Matthew J. Hirn

Diffusion maps for changing data

Abstract 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.

A general theorem of existence of quasi absolutely minimal Lipschitz extensions

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier “quasi” to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.

Wavelet Scattering Regression of Quantum Chemical Energies

We introduce multiscale invariant dictionaries to estimate quantum chemical energies of organic molecules, from training databases. Molecular energies are invariant to isometric atomic displacements, and are Lipschitz continuous to molecular deformations. Similarly to density functional theory (DFT), the molecule is represented by an electronic density function. A multiscale invariant dictionary is calculated with wavelet scattering invariants. It cascades a first wavelet transform which separates scales, with a second wavelet transform which computes interactions across scales. Sparse scattering regressions give state of the art results over two databases of organic planar molecules. On these databases, the regression error is of the order of the error produced by DFT codes, but at a fraction of the computational cost.

Wavelet packets for multi- and hyper-spectral imagery

State of the art dimension reduction and classification schemes in multi- and hyper-spectral imaging rely primarily on the information contained in the spectral component. To better capture the joint spatial and spectral data distribution we combine the Wavelet Packet Transform with the linear dimension reduction method of Principal Component Analysis. Each spectral band is decomposed by means of the Wavelet Packet Transform and we consider a joint entropy across all the spectral bands as a tool to exploit the spatial information. Dimension reduction is then applied to the Wavelet Packets coefficients. We present examples of this technique for hyper-spectral satellite imaging. We also investigate the role of various shrinkage techniques to model non-linearity in our approach.

Frame based kernel methods for automatic classification in hyperspectral data

We propose a new kernel and frame based dimension reducing algorithm by exploiting the synergy between endmembers and kernel based classification schemes. This synergy becomes available by means of utilizing the notions of frames and frame representations.

PHATE: A Dimensionality Reduction Method for Visualizing Trajectory Structures in High-Dimensional Biological Data

With the advent of high-throughput technologies measuring high-dimensional biological data, there is a pressing need for visualization tools that reveal the structure and emergent patterns of data in an intuitive form. We present PHATE, a visualization method that captures both local and global nonlinear structure in data by an information-geometry distance between datapoints. We perform extensive comparison between PHATE and 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. We show that PHATE is applicable to a wide variety of datatypes including mass cytometry, single-cell RNA-sequencing, Hi-C, and gut microbiome data, where it can generate interpretable insights into the underlying systems. Finally, we use PHATE to explore a newly generated scRNA-seq dataset of human germ layer differentiation. Here, PHATE reveals a dynamic picture of the main developmental branches in unparalleled detail.In recent years, dimensionality reduction methods have become critical for visualization, exploration, and interpretation of high-throughput, high-dimensional biological data, as they enable the extraction of major trends in the data while discarding noise. However, biological data contains a type of predominant structure that is not preserved in commonly used methods such as PCA and tSNE, namely, branching progression structure. This structure, which is often non-linear, arises from underlying biological processes such as differentiation, graded responses to stimuli, and population drift, which generate cellular (or population) diversity. We propose a novel, affinity-preserving embedding called PHATE (Potential of Heat-diffusion for Affinity-based Trajectory Embedding), designed explicitly to preserve progression structure in data. PHATE provides a denoised, two or three-dimensional visualization of the complete branching trajectory structure in high-dimensional data. It uses heat-diffusion processes, which naturally denoise the data, to compute cell-cell affinities. Then, PHATE creates a diffusion-potential geometry by free-energy potentials of these processes. This geometry captures high-dimensional trajectory structures, while enabling a natural embedding of the intrinsic data geometry. This embedding accurately visualizes trajectories and data distances, without requiring strict assumptions typically used by path-finding and tree-fitting algorithms, which have recently been used for pseudotime orderings or tree-renderings of cellular data. Furthermore, PHATE supports a wide range of data exploration tasks by providing interpretable overlays on top of the visualization. We show that such overlays can emphasize and reveal trajectory end-points, branch points and associated split-decisions, progression-forming variables (e.g., specific genes), and paths between developmental events in cellular state-space. We demonstrate PHATE on single-cell RNA sequencing and mass cytometry data pertaining to embryoid body differentiation, IPSC reprogramming, and hematopoiesis in the bone marrow. We also demonstrate PHATE on non-single cell data including single-nucleotide polymorphism (SNP) measurements of European populations, and 16s sequencing of gut microbiota.Abstract In the era of ‘Big Data’ there is a pressing need for tools that provide human interpretable visualizations of emergent patterns in high-throughput high-dimensional data. Further, to enable insightful data exploration, such visualizations should faithfully capture and emphasize emergent structures and patterns without enforcing prior assumptions on the shape or form of the data. In this paper, we present PHATE (Potential of Heat-diffusion for Affinity-based Transition Embedding) - an unsupervised low-dimensional embedding for visualization of data that is aimed at solving these issues. Unlike previous methods that are commonly used for visualization, such as PCA and tSNE, PHATE is able to capture and highlight both local and global structure in the data. In particular, in addition to clustering patterns, PHATE also uncovers and emphasizes progression and transitions (when they exist) in the data, which are often missed in other visualization-capable methods. Such patterns are especially important in biological data that contain, for example, single-cell phenotypes at different phases of differentiation, patients at different stages of disease progression, and gut microbial compositions that vary gradually between individuals, even of the same enterotype. The embedding provided by PHATE is based on a novel informational distance that captures long-range nonlinear relations in the data by computing energy potentials of data-adaptive diffusion processes. We demonstrate the effectiveness of the produced visualization in revealing insights on a wide variety of biomedical data, including single-cell RNA-sequencing, mass cytometry, gut microbiome sequencing, human SNP data, Hi-C data, as well as non-biomedical data, such as facebook network and facial image data. In order to validate the capability of PHATE to enable exploratory analysis, we generate a new dataset of 31,000 single-cells from a human embryoid body differentiation system. Here, PHATE provides a comprehensive picture of the differentiation process, while visualizing major and minor branching trajectories in the data. We validate that all known cell types are recapitulated in the PHATE embedding in proper organization. Furthermore, the global picture of the system offered by PHATE allows us to connect parts of the developmental progression and characterize novel regulators associated with developmental lineages.

Bi-stochastic kernels via asymmetric affinity functions

Abstract 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.

Visualizing Transitions and Structure for High Dimensional Data Exploration

With the advent of high-throughput technologies measuring high-dimensional biological data, there is a pressing need for visualization tools that reveal the structure and emergent patterns of data in an intuitive form. We present PHATE, a visualization method that captures both local and global nonlinear structure in data by an information-geometry distance between datapoints. We perform extensive comparison between PHATE and 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. We show that PHATE is applicable to a wide variety of datatypes including mass cytometry, single-cell RNA-sequencing, Hi-C, and gut microbiome data, where it can generate interpretable insights into the underlying systems. Finally, we use PHATE to explore a newly generated scRNA-seq dataset of human germ layer differentiation. Here, PHATE reveals a dynamic picture of the main developmental branches in unparalleled detail.In recent years, dimensionality reduction methods have become critical for visualization, exploration, and interpretation of high-throughput, high-dimensional biological data, as they enable the extraction of major trends in the data while discarding noise. However, biological data contains a type of predominant structure that is not preserved in commonly used methods such as PCA and tSNE, namely, branching progression structure. This structure, which is often non-linear, arises from underlying biological processes such as differentiation, graded responses to stimuli, and population drift, which generate cellular (or population) diversity. We propose a novel, affinity-preserving embedding called PHATE (Potential of Heat-diffusion for Affinity-based Trajectory Embedding), designed explicitly to preserve progression structure in data. PHATE provides a denoised, two or three-dimensional visualization of the complete branching trajectory structure in high-dimensional data. It uses heat-diffusion processes, which naturally denoise the data, to compute cell-cell affinities. Then, PHATE creates a diffusion-potential geometry by free-energy potentials of these processes. This geometry captures high-dimensional trajectory structures, while enabling a natural embedding of the intrinsic data geometry. This embedding accurately visualizes trajectories and data distances, without requiring strict assumptions typically used by path-finding and tree-fitting algorithms, which have recently been used for pseudotime orderings or tree-renderings of cellular data. Furthermore, PHATE supports a wide range of data exploration tasks by providing interpretable overlays on top of the visualization. We show that such overlays can emphasize and reveal trajectory end-points, branch points and associated split-decisions, progression-forming variables (e.g., specific genes), and paths between developmental events in cellular state-space. We demonstrate PHATE on single-cell RNA sequencing and mass cytometry data pertaining to embryoid body differentiation, IPSC reprogramming, and hematopoiesis in the bone marrow. We also demonstrate PHATE on non-single cell data including single-nucleotide polymorphism (SNP) measurements of European populations, and 16s sequencing of gut microbiota.Abstract In the era of ‘Big Data’ there is a pressing need for tools that provide human interpretable visualizations of emergent patterns in high-throughput high-dimensional data. Further, to enable insightful data exploration, such visualizations should faithfully capture and emphasize emergent structures and patterns without enforcing prior assumptions on the shape or form of the data. In this paper, we present PHATE (Potential of Heat-diffusion for Affinity-based Transition Embedding) - an unsupervised low-dimensional embedding for visualization of data that is aimed at solving these issues. Unlike previous methods that are commonly used for visualization, such as PCA and tSNE, PHATE is able to capture and highlight both local and global structure in the data. In particular, in addition to clustering patterns, PHATE also uncovers and emphasizes progression and transitions (when they exist) in the data, which are often missed in other visualization-capable methods. Such patterns are especially important in biological data that contain, for example, single-cell phenotypes at different phases of differentiation, patients at different stages of disease progression, and gut microbial compositions that vary gradually between individuals, even of the same enterotype. The embedding provided by PHATE is based on a novel informational distance that captures long-range nonlinear relations in the data by computing energy potentials of data-adaptive diffusion processes. We demonstrate the effectiveness of the produced visualization in revealing insights on a wide variety of biomedical data, including single-cell RNA-sequencing, mass cytometry, gut microbiome sequencing, human SNP data, Hi-C data, as well as non-biomedical data, such as facebook network and facial image data. In order to validate the capability of PHATE to enable exploratory analysis, we generate a new dataset of 31,000 single-cells from a human embryoid body differentiation system. Here, PHATE provides a comprehensive picture of the differentiation process, while visualizing major and minor branching trajectories in the data. We validate that all known cell types are recapitulated in the PHATE embedding in proper organization. Furthermore, the global picture of the system offered by PHATE allows us to connect parts of the developmental progression and characterize novel regulators associated with developmental lineages.

Computing minimal interpolants in C1,1(Rd)

We consider the following interpolation problem. Suppose one is given a finite set E⊂Rd, a function f:E→R, and possibly the gradients of ff at the points of E. We want to interpolate the given information with a function F∈C1,1(Rd) with the minimum possible value of Lip(∇F). We present practical, efficient algorithms for constructing an F such that Lip(∇F) is minimal, or for less computational effort, within a small dimensionless constant of being minimal.

Sparse endmember extraction and demixing

A novel algorithm for endmember extraction is presented. The approach follows the linear mixture model for hyperspectral data. Endmembers are identified based on sparsity consideration. Theoretical and experimental results suggest the potential of the method.