Searcharxiv⌕ Search

arXiv subjects

Jacobus W. Portegies

Publications and source records attributed to Jacobus W. Portegies.

At least 19 recordsLinked to original sources
arXiv2026

Semicontinuity of capacity under pointed intrinsic flat convergence

The concept of the capacity of a compact set in $\mathbb R^n$ generalizes readily to noncompact Riemannian manifolds and, with more substantial work, to metric spaces (where multiple natural definitions of capacity are possible). Motivated by analytic and geometric considerations, and in particular Jauregui's definition of capacity-volume mass and Jauregui and Lee's results on the lower semicontinuity of the ADM mass and Huisken's isoperimetric mass, we investigate how the capacity functional behaves when the background spaces vary. Specifically, we allow the background spaces to consist of a sequence of local integral current spaces converging in the pointed Sormani--Wenger intrinsic flat sense. For the case of volume-preserving ($\mathcal{VF}$) convergence, we prove two theorems that demonstrate an upper semicontinuity phenomenon for the capacity: one version is for balls of a fixed radius centered about converging points; the other is for Lipschitz sublevel sets. Our approach is motivated by Portegies' investigation of the semicontinuity of eigenvalues under $\mathcal{VF}$ convergence. We include examples to show the semicontinuity may be strict, and that the volume-preserving hypothesis is necessary. Finally, there is a discussion on how capacity and our results may be used towards understanding the general relativistic total mass in non-smooth settings.

math.DG↗
arXiv2024

Topological degree as a discrete diagnostic for disentanglement, with applications to the $Δ$VAE

We investigate the ability of Diffusion Variational Autoencoder ($Δ$VAE) with unit sphere $\mathcal{S}^2$ as latent space to capture topological and geometrical structure and disentangle latent factors in datasets. For this, we introduce a new diagnostic of disentanglement: namely the topological degree of the encoder, which is a map from the data manifold to the latent space. By using tools from homology theory, we derive and implement an algorithm that computes this degree. We use the algorithm to compute the degree of the encoder of models that result from the training procedure. Our experimental results show that the $Δ$VAE achieves relatively small LSBD scores, and that regardless of the degree after initialization, the degree of the encoder after training becomes $-1$ or $+1$, which implies that the resulting encoder is at least homotopic to a homeomorphism.

cs.LG↗
arXiv2022

Neural Langevin Dynamics: towards interpretable Neural Stochastic Differential Equations

Neural Stochastic Differential Equations (NSDE) have been trained as both Variational Autoencoders, and as GANs. However, the resulting Stochastic Differential Equations can be hard to interpret or analyse due to the generic nature of the drift and diffusion fields. By restricting our NSDE to be of the form of Langevin dynamics, and training it as a VAE, we obtain NSDEs that lend themselves to more elaborate analysis and to a wider range of visualisation techniques than a generic NSDE. More specifically, we obtain an energy landscape, the minima of which are in one-to-one correspondence with latent states underlying the used data. This not only allows us to detect states underlying the data dynamics in an unsupervised manner, but also to infer the distribution of time spent in each state according to the learned SDE. More in general, restricting an NSDE to Langevin dynamics enables the use of a large set of tools from computational molecular dynamics for the analysis of the obtained results.

cs.LG↗
arXiv2022

Small time asymptotics of the entropy of the heat kernel on a Riemannian manifold

We give an asymptotic expansion of the relative entropy between the heat kernel $q_Z(t,z,w)$ of a compact Riemannian manifold $Z$ and the normalized Riemannian volume for small values of $t$ and for a fixed element $z\in Z$. We prove that coefficients in the expansion can be expressed as universal polynomials in the components of the curvature tensor and its covariant derivatives at $z$, when they are expressed in terms of normal coordinates. We describe a method to compute the coefficients, and we use the method to compute the first three coefficients. The asymptotic expansion is necessary for an unsupervised machine-learning algorithm called the Diffusion Variational Autoencoder.

math.DG↗
arXiv2022

Quantifying and Learning Linear Symmetry-Based Disentanglement

The definition of Linear Symmetry-Based Disentanglement (LSBD) formalizes the notion of linearly disentangled representations, but there is currently no metric to quantify LSBD. Such a metric is crucial to evaluate LSBD methods and to compare to previous understandings of disentanglement. We propose $\mathcal{D}_\mathrm{LSBD}$, a mathematically sound metric to quantify LSBD, and provide a practical implementation for $\mathrm{SO}(2)$ groups. Furthermore, from this metric we derive LSBD-VAE, a semi-supervised method to learn LSBD representations. We demonstrate the utility of our metric by showing that (1) common VAE-based disentanglement methods don't learn LSBD representations, (2) LSBD-VAE as well as other recent methods can learn LSBD representations, needing only limited supervision on transformations, and (3) various desirable properties expressed by existing disentanglement metrics are also achieved by LSBD representations.

cs.LG↗
arXiv2021

Embedding of $RCD^*(K,N)$ spaces in $L^2$ via eigenfunctions

In this paper we study the family of embeddings $Φ_t$ of a compact $RCD^*(K,N)$ space $(X,d,m)$ into $L^2(X,m)$ via eigenmaps. Extending part of the classical results by Bérard, Bérard-Besson-Gallot, known for closed Riemannian manifolds, we prove convergence as $t\downarrow 0$ of the rescaled pull-back metrics $Φ_t^*g_{L^2}$ in $L^2(X,m)$ induced by $Φ_t$. Moreover we discuss the behavior of $Φ_t^*g_{L^2}$ with respect to measured Gromov-Hausdorff convergence and $t$. Applications include the quantitative $L^p$-convergence in the noncollapsed setting for all $p<\infty$, a result new even for closed Riemannian manifolds and Alexandrov spaces.

math.MG↗
arXiv2020

Universal Approximation in Dropout Neural Networks

We prove two universal approximation theorems for a range of dropout neural networks. These are feed-forward neural networks in which each edge is given a random $\{0,1\}$-valued filter, that have two modes of operation: in the first each edge output is multiplied by its random filter, resulting in a random output, while in the second each edge output is multiplied by the expectation of its filter, leading to a deterministic output. It is common to use the random mode during training and the deterministic mode during testing and prediction. Both theorems are of the following form: Given a function to approximate and a threshold $\varepsilon>0$, there exists a dropout network that is $\varepsilon$-close in probability and in $L^q$. The first theorem applies to dropout networks in the random mode. It assumes little on the activation function, applies to a wide class of networks, and can even be applied to approximation schemes other than neural networks. The core is an algebraic property that shows that deterministic networks can be exactly matched in expectation by random networks. The second theorem makes stronger assumptions and gives a stronger result. Given a function to approximate, it provides existence of a network that approximates in both modes simultaneously. Proof components are a recursive replacement of edges by independent copies, and a special first-layer replacement that couples the resulting larger network to the input. The functions to be approximated are assumed to be elements of general normed spaces, and the approximations are measured in the corresponding norms. The networks are constructed explicitly. Because of the different methods of proof, the two results give independent insight into the approximation properties of random dropout networks. With this, we establish that dropout neural networks broadly satisfy a universal-approximation property.

cs.LG↗
arXiv2020

A Metric for Linear Symmetry-Based Disentanglement

The definition of Linear Symmetry-Based Disentanglement (LSBD) proposed by (Higgins et al., 2018) outlines the properties that should characterize a disentangled representation that captures the symmetries of data. However, it is not clear how to measure the degree to which a data representation fulfills these properties. We propose a metric for the evaluation of the level of LSBD that a data representation achieves. We provide a practical method to evaluate this metric and use it to evaluate the disentanglement of the data representations obtained for three datasets with underlying $SO(2)$ symmetries.

cs.LG↗
arXiv2019

Tropical diagrams of probability spaces

After endowing the space of diagrams of probability spaces with an entropy distance, we study its large-scale geometry by identifying the asymptotic cone as a closed convex cone in a Banach space. We call this cone the tropical cone, and its elements tropical diagrams of probability spaces. Given that the tropical cone has a rich structure, while tropical diagrams are rather flexible objects, we expect the theory of tropical diagrams to be useful for information optimization problems in information theory and artificial intelligence. In a companion article, we give a first application to derive a statement about the entropic cone.

math.DS↗
arXiv2019

Tropical probability theory and an application to the entropic cone

In a series of articles, we have been developing a theory of tropical diagrams of probability spaces, expecting it to be useful for information optimization problems in information theory and artificial intelligence. In this article, we give a summary of our work so far and apply the theory to derive a dimension-reduction statement about the shape of the entropic cone.

cs.IT↗
arXiv2019

Conditioning in tropical probability theory

We define a natural operation of conditioning of tropical diagrams of probability spaces and show that it is Lipschitz continuous with respect to the asymptotic entropy distance.

math.PR↗
arXiv2019

Arrow Contraction and Expansion in Tropical Diagrams

Arrow contraction applied to a tropical diagram of probability spaces is a modification of the diagram, replacing one of the morphisms by an isomorphims, while preserving other parts of the diagram. It is related to the rate regions introduced by Ahlswede and Körner. In a companion article we use arrow contraction to derive information about the shape of the entropic cone. Arrow expansion is the inverse operation to the arrow contraction.

math.PR↗
arXiv2019

Diffusion Variational Autoencoders

A standard Variational Autoencoder, with a Euclidean latent space, is structurally incapable of capturing topological properties of certain datasets. To remove topological obstructions, we introduce Diffusion Variational Autoencoders with arbitrary manifolds as a latent space. A Diffusion Variational Autoencoder uses transition kernels of Brownian motion on the manifold. In particular, it uses properties of the Brownian motion to implement the reparametrization trick and fast approximations to the KL divergence. We show that the Diffusion Variational Autoencoder is capable of capturing topological properties of synthetic datasets. Additionally, we train MNIST on spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a natural dataset like MNIST does not have latent variables with a clear-cut topological structure, training it on a manifold can still highlight topological and geometrical properties.

cs.LG↗
arXiv2017

Continuity of nonlinear eigenvalues in $CD(K,\infty)$ spaces with respect to measured Gromov-Hausdorff convergence

In this note we prove in the nonlinear setting of $CD(K,\infty)$ spaces the stability of the Krasnoselskii spectrum of the Laplace operator $-Δ$ under measured Gromov-Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of $CD^*(K,N)$ metric measure spaces with uniformly bounded diameter. Additionally, we show that every element $λ$ in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial $u$ satisfying the eigenvalue equation $- Δu = λu$.

math.MG↗
arXiv2017

Tropical Limits of Probability Spaces, Part I: The Intrinsic Kolmogorov-Sinai Distance and the Asymptotic Equipartition Property for Configurations

The entropy of a finite probability space $X$ measures the observable cardinality of large independent products $X^{\otimes n}$ of the probability space. If two probability spaces $X$ and $Y$ have the same entropy, there is an almost measure-preserving bijection between large parts of $X^{\otimes n}$ and $Y^{\otimes n}$. In this way, $X$ and $Y$ are asymptotically equivalent. It turns out to be challenging to generalize this notion of asymptotic equivalence to configurations of probability spaces, which are collections of probability spaces with measure-preserving maps between some of them. In this article we introduce the intrinsic Kolmogorov-Sinai distance on the space of configurations of probability spaces. Concentrating on the large-scale geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an asymptotic equivalence relation on sequences of configurations of probability spaces. We will call the equivalence classes \emph{tropical probability spaces}. In this context we prove an Asymptotic Equipartition Property for configurations. It states that tropical configurations can always be approximated by homogeneous configurations. In addition, we show that the solutions to certain Information-Optimization problems are Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai distance. It follows from these two statements that in order to solve an Information-Optimization problem, it suffices to consider homogeneous configurations. Finally, we show that spaces of trajectories of length $n$ of certain stochastic processes, in particular stationary Markov chains, have a tropical limit.

math.MG↗
arXiv2016

On the regular representation of measures

We give sufficient conditions for a parametrised family of probability measures on a Riemannian manifold with boundary to be represented by random maps of class $C^k$. The conditions allow for the probability densities to approach zero towards the boundary of the manifold. We also formulate two obstructions to regular representability.

math.DG↗
arXiv2014

Semicontinuity of eigenvalues under intrinsic flat convergence

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. We define min-max values based on the normalized energy and show that when integral current spaces converge in the intrinsic flat sense without loss of volume, the min-max values of the limit space are larger than or equal to the upper limit of the min-max values of the currents in the sequence. In particular, the infimum of the normalized energy is semicontinuous. On spaces that are infinitesimally Hilbertian, we can define a linear Laplace operator. We can show that semicontinuity under intrinsic flat convergence holds for eigenvalues below the essential spectrum, if the total volume of the spaces converges as well.

math.DG↗