An introduction to finite frames Matthew Fickus Department of - PowerPoint PPT Presentation

An introduction to finite frames Matthew Fickus Department of Mathematics and Statistics Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio July 28, 2015 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.

Outline • Part I: Finite Frames − Motivation − Notation − Terminology • Part II: Unit Norm Tight Frames (UNTFs) − Generalizations of orthonormal bases − Often studied with differential and algebraic geometry − Major open problems • Part III: Equiangular Tight Frames (ETFs) − Optimal packings of lines − Closely related to combinatorial design − Major open problems 1/27

Part I: Finite Frames

A Brief History of Frame Theory 1950s: Frames are introduced to study nonharmonic Fourier series. Infinite-dimensional generalization of standard linear algebra. 1960s-1970s: “Frames” is an obscure term used by harmonic analysts. Time-frequency analysis routinely used in real-world applications. 1980s-1990s: Wavelets (time-scale analysis) invented to address shortcomings of time-frequency analysis. Frame theory used to compare these two competing methods. Frames popularized as “painless nonorthogonal expansions.” 2000s-2010s: Finite frame theory developed to study packing and covering problems in Euclidean geometry. It overlaps with compressed sensing , which is invented to address shortcomings of wavelets. Common theme: In what ways (and to what degree) can nonorthonormal vectors behave like orthonormal vectors? 2/27

Matrix Notation Definition: Let M , N be positive integers and let either F = R or F = C . Given N vectors { ϕ n } N n =1 in F M , consider the � ϕ 1 · · · ϕ N � • M × N synthesis operator Φ = , ϕ ∗   1 . • N × M analysis operator Φ ∗ = .  ,   .  ϕ ∗ N • M × M frame operator ΦΦ ∗ = ϕ 1 ϕ ∗ 1 + · · · + ϕ N ϕ ∗ N , ϕ ∗ 1 ϕ 1 · · · ϕ ∗   1 ϕ N . . ... • N × N Gram matrix Φ ∗ Φ = . .  .   . .  ϕ ∗ N ϕ 1 · · · ϕ ∗ N ϕ N 3/27

Orthonormal Bases Notes: n =1 in F M are orthonormal if and only if Φ ∗ Φ = I . • Vectors { ϕ n } N n =1 are an orthonormal basis for F M if and only if • Vectors { ϕ n } N they’re orthonormal and M = N . • In that case, Φ is square and Φ ∗ = Φ − 1 implying that ∀ x ∈ F M , � N N � x = ΦΦ ∗ x = � ϕ n ϕ ∗ � ( ϕ ∗ x = n x ) ϕ n . n n =1 n =1 • This implies the Pythagorean theorem: ∀ x ∈ F M , � N N � � x � 2 = x ∗ x = x ∗ ΦΦ ∗ x = x ∗ � ϕ n ϕ ∗ � | ϕ ∗ n x | 2 . x = n n =1 n =1 4/27

Finite Frames • Now suppose your real-world application prohibits you from having { ϕ n } N n =1 be an orthonormal basis for F M . • As long as { ϕ n } N n =1 spans F M , you can still “painlessly” expand any x in terms of them: in this case, ΦΦ ∗ is invertible and so � N N � x = ΦΦ ∗ ( ΦΦ ∗ ) − 1 x = ΦΨ ∗ x = � ϕ n ψ ∗ � ( ψ ∗ x = n x ) ϕ n . n n =1 n =1 • This expansion is numerically stable when Φ is well conditioned, i.e. when { ϕ n } N n =1 satisfies a relaxed Pythagorean theorem: N α � x � 2 ≤ x ∗ ΦΦ ∗ x = n x | 2 ≤ β � x � 2 , � | ϕ ∗ ∀ x ∈ F M , n =1 for “close” scalars 0 < α ≤ β < ∞ . Here, we call { ϕ n } N n =1 a frame for F M with lower and upper frame bounds α and β , respectively. 5/27

Tight Frames n =1 is a tight frame for F M if Φ is optimally well • We say { ϕ n } N conditioned, namely when there exists α > 0 such that N α � x � 2 = x ∗ Φ ∗ Φx = � | ϕ ∗ n x | 2 , ∀ x ∈ F M . n =1 • This is equivalent to ΦΦ ∗ = α I , i.e. to when the rows of Φ are orthogonal and have constant norm. • Naimark’s Theorem: Every tight frame is a scalar multiple of an orthogonal projection of an orthonormal basis. 6/27

Example: 6 × 16 Tight Frame   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0   Φ =   0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7/27

Example: 6 × 16 Tight Frame   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0   Φ =   0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7/27

Example: 6 × 16 Tight Frame   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0   Φ =   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0   0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 7/27

Example: 6 × 16 Tight Frame   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0   Φ =   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0   0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 Question: This is one of many tight frames of 16 vectors in R 6 ... can we find others that are even more like orthonormal bases in some sense? 7/27

Example: Better 6 × 16 Tight Frame   + + + + + + + + + + + + + + + + + − + − + − + − + − + − + − + −     + + − − + + − − + + − − + + − −     + − − + + − − + + − − + + − − +     + + + + − − − − + + + + − − − −     + − + − − + − + + − + − − + − +     + + − − − − + + + + − − − − + +     1 + − − + − + + − + − − + − + + −   Φ = √   + + + + + + + + − − − − − − − − 6     + − + − + − + − − + − + − + − +     + + − − + + − − − − + + − − + +     + − − + + − − + − + + − − + + −     + + + + − − − − − − − − + + + +     + − + − − + − + − + − + + − + −     + + − − − − + + − − + + + + − −   + − − + − + + − − + + − + − − + Notation: “ + ” = 1, “ − ” = − 1 8/27

Example: Better 6 × 16 Tight Frame   + + + + + + + + + + + + + + + + + − + − + − + − + − + − + − + −     + + − − + + − − + + − − + + − −     + − − + + − − + + − − + + − − +     + + + + − − − − + + + + − − − −     + − + − − + − + + − + − − + − +     + + − − − − + + + + − − − − + +     1 + − − + − + + − + − − + − + + −   Φ = √   + + + + + + + + − − − − − − − − 6     + − + − + − + − − + − + − + − +     + + − − + + − − − − + + − − + +     + − − + + − − + − + + − − + + −     + + + + − − − − − − − − + + + +     + − + − − + − + − + − + + − + −     + + − − − − + + − − + + + + − −   + − − + − + + − − + + − + − − + 8/27

Example: Better 6 × 16 Tight Frame   + + + + + + + + + + + + + + + + + − + − + − + − + − + − + − + −     1 + + − − + + − − + + − − + + − −   √ Φ =   + − − + + − − + + − − + + − − + 6     + + + + − − − − + + + + − − − −   + − + − − + − + + − + − − + − + 8/27

Example: Better 6 × 16 Tight Frame   + + + + + + + + + + + + + + + + + − + − + − + − + − + − + − + −     1 + + − − + + − − + + − − + + − −   √ Φ =   + − − + + − − + + − − + + − − + 6     + + + + − − − − + + + + − − − −   + − + − − + − + + − + − − + − + Note: All columns are unit norm. 8/27