Lossy Source Coding under a Maximum Distortion Constraint with Decoder Side- Information Jayanth Nayak 1, Ertem Tuncel 2, and Kenneth Rose 1 1 University of California, Santa Barbara 2 University of California, Riverside

Source Coding with Side-Information Rate: Average distortion: Wyner-Ziv ‘76 Maximum Distortion –Prefix-free variable length codes. Encoder Decoder Alice Bob

Wyner-Ziv Converse

Prior work Lossless reconstruction –Graph theoretic formulation: Witsenhausen ‘76 –Variable length coding: Alon and Orlitsky ‘96 –Rate = : Koulgi et al. ‘03 Without side-information –Graph theoretic formulation: Tuncel et al. ‘02

The combinatorial problem Family of D-balls : Family of fan-out sets : –There is no graph on that captures the structure of the problem. Assumptions about set families –Covering: –Independence systems:

Example

The scalar encoder Observed source letter: –Decoder: some set in containing –Encoding strategy: specify a set containing whose intersection with every set in belongs to

Asymptotic rate Complement of w.r.t. : –Family of sets whose intersection with every set in belongs to. –Code = partition into sets from. Encode the partition using a prefix free variable length code

Asymptotic rate Partition into sets from. Minimum asymptotic average rate:

Single letter expression possible? Single letter characterization –Expression involves only scalar quantities. –Not known for the special case of lossless reconstruction. –Derive single letter bounds.

Single letter bounds 2 n  number of sets from needed to cover a high probability subset of –Can be expressed as a mutual information minimization Upper Bound:

Lower Bounds Lower bound: Wyner-Ziv –

Lower Bounds covers with

Coincidence of bounds Given and, when does a nowhere vanishing exist such that Two step coding: Körner, Longo ’73 –First step reconstruction: within D 1 ( ) –Second step reconstruction: within D 2 ( ) ?

Condition for coincidence Graph : = independent sets, = singletons –Reduces to condition for normality of. Result generalizes one of Körner et al. on perfect couples of graphs.

To conclude Maximum distortion source coding with side-information problem defined. Problem recast in a combinatorial framework. Non-single letter expression for the minimum asymptotic rate derived. Single letter bounds presented. Conditions for coincidence of bounds investigated.