Response-Guided Screening and Sketching for Regression on CP-Structured Tensor Covariates

We study regression of a scalar response on a tensor covariate of ambient dimension P = prod_k p_k that is supplied as a rank-R CP decomposition, as in multimodal fusion and polynomial feature expansion. We develop a response-guided pipeline that works only on the CP factors: it screens modes, CP components and coordinates, compresses the retained structure with TensorSketch, and fits ridge regression averaged over randomized repetitions, without forming any P-dimensional object. We compare two screening statistics: multivariate distance correlation, and a cross-covariance energy statistic that extends the cross-covariance tensor E[yX] to subsets of modes. In a multilinear model with independent mean-zero factors, the population energy is positive precisely on the active mode subsets, including multiplicative signals, for which marginal distance correlation has low power. A permutation-calibrated component screen removes redundant CP components. We show that spectral-entropy mode ranking can be computed exactly from the CP factors. For TensorSketch we give a self-contained proof of the bound (3^d-1) m⁻¹ ||X||_F^2 ||Y||_F^2 on the mean squared inner-product error for arbitrary order-d tensors, and an exact formula for the population loss of regression on averaged sketches, which does not vanish with the sample size when the sketch is smaller than the retained tensor. In simulations with observed CP factors, energy screening with component screening attains near-oracle error in most replicates for single-mode and pairwise multiplicative signals, where distance-correlation screening fails on the pairwise signal; for a correctly specified linear tensor model, CP tensor regression is more accurate.Version 1. Submitted to arXiv (stat.ME); the arXiv identifier will be added on announcement. A real-data evaluation is in progress.