Title: Refining Dual Spectral Sparsity in Transformed Tensor Singular Values
Primary area: other topics in machine learning (i.e., none of the above)
Keywords: tensor recovery, tensor nuclear norm, low-rank decomposition, t-SVD

Abstract: The Tensor Nuclear Norm (TNN), derived from the tensor Singular Value Decomposition, is a central low-rank modeling tool that enforces *element-wise* sparsity on frequency-domain singular values and has been widely used in multi-way data recovery for machine learning and computer vision. However, as a direct extension of the matrix nuclear norm, it inherits the assumption of *single-level spectral sparsity*, which strictly limits its ability to capture the *multi-level spectral structures* inherent in real-world data—particularly the coexistence of low-rankness within and sparsity across frequency components. To address this, we propose the tensor $\ell_p$-Schatten-$q$ quasi-norm ($p, q \in (0,1]$), a new metric that enables *dual spectral sparsity control* by jointly regularizing both types of structure. While this formulation generalizes TNN and unifies existing methods such as the tensor Schatten-$p$ norm and tensor average rank, it differs fundamentally in modeling principle by coupling global frequency sparsity with local spectral low-rankness. This coupling introduces significant theoretical and algorithmic challenges.  To tackle these challenges, we provide a theoretical characterization by establishing the first minimax error bounds under dual spectral sparsity, and an algorithmic solution by designing an efficient reweighted optimization scheme tailored to the resulting nonconvex structure. Numerical experiments demonstrate the effectiveness of our method in modeling complex multi-way data.