## Orals 4D : Statistical Methods and Learning II

• Unsupervised One-Class Learning for Automatic Outlier Removal Authors: Wei Liu, Gang Hua, John R. Smith
Outliers are pervasive in many computer vision and pattern recognition problems. Automatically eliminating outliers scattering among practical data collections becomes increasingly important, especially for Internet inspired vision applications. In this paper, we propose a novel one-class learning approach which is robust to contamination of input training data and able to discover the outliers that corrupt one class of data source. Our approach works under a fully unsupervised manner, differing from traditional one-class learning supervised by known positive labels. By design, our approach optimizes a kernel-based max-margin objective which jointly learns a large margin one-class classifier and a soft label assignment for inliers and outliers. An alternating optimization algorithm is then designed to iteratively refine the classifier and the labeling, achieving a provably convergent solution in only a few iterations. Extensive experiments conducted on four image datasets in the presence of artificial and real-world outliers demonstrate that the proposed approach is considerably superior to the state-of-the-arts in obliterating outliers from contaminated one class of images, exhibiting strong robustness at a high outlier proportion up to 60%.
• Novel Methods for Multilinear Data Completion and De-noising Based on Tensor-SVD Authors: Zemin Zhang, Gregory Ely, Shuchin Aeron, Ning Hao, Misha Kilmer
In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4].
• Optimizing Over Radial Kernels on Compact Manifolds Authors: Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash Harandi
We tackle the problem of optimizing over all possible positive definite radial kernels on Riemannian manifolds for classification. Kernel methods on Riemannian manifolds have recently become increasingly popular in computer vision. However, the number of known positive definite kernels on manifolds remain very limited. Furthermore, most kernels typically depend on at least one parameter that needs to be tuned for the problem at hand. A poor choice of kernel, or of parameter value, may yield significant performance drop-off. Here, we show that positive definite radial kernels on the unit $n$-sphere, the Grassmann manifold and Kendall's shape manifold can be expressed in a simple form whose parameters can be automatically optimized within a support vector machine framework. We demonstrate the benefits of our kernel learning algorithm on object, face, action and shape recognition.
• Grassmann Averages for Scalable Robust PCA Authors: Søren Hauberg, Aasa Feragen, Michael J. Black
As the collection of large datasets becomes increasingly automated, the occurrence of outliers will increase -- "big data" implies "big outliers''. While principal component analysis (PCA) is often used to reduce the size of data, and scalable solutions exist, it is well-known that outliers can arbitrarily corrupt the results. Unfortunately, state-of-the-art approaches for robust PCA do not scale beyond small-to-medium sized datasets. To address this, we introduce the Grassmann Average (GA), which expresses dimensionality reduction as an average of the subspaces spanned by the data. Because averages can be efficiently computed, we immediately gain scalability. GA is inherently more robust than PCA, but we show that they coincide for Gaussian data. We exploit that averages can be made robust to formulate the Robust Grassmann Average (RGA) as a form of robust PCA. Robustness can be with respect to vectors (subspaces) or elements of vectors; we focus on the latter and use a trimmed average. The resulting Trimmed Grassmann Average (TGA) is particularly appropriate for computer vision because it is robust to pixel outliers. The algorithm has low computational complexity and minimal memory requirements, making it scalable to big noisy data.'' We demonstrate TGA for background modeling, video restoration, and shadow removal. We show scalability by performing robust PCA on the entire Star Wars IV movie.
• Robust Subspace Segmentation with Block-diagonal Prior Authors: Jiashi Feng, Zhouchen Lin, Huan Xu, Shuicheng Yan
The subspace segmentation problem is addressed in this paper by effectively constructing an \emph{exactly block-diagonal} sample affinity matrix. The block-diagonal structure is heavily desired for accurate sample clustering but is rather difficult to obtain. Most current state-of-the-art subspace segmentation methods (such as SSC~\cite{elhamifar2012sparse} and LRR~\cite{LiuLYSYM13}) resort to alternative structural priors (such as sparseness and low-rankness) to construct the affinity matrix. In this work, we directly pursue the block-diagonal structure by proposing a graph Laplacian constraint based formulation, and then develop an efficient stochastic subgradient algorithm for optimization. Moreover, two new subspace segmentation methods, the block-diagonal SSC and LRR, are devised in this work. To the best of our knowledge, this is the first research attempt to explicitly pursue such a block-diagonal structure. Extensive experiments on face clustering, motion segmentation and graph construction for semi-supervised learning clearly demonstrate the superiority of our novelly proposed subspace segmentation methods.