2020/09/03 by Hongbin Chen, Jiaming Xia, Chen, Hong-Bin +1 · 1 citation
Mathematics · Physics and Astronomy · #82B44 #82D30 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Model Reduction and Neural Networks #Probability (math.PR) #Quantum many-body systems #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2009.01678
openalex publication_date 2020/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. In general, we bound the limit from above by the unique solution to a certain Hamilton-Jacobi equation. Under additional assumptions on the nonlinearity in the equation which is determined explicitly by the model, we identify the limit with the solution. Two notions of solutions, weak solutions and viscosity solutions, are considered, each of which has its own advantages and requires different treatments. For concreteness, we apply our results to a model with i.i.d. entries and symmetric interactions. In particular, for the first order and even order tensor products, we identify the limit and obtain estimates on convergence rates; for other odd orders, upper bounds are obtained.