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Fisher Information (FI) is a quantity ubiquitously measured in such varied areas like metrology, machine learning, and biological complexity. Mathematically, it represents a lower bound in the variance of unknown parameters that are related…

We develop a family of infinite-dimensional Banach manifolds of measures on an abstract measurable space, employing charts that are "balanced" between the density and log-density functions. The manifolds, $(\tilde{M}_{\lambda},\lambda\in…

Let $\phi$ be a polynomial over $K$ (a field of characteristic 0) such that the Hessian of $\phi$ is a nonzero constant. Let $\bar\phi$ be the formal Legendre Transform of $\phi$. Then $\bar\phi$ is well-defined as a formal power series…

The high-dimensional parameter space of deep neural networks -- the neuromanifold -- is endowed with a unique metric tensor defined by the Fisher information. Reliable and scalable computation of this metric tensor is valuable for theorists…

We study information-theoretic phase transitions for the detectability of latent geometry in bipartite random geometric graphs RGGs with Gaussian d-dimensional latent vectors while only a subset of edges carries latent information…

Structured optical beams possess rich spatial features that are commonly characterized using entropic measures of field complexity. However, such measures do not directly quantify the operational usefulness of optical structure for…

We propose to study the Hessian metric of a functional on the space of probability measures endowed with the Wasserstein $2$-metric. We name it transport Hessian metric, which contains and extends the classical Wasserstein-$2$ metric. We…

We prove that the (square root) Fisher information functional is a strong Wasserstein upper gradient of the entropy on non-convex Riemannian domains. This fills a gap in the literature by allowing one to completely dispense from…

Latent space geometry has shown itself to provide a rich and rigorous framework for interacting with the latent variables of deep generative models. The existing theory, however, relies on the decoder being a Gaussian distribution as its…

In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian…

In this paper we study transversely holomorphic foliations of complex codimension one with some hypothesis on the transverse structure.

This paper provides a systematic approach to semiparametric identification that is based on statistical information as a measure of its "quality". Identification can be regular or irregular, depending on whether the Fisher information for…

In this brief paper we revisit the Fisher information content of cosmological power spectra or two-point functions of Gaussian fields in order to comment on the assumption of Gaussian estimators and the use of parameter-dependent covariance…

It is proposed that the mathematical formalism that is most appropriate for the study of spatially non-integrable cosmological models is the transverse geometry of a one-dimensional foliation (congruence) defined by a physical observer. By…

We introduce a positive Hermitian operator, the Fisher operator, and use it to examine a measurement process incorporating unitary dynamics and complete measurements. We develop the idea of information complement, the minimization of which…

We consider an inference on the eigenvalues of the covariance matrix of a multivariate normal distribution. The family of multivariate normal distributions with a fixed mean is seen as a Riemannian manifold with Fisher information metric.…

The Fisher information metric or the Fisher-Rao metric corresponds to a natural Riemannian metric defined on a parameterized family of probability density functions. As in the case of Riemannian geometry, we can define a distance in terms…

We introduce the information geometry module of the Python package Geomstats. The module first implements Fisher-Rao Riemannian manifolds of widely used parametric families of probability distributions, such as normal, gamma, beta,…

We propose a novel Riemannian geometric framework for variational inference in Bayesian models based on the nonparametric Fisher-Rao metric on the manifold of probability density functions. Under the square-root density representation, the…

The importance of the quantum Fisher information metric is testified by the number of applications that this has in very different fields, ranging from hypothesis testing to metrology, passing through thermodynamics. Still, from the rich…