Geometry of the Fisher–Rao metric on the space of smooth densities on a compact manifold

Bruveris, Martins; Michor, Peter W.

doi:10.1002/mana.201600523

Cited by 7 publications

(7 citation statements)

References 13 publications

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“…This is carried out in the context of the exponential Orlicz manifold in [17], where it is applied to the spatially homogeneous Boltzmann equation. Manifolds modelled on the Banach spaces C k b (B; R), where B is an open subset of an underlying (Banach) sample space, are developed in [28], and manifolds modelled on Fréchet spaces of smooth densities are developed in [4,7] and [28].…”

Section: Introductionmentioning

confidence: 99%

A class of non-parametric statistical manifolds modelled on Sobolev space

Newton

2019

Info. Geo.

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We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on R d . The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports the Fisher-Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this "lifts" to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space W 2,1 ; i.e. it maps each of these spaces continuously into itself. It also maps continuously between other fixed-norm spaces with a loss of Lebesgue exponent that increases with the number of derivatives. Some of the results make essential use of a log-Sobolev embedding theorem. Each manifold contains a smoothly embedded submanifold of probability measures. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker-Planck equation) are outlined.

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Section: Introductionmentioning

confidence: 99%

A class of non-parametric statistical manifolds modelled on Sobolev space

Newton

2019

Info. Geo.

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“…This is a convincing evidence that information metric might be the optimal complexity metric for deep network complexity. In [19], it was proved that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the FisherRao metric. This diffeomorphism invariant property perfectly matches the diffeomorphism invariance of spacetime.…”

Section: B G Com Of Deep Networkmentioning

confidence: 99%

Deep network as memory space: complexity, generalization, disentangled representation and interpretability

Dong¹,

Zhou²

2019

Preprint

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By bridging deep networks and physics, the programme of geometrization of deep networks was proposed as a framework for the interpretability of deep learning systems. Following this programme we can apply two key ideas of physics, the geometrization of physics and the least action principle, on deep networks and deliver a new picture of deep networks: deep networks as memory space of information, where the capacity, robustness and efficiency of the memory are closely related with the complexity, generalization and disentanglement of deep networks. The key components of this understanding include:(1) a Fisher metric based formulation of the network complexity; (2)the least action (complexity=action) principle on deep networks and (3)the geometry built on deep network configurations. We will show how this picture will bring us a new understanding of the interpretability of deep learning systems.

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“…In recent related work [5,7], the authors construct a manifold of smooth densities on an underlying finite-dimensional manifold by considering such densities to be smooth sections of the associated volume bundle. (This is a vector bundle of dimension 1 that endows the underlying manifold with an intrinsic notion of volume.)…”

Section: Introductionmentioning

confidence: 99%

“…When restricted to the submanifold of probability measures, these all coincide (modulo scaling) with the Fisher-Rao metric. In [7], they develop the Levi-Civita covariant derivative and carry out a number of extensions and completions of the manifold in order to study its global geometry.…”

Section: Introductionmentioning

confidence: 99%

Manifolds of differentiable densities

Newton

2018

ESAIM: PS

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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 k b with respect to appropriate reference measures. The case k = ∞, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari's α-covariant derivatives for all α ∈ R. By construction, they are C ∞ -embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually (α = ±1) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the α-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the α-divergences are of class C ∞ .

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Geometry of the Fisher–Rao metric on the space of smooth densities on a compact manifold

Cited by 7 publications

References 13 publications

A class of non-parametric statistical manifolds modelled on Sobolev space

A class of non-parametric statistical manifolds modelled on Sobolev space

Deep network as memory space: complexity, generalization, disentangled representation and interpretability

Manifolds of differentiable densities

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