On the conditioning of subensembles Dustin G. Mixon Jubilee of Fourier Analysis and Applications September 20, 2019 1/27
Conditioning Notions of conditioning for vectors ( f i ) i ∈ I in a Hilbert space H : 1/27
Conditioning Notions of conditioning for vectors ( f i ) i ∈ I in a Hilbert space H : ◮ Linearly independent. Quantitative version: Riesz sequence | a i | 2 ≤ � a i f i � 2 ≤ C � � � | a i | 2 ∀ a ∈ ℓ 2 ( I ) c i ∈ I i ∈ I i ∈ I c = C ⇐ ⇒ equal norm and orthogonal 1/27
Conditioning Notions of conditioning for vectors ( f i ) i ∈ I in a Hilbert space H : ◮ Linearly independent. Quantitative version: Riesz sequence | a i | 2 ≤ � a i f i � 2 ≤ C � � � | a i | 2 ∀ a ∈ ℓ 2 ( I ) c i ∈ I i ∈ I i ∈ I c = C ⇐ ⇒ equal norm and orthogonal ◮ Spanning. Quantitative version: Frame c � x � 2 ≤ |� f i , x �| 2 ≤ C � x � 2 � ∀ x ∈ H i ∈ I c = C ⇐ ⇒ tight 1/27
Conditioning Notions of conditioning for vectors ( f i ) i ∈ I in a Hilbert space H : ◮ Linearly independent. Quantitative version: Riesz sequence | a i | 2 ≤ � a i f i � 2 ≤ C � � � | a i | 2 ∀ a ∈ ℓ 2 ( I ) c i ∈ I i ∈ I i ∈ I c = C ⇐ ⇒ equal norm and orthogonal ◮ Spanning. Quantitative version: Frame c � x � 2 ≤ |� f i , x �| 2 ≤ C � x � 2 � ∀ x ∈ H i ∈ I c = C ⇐ ⇒ tight C Riesz = C Frame “how uniformly the energy of ( f i ) i ∈ I is spread” 1/27
Meta-question Various applications point to the same meta-question: Given an ensemble of vectors, what can you say about the conditioning of subensembles? This talk: Important instances, open problems 2/27
Conditioning of subensembles We will consider three types of instances (cf. combinatorics): 3/27
Conditioning of subensembles We will consider three types of instances (cf. combinatorics): ◮ Ramsey type. Conditions for existence of a well-conditioned subensemble (cf. Ramsey’s theorem) 3/27
Conditioning of subensembles We will consider three types of instances (cf. combinatorics): ◮ Ramsey type. Conditions for existence of a well-conditioned subensemble (cf. Ramsey’s theorem) ◮ Symmetric type. Subensembles of symmetric ensembles (cf. clique number of Paley graph) 3/27
Conditioning of subensembles We will consider three types of instances (cf. combinatorics): ◮ Ramsey type. Conditions for existence of a well-conditioned subensemble (cf. Ramsey’s theorem) ◮ Symmetric type. Subensembles of symmetric ensembles (cf. clique number of Paley graph) ◮ Design type. Explicit ensembles with all well-conditioned subensembles (cf. explicit Ramsey graphs) 3/27
Part I Ramsey type Conditions for existence of a well-conditioned subensemble 4/27
A warm up B = subsets ( blocks ) of V of size k such that every point in V is contained in at most r blocks in B 5/27
A warm up B = subsets ( blocks ) of V of size k such that every point in V is contained in at most r blocks in B |B| Lemma. There exist disjoint D ⊆ B such that |D| ≥ ( r − 1) k + 1 . Proof: Iteratively select a block and discard intersecting blocks. � 5/27
A warm up B = subsets ( blocks ) of V of size k such that every point in V is contained in at most r blocks in B |B| Lemma. There exist disjoint D ⊆ B such that |D| ≥ ( r − 1) k + 1 . Proof: Iteratively select a block and discard intersecting blocks. � ◮ ( 1 √ k 1 B ) B ∈D is orthonormal (well conditioned) ◮ r small = ⇒ D large 5/27
A warm up B = subsets ( blocks ) of V of size k such that every point in V is contained in at most r blocks in B |B| Lemma. There exist disjoint D ⊆ B such that |D| ≥ ( r − 1) k + 1 . Proof: Iteratively select a block and discard intersecting blocks. � ◮ ( 1 √ k 1 B ) B ∈D is orthonormal (well conditioned) ◮ r small = ⇒ D large ◮ ( 1 k 1 B ) B ∈B has upper Riesz bound r √ 5/27
Restricted invertibility More general phenomenon: Well-spread unit vectors enjoy a Riesz subensemble. Bourgain, Tzafriri, Israel J. Math., 1987 Spielman, Srivastava, Israel J. Math., 2012 6/27
Restricted invertibility More general phenomenon: Well-spread unit vectors enjoy a Riesz subensemble. Theorem Given n unit vectors with upper Riesz bound C , there exists a subensemble of ≥ ǫ 2 n / C vectors with lower Riesz bound (1 − ǫ ) 2 . Proof gives O ( n 4 ) time algorithm to find subensemble Numerical analysis: column subset selection problem Historically, this served as a stepping stone to the next result Bourgain, Tzafriri, Israel J. Math., 1987 Spielman, Srivastava, Israel J. Math., 2012 6/27
Kadison–Singer Question Does every pure state on { bounded diagonal operators on ℓ 2 } extend uniquely to a pure state on { bounded operators on ℓ 2 } ? Kadison, Singer, Amer. J. Math., 1959 Casazza, Fickus, Tremain, Weber, Operator Theory, Operator Algebras, and Applications, 2006 Bownik, Contemp. Math., 2018 7/27
Kadison–Singer Question Does every pure state on { bounded diagonal operators on ℓ 2 } extend uniquely to a pure state on { bounded operators on ℓ 2 } ? Equivalent: ◮ Paving conjecture ◮ Weaver’s conjecture ◮ Bourgain–Tzafriri conjecture ◮ Feichtinger conjecture ◮ R ǫ conjecture Kadison, Singer, Amer. J. Math., 1959 Casazza, Fickus, Tremain, Weber, Operator Theory, Operator Algebras, and Applications, 2006 Bownik, Contemp. Math., 2018 7/27
Kadison–Singer Question Does every pure state on { bounded diagonal operators on ℓ 2 } extend uniquely to a pure state on { bounded operators on ℓ 2 } ? Equivalent: ◮ Paving conjecture ◮ Weaver’s conjecture ◮ Bourgain–Tzafriri conjecture ◮ Feichtinger conjecture ◮ R ǫ conjecture Answer: Yes (!) Kadison, Singer, Amer. J. Math., 1959 Casazza, Fickus, Tremain, Weber, Operator Theory, Operator Algebras, and Applications, 2006 Bownik, Contemp. Math., 2018 7/27
Kadison–Singer Unit norm tight frames partition into two frames. Casazza, Fickus, M., Tremain, Oper. Matrices, 2011 Marcus, Spielman, Srivastava, Ann. Math., 2015 Bownik, Casazza, Marcus, Speegle, J. Reine Angew. Math., 2019 8/27
Kadison–Singer Unit norm tight frames partition into two frames. KS 2 ( η ): ∃ θ > 0, ∀ finite-dim H , ∀ η -tight frame ( f i ) i ∈ I , � f i � ≤ 1, ∃ partition I 1 ⊔ I 2 = I such that θ � x � 2 ≤ |� f i , x �| 2 ≤ ( η − θ ) � x � 2 � ∀ x ∈ H , j ∈ { 1 , 2 } i ∈ I j Casazza, Fickus, M., Tremain, Oper. Matrices, 2011 Marcus, Spielman, Srivastava, Ann. Math., 2015 Bownik, Casazza, Marcus, Speegle, J. Reine Angew. Math., 2019 8/27
Kadison–Singer Unit norm tight frames partition into two frames. KS 2 ( η ): ∃ θ > 0, ∀ finite-dim H , ∀ η -tight frame ( f i ) i ∈ I , � f i � ≤ 1, ∃ partition I 1 ⊔ I 2 = I such that θ � x � 2 ≤ |� f i , x �| 2 ≤ ( η − θ ) � x � 2 � ∀ x ∈ H , j ∈ { 1 , 2 } i ∈ I j Theorem ◮ KS 2 ( η ) does not hold for η = 2. Casazza, Fickus, M., Tremain, Oper. Matrices, 2011 Marcus, Spielman, Srivastava, Ann. Math., 2015 Bownik, Casazza, Marcus, Speegle, J. Reine Angew. Math., 2019 8/27
Kadison–Singer Unit norm tight frames partition into two frames. KS 2 ( η ): ∃ θ > 0, ∀ finite-dim H , ∀ η -tight frame ( f i ) i ∈ I , � f i � ≤ 1, ∃ partition I 1 ⊔ I 2 = I such that θ � x � 2 ≤ |� f i , x �| 2 ≤ ( η − θ ) � x � 2 � ∀ x ∈ H , j ∈ { 1 , 2 } i ∈ I j Theorem ◮ KS 2 ( η ) does not hold for η = 2. ◮ KS 2 ( η ) holds for η > 4. Casazza, Fickus, M., Tremain, Oper. Matrices, 2011 Marcus, Spielman, Srivastava, Ann. Math., 2015 Bownik, Casazza, Marcus, Speegle, J. Reine Angew. Math., 2019 8/27
Part II Symmetric type Subensembles of symmetric ensembles 9/27
HRT Conjecture For every 0 � = f ∈ L 2 ( R ) and every finite Λ ⊆ R 2 , the ensemble ( e 2 π ibx f ( x − a )) ( a , b ) ∈ Λ is linearly independent. Heil, Ramanathan, Topiwala, Proc. Am. Math. Soc., 1996 Demeter, Zaharescu, J. Math. Anal. Appl., 2012 Bownik, Speegle, Bull. Lond. Math. Soc., 2013 10/27
HRT Conjecture For every 0 � = f ∈ L 2 ( R ) and every finite Λ ⊆ R 2 , the ensemble ( e 2 π ibx f ( x − a )) ( a , b ) ∈ Λ is linearly independent. Solved instances (among several): ◮ Λ ⊆ lattice ◮ Λ = 4 points, 2 on each of 2 parallel lines ◮ f satisfies e cx log x | f ( x ) | → 0 as x → ∞ for each c > 0 Heil, Ramanathan, Topiwala, Proc. Am. Math. Soc., 1996 Demeter, Zaharescu, J. Math. Anal. Appl., 2012 Bownik, Speegle, Bull. Lond. Math. Soc., 2013 10/27
HRT Conjecture For every 0 � = f ∈ L 2 ( R ) and every finite Λ ⊆ R 2 , the ensemble ( e 2 π ibx f ( x − a )) ( a , b ) ∈ Λ is linearly independent. Solved instances (among several): ◮ Λ ⊆ lattice ◮ Λ = 4 points, 2 on each of 2 parallel lines ◮ f satisfies e cx log x | f ( x ) | → 0 as x → ∞ for each c > 0 Open instances (among several): ◮ Λ ⊆ Z × R √ √ ◮ Λ = { (0 , 0) , (1 , 0) , (0 , 1) , ( 2 , 2) } ◮ functions with faster-than-exponential decay Heil, Ramanathan, Topiwala, Proc. Am. Math. Soc., 1996 Demeter, Zaharescu, J. Math. Anal. Appl., 2012 Bownik, Speegle, Bull. Lond. Math. Soc., 2013 10/27
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