matrix exponential decomposition

This paper addresses the weak theoretical foundations of matrix decomposition in machine learning by systematically constructing a self-consistent, comprehensive, and modern-application-oriented pedagogical framework. Methodologically, it grounds the exposition in numerical linear algebra and matrix analysis, unifying classical decompositions—including LU, QR, SVD, and block triangular factorizations—while integrating numerical stability analysis and Hermitian/Hilbert space theory. Crucially, it bridges traditional numerical analysis with deep learning’s backpropagation setting, emphasizing differentiability and computational robustness of decompositions in algorithm design and model optimization. The primary contribution is a compact, dual-purpose (teaching and research) knowledge system that fills critical gaps in both theoretical coherence and machine-learning relevance present in existing literature, thereby providing rigorous mathematical foundations for high-dimensional data modeling and efficient training.

This work extends the classical Kolmogorov–Arnold representation theorem into the quantum domain by investigating the structure of multivariate continuous unitary-valued mappings in a neighborhood of the identity matrix. By integrating local parametrizations of Lie groups, the matrix exponential map, and analysis of skew-Hermitian operators, the paper introduces two novel local quantum Kolmogorov–Arnold representations: one based on an additive decomposition within the exponent of skew-Hermitian maps, and another constructed as a sequential product of finitely many single-variable matrix exponentials leveraging non-commutativity. The study rigorously establishes the exact forms of both representations and demonstrates, via a topological counterexample on SU(2), that they cannot be globally extended—thereby highlighting the inherent optimality and limitations of such local constructions.

本文介绍了一种通过计算交换算子的联合特征向量来解决多项式根和张量分解问题的方法，并分析了其多重结构。

This work addresses the lack of theoretical characterization in existing kernel methods for machine learning regarding the residual structure and energy stability of multichannel signals in complex systems. The authors propose an analytical framework grounded in operator defect identities, introducing the novel concept of “telescopic energy residuals.” By integrating iterative products with a λₙ-relaxed Kaczmarz scheme, they establish admissibility conditions for residuals and derive prior energy bounds. For the first time, this framework incorporates operator defect theory into kernel methods and kernel principal component analysis (KPCA), rigorously proving explicit convergence of generalized algorithms, a residual energy decomposition theorem, and stability criteria under noise. The approach significantly extends infinite-dimensional Kaczmarz theory to broader applications in machine learning.

This work investigates the impact of matrix concatenation operations on singular value spectra, aiming to bridge a theoretical gap in the structural stability of SVDs under concatenation. Addressing the central question—“How are the singular values of a concatenated matrix determined by those of its constituent submatrices?”—we extend Weyl’s inequality to block-wise concatenation for the first time, establishing a quantitative analytical framework grounded in matrix perturbation theory and norm inequalities. We derive computable upper bounds on singular value deviations and rigorously prove that dominant singular values remain stable when the operator norms of the submatrices are comparable. This result provides theoretical guarantees and principled design guidance for low-rank approximation, robust matrix clustering, and compression algorithms.

本文通过构建具有最小支撑的张量基，解决了成对比较理论中加性一致子空间的正交基问题，并提出了新的对数、Saaty和SVD投影公式。

本文通过改进Bailey的四步FFT算法，提出了二进制扩展域上仿射子空间多项式评估的加法FFT技术，利用特定基和分解方法减少了计算中的乘法次数。

Traditional methods for computing elementary transcendental functions rely on differential equations or power series, often suffering from implementation complexity and high memory overhead. This work proposes the first unified fixed-point framework grounded in the Banach contraction mapping principle, modeling the exponential, trigonometric, and natural logarithm functions as the unique fixed points of doubling identities. By constructing residual-based strictly contractive iteration operators, the approach guarantees numerical stability and explicit convergence rates. The resulting floating-point kernel operates without lookup tables or memory accesses, achieving performance in throughput-constrained scenarios that matches or exceeds that of mainstream math libraries; notably, sine and cosine computations are consistently faster across all tested iteration depths.

This study addresses the longstanding challenge of computing generalized inverses for non-square Jacobian matrices in automatic differentiation and solving for preimages in affine spaces. To this end, it proposes the Null-A mode, which employs a compositional algorithm to efficiently compute affine preimages of matrix products. This mode supports dynamic variables and non-square Jacobians, extending reverse-mode automatic differentiation to full affine space solutions while leveraging quasi-local algorithms alongside CPU/GPU parallel acceleration. The primary contribution lies in achieving efficient generalized inverse computation for both scalar functions and aggregated array operations, such as convolutions and attention mechanisms. Ultimately, this work overcomes critical bottlenecks in reverse solving for linearized numerical computations, offering a robust framework that significantly broadens the applicability and computational efficiency of automatic differentiation in complex machine learning architectures.

This study addresses the challenge of optimizing the upper bound of the matrix multiplication exponent ω by reconstructing the combinatorial loss analysis framework to expand the solution space. Furthermore, it innovatively integrates machine learning with the AlphaEvolve evolutionary search algorithm. Through this hybrid optimization strategy, the upper bound of ω is successfully reduced to 2.371177. This achievement not only overcomes existing theoretical bottlenecks but also surpasses previous state-of-the-art records. Consequently, this work establishes a novel algorithmic paradigm and provides critical theoretical support for research on matrix multiplication complexity, significantly advancing progress in the field.