Real Equiangular Frames

Peter G Casazza Department of Mathematics The University of Missouri Columbia Missouri 65 400 Email: pete@mathmissouriedu Real Equiangular Frames (Invited Paper) Dan Redmond Department of Mathematics The University of Missouri Columbia Missouri 65 400 Email: dredmond@mathmissouriedu Janet C Tremain Department of Mathematics The University of Missouri Columbia Missouri 65 400 Email: janet@mathmissouriedu Abstract Real equiangular tight frames can be especially useful in practice because of their structure The problem is that very few of them are known We will look at recent advances on the problem of classifying the equiangular tight frames and as a consequence give a classification of this family of frames for all real Hilbert spaces of dimension less than or equal to 50 I ITRODUCTIO A family of vectors {f i } i I is a frame for a Hilbert space H if there are constants 0 < A B < so that for all f H A f f f i B f i I If A = B this is an A-tight frame and if A = B = this is a Parseval frame If f i = f j for all i j I this is an equal norm frame and if f i = for all i I this is a unit norm frame A unit norm frame with the property that there is a constant c so that f i f j = c for all i j is called an equiangular frame at angle c ote that we are calling this the angle when really it is the arc cos of the angle Frames have traditionally been used in signal processing Today frames have a myriad of applications in mathematics and engineering including sampling theory wavelet theory wireless communication signal processing (including sigmadelta quantization image processing operator theory harmonic analysis pseudodifferential operators sparse approximation data transmission with erasures filter banks geophysics quantum computing distributed processing and more What makes frames such a fundamental tool in these areas is their overcompleteness which allows representations of vectors which are robust to quantization resiliant to additive noise give stable reconstruction after erasures and give greater freedom to capture important signal characteristics For an introduction to frame theory we recommend [5] [8] For an introduction to frames in time frequency analysis (the mathematics of signal processing) see [0] Equiangular tight frames first appeared in discrete geometry [6] but today (especially the complex case) have applications in signal processing communications coding theory and more [3] [7] A detailed study of this class of frames was initiated by Strohmer and Heath [7] and Holmes and Paulsen [4] Holmes and Paulsen [4] show that equiangular tight frames give error correction codes that are robust against two erasures Bodmann and Paulsen [4] analyze arbitrary numbers of erasures for equiangular tight frames Recently Bodmann Casazza Edidin and Balan [3] show that equiangular tight frames are useful for signal reconstruction when all phase information is lost Recently Sustik Tropp Dhillon and Heath [8] made an important advance on this subject (and on the complex version) which we will discuss in detail later Other applications include the construction of capacity achieving signature sequences for multiuser communication systems in wireless communication theory [0] The tightness condition allows equiangular tight frames to achiece the capacity of a Gaussian channel and their equiangularity allows them to satisfy an interference invariance property Equiangular tight frames potentially have many more practical and theoretical applications Unfortunately we know very few of them and so their usefullness is largely untapped In this note we will look at some recent results in the theory of equiangular tight frames and use this to classify all the equiangular tight frames for Hilbert spaces of dimension less than or equal to 50 For notation throughout this paper we will be working with finite dimensional Hilbert spaces of dimension and our frames will have M elements II EQUIAGULAR LIES We cannot construct equiangular tight frames until we have the requisite number of equiangular lines That is we need a set of M lines passing through the origin in R which are equiangular in the sense that if we choose a set of unit length vectors {f m } M m= one on each line then for m n M f m f n is a constant These inner products represent the cosine of the acute angle between the lines The problem of constructing any number (especially the maximal number) of equiangular lines in R is one of the most elementary and at the same time one of the most difficult problems in mathematics After sixty years of research the maximal number of equiangular lines in R is known only for 35 dimensions For a slightly more general view of this topic see Benedetto and Kolesar [] This line of research was started in 948 by Hanntjes [] in the setting of elliptic geometry where he identified the maximal number of equiangular lines in R for n = 3 Later Van Lint and

Seidel [6] classified the largest number of equiangular lines in R for dimensions 7 and at the same time emphasized the relations to discrete mathematics In 973 Lemmens and Seidel [5] made a comprehensive study of real equiangular line sets which is still today a fundamental piece of work Gerzon [5] gave an upper bound for the maximal number of equiangular lines in R : Theorem (Gerzon): If we have M equiangular lines in R then ( + ) M We will see that in most cases there are many fewer lines than this bound gives Also P eumann [5] produced a fundamental result in the area: Theorem (P eumann): If R has M equiangular lines at angle /α and M > then α is an odd integer Finally there is a lower bound on the angle formed by equiangular line sets Theorem 3: If {f m } M m= is a set of norm one vectors in R then max f m f n m n M (M ) () Moreover we have equality if and only if {f m } M m= is an equiangular tight frame and in this case the tight frame bound is M This inequality goes back to Welch [0] Strohmer and Heath [7] and Holmes and Paulsen [4] give more direct arguments which also yields the moreover part For some reason in the literature there is a further assumption added to the moreover part of Theorem 3 that the vectors span R This assumption is not necessary That is equality in inequality already implies that the vectors span the space [7] The status of the equiangular line problem at this point is summarized in the following chart [5] [7] [9] where is the dimension of the Hilbert space M is the maximal number of equiangular lines and these will occur at the angle /α Table I: Maximal equiangular line sets = 3 4 5 6 7 3 4 M = 3 6 6 0 6 8 8 8 30 5 3 3 3 3 3 5 = 5 6 7 8 9 0 M = 36 40 48 48 7 76 90 96 5 5 5 5 5 5 = 3 4 4 43 M = 6 76 76 76 76 344 5 5 5 5 5 7 The in the chart represents two cases which have been reported as solved in the literature but actually are still open III EQUIAGULAR TIGHT FRAMES: THE FUDAMETALS aimark (See [6] []) showed that Parseval frames come from a unique process Theorem 3: The family {f m } M m= is a Parseval frame for R if and only if there is an orthogonal projection P on R M satisfying P e m = f m for all m = M where {e m } M m= is an orthonormal basis for R If {P e m } M m= is an equal norm equiangular frame for R then for all n m M P e m P e n = (I P )e m (I P )e n and {(I P )e m } M m= is an equal norm Parseval frame Also since any tight frame can be renormalized to be a Parseval frame we have Corollary 3: If {f m } M m= is an equiangular tight frame for R with P e m = M f M m then { M (I P )e m} M m= is an equiangular tight frame for R (M ) We call this the complementary equiangular tight frame It follows that equiangular tight frames come in pairs and if M > then M < (M ) So we only need to classify the equiangular tight frames for M > Certain classes of equiangular tight frames always exist Proposition 33: R always has an M = -element equiangular tight frame at angle 0 (ie An orthonormal basis) M = ( + )-element equiangular tight frame at angle / (ie The complementary equiangular tight frame to the M-element equiangular tight frame for R given by g m = for all m = M) So from now on we will always assume that M > + Corollary 34: To have an equiangular tight frame with M elements in R it is necessary that { ( + ) M min or equivalently + + 8 + (M )(M + ) M ( + ) } Proof: The right hand inequality follows from Theorem applied to the frame and to its complementary frame For the left hand inequality we start with M This is equivalent to Simplifying we obtain (M )(M + ) M M M + M + 0 M ( + )M + ( ) =: f(m) ()

Equality occurs in Equation when M = + ± ( + ) 4( ) = + ± 4 + 4 + 4 + 4 = + ± 8 + (3) (4) (5) = + ± + 4 (6) We can discard the negative part of ± in our formula since it would yield that M + + 4 < which contradicts our general assumption that M > + Since the function f(m) is increasing we conclude that our inequality is equivalent to M + + 8 + Proposition 35: If there is an M element equiangular tight frame for R at angle /α then α α Moreover = α if and only if M = + and = α if and only if M = (+) Proof: Recall that Let α = f(x) = (M ) M (x ) x Then f is decreasing and ( ) ( + ) f( + ) = and f = + Since M + we have α = f(m) + IV SOME RECET RESULTS A recent important generalization of Theorem is due to Sustik Tropp Dhillon and Heath [8]: Theorem 4: Let < < M When M a necessary condition for the existence of a real equiangular tight frame is that both of the quantities (M ) (M )(M ) M β = are odd integers When M = it is necessary that be an odd integer and that (M ) equal the sum of two squares ote that the angle /β is the angle for the complementary frame In [8] the authors conjecture that the conditions in Theorem 4 are also sufficient to have an equiangular tight frame It is easy to give a counter-example to this conjecture Example 4: If M = 64 in R 56 then (M ) M = 56 63 64 56 = 7 63 = Also (M )(M ) 8 63 β = = = 3 56 But (M )(M + ) (64 56)(64 56 + ) = = 36 and this is not greater than M = 56 so this class cannot exist by Corollary 34 The following recent result comes from [7]: Theorem 43: If R has an equiangular tight frame with M elements at angle /α then M is an even integer and α divides M Combining Theorems 4 and 43 we see that it is possible to simplify the necessary conditions in Theorem 4 Corollary 44: If we have an equiangular tight frame for R with M elements (M then the following are equivalent: () Both α and β are odd integers () M is even and α is an odd integer which divides M Moreover M = if and only if β Proof: The equivalence of () and () follows immediately from Theorems 4 43 and the observation: (M ) (M )(M ) αβ = = M M For the moreover part we know that α = (M ) M and β = (M )(M ) So α = β if and only if = (M ) ie If and only if M = Example 45: The assumption α divides M is necessary in Corollary 44 Proof: For example if = 7 and M = 40 then M is even (M ) 7 39 M = = 9 3 is odd However (M )(M ) 3 39 β = = = 3 7 3 is not an integer ote also that this satisfies the conditions of our Corollary 34 since ( + ) (M )(M + ) M = 40 min{ } 7 8 3 4 = min{ } = min{378 9} = 9

Example 46: The assumption that M is even is necessary in Corollary 44 Proof: If = 7 and M = 3 7 then both α and β are integers but β is even That is Also β = (M ) M (M )(M ) = 7 50 5 7 = 5 = 50 34 7 = 00 The following result first appeared in [6] with a different proof Proposition 47: If we have M equiangular lines spanning R at angle /α with α > then M α α with equality if and only if the corresponding unit vectors form an equiangular tight frame Proof: By Theorem 3 Hence So and α α Hence since α > 0 M (M ) (M ) M α M α M M(α ) (α ) M α α By Theorem 3 we have equality if and only the corresponding unit vectors give an equiangular tight frame Proposition 48: Let α be an integer Then M = (α ) α if and only if (M ) M = α Proof: We have if and only if if and only if if and only if (M ) M = α M = α M α (α ) = M(α M = (α ) α V CLASSIFYIG EQUIAGULAR TIGHT FRAMES Over the years many authors have made conjectures which were supposed to classify equiangular tight frames All of these were doomed to failure because we cannot construct equiangular tight frames unless we first have the requisite number of equiangular lines The following theorem which summarizes the results presented in this paper is probably the best that can be said about the classification question since it assumes the required number of equiangular lines exist Theorem 5: The following are equivalent: (I) The space R has an equiangular tight frame with M elements at angle /α (II) We have M = (α ) α and there exist M equiangular lines in R at angle /α Moreover in this case we have: () α α () = α if and only if M = + (3) = α if and only if M = (+) (4) M = if and only if α = = a + b If M + then: ab integers (5) α is an odd integer (6) M is even (7) α divides M- (8) β = M α is the angle for the complementary equiangular tight frame Corollary 5: There exists only one equiangular tight frame for R at a fixed angle /α Hence no subset of an equiangular tight frame for R can form an equiangular tight frame for R We have to be careful in interpreting Corollary 5 It just says that any subset of an equiangular tight frame cannot be an equiangular tight frame for the same space For example Tremain [9] has constructed a 8 element equiangular tight frame for R 7 which contains a 6 element subset which is an equiangular tight frame for R 6 Theorem 5 has the advantage that as we discover the maximal number of equiangular lines at an angle /α we can quickly discover the equiangular tight frames which can be derived from this The following table shows how we apply Theorem 5 for the angle /5 In this table we compute for 5 M = (α ) α = 4 5 for 5 5 = 3 We then have an equiangular tight frame for R if and only if M is an even integer and there exist M equiangular lines in R at angle /5 The two question marks again appear because we are not sure yet if the requisite number of equiangular lines exist

Table II: Equiangular tight frames at angle /5 M = 4 5 5 6 4 6 6 9 O 8 7 3 O 4 8 8 7 O 7 9 O 0 6 4 4 O 4 3 O 3 6 4 4 4 O M = 4 5 5 36 4 6 6 9 O 7 5 O 4 8 8 7 O 9 76? 0 96? 6 76 3 76 ow we can list all the equiangular tight frames for dimensions less than or equal to 50 The question marks indicate that we do not know the existence yet of the requisite number of equiangular lines In [8] they give a listing of the equiangular tight frames with fewer than 00 elements Table III: All equiangular tight frames for dimensions 50 M /α 3 6 / 5 5 0 /3 6 6 /3 7 4 / 3 7 8 /3 9 8 / 7 0 6 /5 3 6 /5 5 30 / 9 5 36 /5 9 38 / 37 9 76 /5? 0 96 /5? 8 /9 36 /7 4 / 4 6 /5 76 /5 3 46 / 45 3 76 /5 Proof for Table III: M /α 5 50 /7 7 54 / 53 8 64 /7 3 6 / 6 33 66 / 65 35 0 /7? 36 64 /9 37 74 / 73 37 48 /7? 4 8 /9 4 46 /7 4 88 /7? 43 86 / 85 43 344 /7 45 90 / 89 45 00 /9 45 540 /7? 46 736 /7? 49 98 / 97 ( M α ) represents an M-element equiangular tight frame existing in R at angle /α () (3 6 5 (5 0 /3 (6 6 /3 (7 8 /3 (5 36 /5 ( 6 /5 ( 76 /5 (3 76 /5 (4 46 /7 (43 344 /7) In these cases ( M /α) exist by Table I () (7 4 3 (9 8 7 (3 6 /5 (5 30 9 (9 38 37 ( 4 4 (3 46 45 (5 50 /7 (7 54 53 (8 64 /7 (3 6 6 (33 66 65 (4 8 /9 (43 86 85 (45 90 89 (45 00 /9 (49 98 97 ) These ( M α ) exist by know examples from strongly regular graphs (See Sustik Tropp Dhillon Heath [8]) (3) (0 6 /5 ( 8 /9 ( 36 /7 (36 64 /9) Also there is a strongly regular graph (73 36 7 8) which is know as 73-Paley which shows the existence of 37 74 73 ) These ( M β ) are the complementary equiangular tight frames for the frames given in () (4) = 4 8 4 6 8 4 6 9 30 3 34 38 39 40 44 48 These dimensions have no (M ) M an odd integer The only other case is M = and these dimensions fail to have equiangular tight frames with vectors by [8] since is not the sum of two squares (5) The only other cases where α is an odd integer are ( M) where: (a) (7 5) (b) (5 8) (c) (7 40) (d) (33 45 (33 55 (33 99) (e) (45 50 (45 56) (f) (49 5 (49 55) (g) (50 64) (a (d (f) fail since M is not even (b (c (e (g) fail since they would imply that (3 8 (3 40 (5 50 ( 56) and (4 64) exist as complementary equiangular tight frames But these exceed the maximal number of equiangular lines in their dimensions (See Table I) VI SOME OPE PROBLEMS In this section we make a few conjectures concerning equiangular line sets We have seen that if {f m } M m= is an equiangular tight frame for R at angle /α then α α The corresponding result for equiangular lines should be: Conjecture 6: If M is the maximal number of equiangular lines in R at angle /α and if these lines span R then α α Conjecture 6 would resolve several issues surrounding Table I We know that the maximum number of equiangular lines in

R for 7 3 is 8 and these lines occur at angle /3 But we do not know if these lines can span R for 7 A positive solution to Conjecture 6 would show that these lines must span a 7-dimensional subspace of R A similar result would hold for the 76 equiangular lines in R for 3 4 at angle /5 The case of R 4 is also quite tangled It is known [5] [7] [9] that R 4 has a maximum number of equiangular lines at angle /3 of 8 Again we do not believe that these lines can span R 4 However there are [9] 6 equiangular lines in R 4 at angle /3 which span R 4 The maximal number of equiangular lines in R 4 is not known It can be shown [9] that R 4 also has 8 equiangular lines at angle /5 and these lines span R 4 It can also be shown [7] that the maximal number of equiangular lines in R 4 is 30 and that these lines must occur at angle /5 We have seen that if R has an equiangular tight frame with M elements then M must be even One can ask the same question for maximal equiangular line sets Conjecture 6: The maximal number of equiangular lines in R must be even We know that an upper bound for the maximal number of equiangular lines in R is M = ( + ) and that this number can occur only when = α This raises the problem: Problem 63: For what values of = α with α an odd integer does there exist M = ( + ) equiangular lines in R This problem has been verified for 3 5 Recently Bannai Munemasa and Venkov [] have shown that this conjecture fails for 7 The authors also show that this conjecture fails for infinitely many values of α These examples provide further counter-examples to the conjecture that the conditions in Theorem 4 are sufficient for the existence of equiangular tight frames That is for the extremal cases = α with α an odd integer and M = (+) we have that Also M = (α ) (M ) M = (α )(α + ) So M is even since both α and α+ are even By Corollary 44 we have that both α and β are odd integers It can be shown [7] that an affirmative answer to Conjecture 6 and identifying the class of integers which satisfy Problem 63 would go a long way towards classifying all the maximal number of equiangular line sets But this will have to wait for further progress VII COCLUSIO Today Frame theory has broad applications in mathematics and engineering Equiangular tight frames are especially useful because of their exact structure The problem is that we do not have very many examples of equiangular tight frames We have presented here the state of the art for equiangular tight frames This includes a number of restrictions on their existence which helps explain why there are so few of them We have also seen that the equiangular tight frame problem is heavily dependent on the real equiangular line problem which in itself is one of the deepest and most difficult problems in mathematics and which has yielded a solution for only 35 dimensions in 60 years of research We have seen that it is possible to classify the equiangular tight frames for dimensions less than or equal to 50 Further developments here require further developments on the real equiangular line problem ACKOWLEDGMET P G Casazza and J C Tremain were supported by SF DMS 07046 REFERECES [] E Bannai A Munemasa and B Venkov The nonexistence of certain tight spherical designs St Petersburg Math Jour 6 o 4 (005) 609-65 [] J J Benedetto and J Kolesar Geometric properties of Grassmannian frames for R and R 3 EURASIP J Applied Signal Processing (006 7 pages [3] B Bodman P Casazza D Edidin and R Balan Frames for linear reconstruction without phase Preprint [4] B Bodmann and V Paulsen Frames graphs and erasures Linear Algebra and Applications 404 (005) 8-46 [5] P G Casazza The Art of Frame Theory Taiwanese Jour of Math 4 o (000) 9-0 [6] P G Casazza and J Kovačević Equal norm tight frames with erasures Advances in Compt Math 8 (003) 387-430 [7] P G Casazza D Redmond and J C Tremain Real Equiangular Tight Frames in preparation [8] O Christensen An introduction to frames and Riesz bases Birkhäuser Boston 003 [9] I Daubechies The wavelet transform time-frequency localization and signal analysis IEEE Trans Inform Theory 39 (990) 96-005 [0] K H Gröchenig Foundations of time-frequency analysis Birkhäuser Boston 000 [] J Haantjes Equilateral point-sets in elliptic two- and three-dimensional spaces ieuw Arch Wisk (948) 355-36 [] D Han and D R Larson Frames bases and group representations Birkhauser Boston 000 [3] RW Heath T Strohmer and A J Paulraj On quasi-orthogonal signatures for CDMA IEEE Trans Information Theory 5 o 3 (006) 7-6 [4] R B Holmes and V I Paulsen Optimal frames for erasures Linear Alg and Applications 377 (04) 3-5 [5] P Lemmens and J Seidel Equiangular lines Journal of Algebra 4 (973) 494-5 [6] J H van Lint and J J Seidel Equiangular point sets in elliptic geometry Proc ederl Akad Wetensch Series A 69 (966) 335-348 [7] T Strohmer and R W Heath Grassmannian frames with applications to coding and communication Appl Comp Harmonic Anal 4 o 3 (003) 57-75 [8] M A Sustik J A Tropp I S Dhillon and R W Heath Jr On the existence of equiangular tight frames Linear Alg and Applications 46 o -3 (007) 69-635 [9] JC Tremain Concrete Constructions of Equiangular Line Sets in preparation [0] L R Welch Lower bounds on the maximum cross-correlation of signals IEEE Trans Inform Theory 0 (974) 397-399