Provable Tensor Factorization with Missing Data

We study the problem of low-rank tensor factorization in the presence of missing data. We ask the following question: how many sampled entries do we need, to efficiently and exactly reconstruct a tensor with a low-rank orthogonal decomposition? We propose a novel alternating minimization based method which iteratively refines estimates of the singular vectors. We show that under certain standard assumptions, our method can recover a three-mode n × n × n dimensional rank-r tensor exactly from O(n3/2r5 log4 n) randomly sampled entries. In the process of proving this result, we solve two challenging sub-problems for tensors with missing data. First, in analyzing the initialization step, we prove a generalization of a celebrated result by Szemeredie et al. on the spectrum of random graphs. We show that this initialization step alone is sufficient to achieve the root mean squared error on the parameters bounded by C(r2n3/2(log n)4/|Ω|) from |Ω| observed entries for some constant C independent of n and r. Next, we prove global convergence of alternating minimization with this good initialization. Simulations suggest that the dependence of the sample size on the dimensionality n is indeed tight.