Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure $\text{d}\mu_\phi \propto e^{-\phi} \mathrm{dvol}_g $ on a compact Riemannian manifold. Two estimators of linear functionals of $\mu_\phi $ 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 $\phi$, 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 $\mu_\phi$ 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.

The reliable recovery and uncertainty quantification of a fixed effect function $\mu$ in a functional mixed model, for modeling 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 $\mu$ may be reliably recovered becomes important. We demonstrate that it is possible to recover the size-and-shape of a square-integrable $\mu$ under a Bayesian functional mixed model. The size-and-shape of $\mu$ 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 preserving deviations of $\mu$ from an individual function, while a random linear term and measurement error capture size-and-shape 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.

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.

We propose a statistical framework to analyze 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. We provide a R package that can be used to deploy our framework to identify radiogenomic associations.

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 influenced by the curvature. Focusing on the problem of sanitizing the Fr\'echet mean of a sample of points on a manifold, we exploit the characterization 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.

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 Fr\'echet 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.

Word embeddings are commonly obtained as optimisers of a criterion function f of a text corpus, but assessed on word-task performance using a different evaluation function g of the test data. We contend that a possible source of disparity in performance on tasks is the incompatibility between classes of transformations that leave f and g invariant. In particular, word embeddings defined by f are not unique; they are defined only up to a class of transformations to which f is invariant, and this class is larger than the class to which g is invariant. One implication of this is that the apparent superiority of one word embedding over another, as measured by word task performance, may largely be a consequence of the arbitrary elements selected from the respective solution sets. We provide a formal treatment of the above identifiability issue, present some numerical examples, and discuss possible resolutions.

In multi-hop wireless networks with Carrier Sense Multiple Access (CSMA), unfairness may arise due to a number of reasons. A majority of the existing work focuses on fairness issues due to hidden terminals and the impact on backoff mechanisms. This paper focuses on the unfairness arising due to unequal contention opportunities at a node; some nodes rarely observe an idle channel since two or more interferers that are not in range with each other can transmit together. Contention unfairness is unrelated to hidden terminals. In order to understand the impact of contention and gain insight into developing solutions for contention unfairness, we develop a model from first principles for contention in IEEE 802.11 networks. The accuracy of the model is validated through simulations and the results show that such unfairness is a common phenomenon. Based on the insights gained from the model, we propose and evaluate a distributed scheme that reduces the effect of unfairness due to contention. Simulation results show that the proposed scheme achieves an average improvement of 25% in fairness, with a small reduction in overall throughput.