A Continuity Theory of Source Coding over Networks WeiHsin Gu, Michelle Effros, Mayank Bakshi, and Tracey Ho FLoWS PI Meeting, Washington DC, September 2008

Status Quo Achievable rate regions are hard to characterize. Only solved for some small networks and multicast networks Inner (achievable) and upper (converse) bounds are derived for some example networks Open questions: Are those earlier derived bounds tight? Does single-letter characterization always exist?

New Insights Develop and investigate some abstract properties of the achievable rate regions As functions of probability distribution and distortion vector, are the achievable rate regions continuous? Motivations Understand the existence of 1-letter characterizations Applications: estimation of achievable rate regions Why is it hard? Block length is unbounded in the definition of the achievable rate regions Don’t know much about the achievable rate regions

Formulation and Notation Define a family of network source coding problems which contains the prior example networks as special cases. The family contains four subfamilies –Lossless non-functional –Lossy non-functional –Lossless functional –Lossy functional : Source : Side information : Probability distribution : Distortion vector : Lossy rate region : Lossless rate region

Main Results is concave in – is continuous in when –For any non-functional source coding problem, is continuous in for all For super-source networks, both and are continuous in for all

is continuous in if and only if – is upper semi-continuous in – is lower semi-continuous in We show that the achievable rate regions are inner semi-continuous in for –Lossless non-functional –Lossy non-functional –Lossy functional Continuity w.r.t.

A Possible Solution We conjecture that outer semi-continuity w.r.t. is true when random variables have finite alphabets A possible approach: for any network, consider the super-source network If the above equality is true, then is upper semi- continuous in

Next-Phase Goals Prove or disprove upper semi-continuity w.r.t. for general networks. Characterize a larger family of networks where upper semi-continuity holds Investigate some other useful abstract properties Understand the existence of achievable rate regions