parameterize spd matrices

This work addresses the geometric constraints and computational complexity inherent in generative modeling of symmetric positive-definite (SPD) matrices by introducing “reverse telescoping mapping”—a novel unconstrained coordinate system that decouples SPD matrices into interpretable volume (log-determinant) and shape (relative scale and off-diagonal covariance) components. The proposed representation yields a lossless symbolic encoding of both a matrix and its inverse, reducing the complexity of key operations to O(p²) in the transformed domain and enabling linear interpolation along unit-determinant geodesics. Leveraging this framework, we develop volume–shape disentangled conditional flow matching and intrinsic diffusion models, successfully synthesizing bimodal SPD distributions, generating fMRI brain connectomes, and performing manifold-intrinsic diffusion tasks at scale (p = 200).

This work addresses the challenge of designing metric structures on the manifold of symmetric positive-definite (SPD) matrices that simultaneously preserve geometric naturalness and computational efficiency. Building upon James’ SPD bicone parametrization, the paper introduces—for the first time—a Finsler structure together with a dual Hessian information-geometric framework. The resulting geometric construction yields geodesics that appear as straight lines in an appropriate coordinate system and naturally extends the Hilbert simplex distance to the spectral simplex. Theoretical analysis establishes inequality relations between the proposed metric and classical SPD metrics, thereby broadening the applicability of information geometry and Finsler geometry on SPD manifolds. This framework offers a theoretically rigorous yet practically viable tool for geometric deep learning and modeling of SPD-valued data.

本文研究了一类低参数正交矩阵的黎曼结构，并提出有效算法解决其在深度学习中的计算问题。

This work proposes a unified framework for understanding generalized and randomized inverses of matrix products, revealing their intrinsic structure in subspace geometry and randomized linear algebra. Building upon the geometric relationships among the four fundamental subspaces, we develop a cohesive formulation encompassing the Moore–Penrose pseudoinverse, {1,2}-inverses, and their randomized counterparts, yielding both a universally valid generalized inverse expression and novel randomized inverse representations. The framework elucidates the common structural foundation underlying algorithms such as randomized SVD, Nyström approximation, and CUR decomposition. Moreover, it establishes, for the first time, rigorous error bounds for effective resistance estimation, proving that the error is always an underestimate and providing worst-case spectral bounds. These theoretical insights are applicable to practical problems including sparse sensor placement.

Log-determinant estimation is crucial in Gaussian processes and Bayesian model comparison, yet conventional methods fail under high condition numbers. This work proposes a closed-form estimator based on matrix trace powers: by leveraging derivatives of the moment-generating function of normalized eigenvalues, combined with the log-transform $K(t) = \log M(t)$, local integer-point interpolation, and spectral lower-bound constrained optimization, it enables efficient computation. Requiring only $m = 4$–$8$ trace power evaluations, the method achieves $O(m)$—effectively constant-time—complexity. It also establishes, for the first time, a fundamental limitation: finite positive moments cannot uniformly approximate arbitrary spectral distributions. The resulting verifiable upper and lower bounds, together with tail-sensitivity analysis, offer rigorous error control and failure diagnostics, ensuring both accuracy and reliability even in high-condition-number regimes.

The integration of Symmetric Positive Definite (SPD) matrices into deep learning has historically relied on fixed algebraic Riemannian metrics. Analogous to hand-crafted features in classical machine learning, these static formulations impose rigid geometries limiting network expressivity and adaptability. Recent attempts to parameterize these geometries often violate the axioms of primary matrix functions through unconstrained powers or rank-dependent scaling, inviting spatial folding, loss of global surjectivity, and gradient collapse at spectral singularities. In this paper, we introduce the Spline-Pullback Metric (SPM), instantiated as Spectral-SPM and Cholesky-SPM, marking a paradigm shift from static metric selection to universal geometric approximation. By parameterizing the global diffeomorphism via a rank-invariant, monotonically constrained B-spline, SPM acts as a dense universal approximator for strictly increasing $C^1$ diffeomorphisms and theoretically subsumes existing pullback metrics while enabling localized non-linear spectral modelling. Topologically, SPM provides a globally bijective pullback geometry precluding rank-swapping discontinuities and gradient instabilities. Empirically, SPM achieves a state-of-the-art performance across 3 datasets utilizing Linear Probes, SPDNets, and deep Riemannian ResNets.

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.

本文解决了全局协方差池化中矩阵对数归一化的数值不稳定问题，通过使用正交多项式逼近方法来替代基于特征分解的计算方式。

This work addresses the problem of efficiently performing length-squared sampling on positive semidefinite matrices without prior knowledge of column norms. The authors propose a novel rejection-sampling-based algorithm that exploits the structural properties of positive semidefinite matrices to eliminate the need for any precomputed column norm information. This approach achieves, for the first time, an expected $O(n)$ time complexity for optimal length-squared sampling, substantially outperforming conventional methods that rely on explicit column norm estimates. Empirical evaluations demonstrate that the algorithm matches the performance of existing, more complex techniques in tasks such as Frobenius norm estimation and robust low-rank approximation, offering both theoretical optimality and practical utility.

研究通过多种分析方法揭示了基本数据点(EDPs)在减少数据集大小时对几何结构的保留是部分的，并提供了一种基于矩阵奇异值分解的修正方法。