Geometry in Physics

Quantum metrology holds the promise of an early practical application of quantum technologies, in which measurements of physical quantities can be made with much greater precision than what is achievable with classical technologies. In this Review, the authors collect some of the key theoretical results in quantum parameter estimation by presenting the theory for the quantum estimation of a single parameter, multiple parameters, and optical estimation using Gaussian states. The authors give an overview of results in areas of current research interest, such as Bayesian quantum estimation, noisy quantum metrology, and distributed quantum sensing. The authors address the question of how minimum measurement errors can be achieved using entanglement as well as more general quantum states. This review is presented from a geometric perspective. This has the advantage that it unifies a wide variety of estimation procedures and strategies, thus providing a more intuitive big picture of quantum parameter estimation.Quantum metrology holds the promise of an early practical application of quantum technologies, in which measurements of physical quantities can be made with much greater precision than what is achievable with classical technologies. In this Review, the authors collect some of the key theoretical results in quantum parameter estimation by presenting the theory for the quantum estimation of a single parameter, multiple parameters, and optical estimation using Gaussian states. The authors give an overview of results in areas of current research interest, such as Bayesian quantum estimation, noisy quantum metrology, and distributed quantum sensing. The authors address the question of how minimum measurement errors can be achieved using entanglement as well as more general quantum states. This review is presented from a geometric perspective. This has the advantage that it unifies a wide variety of estimation procedures and strategies, thus providing a more intuitive big picture of quantum parameter estimation.

/pdf/geometric-perspective-on-quantum-parameter-estimation-2he89dngy1.pdf

Geometric perspective on quantum parameter estimation

Assessment of causal influences is a ubiquitous and important subject across diverse research fields. Drawn from consciousness studies, integrated information is a measure that defines integration as the degree of causal influences among elements. Whereas pairwise causal influences between elements can be quantified with existing methods, quantifying multiple influences among many elements poses two major mathematical difficulties. First, overestimation occurs due to interdependence among influences if each influence is separately quantified in a part-based manner and then simply summed over. Second, it is difficult to isolate causal influences while avoiding noncausal confounding influences. To resolve these difficulties, we propose a theoretical framework based on information geometry for the quantification of multiple causal influences with a holistic approach. We derive a measure of integrated information, which is geometrically interpreted as the divergence between the actual probability distribution of a system and an approximated probability distribution where causal influences among elements are statistically disconnected. This framework provides intuitive geometric interpretations harmonizing various information theoretic measures in a unified manner, including mutual information, transfer entropy, stochastic interaction, and integrated information, each of which is characterized by how causal influences are disconnected. In addition to the mathematical assessment of consciousness, our framework should help to analyze causal relationships in complex systems in a complete and hierarchical manner.

https://www.pnas.org/content/pnas/113/51/14817.full.pdf

Unified framework for information integration based on information geometry

This paper considers the relationship between thermodynamics, information and inference. In particular, it explores the thermodynamic concomitants of belief updating, under a variational (free energy) principle for self-organization. In brief, any (weakly mixing) random dynamical system that possesses a Markov blanket-i.e. a separation of internal and external states-is equipped with an information geometry. This means that internal states parametrize a probability density over external states. Furthermore, at non-equilibrium steady-state, the flow of internal states can be construed as a gradient flow on a quantity known in statistics as Bayesian model evidence. In short, there is a natural Bayesian mechanics for any system that possesses a Markov blanket. Crucially, this means that there is an explicit link between the inference performed by internal states and their energetics-as characterized by their stochastic thermodynamics. This article is part of the theme issue 'Harmonizing energy-autonomous computing and intelligence'.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2019.0159

Markov blankets, information geometry and stochastic thermodynamics

Uncertainty computation in deep learning is essential to design robust and reliable systems. Variational inference (VI) is a promising approach for such computation, but requires more effort to implement and execute compared to maximum-likelihood methods. In this paper, we propose new natural-gradient algorithms to reduce such efforts for Gaussian mean-field VI. Our algorithms can be implemented within the Adam optimizer by perturbing the network weights during gradient evaluations, and uncertainty estimates can be cheaply obtained by using the vector that adapts the learning rate. This requires lower memory, computation, and implementation effort than existing VI methods, while obtaining uncertainty estimates of comparable quality. Our empirical results confirm this and further suggest that the weight-perturbation in our algorithm could be useful for exploration in reinforcement learning and stochastic optimization.

/pdf/fast-and-scalable-bayesian-deep-learning-by-weight-27frsotpek.pdf

Fast and Scalable Bayesian Deep Learning by Weight-Perturbation in Adam

Information geometry is a new mathematical discipline which applies the methodology of differential geometry to statistics. Therefore, families of exponential distributions are considered as embedded manifolds, called statistical manifolds. This includes so important families like the multivariate normal or the gamma distributions. Fisher information — well known in information theory — becomes a metric on statistical manifolds. The Fisher information metric enables a hyperbolic structure on the multivariate normal distributions. Information geometry offers new methods for hypothesis testings, estimation theory or stochastic filtering. These can be used in engineering areas like signal processing or video processing or finance.

http://ieeexplore.ieee.org/iel5/6423297/6450609/06450636.pdf

Information geometry and its applications

.The benefits of using multiple unmanned aerial vehicles (UAV) platform to perform certain tasks include increased robustness in the system, greater flexibility, reduced cost, as well as an increase in the speed of task completion. By acting collectively, a team of UAVs can quickly determine the approximate location of intermittently broadcasting RF emitters with simple passive angle-of-arrival sensors. Passive ground emitter location techniques are vital to electronic warfare system (EWS) providing threat warnings in hostile environments. The paper focuses on proposing and evaluating solutions for optimizing the UAV trajectories for localization of a RF emitter using Line of Bearing (LOB) measurements. The optimization approach aims to minimize the size of the target position error ellipse. A framework using only the geometry of the problem is explored to minimize the maximal bounds of the emitters’ locations without using a belief or probability distribution needed for iterative approaches, which requires a good initial guess to ensure convergence on the true position. The effectiveness of the geometric approach is shown by designing and evaluating a global decision-making strategy based on Learning Real-Time A* (LRTA*) algorithm. Through simulations, the search solution is compared to a conventional optimal control solution approach for a two dimensional space case. The LRTA* solution achieves acceptable accuracy with sufficient observation points and it is suitable for real-time purposes. The solution is also attractive for its capability to initialize iterative algorithms if needed.

Real-Time Optimized Trajectories for 2D Emitter Localization using a UAVs team

Data fusion is a key enabler to achieve information superiority in future cross-domain military operations in a system-of-systems context. In this paper, a sensor data fusion system architecture based on containerized apps and connectivity enabled by a data distribution service middleware is presented, which allows hosting collaborative data fusion functions on distributed heterogeneous platforms. Results and insights from an experimental laboratory demonstration and verification up to operator level are summarized.

Felix Opitz

Papers

Information geometry and its applications

Real-Time Optimized Trajectories for 2D Emitter Localization using a UAVs team

Joint All Domain Modular Data Fusion for Distributed Cloud/Fog/Edge Applications