Persistence codebooks for topological data analysis

Zieliński, Bartosz; Lipiński, Michał; Juda, Mateusz; Zeppelzauer, Matthias; Dłotko, Paweł

doi:10.1007/s10462-020-09897-4

Cited by 14 publications

(13 citation statements)

References 45 publications

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“…In practice, such large change may not be significant if finer filtration intervals are chosen but to remedy the instability in the Betti sequence we propose a new stabilized version and prove its stability. In the stabilization, we adopt a similar Gaussian-smoothing approach as in [10,17]. In this paper, we show numerical examples that support our statement and show the validation of the proposed stabilization of the Betti sequence.…”

Section: Introductionsupporting

confidence: 66%

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Instability of the Betti Sequence for Persistent Homology and a Stabilized Version of the Betti Sequence

Johnson¹,

Jung²

2021

Preprint

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Topological Data Analysis (TDA), a relatively new field of data analysis, has proved very useful in a variety of applications. The main persistence tool from TDA is persistent homology in which data structure is examined at many scales. Representations of persistent homology include persistence barcodes and persistence diagrams, both of which are not straightforward to reconcile with traditional machine learning algorithms as they are sets of intervals or multisets. The problem of faithfully representing barcodes and persistent diagrams has been pursued along two main avenues: kernel methods and vectorizations. One vectorization is the Betti sequence, or Betti curve, derived from the persistence barcode. While the Betti sequence has been used in classification problems in various applications, to our knowledge, the stability of the sequence has never before been discussed. In this paper we show that the Betti sequence is unstable under the 1-Wasserstein metric with regards to small perturbations in the barcode from which it is calculated. In addition, we propose a novel stabilized version of the Betti sequence based on the Gaussian smoothing seen in the Stable Persistence Bag of Words for persistent homology. We then introduce the normalized cumulative Betti sequence and provide numerical examples that support the main statement of the paper.2000 Mathematics Subject Classification. 54-08.

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Section: Introductionsupporting

confidence: 66%

“…We now propose a stabilized version of the Betti sequence inspired by the Gaussian smoothing techniques seen in [10,17] and prove its stability with respect to the 1-Wasserstein distance. Definition 3.1.…”

Section: Stablized Betti Sequencementioning

confidence: 99%

Instability of the Betti Sequence for Persistent Homology and a Stabilized Version of the Betti Sequence

Johnson¹,

Jung²

2021

Preprint

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“…As a future direction, we will investigate more sophisticated methods for using ToFU in data augmentation, for example by using ToFU-VAE as a generative model to create training data with desired topological characteristics. Additionally, it will be beneficial to compare ToFU to a larger collection of TDA methods, for example sliced Wasserstein kernels [10] or persistence codebooks [49], on a larger repository of datasets.…”

Section: Variational Autoencodermentioning

confidence: 99%

“…Since the space of PDs lack a Hilbert space structure, they are not directly amenable to commonlyused statistical learning methods. A large body of work sought to remedy this shortcoming by inventing well-behaved Hilbert space representations of PDs [7,1,12,5,4,49]. Other works, notably [46] and [36], derive PD representations that serve as sufficient statistics, thereby ensuring that PD summaries retain all statistically-pertinent information for an inference task.…”

mentioning

confidence: 99%

ToFU: Topology functional units for deep learning

Oballe¹,

Boothe²,

Franaszczuk³

et al. 2022

FoDS

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<p style='text-indent:20px;'>We propose ToFU, a new trainable neural network unit with a persistence diagram dissimilarity function as its activation. Since persistence diagrams are topological summaries of structures, this new activation measures and learns the topology of data to leverage it in machine learning tasks. We showcase the utility of ToFU in two experiments: one involving the classification of discrete-time autoregressive signals, and another involving a variational autoencoder. In the former, ToFU yields competitive results with networks that use spectral features while outperforming CNN architectures. In the latter, ToFU produces topologically-interpretable latent space representations of inputs without sacrificing reconstruction fidelity.</p>

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“…For example, persistence landscapes (Bubenik, 2015) map persistence diagrams into a Banach space, specifically L p space. More examples include persistence image (Adams et al, 2017), generalized persistence landscapes (Berry et al, 2020), persistence path (Chevyrev et al, 2018), persistence codebook (Zelinski et al, 2020), persistence curves (Chung and Lawson, 2019), kernel based methods (Reininghaus et al, 2015;Kusano et al, 2016), and persistent entropy (Chintakunta et al, 2015;Atienza et al, 2019b). These methods have been studied and applied to different applications.…”

Section: Data Analysis With Persistence Diagram and Commonly Considermentioning

confidence: 99%

A Persistent Homology Approach to Heart Rate Variability Analysis With an Application to Sleep-Wake Classification

Chung

et al. 2021

Front. Physiol.

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Persistent homology is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general pipeline to apply persistent homology to study time series, particularly the instantaneous heart rate time series for the heart rate variability (HRV) analysis. The first step is capturing the shapes of time series from two different aspects—the persistent homologies and hence persistence diagrams of its sub-level set and Taken's lag map. Second, we propose a systematic and computationally efficient approach to summarize persistence diagrams, which we coined persistence statistics. To demonstrate our proposed method, we apply these tools to the HRV analysis and the sleep-wake, REM-NREM (rapid eyeball movement and non rapid eyeball movement) and sleep-REM-NREM classification problems. The proposed algorithm is evaluated on three different datasets via the cross-database validation scheme. The performance of our approach is better than the state-of-the-art algorithms, and the result is consistent throughout different datasets.

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Persistence codebooks for topological data analysis

Cited by 14 publications

References 45 publications

Instability of the Betti Sequence for Persistent Homology and a Stabilized Version of the Betti Sequence

Instability of the Betti Sequence for Persistent Homology and a Stabilized Version of the Betti Sequence

ToFU: Topology functional units for deep learning

A Persistent Homology Approach to Heart Rate Variability Analysis With an Application to Sleep-Wake Classification

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