Ping Li

Yunfeng Cai, Ping Li

This paper considers the sparse generalized eigenvalue problem (SGEP), which aims to find the leading eigenvector with at most $k$ nonzero entries. SGEP naturally arises in many applications in machine learning, statistics, and scientific computing, for example, the sparse principal component analysis (SPCA), the sparse discriminant analysis (SDA), and the sparse canonical correlation analysis (SCCA). In this paper, we focus on the development of a three-stage algorithm named {\em inverse-free truncated Rayleigh-Ritz method} ({\em IFTRR}) to efficiently solve SGEP. In each iteration of IFTRR, only a small number of matrix-vector products is required. This makes IFTRR well-suited for large scale problems. Particularly, a new truncation strategy is proposed, which is able to find the support set of the leading eigenvector effectively. Theoretical results are developed to explain why IFTRR works well. Numerical simulations demonstrate the merits of IFTRR.

Jing Wang, Jie Shen, Ping Li

Object proposals are an ensemble of bounding boxes with high potential to contain objects. In order to determine a small set of proposals with a high recall, a common scheme is extracting multiple features followed by a ranking algorithm which however, incurs two major challenges: {\bf 1)} The ranking model often imposes pairwise constraints between each proposal, rendering the problem away from an efficient training/testing phase; {\bf 2)} Linear kernels are utilized due to the computational and memory bottleneck of training a kernelized model. In this paper, we remedy these two issues by suggesting a {\em kernelized partial ranking model}. In particular, we demonstrate that {\bf i)} our partial ranking model reduces the number of constraints from $O(n^2)$ to $O(nk)$ where $n$ is the number of all potential proposals for an image but we are only interested in top-$k$ of them that has the largest overlap with the ground truth; {\bf ii)} we permit non-linear kernels in our model which is often superior to the linear classifier in terms of accuracy. For the sake of mitigating the computational and memory issues, we introduce a consistent weighted sampling~(CWS) paradigm that approximates the non-linear kernel as well as facilitates an efficient learning. In fact, as we will show, training a linear CWS model amounts to learning a kernelized model. Extensive experiments demonstrate that equipped with the non-linear kernel and the partial ranking algorithm, recall at top-$k$ proposals can be substantially improved.