Related papers: A Quasi-isometric Embedding Algorithm
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…
Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great…
Fitting an unknown number of hyperplanes to data is a fundamental yet challenging problem in machine learning, characterized by its non-convexity, non-differentiability, and unknown model order. Existing approaches often struggle with local…
We present a Bayesian approach to identify optimal transformations that map model input points to low dimensional latent variables. The "projection" mapping consists of an orthonormal matrix that is considered a priori unknown and needs to…
There has been a growing interest in statistical inference from data satisfying the so-called manifold hypothesis, assuming data points in the high-dimensional ambient space to lie in close vicinity of a submanifold of much lower dimension.…
In this paper, we introduce a neighbor embedding framework for manifold alignment. We demonstrate the efficacy of the framework using a manifold-aligned version of the uniform manifold approximation and projection algorithm. We show that…
This work constructs Jonson-Lindenstrauss embeddings with best accuracy, as measured by variance, mean-squared error and exponential concentration of the length distortion. Lower bounds for any data and embedding dimensions are determined,…
Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there…
We study representations of data from an arbitrary metric space $\mathcal{X}$ in the space of univariate Gaussian mixtures with a transport metric (Delon and Desolneux 2020). We derive embedding guarantees for feature maps implemented by…
The fields of compressed sensing (CS) and matrix completion have shown that high-dimensional signals with sparse or low-rank structure can be effectively projected into a low-dimensional space (for efficient acquisition or processing) when…
What is the dimension of a network? Here, we view it as the smallest dimension of Euclidean space into which nodes can be embedded so that pairwise distances accurately reflect the connectivity structure. We show that a recently proposed…
We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric $X$. Computing such an embedding (exactly or approximately) is a non-trivial task even when $X$ is the…
Suppose M is a closed submanifold in a Euclidean ball of sufficiently large dimension. We give an optimal bound on the normal curvatures, guaranteeing that M is a sphere. The border cases consist of Veronese embeddings of the four…
The Expectation--Maximization (EM) algorithm is a simple meta-algorithm that has been used for many years as a methodology for statistical inference when there are missing measurements in the observed data or when the data is composed of…
The concept of quasi-isometric embedding maps between $*$-algebras is introduced. We have obtained some basic results related to this notion and similar to quasi-isometric embedding maps on metric spaces, under some conditions, we give a…
High-dimensional data are ubiquitous in contemporary science and finding methods to compress them is one of the primary goals of machine learning. Given a dataset lying in a high-dimensional space (in principle hundreds to several thousands…
Large collections of high-dimensional data have become nearly ubiquitous across many academic fields and application domains, ranging from biology to the humanities. Since working directly with high-dimensional data poses challenges, the…
Spectral embedding finds vector representations of the nodes of a network, based on the eigenvectors of a properly constructed matrix, and has found applications throughout science and technology. Many networks are multipartite, meaning…
The seminal result of Johnson and Lindenstrauss on random embeddings has been intensively studied in applied and theoretical computer science. Despite that vast body of literature, we still lack of complete understanding of statistical…
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…