Joura o Mutivariate Aaysis doi10.1006jmva.2000.1968, avaiabe oie at httpwww.ideaibrary.com o Optima Spherica Decovoutio 1 Peter T. Kim Uiversity o Gueph, Gueph, Otario, Caada ad Ja-Yog Koo Haym Uiversity, Kagwo-Do, Korea Received Jauary 27, 1999 This paper addresses the issue o optima decovoutio desity estimatio o the 2-sphere. Ideed, by usig the trasitive group actio o the rotatio matrices o the 2-dimesioa uit sphere, rotatioa errors ca be itroduced aaogous to the Eucidea case. The resutig desity turs out to be covoutio i the Lie group sese ad so the statistica probem is to recover the true uderyig desity. This recovery ca be doe by decovoutio; however, as i the Eucidea case, the diicuty o the decovoutio turs out to deped o the spectra properties o the rotatioa error distributio. This thereore eads us to deie smooth ad supersmooth casses ad optima rates o covergece are obtaied or these smoothess casses. 2001 Academic Press AMS 1991 subject cassiicatios 62G05; 58G25. Key words ad phrases cosistecy; desity estimatio; miimax; rotatioa harmoics; smooth; super-smooth; Soboev spaces; spherica harmoics. 1. INTRODUCTION Decovoutio techiques have bee show to be o practica use i situatios where the data is idirecty observed. Ideed, i the uderyig desity is a mixture o severa desities, decovoutio aows oe to recover the mai compoets o the mixture; see Eromovich (1997) or a recet exampe o the beeits o circuar decovoutio. Thereore, asymptotic optimay i decovoutio desity estimatio has bee ivestigated i the statistica iterature; see Fa (1991, 1991a, 1993), Eromovich (1997), ad Koo ad Chug (1998). The mai probem ivoves idetiyig the smoothess 1 This research was supported by Korea Research Foudatio Grat (KRF-2000-015- DS0006). 0047-259X01 35.00 Copyright 2001 by Academic Press A rights o reproductio i ay orm reserved.
2 KIM AND KOO o the characteristic uctio o the error distributio ito ordiary smooth or super-smooth casses or which the resutig covergece turs out to be poyomia or ogarithmic, respectivey. The above ivove decovoutio i Eucidea space, however, some recet iterest i o-eucidea decovoutio has appeared i the statistica iterature. Rooij ad Ruymgaart (1991) irst motivated decovoutio o the two dimesioa uit sphere, S 2. Heay ad Kim (1996) ad Heay et a. (1998) work out the techica detais or cosistecy. Ideed, the probem is as oows. I the case o S 2, measuremet error ca be modeed aaogous to Eucidea error by usig the trasitive group actio SO(3)_S 2 S 2, where SO(3) is the space o 3_3 rotatio matrices. The uder appropriate smoothess, rates o covergece are obtaied. It is the atura to ask whether or ot these rates o covergece are optima as deied i Fa (1993) ad Koo (1993). It wi be show that deiitios o ordiary smooth ad super-smooth ca be made through the operator orm o the rotatioa Fourier trasorm o the error distributio. These smoothess casses the ead to poyomia or ogarithmic rates o covergece, respectivey. We ow provide a summary o what is to oow. I Sectio 2, we briey go over the ecessary Fourier toos or the 2-dimesioa uit sphere ad the 3-dimesioa rotatio matrices, as we as the coectios betwee the two. The atter ivoves how covoutio as we as how Fourier trasorms chage covoutio ito idividua Fourier products simiar to the Eucidea case. I Sectio 3, we outie the decovoutio probem o the 2-sphere. I additio, we deie smooth ad super-smooth desities o the space o 3-dimesioa rotatio matrices. This is doe i the Fourier domai usig the operator orm. Foowig this we state the mai resuts. We aso make a coectio with some earier work by Hedriks (1990). The atter obtais upper boud rates o covergece or oparametric desity estimators o compact Riemaia maiods. It oows as a coroary to oe o our mai resuts that i the case o the 2-sphere, this covergece is optima. Sice this area is reativey ew i statistics, i order to motivate the probem urther, we provide exampes o rotatioa error desities i Sectio 4. Two exampes o smooth desities as we as a exampe o a super-smooth desity are itroduced. The atter ivoves the rotatioa versio o the Gaussia distributio, whie the ormer ivoves the rotatioa versio o the Lapace (doube expoetia) distributio as we as a distributio obtaied rom the radom waks o groups iterature. A o these distributios are spectray deied. I Sectio 5 we examie the vo MisesFisher matrix distributio by cacuatig it's rotatioa Fourier trasorm. Oce the cacuatios are
OPTIMAL SPHERICAL DECONVOLUTION 3 compete, we otice that athough the super-smooth deiitio appears appropriate, we ca amost (but ot exacty) get the same power i the expoet o both sides o the iequaities. Cosequety, we ca amost get the same upper ad ower rates o covergece. A proos are coected i Sectio 6. We irst estabish upper bouds ad demostrate that these upper bouds are aso ower bouds by speciyig a subprobem. This method oows the outie o Koo (1993) ad Koo ad Chug (1998), however, oe does eed to accommodate or the spherica geometry i the costructio. It is oud that as ar as the rates o covergece are cocered, aside rom the smoothess cass o the uderyig rotatioa error distributio, these rates oy deped o the dimesio o the 2-sphere. 2. SOME PRELIMINARIES We wi provide a brie overview o Fourier aaysis o SO(3) ad S 2. Most o the materia i expaded orm ca be oud i Tama (1968), Terras (1985), Heay ad Kim (1996), ad Heay et a. (1998). Papers which directy dea with simiar issues ca be oud i Lo ad Eshema (1979) ad Wahba (1981). The we kow Euer age decompositio says, ay g # SO(3) ca amost surey be uiquey represeted by three ages (,, %, ), kow coectivey as the Euer ages, where, #[0,2?), % #[0,?), #[0,2?); see Heay ad Kim (1996) ad Heay et a. (1998) or detais. Cosider the uctio, D q 1 q 2 (,, %, )=e &iq 1, d q 1 q 2 (cos %) e &iq 2, (2.1) where, d q 1 q 2 or &q 1, q 2, =0, 1,... are reated to the Jacobi poyomias; see Lo ad Eshema (1979). The uctios D q 1 q 2,&q 1, q 2, =0, 1,..., are the eigeuctios o the LapaceBetrami operator o SO(3), hece, [- 2+1 D q 1 q 2 &q 1, q 2, =0,1,...] is a compete orthoorma basis or L 2 (SO(3)) with respect to the probabiity Haar measure ad are otherwise kow as the rotatioa harmoics. I additio, i we deie a (2+1)_(2+1) matrix by D (g)=[d q 1 q 2 (g)], (2.2) where &q 1, q 2, 0 ad g # SO(3), these costitute the coectio o iequivaet irreducibe represetatios o SO(3).
4 KIM AND KOO Let # L 2 (SO(3)). We deie the rotatioa Fourier trasorm o SO(3) by q 1 q 2 = SO(3) (g) D q 1 q 2 (g) dg, (2.3) where agai we thik o (2.3) as the matrix etries o the (2+1)_(2+1) matrix =[ q 1 q 2 ], &q 1, q 2, =0, 1,... ad dg is the probabiity Haar measure o SO(3). The rotatioa iversio ca be obtaied by (g)= 0 = 0 q 1, q 2 =& q 1, q 2 =& (2+1) q 1 q 2 D q 1 q 2 (g) (2+1) q 1 q 2 D q 2 q 1 (g &1 ), (2.4) or g # SO(3), where the overbar deotes compex cojugatio. Stricty speakig, (2.4) shoud be iterpreted i the L 2 -sese athough with additioa smoothess coditios, it ca hod poitwise. Spherica Fourier aaysis aso has simiar resuts. Ay poit o S 2 ca be represeted by =(cos, si %, si, si %, cos %) t, where % #[0,?),, #[0,2?) ad superscript t deotes traspose. Let Y ( )=Y (2+1)(&q)! q q (%, P,)=(&1)q q 4?(+q)! (cos %) eiq,, (2.5) where % #[0,?),, #[0,2?), &q, =0, 1,... ad P q ( } ) are the Legedre uctios. We ote that we ca thik o (2.5) as the vector etries to the 2+1 vector Y ( )=[Y q ( )], 0. I this situatio [Y q &q, =0,1,...] orm a compete orthoorma basis over L 2 (S 2 ) ad is sometimes reerred to as the spherica harmoics; see Tama (1968). Let # L 2 (S 2 ). We deie the spherica Fourier trasorm o S 2 by = q ( ) Y q ( ) d, (2.6) S 2
OPTIMAL SPHERICAL DECONVOLUTION 5 where d is the spherica measure o S 2. Agai we thik o (2.6) as the vector etries o the (2+1) vector =[ q ], &q, =0, 1,... The spherica iversio ca be obtaied by ( )= 0 q=& q Y q ( ), (2.7) or # S 2. Agai, stricty speakig, (2.7) shoud be iterpreted i the L 2 -sese athough with additioa smoothess coditios, it ca hod poitwise. I terms o the Fourier basis, the reatio betwee SO(3) ad S 2 ca be described i terms o the Euer ages where Y q (%,,)= (2+1) 4? D q0 (,, %, ), (2.8), #[0,2?), % #[0,?), &q ad =0, 1,... We ote that athough a extra age appears i the right had side o (2.8), it is i act idepedet o. This oows rom (2.1) ad observig that whe q 2 =0, the expressio becomes idepedet o. Oe o the most useu toos o Fourier aaysis is the act that covoutio o two uctios i the Fourier domai turs out to be ordiary matrix mutipicatio. Ideed, et # L 2 (SO(3)) ad h # L 2 (S 2 ). Deie the covoutio, V h( )= SO(3) (u) h(u &1 ) du, (2.9) or # S 2. We have the oowig covoutio property or # L 2 (SO(3)) ad h # L 2 (S 2 ) @= V h h. (2.10) I particuar, or each =0, 1,..., ( @) V h = q j=& qj h j, or a &q, see Lemma 2.1 i Heay et a. (1998), or exercise 25 o Terras (1985, p. 106).
6 KIM AND KOO 3. DECONVOLUTION DENSITY ESTIMATION Cosider the oowig situatio Z==X, (3.1) where = is a SO(3) radom eemet ad Z, X are S 2 radom eemets, with = ad X assumed idepedet. We ote that (3.1) is describig the trasitive group actio SO(3)_S 2 S 2 which cosists o ordiary matrix mutipicatio, where trasitive meas that or ay two, & # S 2, there exists a g # SO(3) such that = g&. Let Z, =, X deote the desities o Z, =, X, respectivey. Through (3.1), the reatio amog the desities ca be described by covoutio, Z = = V X as see by oowig the amiiar correspodig Eucidea resut. We ote that sice = ad X are desity uctios, we have i X ( ) = (u) X (u &1 ) du sup X ( ). (3.2) # S 2 SO(3) # S 2 Now cosider ad or each 0 give by [ ] ad [ X Z X, q respectivey, ad as the matrix [ = =, qj ] or each 0. By (2.10) we ca write X =( = )&1 Z, Z, q ], provided o course that the matrices ( = )&1 exist or a =0, 1,... i a rage o iterest. Statisticay, (3.1) is describig the o-eucidea aaogue o observatios Z made up o the true measuremet X, corrupted by oise =. Our iterest is i the ukow X. It is assumed that = is kow ad that ( = )&1 exists or a rage o 's that cocers us. Sice X is ukow, Z is aso ukow, hece Z is ukow. Nevertheess, we assume that a radom sampe Z 1,..., Z is avaiabe. This wi aow us to costruct a empirica versio Z. By (2.10) a estimator or X is thereore, X =( =) &1, Z, (3.3) or =0, 1,... We ca the produce a oparametric decovoutio desity estimator o X by (2.7), the spherica iversio.
OPTIMAL SPHERICAL DECONVOLUTION 7 3.1. Smooth ad Super-Smooth Errors Decovoutio desity estimatio has bee ivestigated or some time ow ad the degree to which we ca recover the desity X is best characterized i terms o the quaity o smoothess o =. Ideed, oowig Fa (1991a) we wi appropriatey deie the smoothess o = spectray, with a modiicatio. The ecessary modiicatio required comes rom the act that o SO(3), Fourier trasorms are matrices that grow i dimesio. Cosequety, the quaity o smoothess eed to be adapted or this chage ad this ca be doe by regardig covoutio as a operator. Ideed, et E be the (2+1)-dimesioa vector space spaed by [Y q &q] or each =0, 1,... Thus ay h # E ca be writte as h= h Y q=& q q ad through Parseva's idetity, the usua L 2 -orm is &h& 2 2 = h q=& q 2. Now accordig to (2.10), = E E by h= q=& ( j=& =, qjh j) Y q. Agai by Parseva's idetity, & h& 2 = q=& j= h 2 =, qj j, or a 0. Cosequety, we have the operator iequaity, & h& 2 & & = op &h& 2, where & & & = op= sup =!& 2. (3.4)!{0,! # E &!& 2 We wi say that the distributio o = is super-smooth i the rotatioa Fourier trasorm o = satisies &( =) &1 & op Ed &1 0 &; 0 exp( ; #) ad &( = oped 1 ; 1 exp(& ; #) as, (3.5) or some positive costats d 0, d 1, ;, #, costats ; 0 ad ; 1. We wi say that the distributio o = is (ordiary) smooth i the rotatioa Fourier trasorm o = satisies &( = )&1 & op Ed &1 0 ; ad &( = )& oped 1 &; as, (3.6) or some positive costats d 0, d 1 ad oegative costat ;. Exampes o smooth ad super-smooth distributios wi be discussed i Sectio 4. 3.2. Optima Estimatio The empirica Fourier trasorm o S 2 ca be deied by, Z, q =1 j=1 Y q (Z j), (3.7)
8 KIM AND KOO which is a ubiased estimator o Z, q or &q ad =0, 1,... The by (3.3), X, q =1 j=1 s=& = &1, qs Y s (Z j), where &q, =0, 1,... ad or ease o otatio, we write &1= = ( = )&1. Choosig m=m() as eads to the oowig oparametric decovoutio desity estimator o X o S 2, X ( )= m =0 q=&{ 1 j=1 s=& Y (Z = &1, qs s j) = Y q ( ), (3.8) where # S 2. For statistica motivatio, we ca rewrite (3.8) i aother way. Deie K = (, &)= m =0 q, s=& Y q ( ) = &1, qs Y s (&), where &, # S 2. The a aterative way o writig (3.8) is X ( )=1 j=1 K = (, Z j), (3.9) where # S 2. Note that this resembes a ordiary kere estimator i Eucidea space. We woud ike to preset our mai resuts i terms o Soboev spaces. Ideed, o the space C (S 2 ) o iiitey cotiuous dieretiabe uctios o S 2, cosider the so-caed Soboev orm &}& Hs o order s deied i the oowig way. For ay uctio h=, q h q Y et q &h& 2 H s =, q (1+(+1)) s h q 2. (3.10) Oe ca veriy that (3.10) is ideed a orm. Deote by H s (S 2 ) the (vectorspace) competio o C (S 2 ) with respect to (3.10), the Soboev orm o order s. For some ixed costat M>0, et H s (S 2, M) deote the smoothess cass o uctios h # H s (S 2 ) which satisy &h& Hs 1+M.
OPTIMAL SPHERICAL DECONVOLUTION 9 Cosider a ukow distributio P depedig o the desity uctio # H s (S 2, M) ad suppose [b ] is some sequece o positive umbers. This sequece is caed a ower boud or i im c 0 im i i sup P (& & & 2 cb )=1, (3.11) # H s (S 2, M) where the iimum is over a possibe estimators based o Z 1,..., Z. Aterativey, the sequece i questio is said to be a upper boud or i there is a sequece o estimators [ ] such that im c im sup sup # H s (S 2, M) P (& & & 2 cb )=0. (3.12) The sequece o umbers [b ] is caed the optima rate o covergece or i it is both a ower boud ad a upper boud with the associated estimators [, 1], beig caed asymptoticay optima. These deiitios are i the sese o Stoe (1980). The oowig theorems state that the decovoutio desity estimators (3.9) are asymptoticay optima, where the miimax rates o covergece deped o the smoothess characteristics o the error distributio. We wi use the oowig otatio. For sequeces [a ] ad [c ] o positive umbers, et a <<c mea that a c C as. Whe a <<c ad c <<a, we write a c. Theorem 3.1. the Suppose = is smooth. I X # H s (S 2, M) or some s>1, &s(2(s+;)+2) is the optima rate o covergece, where m 1(2(s+;)+2). Theorem 3.2. s>1, the Suppose = is super-smooth. I X # H s (S 2, M) or some (og ) &s; is the optima rate o covergece, where m (og ) 1;. I Sectio 4, a discussio o some possibe error distributios is preseted, however, at this poit et us provide some geera commets about the extreme cases with respect to distributio o the errors =. Ideed, at oe extreme is the Haar measure (uiorm distributio) o SO(3) i which case decovoutio is ot possibe sice = =0 or a >0.
10 KIM AND KOO Oe ca see this by the act that the true measuremets are uiormy corrupted resutig i o hope o beig abe to recover X. I o the other had we cosider poit mass at the uit eemet o SO(3), i.e., $ e, where e deotes the uit eemet i SO(3), the = = SO(3) D (g) $ e (g) dg=d (e)=i 2+1, where I & is the &_& idetity matrix. Thereore & = & op=1 ad this correspods to the smooth case with ;=0 where K $ e (, &)= m =0 q=& Y q ( ) Y q (&), or &, # S 2. Cosequety, i this case, (3.9) woud be X( )= 1 j=1 K $ e (, Z j ), (3.13) where # S 2 ad (3.13) woud be just ordiary oparametric desity estimatio o S 2, sice the observatios are made without error. This act aog with Theorem 3.1 provides the oowig coroary which states that or rate o covergece or oparametric desity estimatio o S 2, the rate obtaied i Theorem 2.1 o Hedriks (1990, p. 834) is optima. Coroary 3.3. Suppose = =$ e. I X # H s (S 2, M) or s>1 the &s(2s+2) is the optima rate o covergece, where m 1(2s+2) as. 4. EXAMPLES OF SMOOTH AND SUPER-SMOOTH DISTRIBUTIONS I this sectio we wi discuss three dieret error distributios with two o them beig smooth ad oe o them beig super-smooth. A o these distributios are characterized spectray. 4.1. Rotatioa Lapace Distributio This distributio is the rotatioa aaogue o the Eucidea Lapace (doube expoetia) distributio ad is discussed i depth i Heay et a. (1998). Athough a cosed orm expressio or SO(3) is avaiabe, see
OPTIMAL SPHERICAL DECONVOLUTION 11 Theorem 3.5 i Heay et a. (1998), it's spectra versio is much more iormative. Ideed i terms o the rotatioa harmoics, = = 0 q=& (1+_ 2 (+1)) &1 (2+1) D qq, (4.1) or some _ 2 >0. Spectray, =, qj=(1+_ 2 (+1)) &1 $ qj, or =0, 1,..., where $ qj =1 i q= j ad is 0 otherwise. As ca be see rom (4.1), this is a exampe o a smooth distributio with ;=2. 4.2. The Rosetha Distributio The ext distributio comes rom a probem i probabiity associated with radom waks o groups; see Diacois (1988) ad Rosetha (1994). Here oe is iterested i perormig radom waks o groups, oowed by estabishig ways i which the measure coverges to the uiorm measure, the so-caed ``mixig''. I terms o the mathematica structure, each movemet i the radom wak is represeted by a covoutio product. The ature i which iite covoutio products coverges to the uiorm measure is aayticay studied usig Fourier methods o the group. The case or SO(N) has bee studied i Rosetha (1994). Borrowig rom his work, we wi cosider the situatio where = is a p-od covoutio product o cojugate ivariat radom measures or a ixed axis, where the p>0 measures the degree o uiormy. For SO(3) take the cojugacy cass o cos % &si % 0 R % =_si % cos % 0&, 0 0 1 or % #(0,?], oowed by takig the uiorm measure over the cojugacy cass o R %. Let = be the p-od covoutio product. Rosetha (1994, p. 407) shows that = =, qj _ si(+12) % p $ (2+1) si %2& qj, or &q, j, =0, 1,..., where 0<%? ad p>0. Sice } si(+12) % (2+1) si %2} 1 (2+1) si %2
12 KIM AND KOO TABLE I Distributio Type ad Rates o Covergece Distributio Smoothess type Covergece rate Lapace Smooth &s(2s+6) Rosetha Smooth &s(2s+2p+2) Gaussia Super-smooth (og ) &s2 wheever 0<%? or =0, 1,..., the accordig to (3.6), = is ordiary smooth with ;= p. For SO(3) or ay ixed SO(N), as the covoutio product idex p, the i various metrics icudig L 2. = (g) dg dg 4.3. Gaussia Distributio The Gaussia distributio ca be soved o geera Riemaia maiod by sovig the appropriate heat equatio. Sice the D q 1, q 2 are the eigeuctios o the Lapacia 2 o SO(3) with eigevaue (+1)2, or &q, =0, 1,..., ater takig accout o the rotatioa symmetries, we ca write the distributio as = = 0 q=& exp(&t(+1))(2+1) D qq, (4.2) or t>0. Cosequety, =, qj =exp(&t(+1)) $ qj, so that it is a exampe o a super-smooth distributio with #=1t ad ;=2. 4.4. Summary o Distributios I Tabe I we summarize the above distributios aog with their smoothess properties. I additio, we state their optima rate o covergece. 5. THE VON MISESFISHER DISTRIBUTION The most widey reereced distributio or directioa data is the vo MisesFisher matrix distributio; see, or exampe, Khatri ad Mardia
OPTIMAL SPHERICAL DECONVOLUTION 13 (1977). Uike the distributios o Sectio 4, this distributio is directy expressed as =(g) (x)=c(}) exp(} tr g &1 x), (5.1) where }>0 is the cocetratio parameter aroud the mea rotatio g # SO(3) ad tr deotes the trace operator; see Khatri ad Mardia (1977). To uderstad the ature o the smoothess o (5.1) we eed to cacuate the rotatioa Fourier trasorm; however, prior to doig so, observe that i g ad } are kow, the we ca assume g=e, the uit eemet o SO(3). This comes rom the observatio that =(g) V X ( )= =(e) V X (g &1 ), (5.2) so that we ca re-oriet the estimatio by g &1 ad estimate the desity usig =(e) istead o =(g). Usig resuts rom the represetatio theory o SO(3), i particuar Schur's emma ad the CebschGorda ormua, the rotatioa Fourier coeiciets o (5.1) ca be cacuated. I Sectio 2, we state that [D =0,1,...] costitute a coectio o iequivaet irreducibe represetatios o SO(3). Two cosequeces o this act are the oowig. First, suppose SO(3) C is a cass uctio, which meas that (gxg &1 )= (x) or a x, g # SO(3). The Schur's emma, see Bro cker ad tom Diek (1985), says that qj = $ qj (2+1) SO(3) (g) /~ (g) dg, (5.3) where &q, j ad / =tr D are the irreducibe characters o SO(3), =0, 1, 2,..., i particuar, the rotatioa Fourier trasorms o cass uctios are a costat mutipe o the idetity matrix. The secod property is that tesor products o irreducibe represetatios, ca be decomposed as a direct sum, the so-caed CebschGorda ormua; see Bro cker ad tom Diek (1985). Ideed, D k D =D k& D k& +1 }}}D k+, (5.4) or a k, =0, 1,... Now or the irreducibe characters we have or a k, =0, 1,... / k } / =/ k& +/ k& +1 +}}}+/ k+, (5.5)
14 KIM AND KOO We are ow ready to discuss the rotatioa Fourier trasorm o (5.1). Assume g=e as i (5.2). Note that =(e) is a cass uctio sice it depeds o / (x)=tr x. Hece by Schur's emma we eed to cacuate SO(3) exp(} tr x) / (x) dx= SO(3) exp(}/ 1 (x)) / (x) dx = k=0 } k k! SO(3) (/ 1 (x)) k / (x) dx, (5.6) or a =0, 1,... By the CauchySchwarz iequaity, oe ca show that (5.6) is bouded, hece takig the Tayor series expasio or the expoetia uctio ad iterchagig itegratio with summatio is justiied. Now by the CebschGorda ormua, we ca write / k 1 = k p=0 ' p, k / p, (5.7) where ' p, k 1 or p0, k1. The set o irreducibe characters [/ =0, 1,...], is a compete orthoorma basis or the space o square itegrabe cass uctios hece /k 1 / =', k, (5.8) where k. Now appyig CauchySchwarz to (5.8) ad otig that / 1 (g) / 1 (e)=3 or a g # SO(3), we cocude that 1', k 3 k, (5.9) or a k ad =0, 1, 2,... The immediate resut is that whe we appy (5.9) to (5.6), the c(}) } (2+1)! =(e), qq 2c(})(3}) (2+1)!, (5.10) or &q as. The upper boud uses the act that or 0<x<2. &1 exp(x)& j=0 x j j!2x!
OPTIMAL SPHERICAL DECONVOLUTION 15 Our iterest is whe, hece we ca appy Stirigs approximatio to (5.10) so that c(})(}e) exp(&(+12) og ) - 2? (2+1) 2c(})(3}e) =(e), qq exp(&(+12) og ), (5.11) - 2? (2+1) as or }>0. Oe ca see that with the appearace o the ogarithm term i the expoet o (5.11), we caot get the same vaue or ; o both sides o (5.11). The cosequece is that the vo MisesFisher distributio is somewhat aomaous i that &( = )&1 & op Ed &1 0 exp( 1+; #) ad &( = oped 1 &32 exp(&#) as, (5.12) or some positive costats ;, d 0, d 1, ad #, hece it is smoother tha super-smooth. We have the oowig resut. Coroary 5.1. Suppose = is distributed accordig to the vo Mises Fisher distributio. I X # H s (S 2, M) or some s>1, the (og ) &s is a ower boud rate o covergece, whie (og ) &s(1+;) is a upper boud rate o covergece or ay ;>0, as. As a aside, it has og bee kow i the directioa statistics iterature that the vo MisesFisher distributio athough cose, is ot the same as the Gaussia distributio. The cacuatios o this sectio aog with Sectio 4.3 show exacty the ature o the dierece betwee the vo Mises Fisher ad Gaussia distributios. As, the characteristic uctio o the Gaussia ad the vo MisesFisher distributios behave ike exp(&t 2 ) ad exp(& og ), respectivey or some t>0. Ceary, the Gaussia distributio has sighty smoother tais which thereore accouts or a sower (but ot by much)
16 KIM AND KOO rate o covergece or the decovoutio desity estimator reative to the vo MisesFisher rotatioa errors. 6. PROOFS We wi prove Theorem 3.1 ad Theorem 3.2 by irst idig upper bouds or the smooth ad super-smooth cases. Foowig this we wi estabish ower bouds or these smoothess casses ad demostrate that the upper ad ower bouds match so that the resutig bouds are optima. The approach o Heay et a. (1998) wi be used or cacuatig the upper bouds, whie the approach o Koo (1993) ad Koo ad Chug (1998) wi be used to id the ower bouds. Forthwith, et M, M 1, M 2,... deote positive costats idepedet o the sampe size ad et C deote a positive costat which may have a dieret vaue at each o its appearaces. 6.1. Upper Bouds Let & & deote the L orm o a uctio o S 2. Lemma 6.1. I h # H s (S 2, M) with s>1, the &h& C(M, s), where C(M, s) is a costat depedig oy o M ad s. Proo. Write h=, q h Y q q. Observe that h( ) 2 \, q 2+\ ( ) 2+ (1+(+1)) s h q (1+(+1)) &s Y q, q (1+M) (1+(+1)) &s (2+1)(4?). I the above, we use the additio ormua, q=& Y ( ) Y (&)=2+1 q q 4? P (cos #(, &)), (6.1) where #(, &) represets the age betwee &, # S 2 ad P (1)=1 are the Legedre poyomias, or a 0. Sice s>1, the series (1+(+1)) &s (2+1) is coverget. K
OPTIMAL SPHERICAL DECONVOLUTION 17 Lemma 6.2. # S 2 Suppose X # H s (S 2, M) with s>1. The Var( ( )) d << X { m2;+2 &1 exp(2m ; #) m &2; 0 +2 &1 smooth super-smooth as. Proo. We ote that Var( ( ))=1 X Var(K = (, Z))1 EK = (, Z) K = (, Z), or # S 2, where Z deotes the radom S 2 eemet =X. By Lemma 6.1 ad (3.2), Z is bouded by a costat C so that S 2 K = (, z) K = (, z) Z(z) dzc S 2 K = (, z) K = (, z) ds m = =0 s=&} q=& 2 Y ( ) q = &1,. qs} Deie ` (&)= q=& Y q( ) Y q(&), where, & # S 2. The s=&} q=& 2 Y ( ) q = &1, =& = qs} &1` & 2 2 & = &1&2 op &` & 2 2 =& = &1&2 op q=& Y q ( ) 2 The secod ie uses the operator iequaity (3.4), the third ie uses Parseva's idetity, whie the ast ie uses the additio ormua (6.1) aog with the act that P (1)=1. Thereore, we have Var( ( )) d C X S 2 m =0 & = &1&2 op (2+1). Now appy the deiitios o smooth ad super-smooth to the ast expressio. K
18 KIM AND KOO Lemma 6.3. Suppose X # H s (S 2, M), where s>1. The & X &E X &2 2 <<m&2s. Proo. Observe that F X ( )&E X ( )= >m q=& X, q Y q ( ). Sice m 2s & X &E X &2 2 =m2s >m q=& X, q 2 >m q=& (1+(+1)) s X, q 2 (1+M) we have the desired resut. K By puttig together Lemma 6.2 ad Lemma 6.3, upper boud estimates ca be estabished as E & & X X& 2 << 2 { m2;+2 &1 +m &2s exp(2m ; #) m &2; 0 +2 &1 +m &2s smooth super-smooth as. Cosequety, choosig m { 12(s+;+1) (og ) 1; smooth super-smooth as optimizes the upper boud rates, respectivey. 6.2. Lower Bouds To show that the upper boud rates are optima rates, we cacuate ower boud rates o covergece ad show that these are the same as the upper bouds. I cacuatig the ower bouds we oow the popuar approach v speciy a subprobem; v use Fao's emma to cacuate the diicuty o the subprobem. Let N be a positive iteger depedig o ad deie V =[(, q) q=0, 1,...,, =N +1,..., 2N ].
OPTIMAL SPHERICAL DECONVOLUTION 19 Deie (Y +Y q &q)- 2 i q>0 is eve ={Y q 0 i q=0 (Y &Y q &q)- 2 i q>0 is odd or (, q)#v. Sice Y q =(&1)q Y, &q q is a rea-vaued uctio or each (, q)#v.let{={()=[{ (, q)#v q ] ad cosider the uctio { =(4?) &12 +M 1 N &s&1 2N =N +1 q=& { q q, (6.2) where M 1 is a positive costat such that 3(7 s ) M 2 1M. Fiay, et F =[ { { # [0, 1] V ], (6.3) where or some give iite set, } wi deote it's cardiaity ad assume that N as. Uder the assumptio that s>1, we have the oowig emma. Lemma 6.4. For suiciety arge, F /H s (S 2, M) ad M &1 2 M 2 or a # F. Proo. Let The { q- 2 i q>0 {~ ={{ q 0 i q=0 (&1) q { q- 2 i q<0. q=0 { q q = q=& {~ q Y q. Usig this act ad appyig the Soboev orm to (6.2), we get & { & 2 H s =1+M 2 1 N &2s&2 2N =N +1 q=& ({~ q )2 (1+(+1)) s 1+M 2 1 N &2s&2 1+7 s M 2 1 N &2 2N =N +1 2N =N N +1 (+1)(1+)(1+(+1)) s (+1) 1+3(7 s ) M 2 1. (6.4)
20 KIM AND KOO Cosequety, we have show that { # H s (S 2, M). Sice s>1, we have the desired resut. K Now et, g # F with { g. By the orthoormaity o q, we have & & g& 2 M 1 N &s&1. (6.5) It oows rom (6.5) ad Lemma 3.1 o Koo (1993) that there exists a F 0 /F such that or a u, v # F 0 with u{v, Deie The &u&v& 2 M 3 N &s ad og( F 0 &1)M 4N 2. (6.6) $#u&v=m 1 N &s&1 & = V $& 2 2 = 2N =N +1 q=& 2N =M 2 N &2(s+1) 1 2N =N +1 q=& ( @ = V $ ) 2 q =N +1 } q=& 2N M 2 1N &2(s+1) & =& 2 op =N +1 2N M 2 N &2(s+1) 1 =N +1 {~ q Y q. 2 {~ =, qj j} j=&1 q=& {~ j 2 & = &2 op (+1). The irst iequaity above is obtaied by the operator iequaity, (3.4), whie the ast ie uses the deiitio o {~. This thereore impies that & = V u& = V v& 2 << 2 { N &2(s+;) exp(&2n ; #) N &2s+2; 1 as. By (3.2) ad Lemma 6.4, we have smooth super-smooth (6.7) M &1 2 = V { M 2. (6.8)
OPTIMAL SPHERICAL DECONVOLUTION 21 Now the KubackLeiber iormatio divergece D( & g) betwee two desities ad g is deied by D( & g)= S 2 og( g) ad ( D( = V & = V g) = V & = V g) 2. (6.9) S 2 = V g Thereore, by (6.7), (6.8) ad Jese's iequaity D( = V u & = V v)<< &2(s+;) {N exp(&2n ; #) N &2s+2; 1 smooth super-smooth (6.10) or a u, v # F 0,as. By Fao's emma, see, or exampe, Birge (1983), Yatrocos (1988), or Koo (1993), i is ay estimator o, the sup # H s (S 2, M) P (& & & 2 >cn &s ) sup # F 0 P (& & & 2 >cn &s ) 1& D( = V u & = V v)+og 2 og( F 0 &1). (6.11) Appy (6.6) ad (6.10) to the ast ie i (6.11). Fiay, et N { 12(s+;+1) (og ) 1; smooth super-smooth as. The it oows or the two smoothess casses that im c 0 im i sup # H s (S 2, M) P (& & &>cn &s )=1, thus estabishig the ower boud. We ca ow use these ower bouds aog with the upper boud resuts which the competes the proos to Theorem 3.1 ad Theorem 3.2. ACKNOWLEDGMENTS Parts o this research were competed whie the irst author was visitig the Departmet o Statistics, Haym Uiversity, ad whie the secod author was visitig the Departmet o Mathematics ad Statistics, Uiversity o Gueph. They thak the two istitutios or their hospitaity durig these visits.
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