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Factoring distributions

X 3. X 2. X 1. X 3. X 7. X 1. X 2. X 7. X 6. X 5. X 4. X 6. X 4. X 5. Factoring distributions. V. Given random variables X 1 ,…,X n Partition variables V into sets A and V n A as independent as possible Formally: Want A* = argmin A I(X A ; X V n A ) s.t. 0<|A|<n

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Factoring distributions

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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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