Remarks, Conjectures and Future Work Sample Clauses

Remarks, Conjectures and Future Work. ‌ This paper investigates the asymptotics of the information score called the expected log-posterior agreement to validate cost functions and algorithms for “parameter rich” combinatorial optimization problems. As subtasks, first we provided rigorous derivations for free energy of Sparse MBP and Xxxxxx QAP, results that had been wanting for some time. However, for general MBP and QAP we do not expect the lower bound to match the upper bound found in Theorem 1. In fact, based on extensive simulation we concluded that there is an additional scaling in the part of linear growth. To establish it, we realize that we need some new techniques to prove lower bounds that we developed. Second, we showed that two second order phase transitions occur for the ex- pected log-posterior agreement. Our analysis and experimental results show three regions of the the expected log-posterior agreement: a high temperature phase with low information, a retrieval phase and a disordered frozen phase. Only the retrieval phase can be used for efficient sampling solutions. While investigating the asymptotics of the log-posterior agreement and free energy we faced a challenging mathematical problem leading to some new research on the interplay between statistical physics and computation. We hope that techniques presented here can be successfully used for a large class of different combinatorial structures and problems. We also have empirically-inspired conjectures for approximating free energy for general problems (i.e. for MBP, QAP, and potentially other problems), which are well supported by our experiments and rigorous analysis for special cases (i.e. conjecture turns into proven asymptotics for Sparse MBP, Xxxxxx QAP), explained below.