Greedy sensor selection: leveraging submodularity

Transcription

1 Grdy snsor slction: lvraging submodularity Manohar Shamaiah Siddhartha Banrj Haris Vikalo Th Univrsity of Txas at Austin Elctrical and Computr Enginring, 201 Spdway, Austin, TX I. ABSTRACT W considr th problm of snsor slction in rsourc constraind snsor ntworks. Th fusion cntr slcts a subst of k snsors from an availabl pool of m snsors according to th imum a postriori or th imum liklihood rul. W cast th snsor slction problm as th imization of a submodular function ovr uniform matroids, and dmonstrat that a snsor slction algorithm achivs prformanc within (1 1 ) of th optimal solution. Th algorithm has a complxity of O(n 3 mk), whr n is th dimnsion of th masurmnt spac. Th complxity of th algorithm is furthr rducd to O(n 2 mk) by xploiting crtain structural faturs of th problm. An application to th snsor slction in linar dynamical systms whr th fusion cntr mploys Kalman filtring for stat stimation is considrd. Simulation rsults dmonstrat th suprior prformanc of th snsor slction algorithm ovr compting tchniqus basd on convx rlaxation. Indx Trms: Submodular functions, Kalman filtr II. INTRODUCTION Snsor ntworks hav attractd much attntion in rcnt yars [1], [2], [3]. A typical fatur of such ntworks is th prsnc of on or mor fusion cntrs, which aggrgat th information from diffrnt snsors. Du to various practical considrations, th snsors ar gnrally rsourc constraind, and hnc thir communication with th fusion cntr is limitd. To this nd, th fusion cntr schduls only a subst of th availabl snsors for transmission in ach tim slot. Diffrnt prformanc mtrics lad to various snsor slction problm formulations. Rgardlss of th formulation, howvr, snsor slction is ssntially combinatorial in natur. Hnc finding th optimal solution is gnrally computationally intnsiv, lading to a numbr of huristics and approximat algorithms. In this papr, w considr th scnario whr th fusion cntr dsirs th imum liklihood (ML) or imum a postriori (MAP) stimat of th unknown vctor. Only a subst of th snsors which acquir th masurmnt is allowd to transmit its masurmnt to th fusion cntr. Du to th high complxity of obtaining th xact solution, Joshi and Boyd [1] propos a huristic approach to th problm basd on convx rlaxation. Howvr, th algorithm proposd in [1] has no guarants on th prformanc; morovr, its complxity is cubic in th total numbr of availabl snsors, which is prohibitiv for larg ntworks. In our work, w show that crtain structural proprtis of th snsor slction problm allow rcasting it as a imization of submodular functions ovr uniform matroids. This allows us to lvrag th rsults of Nmhausr and Wolsy [4], who show that for problms with such a structur, a algorithm rsults in an approximat solution with a prformanc within a (1 1 ) of th optimal. In addition to th prformanc guarants, th algorithm is significantly mor computationally fficint than th convx rlaxation basd approach. In particular, th complxity of th algorithm is O(n 3 mk), i., it scals linarly with th total numbr of snsors. W furthr xploit th structur in th problm to obtain a simplifid algorithm with a complxity of O(n 2 mk). A rlatd snsor slction problm ariss in stat stimation of linar dynamical systms [], [6]. Thr, th fusion cntr mploys Kalman filtr for stat stimation, and th snsor slction problm is concrnd with optimizing a mtric rlatd to th rror covarianc matrix in th Kalman rcursions. Rcntly, convx optimization basd huristics wr proposd to solv this problm []. As an altrnativ, w mploy th aformntiond algorithm and dmonstrat that it significantly outprforms th convx optimization basd solution. Morovr, by xploiting th connction btwn th proposd algorithm and squntial procssing implmntation of Kalman filtr [7], w achiv significant complxity rduction. Intrstingly, th algorithm has an intuitiv intrprtation as a procdur which at ach stp chooss th st of obsrvations in th dirction closst to th imum rror. Th papr is organizd as follows. Th systm modl and th snsor slction problm formulation ar givn in Sction III. Sction IV provids a brif ovrviw of submodular functions and uniform matroids and thn prsnts our analytical and simulation rsults. Snsor slction for stimation in linar dynamical systm using Kalman filtring is prsntd in Sction V, followd by th simulation rsults. Finally, w conclud th papr in Sction VI. III. SYSTEM MODEL AND SENSOR SELECTION PROBLEM Considr th problm of stimating a vctor x R n from k linar masurmnts, corruptd by additiv nois. Th k masurmnts ar to b slctd from th st of m masurmnts, ach acquird at on of th snsors in a ntwork. Th i th masurmnt is givn by y i = a ix + n i (1) whr a i dnots th i th masurmnt vctor and n i rprsnts th zro-man Gaussian nois with varianc σ 2. Following [1], w assum that th fusion cntr knows th masurmnt vctors a i.

2 A. Maximum a postriori critrion Suppos th prior dnsity of x is N(0, Σ x). Th imum a postriori probability (MAP) stimat of x is givn by [8] ˆx MAP = σ X 2 a ia 1 X i + Σ 1 x y ia i, (2) whr S dnots th st of slctd masurmnts. Th stimation rror covarianc matrix is givn by Σ map(s) = σ X 1. 2 a ia i + Σ 1 x (3) An oftn usd scalar masur of th quality of stimation is basd on th volum of η confidnc llipsoid or its man radius [8]. Both of ths ar functions of log dt(σ map(s)). Th snsor slction can thn b posd as th problm of slcting a subst of k n snsors from th st of m snsors, such that th log volum (or th man radius) of th confidnc llipsoid is minimizd. This can b xprssd as th following optimization problm, subjct to S = k. log dt σ X 2 a ia i + Σx 1, (4) whr S dnots th cardinality of th st S. By introducing binary variabls z i {0, 1} to indicat th mmbrship of ach snsor in th slctd subst, (4) can b writtn as X m log dt σ 2 z ia ia i + Σ 1 x, () subjct to z i {0, 1}, z i = k. Th convx rlaxation of th abov optimization problm is givn by subjct to X m log dt σ 2 z ia ia i + Σ 1 x, (6) 0 z i 1, i = 1, 2,..., m z i = k In [1], (6) was solvd by rformulating it as a smi-dfinit program (SDP). Th solution to th SDP may tak fractional valus, in which cas som kind of sorting and rounding nd to b mployd in ordr to obtain th dsird solution. Th complxity of th SDP algorithm scals as O(m 3 ). B. Maximum-Liklihood critrion Th imum-liklihood (ML) critrion lads to th stimator of th form ˆx ML = σ X 2 a ia 1 X i y ia i, (7) with th corrsponding rror covarianc givn by Σ ml (S) = σ 2 X a ia i 1 (8) Th corrsponding snsor slction problm is formulatd as th following optimization, X m log dt z ia ia i, (9) subjct to z i {0, 1} z i = k. Its convx rlaxation lads to th following optimization, X log dt z ia ia i, (10) subjct to 0 z i 1, i = 1, 2,..., m z i = k. Not that th ML stimation is as sam as th MAP stimation in th limit of Σ x = υi, υ (compar formulations () and (9)). Hnc w approximat th ML stimation as th following optimization problm, X m log dt z ia ia i + εi, (11) subjct to z i {0, 1}, z i = k, whr ε > 0 is chosn to b a vry small constant. Th rason for this modification is that a non-zro ε, albit vry small, nsurs applicability of th submodular approach to solving (11), which w dscrib nxt. IV. GREEDY SENSOR SELECTION A. Submodular function imization ovr uniform matroid. In this sction w rviw dfinitions and rsults rlatd to submodular functions and matroids [9]. Dfinition 1 [Submodularity]: Lt S b a finit st and 2 S dnot powr st. A st function f : 2 S R is said to b submodular iff A, B S, f(a B) + f(a B) f(a) + f(b). For finit st S this is quivalnt to A B S, j S\B, f(a + j) f(a) f(b + j) f(b) (12) i.., th function f satisfis th diminishing incrmnts proprty. Th submodular function f is monoton if f(a) f(b), A B. Dfinition 2 [Matroid]: A matroid is a pair M = (S; I) whr I 2 S and 1) B I, A B A I 2) A, B I, A < B x B\A, A + x I A matroid is an abstraction of combinatorial objcts, gnrating notion of linar indpndnc in vctor spacs. If m dnots th cardinality of th st S (i.., S = m), thn for any intgr k m, w can dfin I = {J J S, J k}; M = (S; I) is calld a uniform matroid M m,k.

3 B. Th snsor slction algorithm Th optimization problm (4) is of th form {f(s) S k}. If f(s) is a monoton submodular function thn (4) corrsponds to imization of a submodular function ovr a uniform matroid constraint [9], [4]. For this problm, it was shown in [4] that a algorithm rsults in a solution with th objctiv valu within (1 1 ) of th optimal valu 1. At ach stp th algorithm choss th masurmnt from th availabl masurmnts which imizs th objctiv whn includd with prviously chosn masurmnts. In th following lmma, w show that th objctiv function of th optimization (4) is submodular and hnc th algorithm (formalizd as Algorithm 1) rsults in th guarantd (1 1 ) optimal solution. Lmma 1: f(s) = log dt σ X 2 a i a i + Σ 1 x is a monoton submodular function, and hnc th solution to (4) via Algorithm 1 is (1 1 ) optimal. Morovr, th complxity of Algorithm 1 is O(n 3 mk). Proof: S Appndix. In gnral, k = O(n), i.., th numbr of snsors to b slctd is of th sam ordr as th dimnsion of th unknown vctor, in which cas th complxity of Algorithm 1 b is ssntially O(n 4 m). If th numbr of th availabl masurmnts m is vry larg compard to n (i.., m >> n), th complxity of Algorithm 1 is significantly lowr than th complxity of th convx optimization algorithm (th latr is O(m 3 )). W furthr rduc th complxity of Algorithm 1 by noting that th stp 2 in Algorithm 1 rquirs computation of th dtrminant of a rank 1 matrix. This can b simplifid as follows. Lt M s = σ X 2 a ia i + Σ 1 x, thn i K s log dt M s +a ja j S s j = j S s(a j s a j). (13) Hnc, w nd to propagat (or stor) only Ms 1, which is obtaind using th following rcursion s = s 1 s 1 a k s 1 a k s 1 s a k s 1 s 1 a, (14) k s 1 whr 0 = Σ x and a ks 1 is th optimal vctor chosn by th algorithm at stp s 1. This modification is summarizd as Algorithm 2. Th complxity of Algorithm 2 is O(n 2 mk). First, not that th complxity of computing th quadratic form 13 is O(n 2 ) and thr ar O(m) such computations. Nxt, using 14, w can comput Ms 1 in O(n 2 ) computations (as opposd to O(n 3 ) for dirct matrix invrsion). To s this, dnot r s 1 = s 1 a k s 1, which implis that s 1 a k s 1 a k s 1 s 1 = r s 1r s 1, a product of rank 1 matrics. Th complxity of computing 14 is thus O(n 2 ), and thrfor of slcting ach snsor is O(n 2 m). Finally, thr ar k such itrations. Hnc th ovrall complxity of Algorithm 2 is O(n 2 mk). C. Simulation rsults In this sction, w compar th prformanc of Algorithm 2 and th convx optimization basd algorithm mployd for 1 This has bn rcntly gnralizd to arbitrary matroid constraint[9] Algorithm 1 Basic algorithm 1. Initialization: s = 1, K s = {}, S s = S 2. Dtrmin grdily th nxt masurmnt: σ 2X k s=arg j S s log dt a i a i +a ja j +Σ 1 x i K s 3. Updat th masurmnt st: S s+1 = S s \k s, K s+1 = K s k s s s + 1, go to stp 2 if s k. Algorithm 2 Simplifid algorithm 1. Initialization: s = 1, K s = {}, S s = S, M s = Σ 1 x 2. Dtrmin grdily th nxt masurmnt: k s = arg j S s(a j i a j ) 3. Updat th masurmnt st and M s: S s+1 = S s \k s, K s+1 = K s k s, s s+1, s 1 s 1 a k a s 1 k s 1 s 1 1+a k s 1 s = s 1 a k s 1, go to stp 2 if s k. solving th snsor slction problm (9). Th lmnts of a i ar gnratd as iid Gaussian random variabls N(0, 1 ). To obtain a st of k snsors schduld for n transmission, w solv th modifid ML objctiv (11) with ε = Th prformanc of th grdily slctd snsors is compard using th actual objctiv (9) which thy induc. Figur 1 shows objctiv obtaind by th two algorithms for diffrnt random ralizations of th a i. Hr w st m = 10, n = 20, and k = 20. Th objctiv valu of th Algorithm 2 is almost always bttr than that of th convx optimization algorithm. Figur 2 shows th prformanc comparison of th avrag objctiv vrsus th dimnsion of x (n) for a fixd numbr of snsors k = 30. Th objctiv valu is avragd ovr 100 ralizations of th masurmnt st. Figur 2 shows that, as n approachs k, Algorithm 2 outprforms th convx optimization approach. In Figur 3, w plot th avrag objctiv as a function of th numbr of snsors k for n = 20. It can b infrrd that th Algorithm 2 achivs givn valu of th objctiv function with fwr snsors than th convx rlaxation approach. Howvr, as th numbr of snsors slctd incrass, both th approachs rsult in th sam prformanc. V. APPLICATION: SENSOR SELECTION USING KALMAN FILTER AT THE FUSION CENTER Considr a snsor ntwork in which fusion cntr slcts k snsors for transmitting th masurmnts of a linar dynamical systm. Th fusion cntr mploys Kalman filtr to track th stat of th systm. Th systm modl is givn by x t+1 = A tx t + w t, y t = S th tx t + v t, whr x t R n is th stat vctor, y t R k is th masurmnt vctor, w t and v t ar th Gaussian noiss with covarianc Q t and R t rspctivly. A t R n n is th stat transition matrix and H t R m n is th st of all snsors availabl (m) at tim t. S t R k m is th binary snsor slction matrix at tim t whos nonzro ntris xtract th slctd snsor

4 Objctiv valu n = 20, k = 20, m = 10 Avrag Objctiv k = 30, m = Itration numbr Fig. 1: Instantanous objctiv valus, m = 10, n = 20, k = Dimnsion of vctor x Fig. 3: Objctiv valu vs n, m = 10, k = 30. Avrag Objctiv n = 20, m = 10 objctiv is to choos z t+1(i) so as to minimiz th rror mtric P t+1 t+1. On such critria lads to optimization log dt P 1 t+1 t+1 (18) subjct to z t+1(i) {0, 1}, z t+1(i) = k. Algorithm 3 Snsor slction for Kalman filtr using algorithm 1. Initialization: numbr of snsors slctd (k) Fig. 2: Objctiv valu vs k, m = 10, n = 20. masurmnts that ar transmittd to th fusion cntr. Each row of S t has on and only on nonzro ntry. Each column of S t can hav at most on nonzro ntry. Th goal of th fusion cntr is to dsign th snsor slction matrix S t, so as to minimiz th filtrd rror covarianc. Lt P t+1 t and P t+1 t+1 b th prdiction and filtrd rror covarianc rspctivly at tim instant k + 1. Thn P t+1 t = A tp t t A T t + Qt, (1) P t+1 t+1 = (P 1 t+1 t + HT t+1 R 1 t+1 Z t+1h t+1 ) 1 (16) whr Z t+1 = S T t+1s t+1 R m m is a binary diagonal matrix. Assuming th masurmnts ar indpndnt across snsors (i.., R t is diagonal), w hav P t+1 t+1 = (P 1 t+1 t + HT t+1 R 1 2 t+1 Z t+1r 1 2 t+1 H t+1) 1 = (P 1 t+1 t + CT t+1z t+1 C t+1 ) 1 m = (P 1 t+1 t + X z t+1 (i)c t+1 (i)c t+1(i)) 1, (17) whr C t+1 = R 1 2 Ht+1, c t+1(i) is th i th row of C t+1, and z t+1(i) {0, 1} ar th diagonal ntris of Z t+1. Th s = 1, P s t+1 t+1 = P t+1 t 2. Dtrmin grdily th nxt masurmnt: k s = arg j S s c t+1 (j)p s t+1 t+1 c t+1(j) 3. Covarianc updat: P s+1 t+1 t+1 =P t+1 t+1 s P t+1 t+1 s c t+1(k s)ct+1 (ks)p t+1 t+1 s 1 + c t+1 (ks)p t+1 t+1 s c t+1(k s) 4. Updat th masurmnt st: S s+1 = S s \k s, s s + 1, go to stp 2 if s k. Updat th prdiction stp aftr k stps: P t+1 t+1 = P k t+1 t+1, P t+2 t+1 = A t+1 P t+1 t+1 A T t+1 + Q t+1 Th fusion cntr slcts snsors for transmission at ach tim instant using th abov mtric and thn updats P t+1 t according to (1). Th optimization problm (18) corrsponds to imization of a monoton submodular function subjct to a uniform matroid constraint (in particular, th MAP formulation (4) ) and hnc w can apply th algorithm to solv (18). In particular, w augmnt Algorithm 2 with an additional stp (1). This is formalizd as Algorithm 3. Th stps of Algorithm 3 ar rminiscnt of th squntial procssing in Kalman filtr handling multipl masurmnts [7]. Th stp 2

5 in th Algorithm 3 chooss th masurmnt in th squntial procssing such that it is imally alignd with dirction of th imum rror. This algorithm closly rsmbls th V- Lambda filtring [10]. Not that th intrmdiat covarianc valus P s+1 t+1 t+1 will b usd for th squntial procssing of th transmittd masurmnts. A. Simulation rsults W compar th prformanc of Algorithm 3 with th approach whr th snsor slction at ach stp of Kalman rcursion is obtaind via convx rlaxation of th objctiv function. Elmnts of th masurmnt matrix H t ar gnratd as i.i.d Gaussian random variabls with zro man and varianc 1 n. For th systm dynamics, w assum A = I n n and Q = 9I n n. Th obsrvation nois varianc of ach masurmnt is uniformly distributd in [0. 2], and is known at th fusion cntr. Th rsulting root-man-squar-rror (RMSE) of th stimators is computd by avraging ovr 0 Mont Carlo runs, with a tim horizon T = 20 for ach ralization. Figur 4 shows th prformanc comparison as a function of th numbr of snsors slctd for n = 1. Algorithm 3 prforms significantly bttr than th convx rlaxation basd algorithm, spcially whn th numbr of slctd snsors is clos to n. Figur, shows th RMSE prformanc as a function of n for a fixd numbr of snsors k = 30. As n grows, th prformanc of th convx rlaxation basd approach dtriorats much fastr than that of Algorithm 3. RMSE Q=9, n=1, m= numbr of snsors slctd (k) Fig. 4: Avrag RMSE vrsus th numbr of slctd snsors (m = 100, n = 1). VI. CONCLUSION In this papr, w formulatd imum-a-postriori and imum-liklihood snsor slction problms as optimizations of submodular functions ovr uniform matroids. Rlying on th xisting rsults for optimization of submodular functions, w proposd a snsor slction algorithm which finds a solution achiving objctiv within (1 1 ) from th optimal on. Exploiting th structur of th problm, w simplifid th original algorithm to significantly rduc its complxity. Morovr, w considrd th snsor slction RMSE k = 30, m = Dimnsion of stat vctor Fig. : Avrag RMSE vrsus th dimnsion of th stat vctor x (m = 100, k = 30). problm in th contxt of stat stimation in linar dynamical systms via Kalman filtring. In ach stp of th Kalman filtr algorithm, th problm was formalizd as an optimization of a submodular function ovr uniform matroids and solvd via th algorithm. Simulation rsults dmonstrat that th proposd approach outprforms compting convx rlaxation tchniqus, whil nding smallr computational complxity. REFERENCES [1] S. Joshi and S. Boyd, Snsor Slction via Convx Optimization, IEEE Trans on Signal Procssing, vol. 7, no. 2, pp , [2] H. Rowaihy, S. Eswaran, M. Johnson, D. Vrma, A. Bar-Noy, T. Brown, and T. L. Porta1, A Survy of Snsor Slction Schms in Wirlss Snsor Ntworks, Proc. SPIE, 2007, vol. 662 [3] A. Kraus and C. Gustrin, Nar-optimal Obsrvation Slction using Submodular Functions, Amrican Association for Artificial Intllignc, [4] G. L. Nmhausr and L. A. Wolsy, Bst algorithms for approximating th imum of a submodular st function, Math. Opr. Rsarch, 3(3): , [] J. E. Wimr, B. Sinopoli, and B. H. Krogh, A Rlaxation Approach to Dynamic Snsor Slction in Larg-Scal Wirlss Ntworks, Intrnational Confrnc on Distributd Computing Systms Workshops, [6] N. Moshtagh, L. Chn, and R. Mhra, Optimal Masurmnt Slction For Any-tim Kalman Filtring With Procssing Constraints, CDC, [7] T. Kailath, A. H. Sayd, and B. Hassibi, Linar Estimation, Prntic Hall, [8] Stphn Boyd and Livn Vandnbrgh Convx Optimization,Cambridg Univrsity Prss, 2004 [9] G. Calinscu, C. Chkuri, M. Pal, and J. Vondrak, Maximizing a Monoton Submodular Function subjct to a Matroid Constraint, SICOMP, 2009 [10] Y. Oshman, Optimal snsor slction stratgy for discrttim stat stimators, IEEE Trans on Arospac and Elctronic Systms, vol. 30, no. 2, pp , [11] R. A. Horn and C. R. Johnson Matrix analysis, Cambridg Univrsity Prss, 1990.

6 APPENDIX W first show f(s) = log dt σ X 2 z ia ia i + Σ 1 x is a monoton submodular function 2. Without a loss of gnrality, w assum σ = 1. Monotonicity: Lt A = {a 1, a 2,... a k }, and so F(A) = log dt Σ 1 x + a ia i. (19) Introduc B = {ā 1, ā 2,... ā l }. W nd to show that F(A B) F(A), for all sts B. To s this not that F(A B) F(A) dt Σ 1 x + a i a i + ā i ā i = log dt Σ 1 x + a i a i = log dt I + + `Σ 1 x Thrfor, w nd to show that 1 c = log dt I + `Σx + a ia i Lt M 1 = `Σ 1 x + a i a 1 i ā i ā i 1 a ia i 1 and M 2 =. ā iā i 0 ā iā i. It is asy to s M 2 is positiv smi dfinit. Hnc, w can find a matrix M 3 such that M 2 = M 3M 3. Using this w can writ c = log dt I + M 1 M 3 M 3 = log dt I + M 3 M 1M 3, whr in obtaining th last xprssion w usd th Sylvstr s dtrminant thorm [11] (dt(i + AB) = dt(i + BA)). Sinc M 1 is positiv dfinit, th matrix M 3M 1M 3 is positiv smidfinit. All th ign-valus of I +M 3M 1M 3 ar at last unity, and hnc w obtain th dsird rsult. Submodularity: Hr w prov that f(s) satisfis (12). Lt A = {a 1, a 2,... a k }, B = {a 1, a 2,... a l }, whr l k, i., A B. Choos a gnric lmnt a g / B. W nd to show that F(A a g) F(A) F(B a g) F(B). (20) W can writ whr F(A a g) F(A) F(B a g) + F(B) M 2 = Σ 1 x + = log dt(m 1 + a ga g )dt(m 2) dt(m 2 + a ga g )dt(m 1), M 1 = Σ 1 x + a ia i, a ia i = M 1 + a ia i = M 1 + M 3, M 3 = i=k+1 i=k+1 a ia i So, w nd to prov that W hav dt(m 1 + a g a g)dt(m 2 ) dt(m 2 + a ga g)dt(m 1 ) 1 dt(m 1 + a ga g)dt(m 2) dt(m 2 + a ga g)dt(m 1) = dt(m1 + aga g)dt(m 1 + M 3) dt(m 1 + M 3 + a ga g)dt(m 1) = = Hnc w nd to show that i., a g dt(i + 1 aga g) dt(i + (M 1 + M 3) 1 a ga g) (1 + a g 1 ag) (1 + a g(m 1 + M 3) 1 a g) (1 + a g 1 ag) (1 + a g(m 1 + M 3) 1 a g) 1 1 (M 1 + M 3) 1 a g 0 Sinc for invrtibl positiv dfinit matrics M > N it holds < N 1, th matrix 1 (M 1 + M 3) 1 is positiv smidfinit and hnc th abov inquality holds. This provs th dsird rsult. To complt th proof of Lmma 1, w nxt discuss complxity of Algorithm 1. At stp 2 of Algorithm 1 w valuat th dtrminant of positiv dfinit matrix. Thr ar O(m) such dtrminants to b computd. Th complxity of finding th dtrminant of a n n matrix is, in gnral, O(n 3 ). Whn choosing k snsors, Algorithm 1 has k stps and hnc its complxity is O(n 3 mk). 2 It is to b notd that f(s) is wll dfind for all substs S including th null st.