Search Results (26)

We investigate the information-geometric structure of a self-consistent second-virial approximation with the van der Waals form of the second virial coefficient, using the Fisher information as a generating potential. By constructing the associated Hessian metric and scalar curvature, we obtain a geometric description of thermodynamic fluctuations that complements the standard equation-of-state approach. We show that interaction effects enter the Fisher information selectively through their temperature dependence, with attractive interactions playing the dominant role while excluded-volume contributions remain subleading. The Hessian determinant of the Fisher information changes sign along well-defined crossover loci in the reduced parameter space, separating regions of saddle-like and locally convex curvature of the fluctuation landscape, and diverges as ${\tilde{D}}^{-12}$ at the closure boundary. The scalar curvature diverges at the closure boundary as a simple pole, with a leading term that is universal and independent of the interaction parameters. We derive its exact closed-form expression, which shows that the curvature is negative for weak-to-moderate effective coupling but turns positive in the strong-coupling regime, where the underlying Fisher–Rao metric loses positive-definiteness. The curvature diverges on approach to this metric boundary, which lies strictly inside the mechanically stable region, short of the closure boundary itself. These results demonstrate that Fisher geometry provides a selective probe of interaction-induced fluctuations, capturing how thermal states reorganize under temperature variations rather than encoding phase transitions directly. This approach offers a complementary perspective on interacting systems and suggests new avenues for extending information-geometric methods to more strongly correlated and quantum regimes. Full article

Asymmetric interaction matrices encode the linear coupling structure and directed interaction patterns that arise in mathematical models of complex networks across ecology, finance, and machine learning, yet their geometric structure as points on a statistical manifold has received comparatively little systematic treatment. This paper develops a rigorous information-geometric framework for the manifold ${\mathcal{M}}_n^{+}$ of real $n\times n$ interaction matrices whose symmetric part is negative definite—equivalently, the matrices satisfying the numerical stability condition $\omega \left(A\right)={\lambda }_{\max }\left(S\left(A\right)\right)<0$. The symmetric part $S\left(A\right)=(A+A^{\mathsf{T}})/2$ and the skew-symmetric part $K\left(A\right)=(A-A^{\mathsf{T}})/2$ correspond, respectively, to the metric structure and the torsion of the induced statistical manifold. We construct the natural augmented Riemannian metric g on ${\mathcal{M}}_n^{+}$ as the sum of the Fisher–Rao pullback metric through $-S(·)$ and a Frobenius term on $K(·)$, derive explicit formulae for the sectional curvature in the mixed symmetric–skew plane, and prove that the sectional curvature vanishes if and only if A is normal. The central theoretical result is a curvature-mediated stability theorem: a Fisher–Rao stability margin, derived from the precision representative of A, provides a sharp, computationally accessible certificate for the asymptotic stability of the linear dynamical system $\dot{\mathbf{x}}=A\mathbf{x}$, with the instability boundary lying at infinite Fisher–Rao distance. We further establish an information-geometric reformulation of May’s stability criterion for random ecological networks, a curvature-based covariance regularisation scheme for financial correlation matrices, and a Jacobian stability bound for deep neural networks. All the main results are illustrated with explicit $3\times 3$ and $4\times 4$ numerical examples. Full article

As large language models process ever longer contexts, the positional encoding—most commonly rotary position embeddings (RoPEs)—must remain numerically trustworthy. We give an information-geometric analysis of why the attention matrix becomes ill-conditioned as the sequence length L grows. We first show that the tempting frequency-aliasing explanation fails, because RoPE’s multi-frequency code keeps positions well separated. The mechanism is statistical: at fixed softmax temperature the attention entropy grows like $logL$, and the rows spread their mass over a collision support (inverse participation ratio) that grows polynomially as $L^a$ ($a\approx 0.9$ measured), confining the matrix’s trailing singular values and producing a numerical-rank collapse. We prove, and confirm in 50-digit arithmetic, that the spectral condition number is intrinsically ill-posed here, so the numerical rank is the correct diagnostic. From the exact entropy–temperature identity $\mathrm{d}H/\mathrm{d}\beta =-\beta {\mathrm{Var}}_p\left(e\right)$ we derive the entropy-invariant schedule ${\beta }_{★}\left(L\right)={(L/L_{\mathrm{ref}})}^{c_0}$, Curvature-Corrected RoPE (CC-RoPE), and prove that in the diffuse near-circulant regime it eliminates the collapse at a characterised conditioning–mixing cost, whereas no relative-position-preserving warp can. The two signatures are further confirmed in a trained RoPE model (Pythia-160M); the downstream perplexity effect is framed as a falsifiable prediction. Full article

Graph-based narrative extraction relies on a coherence function to score transitions between events, but the coherence metrics in current use are defined operationally and lack an information-theoretic foundation. We study the composite metric $C=\sqrt{A·T}$, where A is the angular similarity of document embeddings and $T=1-d_{\mathrm{JS}}$ is the topic proximity through the Jensen–Shannon distance of soft cluster memberships, and we provide an information-geometric reading of this metric together with an axiomatic characterization of the geometric-mean combinator. On the product manifold ${\mathbb{S}}^{d-1}\times {\Delta }_{+}^{K-1}$, the negative log-coherence decomposes additively into an angular and a topic cost. Because the Riemannian metric tensor induced by the Jensen–Shannon distance on the simplex is proportional to the Fisher information matrix, the topic component is locally consistent with the Fisher–Rao metric singled out by Chentsov’s theorem. Within a parametric family of combinators (the compensability spectrum), the geometric mean is the unique combinator consistent with four natural axioms (a boundary/veto condition, symmetry, log-additivity, normalization), and the construction also motivates a proper product metric $d_{\times }$ that we use as a reference distance. Experiments on four corpora spanning news and academic domains (40 to 6000 documents), three general-purpose embedding families (GPT-4/ada-002, MPNet, MiniLM-L6) plus citation-aware SPECTER2, and three alternative topic models (LDA, soft k-means, GMM) are consistent with the framework: the Fisher identity holds with $R\ge 0.99$, the geometric mean tracks $d_{\times }$ closely ($\rho =0.999$), and a downstream LLM-as-judge consistency check shows that the geometric mean is not empirically dominated by any alternative combinator or single-channel baseline. Sweeping the compensability spectrum, the bottleneck-coherence gap between extracted storylines and random sequences splits into a symmetric component—maximized at the geometric mean on the four corpora above and a fifth, human-navigation corpus—and a displacement term; a cross-modal case study on a human-curated image narrative reproduces the same effect in a second modality. Together, these results provide an information-geometric justification for the composite coherence metric and articulate the conditions under which the geometric mean is the natural choice. Full article

This paper introduces a geometric framework for analyzing finite-energy signals observed with additive noise by representing them as points on statistical manifolds equipped with the Fisher–Rao metric. Each signal is associated with a parameter vector θ, which defines a unique probability distribution $p\left(\mathit{x}\right|\mathit{\theta })$ on a statistical manifold. We propose a unified approach based on the normal multivariate model to describe a raw signal mixed with additive stationary noise. In the approach considered, the background noise is typically assumed to be stationary, whereas the unknown signal is regarded as deterministic. Leveraging tools from information geometry, we compute geodesic equations for the statistical manifolds. We re-derive known results regarding the multivariate normal models and extend them to the signal processing domain. We show that in some cases, the geodesic equations can be solved to obtain a closed-form expression of the Fisher–Rao distance. This expression corresponds to a minimum bound when the sub-manifold is not geodesic, revealing a fundamental geometric constraint in signal parameter estimation. We introduce the spectral distance function, which characterizes the influence of each spectral component of the signals on the Fisher–Rao distance. Our findings provide theoretical insights for signal clustering and machine learning applications, where geometric distances can characterize classification and estimation tasks. Full article

In this paper, we study the geometric structure induced by the canonical reciprocal cost function and its natural n-dimensional extension. In logarithmic coordinates, the potential depends only on the linear combination $S=\alpha ·t$, and the associated Hessian metric has rank one at every point. The geometry is intrinsically degenerate and effectively one-dimensional, with an $(n-1)$-dimensional null distribution. On the other hand, when the same function is expressed in the original x-coordinates, the corresponding Hessian is generically nondegenerate and defines a pseudo-Riemannian metric away from explicit singular hypersurfaces. We further analyze affine and Levi-Civita geodesics and compare their behavior. In particular, affine geodesics in logarithmic coordinates are globally defined, while in x-coordinates their behavior is restricted by the domain and the singular set. Finally, we relate the construction to symmetrized Itakura–Saito and Bregman divergences, and give a Fisher–Rao realization of the logarithmic Hessian metric. Full article

Non-parametric information geometry has long faced an “intractability barrier”: in the infinite-dimensional setting, the Fisher–Rao metric is a weak Riemannian metric functional that lacks a bounded inverse, rendering classical optimization and estimation techniques computationally inaccessible. This paper resolves this barrier by building the statistical manifold on the Orlicz space $L_0^{\mathsf{\Phi }}\left(P_f\right)$ (the Pistone–Sempi manifold), which provides the necessary exponential integrability for score functions and a rigorous Fréchet differentiability for the Kullback–Leibler divergence. We introduce a novel Structural Decomposition of the Tangent Space ($T_{f}M=S\oplus S^{\perp }$), where the infinite-dimensional space is split into a finite-dimensional covariate subspace (S)—representing the observable system—and its orthogonal complement ($S^{\perp }$). Through this decomposition, we derive the Covariate Fisher Information Matrix (cFIM), denoted as $G_f$, which acts as the computable “Hilbertian slice” of the otherwise intractable metric functional. Key theoretical contributions include proving the Trace Theorem ($H_G\left(f\right)=\mathrm{Tr}\left(G_f\right)$) to identify G-entropy as a fundamental geometric invariant; demonstrating the Geometric Invariance of the Covariate Fisher Information Matrix (cFIM) as a covariant $(0,2)$-tensor under reparameterization; establishing the cFIM as the local Hessian of the KL-divergence; and characterizing the Efficiency Standard through a generalized Cramer–Rao Lower Bound for semi-parametric inference within the Orlicz manifold. Furthermore, we demonstrate that this framework provides a formal mathematical justification for the Manifold Hypothesis, as the structural decomposition naturally identifies the low-dimensional subspace where information is concentrated. By shifting the focus from the intractable global manifold to the tractable covariate geometry, this framework proves that statistical information is not a property of data alone, but an active geometric interaction between the environment (data), the system (covariate subspace), and the mechanism (Fisher–Rao connection). Full article

In parametric statistics, it is well established that the canonical measures of estimator performance—such as bias, variance, and mean squared error—are inherently dependent on the parameterization of the model. Consequently, these quantities describe the behavior of an estimator only relative to a particular parameterization, rather than representing intrinsic properties of either the estimator itself or the underlying probability distribution it seeks to estimate. Some years ago, the authors introduced a framework, termed the intrinsic analysis of point estimation, in which tools from information geometry were employed to construct analogues of classical statistical notions that are intrinsic to both the estimator and the associated probability measure. Within this framework, a contravariant vector field was introduced to define the intrinsic bias, while the squared Riemannian distance naturally emerged as the intrinsic analogue of the classical squared distance. Intrinsic counterparts of the Cramér–Rao inequalities, as well as the Rao–Blackwell and Lehmann–Scheffé theorems, were also established. The present work extends the intrinsic analysis—originally founded on the concept of intrinsic risk, a fundamentally local measure of estimator performance—to an approach that characterizes the estimator over an entire region of the parameter space, thereby yielding an intrinsically global perspective. Building upon intrinsic risk, two indices are proposed to evaluate estimator performance within a bounded region: (i) the integral of the intrinsic risk with respect to the Riemannian volume over the specified region, and (ii) the maximum intrinsic risk attained within that region. The Riemannian volume induced by the Fisher information metric on the manifold associated with the parametric model provides a natural means of averaging the intrinsic risk. Using variational methods, integral inequalities of the Cramér–Rao type are derived for the mean squared integrated Rao distance of the estimators, thereby extending previous contributions by several authors. Furthermore, lower bounds for the maximum intrinsic risk are obtained through corresponding integral formulations. Full article

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. Full article

The main goal of machine learning is the creation of self-learning algorithms in many areas of human activity. It allows a replacement of a person with artificial intelligence in seeking to expand production. The theory of artificial neural networks, which have already replaced humans in many problems, remains the most well-utilized branch of machine learning. Thus, one must select appropriate neural network architectures, data processing, and advanced applied mathematics tools. A common challenge for these networks is achieving the highest accuracy in a short time. This problem is solved by modifying networks and improving data pre-processing, where accuracy increases along with training time. Bt using optimization methods, one can improve the accuracy without increasing the time. In this review, we consider all existing optimization algorithms that meet in neural networks. We present modifications of optimization algorithms of the first, second, and information-geometric order, which are related to information geometry for Fisher–Rao and Bregman metrics. These optimizers have significantly influenced the development of neural networks through geometric and probabilistic tools. We present applications of all the given optimization algorithms, considering the types of neural networks. After that, we show ways to develop optimization algorithms in further research using modern neural networks. Fractional order, bilevel, and gradient-free optimizers can replace classical gradient-based optimizers. Such approaches are induced in graph, spiking, complex-valued, quantum, and wavelet neural networks. Besides pattern recognition, time series prediction, and object detection, there are many other applications in machine learning: quantum computations, partial differential, and integrodifferential equations, and stochastic processes. Full article

A statistical transformation model consists of a smooth data manifold, on which a Lie group smoothly acts, together with a family of probability density functions on the data manifold parametrized by elements in the Lie group. For such a statistical transformation model, the Fisher–Rao semi-definite metric and the Amari–Chentsov cubic tensor are defined in the Lie group. If the family of probability density functions is invariant with respect to the Lie group action, the Fisher–Rao semi-definite metric and the Amari–Chentsov tensor are left-invariant, and hence we have a left-invariant structure of a statistical manifold. In the present work, the general framework of statistical transformation models is explained. Then, the left-invariant geodesic flow associated with the Fisher–Rao metric is considered for two specific families of probability density functions on the Lie group. The corresponding Euler–Poincaré and the Lie–Poisson equations are explicitly found in view of geometric mechanics. Related dynamical systems over Lie groups are also mentioned. A generalization in relation to the invariance of the family of probability density functions is further studied. Full article

Modern day Bayesian imaging problems in astrophysics as well as other scientific areas often result in non-Gaussian and very high-dimensional posterior probability distributions as their formal solution. Efficiently accessing the information contained in such distributions remains a core challenge in modern statistics as, on the one hand, point estimates such as Maximum a Posteriori (MAP) estimates are insufficient due to the nonlinear structure of these problems, while on the other hand, posterior sampling methods such as Markov Chain Monte Carlo (MCMC) techniques may become computationally prohibitively expensive in such high-dimensional settings. To nevertheless enable (approximate) inference in these cases, geometric Variational Inference (geoVI) has recently been introduced as an accurate Variational Inference (VI) technique for nonlinear unimodal probability distributions. It utilizes the Fisher–Rao information metric (FIM) related to the posterior probability distribution and the Riemannian manifold associated with the FIM to construct a set of normal coordinates in which the posterior metric is approximately the Euclidean metric. Transforming the posterior distribution into these coordinates results in a distribution that takes a particularly simple form, which ultimately allows for an accurate approximation with a normal distribution. A computationally efficient approximation of the associated coordinate transformation has been provided by geoVI, which now enables its application to real-world astrophysical imaging problems in millions of dimensions. Full article

We characterise the geometry of the statistical Roegenian manifold that arises from the equilibrium distribution of an income of noninteracting identical economic actors. The main results for ideal income are included in three subsections: partition function in distribution, scalar curvature, and geodesics. Although this system displays no phase transition, its analysis provides an enlightening contrast with the results of Van der Waals Income in Roegenian Economics, where we shall examine the geometry of the economic Van der Waals income, which does exhibit a “monetary policy as liquidity—income” transition. Here we focus on three subsections: canonical partition function, economic limit, and information geometry of the economic Van der Waals manifold. Full article

Approximate Bayesian computation is a likelihood-free inference method which relies on comparing model realisations to observed data with informative distance measures. We obtain functional data that are not only subject to noise along their y axis but also to a random warping along their x axis, which we refer to as the time axis. Conventional distances on functions, such as the L₂ distance, are not informative under these conditions. The Fisher–Rao metric, previously generalised from the space of probability distributions to the space of functions, is an ideal objective function for aligning one function to another by warping the time axis. We assess the usefulness of alignment with the Fisher–Rao metric for approximate Bayesian computation with four examples: two simulation examples, an example about passenger flow at an international airport, and an example of hydrological flow modelling. We find that the Fisher–Rao metric works well as the objective function to minimise for alignment; however, once the functions are aligned, it is not necessarily the most informative distance for inference. This means that likelihood-free inference may require two distances: one for alignment and one for parameter inference. Full article

A geometrical formulation of estimation theory for finite-dimensional $C^{\ast }$-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer–Rao and Helstrom bounds for parametric statistical models with discrete and finite outcome spaces is presented. Full article