Stationary time-vertex signal processing

Area of Science:

• Statistics
• Time Series Analysis
• Graph Signal Processing

Background:

• High-dimensional multivariate processes often exhibit complex structures dependent on graph topologies.
• Existing methods struggle with accurate covariance estimation and efficient recovery for such processes.

Purpose of the Study:

• To introduce a novel definition of stationarity, termed joint stationarity, for graph-dependent processes.
• To demonstrate the benefits of joint stationarity in reducing estimation variance and computational complexity.
• To enable reliable covariance structure learning and efficient MMSE recovery.

Main Methods:

• Definition of time-vertex wide-sense stationarity (joint stationarity) extending beyond product graphs.
• Theoretical analysis of covariance structure learning from single realizations.
• Development of algorithms for MMSE recovery (interpolation, denoising) with near-linear computational time.

Main Results:

• Joint stationarity allows reliable covariance structure learning from a single process realization.
• MMSE recovery problems are solved in nearly linear computational time relative to edges and timesteps.
• Experiments show accuracy improvements in recovering high-dimensional processes on graphs.

Conclusions:

• Joint stationarity offers significant advantages for analyzing and recovering high-dimensional graph-dependent processes.
• The method is robust to approximate graph knowledge and deviations from strict stationarity.
• This framework advances the field of graph signal processing and time series analysis.