Quantum Discriminant Analysis for Dimensionality Reduction and Classification

TL;DR

This paper introduces quantum algorithms that significantly accelerate discriminant analysis and classification tasks in high-dimensional data, outperforming classical methods with exponential speedups in data size and feature dimensions.

Contribution

The paper develops novel quantum algorithms for discriminant analysis and classification, including generalizing quantum linear systems solutions to efficiently handle Hermitian matrix products.

Findings

Quantum algorithms achieve exponential speedup over classical methods.

Efficient implementation of Hermitian chain products with logarithmic complexity.

Practical quantum algorithms for linear and nonlinear discriminant analysis.

Abstract

We present quantum algorithms to efficiently perform discriminant analysis for dimensionality reduction and classification over an exponentially large input data set. Compared with the best-known classical algorithms, the quantum algorithms show an exponential speedup in both the number of training vectors $M$ and the feature space dimension $N$. We generalize the previous quantum algorithm for solving systems of linear equations [Phys. Rev. Lett. 103, 150502 (2009)] to efficiently implement a Hermitian chain product of $k$ trace-normalized $N \times N$ Hermitian positive-semidefinite matrices with time complexity of $O(\log (N))$. Using this result, we perform linear as well as nonlinear Fisher discriminant analysis for dimensionality reduction over $M$ vectors, each in an $N$-dimensional feature space, in time $O(p \text{polylog} (MN)/\epsilon ^{3})$, where $\epsilon $ denotes the tolerance error, and $p$ is the number of principal projection directions desired. We also present a quantum discriminant analysis algorithm for data classification with time complexity $O(\log (MN)/\epsilon^{3})$.