On low-rank tensor train approximability for linear nearest neighbor systems

What the math gives to ML

The paper develops a constructive tensor-train viewpoint for high-dimensional operators assembled from nearest-neighbor interactions, and studies how polynomial and Krylov iterations increase tensor ranks. The transferable asset is not merely TT compression, which is already known, but the fact that locality and partial commutativity can keep ranks of operator polynomials controlled independently of the number of sites. This suggests neural blocks that apply learned polynomial filters of a chain-local operator while truncating TT ranks after every multiplication, giving costs that scale with local dimension and rank rather than exponentially with feature order. The most credible first target is an implicit or recurrent tensorized residual block, with rank growth and accuracy measured against dense and standard tensorized baselines.