Block Decomposition Research Articles

On the uniqueness and computation of commuting extensions

A tuple (Z1,…,Zp) of matrices of size r×r is said to be a commuting extension of a tuple (A1,…,Ap) of matrices of size n×n if the Zi pairwise commute and each Ai sits in the upper left corner of a block decomposition of Zi (here, r and n are two arbitrary integers with n<r). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called “quantum Zeno dynamics.” Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results:(i)Theorems on the uniqueness of commuting extensions for three matrices or more.(ii)Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1.(iii)A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.

Blind image authentication using singular value decomposition with enhanced robustness by augmenting hamming code

Singular value decomposition (SVD) based image authentication has widespread applications for digital media protection including forensic and medical domains due to the high stability and robustness of singular values under small perturbations. However, most of these techniques are non-blind and have high redundancies for the extraction of secret data. The proposed scheme employs a blind extraction strategy by utilizing the intrinsic relationship of coefficient pairs of the U matrix of the host image blocks. This relationship is utilized to hide a secret binary bit after computing a threshold value in constant time. The host image underwent block-wise decomposition and hash-based randomization of blocks, followed by block-wise SVD for embedding secret bits in the U matrix. Hamming coded watermark is embedded for high precision extraction under active attacks. High imperceptibility and strong robustness are observed under attacks when tested across general purpose as well as image datasets and the system outperforms many existing works. Experimental observation shows the average PSNR between the host and marked image and average NCC between embedded and extracted watermark to be 36.98 dB and 0.999 respectively for the SIPI dataset whereas 52.7 dB and 1 respectively for the X-Ray dataset. The system exhibited high robustness under Salt-and-pepper noise (SAPN), Gaussian Noise (GAUN), Cropping (CRP), Median Filtering (MF), Histogram Equalization (HEQ), and Jpeg Compression (JPC) with average NCC values of 0.998, 1, 0.932, 0.999 and 0.967 respectively for X-Ray dataset. The time complexity O (M × N) is observed for both embedding and extraction algorithms for a host image of size M × N.

Spectral flow and the conformal block expansion for strings in AdS3

We present a detailed study of spectrally flowed four-point functions in the SL(2,ℝ) WZW model, focusing on their conformal block decomposition. Dei and Eberhardt conjectured a general formula relating these observables to their unflowed counterparts. Although the latter are not known in closed form, their conformal block expansion has been formally established. By combining this information with the integral transform that encodes the effect of spectral flow, we show how to describe a considerable number of s-channel exchanges, including cases with both flowed and unflowed intermediate states. For all such processes, we compute the normalization of the corresponding conformal blocks in terms of products of the recently derived flowed three-point functions with arbitrary spectral flow charges. Our results constitute a highly non-trivial consistency check, thus strongly supporting the aforementioned conjecture, and establishing its computational power.

Renormalization group flows in AdS and the bootstrap program

We study correlation functions of the bulk stress tensor and boundary operators in Quantum Field Theories (QFT) in Anti-de Sitter (AdS) space. In particular, we derive new sum rules from the two-point function of the stress tensor and its three-point function with two boundary operators. In AdS2, this leads to a bootstrap setup that involves the central charge of the UV limit of the bulk QFT and may allow to follow a Renormalization Group (RG) flow non-perturbatively by continuously varying the AdS radius. Along the way, we establish the convergence properties of the newly discovered local block decomposition of the three-point function.

Split_ Composite: A Radar Target Recognition Method on FFT Convolution Acceleration.

Synthetic Aperture Radar (SAR) is renowned for its all-weather and all-time imaging capabilities, making it invaluable for ship target recognition. Despite the advancements in deep learning models, the efficiency of Convolutional Neural Networks (CNNs) in the frequency domain is often constrained by memory limitations and the stringent real-time requirements of embedded systems. To surmount these obstacles, we introduce the Split_ Composite method, an innovative convolution acceleration technique grounded in Fast Fourier Transform (FFT). This method employs input block decomposition and a composite zero-padding approach to streamline memory bandwidth and computational complexity via optimized frequency-domain convolution and image reconstruction. By capitalizing on FFT's inherent periodicity to augment frequency resolution, Split_ Composite facilitates weight sharing, curtailing both memory access and computational demands. Our experiments, conducted using the OpenSARShip-4 dataset, confirm that the Split_ Composite method upholds high recognition precision while markedly enhancing inference velocity, especially in the realm of large-scale data processing, thereby exhibiting exceptional scalability and efficiency. When juxtaposed with state-of-the-art convolution optimization technologies such as Winograd and TensorRT, Split_ Composite has demonstrated a significant lead in inference speed without compromising the precision of recognition.

Learning Topological Operations on Meshes with Application to Block Decomposition of Polygons

We present a learning based framework for mesh quality improvement on unstructured triangular and quadrilateral meshes. Our model learns to improve mesh quality according to a prescribed objective function purely via self-play reinforcement learning with no prior heuristics. The actions performed on the mesh are standard local and global element operations. The goal is to minimize the deviation of the node degrees from their ideal values, which in the case of interior vertices leads to a minimization of irregular nodes.

Attractors coexistence and representation research of Lotka–Volterra recurrent neural networks by matrix analysis theory

Attractor neural network models have been extensively studied in maintaining brain working memory state, spatial representation, and other brain processing mechanisms. The coexistence of multiple attractors in nonlinear neural network is particularly worth studying. The attractor is a stable state in the dynamic system, and all adjacent states will eventually be pulled to the attractor over time. Attractors can be regarded as the expression of data in low-dimensional state space, and different kinds of attractors can be used to process different information. The coexistence of multiple attractors means that a neural network can store multiple patterns, that is to say different memory mechanisms appear at the same time. Investigating the coexistence and competition of attractors within Lotka–Volterra(LV) recurrent neural networks presents a pivotal avenue for understanding the dynamical underpinnings of complex systems, particularly in the realm of computational neuroscience and ecological modeling. This paper studies the coexistence of different types of attractors in asymmetric Lotka–Volterra recurrent neural networks. Through the weight matrix block and matrix decomposition theory, we obtain the explicit expressions of continuous attractors and discrete attractors. Since neurons have multiple grouping methods, each weight matrix block satisfying certain conditions corresponds to different attractors. Different from traditional symmetric networks, the eigenvalues of the asymmetric weight matrices may be complex, which may correspond to ring attractors. Our research shows that attractors can coexist under certain conditions, depending on the weight matrix and external input. Moreover, through simulation experiments, we show that external input will cause the competition of attractors. Finally, we provide several simulations of attractor coexistence to illustrate our theory.

Parabolic BGG categories and their block decomposition for Lie superalgebras of Cartan type

Parabolic BGG categories and their block decomposition for Lie superalgebras of Cartan type

On Perturbation of Operators and Rayleigh-Schrödinger Coefficients

Let A and E be self-adjoint matrices or operators on ℓ2(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^2({{\\mathbb {N}}})$$\\end{document}, where A is fixed and E is a small perturbation. We study how the eigenvalues of A+E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A+E$$\\end{document} depend on E, with the aim of obtaining second order formulas that are explicitly computable in terms of the spectral decomposition of A and a certain block decomposition of E. In particular we extend the classical Rayleigh-Schrödinger formulas for the one-parameter perturbation A+tE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A+tE$$\\end{document} where t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in {{\\mathbb {R}}}$$\\end{document} varies and E is held fixed, by dropping t and considering E as the variable.

Complex expressional characterizations learning based on block decomposition for temporal knowledge graph completion

A temporal knowledge graph (TKG) is a set of facts associated with different timestamps. TKG completion (TKGC) is the task of inferring unknown facts based on known facts, where mining and understanding the expressional characterization of timestamps from the fact sets are key. Expressional characterizations are complex because of two aspects: type-diversity and dependence-variability. Existing approaches do not comprehensively consider time points and time intervals; hence, facts with different time forms cannot be managed simultaneously. Additionally, these approaches overlook the interdependence and variability of timestamp positions in different types of facts, such as the relative order (e.g., whether the facts belong to the past, present, or future) and relative distance (e.g., the largest distance at which the facts occur), thus hindering the adequate perception of position. Therefore, we propose a novel TKGC model named Complex Expressional Characterizations with Block Decomposition (CEC-BD). This model is based on the block decomposition model and exploits sophisticated interactions for different types of facts by unifying the treatment of both time points and time intervals. In addition, we introduce a time-sensitive approach that injects timestamps with embedded information regarding fact occurrences into the CEC-BD model to capture the relative order and distance. To cluster and order time embeddings more effectively, we add a bias component that is randomly initialized and learned during training and further propose a temporal smoothness scheme. In particular, the model achieved a mean reciprocal rank of 3.05% and 10 times the speedup of state-of-the-art baseline models.

A Technique for Efficient Estimation of Dynamic Structural Equation Models: A Case Study

Dynamic structural equation models (DSEM) are designed for time series analysis of latent structures. Inherent to the application of DSEM is model parameter estimation, which has to be addressed in many applications by a single time series. In this context, however, the methods currently available either lack estimation quality or are computationally inefficient. Given the era of big data, the necessity for a trade-off between these properties may be detrimental to the applicability of DSEM. The paper is aimed at tackling this trade-off by proposing a novel estimator recursioning technique (ER technique) that facilitates the development of computationally efficient raw-data maximum likelihood estimation algorithms through data transformation, covariance matrix block decomposition, likelihood function reduction, and estimator recursioning steps. The ER technique is introduced by applying it to a special case of the general dynamic structural equation model that encompasses a noisy Wiener-process-type structure with input from the factor-analytic model. The resulting algorithm has been verified through a number of numerical experiments as well as implemented in a brand new R package EMLI, which is available on CRAN.

Reinforcement learning for block decomposition of planar CAD models

AbstractThe problem of hexahedral mesh generation of general CAD models has vexed researchers for over 3 decades and analysts often spend more than 50% of the design-analysis cycle time decomposing complex models into simpler blocks meshable by existing techniques. The decomposed blocks are required for generating good quality meshes (tilings of quadrilaterals or hexahedra) suitable for numerical simulations of physical systems governed by conservation laws. We present a novel AI-assisted method for decomposing (segmenting) planar CAD (computer-aided design) models into well shaped rectangular blocks. Even though the simple examples presented here can also be meshed using many conventional methods, we believe this work is proof-of-principle of a AI-based decomposition method that can eventually be generalized to complex 2D and 3D CAD models. Our method uses reinforcement learning to train an agent to perform a series of optimal cuts on the CAD model that result in a good quality block decomposition. We show that the agent quickly learns an effective strategy for picking the location and direction of the cuts and maximizing its rewards. This paper is the first successful demonstration of an agent autonomously learning how to perform this block decomposition task effectively, thereby holding the promise of a viable method to automate this challenging process for more complex cases.

Celestial conformal blocks of massless scalars and analytic continuation of the Appell function F1

In celestial conformal field theory (CCFT), the 4d massless scalars are represented by 2d conformal operators with conformal dimensions h = h¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{h} $$\\end{document} = (1 + iλ)/2. The Mellin transform of 4d massless scalar amplitudes gives the conformal correlators of CCFT. We study the conformal block decomposition of these celestial correlators in CCFT and obtain the explicit blocks. This conformal block decomposition is highly nontrivial, even for the simplest 4d massless scalar amplitude. We use the analytic continuation of the Appell hypergeometric function F1 and the method of monodromy projection of conformal blocks, to achieve this block decomposition. This procedure is consistent with the crossing symmetry, in both the correlator-level and each explicit block-level. We also investigate its behavior in the conformal soft limit and find that the Appell hypergeometric function F1 does not reduce to the Gauss hypergeometric function. This is different from the block decomposition of celestial gluons we studied before, where the Appell hypergeometric function F1 reduces to the Gauss hypergeometric function. This difference comes from the shift of conformal dimensions and is the reason why we adopt the new method here for the block decomposition of celestial massless scalars.

Leader‐following distributed control of networked linear multi‐input systems: A prescribed‐time framework

AbstractThis paper concentrates on the leader‐following distributed control problem for linear multi‐input systems while guaranteeing prescribed finite‐time performance. Compared with the current state of the arts, novel nonsingular coordinate transformations are proposed by using the block decomposition technique and smooth time‐varying scaling functions, such that a prescribed‐time control framework dealing with distinct subsystem dimensions featured by multi‐input systems is acquired. The prescribed‐time boundedness of the closed‐loop system and the control input is proven by introducing nonsingular inverse coordinate transformations. Moreover, the settling time can relax the initial conditions and any other control parameters. Numerical examples are provided to confirm the efficacy of the proposed method.

Hopf PBW-deformations of a new type quantum group Uq(𝔰𝔩2∗) and deformed preprojective algebras

We classify all the Hopf PBW-deformations of a new type quantum group [Formula: see text] from which the classical Drinfeld–Jimbo quantum group [Formula: see text] can arise as an almost unique nontrivial one. Different from the [Formula: see text] case, the category of finite-dimensional [Formula: see text]-modules is non-semisimple. We establish a block decomposition theorem for the category [Formula: see text] of finite-dimensional weight modules of [Formula: see text]. On the level of tensor category, we show that [Formula: see text] (respectively, the category [Formula: see text] of finite-dimensional [Formula: see text]-modules) can be realized via (respectively, deformed) preprojective algebras of Dynkin type [Formula: see text].

Automatic Proper Orthogonal Block Decomposition method for network dynamical systems with multiple timescales

In this work, we introduce a novel reduced order model technique, based on the Proper Orthogonal Decomposition method, for dynamical systems with multiple timescales. The main ideas are to retain the structure of the original model, which is lost in the original POD procedure, while producing a competitive reduction in the number of equations and computational time, and to determine the best structure for the reduced system automatically, via a data-driven analysis of the original model data. For these novel techniques, we present some numerical tests for various behaviors of three different neural network models with multiple timescales, which support the use of these new methods.

Underwater calibration image enhancement based on image block decomposition and fusion

Underwater calibration image enhancement based on image block decomposition and fusion

VASCO: Volume and Surface Co-Decomposition for Hybrid Manufacturing

Additive and subtractive hybrid manufacturing (ASHM) involves the alternating use of additive and subtractive manufacturing techniques, which provides unique advantages for fabricating complex geometries with otherwise inaccessible surfaces. However, a significant challenge lies in ensuring tool accessibility during both fabrication procedures, as the object shape may change dramatically, and different parts of the shape are interdependent. In this study, we propose a computational framework to optimize the planning of additive and subtractive sequences while ensuring tool accessibility. Our goal is to minimize the switching between additive and subtractive processes to achieve efficient fabrication while maintaining product quality. We approach the problem by formulating it as a Volume-And-Surface-CO-decomposition (VASCO) problem. First, we slice volumes into slabs and build a dynamic-directed graph to encode manufacturing constraints, with each node representing a slab and direction reflecting operation order. We introduce a novel geometry property called hybrid-fabricability for a pair of additive and subtractive procedures. Then, we propose a beam-guided top-down block decomposition algorithm to solve the VASCO problem. We apply our solution to a 5-axis hybrid manufacturing platform and evaluate various 3D shapes. Finally, we assess the performance of our approach through both physical and simulated manufacturing evaluations.

Collapsing Embedded Cell Complexes for Safer Hexahedral Meshing

We present a set of operators to perform modifications, in particular collapses and splits, in volumetric cell complexes which are discretely embedded in a background mesh. Topological integrity and geometric embedding validity are carefully maintained. We apply these operators strategically to volumetric block decompositions, so-called T-meshes or base complexes, in the context of hexahedral mesh generation. This allows circumventing the expensive and unreliable global volumetric remapping step in the versatile meshing pipeline based on 3D integer-grid maps. In essence, we reduce this step to simpler local cube mapping problems, for which reliable solutions are available. As a consequence, the robustness of the mesh generation process is increased, especially when targeting coarse or block-structured hexahedral meshes. We furthermore extend this pipeline to support feature alignment constraints, and systematically respect these throughout, enabling the generation of meshes that align to points, curves, and surfaces of special interest, whether on the boundary or in the interior of the domain.

Core blocks for Hecke algebras of type B and sign sequences

We consider the core blocks corresponding to the Hecke algebras of type B over a field of arbitrary characteristic. To each core block, we associate two non-negative integers which determine the indexing of the Specht modules and simple modules in the block, the weight of the block, the multicharge of the algebra (up to a shift) and the block decomposition matrix.