Recovery of Future Data via Convolution Nuclear Norm Minimization

TL;DR

This paper introduces a novel convolution nuclear norm minimization approach for tensor completion, enabling accurate time series forecasting from arbitrary sampled data by leveraging convolutional low-rankness.

Contribution

It proposes a new tensor completion framework using convolutional low-rankness and proves its effectiveness under certain sampling conditions.

Findings

CNNM successfully recovers tensors with arbitrary sampling.

The method outperforms traditional approaches in experiments.

Theoretical bounds relate sampling size to forecast accuracy.

Abstract

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.