Quantum discriminant analysis for dimensionality reduction and classification

• Published in: New journal of physics Vol. 18; no. 7; pp. 73011 - 73020
• Main Authors: Cong, Iris, Duan, Luming
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 06.07.2016
• Subjects: Algorithms; Classification; Complexity; dimensionality reduction; Discriminant analysis; Linear equations; machine learning; Mathematical analysis; Matrix methods; Nonlinear analysis; Physics; quantum algorithms; quantum computation; quantum information; Reduction
• ISSN: 1367-2630; 1367-2630
• DOI: 10.1088/1367-2630/18/7/073011

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 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 polylog ( MN ) / ϵ 3 ) , where ϵ 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 ) / ϵ 3 ) .