Statistical and Computational Efficiency for Smooth Tensor Estimation with Unknown Permutations
研究了在存在未知排列时结构化张量去噪问题，提出一种平滑张量模型族，并证明分块多项式约束最小二乘估计达到极小化最优误差界，同时给出高效的多项式时间Borda计数算法。
We consider the problem of structured tensor denoising in the presence of unknown permutations. Such data problems arise commonly in recommendation systems, neuroimaging, community detection, and multiway comparison applications. Here, we develop a general family of smooth tensor models up to arbitrary index permutations; the model incorporates the popular tensor block models and Lipschitz hypergraphon models as special cases. We show that a constrained least-squares estimator in the block-wise polynomial family achieves the minimax error bound. A phase transition phenomenon is revealed with respect to the smoothness threshold needed for optimal recovery. In particular, we find that a polynomial of degree up to (m−2)(m+1)/2 is sufficient for accurate recovery of order-m tensors, whereas higher degrees exhibit no further benefits. This phenomenon reveals the intrinsic distinction for smooth tensor estimation problems with and without unknown permutations. Furthermore, we provide an efficient polynomial-time Borda count algorithm that provably achieves the optimal rate under monotonicity assumptions. The efficacy of our procedure is demonstrated through both simulations and Chicago crime data analysis. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.