A kernel for multi-parameter persistent homology

Highly resolved spatial data of complex systems encode rich and nonlinear information. Quantification of heterogeneous and noisy data—often with outliers, artifacts, and mislabeled points—such as those from tissues, remains a challenge. The mathematical field that extracts information from the shape of data, topological data analysis (TDA), has expanded its capability for analyzing real-world datasets in recent years by extending theory, statistics, and computation. An extension to the standard theory to handle heterogeneous data is multiparameter persistent homology (MPH). Here we provide an application of MPH landscapes, a statistical tool with theoretical underpinnings. MPH landscapes, computed for (noisy) data from agent-based model simulations of immune cells infiltrating into a spheroid, are shown to surpass existing spatial statistics and one-parameter persistent homology. We then apply MPH landscapes to study immune cell location in digital histology images from head and neck cancer. We quantify intratumoral immune cells and find that infiltrating regulatory T cells have more prominent voids in their spatial patterns than macrophages. Finally, we consider how TDA can integrate and interrogate data of different types and scales, e.g., immune cell locations and regions with differing levels of oxygenation. This work highlights the power of MPH landscapes for quantifying, characterizing, and comparing features within the tumor microenvironment in synthetic and real datasets.

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“…Computational feasability and interpretability distinguish MPH landscapes from other vectorization methods for MPH (compare refs. 46 , 47 ).…”

Section: Resultsmentioning

confidence: 99%

Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors

Vipond

Bull

Macklin

et al. 2021

Proc. Natl. Acad. Sci. U.S.A.

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“…Indicating the two parameters as two elements of b ∈ R 2 , the sublevelsets satisfy the property F bi ⊆ F bj if b i ≤ b j when ≤ denotes elementwise inequality and hence the ordering is partial. Several representations can be obtained for multiparameter persistent homology [38] including vector-space representations such as images [17], landscapes [39], and kernels [40]. In addition to using single-parameter persistence images with atomic distance and atomic charge functions separately, we also use the 2-parameter variant that use the two functions together to augment molecular deep generative models.…”

Section: Tda Augmentation Of Smiles Vaementioning

confidence: 99%

Augmenting Molecular Deep Generative Models with Topological Data Analysis Representations

Schiff¹,

Chenthamarakshan²,

Hoffman³

et al. 2021

Preprint

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Deep generative models have emerged as a powerful tool for learning informative molecular representations and designing novel molecules with desired properties, with applications in drug discovery and material design. Deep generative auto-encoders defined over molecular SMILES strings have been a popular choice for that purpose. However, capturing salient molecular properties like quantumchemical energies remains challenging and requires sophisticated neural net models of molecular graphs or geometry-based information. As a simpler and more efficient alternative, we present a SMILES Variational Auto-Encoder (VAE) augmented with topological data analysis (TDA) representations of molecules, known as persistence images. Our experiments show that this TDA augmentation enables a SMILES VAE to capture the complex relation between 3D geometry and electronic properties, and allows generation of novel, diverse, and valid molecules with geometric features consistent with the training data, which exhibit a varying range of global electronic structural properties, such as a small HOMO-LUMO gap -a critical property for designing organic solar cells. We demonstrate that our TDA augmentation yields better success in downstream tasks compared to models trained without these representations and can assist in targeted molecule discovery.

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“…As these approaches do not depend on the internal linear maps in the persistence modules, they are rather coarse. Other linearizations have been proposed that do depended on the internal maps: Vipond's multiparameter persistence landscapes [77] and Carriére and Blumberg's multiparameter persistence images [26] consider the internal maps of the module only along a fixed direction, whereas the kernel construction of Corbet et al [40] considers the internal maps in all directions. Stability results have been given for each of the last three approaches, though these have some key limitations.…”

Section: Other Approaches To Defining ℓ P -Distances On Multiparamete...mentioning

confidence: 99%

“…On the other hand, [74,Section 7.2] observes that for all p ∈ [1, ∞), 1-parameter persistence landscapes are not Hölder continuous with respect to the Wasserstein distance W p on barcodes and the L p -distance on landscapes. A stability bound for the kernel construction of [40] is given with respect to the matching distance on modules and the L 2 -distance, but the bound involves a constant which depends on the size of the input and can be quite large. While the multiparameter persistence images of [26] are unstable in general, the authors give a partial stability result under special conditions [26, Supplementary Material].…”

Section: Other Approaches To Defining ℓ P -Distances On Multiparamete...mentioning

confidence: 99%

$\ell^p$-Distances on Multiparameter Persistence Modules

Bjerkevik¹,

Lesnick²

2021

Preprint

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Motivated both by theoretical and practical considerations in topological data analysis, we generalize the p-Wasserstein distance on barcodes to multiparameter persistence modules. For each p ∈ [1, ∞], we in fact introduce two such generalizations d p I and d p M , such that d ∞ I equals the interleaving distance and d ∞ M equals the matching distance. We show that d p M ≤ d p I for all p ∈ [1, ∞], extending an observation of Landi in the p = ∞ case. We observe that the distances d p M can be efficiently approximated. Finally, we show that on 1-or 2-parameter persistence modules over prime fields, d p I is the universal (i.e., largest) metric satisfying a natural stability property; our result extends a stability result of Skraba and Turner for the p-Wasserstein distance on barcodes in the 1-parameter case, and is also a close analogue of a universality property for the interleaving distance given by the second author. In a companion paper, we apply some of these results to study the stability of (2-parameter) multicover persistent homology.

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A kernel for multi-parameter persistent homology

Cited by 18 publications

References 37 publications

Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors

Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors

Augmenting Molecular Deep Generative Models with Topological Data Analysis Representations

$\ell^p$-Distances on Multiparameter Persistence Modules

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