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.