On the Query Complexity of Real Functionals Hugo Fre, Walid Gomaa, - PowerPoint PPT Presentation

Outline Introduction Complexity of Norms Query Complexity One Oracle Access On the Query Complexity of Real Functionals Hugo Férée, Walid Gomaa, Mathieu Hoyrup Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Introduction 1 Complexity of Norms 2 Query Complexity 3 One Oracle Access 4 Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Introduction 1 Complexity of Norms 2 Query Complexity 3 One Oracle Access 4 Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Computing with Real Numbers ( q n ) n ∈ N � x if ∀ n , | x − q n | ≤ 2 − n (1) R ∼ ( N → N ) Model: Oracle Turing Machines Finite-time computation − → finite number of queries − → finite knowledge of the input − → continuity Bounding computation time ⇐ ⇒ bounding the computational power the number of queries Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Computing with Real Numbers ( q n ) n ∈ N � x if ∀ n , | x − q n | ≤ 2 − n (1) R ∼ ( N → N ) Model: Oracle Turing Machines Finite-time computation − → finite number of queries − → finite knowledge of the input − → continuity Bounding computation time ⇐ ⇒ bounding the computational power the number of queries Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Computing with Real Numbers Cont’d f ∈ C [ 0 , 1 ] ⇐ ⇒ ∃µ, f Q : modulus of continuity : µ : N → N | x − y | ≤ 2 −µ ( n ) = ⇒ | f ( x ) − f ( y ) | ≤ 2 − n approximation function f Q : Q × N → Q | f Q ( q , n ) − f ( q ) | ≤ 2 − n Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Computing with Real Numbers Cont’d Theorem f : R → R is computable w.r.t. an oracle ⇐ ⇒ f is continuous. Theorem f : R → R is polynomial time computable w.r.t. to an oracle ⇐ ⇒ its modulus of continuity is bounded by a polynomial. Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Introduction 1 Complexity of Norms 2 Query Complexity 3 One Oracle Access 4 Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Dependence of a Norm on a Point F is a norm over C [ 0 , 1 ] f ∈ C [ 0 , 1 ] , α ∈ [ 0 , 1 ] ∆ α implies ∆ F ( f ) Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Dependence of a Norm on a Point Two problems f is continuous, so must change a neighborhood of α , 1 if f = 0 , any change causes F ( f ) = 0 to change to some 2 positive value. Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Dependence of a Norm on a Point Quantitative measure F is always assumed weaker than the uniform norm d F ,α ( n ) = sup { l : ∃ f ∈ Lip 1 , Supp ( f ) ⊆ N ( α, 1 / l ) and F ( f ) > 2 − n } α f ���� 1 / l d F ,α ≤ c 2 n , non-decreasing, unbounded Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Examples The uniform norm The uniform norm is monotonic: d F ,α ( n ) ∼ 2 n F = || . || ∞ = ⇒ (2) Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Examples The L 1 -norm The L 1 -norm is monotonic: n F = || . || 1 = ⇒ d F ,α ( n ) ∼ 2 (3) 2 Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Some Properties of the Dependency Function Proposition α �→ d F ,α ( n ) is continuous 1 F is weaker than G = ⇒ d F ,α ( n ) ≤ d G ,α ( n + k ) 2 Maximal dependence: D F ( n ) = max α ∈ [ 0 , 1 ] d F ,α ( n ) Proposition For F weaker than the uniform norm: n 2 ≤ D F ( n ) ≤ c 2 2 n (4) c 1 2 Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Relevant Points R n , l = { α : d F ,α ( n ) ≥ l } Definition n α is relevant if ∃ c > 0 , ∀ n , d F ,α ( n ) ≥ c · 2 2 � � R = R n , 2 n 2 − k n k � �� � R k Example For || . || ∞ and || . || 1 , R = [ 0 , 1 ]. Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Relevant Points Example Let Q = { q 0 , q 1 , . . . , } be some particular canonical enumeration of the dyadic rationals Define � 2 − i | f ( q i ) | F ( f ) = i Then d F , q i ( n ) ≥ 2 n − i , for n ≥ i d F ,α ( n ) ≤ n 2 ε R = D Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Relevant Points Properties Theorem R is dense. Theorem ⇒ F ( f ) ≤ c . 2 − k . f = 0 on R 2 µ f ( k ) = Corollary ⇒ | F ( f ) − F ( g ) | ≤ 2 − k . f = g on R µ ( k ) = Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access R is Dense (Proof) F ( h 0 ) ≥ 2 − c Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access R is Dense (Proof) F ( h 1 ) ≥ 2 − c − 2 Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access R is Dense (Proof) F ( h 2 ) ≥ 2 − c − 4 Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access R is Dense (Proof) F ( h n ) = F ( h α n , 2 − n − p ) ≥ 2 − c − 2 n = ⇒ F ( 2 n + c ) ≥ 2 n + p ( α n ) → α d F ,α ( 2 n + c ) ≥ 2 n + p − 1 = ⇒ α ∈ R Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Introduction 1 Complexity of Norms 2 Query Complexity 3 One Oracle Access 4 Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Query Complexity Definition Q n : oracle calls of F on x �→ 0 with precision 2 − n . Proposition R n , l ⊆ N ( Q n + 1 , 1 l ) Definition F has a polynomial query complexity if it is computable by a relativized OTM with | Q n | ≤ P ( n ) . Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Query Complexity Cont’d Theorem If F has polynomial query complexity, then almost every point has a polynomial dependency ( d F ,α ∈ P for almost all α ). Theorem If F has polynomial query complexity, then R has Hausdorff dimension 0 . Proposition F has polynomial query complexity 2 n = ⇒ ∃α, d F ,α ( n ) is bounded by a polynomial. Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Query Complexity Char acterizing polynomial time computable norms Theorem F is polynomial time computable w.r.t. an oracle F has polynomial query complexity ⇐ ⇒ R k can be polynomially covered (wrt. k ). ⇐ ⇒ � n 2 − k } = R k = { α ∈ [ 0 , 1 ]: ∀ n , d F ,α ( n ) ≥ 2 R n , 2 n 2 − k n Open question Can it be generalized for any F : C [ 0 , 1 ] → R ? Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity One Oracle Access Introduction 1 Complexity of Norms 2 Query Complexity 3 One Oracle Access 4 Hugo Férée, Walid Gomaa, Mathieu Hoyrup On the Query Complexity of Real Functionals