Canonical correlation analysis (CCA) serves to identify statistical dependencies between pairs of multivariate data. However, its application to high-dimensional data is limited due to considerable computational complexity. As an alternative to the conventional CCA approach that requires polynomial computational time, we propose an algorithm that approximates CCA using quantum-inspired computations with computational time proportional to the logarithm of the input dimensionality. The computational efficiency and performance of the proposed quantum-inspired CCA (qiCCA) algorithm are experimentally evaluated on synthetic and real datasets. Furthermore, the fast computation provided by qiCCA allows directly applying CCA even after nonlinearly mapping raw input data into high-dimensional spaces. The conducted experiments demonstrate that, as a result of mapping raw input data into the high-dimensional spaces with the use of second-order monomials, qiCCA extracts more correlations compared with the linear CCA and achieves comparable performance with state-of-the-art nonlinear variants of CCA on several datasets. These results confirm the appropriateness of the proposed qiCCA and the high potential of quantum-inspired computations in analyzing high-dimensional data.
Canonical correlation analysis (CCA) serves to identify statistical dependencies between pairs of multivariate data. However, its application to high-dimensional data is limited due to considerable computational complexity. As an alternative to the conventional CCA approach that requires polynomial computational time, we propose an algorithm that approximates CCA using quantum-inspired computations with computational time proportional to the logarithm of the input dimensionality. The computational efficiency and performance of the proposed quantum-inspired CCA (qiCCA) algorithm are experimentally evaluated on synthetic and real datasets. Furthermore, the fast computation provided by qiCCA allows directly applying CCA even after nonlinearly mapping raw input data into high-dimensional spaces. The conducted experiments demonstrate that, as a result of mapping raw input data into the high-dimensional spaces with the use of second-order monomials, qiCCA extracts more correlations compared with the linear CCA and achieves comparable performance with state-of-the-art nonlinear variants of CCA on several datasets. These results confirm the appropriateness of the proposed qiCCA and the high potential of quantum-inspired computations in analyzing high-dimensional data.
Principal component analysis (PCA) is a widely used statistical tool for extracting low-dimensional structures underlying multivariate data. However, its application to high-dimensional data is limited due to its large computational time. While the conventional PCA algorithm requires polynomial time, using a quantum-inspired algorithm as a subroutine, we have implemented an algorithm that approximates it with computational time proportional to the logarithm of the input dimensionality. The computational efficiency and performance of the implemented algorithm, quantum-inspired PCA, are experimentally evaluated on synthetic and real datasets.
Abstract Canonical correlation analysis (CCA) is a fundamental technique used to analyze data correlation in various fields, including video and medical data analysis. In this paper, we propose a quantum canonical correlation analysis (QCCA) algorithm. First, we introduce a combined density matrix representation method that transforms CCA into generalized eigenvalue decomposition. Moreover, to address the challenge of performing generalized eigenvalue decomposition in high-dimensional scenarios, we propose a quantum method for extracting the canonical principal axes. In this method, two sets of variables are transformed into a reduced density matrix, so that the product of variable matrices can be accelerated by phase estimation and controlled rotation. Complexity analysis shows that the QCCA algorithm achieves exponential acceleration in variable dimensions n, p and variable size m compared to classical algorithms. The QCCA algorithm serves as a foundation for the subsequent development of quantum algorithms for classification, regression, and other machine learning tasks.
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.
Principal component analysis (PCA) is a widely used statistical tool for extracting low-dimensional structures underlying multivariate data. However, its application to high-dimensional data is limited due to its large computational time. While the conventional PCA algorithm requires polynomial time, using a quantum-inspired algorithm as a subroutine, we have implemented an algorithm that approximates it with computational time proportional to the logarithm of the input dimensionality. The computational efficiency and performance of the implemented algorithm, quantum-inspired PCA, are experimentally evaluated on synthetic and real datasets.
High-dimensional data processing presents substantial computational challenges because increasing feature dimensionality often leads to higher memory requirements, complex similarity calculations, redundant information, and increased learning costs. Conventional dimensionality-reduction and machine-learning approaches remain effective, but their scalability can become constrained when datasets contain large numbers of correlated or heterogeneous attributes. This paper develops a conceptual quantum-inspired computational architecture for efficient high-dimensional data processing by integrating principles associated with quantum machine learning, dimensionality reduction, low-rank computation, tensor representations, and quantum-inspired classical algorithms. The proposed architecture is organized around five functional stages: data encoding and normalization, structure-aware dimensionality reduction, quantum-inspired feature representation, efficient learning, and adaptive output analysis. Principal component analysis provides a theoretical foundation for identifying informative directions while reducing redundant dimensions (Jolliffe, 2016). Quantum machine-learning studies further motivate the use of Hilbert-space representations, quantum-inspired optimization, and compact computational representations (Biamonte et al., 2017; Schuld and Killoran, 2019). The architecture is designed as a hybrid conceptual framework rather than a claim of universal quantum advantage. Its principal contribution is the systematic integration of quantum-inspired computational mechanisms with established classical data-processing methods. The analysis indicates that such an architecture can potentially reduce computational redundancy, improve representation efficiency, and provide a scalable pathway for high-dimensional classification, recommendation, and predictive analytics. Limitations concerning data encoding, computational overhead, hardware dependence, and the distinction between genuine quantum advantage and classical quantum-inspired methods are also examined.
We study the practical performance of quantum-inspired algorithms for recommendation systems and linear systems of equations. These algorithms were shown to have an exponential asymptotic speedup compared to previously known classical methods for problems involving low-rank matrices, but with complexity bounds that exhibit a hefty polynomial overhead compared to quantum algorithms. This raised the question of whether these methods were actually useful in practice. We conduct a theoretical analysis aimed at identifying their computational bottlenecks, then implement and benchmark the algorithms on a variety of problems, including applications to portfolio optimization and movie recommendations. On the one hand, our analysis reveals that the performance of these algorithms is better than the theoretical complexity bounds would suggest. On the other hand, their performance as seen in our implementation degrades noticeably as the rank and condition number of the input matrix are increased. Overall, our results indicate that quantum-inspired algorithms can perform well in practice provided that stringent conditions are met: low rank, low condition number, and very large dimension of the input matrix. By contrast, practical datasets are often sparse and high-rank, precisely the type that can be handled by quantum algorithms.