scholarly journals Linear extension operators between spaces of Lipschitz maps and optimal transport

2020 ◽  
Vol 2020 (764) ◽  
pp. 1-21 ◽  
Author(s):  
Luigi Ambrosio ◽  
Daniele Puglisi
Keyword(s):  
Linear Extension ◽  
Metric Spaces ◽  
Optimal Transport ◽  
Partition Of Unity ◽  
Random Projection ◽  
Random Projections ◽  
Transport Distance ◽  
Lipschitz Maps ◽  
Necessary And Sufficient ◽  
Continuous Operators

AbstractMotivated by the notion of {K\hskip-0.284528pt}-gentle partition of unity introduced in [J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59–95] and the notion of {K\kern-0.284528pt}-Lipschitz retract studied in [S. I. Ohta, Extending Lipschitz and Hölder maps between metric spaces, Positivity 13 (2009), no. 2, 407–425], we study a weaker notion related to the Kantorovich–Rubinstein transport distance that we call {K\kern-0.284528pt}-random projection. We show that {K\kern-0.284528pt}-random projections can still be used to provide linear extension operators for Lipschitz maps. We also prove that the existence of these random projections is necessary and sufficient for the existence of weak{{}^{*}} continuous operators. Finally, we use this notion to characterize the metric spaces {(X,d)} such that the free space {\mathcal{F}(X)} has the bounded approximation propriety.

scholarly journals Multiresolution analysis and wavelets on Vilenkin groups

2008 ◽  
Vol 21 (3) ◽  
pp. 309-325 ◽  
Author(s):  
Yury Farkov
Keyword(s):  
Multiresolution Analysis ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Partition Of Unity ◽  
Necessary And Sufficient Conditions ◽  
Refinable Functions ◽  
Vilenkin Groups ◽  
Orthogonal Wavelets ◽  
The Stability ◽  
Necessary And Sufficient

This paper gives a review of multiresolution analysis and compactly sup- ported orthogonal wavelets on Vilenkin groups. The Strang-Fix condition, the partition of unity property, the linear independence, the stability, and the orthonormality of 'integer shifts' of the corresponding refinable functions are considered. Necessary and sufficient conditions are given for refinable functions to generate a multiresolution analysis in the L2-spaces on Vilenkin groups. Several examples are provided to illustrate these results. .

WEAK SELF-SIMILAR SETS IN SEPARABLE COMPLETE METRIC SPACES

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750021
Author(s):  
R. K. ASWATHY ◽  
SUNIL MATHEW
Keyword(s):  
Metric Spaces ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
Finite Union ◽  
Self Similarity ◽  
Complete Metric Spaces ◽  
Self Similar ◽  
Necessary And Sufficient ◽  
Physical Phenomena

Self-similarity is a common tendency in nature and physics. It is wide spread in geo-physical phenomena like diffusion and iteration. Physically, an object is self-similar if it is invariant under a set of scaling transformation. This paper gives a brief outline of the analytical and set theoretical properties of different types of weak self-similar sets. It is proved that weak sub self-similar sets are closed under finite union. Weak sub self-similar property of the topological boundary of a weak self-similar set is also discussed. The denseness of non-weak super self-similar sets in the set of all non-empty compact subsets of a separable complete metric space is established. It is proved that the power of weak self-similar sets are weak super self-similar in the product metric and weak self-similarity is preserved under isometry. A characterization of weak super self-similar sets using weak sub contractions is also presented. Exact weak sub and super self-similar sets are introduced in this paper and some necessary and sufficient conditions in terms of weak condensation IFS are presented. A condition for a set to be both exact weak super and sub self-similar is obtained and the denseness of exact weak super self similar sets in the set of all weak self-similar sets is discussed.

Fuzzy Lipschitz maps and fixed point theorems in fuzzy metric spaces

2010 ◽  
Vol 161 (8) ◽  
pp. 1117-1130 ◽  
Author(s):  
Gabjin Yun ◽  
Seungsu Hwang ◽  
Jeongwook Chang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Lipschitz Maps

scholarly journals On the representation of metric spaces

1979 ◽  
Vol 20 (3) ◽  
pp. 367-375 ◽  
Author(s):  
G.J. Logan
Keyword(s):  
Algebraic Structure ◽  
Topological Structure ◽  
Dual Space ◽  
Metric Spaces ◽  
Closure Operator ◽  
The Family ◽  
Necessary And Sufficient ◽  
The Relationship

A closure algebra is a set X with a closure operator C defined on it. It is possible to construct a topology τ on MX, the family of maximal, proper, closed subsets of X, and then to examine the relationship between the algebraic structure of (X, C) and the topological structure of the dual space (MX τ) This paper describes the algebraic conditions which are necessary and sufficient for the dual space to be separable metric and metric respectively.

scholarly journals Efficient Implementation of Homomorphic and Fuzzy Transforms in Random-Projection Encryption Frameworks for Cancellable Face Recognition

Electronics ◽  
2020 ◽  
Vol 9 (6) ◽  
pp. 1046 ◽  
Author(s):  
Abeer D. Algarni ◽  
Ghada M. El Banby ◽  
Naglaa F. Soliman ◽  
Fathi E. Abd El-Samie ◽  
Abdullah M. Iliyasu
Keyword(s):  
Face Recognition ◽  
Characteristic Curve ◽  
Similarity Index ◽  
Random Projection ◽  
Personal Identification ◽  
Random Projections ◽  
Person Identification ◽  
Security Systems ◽  
Biometric Data ◽  
Rich Diversity

To circumvent problems associated with dependence on traditional security systems on passwords, Personal Identification Numbers (PINs) and tokens, modern security systems adopt biometric traits that are inimitable to each individual for identification and verification. This study presents two different frameworks for secure person identification using cancellable face recognition (CFR) schemes. Exploiting its ability to guarantee irrevocability and rich diversity, both frameworks utilise Random Projection (RP) to encrypt the biometric traits. In the first framework, a hybrid structure combining Intuitionistic Fuzzy Logic (IFL) with RP is used to accomplish full distortion and encryption of the original biometric traits to be saved in the database, which helps to prevent unauthorised access of the biometric data. The framework involves transformation of spatial-domain greyscale pixel information to a fuzzy domain where the original biometric images are disfigured and further distorted via random projections that generate the final cancellable traits. In the second framework, cancellable biometric traits are similarly generated via homomorphic transforms that use random projections to encrypt the reflectance components of the biometric traits. Here, the use of reflectance properties is motivated by its ability to retain most image details, while the guarantee of the non-invertibility of the cancellable biometric traits supports the rationale behind our utilisation of another RP stage in both frameworks, since independent outcomes of both the IFL stage and the reflectance component of the homomorphic transform are not enough to recover the original biometric trait. Our CFR schemes are validated on different datasets that exhibit properties expected in actual application settings such as varying backgrounds, lightings, and motion. Outcomes in terms standard metrics, including structural similarity index metric (SSIM) and area under the receiver operating characteristic curve (AROC), suggest the efficacy of our proposed schemes across many applications that require person identification and verification.

scholarly journals Increasing the robustness and applicability of full-waveform inversion: An optimal transport distance strategy

2016 ◽  
Vol 35 (12) ◽  
pp. 1060-1067 ◽  
Author(s):  
L. Métivier ◽  
R. Brossier ◽  
Q. Mérigot ◽  
E. Oudet ◽  
J. Virieux
Keyword(s):  
Optimal Transport ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Transport Distance ◽  
Full Waveform

scholarly journals Random projections and Hotelling’s T2 statistics for change detection in high-dimensional data streams

2013 ◽  
Vol 23 (2) ◽  
pp. 447-461 ◽  
Author(s):  
Ewa Skubalska-Rafajłowicz
Keyword(s):  
Discrete Time ◽  
Data Streams ◽  
Random Projection ◽  
Image Sequences ◽  
High Dimensional ◽  
Random Projections ◽  
Random Subspace ◽  
Noisy Image ◽  
Method Of Dimensionality Reduction ◽  
Diagnostic Properties

The method of change (or anomaly) detection in high-dimensional discrete-time processes using a multivariate Hotelling chart is presented. We use normal random projections as a method of dimensionality reduction. We indicate diagnostic properties of the Hotelling control chart applied to data projected onto a random subspace of Rn. We examine the random projection method using artificial noisy image sequences as examples.

scholarly journals Dimensionality reduction for k-distance applied to persistent homology

2021 ◽  
Author(s):  
Shreya Arya ◽  
Jean-Daniel Boissonnat ◽  
Kunal Dutta ◽  
Martin Lotz
Keyword(s):  
Dimensionality Reduction ◽  
Persistent Homology ◽  
Random Projection ◽  
Distance Functions ◽  
Random Projections ◽  
Annual Symposium ◽  
Gaussian Width ◽  
College Park ◽  
Low Dimensional ◽  
Open Question

AbstractGiven a set P of n points and a constant k, we are interested in computing the persistent homology of the Čech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an open question of Sheehy (The persistent homology of distance functions under random projection. In Cheng, Devillers (eds), 30th Annual Symposium on Computational Geometry, SOCG’14, Kyoto, Japan, June 08–11, p 328, ACM, 2014). We show that any linear transformation that preserves pairwise distances up to a $$(1\pm {\varepsilon })$$ ( 1 ± ε ) multiplicative factor, must preserve the persistent homology of the Čech filtration up to a factor of $$(1-{\varepsilon })^{-1}$$ ( 1 - ε ) - 1 . Our results also show that the Vietoris-Rips and Delaunay filtrations for the k-distance, as well as the Čech filtration for the approximate k-distance of Buchet et al. [J Comput Geom, 58:70–96, 2016] are preserved up to a $$(1\pm {\varepsilon })$$ ( 1 ± ε ) factor. We also prove extensions of our main theorem, for point sets (i) lying in a region of bounded Gaussian width or (ii) on a low-dimensional submanifold, obtaining embeddings having the dimension bounds of Lotz (Proc R Soc A Math Phys Eng Sci, 475(2230):20190081, 2019) and Clarkson (Tighter bounds for random projections of manifolds. In Teillaud (ed) Proceedings of the 24th ACM Symposium on Computational Geom- etry, College Park, MD, USA, June 9–11, pp 39–48, ACM, 2008) respectively. Our results also work in the terminal dimensionality reduction setting, where the distance of any point in the original ambient space, to any point in P, needs to be approximately preserved.

Sign in / Sign up

Export Citation Format

Share Document