Finite Frames and Optimal Subspace Packings Matthew Fickus Department of Mathematics and Statistics Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio September 20, 2019 The views expressed in this talk are those of the speaker and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.
Party like it’s 1999...
A Research Problem In the fall of 2000, inspired by a talk by Ed Saff at a conference in Bommerholz and a follow-up question by Hans Feichtinger, John asked me the following question (paraphrased): How is the problem of equally-distributing points on a sphere related to finite unit norm tight frames? This talk is the 2019 progress update. 1/21
Optimal Packings on Spheres
Spherical equidistribution: Thomson vs. Tammes Over all sets of N unit vectors { x n } N n =1 in R D , we can try to: N N 1 � � ◮ minimize � x n − x n ′ � ( Thomson, 1904 ) n =1 n ′ =1 n ′ � = n ◮ maximize min n � = n ′ � x n − x n ′ � ( Tammes, 1930 ) For example, when N = 5, D = 3: 2/21
Solving Tammes in the Simplest Case Theorem: [Rankin 55] When N ≤ D + 1, every solution to Tammes problem is a N -vector regular simplex. Proof: For any unit vectors { x n } N n =1 in R D , � x n − x n ′ � 2 = 2(1 − � x n , x n ′ � ) . Thus, argmax n � = n ′ � x n − x n ′ � = argmin min max n � = n ′ � x n , x n ′ � . Also, { x n } { x n } N 2 N N � � � � � � � 0 ≤ x n = � x n , x n ′ � ≤ N + N ( N − 1) max n � = n ′ � x n , x n ′ � . � � � � n =1 n =1 n ′ =1 Equality only holds ⇔ � N n =1 x n = 0 and � x n , x n ′ � is constant over all n � = n ′ . 3/21
Solving Tammes in the Next Simplest Case Theorem: [Rankin 55] max n � = n ′ � x n , x n ′ � ≥ 0 when N ≥ D + 2. Moreover, for N ≤ 2 D , this bound can be achieved. Example: D = 3, N = 2 , 3 , 4 , 5 , 6: 4/21
Finite Unit-Norm Tight Frames
Notation Let F be either R or C . We usually regard N vectors { ϕ n } N n =1 in F D as the columns of a D × N matrix � � Φ = ϕ 1 . . . ϕ N . Multiplying Φ by its N × D conjugate-transpose Φ ∗ gives its � ϕ 1 , ϕ 1 � · · · � ϕ 1 , ϕ N � . . ... ◮ N × N Gram matrix Φ ∗ Φ = . . . . � ϕ N , ϕ 1 � · · · � ϕ N , ϕ N � N ◮ D × D frame operator ΦΦ ∗ = � ϕ n ϕ ∗ n n =1 In this talk, every ϕ n is unit-norm, meaning the diagonal of Φ ∗ Φ is all ones and ΦΦ ∗ is a sum of rank-one projections. 5/21
Orthonormal Bases (ONBs) n =1 is an ONB for F N then Φ is square and Fact: If { ϕ n } N satisfies Φ ∗ Φ = I . Thus, Φ ∗ = Φ − 1 and so we also have N ΦΦ ∗ = I , x = ΦΦ ∗ x = � � ϕ n , x � ϕ n , ∀ x ∈ F N . i.e., n =1 1 1 1 1 1 1 1 1 ω ω 2 ω 3 ω 4 ω 5 ω 6 1 ω 2 ω 4 ω 6 ω ω 3 ω 5 1 1 ω 3 ω 6 ω 2 ω 5 ω 1 ω 4 ω = exp( 2 π i Example: Φ = 7 ). √ , 7 1 ω 4 ω ω 5 ω 2 ω 6 ω 3 1 ω 5 ω 3 ω ω 6 ω 4 ω 2 1 ω 6 ω 5 ω 4 ω 3 ω 2 ω 6/21
Finite Unit-Norm Tight Frames (FUNTFs) n =1 in F D form a FUNTF for Definition: Unit vectors { ϕ n } N F D if there exists C > 0 such that N ΦΦ ∗ = C I , � C x = ΦΦ ∗ x = � ϕ n , x � ϕ n , ∀ x ∈ F N . i.e., n =1 D since CD = Tr( ΦΦ ∗ ) = Tr( Φ ∗ Φ ) = N . Here, C = N Example: Scaling the any three rows of the previous matrix gives a complex FUNTF(3 , 7). For example, for rows { 1 , 2 , 4 } , 1 ω ω 2 ω 3 ω 4 ω 5 ω 6 Φ = 1 1 ω 2 ω 4 ω 6 ω ω 3 ω 5 ω = exp( 2 π i √ , 7 ) . 3 1 ω 4 ω ω 5 ω 2 ω 6 ω 3 7/21
Some real FUNTFs for R 3 with N = 3 , 4 , 5 , 6 8/21
Relating FUNTFs to the Tammes Problem
A Big Idea from Conway, Hardin, Sloane 96 A unit vector ϕ lifts to a rank-one projection ϕϕ ∗ . The set { ϕϕ ∗ : ϕ ∈ F D , � ϕ � = 1 } is a projective space and lies in the real space of all D × D self-adjoint operators, which is a Hilbert space under the Frobenius inner product � A , B � Fro := Tr( A ∗ B ). Moreover, for unit vectors { ϕ n } N n =1 and any n , n ′ , � ϕ n ϕ ∗ n , ϕ n ′ ϕ ∗ n ′ � Fro = Tr( ϕ n ϕ ∗ n ϕ n ′ ϕ ∗ n ′ ) = |� ϕ n , ϕ n ′ �| 2 , and so the squared-distance between two such projections is: � ϕ n ϕ ∗ n − ϕ n ′ ϕ ∗ n ′ � 2 Fro = Tr[( ϕ n ϕ ∗ n − ϕ n ′ ϕ ∗ n ′ ) 2 ] = 2(1 −|� ϕ n , ϕ n ′ �| 2 ) . 9/21
Applying a Trivial Bound in Projective Space Theorem: [Rankin 56] For any unit vectors { ϕ n } N n =1 in F D , N N N 2 � � |� ϕ n , ϕ n ′ �| 2 D ≤ n =1 n ′ =1 where equality holds if and only if { ϕ n } N n =1 is a FUNTF for F D . Proof: N 2 � � = Tr[( ΦΦ ∗ − N � � ( ϕ n ϕ ∗ n − 1 � D I ) 2 ] 0 ≤ D I ) � � � � Fro n =1 N N |� ϕ n , ϕ n ′ �| 2 − N 2 � � = D . n =1 n ′ =1 10/21
FUNTF Characterization and Construction Theorem: [Benedetto, F 03] When N ≥ D , every local minimizer of the frame potential N N � � |� ϕ n , ϕ n ′ �| 2 n =1 n ′ =1 is a FUNTF (and so is necessarily a global minimizer). Theorem: [Cahill, F, Mixon, Poteet, Strawn 13] Every FUNTF can be explicitly constructed from eigensteps . 11/21
Born Again
Equiangular Tight Frames (ETFs) Theorem: [Strohmer, Heath 03] n =1 in F D satisfy the Welch bound : Any unit vectors { ϕ n } N � N − D max n � = n ′ |� ϕ n , ϕ n ′ �| ≥ D ( N − 1) , and achieve equality ⇔ { ϕ n } N n =1 is an ETF for F D , namely a FUNTF where |� ϕ n , ϕ n ′ �| is constant over all n � = n ′ . Proof: Apply Rankin’s simplex bound to { ϕ n ϕ ∗ n − 1 D I } N n =1 : N N |� ϕ n , ϕ n ′ �| 2 ≤ N + N ( N − 1) max N 2 � � n � = n ′ |� ϕ n , ϕ n ′ �| 2 . D ≤ n =1 n ′ =1 See also: Rankin 56; Welch 74; Conway, Hardin, Sloane 96]. 12/21
Example: A 6-vector ETF for R 3 0 1 1 1 1 1 1 0 1 − 1 − 1 1 2 0 0 Φ ∗ Φ = I + 1 1 1 0 1 − 1 − 1 ΦΦ ∗ = √ 0 2 0 , 1 − 1 1 0 1 − 1 5 0 0 2 1 − 1 − 1 1 0 1 1 1 − 1 − 1 1 0 13/21
Some Remarks on the (Rankin-)Welch Bound ◮ Following [Rankin 55], Rankin studied packing antipodal pairs of points of spheres and discovered the Welch bound about two decades before Welch [Rankin 56]. ◮ The Welch bound is equivalent to Fro ≤ 2 N ( D − 1) n � = n ′ � ϕ n ϕ ∗ n − ϕ n ′ ϕ ∗ n ′ � 2 max (1) D ( N − 1) . In particular, if an ETF( D , N ) exists, then every optimal packing of N lines in F D is necessarily tight. ◮ [Conway, Hardin, Sloane 96] calls (1) the simplex bound n − 1 since it’s achieved ⇔ { ϕ n ϕ ∗ D I } N n =1 is a simplex. They also consider subspaces of dimension > 1. 14/21
More Remarks on the (Rankin-)Welch Bound ◮ (Gerzon) If { ϕ n ϕ ∗ n − 1 D I } N n =1 is a simplex, then N ≤ D 2 when F = C . N ≤ D ( D +1) when F = R , 2 ◮ For larger N , applying Rankin’s other bound n − 1 to { ϕ n ϕ ∗ D I } N n =1 gives the orthoplex bound : 1 max n � = n ′ |� ϕ n , ϕ n ′ �| ≥ √ D . ◮ An ETF with N = D 2 is a SIC-POVM . Zauner has conjectured that these exist for all D [Zauner 99]. ◮ ETFs arise in algebraic coding theory [Grey 62], quantum information theory [Zauner 99], wireless communication [Strohmer, Heath 03], and compressed sensing. 15/21
Equiangular Tight Frames
Harmonic ETFs: Difference Sets Definition: Extracting rows from the character table of a finite abelian group G yields a harmonic frame . Example: G = Z 7 , D = { 1 , 2 , 4 } , 1 ω ω 2 ω 3 ω 4 ω 5 ω 6 Φ = 1 1 ω 2 ω 4 ω 6 ω ω 3 ω 5 ω = exp( 2 π i √ , 3 ) . 3 1 ω 4 ω ω 5 ω 2 ω 6 ω 3 Theorem: [Turyn 65] The harmonic ETF arising from D ⊆ G is an ETF for C D ⇔ D is a difference set for G . Idea: 1 ω 6 n ω 4 n ω n n = 1 = 1 ϕ n ϕ ∗ ω 2 n � ω − n ω − 2 n ω − 4 n � 1 ω 5 n ω n . 3 3 ω 3 n ω 2 n ω 4 n 1 16/21
Steiner ETFs Theorem: [Goethals, Seidel 70] Every balanced incomplete block design (BIBD) with Λ = 1 yields an ETF. 1 1 0 0 0 0 1 1 + − + − 1 0 1 0 to form Example: Combine and + + − − 0 1 0 1 + − − + 1 0 0 1 0 1 1 0 + − + − + − + − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + − + − + − Φ = 1 + + − − 0 0 0 0 + + − − 0 0 0 0 √ . 0 0 0 0 + + − − 0 0 0 0 + + − − 3 + − − + 0 0 0 0 0 0 0 0 + − − + 0 0 0 0 + − − + + − − + 0 0 0 0 17/21
Some Recent Progress on ETFs [Jasper, Mixon, F 14] Every McFarland harmonic ETF is a rotated Steiner ETF. New infinite family of optimal codes. [F, Mixon, Jasper 16] New infinite family of complex ETFs arising from finite projective planes containing hyperovals. [F, Jasper, Mixon, Peterson 18]: Tremain’s construction of an ETF(15 , 36) generalizes. New infinite family of real ETFs. 18/21
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