Testing the manifold hypothesis

Fefferman, Charles; Mitter, Sanjoy K.; Narayanan, Hariharan

doi:10.1090/jams/852

Cited by 285 publications

(203 citation statements)

References 44 publications

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“…Graph Laplacian based methods were proposed in [5,29] and intrinsic dimension estimation was revisited in [35,28]. In a manifold learning context the questions of finite sample efficiency were treated in [26,25,31,1] and a statistical test was proposed [23]. A spectral support estimator was described in [54].…”

Section: Inference Of Low Dimensional Structuresmentioning

confidence: 99%

Data Analysis from Empirical Moments and the Christoffel Function

2020

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Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and approximation theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and approximation theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical moment matrix approach.

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Section: Inference Of Low Dimensional Structuresmentioning

confidence: 99%

Data Analysis from Empirical Moments and the Christoffel Function

2020

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“…If cells in the tissue show some meaningful tendency in terms of markers, they are expected to lie on a low-dimensional surface (manifold) embedded in ℝ m . This idea is known as the manifold hypothesis (34).…”

Section: Cytometry Datamentioning

confidence: 99%

LAVENDER: latent axes discovery from multiple cytometry samples with non-parametric divergence estimation and multidimensional scaling reconstruction

Nakamura

Okada

Setoh

et al. 2019

Preprint

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Computational cytometry methods are now frequently used in flow and mass cytometric data analyses. However, systematic bias-free methodologies to assess inter-sample variability have been lacking, thereby hampering efficient data mining from a large set of samples. Here, we devised a computational method termed LAVENDER (latent axes discovery from multiple cytometry samples with nonparametric divergence estimation and multidimensional scaling reconstruction). It measures the Jensen-Shannon distances between samples using the k-nearest neighbor density estimation and reconstructs samples in a new coordinate space, called the LAVENDER space. The axes of this space can then be compared against other omics measurements to obtain biological information.Application of LAVENDER to multidimensional flow cytometry datasets of 301 Japanese individuals immunized with a seasonal influenza vaccine revealed an axis related to baseline immunological characteristics of each individual. This axis correlated with the proportion of plasma cells and the neutrophil-to-lymphocyte ratio, a clinical marker of the systemic inflammatory response. The same method was also applicable to mass cytometry data with more molecular markers. These results demonstrate that LAVENDER is a useful tool for identifying critical heterogeneity among similar, yet different, single-cell datasets.

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“…The first steps of our method (steps 2 to 9) are based on manifold alignment (Wang and Mahadevan, 2009) and manifold warping (Vu et al, 2012), which based on the manifold hypothesis, describes how the original high-dimensional dataset actually lies on a lower dimensional manifold, which is embedded in the original high-dimensional space (Fefferman et al, 2016). In ManiNetCluster, we project the two networks into a common manifold which preserves the local similarity within each network and which minimizes the distance between two different networks.…”

Section: Manifold Alignment/warpingmentioning

confidence: 99%

ManiNetCluster: A Manifold Learning Approach to Reveal the Functional Linkages Across Multiple Gene Networks

Wang

2018

Preprint

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The coordination of genome encoded function is a critical and complex process in biological systems, especially across phenotypes or states (e.g., time, disease, organism). Understanding how the complexity of genome-encoded function relates to these states remains a challenge. To address this, we have developed a novel computational method based on manifold learning and comparative analysis, ManiNetCluster, which simultaneously aligns and clusters multiple molecular networks to systematically reveal function links across multiple datasets. Specifically, ManiNetCluster employs manifold learning to match local and non-linear structures among the networks of different states, to identify cross-network linkages. By applying ManiNetCluster to the developmental gene expression datasets across model organisms (e.g., worm, fruit fly), we found that our tool significantly better aligns the orthologous genes than existing state-of-the-art methods, indicating the non-linear interactions between evolutionary functions in development. Moreover, we applied ManiNetCluster to a series of transcriptomes measured in the green alga Chlamydomonas reinhardtii, to determine the function links between various metabolic processes between the light and dark periods of a diurnally cycling culture. For example, we identify a number of genes putatively regulating processes across each lighting regime, and how comparative analyses between ManiNetCluster and other clustering tools can provide additional insights. ManiNetCluster is available as an R package together with a tutorial at https://github.com/namtk/ManiNetCluster.

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Testing the manifold hypothesis

Cited by 285 publications

References 44 publications

Data Analysis from Empirical Moments and the Christoffel Function

Data Analysis from Empirical Moments and the Christoffel Function

LAVENDER: latent axes discovery from multiple cytometry samples with non-parametric divergence estimation and multidimensional scaling reconstruction

ManiNetCluster: A Manifold Learning Approach to Reveal the Functional Linkages Across Multiple Gene Networks

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