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

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

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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?

- 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

- 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

- 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

Continuity w.r.t.

- 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

- 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

- 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