Related papers: Dimension reduction for path signatures
Shortcut sets are a vital instrument for reducing the diameter of a static graph and, consequently, its shortest path complexity, which is relevant in numerous subfields of graph theory. We explore the notion of shortcut sets in temporal…
The sparse representation of signals defined on Euclidean domains has been successfully applied in signal processing. Bringing the power of sparse representations to non-regular domains is still a challenge, but promising approaches have…
This chapter discusses the way in which dimensionality reduction algorithms such as diffusion maps and sketch-map can be used to analyze molecular dynamics trajectories. The first part discusses how these various algorithms function, as…
Dimension reduction is a common strategy to study non-linear dynamical systems composed by a large number of variables. The goal is to find a smaller version of the system whose time evolution is easier to predict while preserving some of…
Integration is affected by the curse of dimensionality and quickly becomes intractable as the dimensionality of the problem grows. We propose a randomized algorithm that, with high probability, gives a constant-factor approximation of a…
We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is…
We derive the equations of chains for path geometries on surfaces by solving the equivalence problem of a related structure: sub-Riemannian geometry of signature $(1,1)$ on a contact 3-manifold. This approach is significantly simpler than…
Both Dimensionality Reduction (DR) and Graph Drawing (GD) aim to visualize abstract, non-linear structures, yet rely on different optimization paradigms. This contrast is evident in Multidimensional Scaling (MDS), which typically depends on…
Extracting dense representations for terms and phrases is a task of great importance for knowledge discovery platforms targeting highly-technical fields. Dense representations are used as features for downstream components and have multiple…
This work presents a reduced order modelling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order…
Understanding protein dynamics are essential for deciphering protein functional mechanisms and developing molecular therapies. However, the complex high-dimensional dynamics and interatomic interactions of biological processes pose…
We introduce a unified framework for evaluating dimensionality reduction techniques in spatial transcriptomics beyond standard PCA approaches. We benchmark six methods PCA, NMF, autoencoder, VAE, and two hybrid embeddings on a…
In an era where big and high-dimensional data is readily available, data scientists are inevitably faced with the challenge of reducing this data for expensive downstream computation or analysis. To this end, we present here a new method…
We present a method for obtaining approximate solutions to the problem of optimal execution, based on a signature method. The framework is general, only requiring that the price process is a geometric rough path and the price impact…
High-dimensional representations, such as radial basis function networks or tile coding, are common choices for policy evaluation in reinforcement learning. Learning with such high-dimensional representations, however, can be expensive,…
Symmetry reduction is a well-known approach for alleviating the state explosion problem in model checking. Automatically identifying symmetries in concurrent systems, however, is computationally expensive. We propose a symbolic framework…
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov…
The selection of best variables is a challenging problem in supervised and unsupervised learning, especially in high dimensional contexts where the number of variables is usually much larger than the number of observations. In this paper,…
Numerical computation of shortest paths or geodesics on curved domains, as well as the associated geodesic distance, arises in a broad range of applications across digital geometry processing, scientific computing, computer graphics, and…
Neural Stochastic Differential Equations (Neural SDEs) have emerged as powerful mesh-free generative models for continuous stochastic processes, with critical applications in fields such as finance, physics, and biology. Previous…