patchwise pod encoding

To address the limitations of traditional Haar wavelets on graphs—including poor locality, absence of vanishing moments, and limited compression capability—this paper introduces the samplet transform to graph signal processing for the first time. We propose a novel orthogonal multi-resolution framework based on graph partitioning, Euclidean embedding, and pullback mapping. Specifically, the graph is partitioned via reweighted edge clustering; each subgraph is embedded into a low-dimensional Euclidean manifold using landmark Isomap; samplets with polynomial vanishing moments and strong spatial localization are constructed in the embedded space and then pulled back onto the original graph to yield a sparse, multi-scale representation. The framework rigorously guarantees orthogonality and controlled approximation error, significantly broadening the class of graph signals amenable to efficient compression. Experiments demonstrate superior performance over Haar wavelets in compression efficiency, multi-resolution fidelity, robustness to noise, and scalability.

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.

Traditional proofs of Singular Value Decomposition (SVD) rely on the spectral theorem, lacking geometric intuition and failing to reveal deep connections with machine learning algorithms. This work reconstructs SVD from an ellipsoidal geometry perspective, transforming the recursive process of maximizing stretch into an algorithmic mechanism. By integrating geometric constructions, gradient descent stopping criteria, and duality analysis of kernel methods, it achieves a "proof-as-algorithm" paradigm that derives SVD without presupposing the spectral theorem. Furthermore, this study establishes explicit mappings between SVD and core algorithms such as Principal Component Analysis (PCA) and PageRank, thereby unifying the foundational logic of linear algebra and machine learning. Finally, it provides a pedagogical framework amenable to manual verification alongside a discussion of theoretical boundaries.

For computer experiments involving variables with a grouped additive structure—i.e., no interactions between groups—this paper proposes Group Orthogonal Arrays (GOAs) as a novel design paradigm surpassing conventional space-filling approaches. We establish, for the first time, a systematic theoretical framework for orthogonal arrays tailored to grouped additive models, supporting arbitrary prime-power factor levels and flexible run sizes. Integrating finite-field algebra, combinatorial design theory, and orthogonal array construction techniques, we develop multiple explicit construction algorithms. These yield large-scale, practical GOA tables, substantially expanding the scope of feasible experimental designs. Empirical evaluation demonstrates that GOAs achieve superior within-group projection uniformity compared to state-of-the-art methods and reduce average prediction error in response surface modeling by 12%–28%.

This work proposes a learnable framework for adaptive orthogonal bases that overcomes the rigidity of traditional fixed bases—such as Fourier or wavelet bases—in capturing data-specific structures. The target basis is treated as a point on the Lie manifold of the orthogonal group and is obtained by continuously evolving a reference basis along a path defined by an ordinary differential equation induced by a finite-rank skew-adjoint integral operator, parameterized via neural networks. Theoretically, it is shown that rank-2 generators suffice to densely approximate any orthogonal basis in the operator topology, ensuring both universality and flexibility. Experiments demonstrate successful adaptation of the Fourier basis into data-driven principal components, eigenfunctions of operators, and dynamic modes of physical systems, confirming the method’s effectiveness and broad applicability.

This study investigates the optimality of random embeddings in sketch-and-solve least squares and randomized SVD for low-rank approximation. By integrating tools from random matrix theory, minimax analysis, and rotational invariance principles, the authors establish that random orthogonal matrices achieve minimax optimality within the sketch-and-solve framework, while any rotationally invariant embedding is optimal for randomized SVD. Building on these theoretical insights, they derive the tightest error bounds to date and corroborate their findings through numerical experiments, demonstrating that a variety of commonly used embeddings closely approach the theoretically optimal performance in practice—thereby revealing a striking universality phenomenon across different embedding schemes.

研究解决了3D点云颜色属性压缩问题，通过转换谐波系数并使用正交化方法，在编码前处理系数以提高图像质量。

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 work addresses the challenges of discovering differential equations, approximating functions, and estimating high-dimensional integrals from noisy, non-uniformly sampled data by introducing Sparse Orthogonal Regression Technique (SORT). The method reformulates equation discovery as a spectral coefficient learning problem, directly inferring coefficients of an orthogonal basis expansion from observational data via L1 regularization—without requiring a predefined symbolic library, explicit numerical integration, or inner product evaluations. Its core innovation lies in treating basis function design as the central modeling choice, enabling consistent model order scaling and multi-task reusability. Experiments demonstrate that when the basis functions align with the underlying problem structure, SORT matches or outperforms existing approaches under sparse sampling, noisy derivatives, and representation mismatch, while low-order dominant coefficients remain stable as model complexity increases.

This work addresses the challenge of effectively fusing orthogonally fine-tuned adapters—each specialized for distinct concept and style tasks—without requiring additional training, to enable high-quality multi-attribute image generation. Leveraging the Riemannian manifold structure of Group-and-Shuffle orthogonal matrices, the authors propose a training-free multiplicative adapter fusion method. This approach employs an efficiently approximated geodesic interpolation formula on the manifold and incorporates a spectral restoration transformation to preserve the spectral properties of the fused adapter. The method achieves, for the first time, joint generation using orthogonally fine-tuned adapters without further fine-tuning, successfully synthesizing high-fidelity images that simultaneously embody specified styles and concepts in subject-driven generation tasks, thereby demonstrating its effectiveness and superiority.