Recovery of Future Data via Convolution Nuclear Norm Minimization
This work addresses the fundamental problem of forecasting future data in time series, images, and videos, offering a novel theoretical framework but is incremental in its application of compressed sensing concepts.
The paper tackles time series forecasting by reformulating it as tensor completion with arbitrary sampling and introduces a convex program called Convolution Nuclear Norm Minimization (CNNM) that solves this problem under certain sampling conditions, providing a theoretical answer to the minimum sampling size needed for forecasting.
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.