scholarly journals Persistent homology for low-complexity models

2019 ◽  
Vol 475 (2230) ◽  
pp. 20190081
Author(s):  
Martin Lotz
Keyword(s):  
Compressed Sensing ◽  
Dimension Reduction ◽  
Structural Properties ◽  
Metric Space ◽  
Persistent Homology ◽  
Low Complexity ◽  
Gaussian Width ◽  
Doubling Dimension ◽  
Finite Metric Space

We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is determined by the Gaussian width of a structure associated with the dataset, rather than its size, and such a reduction can be computed efficiently. We further relate the Gaussian width to the doubling dimension of a finite metric space, which appears in the study of the complexity of other methods for approximating persistent homology. We can, therefore, literally replace the ambient dimension by an intrinsic notion of dimension related to the structure of the data.

scholarly journals A Case Study in Low-Complexity ECG Signal Encoding: How Compressing is Compressed Sensing?

2015 ◽  
Vol 22 (10) ◽  
pp. 1743-1747 ◽  
Author(s):  
Valerio Cambareri ◽  
Mauro Mangia ◽  
Fabio Pareschi ◽  
Riccardo Rovatti ◽  
Gianluca Setti
Keyword(s):  
Compressed Sensing ◽  
Low Complexity ◽  
Ecg Signal ◽  
Signal Encoding

scholarly journals Lipschitz-free Spaces on Finite Metric Spaces

2019 ◽  
Vol 72 (3) ◽  
pp. 774-804 ◽  
Author(s):  
Stephen J. Dilworth ◽  
Denka Kutzarova ◽  
Mikhail I. Ostrovskii
Keyword(s):  
Metric Space ◽  
Free Space ◽  
Metric Spaces ◽  
Large Classes ◽  
Complemented Subspace ◽  
Finite Metric Space ◽  
Free Spaces ◽  
Finite Metric Spaces

AbstractMain results of the paper are as follows:(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

scholarly journals A Low-Complexity Compressed Sensing Reconstruction Method for Heart Signal Biometric Recognition

Sensors ◽  
2019 ◽  
Vol 19 (23) ◽  
pp. 5330
Author(s):  
Xiao ◽  
Hu ◽  
Shao ◽  
Li
Keyword(s):  
Compressed Sensing ◽  
Matching Pursuit ◽  
Low Complexity ◽  
Portable Devices ◽  
Reconstruction Method ◽  
Signal Recognition ◽  
Bioelectric Signal ◽  
Increasing Demand ◽  
Feature Values ◽  
Compression Sensing

Biometric systems allow recognition and verification of an individual through his or her physiological or behavioral characteristics. It is a growing field of research due to the increasing demand for secure and trustworthy authentication systems. Compressed sensing is a data compression acquisition method that has been proposed in recent years. The sampling and compression of data is completed synchronously, avoiding waste of resources and meeting the requirements of small size and limited power consumption of wearable portable devices. In this work, a compression reconstruction method based on compression sensing was studied using bioelectric signals, which aimed to increase the limited resources of portable remote bioelectric signal recognition equipment. Using electrocardiograms (ECGs) and photoplethysmograms (PPGs) of heart signals as research data, an improved segmented weak orthogonal matching pursuit (OMP) algorithm was developed to compress and reconstruct the signals. Finally, feature values were extracted from the reconstructed signals for identification and analysis. The accuracy of the proposed method and the practicability of compression sensing in cardiac signal identification were verified. Experiments showed that the reconstructed ECG and PPG signal recognition rates were 95.65% and 91.31%, respectively, and that the residual value was less than 0.05 mV, which indicates that the proposed method can be effectively used for two bioelectric signal compression reconstructions.

scholarly journals TESTING EMBEDDABILITY BETWEEN METRIC SPACES

2009 ◽  
Vol 20 (02) ◽  
pp. 313-329
Author(s):  
CHING-LUEH CHANG ◽  
YUH-DAUH LYUU ◽  
YEN-WU TI
Keyword(s):  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Graph Isomorphism ◽  
Upper And Lower Bounds ◽  
Wrong Answer ◽  
Probability Of Error ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Special Case

Let L ≥ 1, ε > 0 be real numbers, (M, d) be a finite metric space and (N, ρ) be a metric space. A query to a metric space consists of a pair of points and asks for the distance between these points. We study the number of queries to metric spaces (M, d) and (N, ρ) needed to decide whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into N, ρ). When (M, d) is ∊-far from being L-bilipschitz embeddable into (N, ρ), we allow an o(1) probability of error (i.e., returning the wrong answer "L-bilipschitz embeddable"). However, no error is allowed when (M, d) is L-bilipschitz embeddable into (N, ρ). That is, algorithms with only one-sided errors are studied in this paper. When |M| ≤ |N| are both finite, we give an upper bound of [Formula: see text] on the number of queries for determining with one-sided error whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M| = |N|, the above upper bound evaluates to [Formula: see text]. We also prove a lower bound of Ω(|N|3/2) for the special case when |M| = |N| are finite and L = 1, which coincides with testing isometry between finite metric spaces. For finite |M| = |N|, the upper and lower bounds thus match up to a multiplicative factor of at most [Formula: see text], which depends only sublogarithmically in |N|. We also investigate the case when (N, ρ) is not necessarily finite. Our results are based on techniques developed in an earlier work on testing graph isomorphism.

scholarly journals ROUNDNESS PROPERTIES OF ULTRAMETRIC SPACES

2013 ◽  
Vol 56 (3) ◽  
pp. 519-535 ◽  
Author(s):  
TIMOTHY FAVER ◽  
KATELYNN KOCHALSKI ◽  
MATHAV KISHORE MURUGAN ◽  
HEIDI VERHEGGEN ◽  
ELIZABETH WESSON ◽  
...  
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Independent Set ◽  
Ultrametric Space ◽  
Ultrametric Spaces ◽  
Negative Type ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Generalized Roundness

AbstractMotivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space $\mathbb{R}^{n}$ of dimension n but it cannot be isometrically embedded in any Euclidean space $\mathbb{R}^{r}$ of dimension r < n. We use this result as a technical tool to study ‘roundness’ properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class $\mathcal{M}$ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class $\mathcal{M}$ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].

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