Absolute Value Information from IBC perspective

Transcription

1 Absolute Value Information from IBC perspective Leszek Plaskota University of Warsaw RICAM November 7, 2018 (joint work with Paweł Siedlecki and Henryk Woźniakowski) ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 1/12

2 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) y i = L i (f ; y 1,..., y i 1 ) (adaptive) L i ( ; y 1,..., y i 1 ) Λ a class of functionals on F. ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 2/12

3 Classes of information Classes Λ of information in IBC: Λ all Λ std all linear functionals, 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...) ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 3/12

4 Algorithm errors For a set F F of problem instances, In standard IBC: In phase retrieval: e(a n ) = sup f F d ( S(f ), A n (f ) ). d std (g 1, g 2 ) = g 1 g 2. d mod (g 1, g 2 ) = inf g 1 z g 2. z =1 ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 4/12

5 Information ε-complexity We want to compare the powers of Λ and Λ in terms of information ε-complexities: n std (S, Λ; ε) = min { n : there is A n using Λ s.t. e std (A n ) ε }, n mod (S, Λ ; ε) = min { n : there is A n using Λ s.t. e mod (A n ) ε }. ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 5/12

6 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 std( S, Λ all, 4ε ) n mod( S, Λ all, ε ) 3 n std( S, Λ, 1 2 ε). Hence, Λ all and Λ all are roughly of the same power. ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 6/12

7 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 + il 2 Λ. Observe that this holds for Λ all, but not for Λ std. ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 7/12

8 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 ) = +. ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 8/12

9 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. ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 9/12

10 Another way... For S : F G and F F, the problem can be re-defined as recovery of the multi-valued mapping given by S : F 2 G 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, ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 10/12

11 Another way... Algorithm error: e H (A n,m ) = sup f F where d H is the Hausdorff distance, { d H (W, Z) = max sup inf w W z Z d H( S(f ), A n,m (f ) ) w z, sup z Z inf w W } z w. ε-complexity: n H (S, Λ ; ε) = min { n + m : there is A n,m using Λ s.t. e H (A n,m ) ε }. ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 11/12

12 IBC for approximation of multi-valued operators? ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 12/12