Dynamic Tensor Product Regression
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In this work, we initiate the study of Dynamic Tensor Product Regression. One has matrices A_1∈ℝ^n_1× d_1,…,A_q∈ℝ^n_q× d_q and a label vector b∈ℝ^n_1… n_q, and the goal is to solve the regression problem with the design matrix A being the tensor product of the matrices A_1, A_2, …, A_q i.e. min_x∈ℝ^d_1… d_q (A_1⊗…⊗ A_q)x-b_2. At each time step, one matrix A_i receives a sparse change, and the goal is to maintain a sketch of the tensor product A_1⊗…⊗ A_q so that the regression solution can be updated quickly. Recomputing the solution from scratch for each round is very slow and so it is important to develop algorithms which can quickly update the solution with the new design matrix. Our main result is a dynamic tree data structure where any update to a single matrix can be propagated quickly throughout the tree. We show that our data structure can be used to solve dynamic versions of not only Tensor Product Regression, but also Tensor Product Spline regression (which is a generalization of ridge regression) and for maintaining Low Rank Approximations for the tensor product.
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