A new entropy based summary function for topological data analysis

The application of artificial intelligence (AI) to various medical subfields has been a popular topic of research in recent years. In particular, deep learning has been widely used and has proven effective in many cases. Topological data analysis (TDA)—a rising field at the intersection of mathematics, statistics, and computer science—offers new insights into data. In this work, we develop a novel deep learning architecture that we call TopoResNet that integrates topological information into the residual neural network architecture. To demonstrate TopoResNet, we apply it to a skin lesion classification problem. We find that TopoResNet improves the accuracy and the stability of the training process.

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“…In particular, we use where k = 0, 1. Note that persistent entropy was introduced in [43] and is defined as…”

Section: Persistence Statisticsmentioning

confidence: 99%

“…giving rise to a variant of the entropy summary (curve) E(t). The entropy summary and its stability are discussed in [43,45]. In [30], a general stability result for an entire class of PC is given.…”

Section: Persistence Curvesmentioning

confidence: 99%

TopoResNet: A Hybrid Deep Learning Architecture and Its Application to Skin Lesion Classification

Lawson

Chen

et al. 2021

Mathematics

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“…In this work, we are only concerned with persistent homological features from dimensions 0 and 1, and use the python package RIPSER [8] to compute the barcodes. We derive statistical features from persistence barcodes and other persistence-based summaries such as landscapes [9], and entropy [10]. We summarize these below, where x refers to the average, SD the standard deviation, Sk the skewness, and Kurt the kurtosis for the collection of values.…”

Section: Feature Extractionmentioning

confidence: 99%

A Topology Informed Random Forest Classifier for ECG Classification

Ignacio¹,

Bulauan²,

Manzanares³

2020

2020 Computing in Cardiology Conference (CinC)

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This paper accompanies Team Cordi-Ak's entry to the classification of 12-lead ECGs for the Phys-ioNet/Computing in Cardiology Challenge 2020. Our approach leverages mathematically computable topological signatures of 12-lead ECGs as proxy for features informed by medical expertise to train a two-level random forest model in a multi-class classification task. We view ECGs as multivariate time series data and convert different segments and groupings of leads to point cloud embeddings. This stores both local and global structures of ECGs, and encodes periodic information as attractor cycles in high-dimensional space. We then employ topological data analysis on these embeddings to extract topological features based on different summaries available in the literature. We supplement these features with demographic data and statistical moments of RR intervals based on the Pan-Tompkins algorithm for each lead to train the classifier. Our classifier achieved a challenge validation score of 0.304, and a final score of -0.113 on the full hidden test data, placing us 37th out of 41 officially ranked teams that participated in this year's Challenge.

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“…As shown in Table I, they are samples of persistence statistics used in the article. In addition to above statistics measurements, we also consider so called persistence entropy introduced in [26], which is defined as − (b,d)∈Pi p log(p), for i = 0, 1. Persistence entropy can be viewed as the diversity of the lifespans.…”

Section: B Persistence Statistics and Persistence Curvesmentioning

confidence: 99%

TopoResNet: A hybrid deep learning architecture and its application to skin lesion classification

Chung¹,

Hu²,

Lawson³

et al. 2019

Preprint

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Skin cancer is one of the most common cancers in the United States. As technological advancements are made, algorithmic diagnosis of skin lesions is becoming more important. In this paper, we develop algorithms for segmenting the actual diseased area of skin in a given image of a skin lesion, and for classifying different types of skin lesions pictured in a given image. The cores of the algorithms used were based in persistent homology, an algebraic topology technique that is part of the rising field of Topological Data Analysis (TDA). The segmentation algorithm utilizes a similar concept to persistent homology that captures the robustness of segmented regions. For classification, we design two families of topological features from persistence diagrams-which we refer to as persistence statistics (PS) and persistence curves (PC), and use linear support vector machine as classifiers. We also combined those topological features, PS and PC, into ResNet-101 model, which we call TopoResNet-101, the results show that PS and PC are effective in two folds-improving classification performances and stabilizing the training process. Although convolutional features are the most important learning targets in CNN models, global information of images may be lost in the training process. Because topological features were extracted globally, our results show that the global property of topological features provide additional information to machine learning models.

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A new entropy based summary function for topological data analysis

Cited by 17 publications

References 22 publications

TopoResNet: A Hybrid Deep Learning Architecture and Its Application to Skin Lesion Classification

TopoResNet: A Hybrid Deep Learning Architecture and Its Application to Skin Lesion Classification

A Topology Informed Random Forest Classifier for ECG Classification

TopoResNet: A hybrid deep learning architecture and its application to skin lesion classification

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