem. It becomes significantly more difficult when dense groups of indistinguishable objects are present in the scene that cause frequent inter-object interactions and occlusions. We present several practical solutions that tackle the inter-object occlusions for video surveillance applications. In particular, this thesis proposes three methods. First, we propose "reconstruction-tracking," an online multi-camera spatial-temporal data association method for tracking large groups of objects imaged with low resolution. As a variant of the well-known Multiple-Hypothesis-Tracker, our approach localizes the positions of objects in 3D space with possibly occluded observations from multiple camera views and performs temporal data association in 3D. Second, we develop "track linking," a class of offline batch processing algorithms for long-term occlusions, where the decision has to be made based on the observations from the entire tracking sequence. We construct a graph representation to characterize occlusion events and propose an efficient graph-based/combinatorial algorithm to resolve occlusions. Third, we propose a novel Bayesian framework where detection and data association are combined into a single module and solved jointly. Almost all traditional tracking systems address the detection and data association tasks separately in sequential order. Such a design implies that the output of the detector has to be reliable in order to make the data association work. Our framework takes advantage of the often complementary nature of the two subproblems, which not only avoids the error propagation issue from which traditional "detection-tracking approaches" suffer but also eschews common heuristics such as "nonmaximum suppression" of hypotheses by modeling the likelihood of the entire image. The thesis describes a substantial number of experiments, involving challenging, notably distinct simulated and real data, including infrared and visible-light data sets recorded ourselves or taken from data sets publicly available. In these videos, the number of objects ranges from a dozen to a hundred per frame in both monocular and multiple views. The experiments demonstrate that our approaches achieve results comparable to those of state-of-the-art approaches

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal