This paper surveys and evaluates some popular state of the art methods for algorithmic curvature and normal estimation. In addition to surveying existing methods we also propose a new method for robust curvature estimation and evaluate it against existing methods thus demonstrating its superiority to existing methods in the case of significant data noise. Throughout this paper we are concerned with computation in low dimensional spaces (N < 10) and primarily focus on the computation of the Weingarten map and quantities that may be derived from this; however, the algorithms discussed are theoretically applicable in any dimension. One thing that is common to all these methods is their basis in an estimated graph structure. For any of these methods to work the local geometry of the manifold must be exploited; however, in the case of point cloud data it is often difficult to discover a robust manifold structure underlying the data, even in simple cases, which can greatly influence the results of these algorithms. We hope that in pushing these algorithms to their limits we are able to discover, and perhaps resolve, many major pitfalls that may affect potential users and future researchers hoping to improve these methods
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The number of modes in a probability density function is representative of the model's complexity and can also be viewed as the number of existing subpopulations. Despite its relevance, little research has been devoted to its estimation. Focusing on the univariate setting, we propose a novel approach targeting prediction accuracy inspired by some overlooked aspects of the problem. We argue for the need for structure in the solutions, the subjective and uncertain nature of modes, and the convenience of a holistic view blending global and local density properties. Our method builds upon a combination of flexible kernel estimators and parsimonious compositional splines. Feature exploration, model selection and mode testing are implemented in the Bayesian inference paradigm, providing soft solutions and allowing to incorporate expert judgement in the process. The usefulness of our proposal is illustrated through a case study in sports analytics, showcasing multiple companion visualisation tools. A thorough simulation study demonstrates that traditional modality-driven approaches paradoxically struggle to provide accurate results. In this context, our method emerges as a top-tier alternative offering innovative solutions for analysts.
In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts of the function can be inferred accurately and which cannot. For example, it is generally understood that we can identify system parameters accurately only close to detectors, or along ray paths between sources and detectors, because we have "the most information" for these places. Although referenced in many publications, the "information" that is invoked in such contexts is not a well understood and clearly defined quantity. Herein, we present a definition of information density that is based on the variance of coefficients as derived from a Bayesian reformulation of the inverse problem. We then discuss three areas in which this information density can be useful in practical algorithms for the solution of inverse problems, and illustrate the usefulness in one of these areas -- how to choose the discretization mesh for the function to be reconstructed -- using numerical experiments.
Principal Component Analysis (PCA) is a fundamental tool for data visualization, denoising, and dimensionality reduction. It is widely popular in Statistics, Machine Learning, Computer Vision, and related fields. However, PCA is well-known to fall prey to outliers and often fails to detect the true underlying low-dimensional structure within the dataset. Following the Median of Means (MoM) philosophy, recent supervised learning methods have shown great success in dealing with outlying observations without much compromise to their large sample theoretical properties. This paper proposes a PCA procedure based on the MoM principle. Called the \textbf{M}edian of \textbf{M}eans \textbf{P}rincipal \textbf{C}omponent \textbf{A}nalysis (MoMPCA), the proposed method is not only computationally appealing but also achieves optimal convergence rates under minimal assumptions. In particular, we explore the non-asymptotic error bounds of the obtained solution via the aid of the Rademacher complexities while granting absolutely no assumption on the outlying observations. The derived concentration results are not dependent on the dimension because the analysis is conducted in a separable Hilbert space, and the results only depend on the fourth moment of the underlying distribution in the corresponding norm. The proposal's efficacy is also thoroughly showcased through simulations and real data applications.
Gaussian Partial Information Decomposition: Bias Correction and Application to High-dimensional Data
Recent advances in neuroscientific experimental techniques have enabled us to simultaneously record the activity of thousands of neurons across multiple brain regions. This has led to a growing need for computational tools capable of analyzing how task-relevant information is represented and communicated between several brain regions. Partial information decompositions (PIDs) have emerged as one such tool, quantifying how much unique, redundant and synergistic information two or more brain regions carry about a task-relevant message. However, computing PIDs is computationally challenging in practice, and statistical issues such as the bias and variance of estimates remain largely unexplored. In this paper, we propose a new method for efficiently computing and estimating a PID definition on multivariate Gaussian distributions. We show empirically that our method satisfies an intuitive additivity property, and recovers the ground truth in a battery of canonical examples, even at high dimensionality. We also propose and evaluate, for the first time, a method to correct the bias in PID estimates at finite sample sizes. Finally, we demonstrate that our Gaussian PID effectively characterizes inter-areal interactions in the mouse brain, revealing higher redundancy between visual areas when a stimulus is behaviorally relevant.
In this work, we propose a geometric framework for analyzing mechanical manipulation, for example, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. We first review and show that the natural geometric setting is represented by the cotangent bundle and its Lagrangian submanifolds. These are standard concepts in geometric mechanics but usually presented within dynamical frameworks. We review the basic definitions from a static mechanics perspective and show how Lagrangian submanifolds are naturally derived from a first order analysis. Then, via a second order analysis, we derive the Hessian of total energy. As this is not necessarily positive-definite from a control perspective, we propose the use of the squared-Hessian for optimality measures, motivated by insights {derived from both mechanics (Gauss's Principle) and biology (Separation Principle)}. We conclude by showing how such methods can be applied, for example, to the simple case of an elastically driven pendulum. The example is simple enough to allow for analytical solution. However, an extension is further derived and numerically solved, which is more realistically connected with actual robotic manipulation problems.
This paper is devoted to the statistical and numerical properties of the geometric median, and its applications to the problem of robust mean estimation via the median of means principle. Our main theoretical results include (a) an upper bound for the distance between the mean and the median for general absolutely continuous distributions in R^d, and examples of specific classes of distributions for which these bounds do not depend on the ambient dimension d; (b) exponential deviation inequalities for the distance between the sample and the population versions of the geometric median, which again depend only on the trace-type quantities and not on the ambient dimension. As a corollary, we deduce improved bounds for the (geometric) median of means estimator that hold for large classes of heavy-tailed distributions. Finally, we address the error of numerical approximation, which is an important practical aspect of any statistical estimation procedure. We demonstrate that the objective function minimized by the geometric median satisfies a "local quadratic growth" condition that allows one to translate suboptimality bounds for the objective function to the corresponding bounds for the numerical approximation to the median itself, and propose a simple stopping rule applicable to any optimization method which yields explicit error guarantees. We conclude with the numerical experiments including the application to estimation of mean values of log-returns for S&P 500 data.
Unsupervised node clustering (or community detection) is a classical graph learning task. In this paper, we study algorithms, which exploit the geometry of the graph to identify densely connected substructures, which form clusters or communities. Our method implements discrete Ricci curvatures and their associated geometric flows, under which the edge weights of the graph evolve to reveal its community structure. We consider several discrete curvature notions and analyze the utility of the resulting algorithms. In contrast to prior literature, we study not only single-membership community detection, where each node belongs to exactly one community, but also mixed-membership community detection, where communities may overlap. For the latter, we argue that it is beneficial to perform community detection on the line graph, i.e., the graph's dual. We provide both theoretical and empirical evidence for the utility of our curvature-based clustering algorithms. In addition, we give several results on the relationship between the curvature of a graph and that of its dual, which enable the efficient implementation of our proposed mixed-membership community detection approach and which may be of independent interest for curvature-based network analysis.
In the field of state-of-the-art object detection, the task of object localization is typically accomplished through a dedicated subnet that emphasizes bounding box regression. This subnet traditionally predicts the object's position by regressing the box's center position and scaling factors. Despite the widespread adoption of this approach, we have observed that the localization results often suffer from defects, leading to unsatisfactory detector performance. In this paper, we address the shortcomings of previous methods through theoretical analysis and experimental verification and present an innovative solution for precise object detection. Instead of solely focusing on the object's center and size, our approach enhances the accuracy of bounding box localization by refining the box edges based on the estimated distribution at the object's boundary. Experimental results demonstrate the potential and generalizability of our proposed method.
Registration of distant outdoor LiDAR point clouds is crucial to extending the 3D vision of collaborative autonomous vehicles, and yet is challenging due to small overlapping area and a huge disparity between observed point densities. In this paper, we propose Group-wise Contrastive Learning (GCL) scheme to extract density-invariant geometric features to register distant outdoor LiDAR point clouds. We mark through theoretical analysis and experiments that, contrastive positives should be independent and identically distributed (i.i.d.), in order to train densityinvariant feature extractors. We propose upon the conclusion a simple yet effective training scheme to force the feature of multiple point clouds in the same spatial location (referred to as positive groups) to be similar, which naturally avoids the sampling bias introduced by a pair of point clouds to conform with the i.i.d. principle. The resulting fully-convolutional feature extractor is more powerful and density-invariant than state-of-the-art methods, improving the registration recall of distant scenarios on KITTI and nuScenes benchmarks by 40.9% and 26.9%, respectively. The code will be open-sourced.
This work addresses a novel and challenging problem of estimating the full 3D hand shape and pose from a single RGB image. Most current methods in 3D hand analysis from monocular RGB images only focus on estimating the 3D locations of hand keypoints, which cannot fully express the 3D shape of hand. In contrast, we propose a Graph Convolutional Neural Network (Graph CNN) based method to reconstruct a full 3D mesh of hand surface that contains richer information of both 3D hand shape and pose. To train networks with full supervision, we create a large-scale synthetic dataset containing both ground truth 3D meshes and 3D poses. When fine-tuning the networks on real-world datasets without 3D ground truth, we propose a weakly-supervised approach by leveraging the depth map as a weak supervision in training. Through extensive evaluations on our proposed new datasets and two public datasets, we show that our proposed method can produce accurate and reasonable 3D hand mesh, and can achieve superior 3D hand pose estimation accuracy when compared with state-of-the-art methods.
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