On Graduated Optimization for Stochastic Non-Convex Problems

The graduated optimization approach, also known as the continuation method, is a popular heuristic to solving non-convex problems that has received renewed interest over the last decade. Despite its popularity, very little is known in terms of theoretical convergence analysis. In this paper we describe a new first-order algorithm based on graduated optimiza- tion and analyze its performance. We characterize a parameterized family of non- convex functions for which this algorithm provably converges to a global optimum. In particular, we prove that the algorithm converges to an {\epsilon}-approximate solution within O(1/\epsilon^2) gradient-based steps. We extend our algorithm and analysis to the setting of stochastic non-convex optimization with noisy gradient feedback, attaining the same convergence rate. Additionally, we discuss the setting of zero-order optimization, and devise a a variant of our algorithm which converges at rate of O(d^2/\epsilon^4).Comment: 17 page

Orthonormal Expansion l1-Minimization Algorithms for Compressed Sensing

Compressed sensing aims at reconstructing sparse signals from significantly reduced number of samples, and a popular reconstruction approach is $\ell_1$-norm minimization. In this correspondence, a method called orthonormal expansion is presented to reformulate the basis pursuit problem for noiseless compressed sensing. Two algorithms are proposed based on convex optimization: one exactly solves the problem and the other is a relaxed version of the first one. The latter can be considered as a modified iterative soft thresholding algorithm and is easy to implement. Numerical simulation shows that, in dealing with noise-free measurements of sparse signals, the relaxed version is accurate, fast and competitive to the recent state-of-the-art algorithms. Its practical application is demonstrated in a more general case where signals of interest are approximately sparse and measurements are contaminated with noise.Comment: 7 pages, 2 figures, 1 tabl

Generalized Kernel-based Visual Tracking

In this work we generalize the plain MS trackers and attempt to overcome standard mean shift trackers' two limitations. It is well known that modeling and maintaining a representation of a target object is an important component of a successful visual tracker. However, little work has been done on building a robust template model for kernel-based MS tracking. In contrast to building a template from a single frame, we train a robust object representation model from a large amount of data. Tracking is viewed as a binary classification problem, and a discriminative classification rule is learned to distinguish between the object and background. We adopt a support vector machine (SVM) for training. The tracker is then implemented by maximizing the classification score. An iterative optimization scheme very similar to MS is derived for this purpose.Comment: 12 page