Stable rank of C(X)⋊Γ\mathrm{C}(X)\rtimes\Gamma

It is shown that, for an arbitrary free and minimal $\mathbb Z^n$-action on a compact Hausdorff space $X$, the crossed product C*-algebra $\mathrm{C}(X)\rtimes\mathbb Z^n$ always has stable rank one, i.e., invertible elements are dense. This generalizes a result of Alboiu and Lutley on $\mathbb Z$-actions. In fact, for any free and minimal topological dynamical system $(X, \Gamma)$, where $\Gamma$ is a countable discrete amenable group, if it has the uniform Rokhlin property and Cuntz comparison of open sets, then the crossed product C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ has stable rank one. Moreover, in this case, the C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ absorbs the Jiang-Su algebra tensorially if, and only if, it has strict comparison of positive elements

Oracle Based Active Set Algorithm for Scalable Elastic Net Subspace Clustering

State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with $\ell_1$, $\ell_2$ or nuclear norms. $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. $\ell_2$ and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed $\ell_1$, $\ell_2$ and nuclear norm regularizations offer a balance between the subspace-preserving and connectedness properties, but this comes at the cost of increased computational complexity. This paper studies the geometry of the elastic net regularizer (a mixture of the $\ell_1$ and $\ell_2$ norms) and uses it to derive a provably correct and scalable active set method for finding the optimal coefficients. Our geometric analysis also provides a theoretical justification and a geometric interpretation for the balance between the connectedness (due to $\ell_2$ regularization) and subspace-preserving (due to $\ell_1$ regularization) properties for elastic net subspace clustering. Our experiments show that the proposed active set method not only achieves state-of-the-art clustering performance, but also efficiently handles large-scale datasets.Comment: 15 pages, 6 figures, accepted to CVPR 2016 for oral presentatio

Cluster-guided Asymmetric Contrastive Learning for Unsupervised Person Re-Identification

Unsupervised person re-identification (Re-ID) aims to match pedestrian images from different camera views in unsupervised setting. Existing methods for unsupervised person Re-ID are usually built upon the pseudo labels from clustering. However, the quality of clustering depends heavily on the quality of the learned features, which are overwhelmingly dominated by the colors in images especially in the unsupervised setting. In this paper, we propose a Cluster-guided Asymmetric Contrastive Learning (CACL) approach for unsupervised person Re-ID, in which cluster structure is leveraged to guide the feature learning in a properly designed asymmetric contrastive learning framework. To be specific, we propose a novel cluster-level contrastive loss to help the siamese network effectively mine the invariance in feature learning with respect to the cluster structure within and between different data augmentation views, respectively. Extensive experiments conducted on three benchmark datasets demonstrate superior performance of our proposal