Related papers: The Theory behind UMAP?
UMAP is a popular neighborhood-preserving dimension reduction (DR) algorithm. However, its application for graph drawing has not been evaluated. Moreover, a naive application of UMAP to graph drawing would include O(nm) time all-pair…
In 2023 in (3), Uwe finds the explicit form of the map which is which is settled in ZN of finite functional degree and14 discusses how to compute its usual degree w.r.t to the derivative in the linear form, i.e. the product of ones formed…
We prove a universal approximation property (UAP) for a class of ODENet and a class of ResNet, which are simplified mathematical models for deep learning systems with skip connections. The UAP can be stated as follows. Let $n$ and $m$ be…
UMAP (Uniform Manifold Approximation and Projection) is among the most widely used algorithms for non linear dimensionality reduction and data visualisation. Despite its popularity, and despite being presented through the lens of algebraic…
This paper intends to lay the theoretical foundation for the method of functional maps, first presented in 2012 by Ovsjanikov, Ben-Chen, Solomon, Butscher and Guibas in the field of the theory and numerics of maps between shapes. We show…
In this article, we introduce a new parameterized family of topological descriptors, taking the form of candidate decompositions, for multi-parameter persistence modules, and we identify a subfamily of these descriptors, that we call…
MAP inference for general energy functions remains a challenging problem. While most efforts are channeled towards improving the linear programming (LP) based relaxation, this work is motivated by the quadratic programming (QP) relaxation.…
We give a detailed description of the so-called Polynomial Hybrid Monte Carlo (PHMC) algorithm. The effects of the correction factor, which is introduced to render the algorithm exact, are discussed, stressing their relevance for the…
The goal is to modify the known method of mirror descent (MD), proposed by A.S. Nemirovsky and D.B. Yudin in 1979. The paper shows the idea of a new, so-called inertial MD method with the example of a deterministic optimization problem in…
Matrix operations such as matrix inversion, eigenvalue decomposition, singular value decomposition are ubiquitous in real-world applications. Unfortunately, many of these matrix operations so time and memory expensive that they are…
We consider a class of adaptive multilevel domain decomposition-like algorithms, built from a combination of adaptive multilevel finite element, domain decomposition, and partition of unity methods. These algorithms have several interesting…
Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…
This white paper demonstrates that reverse engineering Unidentified Aerial Phenomena (UAP) is NP-complete under classical computational paradigms. By modeling UAP reconstruction as an automaton identification problem with a state…
We present a heuristic strategy for marginal MAP (MMAP) queries in graphical models. The algorithm is based on a reduction of the task to a polynomial number of marginal inference computations. Given an input evidence, the marginals mass…
Optimization methods play a central role in signal processing, serving as the mathematical foundation for inference, estimation, and control. While classical iterative optimization algorithms provide interpretability and theoretical…
The complex dynamics of baker's map and its variants in an infinite-precision mathematical domain have been extensively analyzed in the past five decades. However, their real structure implemented in a finite-precision computer remains…
In the past decade, matrix factorization has been extensively researched and has become one of the most popular techniques for personalized recommendations. Nevertheless, the dot product adopted in matrix factorization based recommender…
The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…
Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the…
FastMap was first introduced in the Data Mining community for generating Euclidean embeddings of complex objects. In this dissertation, we first present FastMap to generate Euclidean embeddings of graphs in near-linear time: The pairwise…