Fast Low-Rank Matrix Learning with Nonconvex Regularization

Low-rank modeling has a lot of important applications in machine learning, computer vision and social network analysis. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better recovery performance. However, the resultant optimization problem is much more challenging. A very recent state-of-the-art is based on the proximal gradient algorithm. However, it requires an expensive full SVD in each proximal step. In this paper, we show that for many commonly-used nonconvex low-rank regularizers, a cutoff can be derived to automatically threshold the singular values obtained from the proximal operator. This allows the use of power method to approximate the SVD efficiently. Besides, the proximal operator can be reduced to that of a much smaller matrix projected onto this leading subspace. Convergence, with a rate of O(1/T) where T is the number of iterations, can be guaranteed. Extensive experiments are performed on matrix completion and robust principal component analysis. The proposed method achieves significant speedup over the state-of-the-art. Moreover, the matrix solution obtained is more accurate and has a lower rank than that of the traditional nuclear norm regularizer.Comment: Long version of conference paper appeared ICDM 201

Socializing the Semantic Gap: A Comparative Survey on Image Tag Assignment, Refinement and Retrieval

Where previous reviews on content-based image retrieval emphasize on what can be seen in an image to bridge the semantic gap, this survey considers what people tag about an image. A comprehensive treatise of three closely linked problems, i.e., image tag assignment, refinement, and tag-based image retrieval is presented. While existing works vary in terms of their targeted tasks and methodology, they rely on the key functionality of tag relevance, i.e. estimating the relevance of a specific tag with respect to the visual content of a given image and its social context. By analyzing what information a specific method exploits to construct its tag relevance function and how such information is exploited, this paper introduces a taxonomy to structure the growing literature, understand the ingredients of the main works, clarify their connections and difference, and recognize their merits and limitations. For a head-to-head comparison between the state-of-the-art, a new experimental protocol is presented, with training sets containing 10k, 100k and 1m images and an evaluation on three test sets, contributed by various research groups. Eleven representative works are implemented and evaluated. Putting all this together, the survey aims to provide an overview of the past and foster progress for the near future.Comment: to appear in ACM Computing Survey

Spectral methods for multimodal data analysis

Spectral methods have proven themselves as an important and versatile tool in a wide range of problems in the fields of computer graphics, machine learning, pattern recognition, and computer vision, where many important problems boil down to constructing a Laplacian operator and finding a few of its eigenvalues and eigenfunctions. Classical examples include the computation of diffusion distances on manifolds in computer graphics, Laplacian eigenmaps, and spectral clustering in machine learning. In many cases, one has to deal with multiple data spaces simultaneously. For example, clustering multimedia data in machine learning applications involves various modalities or ``views'' (e.g., text and images), and finding correspondence between shapes in computer graphics problems is an operation performed between two or more modalities. In this thesis, we develop a generalization of spectral methods to deal with multiple data spaces and apply them to problems from the domains of computer graphics, machine learning, and image processing. Our main construction is based on simultaneous diagonalization of Laplacian operators. We present an efficient numerical technique for computing joint approximate eigenvectors of two or more Laplacians in challenging noisy scenarios, which also appears to be the first general non-smooth manifold optimization method. Finally, we use the relation between joint approximate diagonalizability and approximate commutativity of operators to define a structural similarity measure for images. We use this measure to perform structure-preserving color manipulations of a given image

Content-based image analysis with applications to the multifunction printer imaging pipeline and image databases

Image understanding is one of the most important topics for various applications. Most of image understanding studies focus on content-based approach while some others also rely on meta data of images. Image understanding includes several sub-topics such as classification, segmentation, retrieval and automatic annotation etc., which are heavily studied recently. This thesis proposes several new methods and algorithms for image classification, retrieval and automatic tag generation. The proposed algorithms have been tested and verified in multiple platforms. For image classification, our proposed method can complete classification in real-time under hardware constraints of all-in-one printer and adaptively improve itself by online learning. Another image understanding engine includes both classification and image quality analysis is designed to solve the optimal compression problem of printing system. Our proposed image retrieval algorithm can be applied to either PC or mobile device to improve the hybrid learning experience. We also develop a new matrix factorization algorithm to better recover the image meta data (tag). The proposed algorithm outperforms other existing matrix factorization methods