Quantum Laplacian Eigenmap

TL;DR

This paper introduces a quantum algorithm for Laplacian eigenmaps that significantly accelerates nonlinear dimensionality reduction, leveraging quantum linear algebra techniques for exponential speedup over classical methods.

Contribution

It adapts the Hermitian chain product from quantum linear discriminant analysis to develop an efficient quantum Laplacian eigenmap algorithm.

Findings

Achieves exponential speedup over classical algorithms.

Demonstrates applicability of quantum linear algebra techniques.

Provides a framework for quantum nonlinear dimensionality reduction.

Abstract

Laplacian eigenmap algorithm is a typical nonlinear model for dimensionality reduction in classical machine learning. We propose an efficient quantum Laplacian eigenmap algorithm to exponentially speed up the original counterparts. In our work, we demonstrate that the Hermitian chain product proposed in quantum linear discriminant analysis (arXiv:1510.00113,2015) can be applied to implement quantum Laplacian eigenmap algorithm. While classical Laplacian eigenmap algorithm requires polynomial time to solve the eigenvector problem, our algorithm is able to exponentially speed up nonlinear dimensionality reduction.