A CONSTRAINT GENERATION INTEGER PROGRAMMING APPROACH TO INFORMATION THEORETIC SENSOR RESOURCE MANAGEMENT

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1 A CONSTRAINT GENERATION INTEGER PROGRAMMING APPROACH TO INFORMATION THEORETIC SENSOR RESOURCE MANAGEMENT Jason L. Wams, 1, John W. Fsher III, Aan S. Wsy 1, 1 Laboratory for Informaton and Decson Systems and Computer Scence and Artfca Integence Laboratory Massachusetts Insttute of Technoogy, Cambrdge MA 0139 jwams@aum.mt.edu fsher@csa.mt.edu wsy@mt.edu ABSTRACT Many estmaton probems nvove sensors whch can be actvey controed to ater the nformaton receved and utzed n the underyng nference tas. In ths paper, we dscuss a nove nteger programmng method whch expots the submoduarty of nformaton theoretc estmaton crteron to fnd an effcent souton to constructng an open oop pan for sensor resource management probems nvovng many ndependent objects. The nteger programmng formuaton soves a sequence of smpfed probems; the souton of each forms an upper bound to the fu compexty probem. The updates performed between teratons may be vewed as steps n a constrant generaton process, ensurng that the bound s successvey tghtened. An auxary probem aso provdes a ower bound to the optma souton, and a souton attanng that bound, enabng eary termnaton wth a guaranteed near-optma souton. Computatona experments demonstrate the beneft that the agorthm can provde n varous pannng probems. Index Terms Sequenta decson procedures, sequenta estmaton, tracng 1. INTRODUCTION Actve sensng s motvated by modern sensors whch can be controed to observe dfferent aspects of an underyng probabstc process. For exampe, f we use cameras to trac peope n budngs, we can steer the camera to focus or zoom on dfferent peope or paces; n a sensor networ, we can choose to actvate and deactvate dfferent nodes and dfferent sensng modates wthn a partcuar node; or n a medca dagnoss probem we can choose whch tests to admnster to a patent. In each of these cases, our contro choces mpact the nformaton that we receve n our observaton, and thus the performance acheved n the underyng nference tas. A commony used performance objectve n actve sensng s mutua nformaton (MI) (e.g., [1]). Denotng the quantty that we am to nfer as X and the observaton resutng Ths wor was supported by MIT Lncon Laboratory through ACC PO# from contro choce u as z u, the MI between X and z u s defned as the expected reducton n the entropy produced by the observaton [],.e., I(X; z u ) = H(X) H(X z u ) = H(z u ) H(z u X). 1 Snce H(X) s nvarant to the contro choce u, choosng u to maxmze I(X; z u ) s equvaent to mnmzng the uncertanty n X as measured by the condtona entropy H(X z u ). In ths paper we propose a nove method for addressng a probem structure whch commony arses n mutpe object tracng, smar to that examned n [3]. Suppose that we have a number of objects, numbered {1,..., M}, each of whch can be observed usng any sensor tme sot. We see to construct an open oop pan of whch object to observe wth our sensor (or sensors) n each tme sot. Ths entre pan coud be executed, or the frst few steps coud be executed and an updated pan constructed (so-caed Open Loop Feedbac Contro, [4]). To motvate ths structure, consder a probem n whch we use an arborne sensor to trac objects movng on the ground beneath foage. In some postons, objects w be n cear vew and observatons w yed accurate poston nformaton; n other postons, objects w be obscured by foage and observatons w be essentay unnformatve. Wthn the tme scae of a pannng horzon, objects w move n and out of obscuraton, and t w be preferabe to observe objects durng the porton of tme n whch they are expected to be n cear vew. Our agorthm expots submoduarty, the same property used to obtan performance guarantees for greedy heurstcs n [5, 4]. Submoduarty captures the noton that as we seect more observatons, the vaue of the remanng unseected observatons decreases,.e., the noton of dmnshng returns. Defnton 1. A set functon f s submoduar f f(c A) f(a) f(c B) f(b) B A. It was estabshed n [5] that, assumng that observatons are ndependent condtoned on the quantty to be estmated, MI s a submoduar functon of the observaton seecton set. 1 Note that when we condton on a random varabe (such as a yet unreazed observaton) the condtona entropy nvoves an expectaton over the dstrbuton of that random varabe.

2 The smpe resut that we w utze from submoduarty s that I(x; z C z A ) I(x; z C z B ) B A.. INTEGER PROGRAMMING FORMULATION The emphass of our formuaton s to expot the structure whch resuts n sensor management probems nvovng observaton of mutpe ndependent objects. In addton to the above assumpton that observatons shoud be ndependent condtoned on the state, three new assumptons regardng the objects states must be met for ths structure to arse: (1) the pror dstrbuton over object states must be ndependent; () the objects must evove accordngto ndependent dynamca processes; and (3) the objects must be observed through ndependent observaton processes. The frst two assumptons are not overy mtng n mut-object tracng probems. The thrd, whch s often voated (e.g., due to data assocaton), s made as an approxmaton for pannng purposes. In crcumstances nvovng strong dependency between sma numbers of objects (out of a arge tota number), dependent objects may be consdered as a coectve hyper-object, and ndependence of the hyper-objects remans. When the three assumptons are met, the mutua nformaton reward of observatons of dfferent objects becomes the sum of the ndvdua observaton rewards. Denotng by X = {x 1,..., x N } the jont state (over the N-step pannng horzon) of object, we defne the reward of observaton set A {1,..., N} of object (.e., A represents the subset of tme sots n whch we observe object ) to be: ra = I(X ; za ) (1) where z A are the random varabes correspondng to the observatons of object n the tme sots n A. Under the precedng assumptons, we can wrte the reward of choosng observaton set A for object {1,..., M} as: I(X 1,..., X M ; za 1 1,..., zm A ) M M M = I(X ; za ) = ra () =1 =1 As a sght generazaton, et R denote the set of sensng resources that are avaabe (assumed fnte). The eements of R may correspond to dfferent tme sots of the same sensor, the same tme sot of dfferent sensors, or combnatons of both. We assume that each eement of R may be assgned at most one tas (athough ths can be easy generazed, as shown n [4]). As another sght generazaton, et U = {u 1,..., u L } be the set of eementa observaton actons (assumed fnte) that may be used for object, where each eementa observaton u j corresponds to observng object usng a partcuar mode of a partcuar sensor wthn a partcuar perod of tme. An eementa acton may occupy mutpe resources; et t(u j ) R be the subset of resources consumed by the eementa observaton acton u j. Let S U be the coecton of observaton subsets whch we aow for object. Ths s assumed to tae the form of Eq. (3), consstng smpy of a subsets of U for whch no two eements consume the same resource: S = {A U t(u 1 ) t(u ) = u 1, u A} (3) We denote by t(a) R the set of resources consumed by the actons n set A,.e., t(a) = u A t(u). The probem that we (conceptuay) see to sove s that of seectng the set of observaton actons for each object such that the tota reward s maxmzed subject to the constrant that each resource can be used at most once: max ω A s.t. M ra ω A =1 A S M =1 A S t t(a ) (4a) ω A 1 t R (4b) ωa = 1 {1,..., M} (4c) A S ω A {0, 1}, A S (4d) The bnary ndcator varabes ωa are 1 f the observaton set A s chosen for object and 0 otherwse. The constrants n Eq. (4b) ensure that each resource (e.g., sensor tme sot) s used at most once. The constrants n Eq. (4c) ensure that exacty one observaton set s chosen for any gven object; ths s necessary to ensure that the addtve objectve s the exact reward of correspondng seecton (snce, n genera, ra B ra + r B ). The probem s not a pure assgnment probem, as the observaton subsets A S consume mutpe resources and hence appear n more than one of the constrants defned by Eq. (4b). The probem s actuay a bunde assgnment probem, and conceptuay coud be addressed usng combnatora aucton methods (e.g., [6]). However, generay ths woud requre computaton of ra for every subset A S. If the coectons of observaton sets S, {1,..., M} aow for severa observatons to be taen of the same object, the number of subsets may be combnatoray arge. 3. CONSTRAINT GENERATION APPROACH Ths secton outnes the approach we propose, whch n many practca stuatons can provde an effcent souton of the nteger program n Eq. (4). The agorthm, whch s descrbed n deta n [4], proceeds by sequentay sovng a seres of nteger programs wth progressvey greater compexty. In the mt, we arrve at the fu compexty of the nteger program n Eq. (4), but n many practca stuatons t s possbe to termnate much sooner wth an optma souton. By smutaneousy owerng an upper bound on the optma souton, and

3 rasng a ower bound on the optma souton (whch comes aongsde a souton attanng the ower bound), we can aso termnate eary wth a souton that s guaranteed to be wthn a gven fracton of optmaty. The formuaton may be conceptuay understood as dvdng the coecton of subsets for each object (S ) at teraton nto two coectons: T S and the remander S \T. The subsets n T are those for whch the exact reward has been evauated; we w refer to these as canddate subsets. The reward of each of the remanng subsets (.e., those n S \T ) has not been evauated, but an upper bound to each reward s avaabe. In practce, we w not expcty enumerate the eements n S \T ; rather we use a compact representaton whch mpcty consders a eements on the bass of upper bounds obtaned usng submoduarty. The compact representaton of S \T assocates wth each canddate subset, A T, a subset of observaton actons, B,A ; A may be augmented wth any subset of B,A to generate new subsets that are not n T (but that are n S ). We refer to B,A as an exporaton subset, snce t provdes a mechansm for dscoverng promsng new subsets that shoud be ncorporated nto T+1. The addtona reward for seectng an exporaton subset eement u B,A when the cand- date subset A s aready seected s r u A r A {u} r A. By submoduarty, C B,A, r A C r A + u C r u A Equaty w hod f C 1. To ntaze the probem, we seect T0 = { }, and B0, = U for a. The nteger program that we sove at each stage s: max M ra ω A + (5a) ω A, ω u A =1 s.t. M =1 + A T A T t t(a ) M =1 A T ω A u B,A t t(u) u B,A r u A ω u A ω u A 1 t R (5b) ωa = 1 {1,..., M} (5c) A T ωu A B,A ω A 0, A T u B,A (5d) We assume throughout that B,A A = ; our agorthm for constructng B,A w ensure that ths s the case. ω A {0, 1}, A T (5e) ω u A {0, 1}, A T, u B,A (5f) The souton of the nteger program seects the subset for each object that maxmzes the upper bound, ensurng that the resource constrants (e.g., Eq. (5b)) are satsfed. The observaton subset seected for object s the set A for whch ωa = 1, augmented by any addtona observatons u for whch ωu A = 1. If the subset that the nteger program seects for each object s n T.e., t s a subset whch had been generated and for whch the exact reward had been evauated n a prevous teraton then we have found an optma souton to the orgna probem,.e., Eq. (4). Ths occurs when no more than one exporaton subset eement s chosen for each and every object. Conversey, f the nteger program seects a subset n S \T for one or more objects, then we need to tghten the upper bounds on the rewards of those subsets, e.g., by addng the newy seected subsets to T n the next teraton and evauatng ther exact rewards. Ths occurs when two or more exporaton subset eements are chosen for any object. Each teraton of the optmzaton reconsders a decson varabes, aowng the souton from the prevous teraton to be augmented or reversed n any way. The agorthm used to update the canddate subsets T and exporaton subsets B,A between teratons ensures that the upper bounds are tghtened at each teraton. The agorthm and ts theoretc characterstcs are expored n deta n [4]. At each teraton, a new canddate subset s ntroduced for each object for whch more than two exporaton subset eements were seected. The new canddate subset conssts of the prevousy seected subset A, augmented wth the exporaton subset eement (among those seected) u wth the greatest ncrementa reward. The exporaton subset for A canddate subset s updated such that u s removed,.e., B+1,A = B,A \{u }. The exporaton subset for the new canddate subset A {u } s set to the same subset (.e., B+1,A {u } = B +1,A ) Augmented nteger program In the prevous secton we descrbed a sequence of nteger programs whch form a progressvey tghter upper bound to the souton of the fu compexty nteger program n Eq. (4). In each teraton, we aso sove an augmented nteger program, whch provdes the best souton amongst a soutons for whch the exact reward has been evauated (.e., a ower bound to the reward of the optma souton); ths s formed smpy by addng to Eq. (5) constrants that prevent seecton of more than one exporaton subset eement for any object. The reward of ths augmented nteger program s a nondecreasng functon of teraton number. By combnng ths 3 Actuay, any eements whch cannot be seected aongsde u (e.g., due to resource constrants) are removed from B +1,A {u }.

4 best souton wth the upper bound produced by the prevousy descrbed constrant generaton agorthm, we can termnate when we are wthn a desred toerance of optmaty. 3.. Comments On the surface, our agorthm bears some smarty to the recent wor [7], whch aso soves a sensor resource management probem through an teratve souton of nteger programs. However, the souton methodoogy n [7] s contngent on the cost crteron yedng a reaxaton to a convex mnmzaton. The obvous reaxaton of nformaton theoretc seecton probems s a convex maxmzaton [4] (a geometry for whch few usefu toos exst), hence t s uncear how to appy the approach n [7] when an nformaton theoretc crteron s used. 4. EXPERIMENTAL RESULTS The agorthm was mpemented usng C++, sovng the nteger programs usng ILOG R CPLEX R 10.1 through the caabe brary nterface. Termnaton occurs when the souton of the augmented nteger program s guaranteed to be wthn 95% of optmaty Mutpe object tracng Our frst exampe modes surveance of mutpe objects by a radar patform movng n a fxed racetrac pattern. Observaton nose ncreases when objects become cose to each other: ths s a surrogate for the mpact of data assocaton, athough we do not mode the dependency between objects whch generay resuts. We denote by y the state (.e., poston and veocty) of the sensng patform at tme. There are M objects under trac, the states of whch evove accordng to a nomnay constant veocty mode: x +1 = x + w (6) where w s a dscrete tme zero-mean Gaussan whte nose process wth covarance Q = q (7) wth = 0.01 sec, and q = 0.5. The smuaton runs for 100 tme sots. The nta postons of the objects are dstrbuted unformy on the regon [10, 100] [10, 100]; veocty magntudes are drawn from a Gaussan dstrbuton wth mean 30 and standard devaton 0.5, whe the veocty drectons are dstrbuted unformy on [0, π]. The nta estmates are set to the true state, corrupted by addtve Gaussan nose wth zero mean and standard devaton 0.0 (n poston states) and 0.1 (n veocty states). In each tme sot, the sensor may observe one of the M objects, obtanng ether an azmuth and range observaton, or an azmuth and range rate observaton, each of whch occupes a snge tme sot: [ z,r = z,d = ( tan 1 [x ] y ]3 ([x y ] 1 ) + ([x y ] 3 ) [x y ]1 ) [ ] + d (x 1,..., x M b(x ), y ) 0 v,r 0 1 (8) ( ) tan 1 [x y ]3 [x y ]1 [x y ]1[x y ]+[x y ]3[x y ]4 ([x y ] 1) +([x y ]3) [ + d (x 1,..., x M b(x ), y ) ] v,d (9) where z,r denotes the azmuth/range observaton for object at tme, and z,d denotes the azmuth/range rate (.e., Dopper) observaton. The notaton [a] denotes the -th eement of the vector a; the frst and thrd eements of the object state x and the sensor state y contan the poston n the x-axs and y-axs respectvey, whe the second and fourth eements contan the veocty n the x-axs and y-axs respectvey. The observaton noses v,r and v,d are ndependent whte Gaussan nose processes wth zero mean and ndependent eements. The standard devaton of the nose on the azmuth observatons (σ φ ) s 3 ; the mutper functon b(x, yj ) vares from unty on the broadsde (.e., when the sensor patform headng s perpendcuar to the vector from the sensor to the object) to end-on. The standard devaton of the range observaton (σ r ) s 0.1 unts, whe the standard devaton of the range rate observaton (σ d ) s unts/sec. The functon d(x 1,..., xm ) captures the ncrease n observaton nose when objects are cose together: d (x 1,..., x M ) = ) δ ( ([x xj ] 1) + ([x xj ] 3) j where δ(x) = 10 x for 0 x 10 and δ(x) = 0 otherwse. The state dependent nose s handed n a manner smar to the optma near estmator for bnear systems, n whch we estmate the varance of the observaton nose, and then use ths n a conventona nearzed Kaman fter (for reward evauatons for pannng) and extended Kaman fter (for estmaton). In addton to the opton of these two observatons, the sensor can aso choose a more accurate observaton that taes three tme sots to compete, and s not subject ncreased nose when objects become cosey spaced. The azmuth nose for these observatons n the broadsde aspect has σ φ = 0.6, whe the range nose has σ r = 0.0 unts, and the range rate nose has σ d = unts/sec.

5 Reatve gan Average tme (seconds) Performance n 0 smuatons of 50 objects Horzon ength (tme sots) Average computaton tme to produce pan Horzon ength (tme sots) Fg. 1. Top dagram shows the tota reward for each pannng horzon ength dvded by the tota reward for a snge step pannng horzon, averaged over 0 smuatons. Error bars show the standard devaton of the mean performance estmate. Lower dagram shows the average tme requred to produce pan for dfferent pannng horzon engths. The resuts of the smuaton are shown n Fg. 1. When the pannng horzon s ess than three tme steps, the controer does not have the opton of the three tme step observaton avaabe to t. A moderate gan n performance s obtaned by extendng the pannng horzon from one tme step to three tme steps to enabe use of the onger observaton. The ncrease s roughy doubed as the pannng horzon s ncreased, aowng the controer to antcpate perods when observatons for some objects are poor. As expected, the compexty ncreases exponentay wth the pannng horzon ength. However, usng the agorthm t s possbe to produce a pan for 50 objects over 0 tme sots usng a few seconds n computaton tme. Performng the same pannng through fu enumeraton woud nvove evauaton of the reward of more than dfferent canddate sequences, a computaton whch s ntractabe on any foreseeabe computatona hardware. 4.. Exampe of possbe beneft The scenaro we dscuss here demonstrates the ncrease n performance whch s possbe through ong pannng horzons when observatons occupy dfferent numbers of tme sots. In such crcumstances, agorthms utzng short-term pannng may mae choces that precude seecton of ater observatons that may be arbtrary more vauabe. The scenaro nvoves M = 50 objects observed usng a snge sensor through a near Gaussan observaton mode. The nta dstrbuton of the objects s jonty Gaussan, where a objects are ndependent wth covarance I. In each tme sot, a snge object may be observed through ether of two near Gaussan observatons (.e., of the form z = Hx + v, where v N {0, R}). The frst, whch occupes a snge tme sot, has H,1 = I, and R,1 = I. The second, whch occupes fve tme sots, has H, = I, and R, = r I. The nose varance of the onger observaton, r, vares perodcay wth tme (), accordng to r = 10 mod( 1,5) 1 (the tme ndex commences at = 1). Uness the pannng horzon s suffcenty ong to antcpate the avaabty of the observaton wth nose varance 10 5 severa tme steps ater, the agorthm w seect an observaton wth ower reward, whch precudes seecton of ths ater more accurate observaton. The performance s examned n deta n [4]. As the pannng horzon ncreases from a snge tme sot (.e., myopc) to 50 tme sots, the performance (reward) ncreases by a factor of 4.7. The computaton tme requred to produce a pan for 50 tme sots s on the order of tens of mseconds. Whe ths s an extreme exampe, t ustrates an occason when pannng s hghy benefca (when there are observatons that occupy severa tme sots wth tme varyng rewards), and that the proposed agorthm s abe to effcenty sove arge pannng probems n such a stuaton. An agorthm utzng short-term pannng n such crcumstances may mae choces that precude seecton of ater observatons whch may be arbtrary more vauabe. 5. REFERENCES [1] K.J. Hntz and E.S. McVey, Mut-process constraned estmaton, Systems, Man and Cybernetcs, IEEE Transactons on, vo. 1, no. 1, pp , [] Thomas M. Cover and Joy A. Thomas, Eements of Informaton Theory, John Wey and Sons, New Yor, NY, [3] V. Krshnamurthy and R.J. Evans, Hdden Marov mode mutarm bandts: a methodoogy for beam schedung n muttarget tracng, Sgna Processng, IEEE Transactons on, vo. 49, no. 1, pp , December 001. [4] Jason L. Wams, Informaton Theoretc Sensor Management, Ph.D. thess, Massachusetts Insttute of Technoogy, February 007, Avaabe onne at [5] Andreas Krause and Caros Guestrn, Near-optma nonmyopc vaue of nformaton n graphca modes, n Uncertanty n Artfca Integence, Juy 005. [6] Davd C. Pares and Lye H. Ungar, Iteratve combnatora auctons: Theory and practce, n Proc 17th Natona Conference on Artfca Integence (AAAI), 000, pp [7] Amt S. Chhetr, Darry Morre, and Antona Papandreou- Suppappoa, Sensor resource aocaton for tracng usng outer approxmaton, IEEE Sgna Processng Letters, vo. 14, no. 3, pp , Mar. 007.