Absolute Value Information from IBC perspective Leszek Plaskota - - PowerPoint PPT Presentation

Absolute Value Information from IBC perspective

Leszek Plaskota University of Warsaw RICAM November 7, 2018 (joint work with Paweł Siedlecki and Henryk Wo´ zniakowski)

ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 1/12

Information-Based Complexity

Solution operator: S : F → G, where F linear space, G normed space with · . Approximation: S(f) ∼ An(f) = ϕ(y) where y = (y1, y2, . . . , yn) is information about f, yi = Li(f) (nonadaptive) yi = Li(f; y1, . . . , yi−1) (adaptive) Li(·; y1, . . . , yi−1) ∈ Λ a class of functionals on F.

ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 2/12

Classes of information

Classes Λ of information in IBC:

• Λall

all linear functionals,

• Λstd

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

Algorithm errors

For a set F ⊂ F of problem instances, e(An) = sup

f∈F

d

• S(f), An(f)
• .

In standard IBC: dstd(g1, g2) = g1 − g2. In phase retrieval: dmod(g1, g2) = inf

|z|=1 g1 − z g2.

ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 4/12

Information ε-complexity

We want to compare the powers of Λ and |Λ| in terms of information ε-complexities: nstd(S, Λ; ε) = min

• n : there is An using Λ s.t. estd(An) ≤ ε
• ,

nmod(S, |Λ|; ε) = min

• n : there is An using |Λ| s.t. emod(An) ≤ ε
• .

ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 5/12

Result for Λ = Λall (positive)

Theorem

Let

• the solution operator S : F → G be linear, and
• the class F ⊂ F be convex and balanced.

Then nstd S, Λall, 4ε ≤ nmod S, |Λall|, ε ≤ 3 nstd S, Λ, 1

2ε

• .

Hence, |Λall| and Λall are roughly of the same power.

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Remark

The theorem holds for Λ ⊆ Λall satisfying the following. If L1, L2 ∈ Λ then

• in real case:

L1 + L2 ∈ Λ,

• in complex case:

L1 + L2 ∈ Λ, L1 + iL2 ∈ Λ. Observe that this holds for Λall, but not for Λstd.

ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 7/12

Result for Λ = Λstd (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 f1, f2 ∈ F such that f1, f2 / ∈ ker S and f1 ∗ f2 = 0. Then there is ε0 > 0 such that for all ε ≤ ε0 nmod S, |Λstd|; ε = +∞.

ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 8/12

Recovery of polynomials

Let F = F = Pk be real (algebraic) polynomials f on R with deg f ≤ k − 1. The problem of exact recovery of f ∈ Pk can be solved using:

• k evaluations of f for Λstd,
• 2 k − 1 evaluations of f for |Λstd|.

Note that the assumptions of the last theorem are not satisfied, since f1 ∗ f2 = 0 whenever f1, f2 = 0.

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Another way...

For S : F → G and F ⊂ F, the problem can be re-defined as recovery of the multi-valued mapping S : F → 2G given by S(f) =

• S(f1) : f1 ∈ F, |L(f1)| = |L(f)| for all L ∈ Λ
• .

Algorithm An,m using n functionals from |Λ| returns subsets of G of cardinality m,

ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 10/12

Another way...

Algorithm error: eH(An,m) = sup

f∈F

dH S(f), An,m(f)

• where dH is the Hausdorff distance,

dH(W, Z) = max

• sup

w∈W

inf

z∈Z w − z, sup z∈Z

inf

w∈W z − w

• .

ε-complexity: nH(S, |Λ|; ε) = min

• n + m : there is An,m using |Λ| s.t. eH(An,m) ≤ ε
• .

ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 11/12

IBC for approximation of multi-valued operators?

ABSOLUTE VALUE INFORMATIONFROM IBC PERSPECTIVE 12/12