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arXiv:1510.00113 · New Journal of Physics

Quantum discriminant analysis for dimensionality reduction and classification

Iris Cong, Luming Duan

Published Jul 6, 2016• 133 citations
arXivHTMLPDF

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 (2009 Phys. Rev. Lett. 103 150502 ) to efficiently implement a Hermitian chain product of k trace-normalized N × N Hermitian positive-semidefinite matrices with time complexity of . 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 , where 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 .

DOI
10.1088/1367-2630/18/7/073011

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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 (2009 Phys. Rev. Lett. 103 150502 ) to efficiently implement a Hermitian chain product of k trace-normalized N × N Hermitian positive-semidefinite matrices with time complexity of . 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 , where 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 .
The era of big data has made traditional machine learning algorithms struggle with high-dimensional, nonlinear datasets due to high computational resource consumption, long training times, and limited generalization. Quantum computing, with its exponential parallelism from qubit superposition and entanglement, offers new hope for machine learning. Local Sensitive Discriminant Analysis (LSDA) algorithm is a widely used dimensionality reduction technique in machine learning. However, it is time-consuming when handling large-scale high-dimensional data. This paper proposed a quantum version of LSDA algorithm, which transforms the generalized eigenvalue problem in LSDA into a Hermitian eigenvalue problem, solved using quantum phase estimation and block-encoding techniques. Two post-processing strategies are provided to obtain low-dimensional data for quantum or classical machine learning tasks. The quantum LSDA algorithm obtains polynomial speedups compared to the classical LSDA in terms of the number of data points and dimensionality, showing the potential of quantum technique to boost machine learning efficiency.
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.
In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their respectively classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver that finds the generalized eigenvalues and eigenstates of a matrix pencil (G,S)(\mathcal{G},\mathcal{S})(G,S). As a proof of principle, we implement our algorithm to solve ${2}^{5}\ifmmode\times\else\texttimes\fi{}{2}^{5}$ generalized eigenvalue problems. Finally, our results offer two optional outputs with quantum or classical form, which can be directly applied in another quantum or classical machine learning process.
Abstract Kernel principal component analysis (kernel PCA) is a nonlinear dimensionality reduction technique that employs kernel functions to map data into a high-dimensional feature space, thereby extending the applicability of linear PCA to nonlinear data and facilitating the extraction of informative principal components. However, kernel PCA necessitates the manipulation of large-scale matrices, leading to high computational complexity and posing challenges for efficient implementation in big data environments. Quantum computing has recently been integrated with kernel methods in machine learning, enabling effective analysis of input data within intractable feature spaces. Although existing quantum kernel PCA proposals promise exponential speedups, they impose stringent requirements on quantum hardware that are challenging to fulfill. In this work, we propose a quantum algorithm for kernel PCA by establishing a connection between quantum kernel methods and block encoding, thereby diagonalizing the centralized kernel matrix on a quantum computer. The query complexity is logarithmic with respect to the size of the data vector, D , and linear with respect to the size of the dataset. An exponential speedup could be achieved when the dataset consists of a few high-dimensional vectors, wherein the dataset size is polynomial in log ⁡ ( D ) , with D being significantly large. In contrast to existing work, our algorithm enhances the efficiency of quantum kernel PCA and reduces the requirements for quantum hardware. Furthermore, we have also demonstrated that the algorithm based on block encoding matches the lower bound of query complexity, indicating that our algorithm is nearly optimal. Our research has laid down new pathways for developing quantum machine learning algorithms aimed at addressing tangible real-world problems and demonstrating quantum advantages within machine learning.
Quantum machine learning is a research discipline intersecting quantum algorithms and machine learning. While a number of quantum algorithms with potential speedups have been proposed, it is quite difficult to provide evidence that quantum computers will be useful in solving real-world problems. Our work makes progress towards this goal. In this work we design quantum algorithms for dimensionality reduction and for classification and combine them to provide a quantum classifier that we test on the MNIST data set of handwritten digits. We simulate the quantum classifier, including errors in the quantum procedures, and show that it can provide classification accuracy of 98.598.5%98.5. The running time of the quantum classifier is only polylogarithmic in the dimension and number of data points. Furthermore, we provide evidence that the other parameters on which the running time depends scale favorably, ascertaining the efficiency of our algorithm.
Supervised machine learning is the classification of new data based on already classified training examples. In this work, we show that the support vector machine, an optimized linear and non-linear binary classifier, can be implemented on a quantum computer, with exponential speedups in the size of the vectors and the number of training data examples. At the core of the algorithm is a non-sparse matrix simulation technique to efficiently perform a principal component analysis and matrix inversion of the training data kernel matrix. We thus provide an example of a quantum big feature and big data algorithm and pave the way for future developments at the intersection of quantum computing and machine learning.