A polynomial time algebraic solution to exact marginal inference in Markov Random Field models

This paper develops on algebraic grounds a polynomial time exact linear solution to the hard combinatorial problem of marginal inference in Markov random field (MRF) models under general assumptions. To prove our claim, we first implicitly remodel a MRF joint distribution as the unique solution of some linear identity assuming its clique potential functions (equivalently, its individual conditional distributions) to be specified. Then, by assuming an arbitrary point subset, we relax accordingly such a (global) linear identity for deriving a second linear identity, solely, acting on a polynomial time number of entries (e.g.; local marginals or Fourier frequencies) of a solution. Then, we show, only using linear algebraic techniques, that such an identity enables to capture all the entries necessary for the exact reconstruction of an MRF marginal distribution, thus, allowing to solve for the latter, exactly and in polynomial time, using a standard linear solver. Last, but not least, this paper probably solves, once and for all, the P = NP conjecture.