Hamilton-Jacobi equations for inference of matrix tensor products

Understand

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.