Subspace Kernel Learning on Tensor Sequences

Uncertainty estimation is essential for robust decision-making in the presence of ambiguous or out-of-distribution inputs. Gaussian Processes (GPs) are classical kernel-based models that offer principled uncertainty quantification and perform well on small- to medium-scale datasets. Alternatively, formulating the weight space learning problem under tensor network assumptions yields scalable tensor network kernel machines. However, these assumptions break Gaussianity, complicating standard probabilistic inference. This raises a fundamental question: how can tensor network kernel machines provide principled uncertainty estimates? We propose a novel Bayesian Tensor Network Kernel Machine (LA-TNKM) that employs a (linearized) Laplace approximation for Bayesian inference. A comprehensive set of numerical experiments shows that the proposed method consistently matches or surpasses Gaussian Processes and Bayesian Neural Networks (BNNs) across diverse UCI regression benchmarks, highlighting both its effectiveness and practical relevance.

Recently, tensor decompositions have attracted increasing attention. Fundamentally, different interactions among factors induce distinct tensor decomposition structures (i.e., tensor decomposition). Identifying an appropriate interaction-induced tensor decomposition structure for given data is a fundamental yet challenging problem in tensor modeling, and remains largely under-explored. Existing tensor decomposition structure search methods are typically restricted to a predefined interaction family, such as tensor contraction. To address this problem, we suggest a tensor decomposition structure search framework (I-TSS) from an interaction perspective that can identify either a single structure beyond a predefined interaction family or a mixture of structures involving heterogeneous interaction families. Specifically, we first systematically review existing tensor decomposition structures from an interaction perspective and construct a heterogeneous interaction-induced candidate structure set. Based on it, we propose a unified energy-based rank estimation scheme for heterogeneous interaction-induced candidate structure set. We then introduce the top-$k$ gating mechanism with learnable gating scores to dynamically select a suitable single-candidate structure or combine suitable multi-candidate structures, thereby identifying the tensor decomposition structure in a data-adaptive manner. Theoretically, we derive an approximation error bound for I-TSS, thereby establishing its approximation capability. Extensive experiments on synthetic and real-world datasets demonstrate that I-TSS consistently outperforms state-of-the-art tensor decomposition methods.

Tensorial multi-view clustering (TMC) has achieved strong performance due to its ability to capture high-order correlations across multiple views. Most existing t-SVD-based TMC frameworks apply the Fast Fourier Transform (FFT) along the sample mode to impose frequency-domain low-rank constraints. However, we reveal that this widely adopted design critically relies on an implicit ``periodicity assumption'' induced by the sample arrangement. When samples are ordered by class, neighboring indices tend to be semantically similar, creating artificial local continuity along the sample mode and a favorable spectral structure for FFT-based low-rank regularization. Once this ordering is removed by random permutation, existing t-SVD-based TMC methods suffer severe performance degradation. This strong sensitivity to class ordering conflicts with the permutation-invariant nature of clustering and indicates that part of the reported performance may be attributed to a privileged sample arrangement rather than genuine high-order structure modeling. In this paper, we systematically investigate this phenomenon and its underlying algebraic and spectral mechanisms. To address this fundamental flaw, we further propose a graph-spectral low-rank tensor learning framework based on the Graph Fourier Transform (GFT), which replaces the fixed Fourier basis along the sample mode with a data-driven graph spectral basis, thereby capturing the intrinsic manifold structure without relying on a particular sample ordering. Moreover, we develop an anchor-based variant to address large-scale datasets efficiently. Extensive experiments on various benchmarks validate our findings and demonstrate the competitive or superior performance of the proposed methods compared with state-of-the-art TMC approaches.