Distributed Particle Filters: Stability Results and Graph-based Compression of Weighted Particle Clouds

Transcription

1 Distributed Particle Filters: Stability Results and Graph-based Compression of Weighted Particle Clouds Michael Rabbat Joint with: Syamantak Datta Gupta and Mark Coates (McGill), and Stephane Blouin (DRDC) 1

2 Decentralized Tracking Multiple sensors (e.g., bearings-only) No central processing Communicate, cooperate to estimate object trajectory 2

3 Decentralized Particle Filtering (DPF) Weighted particle cloud representing possible object states at time t {(x i t 1, w i t 1) R d R i = 1,..., N} Approximate posterior p t 1 (x) = 1 N n wt 1δ i x i t 1 (x) i=1 where δ x i t 1 (x) is Dirac delta mass at x i t 1 3

4 Decentralized Particle Filtering (DPF) Predict new positions by sampling from the object dynamic model p(x t x i t 1 ) 4

5 Decentralized Particle Filtering (DPF) New observation y j t j = 1,..., n at sensor 5

6 Decentralized Particle Filtering (DPF) Update particle weight w i t proportional to log p(yt 1,..., yt n x i t t 1 ) n = log p(y j t xi t t 1 ) j=1 Data update/fusion requires communication 6

7 Consensus-Based Decentralized Particle Filtering Survey: Hlinka, Hlawatsch, & Djuric, IEEE SP Mag 2013 Use gossip/consensus algorithms for synchronization 7

8 Gossip Algorithms for Distributed Averaging Seminal work: DeGroot 1974; Tsitsiklis, Bertsekas, & Athans, IEEE TAC 1986 Survey: Dimakis, Kar, Moura, Rabbat, & Scaglione, Proc. IEEE 2010 Communication topology encoded in n n doubly-stochastic matrix P with P j,l > 0 iff nodes j and l communicate directly. Node j begins with initial value z j (0) and repeats iterations z j (k + 1) = n P j,l z j (k) l=1 = P j,j z j (k) + l j P j,l z l (k) 8

9 Gossip Convergence Rates If the communication topology is connected, then j = 1,..., n and z j (k) z δ if lim k z j(k) = 1 n n z l (0) l=1 k log( 1 δ n maxl z l (0) z ). 1 λ 2 (P ) 9

10 Gossip Convergence Rates If the communication topology is connected, then j = 1,..., n and z j (k) z δ if lim k z j(k) = 1 n n z l (0) l=1 k log( 1 δ n maxl z l (0) z ). 1 λ 2 (P ) ring random geometric graph expander = O(n 2 ) n = O log(n) = O(1) 9

11 Consensus-Based Decentralized Particle Filtering Update particle weight w i t proportional to log p(yt 1,..., yt n x i t t 1 ) n = log p(y j t xi t t 1 ) j=1 Data update/fusion requires communication Naïve approach requires communication per particle 10

12 PF Convergence Results and Stability Strong results for stability of centralized particle filters [1 4] Sampling introduces an error. Gossip introduces an error. How do these errors propagate over time? [1] F. LeGland and N. Oudjane, Ann. App. Prob [2] D. Crisan and A. Doucet, IEEE Trans. Sig. Proc., 2002 [3] P. Del Moral, Springer-Verlag, 2004 [4] N. Chopin, Ann. Stat

13 PF Stability: Assumptions Bounded likelihoods: Let L t (x i t) = p(y 1 t,..., y n t x i t) = n j=1 p(yj t xi t) There exists ɛ L (0, 1) s.t. L t (x i t) ɛ L L t (x i t ) i, i = 1,..., N 12

14 PF Stability: Assumptions Bounded likelihoods: Let L t (x i t) = p(y 1 t,..., y n t x i t) = n j=1 p(yj t xi t) There exists ɛ L (0, 1) s.t. L t (x i t) ɛ L L t (x i t ) i, i = 1,..., N Sufficient mixing dynamics: Let M t (x t 1, dx t ) = p(x t x t 1 )dx t Given m > 0, there exists ɛ m (0, 1) s.t., x t, x t, M t,t+m (x t, ) = M t+1 M t+2... M t+m (x t, ) ɛ m M t,t+m (x t, ) 12

15 Bounded gossip error: There exists δ > 0 s.t. log L t (x i t) log L t (x i t) log L t (x i t ) δ i = 1,..., N Note: δ related to number of gossip iterations 13

16 Bounded gossip error: There exists δ > 0 s.t. log L t (x i t) log L t (x i t) log L t (x i t ) δ i = 1,..., N Note: δ related to number of gossip iterations Test function has bounded oscillations: There exists C > 0 s.t. osc(h t ) = sup x,x h t (x) h t (x ) < C E.g., h t (x i t) = x i t x 2 t Then π N (h t ) = 1 N N i=1 wi t x i t x 2 t 13

17 Distributed PF Stability We have uniform stability in the weak-sense L 2 error sense: [ sup E [ π t N π t ](h t ) 2] ( ) 1/2 D ɛ0 N + δ log ɛ L t 0 where D and ɛ 0 are independent of t, and ɛ 0 = 2m ɛ 3 mɛ (2m 1) L. S. Datta Gupta, M. Coates, and M. Rabbat, Error propagation in gossip-based distributed particle filters, IEEE TSIPN,

18 The Challenge: Communication Overhead Need to synchronize all particle weights across sensors Run one gossip instance for each particle? High overhead! Especially since Many particles have low weight Weights typically vary smoothly (Aside: Could communicate measurements... often worse) 15

19 A Graph-Based Approach Idea: Transform coding with a basis adapted to the current particle cloud Use smoothness assumption: nearby particles have similar weight Methodology: Fit a graph to the particle cloud (e.g., k-nearest neighbors) Use eigenbasis of Laplacian to compress log-likelihood weight vector defined over particles Gossip on few low frequency coefficients in Laplacian eigenbasis 16

20 Graph Laplacian Transform Coding k-nn graph over particles Laplacian eigenvectors F define a basis adapted to the particle distribution Examine coefficients in this basis α = F T w Adjacency matrix A Degree matrix D = diag(1 T A) Laplacian L = D A = F ΛF T αt(j ) j 17

21 Graph Laplacian Transform Coding Examine coefficients in this basis Exact weight representation α = F T w N w = F α = α j f j j=1 αt(j ) M N j Approximate by keeping low frequency coefficients (threshold rest to zero) ŵ = M α j f j j=1 18

22 Case Study Two targets with correlated motion 50 time steps 8-dimensional state vector N = 2000 particles T1 Start 4 T2 Start sensors, randomized grid topology Only communicate with nearby neighbors Vary number of gossip averaging iterations Sensors measure bearings only Random number of detections per time step (avg = 1/5 / sensor / step) Additive Gaussian noise, σ = 5 degrees Performance metric: total time-averaged RMSE (ARMSE) 19

23 Communication-Accuracy Tradeoff Varying number of coefficients in graph Laplacian approximation Total ARMSE (Target 1 + Target 2) m = 100 m = 500 m = 1000 m = 1500 m = 1900 Varying number of gossip (averaging) iterations Avg. Scalars per Node per Step x

24 Comparison Total ARMSE (Target 1 + Target 2) Distributed Bootstrap Gaussian Approximation Set Membership Constrained Top m Selective Gossip Graph Laplacian Approx Avg. Scalars per Node per Step 21

25 Conclusion Summary: Stability bound for distributed particle filters Depends on num. particles N, gossip error δ Graph-based approx. to reduce communication overhead Graph captures particle proximity Gives transform adapted to particle geometry Ongoing work: Stability bounds for particle approximation methods Computationally-efficient graph-based approximation via clustering and non-linear reconstruction 22