We measure the out-of-domain uncertainty in the prediction of Neural Networks using a statistical notion called Lens Depth'' (LD) combined with Fermat Distance, which is able to capture precisely the depth'' of a point with respect to a distribution in feature space, without any distributional assumption. Our method also has no trainable parameter. The method is applied directly in the feature space at test time and does not intervene in training process. As such, it does not impact the performance of the original model. The proposed method gives excellent qualitative results on toy datasets and can give competitive or better uncertainty estimation on standard deep learning datasets compared to strong baseline methods.

Our method also has no trainable parameter. The method is applied directly in the feature space at test time and does not intervene in training process.

By defining a Riemannian metric which increases with decreasing probability density, shortest paths naturally follow the data manifold and points are clustered according to the modes of the data. We show that existing methods to estimate Fermat distances, a particular choice of DBD, suffer from poor convergence in both low and high dimensions due to i) inaccurate density estimates and ii) reliance on graph-based paths which are increasingly rough in high dimensions. To address these issues, we propose learning the densities using a normalizing flow, a generative model with tractable density estimation, and employing a smooth relaxation method using a score model initialized from a graph-based proposal. Additionally, we introduce a dimension-adapted Fermat distance that exhibits more intuitive behavior when scaled to high dimensions and offers better numerical properties. Our work paves the way for practical use of density-based distances, especially in high-dimensional spaces.

The Fermat distance has been recently established as a useful tool for machine learning tasks when a natural distance is not directly available to the practitioner or to improve the results given by Euclidean distances by exploding the geometrical and statistical properties of the dataset. This distance depends on a parameter $\alpha$ that greatly impacts the performance of subsequent tasks. Ideally, the value of $\alpha$ should be large enough to navigate the geometric intricacies inherent to the problem. At the same, it should remain restrained enough to sidestep any deleterious ramifications stemming from noise during the process of distance estimation. We study both theoretically and through simulations how to select this parameter.

We analyze the convergence properties of Fermat distances, a family of density-driven metrics defined on Riemannian manifolds with an associated probability measure. Fermat distances may be defined either on discrete samples from the underlying measure, in which case they are random, or in the continuum setting, in which they are induced by geodesics under a density-distorted Riemannian metric. We prove that discrete, sample-based Fermat distances converge to their continuum analogues in small neighborhoods with a precise rate that depends on the intrinsic dimensionality of the data and the parameter governing the extent of density weighting in Fermat distances. This is done by leveraging novel geometric and statistical arguments in percolation theory that allow for non-uniform densities and curved domains. Our results are then used to prove that discrete graph Laplacians based on discrete, sample-driven Fermat distances converge to corresponding continuum operators. In particular, we show the discrete eigenvalues and eigenvectors converge to their continuum analogues at a dimension-dependent rate, which allows us to interpret the efficacy of discrete spectral clustering using Fermat distances in terms of the resulting continuum limit. The perspective afforded by our discrete-to-continuum Fermat distance analysis leads to new clustering algorithms for data and related insights into efficient computations associated to density-driven spectral clustering. Our theoretical analysis is supported with numerical simulations and experiments on synthetic and real image data.

This is a problem because state-of-the-art generative models, like the ones listed above, rely on latent It is challenging for generative models to learn a distribution node embeddings. Such embeddings vary drastically over graphs because of the lack of permutation even under nearly isomorphic graphs [13]. In turn, this invariance: nodes may be ordered arbitrarily across can hamper the fidelity of the graph generation process graphs, and standard graph alignment is combinatorial significantly. Note that this is a much harder setting and notoriously expensive. We propose AlignGraph, a than, e.g., images or text, where inputs have a canonical group of generative models that combine fast and efficient orientation. Finding the correspondence between graph graph alignment methods with a family of deep nodes is a notoriously hard problem [14, 15, 11, 16], and generative models that are invariant to node permutations.

We address the problem of estimating intrinsic distances in a manifold from a finite sample. We prove that the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges a.s. in the sense of Gromov-Hausdorff. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This result is applied to obtain sample persistence diagrams that converge towards an intrinsic persistence diagram. We show that this method outperforms more standard approaches based on Euclidean norm with theoretical results and computational experiments.