High-order processing of singular data

Yomdin, Yosef; Zahavi, Gal

doi:10.2969/aspm/05510173

Cited by 3 publications

(5 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We expect also that taking more than the minimal number of the measurements, and hance of the equations in (4.1) (say, twice the minimal number) can strongly improve the robustness of the solution. This conclusion is supported by recent results in [64,63] where we investigate similar problems for Hermite interpolation and Hermite least-square fitting.…”

Section: A Special Case Of the Inversion Problem Is The "Moment Vanis...supporting

confidence: 89%

See 1 more Smart Citation

Linear versus Non-Linear Acquisition of Step-Functions

2008

View full text Add to dashboard Cite

We address in this paper the following two closely related problems: 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way? 2. How to reconstruct such functions from a small number of measurements? The stress is on a comparison of linear and nonlinear approaches. As a model case we use piecewise-constant functions on [0, 1], in particular, the Heaviside jump function H t = χ [0,t] . Considered as a curve in the Hilbert space L 2 ([0, 1]) it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables.Next we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov's n-width and ǫ-entropy, as well as Temlyakov's (N, m)-width.On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis.Finally, we informally discuss two problems in Computer Imaging which are parallel to the problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other.----------------

show abstract

Section: A Special Case Of the Inversion Problem Is The "Moment Vanis...supporting

confidence: 89%

“…Another approach is based on "non-linear model-net approximation" ( [4,20,63]). In the present paper it was described in a "toy example" of piecewise-constant functions.…”

Section: Digital Imagesmentioning

confidence: 99%

Linear versus Non-Linear Acquisition of Step-Functions

2008

View full text Add to dashboard Cite

show abstract

“…It is well known that the inequality of the form (1.1) or (1.2) may be true also for some sets Z of measure zero and even for certain discrete or finite sets Z. Let us mention here only a couple of the most relevant results in this direction: in [2,3,12,26,31,38,39] such inequalities are provided for Z being a regular grid in [−1, 1] (in [3] trigonometric polynomials are considered). Let us mention [1] where a "dual" problem is considered of interpolation by polynomials of degree higher than the number of the nodes.…”

Section: )mentioning

confidence: 99%

“…Proof: It follows from the bounds on ω d (G 1 s ) computed in Example 1, Section 3. The result of Theorem 2.5 was obtained by a different method in [39,38].…”

Section: Examples In One Dimension and The Favard Boundmentioning

confidence: 99%

Remez-type inequality for discrete sets

Yomdin

2011

Isr. J. Math.

View full text Add to dashboard Cite

show abstract

“…It is well known that the inequality of the form (1.1) or (1.2) may be true also for some sets Z of measure zero and even for certain discrete or finite sets Z. Let us mention here only a couple of the most relevant results in this direction: in [4,9,13] such inequalities are provided for Z being a regular grid in [−1, 1]. In [7] discrete sets Z ⊂ [−1, 1] are studied.…”

Section: Bounding R D (Z) Via Metric Entropymentioning

confidence: 99%

Remez-Type Inequality for Smooth Functions

Yomdin

2014

Springer Proceedings in Mathematics &Amp; Statistics

Self Cite

View full text Add to dashboard Cite

The classical Remez inequality bounds the maximum of the absolute value of a polynomial P (x) of degree d on [−1, 1] through the maximum of its absolute value on any subset Z of positive measure in [−1, 1]. Similarly, in several variables the maximum of the absolute value of a polynomial P (x) of degree d on the unit ball B n ⊂ R n can be bounded through the maximum of its absolute value on any subset Z ⊂ Q n 1 of positive n-measure m n (Z). In [12] a stronger version of Remez inequality was obtained: the Lebesgue n-measure m n was replaced by a certain geometric quantity ω n,d (Z) satisfying ω n,d (Z) ≥ m n (Z) for any measurable Z. The quantity ω n,d (Z) can be effectively estimated in terms of the metric entropy of Z and it may be nonzero for discrete and even finite sets Z.In the present paper we extend Remez inequality to functions of finite smoothness. This is done by combining the result of [12] with the Taylor polynomial approximation of smooth functions. As a consequence we obtain explicit lower bounds in some examples in the Whitney problem of a C k -smooth extrapolation from a given set Z, in terms of the geometry of Z.----------------

show abstract

High-order processing of singular data

Cited by 3 publications

References 13 publications

Linear versus Non-Linear Acquisition of Step-Functions

Linear versus Non-Linear Acquisition of Step-Functions

Remez-type inequality for discrete sets

Remez-Type Inequality for Smooth Functions

Contact Info

Product

Resources

About