Primal-Dual Characterizations of Jointly Optimal Transmission Rate and Scheme for Distributed Sources Bradford D. Boyle Steven Weber bradford@drexel.edu sweber@coe.drexel.edu odeling Modeling & Analysis of Networks Laboratory & Department of Electrical and Computer Engineering nalysis Drexel University, Philadelphia, PA 19104 of etworks Data Compression Conference Snowbird, UT March 27 th , 2014 B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 1 / 27
Introduction Outline 1 Introduction 2 Preliminaries 3 Feasible Set Structural Properties 4 Sufficient Conditions for Characterizing Optimality 5 Conclusion 6 References B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 2 / 27
Introduction Objective Lossless transmission of correlated sources to sink over capacitated network with minimum cost s 2 sources R ( s i ) , h ( s i ) s 1 router/relay s 4 t a s 3 sink f ( a ) , c ( a ) , k ( a ) D ( V, A ) flow cost source cost X X minimize k ( a ) f ( a ) + h ( s ) R ( s ) f ≥ 0 ,R a 2 A s 2 S subject to f ( a ) ≤ c ( a ) a 2 A f ( δ in ( v )) − f ( δ out ( v )) = 0 flow feasibility v 2 N flow supports rate R ( s ) + f ( δ in ( s )) − f ( δ out ( s )) = 0 s 2 S rate feasibility R ( U ) ≥ H ( X U | X U c ) U ⊆ S B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 3 / 27
Introduction Related Work Slepian & Wolf (1973) through Ramamoorthy (2011) Slepian and Wolf (1973) [1] X 1 E • Rates for lossless recovery at D ( X 1 , X 2 ) a single sink using separate X 2 E encoders R ( s 1 ) R ( s 2 ) B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 4 / 27
Introduction Related Work Slepian & Wolf (1973) through Ramamoorthy (2011) Han (1980) [2] • Rates for lossless recovery at a X 1 E a D ( X 1 , X 2 ) single sink using separate X 2 E c encoders cuts • Over capacitated network R ( s 1 ) R ( s 2 ) B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 4 / 27
Introduction Related Work Slepian & Wolf (1973) through Ramamoorthy (2011) Cristescu, Beferull-Lozano, Vetterli (2005) [3] • Rates for lossless recovery at a single sink using separate X 1 E ( X 1 , X 2 ) encoders a D X 2 E k cuts • Over capacitated uncapacitated network R ( s 1 ) • Minimization of nonlinear R ∗ rate and flow objective over feasible (rate, flow) region k R ( s 2 ) B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 4 / 27
Introduction Related Work Slepian & Wolf (1973) through Ramamoorthy (2011) Ramamoorthy (2011) [4] • Rates for lossless recovery at ( X 1 , X 2 ) X 1 E D a single multiple sinks (w/ X 2 E D ( X 1 , X 2 ) c, k identical recovery req.) using separate encoders R ( s 1 ) • Over uncapacitated R ∗ capacitated network • Minimization of nonlinear linear k rate and flow objective over R ( s 2 ) feasible (rate, flow) region Solution approach: dual decomposition with subgradient descent B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 4 / 27
Introduction Related Work . . . & Many Others 1 Draper and Wornell (2004) —achievable lossy coding (Wyner-Ziv) for correlated observations of a single source to a single sink over a sensor network 2 Barros and Servetto (2006) —related formulation/results to Han (1980), pose but don’t solve optimization problem over rate region 3 Ramamoorthy, Jain, Chou, Effros (2006) —distributed source coding of multiple sources over network w/ lossless recovery at multiple receivers (identical recovery req.) 4 Ho, M´ edard, Effros, Koetter (2006) —RLNC to multicast, identifies RLNC error exponents as natural extensions of SW error exponents 5 Han (2011) —extends Han (1980) from one to multiple sinks (identical recovery req.) . . . and many, many more B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 5 / 27
Introduction Summary of Results flow cost source cost X X minimize k ( a ) f ( a ) + h ( s ) R ( s ) f ≥ 0 ,R a 2 A s 2 S subject to f ( a ) ≤ c ( a ) a 2 A f ( δ in ( v )) − f ( δ out ( v )) = 0 flow feasibility v 2 N flow supports rate R ( s ) + f ( δ in ( s )) − f ( δ out ( s )) = 0 s 2 S rate feasibility R ( U ) ≥ H ( X U | X U c ) U ⊆ S R ( s 2 ) Key Results 1 Structure of set of feasible rates R 2 Active & inactive constraints 3 Optimal primal-dual variables from reduced costs O R ( s 1 ) B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 6 / 27
Preliminaries Outline 1 Introduction 2 Preliminaries 3 Feasible Set Structural Properties 4 Sufficient Conditions for Characterizing Optimality 5 Conclusion 6 References B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 7 / 27
Preliminaries Achievable & Supportable Rates R ( s 2 ) R ( s 2 ) Q σ SW P ρ c R ( s 1 ) R ( s 1 ) Slepian-Wolf (1973) Meggido (1974) [5] • Set of achievable rates • Set of supportable rates • Contrapolymatorid associated • Polymatroid associated w/ w/ σ SW ( U ) = H ( X U | X U c ) ρ c ( U ) = c ( min-cut ( U )) Bijective map between source permutations π and vertices R π of (contra)polymatroids (Edmonds 1970) [6] B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 8 / 27
Preliminaries Feasible Rates Han (1980) Sufficiency Example R ( s 2 ) R ( s 2 ) Q σ SW g ≤ h P ρ c R ( s 1 ) R ( s 1 ) g not supermodular • Intersection non-empty iff g ( U ) ≤ h ( U ) σ SW ( U ) ≤ ρ c ( U ) • necessary • Achievability & converse proofs • w/o sub-/supermodularity is for R ∈ R not sufficient B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 9 / 27
Preliminaries Efficient Transmission of Sources to Sink flow cost source cost X X minimize k ( a ) f ( a ) + h ( s ) R ( s ) f ≥ 0 ,R a 2 A s 2 S subject to f ( a ) ≤ c ( a ) a 2 A f ( δ in ( v )) − f ( δ out ( v )) = 0 v 2 N flow feasibility flow supports rate R ( s ) + f ( δ in ( s )) − f ( δ out ( s )) = 0 s 2 S rate feasibility R ( U ) ≥ H ( X U | X U c ) U ⊆ S • Linear program with | A | + | V | − 1 + 2 | S | inequalities • If | S | = O ( | V | ), then the LP is exponential in the size of the graph • Optimal solution ( f ∗ , R ∗ ) will satisfy R ∗ ( S ) = H ( X S ) [2] B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 10 / 27
Feasible Set Structural Properties Outline 1 Introduction 2 Preliminaries 3 Feasible Set Structural Properties 4 Sufficient Conditions for Characterizing Optimality 5 Conclusion 6 References B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 11 / 27
Feasible Set Structural Properties Partial vs. Full Overlap Context: • Han (1980) characterizes empty vs. non-empty R • Q: When are all efficient vertices retained in the intersection? Partial vs. Full Overlap (Sufficient Condition) R ( s 2 ) R ( s 2 ) R ( s 1 ) R ( s 1 ) A: A sufficient condition for full vs. partial overlap R B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 12 / 27
Feasible Set Structural Properties Cross Inequality Proposition Frank & Tardos (1988) [7] cross inequality implies rate region intersection contains both base polytopes. H ( X U ∩ T | X U c ) ≤ ρ c ( T ) − ρ c ( T \ U ) , ∀ T , U ⊆ S Cross inequality specialzed to conditional entropy & min-cut capacity U ρ c ( T ) U \ T T ρ c ( T \ U ) T \ U U c S H ( X U ∩ T | X U c ) ≤ ρ c ( T ) − ρ c ( T \ U ) B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 13 / 27
Feasible Set Structural Properties Full Overlap & Generalized Polymatroids P ρ 0 R ( s ∗ ) B ( P ρ 0 ) O R ( s 2 ) R ( s 1 ) P ρ c R Fujishige (2005) [8] • Cross-inequality satisfied ⇒ R is a generalized polymatroid • R is projection of a base polytope of polymatroid B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 14 / 27
Feasible Set Structural Properties Full Overlap & Generalized Polymatroids R ( s 2 ) Proposition Satisfying cross inequality ⇒ • Extreme points of R are known • The LP � min h ( s ) R ( s ) R R ∈R s ∈ S has an explicitly characterized h | R solution (via Edmonds 1970) R ( s 1 ) Takeaway: Cross inequality ⇒ soln. is SW vertex; network capacities non-binding R = { R : H ( X U | X U c ) ≤ R ( U ) ≤ ρ c ( U ) , U ⊆ S } B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 15 / 27
Feasible Set Structural Properties Active & Inactive Constraints Polyhedral rate region • Slepian & Wolf gives half-space representation • Greedy algorithm gives vertex representation (Edmonds 1970) [6] • A degenerate vertex has > | S | active inequalities R ( s 2 ) R ( s 2 ) R ( s 1 ) R ( s 1 ) degenerate vertex non-degenerate vertex (3 active constraints) (2 active constraints) Q: For which U ⊆ S will SW constraint R ( U ) ≥ H ( X U | X U c ) be (in)active at vertex R π of R SW ? A: Tightness of each SW constraint at each vertex is determinable from π and the source conditional independence (C.I.) structure B. D. Boyle (Drexel MANL) P-D Char. Jointly Opt. Rate & Scheme DCC 2014 16 / 27
Recommend
More recommend
