Computational minimax optimality of tensor MA-PCA and CFA-PCA

Prove that, for order-q tensor observations modeled as X_i = \tilde{\ell}_i \mathscr{U} + \mathcal{Z}_i with a block-structured mean tensor \mathscr{U} and sub-Gaussian noise \mathcal{Z}_i, the tensor moving average PCA (MA-PCA) procedure and the tensor cross-block feature aggregation PCA (CFA-PCA) procedure achieve the computational minimax lower bounds for clustering and signal recovery in the dense and sparse tensor block signal regimes, respectively; equivalently, demonstrate that these two tensor procedures are computationally minimax optimal for the tensor block signal class under which statistical and computational minimax lower bounds have been established.

Background

The paper extends the statistical and computational minimax lower bounds (SMLBs and CMLBs) from vector-valued to tensor-valued data (see Theorem referenced in the Supplementary Material). For vector data, the proposed MA-PCA (dense regime) and CFA-PCA (sparse regime) are shown to achieve the CMLBs, establishing computational minimax optimality.

In the tensor setting, the authors outline tensor versions of MA-PCA and CFA-PCA but do not provide a proof of optimality. They explicitly conjecture that these tensor procedures also attain the CMLBs for clustering and signal recovery in their respective sparsity regimes. Establishing this would close the theoretical gap by proving computational minimax optimality for the tensor case, paralleling the vector results.