In recommendation systems, probabilistic matrix factorization (PMF) is a state-of-the-art collaborative filtering method by determining the latent features to represent users and items. However, two major issues limiting the usefulness of PMF are the sparsity problem and long-tail distribution. Sparsity refers to the situation that the observed rating data are sparse, which results in that only part of latent features are informative for describing each item/user. Long tail distribution implies that a large fraction of items have few ratings. In this work, we propose a sparse probabilistic matrix factorization method (SPMF) by utilizing a Laplacian distribution to model the item/user factor vector. Laplacian distribution has ability to generate sparse coding, which is beneficial for SPMF to distinguish the relevant and irrelevant latent features with respect to each item/user. Meanwhile, the tails in Laplacian distribution are comparatively heavy, which is rewarding for SPMF to recommend the tail items. Furthermore, a distributed Gibbs sampling algorithm is developed to efficiently train the proposed sparse probabilistic model. A series of experiments on Netfilix and Movielens datasets have been conducted to demonstrate that SPMF outperforms the existing PMF and its extended version Bayesian PMF (BPMF), especially for the recommendation of tail items.

Conditional Random Fields (CRF) have been widely used in a variety of computer vision tasks. Conventional CRFs typically define edges on neighboring image pixels, resulting in a sparse graph such that efficient inference can be performed. However, these CRFs fail to model long-range contextual relationships. Fully-connected CRFs have thus been proposed. While there are efficient approximate inference methods for such CRFs, usually they are sensitive to initialization and make strong assumptions. In this work, we develop an efficient, yet general algorithm for inference on fully-connected CRFs. The algorithm is based on a scalable SDP algorithm and the low- rank approximation of the similarity/kernel matrix. The core of the proposed algorithm is a tailored quasi-Newton method that takes advantage of the low-rank matrix approximation when solving the specialized SDP dual problem. Experiments demonstrate that our method can be applied on fully-connected CRFs that cannot be solved previously, such as pixel-level image co-segmentation.

Cascade classifiers are widely used in real-time object detection. Different from conventional classifiers that are designed for a low overall classification error rate, a classifier in each node of the cascade is required to achieve an extremely high detection rate and moderate false positive rate. Although there are a few reported methods addressing this requirement in the context of object detection, there is no principled feature selection method that explicitly takes into account this asymmetric node learning objective. We provide such an algorithm here. We show that a special case of the biased minimax probability machine has the same formulation as the linear asymmetric classifier (LAC) of Wu et al (2005). We then design a new boosting algorithm that directly optimizes the cost function of LAC. The resulting totally-corrective boosting algorithm is implemented by the column generation technique in convex optimization. Experimental results on object detection verify the effectiveness of the proposed boosting algorithm as a node classifier in cascade object detection, and show performance better than that of the current state-of-the-art.