The Sum-Product Theorem: A Foundation for Learning Tractable Models (Supplementary Material)
نویسندگان
چکیده
Let S(X) be a decomposable SPF with size |S| on commutative semiring (R,⊕,⊗, 0, 1), let d = |Xi| for all Xi ∈ X where X = (X1, . . . , Xn), and let the cost of a ⊕ b and a ⊗ b for any elements a, b ∈ R be c. Further, let e denote the complexity of evaluating any unary leaf function φj(Xi) in S and let k = maxv∈Ssum,j∈Ch(v) |Xv\Xj | < n, where Ssum, Sprod, and Sleaf are the sum, product, and leaf nodes in S, respectively, and Ch(v) are the children of v. Then the complexity of computing ⊕ x∈X S(x) is |S| · c+ |Sleaf| · d(e+ c) + |Ssum| · (c+ kdc).
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