Why Tensor Train matters as order grows

For a dense order-N tensor with equal mode size I, dense storage is I^N. At a fixed TT bond rank r, TT storage is ≈ O(N · I · r²).

Exponential in the order against linear in the order, with I and r held fixed.

CP storage is also linear in the order under a fixed global rank. TT’s practical advantage is different: it represents high-order interactions through local bond ranks, rather than through one Tucker core that grows exponentially with the order.