Related papers: Manifold Learning for Dimensionality Reduction: Qu…
Manifold embedding algorithms map high-dimensional data down to coordinates in a much lower-dimensional space. One of the aims of dimension reduction is to find intrinsic coordinates that describe the data manifold. The coordinates returned…
We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are…
In this thesis, we investigate whether quantum algorithms can be used in the field of machine learning for both long and near term quantum computers. We will first recall the fundamentals of machine learning and quantum computing and then…
Quantum machine learning aims to release the prowess of quantum computing to improve machine learning methods. By combining quantum computing methods with classical neural network techniques we aim to foster an increase of performance in…
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…
Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design efficient numerical algorithms. In particular,…
We adapt concepts, methodology, and theory originally developed in the areas of multidimensional scaling and dimensionality reduction for multivariate data to the functional setting. We focus on classical scaling and Isomap -- prototypical…
Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of…
We present several quantum algorithms for performing nearest-neighbor learning. At the core of our algorithms are fast and coherent quantum methods for computing distance metrics such as the inner product and Euclidean distance. We prove…
Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into…
Quantum annealing is typically regarded as a tool for combinatorial optimization, but its coherent dynamics also offer potential for machine learning. We present a model that encodes classical data into an Ising Hamiltonian, evolves it on a…
Entanglement plays a crucial role in quantum physics and is the key resource in quantum information processing. However, entanglement detection and quantification are believed to be hard due to the operational impracticality of existing…
Multidimensional Scaling (MDS) is one of the first fundamental manifold learning methods. It can be categorized into several methods, i.e., classical MDS, kernel classical MDS, metric MDS, and non-metric MDS. Sammon mapping and Isomap can…
Exploratory analysis of high-dimensional data rarely stops at a single embedding. In practice, analysts rerun dimensionality reduction after changing preprocessing, subsets, or hyperparameters, and standard nonlinear methods can quickly…
Clustering and dimensionality reduction have been crucial topics in machine learning and computer vision. Clustering high-dimensional data has been challenging for a long time due to the curse of dimensionality. For that reason, a more…
Gradient descent method, as one of the major methods in numerical optimization, is the key ingredient in many machine learning algorithms. As one of the most fundamental way to solve the optimization problems, it promises the function value…
Unsupervised machine learning is one of the main techniques employed in artificial intelligence. We introduce an algorithm for quantum-assisted unsupervised data clustering using the self-organizing feature map, a type of artificial neural…
We introduce an algorithm for computing geodesics on sampled manifolds that relies on simulation of quantum dynamics on a graph embedding of the sampled data. Our approach exploits classic results in semiclassical analysis and the…
Image classification, a pivotal task in multiple industries, faces computational challenges due to the burgeoning volume of visual data. This research addresses these challenges by introducing two quantum machine learning models that…
Gradient descent is a fundamental algorithm in both theory and practice for continuous optimization. Identifying its quantum counterpart would be appealing to both theoretical and practical quantum applications. A conventional approach to…