Yan Yang

Song Tang, Yan Yang, Zhiyuan Ma et al.

In the classic setting of unsupervised domain adaptation (UDA), the labeled source data are available in the training phase. However, in many real-world scenarios, owing to some reasons such as privacy protection and information security, the source data is inaccessible, and only a model trained on the source domain is available. This paper proposes a novel deep clustering method for this challenging task. Aiming at the dynamical clustering at feature-level, we introduce extra constraints hidden in the geometric structure between data to assist the process. Concretely, we propose a geometry-based constraint, named semantic consistency on the nearest neighborhood (SCNNH), and use it to encourage robust clustering. To reach this goal, we construct the nearest neighborhood for every target data and take it as the fundamental clustering unit by building our objective on the geometry. Also, we develop a more SCNNH-compliant structure with an additional semantic credibility constraint, named semantic hyper-nearest neighborhood (SHNNH). After that, we extend our method to this new geometry. Extensive experiments on three challenging UDA datasets indicate that our method achieves state-of-the-art results. The proposed method has significant improvement on all datasets (as we adopt SHNNH, the average accuracy increases by over 3.0% on the large-scaled dataset). Code is available at https://github.com/tntek/N2DCX.

Yan Yang, Bin Gao, Ya-xiang Yuan

Determinantal varieties -- the sets of bounded-rank matrices or tensors -- have attracted growing interest in low-rank optimization. The tangent cone to low-rank sets is widely studied and underpins a range of geometric methods. The second-order geometry, which encodes curvature information, is more intricate. In this work, we develop a unified framework to derive explicit formulas for both first- and second-order tangent sets to various low-rank sets, including low-rank matrices, tensors, symmetric matrices, and positive semidefinite matrices. The framework also accommodates the intersection of a low-rank set and another set satisfying mild assumptions, thereby yielding a tangent intersection rule. Through the lens of tangent sets, we establish a necessary and sufficient condition under which a nonsmooth problem and its smooth parameterization share equivalent second-order stationary points. Moreover, we exploit tangent sets to characterize optimality conditions for low-rank optimization and prove that verifying second-order optimality is NP-hard. In a separate line of analysis, we investigate variational geometry of the graph of the normal cone to matrix varieties, deriving the explicit Bouligand tangent cone, Fréchet and Mordukhovich normal cones to the graph. These results are further applied to develop optimality conditions for low-rank bilevel programs.