Subspace Learning From Bits

Transcription

1 Subspace Learnng Fro Bs Yueje Ch, Meber, IEEE, Haoyu Fu, Suden Meber, IEEE arxv: v3 [sa.ml 3 Jan 27 Absrac Neworked sensng, where he goal s o perfor coplex nference usng a large nuber of nexpensve and decenralzed sensors, has becoe an ncreasngly aracve research opc due o s applcaons n wreless sensor neworks and nerne-of-hngs. To reduce he councaon, sensng and sorage coplexy, hs paper proposes a sple sensng and esaon fraework o fahfully recover he prncpal subspace of hgh-densonal daa sreas usng a collecon of bnary easureens fro dsrbued sensors, whou ransng he whole daa. The bnary easureens are desgned o ndcae coparson oucoes of aggregaed energy projecons of he daa saples over pars of randoly seleced drecons. When he covarance arx s a low-rank arx, we propose a specral esaor ha recovers he prncpal subspace of he covarance arx as he subspace spanned by he op egenvecors of a properly desgned surrogae arx, whch s provably accurae as soon as he nuber of bnary easureens s suffcenly large. An adapve rank selecon sraegy based on sof hresholdng s also presened. Furherore, we propose a alored specral esaor when he covarance arx s addonally Toeplz, and show relable esaon can be obaned fro a subsanally saller nuber of bnary easureens. Our resuls hold even when a consan fracon of he bnary easureens s randoly flpped. Fnally, we develop a low-coplexy onlne algorh o rack he prncpal subspace when new easureens arrve sequenally. Nuercal exaples are provded o valdae he proposed approach. Index Ters nework sensng, prncpal subspace esaon, subspace rackng, bnary sensng I. INTRODUCTION Neworked sensng, where he goal s o perfor coplex nference usng daa colleced fro a large nuber of nexpensve and decenralzed sensors, has becoe an ncreasngly aracve research opc n recen years due o s applcaons n wreless sensor neworks and nerne-of-hngs. Consder, for exaple, a daa srea whch generaes a zero-ean hghdensonal daa saple x C n a each e, and each sensor ay access a poron of he daa srea. Several an challenges arse when processng he hgh-densonal daa srea: Daa on-he-fly: Due o he hgh rae of daa arrval, each daa saple x ay no be fully sored, and copuaon needs o be accoplshed wh only one pass or a few passes [3 a he sensors o allow fas processng. The auhors are wh he Deparen of Elecrcal and Copuer Engneerng, The Oho Sae Unversy, Colubus, OH 432. Eal: {ch.97, fu.436}@osu.edu. Prelnary resuls have been presened n par a he 24 IEEE Global Conference on Sgnal and Inforaon Processng GlobalSIP) [ and he 26 Asloar Conference on Sgnals, Syses, and Copuers [2. Ths aeral s based upon work suppored by he Ar Force Suer Faculy Fellowshp Progra, by Naonal Scence Foundaon under award nuber ECCS-4629, and by he Ar Force Offce of Scenfc Research under award nuber FA Resource consrans: The sensors usually are powerhungry and resource-led, herefore s hghly desrable o nze he copuaonal and sorage cos a he sensor sde, as well as he councaon overheads o he fuson cener by no ransng all he daa. Dynacs: as new daa saples arrve and/or new sensors ener, s neresng o rack he changes of he nforaon flow a he fuson cener whou sorng all hsory daa n a low-coplexy fashon. A. Conrbuons of hs paper Many praccal daa exhb low-densonal srucures, such ha a sgnfcan proporon of her varance can be capured n he frs few prncpal coponens; and ha he subspace spanned by hese prncpal coponens s he recovery objec of neres raher han he daases heselves. In oher words, he covarance arx of he daa Σ = E[x x H s approxaely) low-rank, rankσ) r, where r n. Ths assupon s wdely applcable o daa such as nework raffc, wdeband specru, survellance, and so on. To reduce he councaon, sensng and sorage coplexy, s of grea neres o consder one-b saplng sraeges a decenralzed sensor nodes [4, [5, [6, where each sensor only ranss a sngle b o he fuson cener raher han all he daa. Our goal n hs paper s hus o desgn a sple ye effcen one-b saplng sraegy o fahfully rereve he prncpal subspace of hgh-densonal daa sreas wh..d. saples when her covarance arces are approxaely) low-rank wh provable perforance guaranees. We focus on he scenaro where each dsrbued sensor ay only access a subse of he whole daa, process he locally, and rans only a sngle b o he fuson cener, who wll esae he prncpal subspace whou referrng o he orgnal daa. I s of neres o reduce he aoun of easureens requred o councae o he fuson cener whle keepng he nuber of sensors as sall as possble o allow fahful recovery of he prncpal subspace. In he proposed one-b saplng schee, each sensor s equpped wh a par of vecors coposed of..d. coplex sandard Gaussan enres, called skechng vecors. A he sensng sage, copares he energy projecons of he daa seen a he sensor ono he par of skechng vecors respecvely, and ranss a sngle b ndcang whch of he wo energy projecons s larger. Ths s equvalen o coparng he energy projecon of a saple covarance arx ono wo randoly seleced rank-one subspaces. A key observaon s ha as long as he nuber of saples seen a he sensor s no oo sall whch we characerze heorecally), he coparson oucoe wll be exacly he sae as f s perfored on he covarance arx or s bes

2 2 low-rank approxaon. By only ransng he coparson oucoe raher han he acual energy easureens, he councaon overhead s nzed o a sngle b whch s ore robus o councaon channel errors and oulers. Moreover, as wll be shown, he energy projecons can be copued exreely sple whou sorng he hsory daa saples, and are always nonnegave, akng he suable for wdeband and opcal applcaons. A he fuson cener, he skechng vecors are assued known, whch s a sandard assupon for decenralzed esaon [7. When he covarance arx s exacly low-rank, we propose a specral esaor, whch esaes he prncpal subspace as he subspace spanned by he op egenvecors of a carefully desgned surrogae arx usng he colleced bnary easureens, whch can be easly copued va a runcaed egenvalue decoposon EVD) wh a low copuaonal coplexy. We show ha, assung all he b easureens are exacly easurng he covarance arx, he esae of he prncpal subspace s provably accurae as long as he nuber of bs s on he order ofnr 3 logn, when he skechng vecors are coposed of..d. sandard coplex Gaussan enres. When he rank s no known a pror, we devse a sof-hresholdng sraegy o adapvely selec he rank, and show obans a slar perforance guaranee. Furherore, we developed a eory-effcen algorh o onlne updae he prncpal subspace esae when he bnary easureens arrve sequenally, whch can be pleened wh a eory requreen on he order of he sze of he prncpal subspace raher han ha of he covarance arx. In any applcaons concernng power) specru esaon, such as array sgnal processng and cognve rados, he covarance arx of he daa can be odeled as a low-rank Toeplz arx [8. I s herefore possble o furher reduce he requred nuber of b easureens by explong he Toeplz consran. We propose o apply he specral esaor o he projecon of he above desgned surrogae arx o s neares Toeplz arx n he Frobenus nor. When he covarance arx s rank-one, provably ads accurae esaon of he prncpal subspace as soon as he nuber of b easureens s on he order of log 4 n. In conras o he scenaro when he Toeplz consran s no explored, hs elnaes he lnear dependency wh n, hus he saple coplexy s grealy reduced. Nuercal sulaons also sugges he algorh works well even n he low-rank seng. Fnally, our resuls connue o hold even when a consan fracon of he bnary easureens s randoly flpped. B. Relaed work Esang he prncpal subspace of a hgh-densonal daa srea fro s sparse observaons has been suded n recen years, bu os exsng work has been focused on recovery of he daa srea [9, [. Recenly, skechng has been prooed as a densonaly reducon ehod o drecly recover he sascs of he daa [ [7. The proposed fraework n hs paper s ovaed by he covarance skechng schee n [5, [6, [7, where a quadrac saplng schee s desgned for low-rank covarance esaon. I s shown n [5 ha a nuber of real-valued quadrac energy) easureens on he order of nr suffces o exacly recover rank-r covarance arces va nuclear nor nzaon, assung he easureen vecors are coposed of..d. sub-gaussan enres. However, ransng hese energy easureens wh hgh precson ay cause unwaned overhead and requre esang he nose level n pracce. Dsrbued esaon of a scalar-valued paraeer fro he one-b quanzaon of s nosy observaons has been consdered n [4, [5, [8, [9. Recenly, one-b copressed sensng [2 [25 and one-b arx copleon [26, [27 have generalzed hs o he esaon of vecor-valued paraeers such as sparse vecors and low-rank arces, where hey a o recover he sgnal of neres fro he sgns of s rando lnear easureens. Our work s relaed o oneb arx copleon as we also consder low-rank srucures of he covarance arces, bu dffers n several poran aspecs. Frs, unlke exsng work, our bnary easureens are no consruced drecly fro he low-rank covarance arx, bu raher a saple covarance arx, herefore we need o carefully jusfy when he bnary easureens accuraely reflec he characersc of he rue covarance arx. Second, he easureen operaors wh respec o he covarance arx ake he for of he dfference of wo rank-one arces, desgned o allow a low-coplexy pleenaon, whch leads o very dfferen resuls fro exsng ones ha assue..d. enres [23. Thrd, we propose sple specral esaors for reconsrucng he prncpal subspace of low-rank covarance arces, and deonsrae boh analycally and hrough sulaons ha obans slar perforance as he ore expensve convex progras usng race nor nzaon [23. Fnally, he specral esaor can be furher alored o he case of low-rank Toeplz covarance arces. Dsrbued wdeband specru sensng for cognve rados s an appealng and ovang applcaon [28. I s recenly proposed o esae he power specral densy of wdeband sgnals va leas-squares esaon fro sub-nyqus saples [29. The frugal sensng fraework [6 consdered he sae esaon proble usng one-b easureens based on coparng he average sgnal power whn a band of neres agans a pre-deerned or adapvely-se hreshold [3. Ther algorh s based on lnear prograng and ay explore paraerc represenaons of he power specral densy. Our work s dfferen fro [6, [3 n several aspecs. Insead of coparng he average sgnal power agans a hreshold whch nroduces he addonal ssue of how o se he hreshold, we copare he average sgnal power beween wo dfferen bands of neres and herefore do no need o se any hreshold. Our algorh explores he low-rank propery of he power specral densy raher han s paraerc represenaon, and does no requre knowng he nose sascs bu explores he concenraon phenoenon of rando arces. Fnally, he paper [3 suded one-b phase rereval, an exenson of he one-b copressed sensng wh phaseless easureens. Despe dfferen ovaons and applcaons, our algorh subsues he scenaro n [3 as a specal case when he covarance arx s assued rank-one.

3 3 C. Organzaon of hs paper and noaons The res of hs paper s organzed as follows. Secon II descrbes he proposed -b saplng fraework and forulaes he prncpal subspace esaon proble. Secon III presens he proposed specral esaors and her perforance guaranees. Secon IV presens an onlne algorh o rack he low-densonal prncpal subspace wh sequenal b easureens. Nuercal exaples are gven n Secon V. Fnally, we conclude n Secon VI and oulne soe fuure drecons. Addonal proofs are provded n he appendx. Throughou hs paper, we use boldface leers o denoe vecors and arces, e.g. a and A. The Heran ranspose of a s denoed by a H, and A, A F, A, TrA) denoe he specral nor, he Frobenus nor, he nuclear nor, and he race of he arx A, respecvely. Denoe T A) = argn T T A F s.. T s Toeplz, as he lnear projecon of A o he subspace of Toeplz arces, and T A) = A T A). Defne he nner produc beween wo arces A,B as A,B = TrB H A). If A s posve sedefne PSD), hen A. The expecaon of a rando varable a s wren as E[a. II. ONE-BIT SAMPLING STRATEGY Le{x } = Cn be a daa srea wh zero-eane[x = and he covarance arx Σ = E[x x H. In hs secon we descrbe he dsrbued one-b saplng fraework for esang he prncpal subspace of Σ based on coparson oucoes of aggregaed energy projecons fro each sensor, as suarzed n Algorh. Algorh One-B Saplng Sraegy Inpu: A daa srea{x } = ; : for each sensor =,..., do 2: Randoly choose wo skech vecors a C n and b C n wh..d. Gaussan enres; 3: Skech an arbrary subsrea ndexed by {l }T = wh wo energy easureens a,x l 2 and b,x l 2, and rans a bnary b o he fuson cener: ) T y,t = sgn a,x T l 2 T b,x T l 2. 4: end for = = Consder a collecon of sensors ha are deployed dsrbuvely o easure he daa srea. Each sensor can access eher a poron or he coplee daa srea. A he h sensor, defne a par of skechng vecors a C n and b C n,, where her enres are..d. sandard coplex Gaussan CN, ). Whou loss of generaly, we assue he h sensor has access o he saples n he daa srea ndexed byl = {l } T = of he sae sze T. Eachh sensor processes he saples locally, naely, for he daa saple x l, akes wo quadrac energy) easureens gven below: u, = a,x l 2, v, = b,x l 2, ) whch are nonnegave and can be easured effcenly n hgh frequency applcaons a a lnear coplexy usng energy deecors. These quadrac easureens are hen averaged over he T saples o oban where U,T = T V,T = T T u, = a H Σ,Ta, = T v, = b H Σ,Tb, = Σ,T = T T x l x H l = s he saple covarance arx seen by he h sensor. I s clear ha U,T = T T U,T + T u,t, and slarly V,T, can be updaed recursvely whou sorng all he hsory daa. A he end of he T saples, he h sensor copares he average energy projecons U,T and V,T, and rans o he fuson cener a sngle b y,t ndcang he oucoe: {, f U,T > V y,t =,T. 3), oherwse The councaon overhead s nal snce only he bnary easureen s ransed raher han he orgnal daa srea. I s also sraghforward o see ha each sensor only needs o sore wo scalars, U,T and V,T. More concreely, defne W = a a H b b H, we can wre 3) as 2) y,t = sgn W,Σ,T ), 4) where sgn ) s he sgn funcon. Inuvely, 4) can be nerpreed as coparng he energy projecon of Σ,T ono wo randoly seleced rank-one subspaces. Fnally, o odel poenal errors occurred durng ranssson, we assue each b has an ndependen flppng probably of p < /2, and he receved b a he fuson cener fro he h sensor s gven as z,t = y,t ǫ,, 5) where Pǫ = ) = p and Pǫ = ) = p are..d. across sensors. As we re neresed n he covarance arx Σ, and noe ha he saple covarance arx Σ,T converges o Σ as T approaches nfny, he bnary easureen a he h sensor also approaches quckly o he followng, y = sgn W,Σ ), 6) as f s easurng he rue covarance arx Σ. In fac, as we ll show n Theore, y and y,t sar o agree very fas for T uch saller han n. For splcy we assue Σ s an exacly rank-r PSD arx, where r n. The exenson o approxaely lowrank case wll be dscussed shorly. Le r Σ = λ k u k u H k = UΛU H, 7) k= where U = [u,u 2,...,u r are he op-r egenvecors of Σ

4 4 and {λ k } r k= are he op-r egenvalues wh λ λ 2 λ r. We furher defne v k, k =,...,n r as he bass vecors spannng he copleen of U. Apparenly no all nforaon abou Σ can be recovered, for exaple, he sgn easureens are nvaran o scalng of he daa saples, and herefore, scalng of he covarance arx. Our goal n hs paper s o recover he prncpal subspace spanned by U C n r fro he colleced bnary easureens. A. How large does T need o be? In he followng proposon, whose proof can be found n Appendx B, we esablsh he saple coplexy of T o guaranee y = y,t f he daa srea follows a Gaussan odel x CN,Σ) wh..d. saples. Proposon. Le < δ. Assue x are..d. Gaussan sasfyng x CN,Σ). Then P[y,T y δ as soon as T > c TrΣ) Σ F log 2 /δ) for soe suffcenly large consan c. In order o guaranee ha all bs are accurae, we need o furher apply a unon bound o Proposon, whch yelds he followng heore. Theore. Le x be..d. x CN,Σ). Le < δ. Wh probably a leas δ, all bnary easureens are exac,.e. y,t = y for =,..., gven ha he nuber of saples observed by each sensor sasfes T > c TrΣ) log 2 Σ F δ for soe suffcenly large consan c. I s worh ephaszng ha Theore holds for any fxed covarance arx Σ. The er TrΣ)/ Σ F easures he effecve rank of Σ, as for PSD arces wh fas specral decays hs er wll be sall [32. If rankσ) = r, hen TrΣ) r Σ F. As soon as T s on he order of rlog 2 all b easureens are accurae wh hgh probably, whch only depends on he nuber of sensors whch n urn depend on he aben denson n as wll be seen fro Theore 2) logarhcally. B. Exenson o approxae low-rank covarance arces When Σ s only approxaely low-rank, denoe s bes rank-r approxaon as Σ r = argn ranka)=r Σ A F, hen f sgn W,Σ ) = sgn W,Σ r ), =,...,, 8) holds, cobned wh Theore, our fraework can be appled o recover he r-densonal prncpal subspace of Σ, by reang he bs as easureens of Σ r. Forunaely, 8) holds wh hgh probably as long as Σ Σ r / Σ F c log)/n s suffcenly sall. The neresed readers are referred o Appendx C for he deals. III. PRINCIPAL SUBSPACE ESTIMATORS AND PERFORMANCE GUARANTEES In hs secon, we frs develop a specral esaor when Σ s a low-rank PSD arx based on runcaed EVD when he ) rank s known exacly; and hen we develop a sof-hresholdng sraegy o adapvely selec he rank when s unknown. Fnally, we alor he specral esaor o he case when Σ s a low-rank Toeplz PSD arx. For he res of hs secon, we assue he bnary easureens y,t = y and z = y ǫ n 5) for all =,,. A. The specral esaor We propose an exreely sple and low-coplexy specral esaor whose coplexy aouns o copung a few op egenvecors of a carefully desgned surrogae arx. To ovae he algorh, consder he specal case when Σ = νν H s a rank-one arx wh ν 2 =. A naural way o recover ν s va he followng: ax ν: ν 2= z W,νν H, 9) = whch as o fnd a rank-one arxνν H ha agrees wh he easured sgns as uch as possble. Snce 9) s equvalen o ax ν: ν νh z W )ν, 2= = s soluon s he op egenvecor of he surrogae arx: J = z W = ǫ sgn W,Σ )W. ) = = More generally, when Σ s rank-r, we recover s prncpal subspace as he subspace spanned by he op-r egenvecors Û C n r of he surrogae arx J n ). Ths procedure s denoed as he specral esaor. B. Saple coplexy of he specral esaor We esablsh he perforance guaranee of he proposed specral esaor by showng ha he prncpal subspace of Σ can be accuraely esaed usng J as soon as s suffcenly large. Ths s accoplshed n wo seps. Defne J = E[J. We frs show ha he prncpal subspace of J s he sae as ha of Σ, and hen show ha J s concenraed around J for suffcenly large. The followng lea accoplshes he frs sep. Lea. The prncpal subspace of J s he sae as ha of Σ wh rankj) = r. For k =,...,r, { uh k Ju ) } r k 2p) ax, +κσ) 9r e κσ), ) and for k =,...,n r, v H k Jv k =, 2) where κσ) = λ /λ r s he condonng nuber of Σ. When r =, he rgh-hand sde of ) equals one. The proof s provded n Appendx D. Lea esablshes ha J and Σ share he sae prncpal subspace, bu her egenvecors ay sll dffer. Followng Lea, he specral

5 5 gap beween he rh egenvalue and he r +)h egenvalue whch s zero) of J s a leas { ) } r α := 2p)ax, +κσ) 9r e κσ), where he frs er exponenal n r) s gher when r s sall whle he second er polynoal n r) s gher when r s large. Indeed, when k =,...,r, u H k Ju k only depends on {λ k } r k= and can be copued exacly once hey are fxed. Fg. plos he derved lower bounds and he exac values of u H k Ju k assung all λ k =. I can be seen ha he polynoal bound on he order of /r s raher accurae excep he leadng consan when r s large. Moreover, fro Fg. confrs ha alhough J preserves he prncpal subspace of Σ, does no preserve he egenvecors and egenvalues. Nex, we show ha for suffcenly large, he arx J s close o s expecaon J. We have he followng lea whose proof can be found n Appendx E. Lea 2. Le < δ <. Then wh probably a leas δ, we have ha ) c n 2n J J log, δ where c s an absolue consan. Our an heore hen edaely follows by applyng an proveen of he Davs-Kahan sn-thea heore [33 n Lea 7, as gven below. Theore 2. Le < δ <. Wh probably a leas δ, here exss an r r orhonoral arx Q such ha ) Û UQ F α c nr 2n log δ { } n +κσ)) r,9e κσ) r 2p) where c s an absolue consan. c nr log ) 2n, δ When κσ) s sall and r s oderae, α scales as /r and we have ha as soon as he nuber of bnary easureens exceeds he order of nr 3 logn, s suffcen o recover o recover he prncpal subspace spanned by he coluns of U wh hgh accuracy. Snce here are a leas nr degrees of freedo o descrbe U, our bound s near-opal up o a polynoal facor wh respec o r and a logarhc facor wh respec o n. I s worh ephaszng ha our resul ndcaes ha he order of requred bnary easureens s uch saller han he aben denson of he covarance arx, and even coparable o he saple coplexy for lowrank arx recovery usng real-valued easureens [34. Furherore, he reconsrucon s robus o rando flppng errors, as long as p < /2. In fac, he error scales nverse proporonally o he expecaon of correc ranssson. C. Adapve Rank Selecon va Sof-hresholdng The perforance guaranee of he specral esaor n Theore 2 requres perfec knowledge of he rank, whch exac exponenal lower bound polynoal lower bound rank r Fg.. The derved lower bounds n Lea copared wh s exac values when all λ k = for dfferen ranks. ay no be readly avalable. One possble sraegy s o sofhreshold he egenvalues of J o selec a proper denson of he prncpal subspace. We ovae hs choce as he soluon o a regularzed convex opzaon proble whose analyss sheds lgh on how o selec he hreshold. To begn wh, followng he raonale of 9), we ay seek o fnd a low-rank PSD arx Σ ha obeys Σ = argax Σ z Σ,W = argax Σ Σ,J, = s.. rankσ) r, Σ F. However, hs forulaon s non-convex due o he rank consran. We herefore consder he followng convex relaxaon by relaxng he rank consran by race nzaon and regularzng he nor of Σ, yeldng: Σ = argn Σ Σ J 2 F +λtrσ), 3) where λ > s he regularzaon paraeer. The above proble 3) ads a closed-for soluon [35. Le he EVD of J be gven as J = n k= λ k û k û H k, where he λ k s are ordered n he descendng order. Then Σ = n k= η kû k û H k, where η k = ax{, λ k λ}. Therefore, lke he specral esaor, he esaed prncpal space are spanned by he egenvecors of Σ correspondng o he nonzero egenvalues, where he rank s now se adapvely va sof-hresholdng by λ. The perforance of 3) can be bounded by he followng lea fro [8, pp Lea 3. Se λ J J, hen he soluon Σ o 3) sasfes: Σ J F 2λ 2r. c Cobned wh Lea 2, f we se λ = n log ) 2n δ, we hen have Σ J c F nr log ) 2n, 4) δ for soe consan c. Therefore, by he Davs-Kahan heore, we agan oban a perforance guaranee on he recovered prncpal subspace ha s qualavely slar o ha n

6 6 Theore 2, excep ha now he specral esaor eploys an adapve sraegy o selec he rank va sof-hresholdng. Reark: I s worhwhle o pause and coen on he dfference beween he convex regularzed algorh 3) wh he algorh proposed by Plan and Vershynn n [23 for oneb arx copleon, whch can be wren as ax Σ y W,Σ, s.. Σ F, TrΣ) r. 5) = Indeed, he wo algorhs are essenally he sae and 4) s also applcable o 5). However, he analyss n [23 assues haw s are coposed of..d. Gaussan enres, wherej can be shown as a scaled verson of Σ, so ha he perforance bound n 4) guaranees ha one can recover he low-rank covarance arx up o a scalng dfference as soon as s on he order of nrlogn. Unforunaely, as n our saplng schee, due o he dependence of he enres of W, J s no longer a scaled varan of Σ as verfed n Fg. ), s no clear wheher s possble o recover he covarance arx n a sraghforward anner. Noneheless, we evaluae he perforance of 5) nuercally n he Secon V for prncpal subspace esaon, and show s coparable o ha of he specral esaor, whle ncurrng a uch hgher copuaonal cos. D. Rank-One Toeplz Subspace Esaon In any applcaons, he covarance arx Σ can be odeled as a low-rank Toeplz PSD arx, and s desrable o furher reduce he saplng coplexy by explong he Toeplz consran. Denoe T J ) as he projecon of J ono Toeplz arces, we can show ha concenraes around he Toeplz arx T J) a a rae uch faser han Lea 2, as soon as scales poly-logarhcally wh respec o n. Lea 4. Wh probably a leas n 9, we have where c 2 s soe consan. T J) T J ) c 2 log2 n, The proof can be found n Appendx F. When Σ s rankone, by Lea, J = 2p)u u H where u s he op egenvecor of Σ, herefore J s also rank-one and Toeplz. Denoe he op egenvecor of T J ) as û, cobnng Lea 4 and Lea 7, we have he followng heore. Theore 3. Assue rankσ) =. Wh probably a leas n 9, here exss θ [,2π) such ha û e jθ u c 3 log 4 n 2p) for soe consan c 3. Therefore, Theore 3 ndcaes ha for he rank-one case, u can be accuraely esaed as long as scales on he The analyss s sraghforward by slghly adapng he arguens, herefore we o he deals. order of log 4 n, and he reconsrucon s robus o rando flppng errors n he receved bs. Ths s a uch saller saple coplexy copared wh Theore 2 when he Toeplz srucure s no exploed, whch requres scales a leas lnearly wh respec o n. Reark: In he general low-rank case, J ay no be Toeplz even when Σ s Toeplz, whch prohbs us fro obanng he perforance guaranee. However, he sulaon suggess ha he specral esaor connues o perfor well n he low-rank case, whose nvesgaor we leave o fuure work. IV. SUBSPACE TRACKING WITH ONLINE BINARY MEASUREMENTS In hs secon, we develop an onlne subspace esaon and rackng algorh for he fuson cener o updae he prncpal subspace esae of he low-rank covarance arx when new bnary easureens arrve sequenally. Ths s parcularly useful when he fuson cener s eory led snce he proposed algorh only requres a eory space on he order of nr, whch s he sze of he prncpal subspace. Essenally we need o updae he prncpal subspace of J fro ha of J gven a rank-wo updae as follows: J = J + y ) a a H b b H 6) We can rewre 6) usng ore general noaons as J = η J +K Λ K H, 7) where 6) can be obaned fro 7) by leng η =, K = [a,b C n 2, andλ = dag[y /, y /). Noe ha gh be of neres o ncorporae an addonal dscounng facor on η o ephasze he curren easureen, by leng η o ake a saller value, as done n [36. Assue he EVD of J can be wren as J = U Π U H where U C n r s orhonoral and Π R r r s dagonal. The goal s o fnd he bes rank-r approxaon of J by updang U and Π. We develop a fas rank-wo updae of he EVD of a syerc arx by nroducng necessary odfcaons of he ncreenal SVD approach n [36, [37. A key dfference fro [36, [37 s ha we do no allow he sze of he prncpal subspace o grow, whch s fxed as r. In he updae we frs copue an expanded prncpal subspace of rank r+2) and hen only keep s r larges prncpal coponens. Le R = I U U H )K and P = orhr ) be he orhonoral coluns spannng he colun space of R. We wre J as J = [ [ η Π U P + [ [ U H K U H H ) [U H P H Λ K R P H R P H := [ [ U H U P Γ P H,

7 7.8 n = 4 convex n = 4 specral n = specral n = 2 specral.8 r = r = 2 r = a) b) Fg. 2. Perforance of he specral esaor for esang he prncpal subspace of low-rank covarance arces. a) wh respec o he nuber of b easureens for n = 4,,2 when r = 3. b) wh respec o he nuber of b easureens for r =,2,3 when n =. where Γ s a sall r+2) r+2) arx whose EVD can be copued easly and yelds Γ = U Π U. Se Π be he op r r sub-arx of Π assung he egenvalues are gven n an absolue descendng order, he prncpal subspace of J can be updaed correspondngly as U := [ U P U I r, where I r s he frs r coluns of he r+2) r+2) deny arx. V. NUMERICAL EXPERIMENTS In he nuercal experens, we frs exane he perforance of he specral esaor n a bach seng n ers of reconsrucon accuracy and robusness o flppng errors, wh coparsons agans he convex opzaon algorh n 5). We hen exane he perforance of he alored specral esaor when he covarance arx s addonally Toeplz. Nex, we exane he perforance of he onlne subspace esaon algorh n Secon IV and apply o he proble of lne specru esaon. Fnally, we exane he effecs of aggregaon over fne saples. A. Recovery for low-rank covarance arces We generae he covarance arx as Σ = UU T, where U R n r s coposed of sandard Gaussan enres. The oneb easureens are hen colleced accordng o 5). Afer he b easureens are colleced, we run he specral esaor usng he consruced J and he convex opzaon algorh 5) assung he rank r of prncpal subspace s known perfecly. The algorh 5) s perfored usng he MOSEK oolbox avalable n CVX [38 and obans an esae Σ, fro whch we exrac s op-r egenvecors. The noralzed ean squared error ) s defned as I ÛÛH )U 2 F / U 2 F, where Û s he esaed prncpal subspace wh orhonoral coluns. Fg. 2 a) shows he wh respec o he nuber of b easureens for dfferen n = 4,,2 when r = 3 averaged over Mone Carlo runs. Gven he hgh coplexy of he convex algorh 5), we only perfor when n = 4. We can see ha s perforance s coparable o ha of he specral esaor. Fg. 2 b) furher exanes he perforance of he specral esaor for dfferen ranks when n =. For he sae nuber of b easureens, he grows gracefully as he rank ncreases. The specral esaor also exhbs a reasonable robusness agans flppng errors. Fg. 3 shows he reconsruced wh respec o he flppng probably p for dfferen nuber of b easureens when n = and r = 3, where he error ncreases as he flppng probably p ncreases = = 2 = 3 = Flppng probably Fg. 3. wh respec o he flppng probably for dfferen nuber of b easureens when n = and r = 3. B. Recovery for low-rank Toeplz covarance arces We generae a rank-r PSD Toeplz arx Σ va s Vanderonde decoposon [39, gven as Σ = V ΛV H, where V = [vθ ),...,vθ r ) C n r s a Vanderonde arx wh vθ k ) = [,e jθ k,...,e jn )θ k T, θ k [,) for k r, and Λ = dag [σ 2,...,σ2 r ) s a dagonal arx descrbng he power of each ode. Fg. 4 depcs he wh respec o he nuber of b easureens for he specral esaors usng eher J

8 8 or T J ) when n = 4, averaged over Mone Carlo sulaons, when r = and r = 3. I s clear ha he alored specral esaor usng T J ) acheves a saller error usng uch fewer bs. Alhough Theore 3 only apples o he rankone case, he sulaon suggess perforance proveens even n he low-rank seng general r = general r = 3 Toeplz r = Toeplz r = Fg. 4. wh respec o he nuber of b easureens for esang he prncpal subspace of low-rank Toeplz covarance arces when n = 4 and r = and r = 3. Mode locaons a) Mode locaons Fg. 5. Exraced ode locaons usng ESPRIT wh respec o he nuber of b easureens usng he proposed specral esaors usng a) T J ) and b) J, when n = 4 and r = 3. We perfor ESPRIT [4 on he esaed subspace o recover he ode locaons {θ k } r k=. Fg. 5 shows he esaed ode locaons vercally wh respec o he nuber of b easureens, wh color ndcang he power of he esaed odes, where he rue ode locaons are se as [θ,θ 2,θ 3 = [.3,.325,.8 and [σ 2,σ2 2,σ2 3 = [,,.5. The specral esaor usng T J ) esaes close-locaed odes and deecs weak odes fro a uch saller nuber of bs. C. Onlne subspace rackng We now exane he perforance of he onlne subspace esaon algorh proposed n Secon IV. Le n = 4 and r = 3. Fg. 6 shows he of prncpal subspace esaon wh respec o he nuber of b easureens, where he resul s averaged over Mone Carlo runs. Copared wh Fg. 2 a), he esaon accuracy s coparable o ha n a bach seng. b) Fg. 6. The of prncpal subspace wh respec o he nuber of b easureens when n = 4 and r = 3 n an onlne seng. The esaon accuracy s coparable o ha n a bach seng. In he sequel, we apply he onlne esaor o he proble of lne specru esaon, n a se up slar o Fg. 5. Noe here we do no explo he addonal Toeplz srucure of Σ. A each new bnary easureen, we frs use he onlne subspace esaon algorh proposed n Secon IV o esae a prncpal subspace of rank r es = 5, hen apply ESPRIT [4 o recover he ode locaons. Fg. 7 shows he esaon resuls for varous paraeer sengs, where he esaes of ode locaons are ploed vercally a each new b easureen. Fg. 7 a) and b) have he sae se of odes, wh wo close locaed frequences separaed by he Raylegh l, /n. When all he odes have srong powers, he odes can be accuraely esaed as depced n a); when one of he close odes s relavely weak, he algorh requres ore easureens o pck up he weak ode, as depced n b). Fg. 7 c) exanes he case when all he odes are well separaed, and one of he s weak. The algorh pcks up a weak ode wh a saller nuber of easureens when he odes are well separaed. Takng hese ogeher, suggess ha he rackng perforance depends on he egengap and he condonng nuber of he covarance arx. D. Perforance wh fne daa saples The above sulaons assue ha he b easureens are exac. We now exane he perforance of he specral esaor assung s easured va 3) usng a fne nuber of daa saples. Le n = and r = 3. Assue here are a collecon of T saples generaed as x = Ua +n, where U s an orhogonal arx noralzed fro a rando arx generaed wh..d. Gaussan enres, a N,I r ) s generaed wh sandard Gaussan enres, and n N,σ 2 I n ) s ndependenly generaed Gaussan enres. In oher words, x NUU T +σ 2 I n ). All sensors easure he sae se of T saples.e. l =, for =,,T ) and councae her b easureens for subspace esaon. Fg. 8 shows he of prncpal subspace esae wh respec o he nuber of b easureens for dfferen nuber of saples T =,2,3,4,5, and 2 averaged over 2 Mone Carlo runs when he saples are a) nose-free

9 Mode locaons.6 a) Mode locaons.9 Mode locaons F = [.,.7,.725; σ 2 = [,, ; b) F = [.,.7,.725; σ 2 = [,,.5; c) F = [.,.5,.7; σ 2 = [,,.3; Fg. 7. Onlne lne specru esaon: he esaed frequency values agans he nuber of b easureens when n = 4 and r = 3. Each e 5 frequences are esaed wh he color bar ndcaes her apludes. The rue frequency profles and her apludes are gven n he subles T = T = 2 T = 3 T = 4 T = 5 T = T = T = T = 2 T = 3 T = 4 T = 5 T = T = a) σ 2 = b) σ 2 =. Fg. 8. The of prncpal subspace esae wh respec o he nuber of b easureens when n = 4 and r = 3 for dfferen nuber of saples for a) nose-free saples and b) nosy saples. wh σ 2 =, and b) nosy wh σ 2 =.. As T ncreases, he decreases as he b easureens ge ore accurae n lgh of Theore. Noe ha he gan dnshes as T s suffcenly large as all b easureens are accurae wh hgh probably. For nosy daa saples, s evden ha ore saples are necessary for he aggregaon procedure o yeld accurae b easureens, and perforance proves as ore saples are averaged. The auhors hank useful feedbacks fro he anonyous revewers ha sgnfcanly prove he qualy of hs paper. The frs auhor hanks Lee Seversky, Lauren Hue and Ma Berger for her hospaly and helpful dscussons durng her say a he Ar Force Research Lab, Roe, New York where par of hs work was accoplshed. She also hanks Yuxn Chen for helpful dscussons. VI. C ONCLUSIONS A PPENDIX A S UPPORTING L EMMAS In hs paper, we presen a low-coplexy dsrbued sensng and cenral esaon fraework o recover he prncpal subspace of low-rank covarance arces fro a sall nuber of one-b easureens based on aggregaed energy coparsons of he daa saples. Specral esaors are proposed wh appealng copuaonal coplexy and heorecal perforance guaranees. In he fuure, s of neres o develop prncpal subspace esaon algorhs fro quanzed easureens beyond he one-b schee exploed n hs paper. ACKNOWLEDGEMENT Lea 5 scalar Bernsen s nequaly wh sub-exponenal nor, [4). Le z,..., zl be ndependen rando varables wh E[zk = and σk2 = E[zk2, and P[ zk > u Ce u/σk PL 2 for soe consans C and σk. Defne σ 2 = k= σk and B = ax k L σk. Then # " L X u2. P zk > u 2 exp 2Cσ 2 + 2Bu k= Lea 6 arx Bernsen s nequaly wh sub-exponenal nor, [42). Le X,, X L be ndependen zero-ean syerc rando arces of denson n n. Suppose

10 σ 2 = L k= E[X kx H k and Xk ψ B alos surely for all k, where ψ s he Orlz nor [32. Then for any τ >, L τ P[ 2 ) X k > τ 2nexp 2σ 2. 8) +2Bτ/3 k= Lea 7 Davs-Kahan, [33). Le A,à Rn n be syerc arces. Denoe V as he subspace spanned by he op r egenvalues of A, and Ṽ as he subspace spanned by he op r egenvalues of Ã. Le δ be he specral gap beween he rh and he r +)h egenvalue of A. Then here exss an r r orhogonal arx Q such ha he wo subspaces V and Ṽ s bounded by { } 2 Ṽ V Q à A F 2 2r à A F ax,. δ δ Lea 8 Hanson-Wrgh nequaly, [43). Le Σ be a fxed n n arx. Consder a rando vecor x = X,...,X n ) where X are ndependen rando varables sasfyng EX = and X ψ2 K. Then for any, we have P [ x H Σx Ex H Σx > 2e cn APPENDIX B PROOF OF PROPOSITION ) 2 K 4 Σ 2, K F 2 Σ Proof: For noaonal splcy, we drop he sensor ndex and le Σ T = T T = x x H and y,t = W,Σ T. Condoned on a and b, we have E[y,T a,b = W,Σ. Le y,t E[y,T a,b = T W,Σ x x H T := T Q, T = = where Q = W,Σ x x H. We ay appeal o he Bernsen-ype nequaly n Lea 6. Frs, E[Q a,b =. Second, Var[Q a,b = Var[ W,x x H a,b = Var [ a H x 2 b H x 2 a,b 2E[ a H x 2 a +2E[ b H x 2 b = 2a H Σa +b H Σb ) := 2B, where x H a a CN,a H Σa ) and x H b b CN,b H Σb ) are wo correlaed Gaussan varables. Thrd, Q = W,Σ x x H a H Σa b H Σb x H a 2 + x H b 2 a H Σa +b H Σb + x H a 2 + x H b 2, Snce x H a 2 /a H Σa and x H b 2 /b H Σb are exponenal rando varables wh paraeer, we have Q ψ 2a H Σa +b H Σb ) := 2B. Assue W,Σ s posve whou loss of generaly, we. have P[ y,t E[y,T a,b > W,Σ a,b W,Σ 2 ) T 2exp 4B+ W,Σ /3) 9) Nex, we can bound he quadrac for a H Σa usng he Hanson-Wrgh nequaly [43 n Lea 8. Snce E[a H Σa = TrΣ), wh probably a leas δ/3, we have a H Σa TrΣ) c Σ F log/δ) for soe consanc. Snce Σ F TrΣ), we havea H Σa c TrΣ)log/δ) wh probably a leas δ/3 for soe absolue consan c. Denoe hs as even G. Furher fro he arguens n [5, Proposon, we have ha c 2 Σ F log/δ) W,Σ c 3 Σ F log/δ) 2) wh probably a leas δ/3 for soe absolue consans c 2 and c 3. Denoe hs as even G 2. Condoned on he even G and G 2, and plug n he above no 9), we have as soon as T c 4 TrΣ) Σ F log 2 /δ) 2) for soe consan c 4, he RHS of 9) can be upper bounded by δ/3. To suarze, assung 2) holds, P[y y,t P[y y,t a,b dµa )dµb ) PG c )+PGc 2 )+ G,G 2 P[ y,t E[y,T a,b > W,Σ a,b dµa )dµb ) δ. Our proposon hen follows. APPENDIX C APPROXIMATE LOW-RANK COVARIANCE MATRICES Whou loss of generaly, assue W,Σ s posve. We wsh W,Σ r s also posve so ha hey have he sae sgn. Noe ha W,Σ r = W,Σ W,Σ Σ r, s suffcen o have for all =,...,. Snce W,Σ Σ r W,Σ 22) W,Σ Σ r W Σ Σ r a b 2 2) Σ Σ r, and fro [44, wh probably a leas 4δ, we have ax{ a 2 2, b 2 2 } n+2 log/δ)). Cobned wh 2), and renang he consans, hen 22) s guaraneed wh probably a leas δ, as long as for soe consan c. Σ Σ r c Σ Flog/δ) n+2 log/δ))

11 APPENDIX D PROOF OF LEMMA Proof: We frs prove 2). For k =,...,n r, [ v H k Jv k 2p) = E r ) r sgn λ k a H u k 2 λ k b H u k 2 = k= a H v k 2 b H v k 2) [ r = E sgn λ k a H u k 2 k= a H v k 2 b H v k 2), [ r = E sgn λ k a H u k 2 k= k= ) r λ k b H u k 2 k= ) r λ k b H u k 2 k= [ E a H v k 2 b H v k 2 =, where he penulae equaon follows fro ha a H v k s and a H u k s are ndependen fro he Gaussany of a, and he las equaly follows fro E [ a H v k 2 b H v k 2 =. We nex prove ). For k =,...,r, [ r u H k Ju k 2p) = E sgn λ k a H u k 2 k= a H u k 2 b H u k 2) [ r ) = E sgn λ k V k V k, k= ) r λ k b H u k 2 k= where V k = a H u k 2 b H u k 2 s are..d. rando varables followng he Laplace dsrbuon wh paraeer. Le E = k k λ k V k, hen by eravely applyng condonal expecaon, [ r ) E sgn λ k V k V k = E[E[sgnλ k V k +E)V k E = ǫ k= Frs, assue ǫ >, = = E[E[sgnV k +ǫ/λ k )V k E = ǫ. E[sgnV k +ǫ/λ k )V k E = ǫ = 2 ǫ/λk v)f Vk v)dv + ǫ/λ k vf Vk v)dv ǫ/λ k vf Vk v)dv = +ǫ/λ k )e ǫ/λ k. Slarly we derve for ǫ <, ogeher we have E[sgnV k +ǫ/λ k )V k E = ǫ = + ǫ /λ k )e ǫ /λ k. I s sraghforward ha + ǫ /λ k )e ǫ /λ k, herefore u H k Ju k 2p). On he oher hand, u H k Ju [ k 2p) = E + E /λ k )e E /λ k [ E[e E /λ k = E e k k λ k V k /λ k. 23) Nex, we provde wo lower bounds on 23), hen ) follows by akng he axu of he wo bounds. The frs lower bound follows sraghforwardly fro E [ e k k λ k V k /λ k [ E = E k k = 2 k k = k k e k k λ k V k /λ k [ e λ k V k /λ k λ k 24) e λ k v/λ k f Vk v)dv, 25) λ k +λ k where 24) follows fro he ndependence of V k s. Cobnng 23) and 25), we oban u H k Ju k 2p) ) r λ k. λ k +λ k +κσ) k k The second lower bound follows fro [ E e k k λ k V k /λ k [ = P e k k λ k V k /λ k h dh ) = P λ k V k λ k k k log dh h P λ k V k λ k κσ) e κσ) k k where he las equaon s obaned by seng h = e κσ). Applyng Lea 5 wh u = κσ), B = 2κΣ), C = and σ 2 = 2rκ 2 Σ), we have P λ k V k λ k k k κσ) κ 2 ) Σ) 2exp 4rκ 2 Σ)+2 2κΣ) exp ) 8r 9r. where he las nequaly follows fro exp 8r) 8r + 28r 2 9r. Cobned wh 23), we oban uh k Ju k 2p)e κσ) /9r). Proof: We wre J J = APPENDIX E PROOF OF LEMMA 2 [z W J := B, = = where B = z W J. To apply Lea 6, we have E[B =, and B z a a H b b H ) + J a b 2 2 +

12 2 where he las nequaly follows fro J 2p) due o Lea. I s obvous ha boh a 2 2 and b 2 2 are chsquared rando varables wh coplex degrees of freedo n, hen B s sub-exponenal wh bounded sub-exponenal nor. We have B ψ Cn for soe consan C. For he varance, we need o copue E[B H B : E[B H B [ = E z a a H b b H ) J) H z a a H b b H ) J) = E [a a H b b H )a a H b b H ) J H J = 2E [ a a H a 2 2 2I J H J = ni J H J, herefore σ 2 = = E[BH B = E[B H B = ax{n, J H J } = n. Applyng Lea 6 we have, τ P[ 2 ) B > τ 2nexp 2n 2. +2Cnτ/3 = Rearrangng wll conclude he proof. APPENDIX F PROOF OF LEMMA 4 Proof: Frs, wre ) T J J) = T [z W J = X, = = where X = T z W J) wh E[X =. Accordng o [5, Lea 5, we have he even { E = T a a H ) } c log 3 2 n) holds wh probably a leas n. Denoe he even } E = { : T W ) c 2 log 3 2 n), whch holds wh probably a leas 2) n. Under he even E, we can bound X as X = T z W J) T z W ) + T J) c 3 log 3 2 n), where we used T J) J. However, hs condonng wll deerorae he zero-ean propery of X. Luckly, we wll show he volaon s sall, and we can sll bound he concenraon n a desrable anner followng a slar reaen n [45, Appendx B. Denoe he condonal expecaon as M = E[X E. We have M c n 8 26) for soe consan c, whch wll be shown a he end of he proof. Also, condoned on E, we have E [ X M) 2 E = [ E X M) 2 E c E[T z W ) 2 E c 3 log 3 n) for soe consanc 3. Applyng he arx Bernsen nequaly n Lea 6, we have ha condoned on E, as long as > clogn, wh probably a leas n 9, we have X M log2 n. = Fnally, f furher sasfes < cn for soe c, we can have ha wh probably a leas n 9, X X M + M c log 2 n 2. = = The proof s coplee afer we prove 26). Usng he law of oal expecaon, we expand T J) as: T J) = E[T z W ) = E[T z W ) E PE ) +E [ T z W )I E c, 27) where I E c s he ndcaor funcon of he even E. Our goal s o bound M where M = E[T z W J) E. Afer soe basc ransforaon, 27) yelds ha M = E[T z W ) E T J) PE c PE c) ) T J) + E [ ) T z W )I E c PE c PE c) )+ E [ ) T z W )I E c. 28) Then wll be suffcen o bound [ E T z W )I E c. By Jensen s nequaly we have E [ [ T z W )I E c E T z W ) I E c 2 E [ T a a H ) IE c. Recall ha accordng o [5, Lea 5, T ) ) P a a H c log 3 2 n) n, by leng = c log 3 2 n) and solvng for n can be alernavely descrbed as P T a a H ) ) ) exp c 2 3 := f) for soe consan. Followng sple calculaons, we have E [ T a a H ) IE c c log 3 2 n)f c log 3 2 n) )+ f)d c n 8 c log 3 2n) for soe consan c when n s large enough. Thus we can bound M as M c/n 8 by pluggng no 28). REFERENCES [ Y. Ch, One-b prncpal subspace esaon, n IEEE Global Conference on Sgnal and Inforaon Processng GlobalSIP), Alana, GA, Dec. 24. [2 H. Fu and Y. Ch, Prncpal subspace esaon for low-rank oeplz covarance arces wh bnary sensng, n Asloar Conference on Sgnals, Syses, and Copuers Asloar), 26. [3 B. Babcock, S. Babu, M. Daar, R. Mowan, and J. Wdo, Models and ssues n daa srea syses, n Proceedngs of he weny-frs ACM SIGMOD-SIGACT-SIGART syposu on Prncples of daabase syses. ACM, 22, pp. 6.

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