Convergence order of the geometric mean errors for Markov-type measures

Zhu, Sanguo

doi:10.1016/j.chaos.2014.11.015

Cited by 2 publications

(2 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Typically, this question is much harder to answer than finding the quantization dimension. So far, the quantization coefficient has been studied for absolutely continuous probability measures ( [6]) and several classes of singular measures, including self-similar and self-conformal [19,29,39,41] measures, Markov-type measures [16,33,29] and self-affine measures on Bedford-McMullen carpets [15,38].…”

Section: Theorem 11 ([6]mentioning

confidence: 99%

See 1 more Smart Citation

Some Recent Developments in Quantization of Fractal Measures

Kesseböhmer

Zhu

2015

Fractal Geometry and Stochastics V

Self Cite

View full text Add to dashboard Cite

Abstract. We give an overview on the quantization problem for fractal measures, including some related results and methods which have been developed in the last decades. Based on the work of Graf and Luschgy, we propose a three-step procedure to estimate the quantization errors. We survey some recent progress, which makes use of this procedure, including the quantization for self-affine measures, Markov-type measures on graph-directed fractals, and product measures on multiscale Moran sets. Several open problems are mentioned.

show abstract

Section: Theorem 11 ([6]mentioning

confidence: 99%

“…In the following we will focus on the L r -quantization problem with r > 0. For the quantization with respect to the geometric mean error, we refer to [8] for rigorous foundations and [37,33,34,40] for more related results.…”

Section: Introductionmentioning

confidence: 99%

Some Recent Developments in Quantization of Fractal Measures

Kesseböhmer

Zhu

2015

Fractal Geometry and Stochastics V

Self Cite

View full text Add to dashboard Cite

show abstract

Quantization dimensions of compactly supported probability measures via Rényi dimensions

Kesseböhmer

Niemann

Zhu

2023

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We provide a complete picture of the upper quantization dimension in terms of the Rényi dimension by proving that the upper quantization dimension of order r > 0 r>0 for an arbitrary compactly supported Borel probability measure ν \nu is given by its Rényi dimension at the point q r q_{r} where the L q L^{q} -spectrum of ν \nu and the line through the origin with slope r r intersect. In particular, this proves the continuity of r ↦ D ¯ r ( \nu ) r\mapsto \overline {D}_{r}(\text {\nu )} as conjectured by Lindsay [Quantization dimension for probability distributions, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University of North Texas]. This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the L q L^{q} -spectrum restricted to ( 0 , 1 ) \left (0,1\right ) . We give sufficient conditions in terms of the L q L^{q} -spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a C 1 \mathcal {C}^{1} -self-conformal iterated function system on R d \mathbb {R}^{d} without any assumption on the separation conditions as well as for inhomogeneous self-similar measures under the inhomogeneous open sets condition. Some known general bounds on the quantization dimension in terms of other fractal dimensions can readily be derived from our new approach, some can be improved.

show abstract

Convergence order of the geometric mean errors for Markov-type measures

Cited by 2 publications

References 17 publications

Some Recent Developments in Quantization of Fractal Measures

Some Recent Developments in Quantization of Fractal Measures

Quantization dimensions of compactly supported probability measures via Rényi dimensions

Contact Info

Product

Resources

About