C-SHIFT normalization method and random networks

• Grant number: 25/05787-6
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: September 01, 2025
• End date: August 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Applied Probability and Statistics
• Principal Investigator: Vladimir Belitsky
• Grantee: Evgenia Vladimirovna Chunikhina
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/13453-5 - Stochastic systems modeling, AP.TEM

Abstract

1) Information Theory in Random Networks: extending the information-theoretical analysis of random self-similar tree networks that I developed in earlier work to other important classes of random self-similar trees. I will work on fundamental problems related to the information content of complex self-similar tree networks. This includes finding the entropy rates for the random self-similar trees induced by the Tokunaga processes and the corresponding random attachment model, the critical hierarchical branching processes, and the Invariant Galton-Watson processes (also known as Stable Galton-Watson processes).2) Genetic Models: extending the C-SHIFT method for offsetting the impact of biases on the covariances that we developed in earlier work. I will work on devising an approach analogous to the C-SHIFT method that would effectively recover the true empirical covariances under multiplicative bias. This extension of the recently developed C-SHIFT normalization method will contribute to the theoretical foundations of data science and enable normalizing covariances for the data collected with RNA-Seq technique. The results of this theoretical work will be tested on synthetic data generated using random network models.3) Compressed Sensing in Sparse Random Networks. Sparsity recovery techniques play a key role in the fields of compressed sensing and statistical signal processing. I plan to further develop the concept of a statistically provable tuning-free approach for row sparsity pattern recovery in noisy conditions. Specifically, I will consider an important theoretical extension to this work: I want to identify the conditions under which non-convex optimization problem and its convex relaxation can lead to the same results.