The high-dimensional data created by high-throughput technologies require visualization tools that reveal data structure and patterns in an intuitive form. We present PHATE, a visualization method ...that captures both local and global nonlinear structure using an information-geometric distance between data points. We compare PHATE to other tools on a variety of artificial and biological datasets, and find that it consistently preserves a range of patterns in data, including continual progressions, branches and clusters, better than other tools. We define a manifold preservation metric, which we call denoised embedding manifold preservation (DEMaP), and show that PHATE produces lower-dimensional embeddings that are quantitatively better denoised as compared to existing visualization methods. An analysis of a newly generated single-cell RNA sequencing dataset on human germ-layer differentiation demonstrates how PHATE reveals unique biological insight into the main developmental branches, including identification of three previously undescribed subpopulations. We also show that PHATE is applicable to a wide variety of data types, including mass cytometry, single-cell RNA sequencing, Hi-C and gut microbiome data.
Due to commonalities in pathophysiology, age-related macular degeneration (AMD) represents a uniquely accessible model to investigate therapies for neurodegenerative diseases, leading us to examine ...whether pathways of disease progression are shared across neurodegenerative conditions. Here we use single-nucleus RNA sequencing to profile lesions from 11 postmortem human retinas with age-related macular degeneration and 6 control retinas with no history of retinal disease. We create a machine-learning pipeline based on recent advances in data geometry and topology and identify activated glial populations enriched in the early phase of disease. Examining single-cell data from Alzheimer's disease and progressive multiple sclerosis with our pipeline, we find a similar glial activation profile enriched in the early phase of these neurodegenerative diseases. In late-stage age-related macular degeneration, we identify a microglia-to-astrocyte signaling axis mediated by interleukin-1β which drives angiogenesis characteristic of disease pathogenesis. We validated this mechanism using in vitro and in vivo assays in mouse, identifying a possible new therapeutic target for AMD and possibly other neurodegenerative conditions. Thus, due to shared glial states, the retina provides a potential system for investigating therapeutic approaches in neurodegenerative diseases.
Diffusion maps for changing data Coifman, Ronald R.; Hirn, Matthew J.
Applied and computational harmonic analysis,
January 2014, 2014-01-00, 20140101, Volume:
36, Issue:
1
Journal Article
Peer reviewed
Open access
Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph ...associated with it changes depending on some set of parameters. We analyze this type of data in terms of the diffusion distance and the corresponding diffusion map. As the data changes over the parameter space, the low dimensional embedding changes as well. We give a way to go between these embeddings, and furthermore, map them all into a common space, allowing one to track the evolution of X in its intrinsic geometry. A global diffusion distance is also defined, which gives a measure of the global behavior of the data over the parameter space. Approximation theorems in terms of randomly sampled data are presented, as are potential applications.
In this short letter we present the construction of a bi-stochastic kernel p for an arbitrary data set X that is derived from an asymmetric affinity function α. The affinity function α measures the ...similarity between points in X and some reference set Y. Unlike other methods that construct bi-stochastic kernels via some convergent iteration process or through solving an optimization problem, the construction presented here is quite simple. Furthermore, it can be viewed through the lens of out of sample extensions, making it useful for massive data sets.
This work examines methods for predicting the partition coefficient (log
P
) for a dataset of small molecules. Here, we use atomic attributes such as radius and partial charge, which are typically ...used as force field parameters in classical molecular dynamics simulations. These atomic attributes are transformed into index-invariant molecular features using a recently developed method called Geometric Scattering for Graphs (GSG). We call this approach “ClassicalGSG” and examine its performance under a broad range of conditions and hyperparameters. We train ClassicalGSG log
P
predictors with neural networks using 10,722 molecules from the OpenChem dataset and apply them to predict the log
P
values from four independent test sets. The ClassicalGSG method’s performance is compared to a baseline model that employs graph convolutional networks (GCNs). Our results show that the best prediction accuracies are obtained using atomic attributes generated with the CHARMM generalized Force Field (CGenFF) and 2D molecular structures.
The ClassicalGSG method utilizes neural networks (NNs) to predict the log
P
values from molecular features generated using a method called Geometric Scattering for Graphs (GSG). The GSG method creates molecular features from the graph representation of the molecule, where each atom has a set of atomic attributes. These atomic attributes are typically used as force field parameters in classical molecular dynamics simulations and include partial charges, Lennard-Jones well depth, Lennard-Jones radius, and atomic type.
Time coupled diffusion maps Marshall, Nicholas F.; Hirn, Matthew J.
Applied and computational harmonic analysis,
November 2018, 2018-11-00, Volume:
45, Issue:
3
Journal Article
Peer reviewed
We consider a collection of n points in Rd measured at m times, which are encoded in an n×d×m data tensor. Our objective is to define a single embedding of the n points into Euclidean space which ...summarizes the geometry as described by the data tensor. In the case of a fixed data set, diffusion maps and related graph Laplacian methods define such an embedding via the eigenfunctions of a diffusion operator constructed on the data. Given a sequence of m measurements of n points, we introduce the notion of time coupled diffusion maps which have natural geometric and probabilistic interpretations. To frame our method in the context of manifold learning, we model evolving data as samples from an underlying manifold with a time-dependent metric, and we describe a connection of our method to the heat equation on such a manifold.
We consider the following interpolation problem. Suppose one is given a finite set $E \subset \mathbb R^d$, a function $f \colon E \to \mathbb R$, and possibly the gradients of $f$ at the points of ...$E$. We want to interpolate the given information with a function $F \in C^{1,1}(\mathbb R^d)$ with the minimum possible value of Lip$(\nabla F)$. We present practical, efficient algorithms for constructing an $F$ such that Lip$(\nabla F)$ is minimal, or for less computational effort, within a small dimensionless constant of being minimal.
The refinability of step functions Hirn, Matthew J.
Proceedings of the American Mathematical Society,
03/2008, Volume:
136, Issue:
3
Journal Article
Peer reviewed
Refinable functions have been widely investigated because of their importance in wavelet theory and multiresolution analysis, as well as because of intrinsic interest. Problems involving refinability ...can be challenging and interesting problems in mathematics. Several papers have investigated refinability of splines and other classes of functions. The purpose of this paper is to develop necessary and sufficient conditions for the refinability of the class of step functions on the real line taking complex values.
As the biomedical community produces datasets that are increasingly complex and high dimensional, there is a need for more sophisticated computational tools to extract biological insights. We present ...Multiscale PHATE, a method that sweeps through all levels of data granularity to learn abstracted biological features directly predictive of disease outcome. Built on a coarse-graining process called diffusion condensation, Multiscale PHATE learns a data topology that can be analyzed at coarse resolutions for high-level summarizations of data and at fine resolutions for detailed representations of subsets. We apply Multiscale PHATE to a coronavirus disease 2019 (COVID-19) dataset with 54 million cells from 168 hospitalized patients and find that patients who die show CD16
CD66b
neutrophil and IFN-γ
granzyme B
Th17 cell responses. We also show that population groupings from Multiscale PHATE directly fed into a classifier predict disease outcome more accurately than naive featurizations of the data. Multiscale PHATE is broadly generalizable to different data types, including flow cytometry, single-cell RNA sequencing (scRNA-seq), single-cell sequencing assay for transposase-accessible chromatin (scATAC-seq), and clinical variables.
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