Revisit CP Tensor Decomposition: Statistical Optimality and Fast Convergence

Canonical Polyadic (CP) tensor decomposition is a fundamental technique for analyzing high-dimensional tensor data. While the Alternating Least Squares (ALS) algorithm is widely used for computing CP decomposition due to its simplicity and empirical success, its theoretical foundation, particularly regarding statistical optimality and convergence behavior, remain underdeveloped, especially in noisy, non-orthogonal, and higher-rank settings. In this work, we revisit CP tensor decomposition from a statistical perspective and provide a comprehensive theoretical analysis of ALS under a signal-plus-noise model. We establish non-asymptotic, minimax-optimal error bounds for tensors of general order, dimensions, and rank, assuming suitable initialization. To enable such initialization, we propose Tucker-based Approximation with Simultaneous Diagonalization (TASD), a robust method that improves stability and accuracy in noisy regimes. Combined with ALS, TASD yields a statistically consistent estimator. We further analyze the convergence dynamics of ALS, identifying a two-phase pattern-initial quadratic convergence followed by linear refinement. We further show that in the rank-one setting, ALS with an appropriately chosen initialization attains optimal error within just one or two iterations.

Tensor Decomposition with Unaligned Observations

This paper presents a canonical polyadic (CP) tensor decomposition that addresses unaligned observations. The mode with unaligned observations is represented using functions in a reproducing kernel Hilbert space (RKHS). We introduce a versatile loss function that effectively accounts for various types of data, including binary, integer-valued, and positive-valued types. Additionally, we propose an optimization algorithm for computing tensor decompositions with unaligned observations, along with a stochastic gradient method to enhance computational efficiency. A sketching algorithm is also introduced to further improve efficiency when using the $\ell_2$ loss function. To demonstrate the efficacy of our methods, we provide illustrative examples using both synthetic data and an early childhood human microbiome dataset.

Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Model

This paper introduces a novel framework called Mode-wise Principal Subspace Pursuit (MOP-UP) to extract hidden variations in both the row and column dimensions for matrix data. To enhance the understanding of the framework, we introduce a class of matrix-variate spiked covariance models that serve as inspiration for the development of the MOP-UP algorithm. The MOP-UP algorithm consists of two steps: Average Subspace Capture (ASC) and Alternating Projection (AP). These steps are specifically designed to capture the row-wise and column-wise dimension-reduced subspaces which contain the most informative features of the data. ASC utilizes a novel average projection operator as initialization and achieves exact recovery in the noiseless setting. We analyze the convergence and non-asymptotic error bounds of MOP-UP, introducing a blockwise matrix eigenvalue perturbation bound that proves the desired bound, where classic perturbation bounds fail. The effectiveness and practical merits of the proposed framework are demonstrated through experiments on both simulated and real datasets. Lastly, we discuss generalizations of our approach to higher-order data.