Fitting Low-Rank Tensors in Constant Time

In this paper, we develop an algorithm that approximates the residual error of Tucker decomposition, one of the most popular tensor decomposition methods, with a provable guarantee. Given an order-K tensor X is an element of R-N1x...xNK, our algorithm randomly samples a constant number s of indices for each mode and creates a "mini" tensor (X) over tilde is an element of R-sx...xs, whose elements are given by the intersection of the sampled indices on X. Then, we show that the residual error of the Tucker decomposition of (X) over tilde is sufficiently close to that of X with high probability. This result implies that we can figure out how much we can fit a low-rank tensor to X in constant time, regardless of the size of X. This is useful for guessing the favorable rank of Tucker decomposition. Finally, we demonstrate how the sampling method works quickly and accurately using multiple real datasets.