相关论文: Geometrical Statistics--Classical and Quantum

Forecasts of statistical constraints on model parameters using the Fisher matrix abound in many fields of astrophysics. The Fisher matrix formalism involves the assumption of Gaussianity in parameter space and hence fails to predict complex…

Quantum Field Theory (QFT) makes predictions by combining assumptions about (1) quantum dynamics, typically a Schrodinger or Liouville equation; (2) quantum measurement, usually via a collapse formalism. Here I define a "classical density…

The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an…

In the present report we discuss measures of classicality/quantumness of states of finite-dimensional quantum systems, which are based on a deviation of quasiprobability distributions from true statistical distributions. Particularly, the…

The Hellinger distance between quantum states is a significant measure in quantum information theory, known for its Riemannian and monotonic properties. It is also easier to compute than the Bures distance, another measure that shares these…

In this note we define geometric classical r-matrices and quantum R-matrices, and show how any geometric classical r-matrix can be quantized to a geometric quantum R-matrix. This is one of the simplest nontrivial examples of quantization of…

The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher-Rao metric and a pair of torsion-free conjugate connections. In recent work,…

Transport-based metrics and related embeddings (transforms) have recently been used to model signal classes where nonlinear structures or variations are present. In this paper, we study the geodesic properties of time series data with a…

In a previous article a relationship was established between the linearized metrics of General Relativity associated with geodesics and the Dirac Equation of quantum mechanics. In this paper the extension of that result to arbitrary curves…

We define metrics on Culler-Vogtmann space, which are an analogue of the Teichmuller metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate…

This paper argues that a class of Riemannian metrics, called warped metrics, plays a fundamental role in statistical problems involving location-scale models. The paper reports three new results : i) the Rao-Fisher metric of any…

We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.

A common feature of methods for analyzing samples of probability density functions is that they respect the geometry inherent to the space of densities. Once a metric is specified for this space, the Fr\'echet mean is typically used to…

Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations…

A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. For a probability distribution, the length along a path between two points can be defined as the amount of probability mass accumulated…

This article presents a novel mathematical formalism for advanced manifold--metric pairs, enhancing the frameworks of geometry and topology. We construct various D-dimensional manifolds and their associated metric spaces using functional…

We propose the generalised Fisher information or the one-parameter extended class of the Fisher information for the case of one random variable. This new form of the Fisher information is obtained from the intriguing connection between the…

The Fisher Information Matrix formalism is extended to cases where the data is divided into two parts (X,Y), where the expectation value of Y depends on X according to some theoretical model, and X and Y both have errors with arbitrary…

Classical Fisher-information asymptotics describe the covariance of regular efficient estimators through the local quadratic approximation of the log-likelihood, and thus capture first-order geometry only. In curved models, including…

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule,…