Streaming Robust PCA

Transcription

1 Sreamig Robus PCA U.N.Niraja Yag Shi The Uiversiy of Califoria a Irvie {u.iraja, shiy4}@uci.edu July 3, 06 Absrac I his paper, we cosider he problem of robus PCA i he sreamig seig wih space cosrais. Specifically, we adop he popular spiked-covariace model ad focus o he rak- case. The problem ca be saed as follows: a ime, we are give a -dimesioal daa vecor x = uz + s + w where u is a fixed vecor, z is a Gaussia radom variable, s is a arbirary sparse perurbaio ad w is he usual dese oise vecor wih a give variace. Wihou sorig samples, we wish o recover u ad subsequely also separae he sparse perurbaio s from each sample. A key challege is ha s is a sparse vecor wih ukow magiude ad suppor. While recovery guaraees for his problem are kow wih sroger assumpios, o he bes of our kowledge, our resul is he firs o obai fiie-sample guaraees while havig he weakes assumpio o he sparse perurbaio, amely, deermiisic suppor, ad a sadard ideifiabiliy assumpio o he low-rak compoe, amely, icoherece. Esseially, our algorihm performs simple ieraive hard-hresholdig followed by sochasic block power mehod. Our algorihm also has he opimal space complexiy of O ad a sample complexiy of O log. Keywords: Robus PCA, Sochasic lock Power Mehod, Ieraive Hard Thresholdig, Low Rak Plus Sparse Model, Olie Algorihm, No-covex Opimizaio. Iroducio The robus PCA problem addresses he followig quesio: suppose we are give a daa marix which is he sum of a ukow low-rak marix ad a ukow sparse marix, ca we recover each of he compoe marices? Despie he ihere o-covexiy of he problem, rece advaces have provided algorihms wih ear-opimal covergece guaraees. However, hese bouds hold oly i he bach seig, ie, whe he eire daa marix is kow. I he prese work, we aalyze robus PCA i he sreamig seig, focusig o he rak- case where we would like o recover he op eigevecor of he rue covariace wihou he perurbaio effec due o sparse corrupios.. Our Coribuio To he bes of our kowledge, we obai he firs-kow covergece guaraees from sreamig robus PCA while havig fiie sample complexiy of O log ad also havig opimal space complexiy of O where is he dimesio; he precise resul is saed i Theorem 3.. The assumpios ha we use are aural ideifiabiliy assumpios used i he bach case as well, he deails of which are preseed i Secio 3. A a high level, our algorihm performs aleraig hard-hresholdig followed by sochasic block power mehod. Two specific improvemes from earlier works are: we have he weaker deermiisic assumpio for he sparse perurbaio We do o eed icoherece of he iermediae updaes i our aalysis.

2 . Relaed work Recely, provable o-covex opimizaio is gaiig a lo of research focus sice a lo of pracically impora problems i big daa aalysis ad large-scale machie learig ca aurally be modeled as o-covex problems. However, wih his represeaioal power comes compuaioal ad saisical iracabiliy, i geeral. I may cases, hese problems ca be solved via algorihms operaig i o-covex spaces ad have bee empirically successful. However, covergece guaraees are less well-kow for such algorihms. Marix facorizaio, a ceral echique i may domais, is a class of o-covex opimizaio problems for which empirically successful algorihms exis. Rece advaces i heoreical machie learig have led o ovel aalysis echiques ad global covergece guaraees for may marix facorizaio problems. Our paper is a aemp o add o his growig collecio. PCA: Pricipal Compoe Aalysis PCA is a ubiquious usupervised learig algorihm ad has a rich hisory. Oja s algorihm is a classical mehod for sreamig PCA []. Though he covergece ad empirical performace were kow, he asympoic covergece rae was firs provided i []. Improvig o he aalysis of [], liear covergece is preseed i [3] bu wih he requireme ha he iiializaio vecor mus have a cosa correlaio wih he rue eigevecor. The covergece of block sochasic power mehod is cosidered i [9] for PCA i he sreamig seig. Recely, a igher aalysis for Oja s algorihm is provided i [6]. Also, Aleco is a SGD algorihm for low-rak marix problems preseed i []. Their aalysis is based o corol of marigales o achieve O ɛ covergece rae where ɛ is he desired umerical error. Marix compleio: Covergece guaraees for o-covex aleraig miimizaio algorihm for marix compleio i he bach seig are kow for example, see [4] ad he refereces herei. Recely, for sreamig marix compleio, covergece guaraees are preseed i []. They assume radom suppor draw accordig o he eroulli model ad he algorihm is esseially he block power mehod performed o a ubiased esimae of he rue covariace. Their sample complexiy is O log ad space complexiy is Or. Isead of receivig icomplee daa vecors a each ime isace, [7] recely obaied guaraees for a olie marix compleio model wherei, a each ime isace, some ery of he marix is revealed. Robus PCA: The covergece of he o-covex aleraig projecios based mehod was aalyzed i [] i he bach seig. Recely, a projeced gradie mehod o facorized marices was preseed i [5]. They also mach he ime complexiy lower boud of Õr i he fully observed seig ad also provide guaraees uder he parially observed seig, however, oly i he bach seig. For he olie seig, he work by He e al [5] preseed a algorihm based o olie l -miimizaio which also had good empirical performace. Usig marigale echiques, he covergece of a olie algorihm for he covex formulaio of he robus PCA is preseed i [3] bu he resul assumes he correcess of he bach covex formulaio. The ReProCS algorihm ad is varias are preseed i [8] ad he refereces herei. They sudy olie robus PCA ad olie marix compleio uder a uified framework bu more resricive assumpios are imposed o he low-rak ad he sparse pars such as a slow chages i he suppor se of he ouliers. Noe ha he refereces meioed above are by o meas exhausive - such a lis is beyod he scope of his paper. They are merely a sample o provide some backgroud ad are represeaive of he sae-of-he-ar resuls. Model ad Algorihm We cosider he popular spiked-covariace model wih sparse perurbaios i -dimesios, ie, x = Az + w + s where A is a ukow r marix of rak-r, s is deermiisic sparse perurbaio wih ukow suppor ad magiude, ad w is idepede Gaussia oise wih a

3 give variace σ I. Give a sample x a ime, we wish o recover s ad wih fiie such samples, we wish o fid he space spaed by he colums of A upo a fixed umerical accuracy ɛ. I oher words, if A = UΣV is he SVD, we wish o fid he eigevecors U. I his paper, we will focus oly o he rak- case, ie, a ime isace, we are give he daa vecor a ime is give by x = uz + s + w. We prese our algorihm for he rak- case i Algorihm. There are hree key loops i Algorihm amely, iermos τ-loop which we call aleraios, middle -loop which we call ieraios, ad 3 he ouermos h-loop which we call epochs. Our algorihm uses radom iiializaio for our eigevecor esimae, which is also very easy i pracice. Iuiively, he τ-loop is performs deoisig via ieraive hard-hresholdig, ie, i solves he opimizaio problem: {ẑ, ŝ } = arg mi a R,b R x ua + b s.. b 0 d h From his, we obai a esimae of he sparse perurbaio vecor ad cosequely, he scalig facor associaed wih u. y subracig his ou, we obai vecor which is close o our desired subspace. Usig a block of such vecors, he -loop accumulaes he sample covariace marix. Fially, he h-loop performs a oisy power mehod updae o he accumulaed covariace marix uil our esimae reach he desired umerical accuracy ɛ wih respec o he rue eigevecor. Noe ha his is effecively a block versio of he usual power mehod bu he key challege is o corol he perurbaio i he sample covariace esimae due o he error iduced by hresholdig ie, ruig oly a fiie umber of aleraios, ad he error i our esimae of he op eigevecor i he curre epoch. Addiioally, we oe ha samples are ever revisied ad hece his is a oe-pass algorihm. Noe ha we do kow Z h i pracice. However, we will see ha from Lemma 3.-4 ad h / Theorem 3.4 ha, a simple humb-rule is o se Z h = C C where C, C > 0 are cosas; also oe ha s max is a kow cosa as described i Secio 3. Algorihm lock Sochasic Power Mehod wih Hard Thresholdig : Ipu: Samples {x,..., x T } R such ha x = uz + s : Oupu: Leadig eigevecor of he deoised samples u H 3: u 0 N 0, I 4: u 0 u 0 u 0 5: for h =,..., H = T do 6: u h 0 7: for = h +,..., T = h do 8: ŝ 0 0 9: for τ =,..., T do τ s max : ζ τ Z h + 5 : ẑ τ u h x ŝ τ : ŝ τ Thresh ζ τ x u h ẑ τ 3: ed for 4: ŝ ŝ T 5: u h u h + x ŝ x ŝ u h 6: ed for 7: u h u h u h 8: ed for 9: reur ẑ,..., ẑ T, ŝ,..., ŝ T, u T/ 3

4 3 Aalysis For simpliciy ad cocreeess, we cosider he aalysis of oly he oiseless rak- case. The oisy case aalysis is very similar excep ha he resul ivolves he oise level σ of w ad coceraio properies associaed wih ha. Noaio, model seup ad assumpios: Firs, for simpliciy ad clariy, we will sar by preseig he rak- case where he dese oise is abse, ie, w = 0. Specifically, he daa vecor a ime is give by x = uz + s. Noe ha for simpliciy we have assumed ha he eigevalue correspodig o he op eigevecor is. Defie u h = ± α h u + α h v h, where u vh ad α h 0,, for every h. Now, we iroduce aural sadard codiios, similar o [] uder which he problem is ideifiable:. Low-rak par: u is µ-icohere, ie, u µ iid ad z N 0,. We oe ha we may relax his geeraive assumpio o z o a radom variable such ha E[z ] = 0, E[z ] =, z Z max almos surely wih a lile care.. Sparse par: we have a deermiisic sparsiy codiio, ie, s 0 d h ad also wihou loss of geeraliy, we assume s smax. We assume d h < mi 0 50Zmax where d h is he umber of o-zeros i s ha appears a epoch h. As a cosequece of his iequaliy, we have he ierpreaio ha as he umber of epochs icreases, we are able o olerae more perurbaio, ie, as u h ges closer o u, he hresholdig become more effecive. Le b i deoe he i h basis vecor i dimesios. Defie he ery-wise hard-hresholdig operaio of a vecor v, deoed as Thresh a v as follows: { for every i, Thresh a b b i i v = v, if b i v > a 0, else µ We defie quaiies Z h = Z max α h + α h α h, µ +3µα h αh ad for all τ, e τ = s ŝ τ. Le e = e T for every ad deoe by E h he marix whose colums are e wih ragig from h + o h wihi epoch h. We oe ha he rue covariace Σ = E[uz uz ] = uu. Le Σ h = h =h + x ŝ x ŝ deoe our esimae of he rue covariace a epoch h ad defie h = Σ Σ h. Our aim is o show ha α h 0 as h grows, wih high probabiliy. We use C, C, C, ec o deoe global cosas. Proof oulie: A a high level, he proof of covergece ivolves aalyzig he hree loops i Algorihm, amely: covergece of iermos τ-loop aleraios, coceraio properies i middle -loop ieraios, ad 3 covergece of ouermos h-loop epochs. We wish re-emphasize ha he coceraio argumes are differe from [9, ] sice we do o have ay radomess assumpios o he suppor of he sparse perurbaio. We ow prese he mai resul ad prese he proof deails i Secios 3., 3., 3.3 ha follow. 3Clog H Theorem 3.. If, H C ɛ 5 log 3, ɛ T > C loghsmax log ɛ, wih probabiliy a leas C H, Algorihm yields a ɛ-close soluio i he sese ha α H ɛ. Succicly, he space complexiy is O, per ieraio ime complexiy O ad he sample complexiy is Õ where we use Õ o suppress log facors. Proof. This follows from Theorems 3., 3.3 ad 3.4 ad he iiializaio propery i Lemma 3.. Lemma 3. quaifies he propery of our iiializaio ha is proved i Lemma [9] bu we provide i here for compleeess. Lemma 3.. The iiializaio give by Seps 3 ad 4 of Algorihm yields a vecor u o such ha α0 = O/ wih probabiliy o. 4

5 3. Covergece of he Iermos Loop The mai resul of his secio is he validiy of he hard-hresholdig operaio, saed as: Theorem 3.. For every, afer T > log smax ɛ aleraios we have e T 4Z h + ɛ wih probabiliy a leas C H. Proof. This follows from Lemmas 3. ad 3.3. Lemma 3.. We have he followig useful shor resuls.. I u h u h u µ α h + α h α h.. u h α h µ + α h + α h α h µ. 3. Whe α h >, u h µ +3µα h. Else, u h 4µ 4. Z h Z max αh.. Proof.. Noe ha u u h = αh ad sigu u h. sigb i u h = sigb i u u h uu h = u α h αh u + α h v h u. Thus, α h u + α h α h v h α h µ + α h α h. Nex, oe ha e τ d h e τ. We also have u h = αh u + α h v h = α h u + α h v h + α h α h u v h = α h µ + α h + α h α h µ 3. Sice α h 0,, we have α h <. The, u h = α h µ + α h + α h α h µ µ + α h + µ α h = µ + α h + µ αh Usig his, whe α h >, we have α h > α h. So, we ge u h µ +3µα h. 4. We ca obai a upper boud o Z h as follows: µ Z h = Z max α h + αh µ α h α h Z max αh + α h y oig ha α h 0, ad µ, we obai Z h Z max αh. Lemma 3.3. If e τ 4Z h +. b i uz u h ẑ τ 6 5 Z h + τ s max τ s max τ+ s max. e τ 4Z h Moreover, Supp e τ Supp e τ Supp s., he we have: 5

6 Proof.. x u h ẑ τ = uz + s u h ẑ τ = b i x u h ẑ τ s = b i uz u h ẑ τ. b i uz u h ẑ τ = b i uz u h u h x ŝ τ ξ = ξ 3 b i uz u h u h uz + b i u h u h eτ Z max max b i u u h uu h + max b i u h u h i i eτ ξ 4 u Z max u h uu h + u h ξ b i uz u h u h uz + e τ e τ ξ 5 µ Z max α h + α h α h + d h u h ξ 6 Z h + e τ ξ 7 0 Z h + 4Z h Z h + τ+ s max e τ τ s max where ξ is by subsiuig ẑ τ = u h x ŝ τ, ξ by defiig e τ = s ŝ τ, ξ 3 by riagle iequaliy, ξ 4 by usig a, b a b, ξ 5 ad ξ 6 by Lemma 3., ξ 7 by iducive hypohesis ha êτ 4Z h + τ s max.. Nex, o complee he iducio over τ, le us calculae ê τ. We have wo cases a Case b i x u h ẑ τ > ζ τ : b i eτ = b i s ŝ τ = b i s x u h ẑ τ = b i uz u h ẑ τ 6 5 Z h + τ+ s max. b Case b i x u h ẑ τ ζ τ : b i ŝ τ = 0 = b i êτ = b i s ad b i x u h ẑ τ = b i uz + s u h ẑ τ ζ τ. So, we have b i e τ = b i s ζ τ + b i uz u h ẑ τ Z h + τ s max + 5 = 76 5 Z h Z h + τ s max 4Z h + τ+ s max τ s max 3. If b i s = 0, he we have b i eτ = I { b i uz u h ẑ τ }>ζτ b i uz u h ẑ τ bu oe ha b i uz u h ẑ τ 6 τ+ 5 Z s h + max < Z h + τ s max = ζ τ 5 This is a coradicio sice he idicaor is iacive a locaio i, so b i eτ = Coceraio Properies i he Middle Loop We aalyze he coceraio properies of may ieraios wihi a sigle epoch, ie, wih eough umber of samples, he covariace wihi a block coceraes. I his sep, i is esseial o show ha he covariace i a sigle epoch coverges o he rue covariace plus a perurbaio ha depeds o he sparse perurbaio ad is decayig as epochs proceed, so ha his esimae may 6

7 he be used for block power mehod updaes. We oe ha he idex i his secio rus from h + o h ad o simplify oaio, we will omi his rage i he summaios. Thus, he mai resul of his secio is: Theorem 3.3. Seig T > log C loghsmax ɛ for every, leig probabiliy 8C H, we have h ɛ + 0d h Zmaxα h. 3Clog H, wih ɛ Proof. We have h = Σ h Σ. Σ h Σ = x ŝ x ŝ uu = uz + e uz + e uu = uu z + u z e + z e u + }{{} }{{} e e }{{}}{{} T erm T erm T erm 3 T erm 4 The secod sep is by riagle iequaliy o he specral orm of he perurbed marix. Now, we boud each of he erms usig similar echiques as [9]. Term-: Usig ail bouds for sub-gaussia radom variables from [4], wih probabiliy C H uu z z uu C log H Term-: As specral orm is sub-muliplicaive, uz e = ue h z u E h z. Now, iid for every i, sice z N 0,, we have b i E hz N 0, σe where σe E h ; his is because var b i E hz = var j= b i E hb j b j z E h. Hece, wih probabiliy C H E h z = b i E h z b E hz H σe log C i= σ where he las lie was obaied by usig he Hoeffdig boud, ie, ail boud for X N 0, σe is give by P r X exp ad oig ha X is half-ormal disribuio saisfyig his boud. Furher simplifyig by subsiuig E h, usig Theorem 3. ad Lemma 3.-4, we ge E h z H E h log H log 8Zmax αh + ɛ C C Dividig boh sides by, we obai E hz log H C 8Zmax αh + ɛ. Term-3: Same as Term-. Term-4: Le ɛ Z h Z max αh. Usig riagle iequaliy, sub-muliplicaive propery, Theorem 3. ad Lemma 3.-4, we have wih probabiliy C H e e e e e. dh E h dh E h d h 4Z h + ɛ d h 5Z h 0dh Z maxα h 7

8 3Clog H Combiig all he erms, leig, we obai, wih probabiliy 8C ɛ H : h = Σ h Σ C log H H + log 8Z max αh + ɛ + 0d h Z C maxα h Noe ha by seig T > log C loghsmax ɛ we have e T 4Z h + ɛ C log H where C is a cosa. Usig his, we have h ɛ + 0d h Zmaxα h. 3.3 Covergece of he Ouermos Loop The goal here is o show ha α h 0 by quaifyig he improveme decrease of α h over α h. The mai resul for his secio is: Theorem 3.4. If H C 5 log 3 ɛ wih probabiliy aleas C H, we obai α H ɛ where C 5 ad C are cosas. Proof. This proof uses similar argumes as Theorem of [9]. Noig u h = u h, decomposig u u h h as u h = u h, u u + u h, v h v h ad similarly for u h, usig Lemma 3.4 ad assumig ɛ < α h, we have, α h = u h, v h = u h u h, v h = u h, v h u = h ɛ + 0dh Zmaxα h u h, v h u h, u + u h, v h αh αh ɛ α h 0d h Zmaxα h + ɛ + 0dh Zmaxα h α h dh Zmax αh αh α h 0. + α h 0d h Zmax αh + αh dh Zmax αh x The secod iequaliy above is obaied by oig ha c+x is a icreasig fucio i x for posiive 0. x ad c. Le C 3 = + 0dh Zmax αh. αh / 0.9 α h 0d h Zmax αh Noe ha if d h < /50Zmax αh, we oe ha he cosa C 3 <. Usig his ad also applyig Lemmas, 6 of [9], we ge α h = C 3 α h α h +C 3 α h C h 3 α 0 C h 3 α 0 C 4 C h 3. Hece, if H log /C 3 C4 ɛ wih probabiliy aleas C, we obai α h ɛ. Noe ha by havig a addiioal facor of log samples, we may boos his o high probabiliy, ie, probabiliy aleas C H. Lemma 3.4. We have he followig upper ad lower bouds.. u h, v h ɛ + 0d h Z maxα h.. u h, u α h ɛ αh α h 0d h Z maxα h. Proof.. Recall ha u, v h = 0. Now, u h, v h = v h Σ h u h = v h Σ + h u h = v h uu u h + v h hu h 0 + v h h u h = ɛ + 0d h Z maxα h 8

9 . Nex, we have lower boud he followig erm sice i would appear i he deomiaor. u h, u = u Σ h u h = u x ŝ x ŝ u h = u uz + e uz + e αh u + α h v h = z + u e αh z + α h e u + α h e v h αh = z + u αh e + z + u e e v h }{{}}{{} T erm 5 T erm 6 Term-5: Wih probabiliy C H, usig he seigs for ad T from Secio 3., ad he upper boud for Term- wih a egaive sig sice his is a absolue value of scalar, z + u e = z + u e + z u e C log H u E h z ɛ 4 u E hz ɛ Term-6: This is similar o specral orm upper bouds i Sep- bu wih a egaive sig, ie, z + u e e v h vh E hz + u E h Eh v h ɛ + 0d hzmaxα h Thus, from Terms-5 ad 6, ad oig α h 4, we have, u h, u α h ɛ ɛ α h + 0d hzmaxα h = αh ɛ α h + α h αh 0d h Z α maxα h h αh α h ɛ 0d h Z α maxα h h 4 Coclusio ad Fuure Work I his paper, we have preseed he firs covergece resul for he robus PCA problem i he sreamig seig uder he mos geeral assumpios compared o previous works meioed i Secio.. Exedig he resuls i his paper o he rak-r case should be possible alog similar lies bu we oe wo pois: a aïve aalysis would lead o loose bouds ad hece care mus be ake i accouig for he r-eigevecors while usig he disace bewee subspaces o rack he progress of he algorihm, ad we suspec ha he covergece guaraee for he rakr aalogue of Algorihm ie, via block sochasic orhogoal ieraio wih hard hresholdig will have a sub-opimal depedece o he codiio umber ad hece oe plausible fix would be cosider he sreamig versio of he sage-wise algorihm i []. We defer hese aalyses of he rak-r case o fuure work. Though he mai focus of his paper was o obai covergece guaraees, we wish o oe ha poeial applicaios iclude real-ime backgroud-foregroud separaio i videos ad real-ime subspace rackig similar o [5]. Ackowledgmes We hak Aru Rajkumar, Gopi Meeakshisudaram ad Aima Aadkumar for helpful discussios, ad Deparmes of ICS, EECS a UCI ad XRCI for heir suppor. 9

10 Refereces [] Akshay alsubramai, Sajoy Dasgupa, ad Yoav Freud. The fas covergece of icremeal pca. I Advaces i Neural Iformaio Processig Sysems, pages , 03. [] Chrisopher De Sa, Kule Olukou, ad Chrisopher Ré. Global covergece of sochasic gradie desce for some o-covex marix problems. arxiv prepri arxiv:4.34, 04. [3] Jiashi Feg, Hua Xu, ad Shuicheg Ya. Olie robus pca via sochasic opimizaio. I Advaces i Neural Iformaio Processig Sysems, pages 404 4, 03. [4] Moriz Hard. Udersadig aleraig miimizaio for marix compleio. I Foudaios of Compuer Sciece FOCS, 04 IEEE 55h Aual Symposium o, pages IEEE, 04. [5] Ju He, Laura alzao, ad Joh Lui. Olie robus subspace rackig from parial iformaio. arxiv prepri arxiv:9.387, 0. [6] Praeek Jai, Chi Ji, Sham M Kakade, Praeeh Nerapalli, ad Aaro Sidford. Sreamig pca: Machig marix bersei ad ear-opimal fiie sample guaraees for ojas algorihm. I 9h Aual Coferece o Learig Theory, pages 47 64, 06. [7] Chi Ji, Sham M Kakade, ad Praeeh Nerapalli. Provable efficie olie marix compleio via o-covex sochasic gradie desce. arxiv prepri arxiv: , 06. [8] ria Lois ad Namraa Vaswai. Olie marix compleio ad olie robus pca. I 05 IEEE Ieraioal Symposium o Iformaio Theory ISIT, pages IEEE, 05. [9] Ioais Miliagkas, Cosaie Caramais, ad Praeek Jai. Memory limied, sreamig pca. I Advaces i Neural Iformaio Processig Sysems, pages , 03. [] Ioais Miliagkas, Cosaie Caramais, ad Praeek Jai. Sreamig pca wih may missig eries. Prepri, 04. [] Praeeh Nerapalli, UN Niraja, Sujay Saghavi, Aimashree Aadkumar, ad Praeek Jai. No-covex robus pca. I Advaces i Neural Iformaio Processig Sysems, pages 7 5, 04. [] Erkki Oja ad Juha Karhue. O sochasic approximaio of he eigevecors ad eigevalues of he expecaio of a radom marix. Joural of mahemaical aalysis ad applicaios, 6:69 84, 985. [3] Ohad Shamir. A sochasic pca ad svd algorihm wih a expoeial covergece rae. I Proc. of he 3s I. Cof. Machie Learig ICML 05, pages 44 5, 05. [4] Roma Vershyi. Iroducio o he o-asympoic aalysis of radom marices. arxiv prepri arxiv:.307, 0. [5] Xiyag Yi, Dohyug Park, Yudog Che, ad Cosaie Caramais. Fas algorihms for robus pca via gradie desce. arxiv prepri arxiv: , 06.