Doug Cochran. 3 October 2011

Transcription

1 ) ) School Electrical, Computer, & Energy Arizona State University (Joint work with Stephen Howard and Bill Moran) 3 October 2011

2 Tenets ) The purpose sensor networks is to sense; i.e., to enable detection, estimation, classification, and tracking To exploit network structure, data at the nodes must be registered Intrinsic data; e.g., clocks, platform orientation Extrinsic data: collected by sensors ω 2 ω 4 ω 3 ω 7 ω 1 ω 5 ω 8 ω 6

3 Registration on a Graph ) Vertex labels: An element a Lie group G is associated with each node a graph modeling a network Edge labels: An element G representing a noisy measurement the difference the values on the vertices at either end Goal: Make the best estimate the connection the true relative fsets between the vertex values v 2 r (1,2) r (4,1) r (4,2) v 4 r (4,3) v 1 v 3

4 ) Denote by f(x θ) a family probability density functions parameterized by elements θ on a manifold M For given data x, f(x θ) defines the likelihood function on M A maximum-likelihood (ML) estimate for θ is defined by ˆθ = arg max θ M f(x θ) The log-likelihood function l(x θ) : M R is defined by l(x θ) = log f(x θ) A natural measure discrimination on M is the Kullback-Leibler (KL) divergence D(f θ f θ ) = f(x θ) log f(x θ) f(x θ ) dx

5 ) Although the KL divergence is not symmetric, locally it gives a metric on M called the Fisher information metric F = E[dl dl] F can also be computed as F = E[ 2 l], for any connection The corresponding volume form is the Jeffreys prior vol F = det F dθ 1 dθ m

6 ) The state the network is a G-valued function x on V (Γ) but it is never directly observed If the network were aligned, the function would be constant (flat connection) What is observed is the connection ω Ω = G m In the absence noise, for an aligned network this is the identity connection If the network is not aligned, a gauge transformation can be found that takes ω to the identity x v g v x v ω e g t(e) ω e g 1 s(e)

7 ) Gauge transformations that fix the identity connection form a normal subgroup that can be quotiented out The noise on the measurements should be a random variable taking values ε Ω and whose density p(ε) is defined with respect to the normalized Haar measure on Ω

8 Example ) v1 e1 e5 v2 e3 v4 e4 e2 v3 Incidence matrix Boundary operator (n m) D =

9 Example ) Coboundary operator (m n) D T = Graph Laplacian Kirchf Matrix (n n) L = DD T = = A det L = 0, but det L = 8 = t(γ) for any (n 1) (n 1) cactor L L.

10 Homological set-up ) Directed graph Γ with vertex set V (Γ) and edge set E(Γ) V (Γ) = n E(Γ) = m Vertex space C 0 (Γ) functions from V (Γ) to R; a real vector space: f = n f j v j j=1 v j (v l ) = δ jl Edge space C 1 (Γ) functions from E(Γ) to R m ω = ω j e j j=1 e j (e l ) = δ jl

11 Homological set-up ) Source and target maps s, t : C 1 (Γ) C 0 (Γ); if e = (v i, v j ), s(e) = v i t(e) = v j Vertex space C 0 (Γ) functions from V (Γ) to R; a real vector space: f = n f j v j j=1 v j (v l ) = δ jl

12 Homological set-up ) Boundary map (incidence matrix) D : C 1 (Γ) C 0 (Γ) by D(e) = t(e) s(e) Coboundary map D T : C 0 (Γ) C 1 (Γ) by D T (v) = t(e j )=v e j s(e j )=v e j C 1 (Γ) D D T C 0 (Γ)

13 Homological set-up ) Cycle space Z(Γ): A cycle is a closed path in Γ; i.e., a sequence vertices L = v 1 v 2... v q where v i is adjacent to v i+1 for i = 1,..., q and v q is adjacent to v 1 1 if e j L and e j is oriented as L z L (e j ) = 1 if e j L and e j is oriented opposite to L 0 otherwise Z(Γ) is the linear subspace C 1 (Γ) spanned by the z L Kirchf s current law (KCL): Z(Γ) = ker D; i.e., Dz = 0 for all z Z(Γ) e j e j = 0 v V (Γ) t(e j )=v s(e j )=v

14 Homological set-up ) Cocycle space B(Γ): The orthogonal complement Z(Γ) in C 1 (Γ) For any z Z(Γ) and ω B(Γ), z, ω C1 = 0 B(Γ) is the image C 0 (Γ) under the coboundary operator; i.e., every ω B(Γ) can be written as ω = D T x for some x C 0 (Γ) This gives Kirchhf s voltage law (KVL) ω1 ω2 ω0 + ω1 ω2 = 0 ω0

15 ) The data has the form r C 1 (Γ) r = ω + ε The true fsets satisfy ω B(Γ) (KVL) The noise ε C 1 (Γ) has probability density 1 f(ε) = ( (2π) m det R exp 1 ) 2 ε, ε R where ε, ε R = ε T R 1 ε

16 ) The conditional density for r given ω is 1 f(ε ω) = ( (2π) m det R exp 1 ) 2 r ω, r ω R The maximum-likelihood estimate ω is ˆω = arg min r ω, r ω R ω B(Γ)

17 ML ) The data may be written as r = ˆω + ˆε with ˆω B(Γ) (KVL) and ˆω, ˆε R = 0 i.e., the residual ˆε satisfies R 1 (ˆε) Z(Γ) (KCL) ˆω 1 ˆω 2 ˆε 0/σ ˆε 1/σ ˆε 2/σ 2 2 = 0 σ 2 2 ˆε 2 ˆω 0 + ˆω 1 ˆω 2 = 0 σ 2 1 ˆε 1 ˆω 0 ˆε 0 σ 2 0

18 ML ) Weighted graph Laplacian L : C 0 (Γ) C 0 (Γ) L = DR 1 D T Not invertible since ker L contains constant functions in C 0 (Γ) Choosing a reference vertex, the fset estimator ˆx W is ˆx = L 1 W D W R 1 r where L W and D W are defined with respect to the reduced vertex set The connection estimate is ˆω = D T W ˆx ˆω = D T W L 1 W D W R 1 r D T W L 1 W D W R 1 is the R-orthogonal projection into B(Γ)

19 d Performance the ML with R = σ 2 I ) For R = σ 2 I, the Fisher information is Its determinant is 1 σ 2 L W det F = t(γ)/σ 2(n 1) t(γ) is the number spanning trees in Γ g 0 g 1 g 2 g 3 g 0 g 1 g 2 g 3 g 0 g 1 g 2 g 3 g 0 g 1 g 2 g 3

20 Performance the ML with R = σ 2 I ) The ML estimator ˆx is unbiased Its covariance is Its determinant is Cˆx = E[(x ˆx)(x ˆx) T ] = σ 2 L 1 W Cˆx = σ 2(n 1) /t(γ)

21 d Performance the ML independent noise unequal variance ) For R = diag(σ 2 1,..., σ2 m), the Fisher information determinant is det F = S 1 σ 2 e j S j where the sum is over all spanning trees S in Γ The estimator ˆx is unbiased and det Cˆx = S g 0 g 1 g 2 g 3 g 0 g 1 σ 2 e j S j g 2 g 3 g 0 1 g 1 1 g 2 g 3 g 0 g 1 g 2 g 3

22 Noise on R If one link can be added to... ) Which these is the best choice?

23 Local One-dimensional noise case ) For R = diag(σ 2 1,..., σ2 m), the ML estimator residual satisfies KCL; i.e., DR 1 (r ˆω) = 0 Thus ˆω = D T x for some x C 0 (Γ) (KVL) So Lx = DR 1 r where L = DR 1 D T is the weighted graph Laplacian Any solution to this equation will serve for the local estimation algorithm

24 Local One-dimensional noise case ) Jacobi s Method gives a locally implementable solution The Laplacian can be written as L = N A where N is the degree map and A is the adjacency map for Γ Jacobi s method entails the iteration x n+1 = N 1 (DR 1 r + Ax n ) i.e., be the average what your neighbors think you are Jacobi s method is guaranteed to converge if L is diagonally dominant; i.e., L ii > i j L ij Unfortunately L ii = i j L ij, so convergence is not guaranteed

25 Other Results Summary ) Extension to G = R d solutions have a tensor structure Von Mises noise on T oscillator phase alignment Includes localizable fast algorithms that converge to near-ml estimates in all cases tested Algorithm: Be the weighted circular mean what your neighbors think you are where the weights come from how well the neighbors are aligned with their neighbors Compact connected abelian Lie groups follow the form R and T, but suitable noise models are a problem Making progress in the non-abelian Lie group setting, including the important special case G = SO(3) New work focusing more on estimation parameters from extrinsic data on a sensor network

26 Other Results Performance Fast s on T ) Mean Circular Error Local Q Eigenvector Local Hybrid ML Global Q Eigenvector Global A Eigenvector TrF κ

27 Other Results Performance Fast s on T ) Mean Circular Error Local Q Eigenvector Local Hybrid ML Global Q Eigenvector Global A Eigenvector TrF κ