Multi-view subspace clustering (MVSC) separates the data with multiple views into multiple clusters, and each cluster corresponds to one certain subspace. Existing tensor-based MVSC methods construct self-representation subspace coefficient matrices of all views as a tensor, and introduce the tensor nuclear norm (TNN) to capture the complementary information hidden in different views. The key assumption is that the data samples of each subspace must be sufficient for subspace representation. This work proposes a nonconvex latent transformed low-rank tensor representation framework for MVSC. To deal with the insufficient sample problem, we study the latent low-rank representation in the multi-view case to supplement underlying observed samples. Moreover, we propose to use data-driven transformed TNN (TTNN), resulting from the intrinsic structure of multi-view samples, to preserve the consensus and complementary information in the transformed domain. Meanwhile, the proposed unified nonconvex low-rank tensor representation framework can better learn the high correlation among different views. To resolve the proposed nonconvex optimization model, we propose an effective algorithm under the framework of the alternating direction method of multipliers and theoretically prove that the iteration sequences converge to the critical point. Experiments on various datasets showcase outstanding performance.

Clustering is one of the most fundamental problems in machine learning and data mining, and many algorithms have been proposed in the past decades. Among them, subspace clustering and spectral clustering are the most famous approaches. In this paper, we provide an explanation for subspace clustering and spectral clustering from the perspective of optimal transport. Optimal transport studies how to move samples from one distribution to another distribution with minimal transport cost, and has shown a powerful ability to extract geometric information. By considering a self optimal transport model with only one group of samples, we observe that both subspace clustering and spectral clustering can be explained in the framework of optimal transport, and the optimal transport matrix bridges the spaces of features and spectral embeddings. Inspired by this connection, we propose a spectral optimal transport barycenter model, which learns spectral embeddings by solving a barycenter problem equipped with an optimal transport discrepancy and guidance of data. Based on our proposed model, we take advantage of optimal transport to exploit both feature and metric information involved in data for learning coupled spectral embeddings and affinity matrix in a unified model. We develop an alternating optimization algorithm to solve the resultant problems, and conduct experiments in different settings to evaluate the performance of our proposed methods.