Diffusion-based kernel methods on Euclidean metric measure spaces

Bermanis, Amit; Wolf, Guy; Averbuch, Amir

doi:10.1016/j.acha.2015.07.005

Cited by 10 publications

(15 citation statements)

References 28 publications

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“…While several similarity kernels are used in practice to construct the diffusion operator P, a standard choice is the Gaussian affinity k ε (x, y) = exp(− x − y 2 /ε) [13,[82][83][84], in which case we denote the diffusion operator P ε , where ε determines the neighborhood radius. This kernel choice is often seen in theoretical and mathematical work due to its established properties on data sampled from locally low dimensional geometries (i.e., data manifolds) [82,85,86].…”

Section: Diffusion Information Geometry For Visualization and Condensmentioning

confidence: 99%

Multiscale PHATE Exploration of SARS-CoV-2 Data Reveals Multimodal Signatures of Disease

Kuchroo

Huang

Wong

et al. 2020

Preprint

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1SummaryThe biomedical community is producing increasingly high dimensional datasets, integrated from hundreds of patient samples, which current computational techniques struggle to explore. To uncover biological meaning from these complex datasets, we present an approach called Multiscale PHATE, which learns abstracted biological features from data that can be directly predictive of disease. Built on a continuous coarse graining process called diffusion condensation, Multiscale PHATE creates a tree of data granularities that can be cut at coarse levels for high level summarizations of data, as well as at fine levels for detailed representations on subsets. We apply Multiscale PHATE to study the immune response to COVID-19 in 54 million cells from 168 hospitalized patients. Through our analysis of patient samples, we identify CD16hi CD66blo neutrophil and IFNγ+GranzymeB+ Th17 cell responses enriched in patients who die. Further, we show that population groupings Multiscale PHATE discovers can be directly fed into a classifier to predict disease outcome. We also use Multiscale PHATE-derived features to construct two different manifolds of patients, one from abstracted flow cytometry features and another directly on patient clinical features, both associating immune subsets and clinical markers with outcome.

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Section: Diffusion Information Geometry For Visualization and Condensmentioning

confidence: 99%

Multiscale PHATE Exploration of SARS-CoV-2 Data Reveals Multimodal Signatures of Disease

Kuchroo

Huang

Wong

et al. 2020

Preprint

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“…For simplicity, the integration notation ∫ ⋅ dy in this paper will refer to the Lebesgue integral ∫ M ⋅ dy over the manifold, instead of the whole space R n . Further, while (for simplicity) such integrals are written without a specific measure one can equivalently, w.l.o.g., replace dx with an appropriate measure representing data sampling distribution over M. Let g(x, y) ≜ exp −∥x − y∥ 2 /ε , x, y ∈ M, ε > 0, define the Gaussian kernel Gf (x) = ∫ g(x, y)f (y)dy used in [3] to capture local neighborhoods from data sampled from M. Following [3] and related work, we define the Gaussian degree q(x) = ∥g(x, ⋅)∥ 1 = ∫ g(x, y)dy and assume it provides a suitable approximation of the distribution (or local density) of data over the manifold M. Finally, given a measure µ over the manifold, an MGC kernel [1], [2] is defined as k µ (x, y) = ∫ g(x, r)g(y, r)dµ(r). Note that while we use a Gaussian kernel for the remainder of this work, the definitions and theorems to follow do not depend on the choice of g, so long as it is a kernel function.…”

Section: A Preliminariesmentioning

confidence: 99%

“…Note that while we use a Gaussian kernel for the remainder of this work, the definitions and theorems to follow do not depend on the choice of g, so long as it is a kernel function. The original MGC construction in [2] considered measures that represent data distribution, and used the constructed MGC kernel to define diffusion maps (see Sec. II-B and [3]) with them.…”

Section: A Preliminariesmentioning

confidence: 99%

Compressed Diffusion

Gigante

Stanley

et al. 2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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Diffusion maps are a commonly used kernel-based method for manifold learning, which can reveal intrinsic structures in data and embed them in low dimensions. However, as with most kernel methods, its implementation requires a heavy computational load, reaching up to cubic complexity in the number of data points. This limits its usability in modern data analysis. Here, we present a new approach to computing the diffusion geometry, and related embeddings, from a compressed diffusion process between data regions rather than data points. Our construction is based on an adaptation of the previously proposed measure-based Gaussian correlation (MGC) kernel that robustly captures the local geometry around data points. We use this MGC kernel to efficiently compress diffusion relations from pointwise to data region resolution. Finally, a spectral embedding of the data regions provides coordinates that are used to interpolate and approximate the pointwise diffusion map embedding of data. We analyze theoretical connections between our construction and the original diffusion geometry of diffusion maps, and demonstrate the utility of our method in analyzing big datasets, where it outperforms competing approaches. † These authors contributed equally. † † These authors contributed equally. § Corresponding

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“…The measure-based Gaussian Correlation (MGC) kernel [4,3] defines the affinities between elements in X, which in this context, it is referred to as the analyzed domain via their relations with the reference dataset M that is referred to as the measure domain. This framework enables to have a flexible representation of X, as long as M , which in some sense characterizes the data, is sufficiently large.…”

Section: Measure-based Gaussian Correlation Kernelmentioning

confidence: 99%

“…In this paper, we focus on deriving a representation that preserves the diffusion distances between multidimensional data points based on the MGC framework [4,3]. This representation is applicable to process efficiently very large datasets by imposing a Markovian diffusion process to define and represent the non-linear relations between multidimensional data points.…”

Section: Introductionmentioning

confidence: 99%

Diffusion representations

Salhov

Bermanis

Wolf

et al. 2018

Applied and Computational Harmonic Analysis

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Diffusion Maps framework is a kernel based method for manifold learning and data analysis that defines diffusion similarities by imposing a Markovian process on the given dataset. Analysis by this process uncovers the intrinsic geometric structures in the data. Recently, it was suggested to replace the standard kernel by a measure-based kernel that incorporates information about the density of the data. Thus, the manifold assumption is replaced by a more general measurebased assumption.The measure-based diffusion kernel incorporates two separate independent representations. The first determines a measure that correlates with a density that represents normal behaviors and patterns in the data. The second consists of the analyzed multidimensional data points.In this paper, we present a representation framework for data analysis of datasets that is based on a closed-form decomposition of the measure-based kernel. The proposed representation preserves pairwise diffusion distances that does not depend on the data size while being invariant to scale. For a stationary data, no out-of-sample extension is needed for embedding newly arrived data points in the representation space. Several aspects of the presented methodology are demonstrated on analytically generated data.

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Diffusion-based kernel methods on Euclidean metric measure spaces

Cited by 10 publications

References 28 publications

Multiscale PHATE Exploration of SARS-CoV-2 Data Reveals Multimodal Signatures of Disease

Multiscale PHATE Exploration of SARS-CoV-2 Data Reveals Multimodal Signatures of Disease

Compressed Diffusion

Diffusion representations

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