Support vector machines (SVMs) are successful modeling and prediction tools with a variety of applications. Previous work has demonstrated the superiority of the SVMs in dealing with the high dimensional, low sample size problems. However, the numerical difficulties of the SVMs will become severe with the increase of the sample size. Although there exist many … Read more
Data are often represented as graphs. Many common tasks in data science are based on distances between entities. While some data science methodologies natively take graphs as their input, there are many more that take their input in vectorial form. In this survey we discuss the fundamental problem of mapping graphs to vectors, and its … Read more
The simulations indicate that the existing hard thresholding technique independent of the residual function may cause a dramatic increase or numerical oscillation of the residual. This inherit drawback of the hard thresholding renders the traditional thresholding algorithms unstable and thus generally inefficient for solving practical sparse optimization problems. How to overcome this weakness and develop … Read more
Previous work regarding low-rank matrix recovery has concentrated on the scenarios in which the matrix is noise-free and the measurements are corrupted by noise. However, in practical application, the matrix itself is usually perturbed by random noise preceding to measurement. This paper concisely investigates this scenario and evidences that, for most measurement schemes utilized in … Read more
This paper studies high-order evaluation complexity for partially separable convexly-constrained optimization involving non-Lipschitzian group sparsity terms in a nonconvex objective function. We propose a partially separable adaptive regularization algorithm using a $p$-th order Taylor model and show that the algorithm can produce an (epsilon,delta)-approximate q-th-order stationary point in at most O(epsilon^{-(p+1)/(p-q+1)}) evaluations of the objective … Read more
Sufficient conditions are provided for a deterministic algorithm for estimating an L1-norm best-fit one-dimensional subspace. To prove the conditions are sufficient, fundamental properties of the L1-norm projection of a point onto a one-dimensional subspace are derived. Also, an equivalence is established between the algorithm, which involves the calculation of several weighted medians, and independently-derived algorithms … Read more
In this paper, we consider the recovery of group sparse signals corrupted by impulsive noise. In some recent literature, researchers have utilized stable data fitting models, like $l_1$-norm, Huber penalty function and Lorentzian-norm, to substitute the $l_2$-norm data fidelity model to obtain more robust performance. In this paper, a stable model is developed, which exploits … Read more
We treat the Feature Selection problem in the Support Vector Machine (SVM) framework by adopting an optimization model based on use of the $\ell_0$ pseudo–norm. The objective is to control the number of non-zero components of normal vector to the separating hyperplane, while maintaining satisfactory classification accuracy. In our model the polyhedral norm $\|.\|_{[k]}$, intermediate … Read more
A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that is constraints whose evaluation and enforcement has negligible cost) under the assumption that the derivative of highest degree is beta-H\”{o}lder continuous. It features a … Read more
We provide sharp worst-case evaluation complexity bounds for nonconvex minimization problems with general inexpensive constraints, i.e.\ problems where the cost of evaluating/enforcing of the (possibly nonconvex or even disconnected) constraints, if any, is negligible compared to that of evaluating the objective function. These bounds unify, extend or improve all known upper and lower complexity bounds … Read more