Recovery of Future Data via Convolution Nuclear Norm Minimization

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.

References in corpus (4)

• GAIN: Missing Data Imputation using Generative Adversarial Nets
• Matrix Completion with Deterministic Sampling: Theories and Methods
• Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case
• Time Series Forecasting via Learning Convolutionally Low-Rank Models

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• Correlating Time Series with Interpretable Convolutional Kernels
• Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery