  This paper studies the problem of time series forecasting (TSF) from the
perspective of compressed sensing. First of all, we convert TSF into a more
inclusive problem called tensor completion with arbitrary sampling (TCAS),
which is to restore a tensor from a subset of its entries sampled in an
arbitrary manner. While it is known that, in the framework of Tucker
low-rankness, it is theoretically impossible to identify the target tensor
based on some arbitrarily selected entries, in this work we shall show that
TCAS is indeed tackleable in the light of a new concept called convolutional
low-rankness, which is a generalization of the well-known Fourier sparsity.
Then we introduce a convex program termed Convolution Nuclear Norm Minimization
(CNNM), and we prove that CNNM succeeds in solving TCAS as long as a sampling
condition--which depends on the convolution rank of the target tensor--is
obeyed. This theory provides a meaningful answer to the fundamental question of
what is the minimum sampling size needed for making a given number of
forecasts. Experiments on univariate time series, images and videos show
encouraging results.
