Vladislav Gennadievich Malyshkin

Mikhail Gennadievich Belov, Victor Victorovich Dubov, Vadim Konstantinovich Ivanov et al.

In this work, we demonstrate experimentally that the execution flow, $I = dV/dt$, is the fundamental driving force of market dynamics. We develop a numerical framework to calculate execution flow from the data using the Radon-Nikodym derivative. A notable feature of this approach is its ability to automatically determine thresholds that can serve as actionable triggers. The technique also determines the characteristic time scale directly from the corresponding eigenproblem. The methodology has been validated on actual market data to support these findings. Additionally, we introduce a framework based on the Christoffel function spectrum, which is invariant under arbitrary non-degenerate linear transformations of input attributes and offers an alternative to traditional principal component analysis (PCA), which is limited to unitary invariance.

Aleksandr Vasilievich Bobyl, Andrei Georgievich Zabrodskii, Mikhail Evgenievich Kompan et al.

Radon--Nikodym approach to relaxation dynamics, where probability density is built first and then used to calculate observable dynamic characteristic is developed and applied to relaxation type signals study. In contrast with $L^2$ norm approaches, such as Fourier or least squares, this new approach does not use a norm, the problem is reduced to finding the spectrum of an operator (virtual Hamiltonian), which is built in a way that eigenvalues represent the dynamic characteristic of interest and eigenvectors represent probability density. The problems of interpolation (numerical estimation of Radon--Nikodym derivatives is developed) and obtaining the distribution of relaxation rates from sampled timeserie are considered. Application of the theory is demonstrated on a number of model and experimentally measured timeserie signals of degradation and relaxation processes. Software product, implementing the theory is developed.

Mikhail Gennadievich Belov, Victor Victorovich Dubov, Alexey Vladimirovich Filimonov et al.

The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $ρ^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} ω^{(l)} F\left(\varrho^{(l)},\sum_s B_s ρ^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,σ)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a mixed unitary quantum channel $A^{OUT}=\sum_s |w_s|^2 \mathcal{U}_s A^{IN} \mathcal{U}_s^{\dagger}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states $φ_l=\mathcal{U} ψ_l$ to a quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping $\varrho^{(l)}=\mathcal{U} ρ^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{ρ^{(l)}} \to \sqrt{\varrho^{(l)}}$ mapping, and on a quantum channel, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more.

Vladislav Gennadievich Malyshkin

A new form of ML knowledge representation with high generalization power is developed and implemented numerically. Initial $\mathit{IN}$ attributes and $\mathit{OUT}$ class label are transformed into the corresponding Hilbert spaces by considering localized wavefunctions. A partially unitary operator optimally converting a state from $\mathit{IN}$ Hilbert space into $\mathit{OUT}$ Hilbert space is then built from an optimization problem of transferring maximal possible probability from $\mathit{IN}$ to $\mathit{OUT}$, this leads to the formulation of a new algebraic problem. Constructed Knowledge Generalizing Operator $\mathcal{U}$ can be considered as a $\mathit{IN}$ to $\mathit{OUT}$ quantum channel; it is a partially unitary rectangular matrix of the dimension $\mathrm{dim}(\mathit{OUT}) \times \mathrm{dim}(\mathit{IN})$ transforming operators as $A^{\mathit{OUT}}=\mathcal{U} A^{\mathit{IN}} \mathcal{U}^{\dagger}$. Whereas only operator $\mathcal{U}$ projections squared are observable $\left\langle\mathit{OUT}|\mathcal{U}|\mathit{IN}\right\rangle^2$ (probabilities), the fundamental equation is formulated for the operator $\mathcal{U}$ itself. This is the reason of high generalizing power of the approach; the situation is the same as for the Schrödinger equation: we can only measure $ψ^2$, but the equation is written for $ψ$ itself.

Mikhail Gennadievich Belov, Vladislav Gennadievich Malyshkin

The problem of an optimal mapping between Hilbert spaces $IN$ of $\left|ψ\right\rangle$ and $OUT$ of $\left|φ\right\rangle$ based on a set of wavefunction measurements (within a phase) $ψ_l \to φ_l$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\sum_{l=1}^{M} ω^{(l)} \left|\langleφ_l|\mathcal{U}|ψ_l\rangle\right|^2$ subject to probability preservation constraints on $\mathcal{U}$ (partial unitarity). The constructed operator $\mathcal{U}$ can be considered as an $IN$ to $OUT$ quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension $\dim(OUT) \times \dim(IN)$ transforming operators as $A^{OUT}=\mathcal{U} A^{IN} \mathcal{U}^{\dagger}$. An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.

Mikhail Gennadievich Belov, Victor Victorovich Dubov, Vadim Konstantinovich Ivanov et al.

We introduce Superstate Quantum Mechanics (SQM) as a theory that considers states in Hilbert space subject to multiple quadratic constraints. Traditional quantum mechanics corresponds to a single quadratic constraint of wavefunction normalization. In its simplest form, SQM considers states in the form of unitary operators, where the quadratic constraints are conditions of unitarity. In this case, the stationary SQM problem is a quantum inverse problem with multiple applications in physics, machine learning, and artificial intelligence. The SQM stationary problem is equivalent to a new algebraic problem that we address in this paper. The SQM non-stationary problem considers the evolution of a quantum system itself, distinct from the explicit time dependence of the Hamiltonian, $H(t)$. Two options for the SQM dynamic equation are considered: (1) within the framework of linear maps from higher-order quantum theory, where 2D-type quantum circuits are introduced to transform one quantum system into another; and (2) in the form of a Gross-Pitaevskii-type nonlinear map. Although no known physical process currently describes such 2D dynamics, this approach naturally bridges direct and inverse quantum mechanics problems, allowing for the development of a new type of computer algorithms. Beyond computer modeling, the developed theory could be directly applied if or when a physical process capable of solving a quantum inverse problem in a single measurement act (analogous to how an eigenvalue arises from a measurement in traditional quantum mechanics) is discovered in the future.

Vladislav Gennadievich Malyshkin

Problems of interpolation, classification, and clustering are considered. In the tenets of Radon--Nikodym approach $\langle f(\mathbf{x})ψ^2 \rangle / \langleψ^2\rangle$, where the $ψ(\mathbf{x})$ is a linear function on input attributes, all the answers are obtained from a generalized eigenproblem $|f|ψ^{[i]}\rangle = λ^{[i]} |ψ^{[i]}\rangle$. The solution to the interpolation problem is a regular Radon-Nikodym derivative. The solution to the classification problem requires prior and posterior probabilities that are obtained using the Lebesgue quadrature[1] technique. Whereas in a Bayesian approach new observations change only outcome probabilities, in the Radon-Nikodym approach not only outcome probabilities but also the probability space $|ψ^{[i]}\rangle$ change with new observations. This is a remarkable feature of the approach: both the probabilities and the probability space are constructed from the data. The Lebesgue quadrature technique can be also applied to the optimal clustering problem. The problem is solved by constructing a Gaussian quadrature on the Lebesgue measure. A distinguishing feature of the Radon-Nikodym approach is the knowledge of the invariant group: all the answers are invariant relatively any non-degenerated linear transform of input vector $\mathbf{x}$ components. A software product implementing the algorithms of interpolation, classification, and optimal clustering is available from the authors.

Vladislav Gennadievich Malyshkin

An important application of Lebesgue integral quadrature arXiv:1807.06007 is developed. Given two random processes, $f(x)$ and $g(x)$, two generalized eigenvalue problems can be formulated and solved. In addition to obtaining two Lebesgue quadratures (for $f$ and $g$) from two eigenproblems, the projections of $f$- and $g$- eigenvectors on each other allow to build a joint distribution estimator, the most general form of which is a density-matrix correlation. Examples of the density-matrix correlation can be a value-correlation $V_{f^{[i]};g^{[j]}}$, similar to a regular correlation concept, and a new one, a probability-correlation $P_{f^{[i]};g^{[j]}}$. If Christoffel function average is used instead of regular average the approach can be extended to an estimation of joint probability of three and more random processes. The theory is implemented numerically; the software is available under the GPLv3 license.

Vladislav Gennadievich Malyshkin

A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussian quadrature groups sums by function argument; it can be viewed as a $n$-point discrete measure, producing the Riemann integral. The Lebesgue quadrature groups sums by function value; it can be viewed as a $n$-point discrete distribution, producing the Lebesgue integral. Mathematically, the problem is reduced to a generalized eigenvalue problem: Lebesgue quadrature value-nodes are the eigenvalues and the corresponding weights are the square of the averaged eigenvectors. A numerical estimation of an integral as the Lebesgue integral is especially advantageous when analyzing irregular and stochastic processes. The approach separates the outcome (value-nodes) and the probability of the outcome (weight). For this reason, it is especially well-suited for the study of non-Gaussian processes. The software implementing the theory is available from the authors.

Vladislav Gennadievich Malyshkin

For Machine Learning (ML) classification problem, where a vector of $\mathbf{x}$--observations (values of attributes) is mapped to a single $y$ value (class label), a generalized Radon--Nikodym type of solution is proposed. Quantum--mechanics --like probability states $ψ^2(\mathbf{x})$ are considered and "Cluster Centers", corresponding to the extremums of $<yψ^2(\mathbf{x})>/<ψ^2(\mathbf{x})>$, are found from generalized eigenvalues problem. The eigenvalues give possible $y^{[i]}$ outcomes and corresponding to them eigenvectors $ψ^{[i]}(\mathbf{x})$ define "Cluster Centers". The projection of a $ψ$ state, localized at given $\mathbf{x}$ to classify, on these eigenvectors define the probability of $y^{[i]}$ outcome, thus avoiding using a norm ($L^2$ or other types), required for "quality criteria" in a typical Machine Learning technique. A coverage of each `Cluster Center" is calculated, what potentially allows to separate system properties (described by $y^{[i]}$ outcomes) and system testing conditions (described by $C^{[i]}$ coverage). As an example of such application $y$ distribution estimator is proposed in a form of pairs $(y^{[i]},C^{[i]})$, that can be considered as Gauss quadratures generalization. This estimator allows to perform $y$ probability distribution estimation in a strongly non--Gaussian case.

Vladislav Gennadievich Malyshkin

For distribution regression problem, where a bag of $x$--observations is mapped to a single $y$ value, a one--step solution is proposed. The problem of random distribution to random value is transformed to random vector to random value by taking distribution moments of $x$ observations in a bag as random vector. Then Radon--Nikodym or least squares theory can be applied, what give $y(x)$ estimator. The probability distribution of $y$ is also obtained, what requires solving generalized eigenvalues problem, matrix spectrum (not depending on $x$) give possible $y$ outcomes and depending on $x$ probabilities of outcomes can be obtained by projecting the distribution with fixed $x$ value (delta--function) to corresponding eigenvector. A library providing numerically stable polynomial basis for these calculations is available, what make the proposed approach practical.

Vladislav Gennadievich Malyshkin

A two--step Christoffel function based solution is proposed to distribution regression problem. On the first step, to model distribution of observations inside a bag, build Christoffel function for each bag of observations. Then, on the second step, build outcome variable Christoffel function, but use the bag's Christoffel function value at given point as the weight for the bag's outcome. The approach allows the result to be obtained in closed form and then to be evaluated numerically. While most of existing approaches minimize some kind an error between outcome and prediction, the proposed approach is conceptually different, because it uses Christoffel function for knowledge representation, what is conceptually equivalent working with probabilities only. To receive possible outcomes and their probabilities Gauss quadrature for second--step measure can be built, then the nodes give possible outcomes and normalized weights -- outcome probabilities. A library providing numerically stable polynomial basis for these calculations is available, what make the proposed approach practical.

Vladislav Gennadievich Malyshkin

For an image pixel information can be converted to the moments of some basis $Q_k$, e.g. Fourier-Mellin, Zernike, monomials, etc. Given sufficient number of moments pixel information can be completely recovered, for insufficient number of moments only partial information can be recovered and the image reconstruction is, at best, of interpolatory type. Standard approach is to present interpolated value as a linear combination of basis functions, what is equivalent to least squares expansion. However, recent progress in numerical stability of moments estimation allows image information to be recovered from moments in a completely different manner, applying Radon-Nikodym type of expansion, what gives the result as a ratio of two quadratic forms of basis functions. In contrast with least squares the Radon-Nikodym approach has oscillation near the boundaries very much suppressed and does not diverge outside of basis support. While least squares theory operate with vectors $<fQ_k>$, Radon-Nikodym theory operates with matrices $<fQ_jQ_k>$, what make the approach much more suitable to image transforms and statistical property estimation.