Information-Theoretic Models for Physical Observables

This work addresses J.A. Wheeler's critical idea that all things physical are information-theoretic in origin. In this paper, we introduce a novel mathematical framework based on information geometry, using the Fisher information metric as a particular Riemannian metric, defined in the parameter space of a smooth statistical manifold of normal probability distributions. Following this approach, we study the stationary states with the time-independent Schrödinger's equation to discover that the information could be represented and distributed over a set of quantum harmonic oscillators, one for each independent source of data, whose coordinate for each oscillator is a parameter of the smooth statistical manifold to estimate. We observe that the estimator's variance equals the energy levels of the quantum harmonic oscillator, proving that the estimator's variance is definitively quantized, being the minimum variance at the minimum energy level of the oscillator. Interestingly, we demonstrate that quantum harmonic oscillators reach the Cramér-Rao lower bound on the estimator's variance at the lowest energy level. In parallel, we find that the global probability density function of the collective mode of a set of quantum harmonic oscillators at the lowest energy level equals the posterior probability distribution calculated using Bayes' theorem from the sources of information for all data values, taking as a prior the Riemannian volume of the informative metric. Interestingly, the opposite is also true, as the prior is constant. Altogether, these results suggest that we can break the sources of information into little elements: quantum harmonic oscillators, with the square modulus of the collective mode at the lowest energy representing the most likely reality, supporting A. Zeilinger's recent statement that the world is not broken into physical but informational parts.

Fast Characterization of Inducible Regions of Atrial Fibrillation Models With Multi-Fidelity Gaussian Process Classification

Computational models of atrial fibrillation have successfully been used to predict optimal ablation sites. A critical step to assess the effect of an ablation pattern is to pace the model from different, potentially random, locations to determine whether arrhythmias can be induced in the atria. In this work, we propose to use multi-fidelity Gaussian process classification on Riemannian manifolds to efficiently determine the regions in the atria where arrhythmias are inducible. We build a probabilistic classifier that operates directly on the atrial surface. We take advantage of lower resolution models to explore the atrial surface and combine seamlessly with high-resolution models to identify regions of inducibility. We test our methodology in 9 different cases, with different levels of fibrosis and ablation treatments, totalling 1,800 high resolution and 900 low resolution simulations of atrial fibrillation. When trained with 40 samples, our multi-fidelity classifier that combines low and high resolution models, shows a balanced accuracy that is, on average, 5.7% higher than a nearest neighbor classifier. We hope that this new technique will allow faster and more precise clinical applications of computational models for atrial fibrillation. All data and code accompanying this manuscript will be made publicly available at: https://github.com/fsahli/AtrialMFclass.

Performance enhancement of facial electromyogram-based facial-expression recognition for social virtual reality applications using linear discriminant analysis adaptation

Recent studies have indicated that facial electromyogram (fEMG)-based facial-expression recognition (FER) systems are promising alternatives to the conventional camera-based FER systems for virtual reality (VR) environments because they are economical, do not depend on the ambient lighting, and can be readily incorporated into existing VR headsets. In our previous study, we applied a Riemannian manifold-based feature extraction approach to fEMG signals recorded around the eyes and demonstrated that 11 facial expressions could be classified with a high accuracy of 85.01%, with only a single training session. However, the performance of the conventional fEMG-based FER system was not high enough to be applied in practical scenarios. In this study, we developed a new method for improving the FER performance by employing linear discriminant analysis (LDA) adaptation with labeled datasets of other users. Our results indicated that the mean classification accuracy could be increased to 89.40% by using the LDA adaptation method (p < .001, Wilcoxon signed-rank test). Additionally, we demonstrated the potential of a user-independent FER system that could classify 11 facial expressions with a classification accuracy of 82.02% without any training sessions. To the best of our knowledge, this was the first study in which the LDA adaptation approach was employed in a cross-subject manner. It is expected that the proposed LDA adaptation approach would be used as an important method to increase the usability of fEMG-based FER systems for social VR applications.

Geometry-aware manipulability learning, tracking, and transfer

Body posture influences human and robot performance in manipulation tasks, as appropriate poses facilitate motion or the exertion of force along different axes. In robotics, manipulability ellipsoids arise as a powerful descriptor to analyze, control, and design the robot dexterity as a function of the articulatory joint configuration. This descriptor can be designed according to different task requirements, such as tracking a desired position or applying a specific force. In this context, this article presents a novel manipulability transfer framework, a method that allows robots to learn and reproduce manipulability ellipsoids from expert demonstrations. The proposed learning scheme is built on a tensor-based formulation of a Gaussian mixture model that takes into account that manipulability ellipsoids lie on the manifold of symmetric positive-definite matrices. Learning is coupled with a geometry-aware tracking controller allowing robots to follow a desired profile of manipulability ellipsoids. Extensive evaluations in simulation with redundant manipulators, a robotic hand and humanoids agents, as well as an experiment with two real dual-arm systems validate the feasibility of the approach.

[Detection algorithm of paroxysmal atrial fibrillation with sparse coding based on Riemannian manifold]

In order to solve the problem that the early onset of paroxysmal atrial fibrillation is very short and difficult to detect, a detection algorithm based on sparse coding of Riemannian manifolds is proposed. The proposed method takes into account that the nonlinear manifold geometry is closer to the real feature space structure, and the computational covariance matrix is used to characterize the heart rate variability (RR interval variation), so that the data is in the Riemannian manifold space. Sparse coding is applied to the manifold, and each covariance matrix is represented as a sparse linear combination of Riemann dictionary atoms. The sparse reconstruction loss is defined by the affine invariant Riemannian metric, and the Riemann dictionary is learned by iterative method. Compared with the existing methods, this method used shorter heart rate variability signal, the calculation was simple and had no dependence on the parameters, and the better prediction accuracy was obtained. The final classification on MIT-BIH AF database resulted in a sensitivity of 99.34%, a specificity of 95.41% and an accuracy of 97.45%. At the same time, a specificity of 95.18% was realized in MIT-BIH NSR database. The high precision paroxysmal atrial fibrillation detection algorithm proposed in this paper has a potential application prospect in the long-term monitoring of wearable devices.

Gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces and applications

In this paper local and global gradient estimates are obtained for positive solutions to the following nonlinear elliptic equationΔfu+p(x)u+q(x)uα=0, on complete smooth metric measure spaces (MN,g,e−fdv) with ∞-Bakry-Émery Ricci tensor bounded from below, where α is an arbitrary real constant, p(x) and q(x) are smooth functions. As an application, Liouville-type theorems for various special cases of the equation are recovered. Furthermore, we discuss nonexistence of smooth solution to Yamabe type problem on (MN,g,e−fdv) with nonpositive weighted scalar curvature.

Statistical regression analysis of functional and shape data

We develop a multivariate regression model when responses or predictors are on nonlinear manifolds, rather than on Euclidean spaces. The nonlinear constraint makes the problem challenging and needs to be studied carefully. By performing principal component analysis (PCA) on tangent space of manifold, we use principal directions instead in the model. Then, the ordinary regression tools can be utilized. We apply the framework to both shape data (ozone hole contours) and functional data (spectrums of absorbance of meat in Tecator dataset). Specifically, we adopt the square-root velocity function representation and parametrization-invariant metric. Experimental results have shown that we can not only perform powerful regression analysis on the non-Euclidean data but also achieve high prediction accuracy by the constructed model.

Covariance matrix based fall detection from multiple wearable sensors

Falls are among the critical accidents experienced by elderly people and patients carrying some diseases. Subsequently, the detection and prevention of falls have become a hot research and industrial topic. This is due to the fact that falls are behind numerous irreversible injuries, or even death, and are weighting on the budgets of the health services. Automatic fall detection is one of the proposed solutions which aim at monitoring people who are likely to fall. Such solutions mitigate the fall impact by taking a quick action, e.g. in case of a fall occurrence, an alert is sent to the hospital. In this paper, we propose a new fall detection system relying on different signals acquired with multiple wearable sensors. Our system makes use of the covariance of the raw signals and the nearest neighbor classifier. Besides feature extraction, we also employ the covariance matrix as a straightforward mean for fusing signals from multiple sensors, to enhance the classification performance. Evaluation on two publicly available fall datasets, namely CogentLabs and DLR, demonstrates that the proposed approach is efficient when exploiting a single sensor as well as when fusing data from multiple sensors. Geodesic metrics are found to provide a higher fall detection accuracy than the Euclidean metric. The best obtained classification accuracies are 92.51% and 98.31% for CogentLabs and DLR datasets, respectively.

The obstacle problem for conformal metrics on compact Riemannian manifolds

We prove a priori estimates up to their second order derivatives for solutions to the obstacle problem of curvature equations on Riemannian manifolds ( M n , g ) arising from conformal deformation. With the a priori estimates the existence of a C 1 , 1 solution to the obstacle problem with Dirichlet boundary value is obtained by approximation.

Global Weak Rigidity of the Gauss-Codazzi-Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with corresponding different metrics.

Volume comparison for C 1 , 1 -metrics

The aim of this paper is to generalize certain volume comparison theorems (Bishop-Gromov and a recent result of Treude and Grant, Ann Global Anal Geom, 43:233-251, 2013) for smooth Riemannian or Lorentzian manifolds to metrics that are only C 1 , 1 (differentiable with Lipschitz continuous derivatives). In particular we establish (using approximation methods) a volume monotonicity result for the evolution of a compact subset of a spacelike, acausal, future causally complete (i.e., the intersection of any past causal cone with the hypersurface is relatively compact) hypersurface with an upper bound on the mean curvature in a globally hyperbolic spacetime with a C 1 , 1 -metric with a lower bound on the timelike Ricci curvature, provided all timelike geodesics starting in this compact set exist long enough. As an intermediate step, we also show that the cut locus of such a hypersurface still has measure zero in this regularity-generalizing the well-known result for smooth metrics. To show that these volume comparison results have some very nice applications, we then give a proof of Myers' theorem, of a simple singularity theorem for globally hyperbolic spacetimes, and of Hawking's singularity theorem directly in this regularity.