Quantum algorithms for SVD-based data representation and analysis

Abstract

This paper narrows the gap between previous literature on quantum linear algebra and practical data analysis on a quantum computer, formalizing quantum procedures that speed-up the solution of eigenproblems for data representations in machine learning. The power and practical use of these subroutines is shown through new quantum algorithms, sublinear in the input matrix’s size, for principal component analysis, correspondence analysis, and latent semantic analysis. We provide a theoretical analysis of the run-time and prove tight bounds on the randomized algorithms’ error. We run experiments on multiple datasets, simulating PCA’s dimensionality reduction for image classification with the novel routines. The results show that the run-time parameters that do not depend on the input’s size are reasonable and that the error on the computed model is small, allowing for competitive classification performances.

Quantum Eigenfaces: Linear Feature Mapping and Nearest Neighbor Classification with Outlier Detection

“…We focus on the classification steps and consider that the training has been performed on a classical computer; i.e., the quantum computer is only used in the prediction stage, whose time efficiency is usually more relevant in industrial applications. However, the interested reader can consult Bellante et al [3,Theorems 10,12] for a method to extract the top-k principal components and the total retained amount of variance from a matrix stored in a quantum accessible data structure.…”

“…First, we assessed the utility of the norm-based outlier detection step in the classical scenario. Secondly, we studied the performance of 3 Code available at https://github.com/WilliamBonvini/quantum-eigenfaces the quantum algorithm at increasing values of error ϵ. Lastly, we compared the quantum running times for each value of ϵ (as defined in Eq. 13) with the classical counterpart O(mk + pk).…”