This work introduces Riemannian Orthogonally Decoupled Regularization (RODR) to reformulate the optimization trajectory by disentangling the normal (fitting) and tangential (distribution) components and establishes a generic and interpretable framework for disentangled geometric optimization in point cloud processing.
Abstract
Point cloud denoising is essentially a geometric recovery task that aims to reconstruct the intrinsic structure of a smooth 2D Riemannian manifold embedded in R^3 from noisy, discrete ambient-space samples. Despite the remarkable progress of modern manifold-aware encoders and generative transport models in geometric representation learning, a fundamental objective-geometry mismatch remains underexplored. Theoretically, we identified that this mismatched coupling leads to geometric gradient interference, where conflicting optimization objectives result in structural degradation and point clustering. We introduce Riemannian Orthogonally Decoupled Regularization (RODR) to reformulate the optimization trajectory by disentangling the normal (fitting) and tangential (distribution) components. Guided by a vector-attention and entropy-aware adaptive strategy, RODR effectively preserves high-fidelity geometric details while maintaining sampling uniformity. Experiments demonstrate that RODR reaches performance comparable to state-of-the-art baselines and suggests improved distribution regularity and reduced local aggregation effectively. Our work establishes a generic and interpretable framework for disentangled geometric optimization in point cloud processing.
In Geometric Deep Learning, inductive biases serve two primary functions: enforcing manifold constraints and embedding relational priors. Currently, representation learning on SPD manifolds frequently relies on pullback Euclidean metrics, such as the Log-Euclidean Metric, to satisfy the former. While computationally efficient in avoiding domain boundary violations, these metrics induce a flat geometry that may fail to capture the intrinsic relational priors of datasets. While metrics such as the Poincar\'e metric are widely utilized to induce domain-aligned relational priors, generalizing them from standard vector representations to the SPD manifold has remained a challenge. To bridge this gap, we introduce a Nested Inductive Bias framework that utilizes a two-stage diffeomorphic composition to formally pull back non-Euclidean target geometries onto the SPD manifold. This framework enables the construction of curvature-aligned Riemannian classifiers that simultaneously respect matrix constraints and the latent relational geometry of the data. Empirical evaluations on kinematic and signal processing benchmarks, together with synthetic experiments, demonstrate that deep manifold networks experience degradation in class separability unless the metric curvature aligns with the intrinsic data distribution. Furthermore, for standard vectorized architectures, we propose the Rational Conformal Metric (RCM), designed to establish state-of-the-art geometric robustness against outliers by bounding the representation space.
Deep representation learning has primarily focused on how features evolve across network layers, while largely overlooking the structured geometry embedded in network parameters. We introduce a dual-manifold perspective in which each convolutional layer contains two coupled geometric spaces: a Kernel Manifold induced by convolutional filters and a Data Manifold characterized by intermediate feature representations. Because these manifolds share the same channel space, parameter geometry can provide complementary structural information to guide feature evolution. Based on this insight, we propose Kernel-Guided Feature Transform (KGFT), a lightweight module that derives a geometric guidance matrix from the kernel Gram matrix and uses it to transform the covariance structure of feature representations. Unlike conventional attention mechanisms that reweight feature responses, KGFT explicitly reshapes feature relationships by transferring geometric information from the kernel manifold to the data manifold. To accommodate network hierarchy, we further introduce Exploit and Explore modes with a depth-aware scheduling strategy and a learnable guidance strength that adaptively controls the contribution of geometric transformation. This design promotes geometric alignment in shallow layers while encouraging feature diversity in deeper layers, without imposing excessive constraints on representation learning. Theoretical analysis establishes the validity of the proposed transformation and characterizes its effect on feature covariance. Extensive experiments across CNN- and Transformer-based architectures, including ResNet, ViT, and LLaMA-7B, demonstrate consistent improvements on image classification and arithmetic reasoning tasks, validating the generality and effectiveness of kernel-guided dual-manifold representation learning. Code will be publicly available.
Wencong Zhang, Yue Zhang, Meiyan Huang et al.· 0 citations
Divergence-based regularization and Sharpness-Aware Minimization (SAM) are two prominent approaches for improving generalization in deep learning, both motivated by robustness to perturbations. However, their relationship has remained largely unexplored. Building on classical second-order expansions of $f$-divergences, we show that the two methods are locally consistent under parameter-space perturbations: both induce curvature-sensitive penalties, with divergence regularization yielding a Fisher-weighted quadratic form and SAM penalizing sharpness through the dominant Hessian eigenvalue. For negative log-likelihood objectives with exponential-family output distributions, this correspondence becomes especially transparent, since the Fisher and Gauss-Newton matrices coincide. We further show that the same local geometric perspective extends to input-space perturbations, where divergence-based regularization is defined through transformations of the input. In this setting, the regularizer induces a pullback quadratic form on the input space, providing a more general perturbation framework than standard SAM while preserving the same local sensitivity interpretation. To validate the analysis empirically, we use the asymmetric $\alpha$-skew Jensen-Shannon divergence (JSD) family as a controlled testbed. Its local curvature coefficient scales as $\alpha(1-\alpha)$ and is maximized at the symmetric point $\alpha=\tfrac12$, which recovers the standard JSD. Loss-landscape visualizations in the input-perturbation regime show that stronger induced curvature penalization is associated with flatter local minima. Experiments on four benchmark datasets further demonstrate that both accuracy and negative log-likelihood are consistently best near this regime of maximal curvature penalization.
Several geometry-aware approaches to low-rank adaptation have emerged for parameter-efficient fine-tuning of large pre-trained models. These methods aim to take full advantage of the geometric structure of low-rank manifolds for improving the efficiency in subspace utilization and reducing redundancy by enforcing orthogonality constraints during optimization. The strong empirical results of these techniques have motivated further study into whether predictions from such geometry-based adaptation methods could be overconfident. In this paper, we build on the singular value decomposition factorization of adapters to develop a framework based on Stein variational gradient descent (SVGD). In this formulation, the low-rank matrices are transported along the Stiefel manifold to match the targeted distributions while retaining their crucial geometric structure. Since this geometry-aware SVGD approach provides multiple solutions during inference, it supports uncertainty quantification and produces better-calibrated adapters on the Stiefel manifold. Extensive experiments show that our method delivers strong model calibration and attains higher prediction accuracy than SVGD and related uncertainty estimation methods that are formulated in Euclidean space.
Quang-Duy Tran, Trung Le, Bao Duong et al.· 0 citations
The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network localization, molecular conformation, and manifold learning. In this paper, we propose a Riemannian optimization framework for solving the EDMC problem by formulating it as a low-rank matrix completion task over the space of positive semi-definite Gram matrices. The available distance measurements are encoded as expansion coefficients in a non-orthogonal basis, and optimization over the Gram matrix implicitly enforces geometric consistency through nonnegativity and the triangle inequality, a structure inherited from classical multidimensional scaling. Under a Bernoulli sampling model for observed distances, we prove that Riemannian gradient descent on the manifold of rank-<inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> matrices locally converges linearly with high probability when the sampling probability satisfies <inline-formula> <tex-math notation="LaTeX">$p\geq {\mathcal {O}} (\nu ^{2} r^{2}\log (n)/n)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\nu $ </tex-math></inline-formula> is an EDMC-specific incoherence parameter. Furthermore, we provide an initialization candidate using a one-step hard thresholding procedure that yields convergence, provided the sampling probability satisfies <inline-formula> <tex-math notation="LaTeX">$p \geq {\mathcal {O}} (\nu r^{3/2}\log ^{3/4}(n)/n^{1/4})$ </tex-math></inline-formula>. A key technical contribution of this work is the analysis of a symmetric linear operator arising from a dual basis expansion in the non-orthogonal basis, which requires analysis of a second order degenerate U-statistic to establish an optimal restricted isometry property in the presence of coupled terms. Empirical evaluations on synthetic data demonstrate that our algorithm achieves competitive performance relative to state-of-the-art methods. Moreover, we provide a geometric interpretation of matrix incoherence tailored to the EDMC setting and provide robustness guarantees for our method. Due to space constraints, the complete proofs of the convergence bounds and technical lemmas are provided in the extended pre-print 2508.00091, and the appendices containing said results are also available online at IEEE Xplore.
We introduce the Intrinsic Hybrid Latent Diffusion Model (ILDM), a generative framework that integrates probabilistic dimensionality reduction with geometry-aware diffusion on unknown manifolds. While diffusion models (DMs) have achieved state-of-the-art results in high-dimensional data synthesis, they rely on large training datasets and ignore intrinsic geometric structure. Latent diffusion models (LDMs) address the high dimensionality by learning a latent space, but they typically impose a Euclidean structure, failing to capture the underlying manifold geometry, especially problematic in data-sparse regimes. ILDM addresses these limitations by interpreting the latent space as a chart of an unknown Riemannian manifold, with geometry and uncertainty quantified through a probabilistic decoder. The forward process is a hybrid diffusion that switches between Riemannian and Euclidean dynamics based on local uncertainty, where the Riemannian component is governed by a probabilistic metric tensor derived from the decoder. To learn the generative dynamics, we introduce an approximate denoising score matching method tailored to the hybrid diffusion setting, enabling a backward process defined by hybrid Langevin dynamics. Experiments on COIL-100, MNIST, and cardiac MRI datasets demonstrate that ILDM significantly improves generation quality, achieving lower FID and LPIPS scores compared to standard diffusion and latent diffusion models.
Yizhu Wang, Mu Niu, Xiaochen Yang· 0 citations
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