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Calculating Rate Regions for Network Coding and Distributed Storage Via Polyhedral Projection and Representable Matroid Enumeration

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Calculating Rate Regions for Network Coding and Distributed Storage Via Polyhedral Projection and Representable Matroid Enumeration Calculating rate regions for network coding and distributed storage via polyhedral projection and representable matroid enumeration John MacLaren Walsh Department of Electrical and Computer Engineering Drexel University Philadelphia, PA [email protected] Thanks to NSF CCF-1016588, NSF CCF-1053702, & AFOSR FA9550-12-1-0086. 1 Entropic Vectors and Polyhedral Computation (Development Team) Jayant Apte Congduan Li Daniel Venutolo efficient parallel rate regions & computing non-Shannon infinite precision rate delay tradeoffs inequalities polyhedral computation Prof. Steven Weber Drexel University 2 Outline 1. Bounding Rate Regions in Principle/Theory (a) Distributed Storage/Network Coding Regions & Entropic Vectors (b) Outer Bounds: Shannon/Non-Shannon (c) Inner Bounds: Linear Polymatroids/ Representable Matroids 2. Calculating Polyhedral Rate Regions: Why is na¨ıve? " (a) Di↵erence Between Descriptions (b) Projecting Polyhedra (c) Exploiting Symmetries 3. Linear Polymatroid Enumeration for Rate Regions (a) Review of Non-isomorphic Matroid Enumeration (b) Non-isomorphic Representable Matroid Enumeration (c) Non-isomorphic Extremal Representable Matroid Enumeration (d) Network Constrained Linear Polymatroid Enumeration 4. Related Current & Future Work Outline 1. Bounding Rate Regions in Principle/Theory (a) Distributed Storage/Network Coding Regions & Entropic Vectors (b) Outer Bounds: Shannon/Non-Shannon (c) Inner Bounds: Linear Polymatroids/ Representable Matroids 2. Calculating Polyhedral Rate Regions: Why is na¨ıve? " (a) Di↵erence Between Descriptions (b) Projecting Polyhedra (c) Exploiting Symmetries 3. Linear Polymatroid Enumeration for Rate Regions (a) Review of Non-isomorphic Matroid Enumeration (b) Non-isomorphic Representable Matroid Enumeration (c) Non-isomorphic Extremal Representable Matroid Enumeration (d) Network Constrained Linear Polymatroid Enumeration 4. Related Current & Future Work Networking Coding and Distributed Storage Rate Regions Network Coding Distributed Storage (MDCS DSCSC) . Out(i) . In(i) . Ys i Yj H(Xj) Zl Rl s t Sj El Dm Fm ωs β(t) . e Ue . Ul . Vm . ⊂ S Re A B ⊂ S × E ⊂ E × D D S ¯ T S E¯ (Roughly) intersect ΓN⇤ with (Roughly) intersect ΓN⇤ with hYs !s,s hY = hYs , hYj ,j = hYj ≥ 2S S 2S s j X2S X2S hU Y =0,s hZ (Y ,j U ) =0,l Out(s)| s 2S l| j 2 l 2E hU U =0,i ( ) h(Y ,j F ) (Z ,l V ) =0,m Out(i)| In(i) 2V\ S[T j 2 m | l 2 m 2D hU Re,e hY >H(Xj ),j e 2E j 2S hY U =0,t hZ Rl,l β(t)| In(t) 2T l 2E and project onto !s,Re and project onto H(Xi),Rl { } ¯ Substitute inner/outer bounds for ΓN⇤ to get inner/outer bounds for rate region 5 ¯ Region of Entropic Vectors ΓN⇤ – What is it? 1. X =(X1,...,XN ) N discrete RVs 2. every subset X =(Xi,i ) H(XY ) A 2A A✓ 1,...,N [N] has joint entropy { }⌘ h(X ). A H(Y ) 2N 1 3. h =(h(X ) [N]) R − en- A |A ✓ 2 tropic vector Example: for N =3, h = • H(X) (h1,h2,h3,h12,h13,h23,h123). 2N 1 REV for N =2: 4. a ho R − is entropic if joint PMF 2 9 pX s.t. h(pX )=ho. H(X) H(XY ) 5. Region of entropic vectors =Γ⇤ H(Y ) H(XY ) N 6. Closure Γ¯⇤ is a convex cone [1]. H(XY ) H(X)+H(Y ) N Γ¯⇤ is an unknown non-polyhedral convex cone for N 4. N ≥ determining capacity regions of all networks under network coding complete ¯ , characterization of ΓN⇤ [2] Outline 1. Bounding Rate Regions in Principle/Theory (a) Distributed Storage/Network Coding Regions & Entropic Vectors (b) Outer Bounds: Shannon/Non-Shannon (c) Inner Bounds: Linear Polymatroids/ Representable Matroids 2. Calculating Polyhedral Rate Regions: Why is na¨ıve? " (a) Di↵erence Between Descriptions (b) Projecting Polyhedra (c) Exploiting Symmetries 3. Linear Polymatroid Enumeration for Rate Regions (a) Review of Non-isomorphic Matroid Enumeration (b) Non-isomorphic Representable Matroid Enumeration (c) Non-isomorphic Extremal Representable Matroid Enumeration (d) Network Constrained Linear Polymatroid Enumeration 4. Related Current & Future Work ¯ Bounding the Region of Entropic Vectors ΓN⇤ from the Outside Shannon Outer Bound: Γ . = region of poly- • N matroid rank functions = entropy is submodular: I(X ; X X ) 0 , , A B| C ≥ 8A B C ¯ ¯ Γ2 = Γ2⇤, Γ3 = Γ3⇤. Γ = Γ¯⇤ ,N 4 Γ¯⇤ non-polyhedral convex cone N 6 N ≥ N Non-Shannon Outer Bounds:[3, 4, 5, 6, 7, 8, 9] • Yeung & Zhang, Dougherty & Freiling & Zeger, Matus Start with 4 unconstr. r.v.s add rv. obeying distr. match & Markov. cond. Intersect Γ for N 5 w/ Markov & distr. match N ≥ Project back to orig. 4 unconstr. vars. ΓN Shannon Outer Bound obtain new information inequalities this way! N Non-Shannon Outer Bound Z¯ Region of Entropic Vectors overall: Shannon linear eq./ineq. project ΓN⇤ ! ! H(XY ) Subspace Ranks Bound H(XY ) SN T H(Y ) q GF(q)-Representable Matroid Constraints Projection MN Bound H(Y ) H(Y ) Φ4 binary entropic vectors H(X) conv(Φ4) convex hull H(X) H(X) 8 Outline 1. Bounding Rate Regions in Principle/Theory (a) Distributed Storage/Network Coding Regions & Entropic Vectors (b) Outer Bounds: Shannon/Non-Shannon (c) Inner Bounds: Linear Polymatroids/ Representable Matroids 2. Calculating Polyhedral Rate Regions: Why is na¨ıve? " (a) Di↵erence Between Descriptions (b) Projecting Polyhedra (c) Exploiting Symmetries 3. Linear Polymatroid Enumeration for Rate Regions (a) Review of Non-isomorphic Matroid Enumeration (b) Non-isomorphic Representable Matroid Enumeration (c) Non-isomorphic Extremal Representable Matroid Enumeration (d) Network Constrained Linear Polymatroid Enumeration 4. Related Current & Future Work ¯ Bounding ΓN⇤ from the Inside, 1: Representable Matroids Conceptual Inner Bound Creation Process 2N 1 Matroids: r ΓN Z − ,r( ) all non-isomorphic matroids 2 \ A |A| w/ ground set size N All non-isomorphic matroids for N 9 [10] ’08 • 2N 1 Minor Exclusion: GF (q)-Representable Matroid: r ΓN Z − Purge any w/ U 2 , 4 . M N 2 \ s.t. A GF (q) ⇥ s.t. r( ) = rank(A:, ) 9 2 A A Add isomorphs repr. matroid = scaled EV!: u (GF (q)M ) • ⇠U (permutations) to list by mapping elements in ground X = uA h = r( ) log2 q set to network variables ) A A Key: representability no forbidden minors: Constraint Exclusion: • , Purge vectors not obeying – complete small list known for q 2, 3, 4 2{ } network constraints from list [11, 12, 13, 14, 15] eg.:GF (2) repr. no , Projection: Project rays onto U(2, 4) minor (Tutte 1958) capacity/rate variables, & q remove non-extremal rays Γ¯⇤ bound: Γ conic hull of GF (q)-repr. ma- • N N Result: extreme ray troids. representation of rate region polyhedral inner bound 10 ¯ Bounding ΓN⇤ from the Inside, 2: Inner Bounds from Subspace Ranks 2N 1 Subspace Bounds: r ΓN Z − projec- 2 \ tions of representable matroids, N 0 N,parti- N ≥ tion 1,...,N0 = , = n = k { } n=1 Gn Gn \Gk ; 6 Build inner bound for Γ¯ : S N ( N⇤ r( ) = rank([A:, n n ]) (1) S A G | 2A 1. Obtain Γq using, e.g., N 0 r( ) =scaledEV!=linear polymatroid method from previous slide, · 2. project (remove all but en- Xn = uA:, n h = r( ) log2 q G ) A A tropies where each element in N : conic hull of all subspace ranks S Xn appears together) : Γ Ingleton’s [16, 17, 18] S4 4\ H(XY ) H(XY ) H(Y ) I(X ; X )+I(X ; X X )+I(X ; X X ) I(X ; X ) 0 1 2 3 4| 1 3 4| 2 − 3 4 ≥ Constraints Projection H(Y ) H(Y ) H(X) recently characterized by DFZ + Kinser [19, H(X) H(X) S5 20] unknown for N 6, but can inner bounded sound familiar??? SN ≥ by projecting Γq (see right) Shannon lin. project N ! ! 11 T Outline 1. Bounding Rate Regions in Principle/Theory (a) Distributed Storage/Network Coding Regions & Entropic Vectors (b) Outer Bounds: Shannon/Non-Shannon (c) Inner Bounds: Linear Polymatroids/ Representable Matroids 2. Calculating Polyhedral Rate Regions: Why is na¨ıve? " (a) Di↵erence Between Descriptions (b) Projecting Polyhedra (c) Exploiting Symmetries 3. Linear Polymatroid Enumeration for Rate Regions (a) Review of Non-isomorphic Matroid Enumeration (b) Non-isomorphic Representable Matroid Enumeration (c) Non-isomorphic Extremal Representable Matroid Enumeration (d) Network Constrained Linear Polymatroid Enumeration 4. Related Current & Future Work Why this is na¨ıve : Di↵erence Between Descriptions Inner Bounds Outer Bounds 6 Shannon Outer Bound 5 Binary Inner Bound 10 10 4 5 4 >3.3*10 Inequalities 10 10 Inequalities 4 Extreme rays 3 10 10 Extreme rays 3 2 10 10 2 1 10 10 # in log scale 1 # in log scale 10 0 10 0 3 4 5 10 # of RVs 3 4 5 # of RVs Small # of Extreme Rays Small # of Inequalities HUGE # of Inequalities HUGE # of Extreme Rays 13 Outline 1. Bounding Rate Regions in Principle/Theory (a) Distributed Storage/Network Coding Regions & Entropic Vectors (b) Outer Bounds: Shannon/Non-Shannon (c) Inner Bounds: Linear Polymatroids/ Representable Matroids 2. Calculating Polyhedral Rate Regions: Why is na¨ıve? " (a) Di↵erence Between Descriptions (b) Projecting Polyhedra (c) Exploiting Symmetries 3. Linear Polymatroid Enumeration for Rate Regions (a) Review of Non-isomorphic Matroid Enumeration (b) Non-isomorphic Representable Matroid Enumeration (c) Non-isomorphic Extremal Representable Matroid Enumeration (d) Network Constrained Linear Polymatroid Enumeration 4. Related Current & Future Work Why this is na¨ıve : Projecting Polyhedra – Options[7, 21, 22, 23, 24, 25, 26] Extreme Ray/Point Fourier Motzkin Benson's Outer Convex Hull Method Projection Elimination Approximation Algorithm Successively only deals with Project Extreme Eliminates works directly Pareto frontier in Rays & Points Variables (slow) in projected projected space inequalities space of N dimen. Redundancy polyhedron Removal (convex hull) N-1 dim.
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