DTM-Based Filtrations

Anai, Hirokazu; Chazal, Frédéric; Glisse, Marc; Ike, Yuichi; Inakoshi, Hiroya; Tinarrage, Raphaël; Umeda, Yohtaro

doi:10.1007/978-3-030-43408-3_2

“…However, the distance function δ X is extremely sensitive to outliers and noise ("even one outlier is deadly", or, in the language of robust statistics, the distance function has breakdown point zero [39]). To circumvent this issue, [38] propose to rather consider distance-to-ameasure (DTM) δ X,m as the filtration function, which is defined as the average distance from a given number of nearest neighbours in X (and is thus a smooth version of the distance function) [40]. The number of neighbours that are considered is determined by the parameter m, which represents the percentage of the total number of point cloud X points.…”

Section: Plos Onementioning

confidence: 99%

Noise robustness of persistent homology on greyscale images, across filtrations and signatures

Turkeš

¹

,

Nys

²

,

Verdonck

³

et al. 2021

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Topological data analysis is a recent and fast growing field that approaches the analysis of datasets using techniques from (algebraic) topology. Its main tool, persistent homology (PH), has seen a notable increase in applications in the last decade. Often cited as the most favourable property of PH and the main reason for practical success are the stability theorems that give theoretical results about noise robustness, since real data is typically contaminated with noise or measurement errors. However, little attention has been paid to what these stability theorems mean in practice. To gain some insight into this question, we evaluate the noise robustness of PH on the MNIST dataset of greyscale images. More precisely, we investigate to what extent PH changes under typical forms of image noise, and quantify the loss of performance in classifying the MNIST handwritten digits when noise is added to the data. The results show that the sensitivity to noise of PH is influenced by the choice of filtrations and persistence signatures (respectively the input and output of PH), and in particular, that PH features are often not robust to noise in a classification task.

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“…For example, if

is a point cloud in

, thanks to the Nerve theorem, the filtration

encodes the topology of the whole family of unions of balls

, as r goes from 0 to + ∞ . As the notion of filtration is quite flexible, many other filtrations have been considered in the literature and can be constructed on the top of data, such as the so-called witness complex popularized in tda by De Silva and Carlsson (2004) , the weighted Rips filtrations Buchet et al (2015b) , or the so-called DTM filtrations Anai et al (2019) that allow us to handle data corrupted by noise and outliers.…”

Section: Persistent Homologymentioning

confidence: 99%

An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

Chazal

¹

,

Michel

²

2021

Front. Artif. Intell.

Self Cite

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With the recent explosion in the amount, the variety, and the dimensionality of available data, identifying, extracting, and exploiting their underlying structure has become a problem of fundamental importance for data analysis and statistical learning. Topological data analysis (tda) is a recent and fast-growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. It proposes new well-founded mathematical theories and computational tools that can be used independently or in combination with other data analysis and statistical learning techniques. This article is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of tda for nonexperts.

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“…However, the distance function δ X is extremely sensitive to outliers and noise ("even one outlier is deadly", or, in the language of robust statistics, the distance function has breakdown point zero [19]). To circumvent this issue, [18] propose to rather consider distance-to-a-measure (DTM) δ X,m as the filtration function, which is defined as the average distance from a given number of nearest neighbours in X (and is thus a smooth version of the distance function) [4]. The number of neighbours that are considered is determined by the parameter m, which represents the percentage of the total number of point cloud X points.…”

Section: Filtrationsmentioning

confidence: 99%

“…The greyscale pixel values are clipped to the interval [0, 255] 4. For 0-dimensional homology, we truncate the death value of infinite intervals to the maximum finite death value for the given filtration function, across all transformations 5.…”

mentioning

confidence: 99%

Noise robustness of persistent homology on greyscale images, across filtrations and signatures

Turkeš,

Nys,

Verdonck

et al. 2021

Preprint

0

View full text Add to dashboard Cite

Topological data analysis is a recent and fast growing field that approaches the analysis of datasets using techniques from (algebraic) topology. Its main tool, persistent homology (PH), has seen a notable increase in applications in the last decade. Often cited as the most favourable property of PH and the main reason for practical success are the stability theorems that give theoretical results about noise robustness, since real data is typically contaminated with noise or measurement errors. However, little attention has been paid to what these stability theorems mean in practice. To gain some insight into this question, we evaluate the noise robustness of PH on the MNIST dataset of greyscale images. More precisely, we investigate to what extent PH changes under typical forms of image noise, and quantify the loss of performance in classifying the MNIST handwritten digits when noise is added to the data. The results show that the sensitivity to noise of PH is influenced by the choice of filtrations and persistence signatures (respectively the input and output of PH), and in particular, that PH features are often not robust to noise in a classification task.

show abstract

DTM-Based Filtrations

Cited by 18 publications

References 12 publications

Noise robustness of persistent homology on greyscale images, across filtrations and signatures

Noise robustness of persistent homology on greyscale images, across filtrations and signatures

An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

Noise robustness of persistent homology on greyscale images, across filtrations and signatures

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