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Efficient Algorithms for Minimizing Submodular Functions in Factorization Problems

This paper discusses the problem of partitioning random variables into sets while maintaining independence. We focus on minimizing mutual information through the function z(A), highlighting its submodular properties. We present Queyranne’s algorithm, a combinatorial and strongly polynomial approach for solving the minimization problem for symmetric submodular functions, achieving a runtime of O(n^3). Additionally, we analyze the application of pendent pairs in optimizing variable separation, illustrating their significance in achieving efficient solutions.

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Efficient Algorithms for Minimizing Submodular Functions in Factorization Problems

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  1. X3 X2 X1 X3 X7 X1 X2 X7 X6 X5 X4 X6 X4 X5 Factoring distributions V • Given random variables X1,…,Xn • Partition variables V into sets A and VnA as independent as possible Formally: Want A* = argminA I(XA; XVnA) s.t. 0<|A|<n where I(XA,XB) = H(XB) - H(XBj XA) A VnA

  2. Example: Mutual information • Given random variables X1,…,Xn • z(A) = I(XA; XVnA) = H(XVnA) – H(XVnA |XA)=z(V\A) Lemma: Mutual information z(A) is submodular z(A [ {s}) – z(A) = H(Xsj XA) – H(Xsj XVn(A[{s}) ) s(A) = z(A[{s})-z(A) monotonically nonincreasing z submodular  Nondecreasing in A Nonincreasing in A:AµB ) H(Xs|XA) ¸ H(Xs|XB)

  3. Queyranne’s algorithm[Queyranne ’98] Theorem: There is a fully combinatorial, strongly polynomial algorithm for solving A* = argminA z(A) s.t. 0<|A|<nfor symmetric submodular functions z • Runs in time O(n3) [instead of O(n8)…]

  4. V A* u t V V A* A* u u t t Why are pendent pairs useful? • Key idea: Let (t,u) pendent, A* = argmin z(A) Then EITHER • t and u separated by A*, e.g., u2A*, tA*. But then A*={u}!! OR • u and t are not separated by A* Then we can merge u and t…

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