  Recently,~\citet{liu:arxiv:2019} studied the rather challenging problem of
time series forecasting from the perspective of compressed sensing. They
proposed a no-learning method, named Convolution Nuclear Norm Minimization
(CNNM), and proved that CNNM can exactly recover the future part of a series
from its observed part, provided that the series is convolutionally low-rank.
While impressive, the convolutional low-rankness condition may not be satisfied
whenever the series is far from being seasonal, and is in fact brittle to the
presence of trends and dynamics. This paper tries to approach the issues by
integrating a learnable, orthonormal transformation into CNNM, with the purpose
for converting the series of involute structures into regular signals of
convolutionally low-rank. We prove that the resulted model, termed
Learning-Based CNNM (LbCNNM), strictly succeeds in identifying the future part
of a series, as long as the transform of the series is convolutionally
low-rank. To learn proper transformations that may meet the required success
conditions, we devise an interpretable method based on Principal Component
Purist (PCP). Equipped with this learning method and some elaborate data
argumentation skills, LbCNNM not only can handle well the major components of
time series (including trends, seasonality and dynamics), but also can make use
of the forecasts provided by some other forecasting methods; this means LbCNNM
can be used as a general tool for model combination. Extensive experiments on
100,452 real-world time series from TSDL and M4 demonstrate the superior
performance of LbCNNM.
