The paper reports a method for compressed representation of matrix data on the principles of quantum theory. The method is formalized as complex-valued matrix factorization based on standard singular value decomposition. The developed approach establishes a bridge between standard methods of semantic data analysis and quantum models of cognition and decision. According to the quantum theory, real-valued observable quantities are generated by wavefunctions being complex-valued vectors in multidimensional Hilbert-space. Wavefunctions are defined as superpositions of basis vectors encoding composition of semantic factors. Basis vectors are found by singular value decomposition of the initial data matrix transformed to a real-valued amplitude form. Phase-dependent superposition amplitudes are found to optimize approximation of the source data. The resulting model represents the observed real-valued data as generated from a small number of basis wavefunctions superposed with complex-valued coefficients. The method is tested for random matrices of sizes from 3 × 3 to 12 × 12 and dimensionality of latent Hilbert-space from 2 to 4. The best approximation is achieved by encoding latent factors in normalized complex-valued amplitude vectors interpreted as wavefunctions generating the data. In terms of approximation fitness, the developed method surpasses standard truncated SVD of the same dimensionality. The mean advantage over the considered range of parameters is 22 %. The method permits cognitive interpretation in accord with the existing quantum models of cognition and decision. The method can be integrated in the algorithms of semantic data analysis including natural language processing. In these tasks, the obtained improvement of approximation translates to the increased precision of similarity measures, principal component analysis, advantage in classification, and document ranking methods. Integration with quantum models of cognition and decision is expected to boost methods of artificial intelligence and machine learning improving imitation of natural thinking.
The paper reports a method for compressed representation of matrix data on the principles of quantum theory. The method is formalized as complex-valued matrix factorization based on standard singular value decomposition. The developed approach establishes a bridge between standard methods of semantic data analysis and quantum models of cognition and decision. According to the quantum theory, real-valued observable quantities are generated by wavefunctions being complex-valued vectors in multidimensional Hilbert-space. Wavefunctions are defined as superpositions of basis vectors encoding composition of semantic factors. Basis vectors are found by singular value decomposition of the initial data matrix transformed to a real-valued amplitude form. Phase-dependent superposition amplitudes are found to optimize approximation of the source data. The resulting model represents the observed real-valued data as generated from a small number of basis wavefunctions superposed with complex-valued coefficients. The method is tested for random matrices of sizes from 3 × 3 to 12 × 12 and dimensionality of latent Hilbert-space from 2 to 4. The best approximation is achieved by encoding latent factors in normalized complex-valued amplitude vectors interpreted as wavefunctions generating the data. In terms of approximation fitness, the developed method surpasses standard truncated SVD of the same dimensionality. The mean advantage over the considered range of parameters is 22 %. The method permits cognitive interpretation in accord with the existing quantum models of cognition and decision. The method can be integrated in the algorithms of semantic data analysis including natural language processing. In these tasks, the obtained improvement of approximation translates to the increased precision of similarity measures, principal component analysis, advantage in classification, and document ranking methods. Integration with quantum models of cognition and decision is expected to boost methods of artificial intelligence and machine learning improving imitation of natural thinking.
Abstract This paper presents a method for semantic data analysis by complex-valued matrix decomposition. The method is based on the quantum model of contextual decision-making, according to which observable probabilities are generated by the qubit states, representing the subjective meaning of the contexts relative to the basis decision. In the simplest three-context case, one of these qubits is decomposed into a superposition of the remaining two, mathematically encoding semantic relations between the three contexts. For use in data analysis, this model is translated to the matrix form, in which the rows and columns correspond to the contexts and samples of the experiment. The observable real-valued data then emerge from a complex-valued amplitude matrix, decomposed to a product of a real basis matrix and complex-valued matrix of superposition coefficients. This decomposition reveals stable process-semantic relations between the considered contexts, not captured by other methods of analysis. As a result, the data are approximated with higher precision and fewer parameters than singular and nonnegative matrix decompositions, truncated to the same dimension. The model is experimentally tested in the descriptive and prognostic regimes. The result opens prospects for development of nature-like computational architectures on novel logical grounds.
The paper presents a method for semantic data analysis by means of complex-valued matrix decomposition. The method is based on the quantum model of contextual decision-making, according to which observable probabilities are generated by qubit states, representing subjective meaning of the contexts relative to the basis decision. In the simplest three-context case, one of these qubits is decomposed to superposition of the remaining two, mathematically encoding semantic relations between the three contexts. For use in data analysis this model is translated to the matrix form, in which rows and columns correspond to the contexts and instances of experiment. The observable real-valued data then emerge from a complex-valued amplitude matrix, decomposed to a product of a real basis matrix and complex-valued matrix of superposition coefficients. This decomposition reveals stable process-semantic relations between the considered contexts, not captured by other methods of analysis. As a result, the data are approximated with higher precision and fewer parameters than singular and non-negative matrix decompositions, truncated to the same dimension. The model is experimentally approved in descriptive and prognostic regimes. The result opens prospects for development of nature-like computational architectures on novel logical grounds.
real-valued calculations in artificial neural networks captures the amplitudes of neural signals but neglects their phases, which are critical parameters controlling the composition of cognitive waves in natural brains. To address this limitation, we present a complex-valued modification of singular matrix decomposition, a conceptual precursor to the tensor algebra underlying modern neural networks. In this approach, we generalize the diagonal matrix of singular values to a self-adjoint complex matrix, analogous to the density matrix in quantum theory. Within low-dimensional “semantic” spaces, the additional non-diagonal elements of the complex matrix account for the non-stationary logic of the cognitive systems that generate the data. As in the standard formulation, the density matrix is sandwiched between two orthogonal real-valued matrices, unfolding semantic regularities into the space of observable events. The squared modulus of the resulting set of complex-valued amplitudes then produces observable real-valued data, following the quantum-mechanical Born rule. By introducing a minor increase in the number of parameters, our method significantly enhances the precision of classical singular value decomposition. This improvement highlights the efficiency of wave-like and quantum-inspired principles in natural cognition, as expressed in the proposed algebra. The method provides new opportunities for semantic data analysis and offers pathways to advance modern neural network architectures.
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.
Singular Value Decomposition (SVD) is one of the most useful techniques for analyzing data in linear algebra. SVD decomposes a rectangular real or complex matrix into two orthogonal matrices and one diagonal matrix. In this work we introduce a new approach to improve the preciseness of the standard Quantum Fourier Transform. The presented Quantum-SVD algorithm is based on the singular value decomposition mechanism. While the complexity of the proposed scheme is the same as the standard Quantum Fourier Transform, the precision of the Quantum-SVD approach is some orders higher. The Quantum-SVD approach also exploits the benefits of quantum searching.
Quantum-inspired singular value decomposition (SVD) is a technique to perform SVD in logarithmic time with respect to the dimension of a matrix, given access to the matrix embedded in a segment-tree data structure. The speedup is possible through the efficient sampling of matrix elements according to their norms. Here, we apply it to extreme learning which is a machine learning framework that performs linear regression using random feature vectors generated through a random neural network. The extreme learning is suited for the application of quantum-inspired SVD in that it first requires transforming each data to a random feature during which we can construct the data structure with a logarithmic overhead with respect to the number of data. We implement the algorithm and observe that it works order-of-magnitude faster than the exact SVD when we use high-dimensional feature vectors. However, we also observe that, for random features generated by random neural networks, we can replace the norm-based sampling in the quantum-inspired algorithm with uniform sampling to obtain the same level of test accuracy due to the uniformity of the matrix in this case. The norm-based sampling becomes effective for more non-uniform matrices obtained by optimizing the feature mapping. It implies the non-uniformity of matrix elements is a key property of the quantum-inspired SVD. This work is a first step toward the practical application of the quantum-inspired algorithm.