Local conformal autoencoder for standardized data coordinates.

We propose a local conformal autoencoder (LOCA) for standardized data coordinates. LOCA is a deep learning-based method for obtaining standardized data coordinates from scientific measurements. Data observations are modeled as samples from an unknown, nonlinear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized, latent variables. We assume a repeated measurement sampling strategy, common in scientific measurements, and present a method for learning an embedding in [Formula: see text] that is isometric to the latent variables of the manifold. The coordinates recovered by our method are invariant to diffeomorphisms of the manifold, making it possible to match between different instrumental observations of the same phenomenon. Our embedding is obtained using LOCA, which is an algorithm that learns to rectify deformations by using a local z-scoring procedure, while preserving relevant geometric information. We demonstrate the isometric embedding properties of LOCA in various model settings and observe that it exhibits promising interpolation and extrapolation capabilities, superior to the current state of the art. Finally, we demonstrate LOCA's efficacy in single-site Wi-Fi localization data and for the reconstruction of three-dimensional curved surfaces from two-dimensional projections.

Reconstruction of normal forms by learning informed observation geometries from data.

The discovery of physical laws consistent with empirical observations is at the heart of (applied) science and engineering. These laws typically take the form of nonlinear differential equations depending on parameters; dynamical systems theory provides, through the appropriate normal forms, an "intrinsic" prototypical characterization of the types of dynamical regimes accessible to a given model. Using an implementation of data-informed geometry learning, we directly reconstruct the relevant "normal forms": a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics. Interestingly, the state variables and the parameters of these realizations are inferred from the empirical observations; without prior knowledge or understanding, they parametrize the dynamics intrinsically without explicit reference to fundamental physical quantities.

Intrinsic map dynamics exploration for uncharted effective free-energy landscapes.

We describe and implement a computer-assisted approach for accelerating the exploration of uncharted effective free-energy surfaces (FESs). More generally, the aim is the extraction of coarse-grained, macroscopic information from stochastic or atomistic simulations, such as molecular dynamics (MD). The approach functionally links the MD simulator with nonlinear manifold learning techniques. The added value comes from biasing the simulator toward unexplored phase-space regions by exploiting the smoothness of the gradually revealed intrinsic low-dimensional geometry of the FES.

A REMARK ON THE ARCSINE DISTRIBUTION AND THE HILBERT TRANSFORM.

It is known that if (p n ) n ∈ℕ is a sequence of orthogonal polynomials in L 2([-1,1],w(x)dx), then the roots are distributed according to an arcsine distribution π -1(1 - x 2)-1 dx for a wide variety of weights w(x). We connect this to a result of the Hilbert transform due to Tricomi: if f(x)(1 - x 2)1/4 ∈ L 2(-1,1) and its Hilbert transform Hf vanishes on (-1,1), then the function f is a multiple of the arcsine distribution f ( x ) = c 1 - x 2 χ ( - 1 , 1 ) where c ∈ ℝ . We also prove a localized Parseval-type identity that seems to be new: if f(x)(1-x 2)1/4 ∈ L2(-1, 1) and f ( x ) 1 - x 2 has mean value 0 on (-1, 1), then ∫ - 1 1 ( H f ) ( x ) 2 1 - x 2 d x = ∫ - 1 1 f ( x ) 2 1 - x 2 d x . .

Empirical intrinsic geometry for nonlinear modeling and time series filtering.

In this paper, we present a method for time series analysis based on empirical intrinsic geometry (EIG). EIG enables one to reveal the low-dimensional parametric manifold as well as to infer the underlying dynamics of high-dimensional time series. By incorporating concepts of information geometry, this method extends existing geometric analysis tools to support stochastic settings and parametrizes the geometry of empirical distributions. However, the statistical models are not required as priors; hence, EIG may be applied to a wide range of real signals without existing definitive models. We show that the inferred model is noise-resilient and invariant under different observation and instrumental modalities. In addition, we show that it can be extended efficiently to newly acquired measurements in a sequential manner. These two advantages enable us to revisit the Bayesian approach and incorporate empirical dynamics and intrinsic geometry into a nonlinear filtering framework. We show applications to nonlinear and non-Gaussian tracking problems as well as to acoustic signal localization.

Rapid fluctuations in functional connectivity of cortical networks encode spontaneous behavior.

Experimental work across species has demonstrated that spontaneously generated behaviors are robustly coupled to variations in neural activity within the cerebral cortex. Functional magnetic resonance imaging data suggest that temporal correlations in cortical networks vary across distinct behavioral states, providing for the dynamic reorganization of patterned activity. However, these data generally lack the temporal resolution to establish links between cortical signals and the continuously varying fluctuations in spontaneous behavior observed in awake animals. Here, we used wide-field mesoscopic calcium imaging to monitor cortical dynamics in awake mice and developed an approach to quantify rapidly time-varying functional connectivity. We show that spontaneous behaviors are represented by fast changes in both the magnitude and correlational structure of cortical network activity. Combining mesoscopic imaging with simultaneous cellular-resolution two-photon microscopy demonstrated that correlations among neighboring neurons and between local and large-scale networks also encode behavior. Finally, the dynamic functional connectivity of mesoscale signals revealed subnetworks not predicted by traditional anatomical atlas-based parcellation of the cortex. These results provide new insights into how behavioral information is represented across the neocortex and demonstrate an analytical framework for investigating time-varying functional connectivity in neural networks.

Nonlinear intrinsic variables and state reconstruction in multiscale simulations.

Finding informative low-dimensional descriptions of high-dimensional simulation data (like the ones arising in molecular dynamics or kinetic Monte Carlo simulations of physical and chemical processes) is crucial to understanding physical phenomena, and can also dramatically assist in accelerating the simulations themselves. In this paper, we discuss and illustrate the use of nonlinear intrinsic variables (NIV) in the mining of high-dimensional multiscale simulation data. In particular, we focus on the way NIV allows us to functionally merge different simulation ensembles, and different partial observations of these ensembles, as well as to infer variables not explicitly measured. The approach relies on certain simple features of the underlying process variability to filter out measurement noise and systematically recover a unique reference coordinate frame. We illustrate the approach through two distinct sets of atomistic simulations: a stochastic simulation of an enzyme reaction network exhibiting both fast and slow time scales, and a molecular dynamics simulation of alanine dipeptide in explicit water.

Robust Estimation of Position-Dependent Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories.

Confinement breaks translational and rotational symmetry in materials and makes all physical properties functions of position. Such spatial variations are key to modulating material properties at the nanoscale, and characterizing them accurately is therefore an intense area of research in the molecular simulations community. This is relatively easy to accomplish for basic mechanical observables. Determining spatial profiles of transport properties, such as diffusivity, is, however, much more challenging, as it requires calculating position-dependent autocorrelations of mechanical observables. In our previous paper (Domingues, T.S.; Coifman, R.; Haji-Akbari, A. J. Phys. Chem. B 2023, 127, 5273 10.1021/acs.jpcb.3c00670), we analytically derive and numerically validate a set of filtered covariance estimators (FCEs) for quantifying spatial variations of the diffusivity tensor from stochastic trajectories. In this work, we adapt these estimators to extract diffusivity profiles from MD trajectories and validate them by applying them to a Lennard-Jones fluid within a slit pore. We find our MD-adapted estimator to exhibit the same qualitative features as its stochastic counterpart, as it accurately estimates the lateral diffusivity across the pore while systematically underestimating the normal diffusivity close to hard boundaries. We introduce a conceptually simple and numerically efficient correction scheme based on simulated annealing and diffusion maps to resolve the latter artifact and obtain normal diffusivity profiles that are consistent with the self-part of the van Hove correlation functions. Our findings demonstrate the potential of this MD-adapted estimator in accurately characterizing spatial variations of diffusivity in confined materials.

Robust Estimation of Position-Dependent Anisotropic Diffusivity Tensors from Stochastic Trajectories.

Materials under confinement can possess properties that deviate considerably from their bulk counterparts. Indeed, confinement makes all physical properties position-dependent and possibly anisotropic, and characterizing such spatial variations and directionality has been an intense area of focus in experimental and computational studies of confined matter. While this task is fairly straightforward for simple mechanical observables, it is far more daunting for transport properties such as diffusivity that can only be estimated from autocorrelations of mechanical observables. For instance, there are well established methods for estimating diffusivity from experimentally observed or computationally generated trajectories in bulk systems. No rigorous generalizations of such methods, however, exist for confined systems. In this work, we present two filtered covariance estimators for computing anisotropic and position-dependent diffusivity tensors and validate them by applying them to stochastic trajectories generated according to known diffusivity profiles. These estimators can accurately capture spatial variations that span over several orders of magnitude and that assume different functional forms. Our kernel-based approach is also very robust to implementation details such as the localization function and time discretization and performs significantly better than estimators that are solely based on local covariance. Moreover, the kernel function does not have to be localized and can instead belong to a dictionary of orthogonal functions. Therefore, the proposed estimator can be readily used to obtain functional estimates of diffusivity rather than a tabulated collection of pointwise estimates. Nonetheless, the susceptibility of the proposed estimators to time discretization is higher at the immediate vicinity of hard boundaries. We demonstrate this heightened susceptibility to be common among all covariance-based estimators.

Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps.

Nonlinear independent component analysis is combined with diffusion-map data analysis techniques to detect good observables in high-dimensional dynamic data. These detections are achieved by integrating local principal component analysis of simulation bursts by using eigenvectors of a Markov matrix describing anisotropic diffusion. The widely applicable procedure, a crucial step in model reduction approaches, is illustrated on stochastic chemical reaction network simulations.

Doubly Stochastic Normalization of the Gaussian Kernel Is Robust to Heteroskedastic Noise.

A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When the data points reside in Euclidean space, a widespread approach is to from an affinity matrix by the Gaussian kernel with pairwise distances, and to follow with a certain normalization (e.g. the row-stochastic normalization or its symmetric variant). We demonstrate that the doubly-stochastic normalization of the Gaussian kernel with zero main diagonal (i.e., no self loops) is robust to heteroskedastic noise. That is, the doubly-stochastic normalization is advantageous in that it automatically accounts for observations with different noise variances. Specifically, we prove that in a suitable high-dimensional setting where heteroskedastic noise does not concentrate too much in any particular direction in space, the resulting (doubly-stochastic) noisy affinity matrix converges to its clean counterpart with rate m -1/2, where m is the ambient dimension. We demonstrate this result numerically, and show that in contrast, the popular row-stochastic and symmetric normalizations behave unfavorably under heteroskedastic noise. Furthermore, we provide examples of simulated and experimental single-cell RNA sequence data with intrinsic heteroskedasticity, where the advantage of the doubly-stochastic normalization for exploratory analysis is evident.

Detecting consistent common lines in cryo-EM by voting.

The single-particle reconstruction problem of electron cryo-microscopy (cryo-EM) is to find the three-dimensional structure of a macromolecule given its two-dimensional noisy projection images at unknown random directions. Ab initio estimates of the 3D structure are often obtained by the "Angular Reconstitution" method, in which a coordinate system is established from three projections, and the orientation of the particle giving rise to each image is deduced from common lines among the images. However, a reliable detection of common lines is difficult due to the low signal-to-noise ratio of the images. In this paper we describe a global self-correcting voting procedure in which all projection images participate to decide the identity of the consistent common lines. The algorithm determines which common line pairs were detected correctly and which are spurious. We show that the voting procedure succeeds at relatively low detection rates and that its performance improves as the number of projection images increases. We demonstrate the algorithm for both simulative and experimental images of the 50S ribosomal subunit.

Two-sample statistics based on anisotropic kernels.

The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between [Formula: see text] data points and a set of [Formula: see text] reference points, where [Formula: see text] can be drastically smaller than [Formula: see text]. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as [Formula: see text], and a finite-sample lower bound of the testing power is obtained. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.

Author Correction: Visualizing structure and transitions in high-dimensional biological data.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Visualizing structure and transitions in high-dimensional biological data.

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.

Graph Laplacian tomography from unknown random projections.

We introduce a graph Laplacian-based algorithm for the tomographic reconstruction of a planar object from its projections taken at random unknown directions. A Laplace-type operator is constructed on the data set of projections, and the eigenvectors of this operator reveal the projection orientations. The algorithm is shown to successfully reconstruct the Shepp-Logan phantom from its noisy projections. Such a reconstruction algorithm is desirable for the structuring of certain biological proteins using cryo-electron microscopy.

Data-Driven Tree Transforms and Metrics.

We consider the analysis of high dimensional data given in the form of a matrix with columns consisting of observations and rows consisting of features. Often the data is such that the observations do not reside on a regular grid, and the given order of the features is arbitrary and does not convey a notion of locality. Therefore, traditional transforms and metrics cannot be used for data organization and analysis. In this paper, our goal is to organize the data by defining an appropriate representation and metric such that they respect the smoothness and structure underlying the data. We also aim to generalize the joint clustering of observations and features in the case the data does not fall into clear disjoint groups. For this purpose, we propose multiscale data-driven transforms and metrics based on trees. Their construction is implemented in an iterative refinement procedure that exploits the co-dependencies between features and observations. Beyond the organization of a single dataset, our approach enables us to transfer the organization learned from one dataset to another and to integrate several datasets together. We present an application to breast cancer gene expression analysis: learning metrics on the genes to cluster the tumor samples into cancer sub-types and validating the joint organization of both the genes and the samples. We demonstrate that using our approach to combine information from multiple gene expression cohorts, acquired by different profiling technologies, improves the clustering of tumor samples.

Geometric diffusions for the analysis of data from sensor networks.

Harmonic analysis on manifolds and graphs has recently led to mathematical developments in the field of data analysis. The resulting new tools can be used to compress and analyze large and complex data sets, such as those derived from sensor networks or neuronal activity datasets, obtained in the laboratory or through computer modeling. The nature of the algorithms (based on diffusion maps and connectivity strengths on graphs) possesses a certain analogy with neural information processing, and has the potential to provide inspiration for modeling and understanding biological organization in perception and memory formation.

Describing the performance of U.S. hospitals by applying big data analytics.

Public reporting of measures of hospital performance is an important component of quality improvement efforts in many countries. However, it can be challenging to provide an overall characterization of hospital performance because there are many measures of quality. In the United States, the Centers for Medicare and Medicaid Services reports over 100 measures that describe various domains of hospital quality, such as outcomes, the patient experience and whether established processes of care are followed. Although individual quality measures provide important insight, it is challenging to understand hospital performance as characterized by multiple quality measures. Accordingly, we developed a novel approach for characterizing hospital performance that highlights the similarities and differences between hospitals and identifies common patterns of hospital performance. Specifically, we built a semi-supervised machine learning algorithm and applied it to the publicly-available quality measures for 1,614 U.S. hospitals to graphically and quantitatively characterize hospital performance. In the resulting visualization, the varying density of hospitals demonstrates that there are key clusters of hospitals that share specific performance profiles, while there are other performance profiles that are rare. Several popular hospital rating systems aggregate some of the quality measures included in our study to produce a composite score; however, hospitals that were top-ranked by such systems were scattered across our visualization, indicating that these top-ranked hospitals actually excel in many different ways. Our application of a novel graph analytics method to data describing U.S. hospitals revealed nuanced differences in performance that are obscured in existing hospital rating systems.

Heterogeneity in Early Responses in ALLHAT (Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial).

Randomized trials of hypertension have seldom examined heterogeneity in response to treatments over time and the implications for cardiovascular outcomes. Understanding this heterogeneity, however, is a necessary step toward personalizing antihypertensive therapy. We applied trajectory-based modeling to data on 39 763 study participants of the ALLHAT (Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial) to identify distinct patterns of systolic blood pressure (SBP) response to randomized medications during the first 6 months of the trial. Two trajectory patterns were identified: immediate responders (85.5%), on average, had a decreasing SBP, whereas nonimmediate responders (14.5%), on average, had an initially increasing SBP followed by a decrease. Compared with those randomized to chlorthalidone, participants randomized to amlodipine (odds ratio, 1.20; 95% confidence interval [CI], 1.10-1.31), lisinopril (odds ratio, 1.88; 95% CI, 1.73-2.03), and doxazosin (odds ratio, 1.65; 95% CI, 1.52-1.78) had higher adjusted odds ratios associated with being a nonimmediate responder (versus immediate responder). After multivariable adjustment, nonimmediate responders had a higher hazard ratio of stroke (hazard ratio, 1.49; 95% CI, 1.21-1.84), combined cardiovascular disease (hazard ratio, 1.21; 95% CI, 1.11-1.31), and heart failure (hazard ratio, 1.48; 95% CI, 1.24-1.78) during follow-up between 6 months and 2 years. The SBP response trajectories provided superior discrimination for predicting downstream adverse cardiovascular events than classification based on difference in SBP between the first 2 measurements, SBP at 6 months, and average SBP during the first 6 months. Our findings demonstrate heterogeneity in response to antihypertensive therapies and show that chlorthalidone is associated with more favorable initial response than the other medications.