Absolute Value Information from IBC perspective Leszek Plaskota University of Warsaw RICAM November 7, 2018 (joint work with Paweł Siedlecki and Henryk Wo´ zniakowski) A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 1/12
Information-Based Complexity Solution operator: S : F → G , where F linear space, G normed space with � · � . Approximation: S ( f ) ∼ A n ( f ) = ϕ ( y ) where y = ( y 1 , y 2 , . . . , y n ) is information about f , y i = L i ( f ) (nonadaptive) = L i ( f ; y 1 , . . . , y i − 1 ) y i (adaptive) L i ( · ; y 1 , . . . , y i − 1 ) ∈ Λ a class of functionals on F . A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 2/12
Classes of information Classes Λ of information in IBC: • Λ all all linear functionals, • Λ std function values only. Different class in phase retrieval : • | Λ | = { | L | : L ∈ Λ } for given Λ . Information | Λ | was used in exact recovery in Hilbert spaces, up to the phase shift, e.g., Cahill, Casazza, Daubechies (2016). (Applications in signal reconstruction, audio processing...) A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 3/12
Algorithm errors For a set F ⊂ F of problem instances, � � e ( A n ) = sup d S ( f ) , A n ( f ) . f ∈F In standard IBC: d std ( g 1 , g 2 ) = � g 1 − g 2 � . In phase retrieval: d mod ( g 1 , g 2 ) = inf | z | = 1 � g 1 − z g 2 � . A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 4/12
Information ε -complexity We want to compare the powers of | Λ | Λ and in terms of information ε -complexities: n std ( S , Λ ; ε ) n : there is A n using Λ s.t. e std ( A n ) ≤ ε � � = min , n mod ( S , | Λ | ; ε ) n : there is A n using | Λ | s.t. e mod ( A n ) ≤ ε � � = min . A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 5/12
Result for Λ = Λ all (positive) Theorem Let • the solution operator S : F → G be linear, and • the class F ⊂ F be convex and balanced. Then � ≤ n mod � � ≤ 3 n std � n std � S , Λ all , 4 ε S , | Λ all | , ε S , Λ , 1 � . 2 ε Hence, | Λ all | and Λ all are roughly of the same power. A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 6/12
Remark The theorem holds for Λ ⊆ Λ all satisfying the following. If L 1 , L 2 ∈ Λ then • in real case: L 1 + L 2 ∈ Λ , • in complex case: L 1 + L 2 ∈ Λ , L 1 + i L 2 ∈ Λ . Observe that this holds for Λ all , but not for Λ std . A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 7/12
Result for Λ = Λ std (negative) Theorem Let • F be a linear space of (real or complex valued) functions, • the class F ⊂ F be convex and balanced, • the solution operator S : F → G be linear. Suppose there are two functions f 1 , f 2 ∈ F such that f 1 , f 2 / ∈ ker S and f 1 ∗ f 2 = 0. Then there is ε 0 > 0 such that for all ε ≤ ε 0 � = + ∞ . n mod � S , | Λ std | ; ε A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 8/12
Recovery of polynomials Let F = F = P k be real (algebraic) polynomials f on R with deg f ≤ k − 1. The problem of exact recovery of f ∈ P k can be solved using: • k evaluations of f for Λ std , • 2 k − 1 evaluations of f for | Λ std | . Note that the assumptions of the last theorem are not satisfied, since f 1 ∗ f 2 � = 0 whenever f 1 , f 2 � = 0. A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 9/12
Another way... For S : F → G and F ⊂ F , the problem can be re-defined as recovery of the multi-valued mapping S : F → 2 G given by S ( f ) = � S ( f 1 ) : f 1 ∈ F , | L ( f 1 ) | = | L ( f ) | for all L ∈ Λ � . Algorithm A n , m using n functionals from | Λ | returns subsets of G of cardinality m , A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 10/12
Another way... Algorithm error: e H ( A n , m ) = sup d H � S ( f ) , A n , m ( f ) � f ∈F where d H is the Hausdorff distance , � � d H ( W , Z ) = max z ∈ Z � w − z � , sup w ∈ W � z − w � sup inf inf . w ∈ W z ∈ Z ε -complexity: n H ( S , | Λ | ; ε ) n + m : there is A n , m using | Λ | s.t. e H ( A n , m ) ≤ ε � � = min . A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 11/12
IBC for approximation of multi-valued operators? A BSOLUTE V ALUE I NFORMATIONFROM IBC PERSPECTIVE 12/12
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