arXiv research

Study curvature and torsion in Gaussian distribution's dual coordinate system.

Defines vector Laplacian on statistical manifolds.

We study the family of $α$-connections of Amari-Chentsov on the homogeneous space $\mathcal{D}(M)/\mathcal{D}_μ(M)$ of diffeomorphisms modulo volume-preserving diffeomorphims of a compact manifold $M$. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems.

Characterizes connections on multivariate normal distributions.

Dual affine connections on Riemannian manifolds have played a central role in the field of information geometry since their introduction by Amari. Here I would like to extend the notion of dual connections to general vector bundles with an inner product, in the same way as a unitary connection generalizes a metric affi…

A new Weyl prior is proposed for Bayesian statistics, offering a more canonical choice for parameter α.

We review basic notions in the field of information geometry such as Fisher metric on statistical manifold, $α$-connection and corresponding curvature following Amari's work . We show application of information geometry to asymptotic statistical inference.

We develope a new and general notion of parametric measure models and statistical models on an arbitrary sample space $Ω$ which does not assume that all measures of the model have the same null sets. This is given by a diffferentiable map from the parameter manifold $M$ into the set of finite measures or probability me…

New connections found on zero-mean multivariate normal distributions.

We deal with finite dimensional differentiable manifolds. All items are concerned with are differentiable as well. The class of differentiability is $C^\infty$. A metric structure in a vector bundle $E$ is a constant rank symmetric bilinear vector bundle homomorphism of $E\times E$ in the trivial bundle line bundle. We…

Introduces new Finsler metrics and connects them to information geometry.

New geometric interpretation of Amari-Cencov α-connections on probability densities.

Improved Bayesian inference using power priors with historical data.

New insights into Markov chain geometry via positive transition measures.

Characterizes connections on normal distributions manifold.

The paper extends gradient flow and relaxation studies to non-flat Riemannian manifolds.

A new geometric structure for singular models is introduced.

Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. Thi…

The extended Kalman filter is perhaps the most standard tool to estimate in real time the state of a dynamical system from noisy measurements of some function of the system, with extensive practical applications (such as position tracking via GPS). While the plain Kalman filter for linear systems is well-understood, th…

Study classifies submanifolds in probability simplex.

On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q spaces for 1/p+1/q=1, we show that we can introduce the α-divergence, for αin (…

The Conant-Ashby theorem is verified for hypergraph observers, leading to unique learning rules.

Noise causes learning plateaus in neural networks.

The paper explores how information geometry impacts classical CR inequalities.

We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case $k=\infty$, in which the manifolds are modelled on Fréchet spaces, is included.…

Optimization algorithms that leverage gradient covariance information, such as variants of natural gradient descent (Amari, 1998), offer the prospect of yielding more effective descent directions. For models with many parameters, the covariance matrix they are based on becomes gigantic, making them inapplicable in thei…

We cast Amari's natural gradient in statistical learning as a specific case of Kalman filtering. Namely, applying an extended Kalman filter to estimate a fixed unknown parameter of a probabilistic model from a series of observations, is rigorously equivalent to estimating this parameter via an online stochastic natural…

More than thirty years ago, Charnes, Cooper and Schinnar (1976) established an enlightening contact between economic production functions (EPFs) -- a cornerstone of neoclassical economics -- and information theory, showing how a generalization of the Cobb-Douglas production function encodes homogeneous functions. As ex…

This paper studies moduli spaces of statistical structures on Lie groups.

In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove that the rays of level-sets defined by a pseudo-distance are generated by the sum of two tangent vectors. By relying on these vector…

Paper studies statistical manifolds with logarithmic divergences.

Study reveals how model volume affects learning curves in machine learning.

The parametric complexity is the key quantity in the minimum description length (MDL) approach to statistical model selection. Rissanen and others have shown that the parametric complexity of a statistical model approaches a simple function of the Fisher information volume of the model as the sample size $n$ goes to in…

Introduces a new geometric framework for probability distributions.

The paper introduces natural α-embeddings for item representations.

Study of generalized Csiszár divergences and their application to Cramér-Rao bounds.

Paper finds local normal forms for wavefronts in flat coordinates.

One way to avoid overfitting in machine learning is to use model parameters distributed according to a Bayesian posterior given the data, rather than the maximum likelihood estimator. Stochastic gradient Langevin dynamics (SGLD) is one algorithm to approximate such Bayesian posteriors for large models and datasets. SGL…

Quantum connections replace metrics with operator inner products.

Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections between the geometric properties of the induced manifold and statistical properti…

Introduces new geometric framework for probability densities on manifolds.

Proves a general connected sum formula for families Seiberg-Witten invariants.

Proposes a new stochastic optimization method for MLR models.

Derives an integral formula for G2-structures.

Derives integral formulae on weighted manifolds.

Paper derives trace formula for magnetic Laplacian at zero energy.

The Gauss formula is extended to various Laplacians on submanifolds.

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…