Decomposition Problem Research Articles

On the cryptosystems based on two Eulerian transformations defined over the commutative rings Z2s, s>1

We suggest the family of ciphers sEn, n = 2, 3, ... with the space of plaintexts (Z2^s)^n, s > 1, such that the encryption map is the composition of kinds G = G1 A1 G2 A2, where Ai are the affine transformations from AGLn(Z2^s) preserving the variety (Z*2^s)^n. Eulerian endomorphisms Gi, i = 1, 2, of K[x1, x2, ..., xn] move xi to the monomial term M x1^d(1) x2^d(2) ... xn^d(n), M in Z2^s, and act on (Z2^s)^n as bijective transformations. The cipher is converted to a protocol-supported cryptosystem. Protocols of Noncommutative Cryptography implemented on the platform of Eulerian endomorphisms are used for the delivery of Gi and Ai from Alice to Bob. One can use twisted Diffie-Hellman protocols, which security rests on the complexity of the Conjugacy Power problem, or the hidden tame homomorphism protocol, which security rests on the word decomposition problem. Instead of delivering Gi, Alice and Bob can elaborate these transformations via the inverse twisted Diffie-Hellman protocol, implemented on the platform of tame Eulerian transformations of (Z*2^s)^n. The cost of a single protocol is O(n^3), and the cost of computing the reimage of the used nonlinear map is O(n^2). So, the verification of nt, t ≥ 1, signatures takes time O(nt + 2). Instead of the inverse twisted Diffie-Hellman protocol, correspondents can use the inverse hidden tame homomorphism protocol, which rests on the complexity of word decomposition for tame Eulerian transformations. We use natural bijections between Z2^s and Z2^(s-1), Z2^s and finite field F2^(s-1), and Z2^s and Boolean ring B(s-1) of order 2^(s-1) to modify the family of ciphers or cryptosystems via the change of AGLn(Z2^s) for AGLn(K), where K is one of the rings Z2^(s-1), F2^(s-1), or B(s-1). New ciphers are defined via the multiplication of two different commutative rings Z2^s and K. This does not allow treating them as stream ciphers of multivariate cryptography and using corresponding cryptanalytic techniques. An adversary is not able to use known cryptanalytical methods such as linearization attacks. We discuss the option of changing the elements of AGLn(Z2^s) or AGLn(K) for nonlinear multivariate transformations F of (Z2^s)^n or K^n with a symmetric trapdoor accelerator T, i.e., a piece of information such that the knowledge of T allows computing the value F(p) for an arbitrarily chosen p in P in time O(n^2) and solving the equation of the form F(x) = c for each c in C in time O(n^2).

Short-term prediction of power outages in electrical distribution networks

Predictive maintenance and reliability engineering are critical in industrial settings to enhance equipment performance and minimize unplanned downtime. This research, conducted within the machine learning framework, presents innovative solutions to the challenging problem of equipment failure prediction. The study creatively utilizes extensive datasets, including equipment records, weather conditions, and maintenance logs, to develop robust predictive models. Two distinct machine learning models are established for equipment and cables/lines, addressing the intricacies of class imbalances and missing data attributes. Model refinement, feature engineering, and interdisciplinary collaboration enhance predictive accuracy, precision, and recall. Notably, this research highlights the creative application of engineering knowledge and data science techniques, reasoning about complex equipment systems, and the importance of problem decomposition. The outcomes underscore the potential for real-time predictive maintenance in industrial contexts, offering substantial cost savings and improved equipment reliability. This research contributes to the evolving field of predictive maintenance and paves the way for future innovations in reliability engineering.

Technical framework and empirical analysis of resource and environmental carrying capacity evaluation in karst regions under the constraint of ecological civilization

Evaluating resource and environmental carrying capacity (RECC) within the framework of ecological civilization is essential for reconciling development with ecological preservation and optimizing land-use patterns. This evaluation through the lens of ecological civilization is crucial for effective resource allocation. This article introduces a technical framework specifically designed for analyzing and evaluating resource and environmental carrying capacity in karst regions, focusing on Anshun City in Guizhou Province as a case study. The framework follows a structured approach, i.e., “problem identification, problem decomposition, and decision-making assessment,” which proves effective for these evaluations. Furthermore, the integrated evaluation index system incorporating resources, environment, and socioeconomic factors under the principles of ecological civilization aligns with contemporary developmental requirements while reflecting the distinctive characteristics of the resources and environment in karst regions. This alignment enhances the specificity and relevance of the evaluation methodology. The application of the obstacle degree model strengthened the utility and depth of the evaluation results. Additionally, Pearson’s correlation analysis clarified the complex interrelationships among various index factors, highlighting the significant role of unique local resources in the carrying capacity of karst regions. Additionally, the article reveals a delay in the socioeconomic development of karst regions, which adversely impacts their carrying capacity. Thus, promoting regional economic growth is crucial for enhancing the overall carrying capacity. In light of these findings, the article proposes practical strategies to enhance resource and environmental carrying capacity. These include establishing technical standards for evaluation, accelerating industrial transfers focused on technological innovation, and enhancing market mechanisms for resource allocation.

Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets

AbstractWe study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.

Using Artificial Intelligence for Competitive Procurements: Legal Regulation Issues

Objective: to substantiate the promising directions of legal regulation of relations in the use of artificial intelligence technologies in competitive (commercial and public) procurement.Methods: the study was conducted using induction, synthesis, analogy, decomposition of problems and generalization of conclusions. The reasoning was based on the experience of a complex procurement of high-tech equipment. This real-life example was considered as an experimental model for the study and subsequent prediction of the potential use of artificial intelligence technologies in competitive procurement procedures.Results: advantages and potential risks of using artificial intelligence technologies in procurement work were formulated; recommendations on regulating such use were given. The authors highlighted recommendations of general legal nature concerning the legal personality and delictual capacity of artificial intelligence and proposed the wordings for new norms and options for regulating the use of new procurement tools. It was proved that artificial intelligence technologies, if used thoughtfully, may not only improve the work quality and significantly reduce organizational costs, but also help to develop the basic principles of regulated procurement: transparency of procedures, development of competition for contracts between qualified suppliers, reasonableness of decisions, and economic efficiency of the customer’s expenditures.Scientific novelty: despite a large number of works devoted to both the problems of artificial intelligence in general and its use in procurement in particular, the article considers this topic on the basis of mainly inductive reasoning, built on handling a particular case and experience of complex procurement for knowledge-intensive research, refracted through the prism of essential correlation between the basic concepts of “digitalization”, “automation”, “robotization” and so on.Practical significance: the directions of using artificial intelligence described in this paper can be implemented by corporate and, in the future, by public customers to improve the quality of their procurement. At the same time, the recommendations on the normative regulation of such innovation seem to be in demand both at the legislative and local levels.

Complexity Results on the Decomposition of a Digraph into Directed Linear Forests and Out-Stars

We consider two decomposition problems in directed graphs. We say that a digraph is $k$-bounded for some $k \in \mathbb{Z}_{\geq 1}$ if each of its connected components contains at most $k$ arcs. For the first problem, a directed linear forest is a collection of vertex-disjoint directed paths and we consider the problem of decomposing a given digraph into a $k$-bounded and an $\ell$-bounded directed linear forest for some fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. We give a full dichotomy for this problem by showing that it can be solved in polynomial time if $k+\ell \leq 3$ and is NP-complete otherwise. This answers a question of Campbell, Hörsch, and Moore. For the second problem, we say that an out-galaxy is a vertex-disjoint collection of out-stars. Again, we give a full dichotomy of when a given digraph can be edge-decomposed into a $k$-bounded and an $\ell$-bounded out-galaxy for fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. More precisely, we show that the problem can be solved in polynomial time if $\min\{k,\ell\}\in \{1,\infty\}$ and is NP-complete otherwise.

On Hidden Rank Deficiency in MCR Problems

ABSTRACTPure component decomposition problems in chemometrics can be classified into rank‐regular and rank‐deficient problems. Rank‐deficient problems are characterized by a spectral data matrix that has a lower rank than the number of chemical species. However, it is possible that there exists rank‐regular factorization of the spectral data matrix, but none of these solutions can be interpreted chemically, and only a solution of the MCR problem with rank deficiency is chemically meaningful. Then we say that the underlying problem suffers from a hidden rank deficiency. In this paper, MCR problems with hidden rank deficiency are introduced and analyzed with several examples for problems of rank 2 and rank 3. The area of feasible solutions is determined with the help of additional constraints on the solution.

PARALLEL COMPUTING

Parallel computing is a computational technique in which two or more tasks or instructions can be processed concurrently by means of several processors resulting into enhanced computational efficiency as well as decreased time for solving complex problems. As an approach to solve a particular problem, parallel computing is an attempt that decomposes a problem into constituent subproblems to be solved concurrently; it uses multiple cores, hardware systems to gain advantage in speed The use of parallel computing is prominent in sciences that call for massive computations such as simulations, data mining, artificial intelligence and HPC. Key Words- Parallel computing, Concurrent processing, Computational efficiency, Problem decomposition, Subproblems, Simulations, Data mining, Artificial intelligence, High-performance computing (HPC), Multiple processors, Multiple cores, Hardware systems

Decomposition of a parallel automaton into a net of sequential automata and low power state assignment of them at asynchronous implementation

Objectives. The problems of decomposition of a parallel automaton into a net of sequential automata at synchronous realization and low power race free state assignment of them are considered. The objective of the paper is to investigate the possibilities of applying decomposition in state assignment of partial states in order to decrease the problem dimension taking into account the peculiar properties of the asynchronous realization.Methods. The given parallel automaton is decomposed into a net of sequential asynchronous automata whose states are assigned then with ternary vectors. The power consumption lowering of the designed device is achieved by lowering the intensity of its memory elements switching that is appreciated by probabilities of transitions between the states of the automaton. The state assignment is reduced to the problem of minimal weighted cover. The probabilities of transitions between sets are calculated by means of solving a system of linear equations according to the Chapmann – Kolmogorov method.Results. A method to construct a net of sequential asynchronous automata that realizes the given parallel automaton is described. The paper touches upon the problem of minimization of interconnections in the net.Conclusion. Applying parallel automaton decomposition allows decreasing the dimension of the laborious problem of state assignment. The proposed method is intended for application in computer aided systems for design of discrete devices.

Toward subtask-decomposition-based learning and benchmarking for predicting genetic perturbation outcomes and beyond.

Deciphering cellular responses to genetic perturbations is fundamental for a wide array of biomedical applications. However, there are three main challenges: predicting single-genetic-perturbation outcomes, predicting multiple-genetic-perturbation outcomes and predicting genetic outcomes across cell lines. Here we introduce Subtask Decomposition Modeling for Genetic Perturbation Prediction (STAMP), a flexible artificial intelligence strategy for genetic perturbation outcome prediction and downstream applications. STAMP formulates genetic perturbation prediction as a subtask decomposition problem by resolving three progressive subtasks in a problem decomposition manner, that is, identifying postperturbation differentially expressed genes, determining the expression change directions of differentially expressed genes and finally estimating the magnitudes of gene expression changes. STAMP exhibits a substantial improvement over the existing approaches on three subtasks and beyond, including the ability to identify key regulatory genes and pathways on small samples and to reveal precise genetic interactions of diverse types.

Schwarz decomposition for parallel minimum lap-time problems: evaluating against ADMM

The Minimum Lap Time Problem (MLTP) remains a significant area of research, particularly in the motorsport context. This form of Optimal Control Problem (OCP) aims to minimise lap times on a specific track with a given vehicle. Various complexities in both vehicle and track models are employed across the literature to address optimal trajectory planning. While previous works have tackled MLTP as a singular task using a serial approach, the increasing model complexity and horizon length demands the utilisation of parallelisation techniques. This paper introduces a novel application of the Overlapping Schwarz Decomposition algorithm to address the MLTP. The algorithm divides the problem into smaller sub-problems based on different sectors of a track, distributing them among multiple processors. We validate and compare the Schwarz approach against a serial approach and the Alternating Direction Method of Multipliers (ADMM) in solving MLTP with over 2.5 million variables. Despite the general efficiency improvement of parallelisation compared to the serial approach, the Schwarz algorithm demonstrates superior speed, accuracy and robustness compared to ADMM. As a result of our findings, it emerges as the preferred choice when large-scale MLTPs need to be solved.

Analysis of students' computational thinking skills in solving algebraic problems in cultural contexts

This research aims to find out about students' computational thinking abilities in solving algebraic problems in cultural contests. This research is qualitative research carried out at Yogyakarta Middle School. The participants involved were 31 students in the even semester of grade 7. The data sources in this research were the results of students' computational thinking tests, interviews with students who were selected based on their communication skills and the results of test answers. Data instrument testing uses validity and reliability tests with data validity using triangulation techniques. Data analysis in this study used the pullulation technique between student test results and student interview results, then data reduction, data categorization, power display and conclusion drawing were carried out. The results of this research are that students in the high category can complete all indicators of computational thinking and problem-solving abilities. Students in the medium category can only complete two indicators of computational thinking and problem-solving abilities. Students in the low category have not been able to complete all indicators of computational thinking and problem-solving abilities. The study reveals a significant variance in the computational thinking abilities among junior high school students in Piyungan Development class VIII when solving algebra problems within a cultural context. High-achieving students demonstrated proficiency across all indicators of computational thinking, effectively navigating through various stages from abstraction to solution verification. In contrast, students categorized as moderate showed partial success, managing only a subset of the indicators, while struggling with the more complex stages of problem decomposition and implementation. Meanwhile, students classified in the low category were unable to achieve any indicators of computational thinking and problem-solving skills. This indicates a clear need for targeted educational interventions to enhance computational thinking among students, particularly those who are struggling, to ensure all students can develop the necessary skills to solve problems effectively.

Accurate Flow Decomposition via Robust Integer Linear Programming.

Minimum flow decomposition (MFD) is a common problem across various fields of Computer Science, where a flow is decomposed into a minimum set of weighted paths. However, in Bioinformatics applications, such as RNA transcript or quasi-species assembly, the flow is erroneous since it is obtained from noisy read coverages. Typical generalizations of the MFD problem to handle errors are based on least-squares formulations or modelling the erroneous flow values as ranges. All of these are thus focused on error handling at the level of individual edges. In this paper, we interpret the flow decomposition problem as a robust optimization problem and lift error-handling from individual edges to solution paths. As such, we introduce a new minimum path-error flow decomposition problem, for which we give an Integer Linear Programming formulation. Our experimental results reveal that our formulation can account for errors significantly better, by lowering the inaccuracy rate by 30-50% compared to previous error-handling formulations, with computational requirements that remain practical.

Using flow-through reactor to enhance ferric ions electrochemical regeneration in electro-fenton for wastewater treatment

Electrochemical regeneration of ferric ions catches more and more attention since it could decrease iron sludge to avoid the sludge problem while it could simultaneously obtain similar robust removal efficiency to refractory organic contaminants in wastewater treatment as the normal Fenton agent method. However, the electrochemical regeneration of ferric ions in aqueous solution was very slow. In this paper, the mass transfer limit of ferric ions in electro-Fenton reaction was evaluated. Koutecky-Levich equation reveals an extremely low diffusion coefficient (D0) value of ferric ions since its complex coordination of [Fe(HO)x(H2O)6-x]3-x. The D0 value was only 2.70 × 10−6 cm2·s−1. Flow-through reactor was therefore introduced in which the ferric ions was designed to penetrate through the 73.1 μm porous holes in the graphite fiber electrode. The μm-scaled confinement of ferric ions diffusion inside the holes was proved to successfully enhance the reduction current of ferric ions by more than 200 % since the diffusion distance of the ferric ions was significantly decreased in the flow-through reactor. However, besides the benefit of the flow-through reactor, the ferric ions reduction electro-Fenton (FeRR electro-Fenton) in flow-through still faces both the pH limit and H2O2 decomposition problems. Hydrogen evolution reaction (HER) could also cause the decrease of pH which exceeded the optimal pH window for Fenton and therefore destroyed the electro-Fenton reaction consequently although the electrochemical reaction of FeRR was 770 mV prior to HER reaction. The regeneration of Fe (II) process simultaneous destruction of H2O2 since H2O2 e-reduction was 120 mV prior to FeRR reaction, which resulted in only very low H2O2 concentration suitable for FeRR electro-Fenton. Even using stepwise addition of H2O2 in electro-Fenton, the decomposition of H2O2 still could not be avoided. Although the decomposition of H2O2 quite limited the application of FeRR electro-Fenton in real application, FeRR electro-Fenton still supports the enhancement removal efficiency of the refractory organic contaminants under low organic contaminants application.

Arithmetic branching law and generic 𝐿-packets

Let G G be a classical group defined over a local field F F of characteristic zero. For any irreducible admissible representation π \pi of G ( F ) G(F) , which is of Casselman-Wallach type if F F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F F . In particular, if π \pi has a generic local L L -parameter, we introduce the spectral first occurrence index f s ( π ) {\mathfrak {f}}_{\mathfrak {s}}(\pi ) and the arithmetic first occurrence index f a ( π ) {\mathfrak {f}}_{{\mathfrak {a}}}(\pi ) of π \pi and prove in this paper that f s ( π ) = f a ( π ) {\mathfrak {f}}_{\mathfrak {s}}(\pi )={\mathfrak {f}}_{{\mathfrak {a}}}(\pi ) . Based on the theory of consecutive descents of enhanced L L -parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L L -packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality.

Matheuristic approaches for multi-visit drone routing problem to prevent forest fires

Forest fires draw more attention as the impact of elements that threaten nature increases, such as the thinning of the ozone layer, and global warming. The prevention of forest fires is extremely important for the protection of natural life, and the provision of a healthy world to future generations. Some of the methods used for the prevention of forest fires are observation towers, unmanned aerial vehicles, images taken from satellites, and detectors. Noticing the fire as soon as it starts and intervening in the fire prevents the fire from spreading and causing major and negative consequences. The degree of fire sensitivity of forest areas may vary depending on factors such as the climate of the region, topographic structure, humidity ratio, vegetation, tree species, and density. Observing regions with high fire sensitivity more frequently than regions with low fire sensitivity will prevent the spread of fires faster by detecting them. Different from the literature, in this study, the degree of sensitivity of fire-sensitive areas are taken into account. Visit frequency is determined according to the degree of fire sensitivity. Due to the complexity of the constraints, the mathematical model can not reach the optimal solution in a short time. To solve larger problem decomposition based matheuristic approaches are proposed. Matheuristic algorithms are compared using different parameters for samples of different sizes.

Three layered sparse dictionary learning algorithm for enhancing the subject wise segregation of brain networks

Independent component analysis (ICA) and dictionary learning (DL) are the most successful blind source separation (BSS) methods for functional magnetic resonance imaging (fMRI) data analysis. However, ICA to higher and DL to lower extent may suffer from performance degradation by the presence of anomalous observations in the recovered time courses (TCs) and high overlaps among spatial maps (SMs). This paper addressed both problems using a novel three-layered sparse DL (TLSDL) algorithm that incorporated prior information in the dictionary update process and recovered full-rank outlier-free TCs from highly corrupted measurements. The associated sequential DL model involved factorizing each subject’s data into a multi-subject (MS) dictionary and MS sparse code while imposing a low-rank and a sparse matrix decomposition restriction on the dictionary matrix. It is derived by solving three layers of feature extraction and component estimation. The first and second layers captured brain regions with low and moderate spatial overlaps, respectively. The third layer that segregated regions with significant spatial overlaps solved a sequence of vector decomposition problems using the proximal alternating linearized minimization (PALM) method and solved a decomposition restriction using the alternating directions method (ALM). It learned outlier-free dynamics that integrate spatiotemporal diversities across brains and external information. It differs from existing DL methods owing to its unique optimization model, which incorporates prior knowledge, subject-wise/multi-subject representation matrices, and outlier handling. The TLSDL algorithm was compared with existing dictionary learning algorithms using experimental and synthetic fMRI datasets to verify its performance. Overall, the mean correlation value was found to be 26%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$26\\%$$\\end{document} higher for the TLSDL than for the state-of-the-art subject-wise sequential DL (swsDL) technique.

Super-localized orthogonal decomposition for convection-dominated diffusion problems

This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm, the Galerkin projection onto this generalized finite element space even yields ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-independent error bounds, ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.

Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems

Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems

Parametric channel estimation for RIS-assisted mmWave MIMO-OFDM systems with low pilot overhead

We consider the channel estimation problem in a millimeter-wave (mmWave) multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) system aided by a passive reconfigurable intelligent surface (RIS). Initially, leveraging the inherent sparsity of mmWave channels, we decouple the channel estimation into the recovery of multipath parameters, including spatial frequencies of angles, delays, and complex gains. Subsequently, we employ two training schemes: the first involves a full row rank constrained RIS phase shift pattern, and the second incorporates a Kronecker structure constrained RIS phase shift pattern. These schemes reform channel estimation into estimating parameters from two fifth-order tensor signals, which admit constrained Canonical Polyadic decomposition. Moreover, both tensor signals share Vandermonde-constrained factor matrices, enabling algebraic algorithms for tensor decomposition problems and recovering channel multipath parameters. Theoretical analyses show that the first training scheme has a lower signal processing complexity for channel estimation, while the second attains a higher estimation accuracy. Also, we prove that our proposed two schemes have the same minimum pilot overhead, which is as low as proportional to the product of the number of paths in the two MIMO channels (transmitter-RIS and RIS-receiver). Numerical results validate the effectiveness of our proposed methods.