arXiv subjects

Karthik Bharath

Publications and source records attributed to Karthik Bharath.

At least 19 recordsLinked to original sources

arXiv2026

Rolled Gaussian process models for curves on manifolds

Given a planar curve, imagine rolling a sphere along that curve without slipping or twisting, and by this means tracing out a curve on the sphere. It is well known that such a rolling operation induces a local isometry between the sphere and the plane so that the two curves uniquely determine each other, and moreover, the operation extends to a general class of manifolds in any dimension. We use rolling to construct an analogue of a Gaussian process on a manifold starting from a Euclidean Gaussian process with mean $m$ and covariance $K$, and refer to it as a rolled Gaussian process parameterized by $m$ and $K$. The resulting model is generative, and is amenable to statistical inference given data as curves on a manifold. We identify conditions on the manifold under which the rolling of $m$ equals the Fréchet mean of the rolled Gaussian process, propose computationally simple estimators of $m$ and $K$, and derive their rates of convergence. We illustrate with examples on the unit sphere, symmetric positive-definite matrices, and with a robotics application involving 3D orientations.

math.ST↗

arXiv2025

Second-order spatial analysis of shapes of tumor cell nuclei

Intra-tumor heterogeneity driving disease progression is characterized by distinct growth and spatial proliferation patterns of cells and their nuclei within tumor and non-tumor tissues. A widely accepted hypothesis is that these spatial patterns are correlated with morphology of the cells and their nuclei. Nevertheless, tools to quantify the correlation, with uncertainty, are scarce, and the state-of-the-art is based on low-dimensional numerical summaries of the shapes that are inadequate to fully encode shape information. To this end, we propose a marked point process framework to assess spatial correlation among shapes of planar closed curves, which represent cell or nuclei outlines. With shapes of curves as marks, the framework is based on a mark-weighted $K$ function, a second-order spatial statistic that accounts for the marks' variation by using test functions that capture only the shapes of cells and their nuclei. We then develop local and global hypothesis tests for spatial dependence between the marks using the $K$ function. The framework is brought to bear on the cell nuclei extracted from histopathology images of breast cancer, where we uncover distinct correlation patterns that are consistent with clinical expectations.

stat.AP↗

arXiv2025

Sampling and estimation on manifolds using the Langevin diffusion

Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure $\text{d}μ_ϕ\propto e^{-ϕ} \mathrm{dvol}_g $ on a compact Riemannian manifold. Two estimators of linear functionals of $μ_ϕ$ based on the discretized Markov process are considered: a time-averaging estimator based on a single trajectory and an ensemble-averaging estimator based on multiple independent trajectories. Imposing no restrictions beyond a nominal level of smoothness on $ϕ$, first-order error bounds, in discretization step size, on the bias and variance/mean-square error of both estimators are derived. The order of error matches the optimal rate in Euclidean and flat spaces, and leads to a first-order bound on distance between the invariant measure $μ_ϕ$ and a stationary measure of the discretized Markov process. This order is preserved even upon using retractions when exponential maps are unavailable in closed form, thus enhancing practicality of the proposed algorithms. Generality of the proof techniques, which exploit links between two partial differential equations and the semigroup of operators corresponding to the Langevin diffusion, renders them amenable for the study of a more general class of sampling algorithms related to the Langevin diffusion. Conditions for extending analysis to the case of non-compact manifolds are discussed. Numerical illustrations with distributions, log-concave and otherwise, on the manifolds of positive and negative curvature elucidate on the derived bounds and demonstrate practical utility of the sampling algorithm.

math.ST↗

arXiv2025

Regression graphs and sparsity-inducing reparametrizations

That parametrization and sparsity are inherently linked raises the possibility that relevant models, not obviously sparse in their natural formulation, exhibit a population-level sparsity after reparametrization. In covariance models, positive-definiteness enforces additional constraints on how sparsity can legitimately manifest. It is therefore natural to consider reparametrization maps in which sparsity respects positive definiteness. The main purpose of this paper is to provide insight into structures on the physically-natural scale that induce and are induced by sparsity after reparametrization. The richest of the four structures initially uncovered can be generated, under a causal ordering, by the joint-response graphs studied by Wermuth & Cox (2004), while the most restrictive is that induced by sparsity on the scale of the matrix logarithm, studied by Battey (2017). The Iwasawa decomposition of the general linear group, combined with the graphical-models interpretation, points to a class of reparametrizations for the chain-graph models (Andersson et al. 2001), with undirected and directed acyclic graphs as special cases. An important insight is the interpretation of approximate zeros, explaining the modelling implications of enforcing sparsity after reparameterization: in effect, the relation between two variables would be declared null if relatively direct regression effects were negligible and others manifested through long paths. The insights have a bearing on methodology, some aspects of which are presented. A detailed simulation uses the theoretical insights to further explore regimes under which reparametrization is beneficial.

math.ST↗

arXiv2025

Geometry-driven Bayesian Inference for Ultrametric Covariance Matrices

Ultrametric matrices are a class of covariance matrices that arise in latent tree models. As a parameter space in a statistical model, the set of ultrametric matrices is neither convex nor a smooth manifold. Focus in the literature has hitherto been restricted to estimation through projections and relaxation-based techniques, and inferential methods are lacking. Motivated by this, we establish a bijection between the set of positive definite ultrametric matrices and the set of rooted, leaf-labeled trees equipped with the stratified geometry of the well-known phylogenetic treespace. Using the pullback geometry under the bijection and by adapting sampling algorithms in Bayesian phylogenetics, we develop algorithms to sample from the posterior distribution on the set of ultrametric matrices in a Bayesian latent tree model where the tree may be binary or multifurcating. We demonstrate the utility of the algorithms in simulation studies, and illustrate them on a pre-clinical cancer application to quantify uncertainty about treatment trees that identify treatments with high mechanism similarity that target correlated pathways.

stat.ME↗

arXiv2024

Empirical likelihood for Fréchet means on open books

Empirical Likelihood (EL) is a type of nonparametric likelihood that is useful in many statistical inference problems, including confidence region construction and $k$-sample problems. It enjoys some remarkable theoretical properties, notably Bartlett correctability. One area where EL has potential but is under-developed is in non-Euclidean statistics where the Fréchet mean is the population characteristic of interest. Only recently has a general EL method been proposed for smooth manifolds. In this work, we continue progress in this direction and develop an EL method for the Fréchet mean on a stratified metric space that is not a manifold: the open book, obtained by gluing copies of a Euclidean space along their common boundaries. The structure of an open book captures the essential behaviour of the Fréchet mean around certain singular regions of more general stratified spaces for complex data objects, and relates intimately to the local geometry of non-binary trees in the well-studied phylogenetic treespace. We derive a version of Wilks' theorem for the EL statistic, and elucidate on the delicate interplay between the asymptotic distribution and topology of the neighbourhood around the population Fréchet mean. We then present a bootstrap calibration of the EL, which proves that under mild conditions, bootstrap calibration of EL confidence regions have coverage error of size $O(n^{-2})$ rather than $O(n^{-1})$.

math.ST↗

arXiv2024

Probabilistic size-and-shape functional mixed models

The reliable recovery and uncertainty quantification of a fixed effect function $μ$ in a functional mixed model, for modelling population- and object-level variability in noisily observed functional data, is a notoriously challenging task: variations along the $x$ and $y$ axes are confounded with additive measurement error, and cannot in general be disentangled. The question then as to what properties of $μ$ may be reliably recovered becomes important. We demonstrate that it is possible to recover the size-and-shape of a square-integrable $μ$ under a Bayesian functional mixed model. The size-and-shape of $μ$ is a geometric property invariant to a family of space-time unitary transformations, viewed as rotations of the Hilbert space, that jointly transform the $x$ and $y$ axes. A random object-level unitary transformation then captures size-and-shape \emph{preserving} deviations of $μ$ from an individual function, while a random linear term and measurement error capture size-and-shape \emph{altering} deviations. The model is regularized by appropriate priors on the unitary transformations, posterior summaries of which may then be suitably interpreted as optimal data-driven rotations of a fixed orthonormal basis for the Hilbert space. Our numerical experiments demonstrate utility of the proposed model, and superiority over the current state-of-the-art.

stat.ME↗

arXiv2024

Topo-Geometric Analysis of Variability in Point Clouds using Persistence Landscapes

Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects known as persistence diagrams. To better aid statistical analysis, a functional representation of the diagrams, known as persistence landscapes, enable use of functional data analysis and machine learning tools. Topological and geometric variabilities inherent in point clouds are confounded in both persistence diagrams and landscapes, and it is important to distinguish topological signal from noise to draw reliable conclusions on the structure of the point clouds when using persistence homology. We develop a framework for decomposing variability in persistence diagrams into topological signal and topological noise through alignment of persistence landscapes using an elastic Riemannian metric. Aligned landscapes (amplitude) isolate the topological signal. Reparameterizations used for landscape alignment (phase) are linked to a resolution parameter used to generate persistence diagrams, and capture topological noise in the form of geometric, global scaling and sampling variabilities. We illustrate the importance of decoupling topological signal and topological noise in persistence diagrams (landscapes) using several simulated examples. We also demonstrate that our approach provides novel insights in two real data studies.

stat.ME↗

arXiv2023

A diffusion approach to Stein's method on Riemannian manifolds

We detail an approach to develop Stein's method for bounding integral metrics on probability measures defined on a Riemannian manifold $\mathbf M$. Our approach exploits the relationship between the generator of a diffusion on $\mathbf M$ with target invariant measure and its characterising Stein operator. We consider a pair of such diffusions with different starting points, and through analysis of the distance process between the pair, derive Stein factors, which bound the solution to the Stein equation and its derivatives. The Stein factors contain curvature-dependent terms and reduce to those currently available for $\mathbb R^m$, and moreover imply that the bounds for $\mathbb R^m$ remain valid when $\mathbf M$ is a flat manifold

math.PR↗

arXiv2022

Shape And Structure Preserving Differential Privacy

It is common for data structures such as images and shapes of 2D objects to be represented as points on a manifold. The utility of a mechanism to produce sanitized differentially private estimates from such data is intimately linked to how compatible it is with the underlying structure and geometry of the space. In particular, as recently shown, utility of the Laplace mechanism on a positively curved manifold, such as Kendall's 2D shape space, is significantly influences by the curvature. Focusing on the problem of sanitizing the Fréchet mean of a sample of points on a manifold, we exploit the characterisation of the mean as the minimizer of an objective function comprised of the sum of squared distances and develop a K-norm gradient mechanism on Riemannian manifolds that favors values that produce gradients close to the the zero of the objective function. For the case of positively curved manifolds, we describe how using the gradient of the squared distance function offers better control over sensitivity than the Laplace mechanism, and demonstrate this numerically on a dataset of shapes of corpus callosa. Further illustrations of the mechanism's utility on a sphere and the manifold of symmetric positive definite matrices are also presented.

stat.ML↗

arXiv2022

Spatially Penalised Registration of Multivariate Functional Data

Registration of multivariate functional data involves handling of both cross-component and cross-observation phase variations. Allowing for the two phase variations to be modelled as general diffeomorphic time warpings, in this work we focus on the hitherto unconsidered setting where phase variation of the component functions are spatially correlated. We propose an algorithm to optimize a metric-based objective function for registration with a novel penalty term that incorporates the spatial correlation between the component phase variations through a kriging estimate of an appropriate phase random field. The penalty term encourages the overall phase at a particular location to be similar to the spatially weighted average phase in its neighbourhood, and thus engenders a regularization that prevents over-alignment. Utility of the registration method, and its superior performance compared to methods that fail to account for the spatial correlation, is demonstrated through performance on simulated examples and two multivariate functional datasets pertaining to EEG signals and ozone concentrations. The generality of the framework opens up the possibility for extension to settings involving different forms of correlation between the component functions and their phases.

stat.ME↗

arXiv2022

Probabilistic Learning of Treatment Trees in Cancer

Accurate identification of synergistic treatment combinations and their underlying biological mechanisms is critical across many disease domains, especially cancer. In translational oncology research, preclinical systems such as patient-derived xenografts (PDX) have emerged as a unique study design evaluating multiple treatments administered to samples from the same human tumor implanted into genetically identical mice. In this paper, we propose a novel Bayesian probabilistic tree-based framework for PDX data to investigate the hierarchical relationships between treatments by inferring treatment cluster trees, referred to as treatment trees (Rx-tree). The framework motivates a new metric of mechanistic similarity between two or more treatments accounting for inherent uncertainty in tree estimation; treatments with a high estimated similarity have potentially high mechanistic synergy. Building upon Dirichlet Diffusion Trees, we derive a closed-form marginal likelihood encoding the tree structure, which facilitates computationally efficient posterior inference via a new two-stage algorithm. Simulation studies demonstrate superior performance of the proposed method in recovering the tree structure and treatment similarities. Our analyses of a recently collated PDX dataset produce treatment similarity estimates that show a high degree of concordance with known biological mechanisms across treatments in five different cancers. More importantly, we uncover new and potentially effective combination therapies that confer synergistic regulation of specific downstream biological pathways for future clinical investigations. Our accompanying code, data, and shiny application for visualization of results are available at: https://github.com/bayesrx/RxTree.

stat.AP↗

arXiv2021

Differential Privacy Over Riemannian Manifolds

In this work we consider the problem of releasing a differentially private statistical summary that resides on a Riemannian manifold. We present an extension of the Laplace or K-norm mechanism that utilizes intrinsic distances and volumes on the manifold. We also consider in detail the specific case where the summary is the Fréchet mean of data residing on a manifold. We demonstrate that our mechanism is rate optimal and depends only on the dimension of the manifold, not on the dimension of any ambient space, while also showing how ignoring the manifold structure can decrease the utility of the sanitized summary. We illustrate our framework in two examples of particular interest in statistics: the space of symmetric positive definite matrices, which is used for covariance matrices, and the sphere, which can be used as a space for modeling discrete distributions.

math.ST↗

arXiv2021

Tangent functional canonical correlation analysis for densities and shapes, with applications to multimodal imaging data

It is quite common for functional data arising from imaging data to assume values in infinite-dimensional manifolds. Uncovering associations between two or more such nonlinear functional data extracted from the same object across medical imaging modalities can assist development of personalized treatment strategies. We propose a method for canonical correlation analysis between paired probability densities or shapes of closed planar curves, routinely used in biomedical studies, which combines a convenient linearization and dimension reduction of the data using tangent space coordinates. Leveraging the fact that the corresponding manifolds are submanifolds of unit Hilbert spheres, we describe how finite-dimensional representations of the functional data objects can be easily computed, which then facilitates use of standard multivariate canonical correlation analysis methods. We further construct and visualize canonical variate directions directly on the space of densities or shapes. Utility of the method is demonstrated through numerical simulations and performance on a magnetic resonance imaging dataset of Glioblastoma Multiforme brain tumors.

stat.ME↗

arXiv2021

Tumor Radiogenomics with Bayesian Layered Variable Selection

We propose a statistical framework to integrate radiological magnetic resonance imaging (MRI) and genomic data to identify the underlying radiogenomic associations in lower grade gliomas (LGG). We devise a novel imaging phenotype by dividing the tumor region into concentric spherical layers that mimics the tumor evolution process. MRI data within each layer is represented by voxel--intensity-based probability density functions which capture the complete information about tumor heterogeneity. Under a Riemannian-geometric framework these densities are mapped to a vector of principal component scores which act as imaging phenotypes. Subsequently, we build Bayesian variable selection models for each layer with the imaging phenotypes as the response and the genomic markers as predictors. Our novel hierarchical prior formulation incorporates the interior-to-exterior structure of the layers, and the correlation between the genomic markers. We employ a computationally-efficient Expectation--Maximization-based strategy for estimation. Simulation studies demonstrate the superior performance of our approach compared to other approaches. With a focus on the cancer driver genes in LGG, we discuss some biologically relevant findings. Genes implicated with survival and oncogenesis are identified as being associated with the spherical layers, which could potentially serve as early-stage diagnostic markers for disease monitoring, prior to routine invasive approaches.

stat.ME↗

arXiv2021

RADIOHEAD: Radiogenomic Analysis Incorporating Tumor Heterogeneity in Imaging Through Densities

Recent technological advancements have enabled detailed investigation of associations between the molecular architecture and tumor heterogeneity, through multi-source integration of radiological imaging and genomic (radiogenomic) data. In this paper, we integrate and harness radiogenomic data in patients with lower grade gliomas (LGG), a type of brain cancer, in order to develop a regression framework called RADIOHEAD (RADIOgenomic analysis incorporating tumor HEterogeneity in imAging through Densities) to identify radiogenomic associations. Imaging data is represented through voxel intensity probability density functions of tumor sub-regions obtained from multimodal magnetic resonance imaging, and genomic data through molecular signatures in the form of pathway enrichment scores corresponding to their gene expression profiles. Employing a Riemannian-geometric framework for principal component analysis on the set of probability densities functions, we map each probability density to a vector of principal component scores, which are then included as predictors in a Bayesian regression model with the pathway enrichment scores as the response. Variable selection compatible with the grouping structure amongst the predictors induced through the tumor sub-regions is carried out under a group spike-and-slab prior. A Bayesian false discovery rate mechanism is then used to infer significant associations based on the posterior distribution of the regression coefficients. Our analyses reveal several pathways relevant to LGG etiology (such as synaptic transmission, nerve impulse and neurotransmitter pathways), to have significant associations with the corresponding imaging-based predictors.

stat.ME↗

arXiv2021

Joint quasiprobability distribution on the measurement outcomes of MUB-driven operators

We propose a method to define quasiprobability distributions for general spin-$j$ systems of dimension $n=2j+1$, where $n$ is a prime or power of prime. The method is based on a complete set of orthonormal commuting operators related to Mutually Unbiased Bases which enable (i) a parameterisation of the density matrix and (ii) construction of measurement operators that can be physically realised. As a result we geometrically characterise the set of states for which the quasiprobability distribution is non-negative, and can be viewed as a joint distribution of classical random variables assuming values in a finite set of outcomes. The set is an $(n^2-1)$-dimensional convex polytope with $n+1$ vertices as the only pure states, $n^{n+1}$ number of higher dimensional faces, and $n^3(n+1)/2$ edges.

quant-ph↗

arXiv2020

Variograms for spatial functional data with phase variation

Spatial, amplitude and phase variations in spatial functional data are confounded. Conclusions from the popular functional trace variogram, which quantifies spatial variation, can be misleading when analysing misaligned functional data with phase variation. To remedy this, we describe a framework that extends amplitude-phase separation methods in functional data to the spatial setting, with a view towards performing clustering and spatial prediction. We propose a decomposition of the trace variogram into amplitude and phase components and quantify how spatial correlations between functional observations manifest in their respective amplitude and phase components. This enables us to generate separate amplitude and phase clustering methods for spatial functional data, and develop a novel spatial functional interpolant at unobserved locations based on combining separate amplitude and phase predictions. Through simulations and real data analyses, we found that the proposed methods result in more accurate predictions and more interpretable clustering results.

stat.ME↗