Tensor Methods

A tensor is a multidimensional array, but in computational many-body theory the important question is not its order—it is whether the apparent high-dimensional complexity contains compressible structure. Low-rank decompositions replace a dense object with smaller factors that can be stored, contracted, and evaluated efficiently.

Why compression matters

An order-$d$ tensor with mode size $n$ contains $n^d$ entries. This exponential growth is the curse of dimensionality. A tensor-train representation instead writes

$$ A(i_1,\ldots,i_d)\approx G_1(i_1)G_2(i_2)\cdots G_d(i_d), $$

where each $G_k(i_k)$ is a small matrix and the connecting dimensions are the TT ranks. When those ranks remain moderate, storage changes from exponential in $d$ to approximately $O(dnr^2)$.

Methods in this notebook