Refining Dual Spectral Sparsity in Transformed Tensor Singular Values

The Tubal Nuclear Norm (TNN), derived from the tensor Singular Value Decomposition (t-SVD), is a widely used low-rank modeling tool that promotes sparsity of frequency-domain singular values. However, as a direct extension of the matrix nuclear norm, TNN applies a uniform element-wise penalty to transformed singular values, without explicitly distinguishing sparsity across frequency components from low-rankness within each component. This can be restrictive for real-world tensor data that exhibit multi-level spectral structures, where spectral energy is concentrated in a subset of frequency components while active components remain low-rank. To overcome this limitation, we propose the tensor $\ell_p$-Schatten-$q$ quasi-norm ($p,q\in(0,1]$), which enables explicit control of dual spectral sparsity by jointly regularizing inter-frequency sparsity and intra-frequency low-rankness. This formulation includes TNN as a special case and subsumes several existing tensor regularizers by coupling global frequency sparsity with local spectral low-rankness, yielding a more flexible modeling principle. We establish minimax error bounds under the proposed dual spectral sparsity model, develop a reweighted optimization algorithm for the resulting nonconvex problem, and demonstrate its effectiveness and robustness on noisy and Poisson tensor completion as well as image clustering tasks.

Modern AI systems often organize data as large multi-dimensional arrays, called tensors. Examples include videos, hyperspectral images, medical scans, and even the internal memory states used in large language models. A common strategy is to compress these tensors by assuming that their information is concentrated in a simpler hidden structure. Our work studies a new type of structure that appears in transformed tensor representations. We observe that useful information is often concentrated in only a small number of transformed components, while each important component itself still contains strong internal redundancy. To model this behavior, we introduce a new mathematical regularization framework that jointly captures these two effects. We also develop a practical optimization algorithm and provide theoretical results describing the statistical difficulty of recovering tensors with this type of structure. Experiments on image recovery, scientific imaging, and clustering tasks show that the proposed approach can recover tensor data more accurately than several widely used existing methods, particularly when observations are limited or noisy.