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.
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.
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 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.
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.
Difficulties in prognosis of human behavior are due to the complexity of our cognition, routinely breaking the boundaries of classical rationality. The paper solves this problem for the simplest kind of such “irrational” behavior in which a single binary decision is made in three related contexts. Subjective meanings of these contexts relative to the basis decision alternative are represented by three qubit states, borrowed from quantum theory. These states are bound together by linear superpositions, which encode semantic composition of the contexts in the subject’s mind. The resulting theory supplements classical probabilistic model with nonlinear interference factor, accounting for the “irrational”, emotionally-semantic side of intelligence. This model is built for different realizations of two classic experiments used to study behavioral irrationality: the prisoner’s dilemma and the two-stage gambling task. In 24 such realizations, the interference phase is shown to fall in a narrow range of values, encoding regularities of semantic composition of contexts. Extrapolation of this regularity to novel experiments allows using the model in prognostic mode. This possibility is tested on the task of probabilistic prediction of target decision based on the same probability in two other contexts. For the prisoner’s dilemma and the two-stage gambling task the such prognosis has relative errors of 9 and 11% respectively. The proposed approach allows for putting other quantum models of cognition and decision to predictive and interpretable use, whereas its principles also apply to modeling of decisions with larger sets of contexts and behavioral options. By formalizing a novel type of semantic regularities behind “irrational” thinking, models of the present type open prospects for empowering the existing means for socio-economical analytics and prediction.
Difficulties in algorithmic simulation of natural thinking point to the inadequacy of information encodings used to this end. The promising approach to this problem represents information by the qubit states of quantum theory, structurally aligned with major theories of cognitive semantics. The paper develops this idea by linking qubit states with color as fundamental carrier of affective meaning. The approach builds on geometric affinity of Hilbert space of qubit states and color solids, used to establish precise one-to-one mapping between them. This is enabled by original decomposition of qubit in three non-orthogonal basis vectors corresponding to red, green, and blue colors. Real-valued coefficients of such decomposition are identical to the tomograms of the qubit state in the corresponding directions, related to ordinary Stokes parameters by rotational transform. Classical compositions of black, white and six main colors (red, green, blue, yellow, magenta and cyan) are then mapped to analogous superposition of the qubit states. Pure and mixed colors intuitively map to pure and mixed qubit states on the surface and in the volume of the Bloch ball, while grayscale is mapped to the diameter of the Bloch sphere. Herewith, the lightness of color corresponds to the probability of the qubit’s basis state «1», while saturation and hue encode coherence and phase of the qubit, respectively. The developed code identifies color as a bridge between quantum-theoretic formalism and qualitative regularities of the natural mind. This opens prospects for deeper integration of quantum informatics in semantic analysis of data, image processing, and the development of nature-like computational architectures.
The paper presents quantum model of subjective text perception based on binary cognitive distinctions corresponding to words of natural language. The result of perception is quantum cognitive state represented by vector in the qubit Hilbert space. Complex-valued structure of the quantum state space extends the standard vector-based approach to semantics, allowing to account for subjective dimension of human perception in which the result is constrained, but not fully predetermined by input information. In the case of two distinctions, the perception model generates a two-qubit state, entanglement of which quantifies semantic connection between the corresponding words. This two-distinction perception case is realized in the algorithm for detection and measurement of semantic connectivity between pairs of words. The algorithm is experimentally tested with positive results. The developed approach to cognitive modeling unifies neurophysiological, linguistic, and psychological descriptions in a mathematical and conceptual structure of quantum theory, extending horizons of machine intelligence.