Quantum singular-value decomposition

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.