Bijective Transformation Research Articles

Efficient solutions for vector optimization problem on an extended interval vector space and its application to portfolio optimization

In this paper, a generalized interval vector space is investigated and defined as an ordered relation in the form of a bijective linear transformation of its onto a real vector space. The ordered relation is utilized to formulate an interval optimization problem in the same manner as a classical multi-objective programming problem. A methodology that addresses the existence of efficient solutions for the multi-objective interval optimization problem has also been discussed. Various numerical examples are described to illustrate and substantiate all developed concepts. Furthermore, the multi-objective portfolio rebalancing problem for a time horizon is designed based on the developed interval vector optimization. An algorithm using exhaustive solution technology has been proposed to achieve an efficient investment strategy. Finally, its applicability and efficacy are analyzed using Bombay Stock Exchange India data sets.

On the Bit-Slice representations of some nonlinear bijective transformations

Изучаются способы построения эффективных Bit-Slice (схемных) представлений некоторых классов нелинейных биективных преобразований, имеющих высокие криптографические качества. Для некоторых биективных преобразований пространства $V_8$ из этих классов мы находим (путем объединения аналитических методов с программным обеспечением с открытым исходным кодом) представления логических схем с малым количеством вентилей с помощью двоичных логических операций $\mathtt{AND, XOR, OR}$ и $\mathtt{NOT}$. В частности, для подстановки, используемой в российском криптографическом стандарте ГОСТ Р 34.12-2015 «Кузнечик», мы получаем схемную реализацию, которая в общей сложности использует 176 двоичных логических операций. Новое представление требует на 51 логическую операцию меньше, чем ранее известное (требующее 227 поразрядных логических операций, см., например, [1]).

Chow's theorem for Hilbert Grassmannians as a Wigner-type theorem

Let H be an infinite-dimensional complex Hilbert space. Denote by G ∞ ( H ) the Grassmannian formed by closed subspaces of H whose dimension and codimension both are infinite. We say that X , Y ∈ G ∞ ( H ) are ortho-adjacent if they are compatible and X ∩ Y is a hyperplane in both X, Y. A subset C ⊂ G ∞ ( H ) is called an A-component if for any X , Y ∈ C the intersection X ∩ Y is of the same finite codimension in both X, Y and C is maximal with respect to this property. Let f be a bijective transformation of G ∞ ( H ) preserving the ortho-adjacency relation in both directions. We show that the restriction of f to every A-component is induced by a unitary or anti-unitary operator or it is the composition of the orthocomplementary map and a map induced by a unitary or anti-unitary operator. Note that the restrictions of f to distinct A-components can be related to different operators.

An experimental study of neural estimators of the mutual information between random vectors modeling power spectrum features

Mutual information (MI) quantifies the statistical dependency between a pair of random variables and plays a central role in signal processing and data analysis. Recent advances in machine learning have enabled the estimation of MI from a dataset using the expressive power of neural networks. In this study, we conducted a comparative experimental analysis of several existing neural estimators of MI between random vectors that model power spectrum features. We explored alternative models of power spectrum features by leveraging information-theoretic data processing inequality and bijective transformations. Empirical results demonstrated that each neural estimator of MI covered in this study has its limitations. In practical applications, we recommend the collective use of existing neural estimators in a complementary manner for the problem of estimating MI between power spectrum features.

YANG2UML: Bijective Transformation and Simplification of YANG to UML

YANG2UML: Bijective Transformation and Simplification of YANG to UML

A Kernel Measure of Dissimilarity between M Distributions

Given M ≥ 2 distributions defined on a general measurable space, we introduce a nonparametric (kernel) measure of multi-sample dissimilarity (KMD) — a parameter that quantifies the difference between the M distributions. The population KMD, which takes values between 0 and 1, is 0 if and only if all the M distributions are the same, and 1 if and only if all the distributions are mutually singular. Moreover, KMD possesses many properties commonly associated with f-divergences such as the data processing inequality and invariance under bijective transformations. The sample estimate of KMD, based on independent observations from the M distributions, can be computed in near linear time (up to logarithmic factors) using k-nearest neighbor graphs (for k ≥ 1 fixed). We develop an easily implementable test for the equality of M distributions based on the sample KMD that is consistent against all alternatives where at least two distributions are not equal. We prove central limit theorems for the sample KMD, and provide a complete characterization of the asymptotic power of the test, as well as its detection threshold. The usefulness of our measure is demonstrated via real and synthetic data examples; our method is also implemented in an R package.

Varieties of Systems of DEF Generated by Isomorphic Transformations

Despite more than a century of origin and development, the theory of discrete exponential function (DEF) systems continues to attract the attention of mathematicians and application specialists in various fields of science and technology. One of the most successful applications of the DEF systems is the spectral processing of discrete signals based on fast Fourier transform (FFT) algorithms in the DEF bases. The construction of structural schemes of FFT algorithms is preceded, as a rule, by the factorization of the DEF matrices. The main problem encountered when factorizing DEF matrices is that the elements of such matrices are the degrees of phase multipliers, which are complex-valued quantities. In this connection, the computational complexity of factorization of DEF matrices may be too large, especially when the number of components of the matrix order decomposition is large. In this paper, we propose a relatively simple method of mutually unambiguous transition from complex-valued DEF matrices to matrices whose elements are natural numbers equal to the degree indices of phase multipliers in the canonical DEF matrices. Through this bijective transformation, the factorization of DEF matrices becomes significantly more manageable, streamlining the overall process of factorization.

A classification of S-boxes generated by orthogonal cellular automata

Most of the approaches published in the literature to construct S-boxes via Cellular Automata (CA) work by either iterating a finite CA for several time steps, or by a one-shot application of the global rule. The main characteristic that brings together these works is that they employ a single CA rule to define the vectorial Boolean function of the S-box. In this work, we explore a different direction for the design of S-boxes that leverages on Orthogonal CA (OCA), i.e. pairs of CA rules giving rise to orthogonal Latin squares. The motivation stands on the facts that an OCA pair already defines a bijective transformation, and moreover the orthogonality property of the resulting Latin squares ensures a minimum amount of diffusion. We exhaustively enumerate all S-boxes generated by OCA pairs of diameter 4≤d≤6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4 \\le d \\le 6$$\\end{document}, and measure their nonlinearity. Interestingly, we observe that for d=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=4$$\\end{document} and d=5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=5$$\\end{document} all S-boxes are linear, despite the underlying CA local rules being nonlinear. The smallest nonlinear S-boxes emerges for d=6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=6$$\\end{document}, but their nonlinearity is still too low to be used in practice. Nonetheless, we unearth an interesting structure of linear OCA S-boxes, proving that their Linear Components Space is itself the image of a linear CA, or equivalently a polynomial code. We finally classify all linear OCA S-boxes in terms of their generator polynomials.

Catastrophe Bond Diversification Strategy Using Probabilistic–Possibilistic Bijective Transformation and Credibility Measures in Fuzzy Environment

The variety of catastrophe bond issuances can be used for portfolio diversification. However, the structure of catastrophe bonds differs from traditional bonds in that the face value and coupons depend on triggering events. This study aims to build a diversification strategy model framework using probabilistic–possibilistic bijective transformation (PPBT) and credibility measures in fuzzy environments based on the payoff function. The stages of modeling include identifying the trigger distribution; determining the membership degrees for the face value and coupons using PPBT; calculating the average face value and coupons using the fuzzy quantification theory; formulating the fuzzy variables for the yield; defining the function of triangular fuzzy membership for the yield; defining the credibility distribution for the triangular fuzzy variables for the yield; determining the expectation and total variance for the yield; developing a model of the catastrophe bond diversification strategy; the numerical simulation of the catastrophe bond strategy model; and formulating a solution to the simulation model of the diversification strategy using the sequential method, quadratic programming, transformation, and linearization techniques. The simulation results show that the proposed model can overcome the self-duality characteristic not possessed by the possibilistic measures in the fuzzy variables. The results obtained are expected to contribute to describing the yield uncertainty of investing in catastrophe bond assets so that investors can make wise decisions.

Ordinal optimization through multi-objective reformulation

We analyze combinatorial optimization problems with ordinal, i.e., non-additive, objective functions that assign categories (like good, medium and bad) rather than cost coefficients to the elements of feasible solutions. We review different optimality concepts for ordinal optimization problems and discuss their similarities and differences. We then focus on two prevalent optimality concepts that are shown to be equivalent. Our main focus lies on the investigation of a bijective linear transformation that transforms ordinal optimization problems to associated standard multi-objective optimization problems with binary cost coefficients. Since this transformation preserves all properties of the underlying problem, problem-specific solution methods remain applicable. A prominent example is dynamic programming and Bellman’s principle of optimality, that can be applied, e.g., to ordinal shortest path and ordinal knapsack problems. We investigate the interrelation between scalarization techniques and methods based on the hypervolume indicator when applied to the ordinal and the transformed problem, respectively. Furthermore, we extend our results to multi-objective optimization problems that combine ordinal and real-valued objective functions.

Interaction Control of a Robotic Manipulator With the Surface of Deformable Object

Robotic manipulation of deformable objects has drawn the attention of researchers over the past few years and is associated with a large spectrum of new application perspectives. In this article, we present an efficient integrated motion planning framework to effectively and accurately control a robotic manipulator executing interactive tasks on the surface of a deformable object. The proposed interactive motion planning framework is based on a mesh representation of the object, integrating three efficient preprocessing algorithmic steps, including visual object segmentation, finite element method deformation tracking, and local mesh parameterization. The use of barycentric coordinates, defined on the mesh triangles, enables the establishment of bijective transformations between the deformable part of an object surface and its planar (static and dynamic) parameterized mapping. By merging these spatial transformations with the preprocessing steps, in combination with an active stiffness scheme for robot manipulator control, we are able to achieve accurate and reactive motion planning of interactive trajectories, even under large and persistent visual occlusions (such as due to the presence of the robot in the visual scene). An extensive experimental evaluation study is presented, involving a robotic manipulator in interaction with a hemispherical model of controllable periodic active deformation, which permits precise ground truth derivation. Motion planning accuracy is evaluated in comparison with our previous direct vision-based approach, showing clearly superior performance of the proposed approach under all experimental conditions. The performance of the proposed framework is also further highlighted in tasks involving physical point tracking, interactive programming by human demonstration, as well as contact force regulation.

On a preserver problem concerning Kubo-Ando means of positive operators

We study bijective transformations ϕ of the set of positive operators on a Hilbert space such that given a Kubo-Ando mean σ with certain properties, ‖AσB‖=‖ϕ(A)σϕ(B)‖ holds for every pair A,B of positive operators. It is shown that whenever the Hilbert space is finite dimensional, then ϕ is necessarily conjugation by a fixed unitary or antiunitary operator. Coupled with earlier results, this completely resolves the above preserver problem in the finite dimensional case. Most ideas of the proof carry over to the infinite dimensional setting, thus some partial results are obtained in that case as well.

Boundary-free estimators of the mean residual life function for data on general interval

We propose two new kernel-type estimators of the mean residual life function m X of bounded or half-bounded interval-supported distributions. Though not as severe as the boundary problems in kernel density estimation, eliminating the boundary bias problems that occur in the naive kernel estimator of the mean residual life function is needed. In this article, we utilize the property of bijective transformation. Furthermore, our proposed methods preserve the mean value property, which cannot be done by the naive kernel estimator. Some simulation results showing the estimators’ performances and real data analysis will be presented in the last part of this article.

РЕАЛІЗАЦІЯ ПІДСТАНОВОК ДОВІЛЬНОЇ РОЗРЯДНОСТІ НА БАЗІ КОМБІНОВАНИХ КАСКАДІВ КОНСТРУКТИВНИХ МОДУЛІВ

The most crucial aspects of permutations are their speed and ease of implementation. This article examines the implementation of arbitrary bitness permutations in computer engineering using a particular class of combination structures with linear complexity, namely, combined cascades of structural units. The reflection formed by this linear structure is identical to that of the corresponding Mealy finite state machine, which allows for the exploration of the properties of structural units and the cascade in the context of the theory of digital automata. The purpose of this permutation is to convert large volumes of data using hardware or software quickly and simply that can be used in various research fields. The paper investigates the bijectivity and equivalence of the reflection and presents an algorithm for constructing finitestate machines for both direct and inverted permutations, along with examples of state and output table construction. The article also provides examples of hardware implementation using field-programmable gate arrays and estimates the size of state and output tables for software implementation. The theoretical speed of bijective reflection transformations is calculated for both field-programmable gate arrays and software implementation, and the paper compares the speed of software implementations based on combined and one-dimensional cascades of constructive units. The practical verification of processing speed is made with software implementation. Finally, the article proposes areas of application for this arbitrary bitness permutation.

A Multi-Centric Evaluation of AI-Driven Synthetic CT Generation Form Low Field Magnetic Resonance Imaging

<h3>Purpose/Objective(s)</h3> Magnetic Resonance is an essential modality in the context of radiotherapy primarily for providing additional information with respect to tumor functional information leading to better target delineation (brain, prostate, etc.) and secondary in the context of emergence of magnetic resonance guided radiotherapy. However, in both cases computed tomography acquisition remains necessary for dosimetric purposes hampering the workflow, introducing additional cost and dose simulation/optimization discrepancies (due to the registration issues between MR & CT) while being associated with additional patient toxicity. This study aims to investigate in a retrospective manner the relevance deep learning synthetic CT generation as an alternative for simulation and planning for pelvic tumors treated with low field (0.35T) MRgRT. <h3>Materials/Methods</h3> In the context of this study, a cycle generative adversarial neural network (GAN) deep learning architecture was trained to determine a bijective transformation between a low field (0.35T) MR and the associated computed tomography scan acquisition. The training set involved 350 pairs of weakly aligned data of pelvis cases. A retrospective cohort involving 20 test cases coming from eight different institutions (US: 2, EU: 5, AS: 1) involving different CT vendors was considered for testing. <h3>Results</h3> Reconstruction performance was assessed using the OARs used for treatment. The observed average scaled (between maximum and minimum HU values of the CT) difference between the reconstructed and the CT used for planning was 35.08 with significant discrepancies across organs. Femoral heads were the most reliably reconstructed (4.51 & 4.77) while rectum and sigmoid were the less precise ones (53.13 & 51.48). In terms of qualitative evaluation, the presence of fiducial markers heavily penalizes the reconstruction due to the implicit propagation of errors associated with the "convolution" nature of deep learning. The presence of air bubbles visible on the CT (sigmoid and rectum) is also an issue since these elements are marginally perceptible in the MR and therefore can hardly be predicted from generator. Detailed reconstruction results per organ are appended. <h3>Conclusion</h3> This retrospective multi-centric study is a first step toward assessing the potential of a fully low field MR-based treatment planning workflow that eliminates the need of CT acquisition. Integration of the synthetic CT in the simulation/optimization and assessment of the induced dosimetric errors are necessary steps to determine the clinical relevance of our preliminary findings.

Invertible Rescaling Network and Its Extensions

Image rescaling is a commonly used bidirectional operation, which first downscales high-resolution images to fit various display screens or to be storage- and bandwidth-friendly, and afterward upscales the corresponding low-resolution images to recover the original resolution or the details in the zoom-in images. However, the non-injective downscaling mapping discards high-frequency contents, leading to the ill-posed problem for the inverse restoration task. This can be abstracted as a general image degradation–restoration problem with information loss. In this work, we propose a novel invertible framework to handle this general problem, which models the bidirectional degradation and restoration from a new perspective, i.e. invertible bijective transformation. The invertibility enables the framework to model the information loss of pre-degradation in the form of distribution, which could mitigate the ill-posed problem during post-restoration. To be specific, we develop invertible models to generate valid degraded images and meanwhile transform the distribution of lost contents to the fixed distribution of a latent variable during the forward degradation. Then restoration is made tractable by applying the inverse transformation on the generated degraded image together with a randomly-drawn latent variable. We start from image rescaling and instantiate the model as Invertible Rescaling Network, which can be easily extended to the similar decolorization–colorization task. We further propose to combine the invertible framework with existing degradation methods such as image compression for wider applications. Experimental results demonstrate the significant improvement of our model over existing methods in terms of both quantitative and qualitative evaluations of upscaling and colorizing reconstruction from downscaled and decolorized images, and rate-distortion of image compression. Code is available at https://github.com/pkuxmq/Invertible-Image-Rescaling.

Improving bridge estimators via f-GAN

Bridge sampling is a powerful Monte Carlo method for estimating ratios of normalizing constants. Various methods have been introduced to improve its efficiency. These methods aim to increase the overlap between the densities by applying appropriate transformations to them without changing their normalizing constants. In this paper, we first give a new estimator of the asymptotic relative mean square error (RMSE) of the optimal Bridge estimator by equivalently estimating an f-divergence between the two densities. We then utilize this framework and propose f-GAN-Bridge estimator (f-GB) based on a bijective transformation that maps one density to the other and minimizes the asymptotic RMSE of the optimal Bridge estimator with respect to the densities. This transformation is chosen by minimizing a specific f-divergence between the densities. We show f-GB is optimal in the sense that within any given set of candidate transformations, the f-GB estimator can asymptotically achieve an RMSE lower than or equal to that achieved by Bridge estimators based on any other transformed densities. Numerical experiments show that f-GB outperforms existing methods in simulated and real-world examples. In addition, we discuss how Bridge estimators naturally arise from the problem of f-divergence estimation.

Commutativity preserving transformations on conjugacy classes of compact self-adjoint operators

Let H be a complex Hilbert space of dimension not less than 3 and let C be a conjugacy class of compact self-adjoint operators on H. Suppose that the dimension of the kernels of operators from C is not less than the dimension of their ranges. In the case when C is formed by operators of finite rank k and dim⁡H=2k, we require that k≥4. We show that every bijective transformation of C preserving the commutativity in both directions is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces of the same dimension.

Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models

We introduce in this work the normalizing field flows (NFF) for learning random fields from scattered measurements. More precisely, we construct a bijective transformation (a normalizing flow characterizing by neural networks) between a Gaussian random field with the Karhunen-Loève (KL) expansion structure and the target stochastic field, where the KL expansion coefficients and the invertible networks are trained by maximizing the sum of the log-likelihood on scattered measurements. This NFF model can be used to solve data-driven forward, inverse, and mixed forward/inverse stochastic partial differential equations in a unified framework. We demonstrate the capability of the proposed NFF model for learning non-Gaussian processes and different types of stochastic partial differential equations.

A Probabilistic Forecast-Driven Strategy for a Risk-Aware Participation in the Capacity Firming Market

This paper addresses the energy management of a grid-connected renewable generation plant coupled with a battery energy storage device in the capacity firming market, designed to promote renewable power generation facilities in small non-interconnected grids. The core contribution is to propose a probabilistic forecast-driven strategy, modeled as a min-max-min robust optimization problem with recourse. It is solved using a Benders-dual cutting plane algorithm and a column and constraints generation algorithm in a tractable manner. A dynamic risk-averse parameters selection strategy based on the quantile forecasts distribution is proposed to improve the results. A secondary contribution is to use a recently developed deep learning model known as normalizing flows to generate quantile forecasts of renewable generation for the robust optimization problem. This technique provides a general mechanism for defining expressive probability distributions, only requiring the specification of a base distribution and a series of bijective transformations. Overall, the robust approach improves the results over a deterministic approach with nominal point forecasts by finding a trade-off between conservative and risk-seeking policies. The case study uses the photovoltaic generation monitored on-site at the University of Li\`ege (ULi\`ege), Belgium.