Concentration for matrix martingales in continuous time and microscopic activity of social networks

2014/12/24 by Emmanuel Bacry, Bacry, Emmanuel, Stéphane Gaïffas +4
Mathematics · #Advanced Algebra and Geometry #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Mathematical Analysis and Transform Methods #Probability (math.PR) #Random Matrices and Applications #advanced mathematical theories #math.PR #stat.ML

paper · pdf · doi:10.48550/arxiv.1412.7705

openalex publication_date 2014/12/24 · arxiv created 2016/10/27 · arxiv updated 2016/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermartingale property of the trace exponential proved within the framework of stochastic calculus. We provide also several examples that illustrate the fact that our results allow us to recover easily several formerly obtained sharp bounds for discrete time matrix martingales.

Citations

• Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
• A Simpler Approach to Matrix Completion
• Robust principal component analysis?

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