Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices from equiangular tight frames Lenny Fukshansky Claremont McKenna College (joint work with Albrecht B¨ ottcher, Stephan Garcia, Hiren Maharaj, Deanna Needell) Computational Challenges in the Theory of Lattices ICERM Brown April 23 - 27, 2018
Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices: basic notions A lattice Λ ⊂ R k of rank m , 1 ≤ m ≤ k , is a free Z -module of rank m , which is the same as a discrete co-compact subgroup of V := span R Λ. If m = k , i.e. V = R k , we say that Λ is a lattice of full rank in R k . Hence Λ = span Z { a 1 , . . . , a m } = A Z m , where a 1 , . . . , a m ∈ R k are R -linearly independent basis vectors for Λ and A = ( a 1 . . . a m ) is the corresponding k × m basis matrix.
Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices: basic notions A lattice Λ ⊂ R k of rank m , 1 ≤ m ≤ k , is a free Z -module of rank m , which is the same as a discrete co-compact subgroup of V := span R Λ. If m = k , i.e. V = R k , we say that Λ is a lattice of full rank in R k . Hence Λ = span Z { a 1 , . . . , a m } = A Z m , where a 1 , . . . , a m ∈ R k are R -linearly independent basis vectors for Λ and A = ( a 1 . . . a m ) is the corresponding k × m basis matrix. The determinant of Λ is � det Λ := det( A ⊤ A ) , which is equal to the volume (quotient Lebesgue measure) of V / Λ.
Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices: minimal vectors Minimal norm of a lattice Λ is | Λ | = min {� x � : x ∈ Λ \ { 0 }} , where � � is Euclidean norm. The set of minimal vectors of Λ is S (Λ) = { x ∈ Λ : � x � = | Λ |} .
Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices: minimal vectors Minimal norm of a lattice Λ is | Λ | = min {� x � : x ∈ Λ \ { 0 }} , where � � is Euclidean norm. The set of minimal vectors of Λ is S (Λ) = { x ∈ Λ : � x � = | Λ |} . • A lattice Λ is well-rounded (WR) if span R Λ = span R S (Λ).
Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices: minimal vectors Minimal norm of a lattice Λ is | Λ | = min {� x � : x ∈ Λ \ { 0 }} , where � � is Euclidean norm. The set of minimal vectors of Λ is S (Λ) = { x ∈ Λ : � x � = | Λ |} . • A lattice Λ is well-rounded (WR) if span R Λ = span R S (Λ). • If rk Λ > 4, a strictly stronger condition is that Λ is generated by minimal vectors , i.e. Λ = span Z S (Λ).
Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices: minimal vectors Minimal norm of a lattice Λ is | Λ | = min {� x � : x ∈ Λ \ { 0 }} , where � � is Euclidean norm. The set of minimal vectors of Λ is S (Λ) = { x ∈ Λ : � x � = | Λ |} . • A lattice Λ is well-rounded (WR) if span R Λ = span R S (Λ). • If rk Λ > 4, a strictly stronger condition is that Λ is generated by minimal vectors , i.e. Λ = span Z S (Λ). • It has been shown by Conway & Sloane (1995) and Martinet & Sch¨ urmann (2011) that there are lattices of rank ≥ 10 generated by minimal vectors which do not contain a basis of minimal vectors .
Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices: eutaxy and perfection Let m = rk Λ and S (Λ) = { x 1 , . . . , x n } be the set of minimal vectors of the lattice Λ.
Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices: eutaxy and perfection Let m = rk Λ and S (Λ) = { x 1 , . . . , x n } be the set of minimal vectors of the lattice Λ. This lattice is called eutactic if there exist positive real numbers c 1 , . . . , c n such that n � v � 2 = � c i ( v , x i ) 2 i =1 for every vector v ∈ span R Λ, where ( · , · ) is the usual inner product. If c 1 = · · · = c n , we say that Λ is strongly eutactic .
Lattice notation Equiangular frame lattices Other tight frames Conclusion Lattices: eutaxy and perfection Let m = rk Λ and S (Λ) = { x 1 , . . . , x n } be the set of minimal vectors of the lattice Λ. This lattice is called eutactic if there exist positive real numbers c 1 , . . . , c n such that n � v � 2 = � c i ( v , x i ) 2 i =1 for every vector v ∈ span R Λ, where ( · , · ) is the usual inner product. If c 1 = · · · = c n , we say that Λ is strongly eutactic . This lattice is called perfect if the set of symmetric matrices { x i x ⊤ i : x i ∈ S (Λ) } spans the space of m × m symmetric matrices.
Lattice notation Equiangular frame lattices Other tight frames Conclusion Packing density The packing density of a lattice Λ of rank m is defined as δ (Λ) = ω m | Λ | m 2 m det Λ , where ω m is the volume of a unit ball in span R Λ.
Lattice notation Equiangular frame lattices Other tight frames Conclusion Packing density The packing density of a lattice Λ of rank m is defined as δ (Λ) = ω m | Λ | m 2 m det Λ , where ω m is the volume of a unit ball in span R Λ. Space of full-rank lattices in R k is identified with GL k ( R ) / GL k ( Z ), and δ is a continuous function on this space.
Lattice notation Equiangular frame lattices Other tight frames Conclusion Packing density The packing density of a lattice Λ of rank m is defined as δ (Λ) = ω m | Λ | m 2 m det Λ , where ω m is the volume of a unit ball in span R Λ. Space of full-rank lattices in R k is identified with GL k ( R ) / GL k ( Z ), and δ is a continuous function on this space. A lattice is called extremal if it is a local maximum of the packing density function in its dimension.
Lattice notation Equiangular frame lattices Other tight frames Conclusion Packing density The packing density of a lattice Λ of rank m is defined as δ (Λ) = ω m | Λ | m 2 m det Λ , where ω m is the volume of a unit ball in span R Λ. Space of full-rank lattices in R k is identified with GL k ( R ) / GL k ( Z ), and δ is a continuous function on this space. A lattice is called extremal if it is a local maximum of the packing density function in its dimension. Theorem 1 (G. Voronoi, 1908) A lattice is extremal if and only if it is perfect and eutactic.
Lattice notation Equiangular frame lattices Other tight frames Conclusion What is a frame? A spanning set { f 1 , . . . , f n } ⊂ R k , n ≥ k , is called a frame if there exist constants γ 1 , γ 2 ∈ R such that for every x ∈ R k , n γ 1 � x � 2 ≤ ( x , f j ) 2 ≤ γ 2 � x � 2 , � j =1 where ( , ) stand for the usual dot-product.
Lattice notation Equiangular frame lattices Other tight frames Conclusion What is a frame? A spanning set { f 1 , . . . , f n } ⊂ R k , n ≥ k , is called a frame if there exist constants γ 1 , γ 2 ∈ R such that for every x ∈ R k , n γ 1 � x � 2 ≤ ( x , f j ) 2 ≤ γ 2 � x � 2 , � j =1 where ( , ) stand for the usual dot-product. • A frame is called unit if � f j � = 1 for every 1 ≤ j ≤ n .
Lattice notation Equiangular frame lattices Other tight frames Conclusion What is a frame? A spanning set { f 1 , . . . , f n } ⊂ R k , n ≥ k , is called a frame if there exist constants γ 1 , γ 2 ∈ R such that for every x ∈ R k , n γ 1 � x � 2 ≤ ( x , f j ) 2 ≤ γ 2 � x � 2 , � j =1 where ( , ) stand for the usual dot-product. • A frame is called unit if � f j � = 1 for every 1 ≤ j ≤ n . • A frame is called tight if γ 1 = γ 2 .
Lattice notation Equiangular frame lattices Other tight frames Conclusion What is a frame? A spanning set { f 1 , . . . , f n } ⊂ R k , n ≥ k , is called a frame if there exist constants γ 1 , γ 2 ∈ R such that for every x ∈ R k , n γ 1 � x � 2 ≤ ( x , f j ) 2 ≤ γ 2 � x � 2 , � j =1 where ( , ) stand for the usual dot-product. • A frame is called unit if � f j � = 1 for every 1 ≤ j ≤ n . • A frame is called tight if γ 1 = γ 2 . • A frame is called equiangular if | ( f i , f j ) | = c for all 1 ≤ i � = j ≤ n , for some constant c ∈ [0 , 1],
Lattice notation Equiangular frame lattices Other tight frames Conclusion Equiangular tight frames (ETFs) In this talk we will be especially concerned with unit equiangular tight frames, abbreviated ETFs.
Lattice notation Equiangular frame lattices Other tight frames Conclusion Equiangular tight frames (ETFs) In this talk we will be especially concerned with unit equiangular tight frames, abbreviated ETFs. ETFs generalize the notion of an orthonormal basis, while redun- dancy of an overdetermined spanning set allows for better recovery of information in case of errors: we can think of “coordinates” with respect to such an overdetermined set as extra “frequencies” that can help recover information in case of erasures in transmission.
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