Hamilton–Jacobi equations for inference of matrix tensor products

We study the high-dimensional limit of free energy associated with inference problem finite-rank matrix tensor products. In general, we bound from above by unique solution to a certain Hamilton-Jacobi equation. Under additional assumptions on nonlinearity in equation which is determined explicitly model, identify solution. Two notions solutions, weak solutions and viscosity are considered, each has its own advantages requires different treatments. For concreteness, apply our results model i.i.d. entries symmetric interactions. particular, for first order even products, obtain estimates convergence rates; other odd orders, upper bounds obtained.