R 2 and R 3 John J. Benedetto and Joseph D. Kolesar

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1 Geometric Properties of Grassmannian Frames for R R 3 John J Benedetto Joseph D Kolesar Abstract Grassmannian frames are frames satisfying a minma correlation criterion We translate a geometrically intuitive approach for two three dimensional Euclidean space R R 3 into a new analytic method which is used to classify many Grassmannian frames in this setting The method associated algorithm decrease the maimum frame correlation hence give rise to the construction of specific eamples of Grassmannian frames Many of the results are known by other techniques even more generally so that this paper can be viewed as tutorial However our analytic method is presented with the goal of developing it to address unresovled problems in d-dimensional Hilbert spaces which serve as a setting for sherical codes erasure channel modeling other aspects of communications theory I ITRODUCTIO A finite frame k } k Rd where R d is d-dimensional Euclidean space is characterized by the property that its span is R d see [] The norm of R d is the usual Euclidean distance Given a finite frame for R d with elements we would like to measure the correlation between frame elements in particular to decide when the correlation is small We consider the following metric which is similar to an l norm [] Definition I Let d let Xd k} k be a subset of R d with each k The maimum correlation of Xd M X d is defined as M X d ma k l k l ote that because we consider the absolute value of the inner product rather than just the inner product if the angle between a pair of vectors is closer to 9 then the pair is less correlated while if the angle is closer to or 8 then the pair is more correlated Thus we are measuring the smaller angle between the lines one dimensional subspaces spanned by these vectors We could instead consider an l l or l p - type norm to measure correlation ie M p X d k l p k l /p I or even weighted versions of I See [3] for a discussion of the case p Address: Mathematics Department University of Maryl College Park MD 74 jjb@mathumdedu jkolesar@mathumdedu Both authors gratefully acknowledge support from SF DMS Grant 3979 OR Grant We also greatly appreciate the perspicacious constructive observations by the three referees Fi d with d Our goal is to construct element unit norm frames Xd with smallest maimum correlation M X d ie unit norm frames that are maimally spread apart To this end we make the following definition Definition I Let d A sequence Ud u k} k Rd of unit norm vectors is an d-grassmannian frame if it is a frame if M U d inf M X d } I where the infimum is taken over all unit norm -element frames for R d A compactness argument shows that Grassmannian frames eist see the Appendi but constructing Grassmannian frames is challenging [4] [] [6] As is described in [] the concept of Grassmannian frames is related to several other areas of mathematics engineering for eample packings in Grassmannian spaces spherical codes designs the construction of equiangular lines strongly regular graphs reduction of losses associated with packet based communications systems such as the Internet [7] [8] [9] In this paper we give an analytic construction of Grassmannian frames in R d when d 3 The first treatment of this construction problem in the case d 3 is found in [] There are etensive computational theoretical results in [4] which approach this construction problem from a sphere packing point of view The relevance of such constructions was brought to the attention of the frame community in [] After stating some technical preliminaries in Section II we characterize all -Grassmannian frames in Section III In Section IV we state prove a modest generalization of a theorem given in [] which provides a lower bound for M X d Section V is devoted to the construction of 4 3-Grassmannian frames from first principles rather than using the theorem in Section IV Conveity arguments are used in Section VI to construct eamples of 3-Grassmannian frames In Section VII we construct a 6 3-Grassmannian frame using the theorem in Section IV The techniques used in the constructions of Sections V VI VII were developed in part to fathom the geometrical ideas of Fejes-Tóth [] Our presentation is technical we believe necessarily so Some of the technicalities are routine but they are included since going from one step to the net without ehibiting them sometimes seemed mysterious On the other h some of our techniques may very well be useful in developing more general methods of finding frames with small correlations in applicable comple higher dimensional settings For eample see the new techniques used in proving Propositions VI3 VI4

2 Lemmas VI VI6 Our notation is stard but we do mention that means implies means if only if II PRELIMIARIES In this section we collect some definitions theorems used in the sequel The transpose of a vector or matri A is denoted by A T ; the Hermitian transpose of a vector or matri B with comple entries is denoted by B the conjugate transpose of B ie B B T A d d matri U with real entries is orthogonal if the columns of U are orthonormal ie U T U I d where U T is the transpose of U I d is the d d identity matri If U is orthogonal ie U SO d then for any y R d U U Uy y The torus is T π R/πZ We take any fied half-open interval of length π to be a representative of T π The unit sphere in R d is S d R d : } A set } of unit norm vectors is equiangular if there is an α [ ] such that k l α when k l A d d matri A with real entries is symmetric if A T A The spectral theorem for symmetric matrices is the following see [] We use it in Theorem IV Theorem II Spectral theorem A d d symmetric matri A over R has the following properties: i A has d real eigenvalues counting multiplicities ii The dimension of the eigenspace for each eigenvalue λ equals the multiplicity of λ as a root of the characteristic equation deta λi iii The eigenspaces are mutually orthogonal in the sense that eigenvectors corresponding to different eigenvalues are orthogonal iv A is orthogonally diagonalizable ie there is an orthonormal basis of eigenvectors for A We now state some basic definitions of frame theory [] [3] [3] [4] [] Let H be a separable Hilbert space let X n : n I} H where I is a countable indeing set Consider the following map associated with the set X: L : H l I y y n } n I If L is a well-defined linear map ie if n I y n < for any y H then L is a Bessel map X is a Bessel sequence The adjoint of L is the map L : l I H c[n]} n I n I c[n] n If L is a Bessel map the corresponding frame operator is the map S : H H defined as L L Thus for any y H Sy L Ly n I y n n As such y n I y n S n The Grammian operator is the map G : l I l I defined as G LL Both S G are positive hence self-adjoint operators A Bessel sequence X is a frame for H if there eist constants A B with < A B < such that for any y H A y y n B y n I Thus given any frame we have four natural maps: L L S G If the indeing set I is finite then X is called a finite frame Also if A B then X is called a tight frame or if we wish to emphasize the bound an A-tight frame Throughout this paper we shall use that fact that any finite set of vectors forms a frame for its span with the frame bounds being the largest smallest eigenvalues of the frame operator Since any finite set of vectors automatically has an upper frame bound by Cauchy-Schwarz the fact that any finite set is a frame for its span is a consequence of the following result Proposition II The following three statements are equivalent: i n } n is a frame for R d ; ii span n } n Rd iii A > such that y R d A y y n n III TWO DIMESIOAL GRASSMAIA FRAMES We classify all -Grassmannian frames The idea for the following proof is illustrated in Figure In fact in order to maimize the minimum angle between pairs of vectors the vectors we must be equally spaced Theorem III -Grassmannian Let X X k } k be a collection of unit vectors in R Then we have the lower bound cosπ/ M X Furthermore X is an -Grassmannian frame if only if there is P SO a sequence ε k } k ±} such that } PεX : Pε k k : k X cosπk/ : k sinπk/ } III Proof: Let δ T let δ T Since y y we note that changing the sign of any k X does not effect the value of M X Thus by changing the sign on k when necessary we may assume k v S : v δ } Also since rotations preserve inner products applying a rotation to all the vectors in X does not effect M X Thus rotating by φ where φ min k cos k δ reordering if necessary we may assume δ T 3 III For k let θ k be the angle between k k+ let θ be the angle between the negative -ais ie θ k cos k+ k θ cos δ see Figure for an eample when 6

3 3 6 θ θ 4 θ 3 θ 6 θ Fig An eample of the reordering induced by the inequalities on the inner products in III for 6 Then because of the above reordering θ k for k k θ k π Thus for l < k k k l cos θ j where jl k min θ k θ j π θ k Furthermore cosθ has a maimum on [ π] at θ θ π cosθ is monotone decreasing on [ π/] monotone increasing on [π/ π] Hence k M X ma k l cos θ j jl } ma cosπ θ cos min θ k k cos min k θ k Therefore in order to minimize M X we must choose positive numbers α α which sum to π which minimize cosmin k α k hence which maimize the epression min α k k jl θ 4 3 III3 ow we claim that if α α maimize III3 then α α We prove this implication by contraposition ie assume it is not the case that α α Then there is an m } so that if we list α α by size then only the first m are equal the m + st is strictly larger than the mth ie α k α k α km < α km+ α k Let ν α km+ α km for j define the sequence β kj as α kj + ν m for j m β kj α kj ν for j m + α kj for j m + ow the new set β k β k β km β km+ β k has a strictly larger minimum angle than the original since for j min α k α k < α k + ν k m β k β j We see that the original αs do not maimize III3 So by contraposition we have that if the αs maimize III3 then they must all be equal Finally if α is the common value then k α k α π therefore α π/ Thus π/ min k θ K therefore cosπ/ cos min θ K M X k et we prove that any -Grassmannian frame is up to a sign change the first adjacent vertices of a regular gon If X is an -Grassmannian frame then using the above argument we see that we can choose ε k } ±} P SO so that the frame PεX Pε k k : k X} is in the closed upper half plane with one of the vectors being T ; M X M PεX cos min θ k k where θ k is the angle between the kth k + st adjacent vectors in PεX reindeing may be necessary Since an -Grassmannian frame minimizes the -correlation M X the above argument also shows that θ θ π/ Therefore the angle between adjacent vectors in PεX is π/ we have proved the forward direction of the equivalence To show the reverse implication we note that if then PεX cosπk/ sinπk/ : k } M X M PεX cos min θ k cosπ/ k Hence X is -Grassmannian since it achieves the lower bound QED otice that for odd if we change the sign on the the th roots of unity below the real ais then we obtain the frame described in the above claim with ε k ie with all vectors in the upper half plane a common angle of π/ between

4 4 adjacent vectors Hence for odd the th roots of unity are -Grassmannian Furthermore for even the th roots of unity do not form an -Grassmannian frame because ζ ζ are both th roots If we identify ζ ζ then we obtain an / -Grassmannian frame IV A LOWER BOUD FOR M It is more difficult to construct a Grassmannian frame in R 3 for > 3 than in R Thus we first derive a lower bound for the maimum correlation between frame elements of an -element frame for R d see [] [] for superb treatments although we have felt compelled to spell-out all details Such lower bounds are useful in coding theory [6] we first learned of them in [] We shall need the following lemma whose proof can be found in the Appendi Lemma IV Let H n be the n n matri with on the main diagonal β elsewhere let C n be the n n matri defined by β if i j [C n ] ij [H n ] ij otherwise where [H n ] ij is the i jth entry of the matri H n Then deth n + n β β n detc n β β n IV IV Theorem IV Let d let Xd be an -element subset of S d let dim span Xd Then M X d IV3 where equality holds in IV3 if only if a Xd is equiangular b Xd is a tight frame for its span with frame bounds A B Furthermore if > dd+ then Xd is not equiangular hence equality cannot hold in IV3 Proof: a First we prove the inequality IV3 Since the Grammian matri G is symmetric the spectral theorem applies so G has eigenvalues λ j counted with multiplicity ordered by size ie λ λ λ Furthermore since rankg only the first of these eigenvalues are nonzero Hence k λ k TraceG k k k ow set e k λ k Then k e k λ k d k k so k λ k k k + e k d + + k k e k e k + k with equality if only if e k for k ie if only if λ k for k et the eigenvalues of G are λ λ λ so that if g k is the kth column of G then by matri multiplication we have k k l λ k TraceG k l gk T g k k e k IV4 Since G is symmetric k l l k so that by IV4 we compute k l kl k l k l + k<l k l + k>l + k l k<l + ma k l } k l Therefore solving for the ma in IV we have k l IV M X d IV6 d For future reference we note that d implies d Hence IV3 remain true when we replace d with d b et we prove that equality holds in IV3 if only if Xd is equiangular is a tight frame for its span : Suppose M Xd d Then IV becomes k l k l which implies from IV4 that k λ k ; as we saw above equality in this sum implies that λ k for k The frame bounds for Xd are the largest smallest nonzero eigenvalues hence A / B so Xd is a tight frame for its span

5 To see that Xd gives hence is also equiangular we notice that IV + k<l k<l k l Our assumption ma k l k l for any k l k l IV7 d implies that f 3 k l ε kl where ε kl Thus IV7 is d ε kl k<l d ε kl k<l k<l ε kl Therefore k<l ε kl since the ε kl are non-negative we can assert that ε kl for k < l Also since G is symmetric ε kl for all k l hence Xd is equiangular with k l d d : ow assume Xd is equiangular is a tight frame for its span with frame bound A B Then there is an α [ ] such that k l α for k l ow since Xd is tight λ k for k zero otherwise Hence IV4 IV imply k λ k k l k l + α Thus solving for α we see that equality holds in IV3 c Finally to prove > dd+ implies Xd is not equiangular we prove the contrapositive using Lemma IV the following argument cf [7] Assume Xd is equiangular Let P k : R d R d be the projection of onto the line spanned by k ie P k k k Let V be the vector space of symmetric linear mappings R d R d Then dimv dd+ the map : V V R given by C D TraceCD is an inner product on V Since Xd is equiangular there is an α [ ] such that k l ±α for k l Furthermore α implies since the elements of Xd are assumed to be distinct of unit norm Thus for d we have < 3 dd+ Therefore we may assume α [ ow P k P l k l if k l α if k l Hence the Grammian of the set P P } V is if k l [G] kl [ P k P l ] kl α if k l Fig The function f cos π/ on [7] where [G] kl is the k lth entry of the matri G Thus Lemma IV applies with G H β α Consequently if α [ then detg + α α Therefore G is invertible has full rank Finally since rankg ranks we have that rankg dimspan P P } dd + dimv We have proved that if Xd is equiangular then dd + /; so by contraposition we have proven the result QED Remark: Theorem III shows that M X cosπ/ while Theorem IV shows M X Using stard calculus techniques we can show that the equality in Theorem III is an improvement over the bound in Theorem IV for all > 3 Let f cos π/ see Figure so that f π 3 Hence f π sin π π cos π + sin π + f > sinπ/ > π For [3 6] π sin 3 9 8π π 3 so f is increasing for [3 6]; since f3 we have that f for [3 6] Furthermore for

6 6 Optimal bound Bound from Theorem III q cosπ/ TABLE I IMPROVEMET OF THE OPTIMAL BOUD DERIVED I THEOREM IV FOR 36 [6 π π if THE CASE OF -GRASSMAIA FRAMES Also sin π 36 π π sin < π if only Thus f is increasing for [3 7] hence it is greater that zero on that same interval We also note that for > π < π Further sin π π when π sin > 393 π Hence f is decreasing on the interval [4 since lim f we have that f > for [4 Finally we check that f < on the interval [7 4] f7 f4 > ; thus f > on [7 4] In summary we have shown that f > on 3 that f > on [7 Therefore π cos > for > 3 we see that Theorem III is an improvement over Theorem IV in the case d see Table I In light of Theorem IV we make the following definition Definition IV3 Let d with d dd+ Let Xd k} k be a frame for Rd with k We call Xd an optimal Grassmannian frame if Xd satisfies IV3 with equality ie M X d d d In R since d dd+ 3 only frames with 3 elements can be optimal Grassmannian Since cosπ/ / cosπ/3 / 3 /3 both - 3 -Grassmannian frames are optimal The same phenomenon does not happen in three dimensions Table II lists the Grassmannian bound which Optimal bound Grassmannian bound q 3 3 min M `X TABLE II BOUDS FOR -ELEMET FRAMES I R 3 WITH POTETIAL OF BEIG OPTIMAL GRASSMAIA will be proven below the optimal bound for the only s with the possibility of being optimal By inspecting Table II we notice that 3-Grassmannian frames are not optimal while Grassmannians are optimal V 4 3-GRASSMAIA FRAMES In this section the net two we shall derive the bounds for three dimensional Grassmannian frames with First note that if 3 if X is any orthonormal basis for R 3 then M X Hence any orthonormal basis is Grassmannian in fact X is trivially optimal Grassmannian et consider 4 We need the following two lemmas which are necessary to rigorize Fejes Tóth s ideas in [] In particular Lemma V is intuitively elementary when we consider the fact that the l -norm is conve Q is conve C is the set of etreme points of Q Lemma V Let a R d let v v v d } R d Set d Q a + s j v j : s j [ ] j d C a + ε j v j : ε j } j choose c C such that c ma c l : c l C} where l d Then for any v Q \ C v < c Proof: Let v Q \C so that v a+ d j s jv j Since v / C there is an m such that s j s jm s jm+ s jd } For i m set t i s ji for i > m set ε i s ji so t i ε i } ow let w v for each i m recursively let w i w i + ε i t i v ji where if v ji w i > ε i if v ji w i By induction on i we show that v w < w < < w m

7 7 ote that by construction of ε i we have that w m C hence w m c Also note that for i m w i w i + ε i t i w i v ji + ε i t i v ji V We begin the induction with the base case i Inspecting V with i we have 3 cases Case w v j > Then ε so ε t > t > ε t w v j > Hence by V w > w Case w v j < Then ε so that ε t < t < therefore ε t w v j > Hence by V w > w Case 3 w v j Then ε w + t v j y where for k m d y k a + s ji v ji ik By the Pythagorean theorem w + t v j y hence w y t v j V Thus ε t < imply t v j < ε v j by equation V w < y + w + ε t v j w Consequently in every case w < w ow for the induction step If < i m if we have w < w < < w then repeating the above argument with w w y replaced with w i w i y i+ respectively we have w i < w i Finally since w m C we obtain v < w m c QED Lemma V Let b y y y 3 } S If b y b y b y 3 are not all equal then there is a constructible c R 3 such that ma b y k : k 3} V3 c > ma c y k : k 3} c Furthermore c y c c y c c 3 y Proof: Case y y y 3 } S is linearly dependent Then there is a a a 3 R 3 with at least one actually two a k such that a y + a y + a 3 y 3 Therefore dim kery where Y is a 3 3 matri with columns y j Hence dimspany ranky We can choose c span Y so that k 3 b y k > c c y k c c 4 y c 3 c Fig 3 An eample showing the points ±c k k 4 their relationship to the vectors y y y 3 ote y lies on the plane with vertices c c c 3 c 4 } since by assumption we know that the b y k cannot all be equal hence cannot all equal zero Case y y y 3 } S is linearly independent Let Y be the 3 3 matri whose columns are y j Then Y T is invertible Let c c 4 be the columns of the following 3 4 matri product [ c c c 3 c 4 c y 3 y c 3 c 4 ] Y T V4 see Figure 3 otice that c Y T Y T + + c + c 3 + c 4 Let c c c 4 } have the property that c ma c c 4 } We now show that this procedure of matri multiplication followed by taking the maimum gives rise to a vector c R 3 which satisfies the conclusion of Lemma V For j 3 define v j c j+ c set 3 Q c + s j v j : s j [ ] C c + j 3 ε j v j : ε j } j Identify a point in C with a vector ε ε ε 3 For eample if ε ε ε 3 then v c + v + v 3 Observe c y

8 8 that since c c + c 3 + c 4 we have the following bijection between C ±c ±c ±c 3 ±c 4 }: c c c 3 c 4 c 4 c 3 c c In particular c ma c : c C} ow if H v R 3 : v y k for k 3 } then Q H To see this we check both containments but first note that V4 with j implies Y T c yt c y T c y3 Tc ie y c y c y 3 c similarly for the other c j Q H: Let v Q Then 3 v y k c y k + s j v j y k 3 + s j v j y k j j whereas if k j v j y k c j+ c y k if k j Thus v j y k s k so s k [ ] implies that s k [ ] hence v j y k H Q: Equivalently we prove the inclusion Q c H c of complements Let v / Q First v v v 3 are the images of T T T respectively under the transformation Y T Hence v v v 3 } is a basis for R 3 Thus there are unique elements s s s 3 R 3 such that v c s v + s v + s 3 v 3 ie v c + s v + s v + s 3 v 3 Because of the uniqueness of the s j s v / Q implies there is a j 3} such that s j / [ ] ow v y j s j so s j / [ ] implies s j Thus v y j > so v / H Therefore we have proven H Q We complete the proof of Lemma V as follows For the set b y y y 3 } S let b y kb ma b y b y b y 3 } set λ b b y kb Then for any k 3 b y k b y k λ b b y kb b y k b b y kb b so λ b Q ow v C implies that the elements of v y k : k 3} are all equal By the contrapositive of this implication we see that the assumption that the elements of b y k : k 3} are not all equal implies b λ b / C Thus we have shown b λ b Q \ C Therefore by Lemma V we have b λ b < c ; hence b S implies λ b λ b b b < c We conclude that ma b y k : k 3} λ b > c c ma c y k : k 3} λ b so Lemma V is proved QED With these two lemmas we can prove the following theorem Theorem V3 43-Grassmannian Let U u u u 3 u 4 } S R 3 If U is 4 3-Grassmannian then U is equiangular ie u k u l β [ ] whenever k l Proof: We show the contrapositive of the above implication viz we prove that if U is not equiangular then there is a 4 element set X S such that M X < M U Hence U does not have minimal maimum correlation therefore it is not 4 3-Grassmannian Suppose U u u u 3 u 4 } is not equiangular Then there is an m 3 4} such that if k k k 3 are the remaining indices we have ma u m u k u m u k u m u k3 } M U u m u k u m u k u m u k3 are not all equal Applying Lemma V with b u m y y y 3 } u k u k u k3 } there is c R 3 such that ma u m u ki : i 3 } } c > ma c u k i : i 3 Let m c c see step in Figure 4 Since we have only moved the point u m to m the remaining correlations are unaffected since they do not involve u m Thus M U ma u m u ki : i 3 } > ma m u ki : i 3 } : α V ow either M m u k u k u k3 } α or there is an m 3 4} \ m } such that if m j j are the remaining indices then M U ma u m u j u m u j } V6

9 9 u k 3 Step u k u m 3 4 u j 4 Step u m m 4 4 u j 4 4 Step 3 m m m3 48 Step 4 m 48 m 4 u k3 u j 4 Fig 4 An eample of the four steps in proving Theorem V3 A number net to an edge represents the inner product of the two boundary points of the edge u j 4 u j u m m u m u j u m u j are not all equal V7 where V7 follows from V In this case we apply Lemma V to b u m y y y 3 } u j u j m } Thus there is a c R 3 such that ma u m m u m u j u m u j } ma u m u j u m u j } } c > ma c m c c u j c c u j Let m c c see step 3 in Figure 4 Thus M U ma u m u j u m u j } > ma m m m u j m u j } V8 : α Therefore V V8 imply M U > ma ma α α } m u j m u j m m m u j m u j V9 because j j k k k 3 } So either M m m u j u j } ma α α } or else M U u j u j In the latter case V8 implies that u j u j u j m u j m are not all equal so we apply Lemma V to b u j y y y 3 } u j m m } Thus there is c R 3 such that ma u j m u j m u j u j } u j u j c c > ma c m c m c c u j V Let m3 c c let X m m m3 u j } Then V9 V imply M U V } m3 > ma m m3 m m3 u j m m m u j m u j M X QED et we show that if a four element set is equiangular then the vectors are parallel to the diagonals of a cube or to four of the diagonals of an icosahedron Theorem V4 If u u u 3 u 4 S u k u l α for k l 4} with k l then α 3 or α

10 w w 3 w w 4 w 3 w 4 case α α α impossible case α α α α / case 3 α α α α / case 4 α α α α /3 TABLE III FOUR MAI CASES I THE PROOF OF THEOREM V4 Proof: Since sign changes rotations do not effect inner products let P be an element of SO 3 which rotates u to δ 3 T For k 3 4 let ε k sign P k δ 3 let Q SO 3 so that Q fies δ 3 Q rotates ε k P to the positive z-plane ie Qε k P δ Qε k P δ Then for k l α u k u l ε k QPu k ε l QPu l w k w l where w k ε k QPu k By the choice of ε k for k 3 4 we have α w w k δ 3 w k so that the third component of w k is α Also δ w w S so that the first component of w is α Therefore we have w T w α α T w 3 3 y 3 α T w 4 4 y 4 α T V We now have four cases see Table III where both case 3 have three subcases which by relabeling can be reduced to the considered case Case For k 3 4 w w k α implies k α α α α α + α V3 Then V3 w k imply + α α y k ± V4 + α In addition V3 w 3 w 4 α imply α α y 3 y 4 + α V Combining V4 V we have + α α + α α α + α hence case is impossible Case ow w w 3 α implies 3 α α α α + α C \ R; V6 w w 4 α implies 4 V6 V7 imply α α α y 3 y 4 α Then V6 w 3 imply y 3 α α α + V7 w 4 imply y 4 α + α α Finally V8 V9 V imply V7 V8 α + α V9 α + α α + V α α α + α α ± Since α is assumed to be positive we have proven case Case 3 For k 3 4 w w k α implies k Then V w k imply y k α + α α Hence V w 3 w 4 α imply Combining V V3 we have α α + α α α α V α α + V α y 3 y 4 α 3α α V3 α α + yk α y 3 y 4 α 3α α α ± since α is positive we have proven case 3 Case 4 This is the same as case 3 ecept that w 3 w 4 α V imply y 3 y 4 αα + α Combining V V4 we have α α + y αα + k y 3 y 4 α α α 3 we have proven case 4 Therefore the theorem is proved QED V4 By Theorem V4 since > 3 we see that the 4 3- Grassmannian bound is 3 which is also seen to be optimal by inspection

11 VI 3-GRASSMAIA FRAMES We first introduce some ideas from conve analysis [8] [9] [] Definition VI A set A R n is conve if for any A for any λ [ ] λ + λ A A point A is an etreme point of A if whenever λ + λ where < λ < A then Given a set A R n the conve hull of A is HullA m m λ j j : λ j λ j > j A m j j There is the following relationship between etreme points conve hulls conve sets [] Theorem VI A nonempty bounded conve set in R d is the conve hull of its set of etreme points We need the following two conveity propositions to prove Lemma VI which in turn is used to prove Lemma VI6 the key lemma for computing the 3-Grassmannian bound in Theorem VI7 Proposition VI3 Let d let Y y y } S d R d assume spany R d Let Q v R d : v y k for k } let C be the set of etreme points of Q Then a Q is a bounded conve set b If v C then there are at least d distinct integers k k d } such that v y ki for i d c cardc d d < Proof: a First to show Q is conve let Q Then for any λ [ ] for any k } λ + λ y k λ y k + λ y k λ + λ et we show Q is bounded Since spany R d Y is a frame for R d Let S be the associated frame operator let A B be the lower upper frame bounds respectively S is invertible so we can define v j S y j for j Then we have v j S y j S y j A ow for any R d S S y j S y j y j v j j j j j Thus given Q y j v j y j v j v j A j b We prove the contrapositive Assume v y k for fewer than d vectors in Y ie by relabeling assume there is an m such that v y k for k with k m v y k < for k m + We now show that v is not an etreme point by constructing Q with such that there is a λ for which v λ + λ Let Ỹ span y y m } where Ỹ is empty if m Since by assumption m < d we have dimỹ < d Let z Ỹ S d define β ma v y k : k m + } By the choice of m we have β < Set v + β z v β z otice that β < implies β z β > hence Furthermore if λ then λ + λ v + β z + 4 v β z v 4 Finally we check that are in Q For k m l we have l y k v y k ± β z y k v y k ; for k m + l we have l y k v y k ± β z y k v y k + β z y k β + β z y k + β < Thus v Q \ C c If v is an etreme point of Q ie v C then v must satisfy at least d of the equations which define Q Therefore we count the number of ways we can pick d distinct elements y k from Y to satisfy the d equations v y k There are d d-element subsets of } because of the absolute value there are two choices for the equation each v can satisfy namely v y k or v y k ote if any one of the remaining d inequalities is not satisfied by v then v is not an etreme point This proves that we can have fewer than d d etreme points for a given arrangement of y k s QED Under the same hypotheses of Proposition VI3 we have the following result Proposition VI4 Let d Y Q C be as in Proposition VI3 let c C have the property that c ma c : c C} Then for any v Q \ C v < c

12 Proof: Let v Q\C Then there is a λ there are Q with such that v λ + λ Consider the function f : R R defined by fλ λ + λ We first verify that f is uniformily continuous on R Let ε > ε be given choose δ < Then for all λ for which λ λ < δ we have fλ fλ λ + λ λ + λ λ + λ λ λ λ λ λ λ < δ < ε ow set gλ fλ Then gλ is also continuous on R it is the parabola gλ λ + λ λ + λ λ λ We compute g λ λ + so that g λ at Furthermore for all λ R λ g λ > VI Hence g attains a minimum at λ for all λ λ we have gλ > gλ ow if we restrict g to [ ] then g achieves its maimum minimum on [ ] Thus if λ [ ] then by VI the fact that g describes a parabola we have if λ / [ ] then min gλ gλ λ [] ma gλ ma g g}; λ [] min gλ min g g} λ [] ma gλ ma g g} λ [] In either case the maimum of g occurs at one of the end points Furthermore at interior points g is strictly less than the maimum value ow since v g λ for some λ VI implies v g λ < ma gλ ma g g} ma } λ [] Thus we have shown that hence v Q \ C Q such that v < v Q \ C v < sup : Q} VI Q is a bounded closed set so that by continuity of the supremum in VI is achieved on Q whereas VI also shows that this supremum is not achieved on Q \ C Thus sup : Q} sup : C} c Therefore for any v Q \ C we have v < sup : Q} c QED Following the basic geometric idea in [] but using the previous propositions which can be implemented as eplicit algorithms we can reduce the correlation of a given frame We proceed as follows Lemma VI Let U b y y y 3 y 4 } S R 3 let α M U Assume b y < α b y < α Then there eists c R 3 such that c c y k < α for k 3 4 Proof: If both b y 3 < α b y 4 < α then take c b Otherwise without loss of generality assume b y 3 α We have cases Case dimspan y y 4 } < 3 Then similar to Lemma V choose c span y y 4 } By Theorem IV c c y k < 6 α Case span y y 4 } R 3 Let Q v R d : v y k k 4 } let C be the set of etreme points of Q By Proposition VI3 Q is bounded conve C is finite Let c be a point in C of maimum norm Then by assumption b α y < b α y < b α y 3 b α y 4 b which shows that α can satisfy with equality at most two of the four equations which define Q Then by Proposition VI3 b α is not an etreme point of Q Hence by Proposition VI4 α b α < c Therefore since c C Q we have c y k c c y k c < α for k 3 4 QED

13 3 Lemma VI6 Let U u u } be a 3- Grassmannian frame let α M U Then for any j there are distinct j j j 3 } \ j} such that u j u jk α for k 3 Proof: We prove the contrapositive By relabeling if necessary without loss of generality assume u u < α u u 3 < α We use Lemma VI to construct a new set W for which M W < α This shows U is not 3- Grassmannian First let b u y y 4 } u u } apply Lemma VI Then there is a c R 3 such that c c u k < α for k 3 4 Second consider the set Ũ : u u } We have two cases Case There eist j k 3 4 } with j k for which u j u k < α For ease in notation by relabeling if necessary we assume j k 3 In this case we can apply Lemma VI with b u y y 4 } c c u 3 u 4 u } construct c R 3 such that c c c < α c c ma c u k : k 3 4 } < α ow we can apply Lemma VI to the remaining points produce a frame with a strictly smaller value of M In fact since ci c u i 3 < α for i we let b u3 c y y 4 } c c c u 4 u } Then by Lemma VI there is a c 3 R d such that c3 c 3 c i < α for i c i c3 ma c 3 u k : k 4 } < α Finally apply Lemma VI one last time to b u 4 } c y y 4 } c c c c 3 c 3 u obtain c 4 R d for which c4 c 4 c i < α for i 3 c i c4 c 4 u < α } Thus if we let W c c c c c 3 c c 4 3 c u 4 then by construction for any i j 4} i j we have c i c c i c < α c i c u i < α Hence M W < α M U so U is not 3-Grassmannian This finishes Case Case Ũ is equiangular Since Ũ has four elements Theorem V4 implies α /3 or α / If α /3 then set c β ma c u k : k 3 4 } Thus by construction of c we have β < 3 } } c M c Ũ ma 3 β 3 whereas Theorem IV with d 3 implies } c M 6 c Ũ 3 a contradiction Thus α ; u u k < α for k 3 4 u k u j α for k j k j 3 4 } We seek to find a contradiction Without loss of generality the setting can reduce to the following general position by using rotations sign changes as in Theorem V4: u T u 3 α α T u 4 u p p p 3 p 4 } where T α + α α p α + α α + α π α cos T π α sin α T + α α + α p α α α α 4π α cos T 4π α sin α + α α + α p 3 α α α α α cos 4π α sin 4π T α T α + α α p 4 α + α α + α α cos π α sin π T α T Therefore if cosπ/ sinπ/ A sinπ/ cosπ/

14 4 3 8 p γ β 6 p 4 p 3 p p p p 3 p 8 6 p Fig Top figure is the function γ γ bottom figure is the function d dβ γ γ We can see that the original function is strictly increasing 8 p Fig 6 Ten intervals on T π corresponding to the points p u 3 p p 4 p u 3 then A k p p p p 3 p 4 p σ p σ p σ p σ3 p σ4 where σn n + k mod If we set β u u < α then by changing the sign of u if necessary since u we may assume u β cost β sint β T for some fied t [ π π Hence u u 3 < α α β cost + αβ < α α + β < cost α < α β α cos β < t < cos + β }} γ β We observe that d γ β γ β 6π π π π 3 π 4π π π 3 VI3 β β + β β }} γ β VI4 if β α if β if β α if β that dβ γ γ β > for β α see Figure Thus π γ β γ β < 6π when β [ α Hence for a fied β [ α γ β < γ β + 6π for k 3 4 we have where α > u p k A k u A k p k A k u p A k u u 3 A k u β cos t πk β sin t πk T β Therefore by VI4 α > u p k γ β t πk γ β VI for k 3 4 These inequalities define ten intervals on the torus T π If we plot these ten intervals on T π we note that no set of three of them overlap see Figure 6 This assertion can also be seen since γ β t πk γ β γ β t πk < γ β + π t [ γ + εk γ + εk + }} P k [ γ + εk γ + εk }} k where γ γ β ε π k 3 4 Thus 4 k P k is a disjoint cover of T π \[γ ε γ 4 k k is a disjoint cover of T π \[ γ + ε γ Hence t can be in at most two of the ten sets P k k ow by assumption u u 3 u p < α Also u u 4 < α u u < α where u 4 u p p p 3 p 4 } Thus VI implies t lies in three of the ten intervals represented in Figure 6 a

15 contradiction Consequently Ũ cannot be equiangular QED Finally using Lemma VI6 we have the following result Theorem VI7 If U S R 3 is 3-Grassmannian then M U Proof: Let α M U consider the graph whose vertices are u u whose edges are defined as follows: for any pair of points u k u j U with k j an edge connects u k u j if only if u k u j α We call the number of edges emanating from a verte u k the degree of u k denoted deg u k Then Lemma VI6 implies that deg u k k 3 k Since each edge connects two vertices the sum of the degrees must be an even number Thus at least one verte u j must have degree 4 ie there is a j } such that u j u ji α for i 4 where j j j 3 j 4 } 3 4 }\j} By relabeling if necessary we may assume u u k α for k 3 4 u u k α for k 3 4 Furthermore we can reduce to the general position used in Theorem V4 ie assume u T u α α T u 3 3 y 3 α T u 4 4 y 4 α T u y α T We have two cases Case u 3 u 4 α Then the subset Ũ u u u 3 u 4 } is equiangular hence Theorem V4 implies α 3 or However just as in Lemma VI6 α 3 implies that 3 M U 6 so α Case u 3 u 4 < α Then since each verte must be of degree 3 we have that u 3 u u 4 u each equals α Thus if we remove the absolute values we have the following equations: u u 3 ±α u u 4 ±α u 3 u ±α u 4 u ±α This gives 4 6 possible cases Of these 6 cases 7 lead to contradictions the remaining 9 fall into types; but each implies that u 3 u 4 u are three of the four points T α + α α α + α ± α + α T + α α + α α α ± α α which are the positive endpoints on the remaining 4 diagonals of an icosahedron Hence in each case α / QED The 3-Grassmannian frame is the first eample of a non-optimal Grassmannian frame since > 6 Hence by Theorem IV the 3-Grassmannian frame is the first three dimensional eample of a Grassmannian frame which is not tight VII 6 3-GRASSMAIA FRAMES The 6 3-Grassmannian bound can be calculated as a consequence of Theorem IV Theorem VII If U u u 6 } S is 6 3- Grassmannian then M U / Proof: Set α consider the set W with vertices w T T w α α T α + α α w 3 α + α α + α T α + α α w 4 α + α α + α T + α α + α w α α α α T + α α + α w 6 α α α α ote that ±W are the twelve vertices of an icosahedron For k l we compute w k w l Furthermore by Theorem IV if U is a 6 element subset of S then 6 3 M U 36 M W Thus W is a 6 3-Grassmannian frame QED otice the 6 3 Grassmannian arrangement is so good that when you remove a vector from it it remains Grassmannian when we remove two vectors from it it is still a local minimum of M In [4] Conway Hardin Sloane have found that there are other instances of this in higher dimensions particularly when the symmetry group of the frame has a large number of elements

16 6 APPEDIX i We show that Grassmannian frames eist First we define the function f : S d S } d [ ] } times f M k } k et we check that f is continuous on X : R d R d times Consider the norm on X defined by k } k k X k let k } k X be fied set R ma k k } let ε > be given Clearly R Choose δ such that < δ < +ε R ie R δ + Rδ < ε Then whenever y k } k k} k < δ we have that for every j X } y j j y k k y k } k k} k < δ X k Therefore for each j there is an α j R d with α j < δ such that y j j + α j Thus fy y f ma k l + k α l + α k l + α k α l } k l ma k l } k l ma k l + k α l + α k l + α k α l } k l ma k l k l } < ma k l k l } + Rδ + Rδ ma k l k l } Rδ + Rδ < ε Hence f is continuous on the compact set S d S d times so f achieves its absolute maimum absolute minimum on this set Thus we know that d- Grassmannian frames eist for any d et we must check that if Ud solves I then U d is a unit norm frame for R d but this a tautology since by compactness Ud is one of the frames over which we are taking the infimum ii We now give a proof of Lemma IV used in Section IV Proof: [Proof of Lemma IV] We proceed by induction Let Pn be the statement deth n + n β β n detc n β β n For n H C β so deth detc β; this is P et assume Pn Using the cofactor epansion of the determinant we first note that the -cofactor of H n+ C n+ is deth n Also note that for j n + the j-cofactor of both H n+ C n+ is +j det B j n where B j n can be defined recursively as B n C n B j n B j n j B n for j n + where is B n j with the jth j st rows interchanged Since det is multilinear interchanging a row changes the sign of the determinant Hence +j detb j n detc n for j n + Using the induction hypothesis the cofactor epansion we compute n+ deth n+ deth n + β +j detb n j j deth n nβ detc n + n β β n nβ β n + nβ β β n detc n+ β deth n nβ detc n β + n β β n nβ β n β β β n ; this is Pn+ so the result follows by induction QED REFERECES [] O Christensen An Introduction to Frames Riesz Bases Birkhäuser Boston MA [] T Strohmer R W Heath Jr Grassmannian frames with applications to coding communication Appl Comput Harmon Anal vol 4 no 3 pp [3] J J Benedetto M Fickus Finite normalized tight frames Adv Comput Math vol 8 no 4 pp [4] J H Conway R H Hardin J A Sloane Packing lines planes etc: packings in Grassmannian spaces Eperiment Math vol no pp [] Y C Eldar H Bölcskei Geometrically uniform frames IEEE Trans Inform Theory vol 49 no 4 pp [6] J A Tropp I S Dhillion R W Heath T Strohmer Designing structured tight frames via an alternating projection method accepted Institute of Electrical Electronics Engineers Transactions on Information Theory [7] V K Goyal J Kovačević J A Kelner Quantized frame epansions with erasures Appl Comput Harmon Anal vol no 3 pp 3 33 [8] P G Casazza J Kovačević Equal-norm tight frames with erasures Adv Comput Math vol 8 no -4 pp [9] R Holmes V Paulsen Optimal frames for erasures Linear Algebra its Applications vol 377 pp 3 4 [] L Fejes-Tóth Distribution of points in the elliptic plane Acta Math Acad Sci Hungar vol 6 pp [] D Lay Linear Algebra its Applications Addison Wesley Boston MA 3 [] R J Duffin A C Schaeffer A class of nonharmonic Fourier series Trans Amer Math Soc vol 7 pp [3] J J Benedetto M W Frazier Eds Wavelets: Mathematics Applications CRC Press Boca Raton FL 994 [4] I Daubechies Ten Lectures on Wavelets SIAM Philadelphia PA 99 [] M Rosenfeld In praise of the Gram matri in The Mathematics of Paul Erdős II vol 4 of Algorithms Combin pp Springer Berlin 997 [6] L R Welch Lower bounds on the maimum cross-correlation of signals IEEE Trans Inform Theory vol pp

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