Questions tagged [information-geometry]

Information geometry is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. This is done by taking probability distributions for a statistical model as the points of a Riemannian manifold, forming a statistical manifold. The Fisher information metric provides the Riemannian metric.

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1answer
188 views

Which books should I read in order to be prepared to study information geometry?

At the moment, I am preparing my master's thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information ...
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55 views

Manifold structure of Gaussian mixtures

Fix $l$ a positive integer. Let $\mathcal{M}$ denote the set of Gaussian mixtures of the form $$ \sum_{i=1}^l k_i \mu_i, $$ where $\mu_i $ is a non-degenerate Gaussian measure on $\mathbb{R}^k$ and ...
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38 views

Bounding the total variation metric between Gaussian mixtures

Let $\mathcal{P}(\mathbb{R}^d)$ the space of probability measures on $\mathbb{R}^d$ with total variation metric $\delta$, fix $k \in \mathbb{N}$, and let $\mathcal{P}'\subset \mathcal{P}(\mathbb{R}^d)$...
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Inequality on the Kullback-Leibler divergence

Let us define the arithmetic, geometric, and harmonic means of $x,y \in \mathbb{R}$ weighted by $\alpha =(\alpha_x,\alpha_y) \in [0,1]$, respectively as \begin{equation} a_\alpha(x,y) = \frac{\...
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Relation between the covariance of a random variable taking values in an embedded submanifold and the covariance matrix in the ambient Euclidean space

Let $M^m \subset \mathbb{R}^d, m < d $ be an $m$-dimensional embedded submanifold. Let $X: \Omega \to M^m$ be a manifold valued random variable. Then we've apparently two different notions of ...
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185 views

Pythagorean theorems for other distances

Question The usual projection in $\mathbb{R}^n$ on a subspace can be defined as the point that minimizes the squared distance to the subspace. I'll call the Pythagorean theorem the easy fact that, ...
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93 views

Connecting Wasserstein distance with mutual information?

Suppose I have Markov chains: $$X \rightarrow f(X) \rightarrow g(X)$$ $$Y \rightarrow f(Y) \rightarrow g(Y)$$ where it is known that minimizing the $\mathbb{E}(g(X)) - \mathbb{E}(g(Y))$ minimizes the ...
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1answer
91 views

Does a 1-Lipschitz function preserve mutual information between two random variables?

Suppose we have a 1-Lipschitz function $f$ such that 1-Lipschitzness is preserved, with $D_A(f(X), f(Y)) \leq D_B(X, Y)$ for some metric spaces $A$ and $B$. Does this also imply that $I(f(X); f(Y)) = ...
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286 views

Introduction to information geometry and/or geometric control theory

Some background: I'vebeen searching for a research project to work through my grad studies and I found information geometry like a strong candidate but the amount of work out there is overwhelming. I ...
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123 views

Is there any geometric interpretation for the trace of Fisher information matrix?

Consider a parametric family $p_\theta(x)$ of distributions, with parameter $\theta \in \Theta \subseteq \mathbb R^p$. If the mapping $\theta \mapsto p_\theta(x)$ is continuously differentiable at $\...
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Relation between information geometry and geometric deep learning

Disclaimer: This is a cross-post from a very similar question on math.SE. I allowed myself to post it here after reading this meta post about cross-posting between mathoverflow and math.SE, I did ...
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1answer
115 views

Relationship between $\alpha$-divergences?

I am working with $\alpha$-divergences and was wondering how understand the relationship between the definitions of Renyi and Amari? Renyi: $D_{\alpha}[p||q] = \frac{1}{\alpha - 1} \log \int p^{\...
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Convexity of exponential family

It is known that (given a $\sigma$-finite Borel reference measure $\nu$ on $\mathbb{R}$) the parameter space of an exponential family is convex in Euclidean space. However, my question is, for an the ...
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0answers
41 views

Variational inference: Does the natural gradient follow (Fisher-Rao) geodesics locally?

Amari's natural gradient descent is a well-known optimisation algorithm for functionals defined on statistical manifolds. It consists of preconditioning the vanilla gradient descent update rule with ...
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78 views

Binary search extension for determining a hyperplane splitting a set of points in $\mathbb{R}^d$

We are given a set $S$ of $n$ points in $\mathbb{R}^d$ and a (hidden) vector $\mathbf{w}\in\mathbb{R}^d$, where each point $\mathbf{v}\in S$ is associated with a binary label equal to the sign of $\...
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121 views

Jensen-Shannon Divergence of Sample Distributions

Given normal distributions with a single positional and variation parameter each, $p_1=\big[\mu_1, \sigma_1\big]$, $p_2=\big[\mu_2, \sigma_2\big]$, we define their Jensen-Shannon divergence as: $$ \...
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2answers
554 views

Approximation of Wasserstein distance between $p_\theta$ and $p_{\theta + d\theta}$

Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid $$\operatorname{KL}(p_\theta\...
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1answer
122 views

Interpolation inequality related to the 5/3-Laplace operator

I'm having trouble with an estimate that would be helpful in information geometry. The background is the following. Suppose we have a smooth positive function $g:X \to \mathbb{R}^+$ where $X$ is a ...
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Reviews of Probability in High Dimension not by Van Handel

I'm completely in love with Ramon van Handel's lecture notes Probability in High Dimension and I would like to find more learning resources. Lecture notes or reviews would be ideal as anything in this ...
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0answers
131 views

Parametric distances on product spaces of measures

Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you. Let $X$ be a topological ...
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151 views

Distance measures that preserve Pythagoras' theorem but break the triangle inequality

In information geometry, we can think of the Kullback-Leibler divergence as being "something like a squared distance." The sense of this is that if we have three probability measures, $P$, $Q$ and $R$...
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1answer
153 views

Is Bregman divergence independent of coordinates?

Question Is Bregman divergence free of coordinates? Although it is invariant w.r.t. which local affine coordinate you take, is it possible to prove that it does not change w.r.t. an arbitrary change ...
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138 views

Covariance operator analogue for manifolds and respective measure manifolds

Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
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197 views

Partial information decomposition for tangle machines

In (Williams and Beer, 2010), they define the partial information decomposition (PID) as a generalization of Shannon's Mutual Information for multiple information sources. Their key insight is that ...
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Minimizer of a class of SDEs

Setup Let $\mathscr{H}$ be a separable Hilbert space, $\mathcal{X}\triangleq \langle \Omega,\mathscr{F},\mathscr{F}_t,\mathbb{P}\rangle$ be a stochastic base and $X_t$ be an $H$-valued stochastic ...
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1answer
222 views

How much can KL divergence decrease by diluting the reference distribution

Let $\Omega$ be a countable set and $\mu,\nu\colon\Omega\to[0,1]$ be distributions on $\Omega$, that is we have $\sum_{x\in\Omega}\mu(x)=1$ and likewise for $\nu$. The Kullback-Leibler divergence of $\...
2
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1answer
184 views

(quasi)metric on Riemannian manifolds via Brownian Motion?

Given points $a$ and $b$ on a Riemannian manifold $M$, I would like a (quasi)metric that corresponds to some property of Brownian Motion from $a$ to $b$ (or rather, to $\epsilon$-ball $B = \{ x : |x - ...
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342 views

About optimization with Renyi divergence

Can someone link me to some pedagogic example of computing the Renyi divergence between two discrete/continuous distributions? Like examples where someone has been able to obtain a neat closed form or ...
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0answers
198 views

Relation between Aitchison Distance on a Simplex and Geodesic distance on the multinomial manifold [closed]

I am trying to understand the difference/relation between the Aitchison distance on a simplex $$\left[ \sum^D_{k=1} (\log{\frac{x_{ik}}{g(\mathbf{x}_i)}} - \log{\frac{x_{jk}}{g(\mathbf{x}_j)}})^2 \...
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1answer
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Research situation in the field of Information Geometry

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field. I have read (4) and parts of (3). ...