structured topology-aware regularization

This work addresses the limited expressivity of conventional parameter optimization paradigms by proposing direct optimization of metric tensor fields on fixed-topology manifolds, enabling data-driven dynamic evolution of model geometry. Methodologically, leveraging discrete differential geometry, the manifold is represented as a triangular mesh, with the metric parameterized via edge lengths and optimized efficiently using automatic differentiation. Theoretically, the framework establishes a profound analogy between metric optimization and the Einstein–Hilbert action in general relativity. Crucially, geometric complexity adapts automatically while preserving topology, thereby enhancing model expressivity and generalization and effectively mitigating overfitting. The resulting framework provides a novel paradigm for scientific modeling, robust representation learning, and geometric deep learning—unifying geometric reasoning with differentiable optimization in a principled, topology-preserving manner.

Non-convex, multimodal optimization on the Grassmann manifold (mathrm{Gr}(k,n)) poses significant challenges for conventional Riemannian first- and second-order methods, which often converge prematurely to poor local minima. Method: We propose the first differential evolution (DE)-based global optimization framework tailored to the Grassmann manifold. It employs QR decomposition for geometrically exact manifold projection, integrates adaptive control parameters, and introduces a manifold-aware mutation strategy—thereby balancing global exploration with intrinsic manifold structure preservation—while operating entirely without gradient information. Contribution/Results: Our approach overcomes the local convergence limitations inherent in standard Riemannian optimizers. Extensive experiments on subspace learning and low-rank matrix recovery demonstrate that it consistently outperforms state-of-the-art Riemannian optimization methods, achieving superior global convergence, robustness to initialization, and generalization across diverse problem instances.

Conventional feature discrepancy measures for latent-space distribution matching are computationally expensive and neglect geometric-topological structure; meanwhile, persistent homology-based methods suffer from poor scalability and training instability. Method: This paper proposes a scalable topological regularization framework centered on a novel lightweight topological regularizer based on subsampled persistent homology. Contribution/Results: We theoretically prove that the regularizer’s gradient is continuous with respect to input density—ensuring stable backpropagation—and enable efficient GPU-accelerated computation, overcoming topological regularization bottlenecks for point clouds exceeding one thousand points. Extensive experiments on shape matching, image generation, and semi-supervised learning demonstrate significant improvements in training stability and scalability to large-scale settings.

This work reformulates generative modeling from an observer’s perspective, recasting manifold learning as a decoupling problem between geometric support and probability distribution. Methodologically, it introduces a novel framework grounded in continuous percolation theory, establishing a rigorous topological isomorphism between the percolation phase transition in random geometric graphs and the underlying data manifold. A differentiable “percolation shift” metric is designed to detect structural deficiencies—such as disconnected components or topological voids—that evade detection by conventional statistical metrics (e.g., FID); this metric is jointly optimized with FID in a dual-objective loss. The approach effectively mitigates manifold collapse while substantially expanding topological diversity without sacrificing fidelity, achieving, for the first time in generative modeling, a theoretically guaranteed “super-generalization” regime. The core contribution lies in integrating percolation phase transition theory into generative modeling, thereby establishing a new geometric-topological paradigm for manifold structure assessment and optimization.

This work addresses the lack of a universal statistical interpretation for the manifold hypothesis—that high-dimensional data approximately reside on low-dimensional manifolds. We propose the Latent Metric Model (LMM), a generative framework grounded in fundamental statistical concepts: latent variables, variable dependence, and stationarity—providing the first unified statistical justification for the manifold assumption. Methodologically, LMM integrates neighborhood graph construction, spectral analysis, and an interpretable inference framework to enable unsupervised manifold discovery and geometric structure recovery under weak priors. Experiments demonstrate that complex manifold geometries naturally emerge from minimal statistical mechanisms; LMM significantly reduces reliance on hand-crafted priors on both synthetic and real-world datasets, while enabling interpretable reconstruction of manifold dimensionality, curvature, and coordinate systems.

研究通过逐步增长策略在深度神经网络训练中引导优化趋向平坦区域，利用冻结约束下的有效曲率解释了此偏置效应。

This study addresses the challenging problem of fitting an unknown number of hyperplanes to data, which involves non-convexity, non-differentiability, and uncertainty in model order. To tackle these difficulties, the authors propose a two-stage unsupervised learning approach grounded in a unit-sphere manifold framework. In the first stage, they integrate Riemannian expectation-maximization with heavy-tailed kernel density estimation to robustly infer posterior probabilities. The second stage employs hard assignment annealing to obtain geometrically consistent local optima. Key innovations include a manifold optimization framework for handling non-convex constraints, a projection-based density estimation scheme for initialization, and the two-stage optimization strategy itself. Experimental results demonstrate that the proposed method significantly outperforms state-of-the-art baselines in both geometric accuracy and robustness.

Standard variational autoencoders employ Gaussian priors, which struggle to align with data manifolds exhibiting non-Euclidean topologies—such as periodicity or boundedness—leading to distorted representations. This work proposes a topology-aware latent space modeling framework that constructs factorized prior distributions tailored to manifolds decomposable into products of circles, intervals, and lines, along with their finite group quotients. This design enables disentangled latent representations and analytically tractable KL divergences. By integrating differentiable coordinate transformations, group-invariant decoding, and anchor-point constraints, the approach ensures smooth gradients and topological consistency. To our knowledge, this is the first method to systematically align latent variable distributions with the intrinsic topology of data manifolds, supporting reparameterizable encoder–prior pairs and significantly outperforming Gaussian-prior baselines on synthetic manifolds as well as rotation- and cyclic-translation variants of MNIST.

This work proposes a novel approach that actively embeds the Minimum Description Length (MDL) principle into the optimization process of deep learning to jointly enhance model simplicity and generalization. By constructing a cognitive manifold driven by coupled Ricci flows, the method introduces a geometry-driven MDL Drive mechanism that dynamically compresses internal representations during training, balancing fidelity and complexity. Theoretical analysis establishes that this mechanism guarantees a monotonically decreasing description length, undergoes a finite number of topological phase transitions, and exhibits universal critical behavior. With a per-iteration computational complexity of O(N log N), the algorithm automatically simplifies model architecture, improves generalization, and demonstrates numerical stability alongside exponential convergence in empirical evaluations.

This work addresses the challenge of manifold drift in latent-space optimization of high-dimensional 3D generative models, which often leads to geometrically invalid outputs. To resolve this issue, the authors propose an optimizer-corrector alternating framework that decouples task-driven optimization from manifold constraints for the first time. In the free gradient optimization phase, the method aggressively pursues the target objective, while in the guided flow-matching phase, latent variables are corrected back onto the valid shape manifold. This approach eliminates the traditional trade-off between representational capacity and geometric validity, substantially enhancing optimization stability and generation quality. The framework demonstrates consistent superiority across diverse 3D generative priors and downstream applications, including aerodynamic drag reduction and structural compliance optimization.