Computation of Joint Moment Functions on Convolutional Factor Graphs

Theorem 1 For a cycle-free convolutional factor graph representing function f , the summary message pi(xi) at each variable node i computed using the algorithm above is precisely M ; (xi). Theorem 2 Suppose that f ( x ) is the joint pdf of random Then, for any subset variables X = { X 1 , X z , . . . X,}. C C I and non-negative integer n-vector s , the expectation E [ X ” I X c = xc]of the product X ” conditioned upon X c = xc
c
n,,,
nzEI
When the algorithm converges, we compute the sum) t each variable node i as pi(zi) = mary message p i ( % %a z:~ p j + i ( z z ) where stands for convolution.
ISlT 2002, Lausanne, Switzerland, June 30 -July 5,2002
Computation of Joint Moment Functions on Convolutional Factor Graphs
Yongyi Mao
Department of Electrical & Computer Engineering, University of Toronto yymaoQcomm.utoronto.ca
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YБайду номын сангаас”B)h(xAnB
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Y A n B , ZB\A),
where, for all i E I, yz takes on values in the same set as 2 % . Let s = ( S I , . . . , ) s , be a vector with non-negative integer components. We let x; denote denote x : ” Likewise, for 5:’. Given a function g A ( x A ) , a nonA I , z> denotes negative integer vector s and a subset C A, we define the joint sth moment functzon of g A at xc as
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0-7803-7501 -7/02/$17.00@2002 IEEE.
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In the convolutional factor graph representing a global function f ,we let set I index the set of variable nodes and we let set J index the set of factor nodes. A node v has neighbors N ( v ) and degree d(v). For any non-negative integer m, let Z ( m ) denote the set {0,1,.. . ,m}. Given a positive integer let Ze((m) denote the l-fold Cartesian product of Z ( m ) with itself. Let P ( T ) denote the set {(a1,a2,. . . , a t ) E Z e ( m ) : E : = , ai = m}. For a non-negative integer vector ( m l , m2,. . . ,me), let W(m1, m2,. . . ,me) be the Cartesian product T(m1) x Z ( m 2 )x REFERENCES . . . x Z ( m e ) . Let Q be the power set of I U J , and F be the sort- [I] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor ing map that takes any nonempty element A E Q t o the vector graphs and the sum-product algorithm,” IEEE Duns. Inform. Theory, vol. 47, no. 2, pp. 498-519, Feb 2001. a ( A ) = ( o l ( A )F , ~ ( A ). ,. . , U I A ( I A ) )having the property that m ( A ) 5 az(A) 5 . . . 5 o ~ A I ( A ) Given . an n-vector s and a [2] Y. Mao and F. R. Kschischang, “On factor graphs and the Fourier transform,” in Proc. ISIT 2001, Washington, D.C., subset A I , S A is the IAl-vector sol(^), s b z ( a ).,. . , s o l A l ( ~ ) ) . Jun. 2001, p. 224. Finally, for any A E Q and any vector a of IAl elements, define the map p that takes the pair (A,a) to the n-vector p(A,a) [3] Y. Mao and F. R. Kschischang, “On factor graphs and the Fourier transform,” in preparation. with the property that the ai(A)th component of p(A,a) is
I. INTRODUCTION
Just as a conventional (multiplicative) factor graph [l]represents the multiplicative factorization of a function as a product of local factors, a convolutional factor graph [2,3] represents the convolutional factorization of a function. A convolutional factor graph may be used to represent the characteristic function of a code, the joint probability density function (pdf) of multiple random variables, etc. Let function f have n arguments, 2 1 , x2,.. . , xn, indexed by the set I = { 1 , 2 , . . . , n } . If A I , let Z A denote the set of have ) variables {xz : z E A}. Two functions g ( Z A ) and ~ ( z B a convolution g * h defined (using an obvious notation) as
is
M ;( z ~ ) / M ~ ~ ~ ~ - ~ )
11. THEALGORITHM
111. CONCLUSION
Iterative algorithms on graphical models are of current research interest. In this paper, we show that for a function represented by a convolutional factor graph, its joint moment functions can be computed by a message-passing algorithm on the graph, without explicitly computing the function itself; when the function represented by the graph is a joint pdf, these joint moment functions are effectively conditional expectations. It is also worth noting that, as an application of factor graph duality, the algorithm translates t o a new message-passing algorithm on multiplicative factor graphs.
g(ZA\B,
YAnB
a,, for all z E { 1 , 2 , . . . , IAI}, while the other components of p(A, a ) are all zero. Assume variable node i and factor node j are neighbors. is a The message passed from j to i , denoted by p J + 2 ( x 2 ) function of the single variable x,, while the message passed P : ~ , p ! 2 J ,. . . ,p!:;), of from z t o j is a vector, denoted by ( sz 1 scalar values. The messages are computed as follows.