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Item type:Publication, Access status: with files , On the natural gradient of the evidence lower boundThis article studies the Fisher-Rao gradient, also referred to as the natural gradient, of the evidence lower bound (ELBO) which plays a central role in generative machine learning. It reveals that the gap between the evidence and its lower bound, the ELBO, has essentially a vanishing natural gradient within unconstrained optimization. As a result, maximization of the ELBO is equivalent to minimization of the Kullback-Leibler divergence from a target distribution, the primary objective function of learning. Building on this insight, we derive a condition under which this equivalence persists even when optimization is constrained to a model. This condition yields a geometric characterization, which we formalize through the notion of a cylindrical model .Publicationtype: Journal Article20 10 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , A concise mathematical description of active inference in discrete timeIn this paper we present a concise mathematical description of active inference in discrete time. The main part of the paper serves as a basic introduction to the topic, including a detailed example of the action selection mechanism. The appendix discusses the more subtle mathematical details, targeting readers who have already studied the active inference literature but struggle to make sense of the mathematical details and derivations. Throughout, we emphasize precise and standard mathematical notation, ensuring consistency with existing texts and linking all equations to widely used references on active inference. Additionally, we provide Python code that implements the action selection and learning mechanisms described in this paper and is compatible with pymdp environments.Publicationtype: Review Article34 18 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , On the Natural Gradient of the Evidence Lower Bound(2025); ; This article studies the Fisher-Rao gradient, also referred to as the natural gradient, of the evidence lower bound (ELBO) which plays a central role in generative machine learning. It reveals that the gap between the evidence and its lower bound, the ELBO, has essentially a vanishing natural gradient within unconstrained optimization. As a result, maximization of the ELBO is equivalent to minimization of the Kullback-Leibler divergence from a target distribution, the primary objective function of learning. Building on this insight, we derive a condition under which this equivalence persists even when optimization is constrained to a model. This condition yields a geometric characterization, which we formalize through the notion of a cylindrical model.Publicationtype: Journal Article17 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Inversion of Bayesian networksVariational autoencoders and Helmholtz machines use a recognition network (encoder) to approximate the posterior distribution of a generative model (decoder). In this paper we establish some necessary and some sufficient properties of a recognition network so that it can model the true posterior distribution exactly. These results are derived in the general context of probabilistic graphical modelling / Bayesian networks, for which the network represents a set of conditional independence statements. We derive both global conditions, in terms of d-separation, and local conditions for the recognition network to have the desired qualities. It turns out that for the local conditions the perfectness property (for every node, all parents are joined) plays an important role.Publicationtype: Journal Article45 73 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Invariance properties of the natural gradient in overparametrised systemsThe natural gradient field is a vector field that lives on a model equipped with a distinguished Riemannian metric, e.g. the Fisher–Rao metric, and represents the direction of steepest ascent of an objective function on the model with respect to this metric. In practice, one tries to obtain the corresponding direction on the parameter space by multiplying the ordinary gradient by the inverse of the Gram matrix associated with the metric. We refer to this vector on the parameter space as the natural parameter gradient. In this paper we study when the pushforward of the natural parameter gradient is equal to the natural gradient. Furthermore we investigate the invariance properties of the natural parameter gradient. Both questions are addressed in an overparametrised setting.Publicationtype: Journal Article61 232 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Bures–Wasserstein geometry for positive-definite Hermitian matrices and their trace-one subsetIn his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures–Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.Publicationtype: Journal ArticleScopus© Citations 5 69 74 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Parametrisation Independence of the Natural Gradient in Overparametrised Systems(2021-07); In this paper we study the natural gradient method for overparametrised systems. This method is based on the natural gradient field which is invariant with respect to coordinate transformations. One calculates the natural gradient of a function on the manifold by multiplying the ordinary gradient of the function by the inverse of the Fisher Information Matrix (FIM). In overparametrised models, the FIM is degenerate and therefore one needs to use a generalised inverse. We show explicitly that using a generalised inverse, and in particular the Moore-Penrose inverse, does not affect the parametrisation independence of the natural gradient. Furthermore, we show that for singular points on the manifold the parametrisation independence is not even guaranteed for non-overparametrised models.Publicationtype: Conference Paper63