CS774. Markov Random Field : Theory and Application Lecture 20 Kyomin Jung KAIST Nov

Remind: X is a positive binary MRF if for some, where is the set of the cliques of G. Learning positive binary MRF Note that is dependent on at most |C| many variables. Now we consider the problem of learning under the condition that we can check P[X=x] values for polynomially many x’s.

Note that Learning positive binary MRF Given P[X=x], the expression of is not unique : one can add a constant to. Hence we want to learn one such set of (or want to learn up to constant addition.)

Pseudo Boolean function A function f is called a Pseudo-Boolean function if f is defined on and its value is in R. A pseudo-Boolean function is of order k if f can be expressed as If one can learn a pseudo-Boolean function of order k from function queries, one can learn the MRF of order k from the probability queries.

Relation with Fourier transform f can be expressed by the Fourier coefficients (with Walsh functions):

The underlying graph G is said to have linkage  If there is correlation among the variables of H.  (for any expression, there is j so that H belongs to the support set of ) The hyper-graph consisting of all such H’s is called the linkage graph of f. Linkage graph corresponds to the underlying graph G of the MRF.

Learning the Linkage Graph The following linkage test function tests whether there is a linkage among Linkage among H? H

Property of the Linkage Test A subset H of [n] is a hyper-edge of if and only if for some string x. For an order k function f, and a hyperedge H of order j in, the probability that for x chosen uniformly at random from is at least Linkage among H? H

Learning the Fourier coefficients If and for all, is called a maximal non-zero Fourier coefficient of f. For any H, is a maximal non-zero Fourier coefficient of f if and only if H is a maximal hyperedge of. For a maximal hyperedge H, For any subset H, From these relations, we can learn all the non-zero, which enables us to learn f.