Multi-Functional Compression with Side Information

Soheil Feizi-Khankandi, Muriel Médard

Abstract: In this paper, we consider the problem of multifunctional compression with side information. The problem is how we can compress a source X so that the receiver is able to compute some deterministic functions f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (X,Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), ..., f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> (X,Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> ), where Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , 1 ? i ? m, are available at the receiver as side information. In, Wyner and Ziv considered this problem for the special case of m = 1 and f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (X, Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ) = X and derived a rate-distortion function. Yamamoto extended this result in to the case of having one general function f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (X,Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ) . Both of these results were in terms of an auxiliary random variable. For the case of zero distortion, in, Orlitsky and Roche gave an interpretation of this variable in terms of properties of the characteristic graph which led to a particular coding scheme. This result was extended in by providing an achievable scheme based on colorings of the characteristic graph. In a recent work, reference has considered this problem for a general tree network where intermediate nodes are allowed to perform some computations. These previous works only considered the case where the receiver only wants to compute one function (m = 1). Here, we want to consider the case in which the receiver wants to compute several functions with different side information random variables and zero distortion. Our results do not depend on the fact that all functions are desired in one receiver and one can apply them to the case of having several receivers with different desired functions (i.e., functions are separable). We define a new concept named the multi-functional graph entropy which is an extension of the graph entropy defined by Korner in. We show that the minimum achievable rate for this problem is equal to the conditional multi-functional graph entropy of random variable X given side informations. We also propose a coding scheme based on graph colorings to achieve this rate.