orthogonal subspace projection

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。

Traditional neural networks suffer from limited interpretability and weak theoretical foundations. Method: This paper proposes a novel machine learning paradigm grounded in infinite-dimensional Hilbert spaces, centering on linear operators. It integrates reproducing kernel Hilbert spaces (RKHS), spectral operator learning, wavelet representations, scattering transforms, and Koopman operator theory to formulate learning tasks as sampling, approximation, and dynamical inference in infinite-dimensional function spaces. Contribution/Results: We establish the first unified Hilbert-space-theoretic framework bridging spectral learning and symbolic reasoning. The approach significantly enhances mathematical rigor and model interpretability by grounding learning in well-defined functional-analytic principles. Moreover, it provides a rigorous mathematical foundation and new methodological pathways for deep interdisciplinary integration between signal processing and machine learning—enabling principled analysis of structured data, hierarchical feature extraction, and nonlinear dynamical system modeling.

This paper studies the low-dimensional $k$-Orthogonal Vectors ($k$-OV) problem: given $k$ families $A_1,dots,A_k subseteq 2^{[d]}$, each of size $n$, decide whether there exist $a_i in A_i$ such that $igcap_i a_i = emptyset$. We present the first randomized algorithm for $k$-OV based on combinatorial design and probabilistic analysis, establishing—for any fixed $k$—the existence of $varepsilon_k > 0$ such that $k$-OV is solvable in $O(2^{(1-varepsilon_k)d} n)$ time. In particular, for $k=2$, our algorithm runs in $O(1.16^d n)$ time, improving upon all prior deterministic and randomized algorithms. Our approach centers on a succinct rank-decomposition framework capturing the problem’s combinatorial structure, augmented by computer-assisted parameter optimization. Under the Set Cover Conjecture, our bound matches the current best asymptotic lower bound, achieving both theoretical optimality and practical computability.

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 paper addresses the universal approximation of continuous (including nonlinear) operators on Banach spaces. Methodologically, it introduces a novel learning framework based on orthogonal polynomial projections—marking the first integration of Leray–Schauder mapping theory into operator approximation theorems, synergizing Banach-space operator analysis with spectral approximation techniques in $L^p$ spaces. Specifically, in $L^p$ (notably $L^2$), it establishes a two-stage operator learning paradigm: “learnable projection” followed by “finite-dimensional mapping.” Theoretical contributions include: (1) a proof of universal approximation capability for the framework on arbitrary Banach spaces; (2) explicit sufficient conditions ensuring high-precision operator approximation in $L^2$; and (3) the first rigorous, unified mathematical foundation for operator neural networks.

This work addresses the dimensionality explosion in Tensor Product Representations (TPRs) during deep recursive symbolic reasoning and the compromised capacity and structural fidelity of existing vector-symbolic architectures that rely on superposition within fixed-dimensional spaces. To overcome these limitations, the paper introduces Orthogonal Subspace Carving (OSC), a mechanism that projects filler vectors into the null space of role bases and aggregates them into a fixed-order tensor, thereby decoupling recursive binding from both tensor order and structural depth. OSC enables deep recursion under constant memory overhead, achieving substantially higher memory efficiency in high-superposition regimes—where component vector dimensions are orders of magnitude smaller than the memory tensor—while preserving structural fidelity and scalability. Furthermore, the study reveals TPR as a special case of binding in Clifford algebra and provides a Clifford-algebraic formulation of OSC.

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.

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。

本文研究了关系数据库中线性代数运算的数值稳定性问题，提出投影向后稳定性和数据库条件数概念以解决传统方法失效的问题。

This work addresses the computational and memory bottlenecks of traditional Grassmannian kernel methods, which require constructing full Gram matrices and thus struggle with high-dimensional subspace data. To overcome these limitations, the authors propose a scalable kernel approximation framework based on random rank-one projections combined with bounded nonlinear transformations—either periodic or binary—that yield compact one-bit subspace feature representations. This approach enables continuous interpolation between the inverse Binet–Cauchy kernel and Gaussian-like kernels while effectively preserving the intrinsic geometry of subspaces. The method substantially reduces computational, memory, and storage costs. Experimental results on synthetic data and the ETH-80 classification benchmark demonstrate that the proposed technique accurately maintains Grassmannian geometric relationships with high fidelity, confirming its efficiency and practical utility.