Related papers: A Large Scale Approach to Decomposition Spaces
Geometric relational embeddings map relational data as geometric objects that combine vector information suitable for machine learning and structured/relational information for structured/relational reasoning, typically in low dimensions.…
In this article we study atomic and molecular decompositions in $2$-microlocal Besov and Triebel--Lizorkin spaces with variable integrability. We show that, in most cases, the convergence implied in such decompositions holds not only in the…
Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure…
Partitionings (or segmentations) divide a given domain into disjoint connected regions whose union forms again the entire domain. Multi-dimensional partitionings occur, for example, when analyzing parameter spaces of simulation models,…
Studying how embeddings are organized in space not only enhances model interpretability but also uncovers factors that drive downstream task performance. In this paper, we present a comprehensive analysis of topological and geometric…
This paper introduces a new shape-matching methodology, combinative matching, to combine interlocking parts for geometric shape assembly. Previous methods for geometric assembly typically rely on aligning parts by finding identical surfaces…
In the present paper, we investigate whether an embedding of a decomposition space $\mathcal{D}\left(\mathcal{Q},L^{p},Y\right)$ into a given Sobolev space $W^{k,q}(\mathbb{R}^{d})$ exists. As special cases, this includes embeddings into…
Borrowing elementary ideas from solid mechanics and differential geometry, this presentation shows that the volume swept by a regular solid undergoing a wide class of volume-preserving deformations induces a rather natural metric structure…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
This paper provides an introduction to decomposition spaces and 2-Segal spaces, unifying the two perspectives. We begin by defining decomposition spaces using the active-inert factorization system on the simplicial category, and show their…
We review work on `decomposition,' a property of two-dimensional theories with 1-form symmetries and, more generally, d-dimensional theories with (d-1)-form symmetries. Decomposition is the observation that such quantum field theories are…
Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…
Embedding of large but redundant data, such as images or text, in a hierarchy of lower-dimensional spaces is one of the key features of representation learning approaches, which nowadays provide state-of-the-art solutions to problems once…
We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be…
Our goal here is to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together. We pay special attention to the nearest orthogonal…
We apply a number of atomic decomposition schemes across the standard QM7 dataset -- a small model set of organic molecules at equilibrium geometry -- to inspect the possible emergence of trends among contributions to atomization energies…
The distribution of the deformations of elementary cells is studied in an abstract lattice constructed from the existence of the empty set. One combination rule determining oriented sequences with continuity of set-distance function in such…
Concepts play a central role in many applications. This includes settings where concepts have to be modelled in the absence of sentence context. Previous work has therefore focused on distilling decontextualised concept embeddings from…
Roughly speaking, let us say that a map between metric spaces is large scale conformal if it maps packings by large balls to large quasi-balls with limited overlaps. This quasi-isometry invariant notion makes sense for finitely generated…
Molecules have seemed like a natural fit to deep learning's tendency to handle a complex structure through representation learning, given enough data. However, this often continuous representation is not natural for understanding chemical…