Concept

Tensor train — where it appears

A tensor written as a chain of three-index cores, one per index, so that each entry is a product of small matrices. Its ranks are those of the reshapes that cut the index list, and its storage is linear in the number of indices rather than exponential.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

The format that does not notice the dimension

A Tucker core is r^d numbers, so the format that repaired the definition still cannot go past five indices. Cutting between the indices rather than across them gives d − 1 ranks instead of d, storage linear in the number of indices, and a family whose ranks are two everywhere by an addition formula.

tensor · tensor train

The order the products are taken in

The sparsity field's first essay says the elimination order decides the memory. This is the same sentence about arithmetic: a contraction of several tensors over shared indices has one value and many evaluation orders, and on the inner product of two trains they differ by a factor of two million.

cost · contraction

An iterate that must be made smaller

Applying a Kronecker-sum operator to a low-rank iterate multiplies its ranks by d and adding two of them adds their ranks, so a solver in a compressed format cannot keep what it produces. Every step is followed by a truncation — and whether that truncation is a floor on the residual depends on the right-hand side rather than on the truncation.

iterative · low rank iteration

Named alongside it

The objects these essays reach for when they reach for this one.

Curse of dimensionality Low-rank approximation Alternating least squares Block methods Conjugate gradients Contraction order Elimination order Exact ground truth Fill-reducing ordering Flop count Higher-order SVD Khatri–Rao product

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