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Most state-of-the-art deep geometric learning single-view reconstruction approaches rely on encoder-decoder architectures that output either shape parametrizations or implicit representations. However, these representations rarely preserve…
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the…
We consider a model of D-dimensional tethered manifold interacting by excluded volume in R^d with a single point. By use of intrinsic distance geometry, we first provide a rigorous definition of the analytic continuation of its perturbative…
In this paper, we propose an algebraic approach to upgrade a projective reconstruction to a Euclidean one, and aim at computing the rectifying homography from a minimal number of 9 segments of known length. Constraints are derived from…
High-dimensional data with intrinsic low-dimensional structure is ubiquitous in machine learning and data science. While various approaches allow one to learn a data manifold with a Riemannian structure from finite samples, performing…
The Persistent Homology Transform (PHT) was introduced in the field of Topological Data Analysis about 10 years ago, and has since been proven to be a very powerful descriptor of Euclidean shapes. The PHT consists of scanning a shape from…
We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied…
The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions…
The study of embeddings of smooth manifolds into Euclidean and projective spaces has been for a long time an important area in topology. In this paper we obtain improvements of classical results on embeddings of smooth manifolds, focusing…
We propose a novel algorithm for supervised dimensionality reduction named Manifold Partition Discriminant Analysis (MPDA). It aims to find a linear embedding space where the within-class similarity is achieved along the direction that is…
Manifold learning methods are an invaluable tool in today's world of increasingly huge datasets. Manifold learning algorithms can discover a much lower-dimensional representation (embedding) of a high-dimensional dataset through non-linear…
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…
This paper presents a new scalable algorithm for cross-modal similarity preserving retrieval in a learnt manifold space. Unlike existing approaches that compromise between preserving global and local geometries, the proposed technique…
We initiate a systematic study of intrinsic dimensional versions of classical functional inequalities which capture refined properties of the underlying objects. We focus on model spaces: Euclidean space, Hamming cube, and manifolds of…
Deep learning methods have played a more and more important role in hyperspectral image classification. However, the general deep learning methods mainly take advantage of the information of sample itself or the pairwise information between…
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…
Manifold learning is a fundamental task at the core of data analysis and visualisation. It aims to capture the simple underlying structure of complex high-dimensional data by preserving pairwise dissimilarities in low-dimensional…
We propose a method for learning topology-preserving data representations (dimensionality reduction). The method aims to provide topological similarity between the data manifold and its latent representation via enforcing the similarity in…
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…
In this paper we consider adaptive sampling's local-feature size, used in surface reconstruction and geometric inference, with respect to an arbitrary landmark set rather than the medial axis and relate it to a path-based adaptive metric on…