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新论文 · v1arXiv:2611.13337

VC-Dimension and Generalization in Geometric Machine Learning with Applications to Active Learning

Gokul Bhusal, Delanna Do, Ekaterina Merkurjev, Kevin Miller, Stephan Wojtowytsch

原文摘要

We compute the VC-dimension of metric ball classifiers in the model spaces of Riemannian geometry (Euclidean, spherical, hyperbolic) and present upper and lower bounds on the VC-dimension of metric balls in several other situations, including general normed vector spaces (potentially infinite VC-dimension), vector spaces with $\ell^p$-norms (VC-dimension grows at least linearly in the dimension, faster than Euclidean for $p=1$), flat tori, and general Riemannian manifolds (arbitrarily large VC-dimension after a conformal change). In several situations where the VC-dimension is large or even infinite, we prove generalization bounds under additional assumptions on the data distribution using covering arguments. Both in the VC-dimension and covering setting, we consider data on low-dimensional subsets of high-dimensional Euclidean spaces. As an application, we consider coverage-based algorithms in active learning.

stat.ML · cs.LG · math.DG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2609.20159

Special Lagrangian cones in Deep Learning

Tejas Kotwal, Govind Menon

原文摘要

We introduce a matrix generalization of the cone of Harvey and Lawson and prove that it is an exact special Lagrangian manifold. We further show that it belongs to a family of exact special Lagrangian manifolds that foliate the balanced manifold arising in deep learning.

math.DG · cs.LG · math.SG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.13040

On the Concavity of Rényi Entropy and Rényi Entropy Power along Heat Flows on Manifolds

Hao Wu, Lei Yu

原文摘要

In this paper, we establish the concavity of Rényi entropy and Rényi entropy power along heat flow on Euclidean space of arbitrary dimension, for Rényi orders $α\in[1,2]$. More generally, we prove the corresponding result along the heat flow generated by the Witten Laplacian on complete Riemannian manifolds under a suitable curvature-dimension condition. Our argument builds on novel functional inequalities that were recently developed for the study of Tsallis entropy concavity along the heat flow. We extend both these inequalities and the resulting concavity theorem from the Euclidean setting to the Riemannian manifold setting. In addition, we establish the complete monotonicity of Tsallis entropy of order $2$ along the heat flow and derive its spectral decomposition.

math.DG · cs.IT · math.PR

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2603.28764

Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds

N Alex Cayco-Gajic, Arthur Pellegrino

原文摘要

Similarity measures such as centred kernel alignment (CKA) or representational similarity analysis (RSA) are widely used to compare the representational geometries employed by different neural networks to solve the same task. Yet, because existing methods compare the extrinsic geometry of representations in state space, rather than their intrinsic geometry, they may fail to capture subtle yet crucial distinctions between fundamentally different computations. Here, we introduce metric similarity analysis (MSA), a novel method which leverages tools from Riemannian geometry to compare the intrinsic geometry of neural representations under the manifold hypothesis. We show that MSA can be used to (i) disentangle neural computations of deep networks in rich vs. lazy learning regimes, (ii) compare the dynamics of recurrent neural networks and state-space models, and (iii) investigate the statistical manifolds of diffusion models. Overall, we introduce a mathematically grounded and broadly applicable framework to understand the mechanisms behind neural computations by contrasting their intrinsic geometries.

cs.LG · cs.AI · math.DG · q-bio.NC

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.10466

The Fast Tree Transform

Paul Burchard

原文摘要

The Fast Tree Transform is a locally-aggregated transform defined for arbitrary graph geometries that generalizes both the Hadamard frequency transform and the Haar wavelet transform on the circle. The pure frequency part of the transform approximately diagonalizes the graph Laplacian; this fact makes it useful as a preconditioner in geometric optimization problems. The full transform provides an only slightly-redundant representation of signals on the graph that allows sparse representation of signals that include both edges and textures.

cs.LG · math.DG · math.SP

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.08421

数据驱动的微分方程系统内在李点对称性发现

Data-driven discovery of intrinsic Lie point symmetries of differential equation systems

Ilker Gurcan, Kadir Ustun, Ramazan Gokberk Cinbis

AI 速读 基于摘要

提出一种算子理论框架，从采样射流空间数据中直接发现微分方程系统的内在李点对称生成元，无需符号确定方程或闭式系统表示。

cs.LG · math.DG · math.DS · math.GR · math.NA

arXiv ↗PDF ↗
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Discovering continuous symmetries directly from differential-equation data can reveal geometric structure for model reduction and scientific machine learning without symbolic determining equations or a closed-form representation of the governing system. We formulate intrinsic Lie point-symmetry discovery as an operator-theoretic problem on sampled jet-space data. Admissible generators are identified from a sampled determining operator and separated from redundant, solution-dependent, and geometrically trivial directions. The framework combines prolongation, sequential subspace restrictions, and a fixed nonlinear dictionary linear in trainable coefficients. A rank-stabilization preflight test diagnoses systems whose represented intrinsic symmetry algebra does not stabilize under dictionary refinement. On a controlled differential-equation benchmark with an analytically known 11-dimensional represented intrinsic symmetry algebra, the method recovers the complete represented generator set, reproduces its Lie-bracket structure to numerical precision, and approximates a known generator beyond the training domain. Here we show that explicit intrinsic continuous-symmetry generators can be recovered from data obtained from exact or approximate differential systems.

版本更新 · v6arXiv:2603.13535

From Proxy to Certificate: Two-Sided Ollivier-Ricci Bounds from Balanced Forman Curvature

Giorgio Micaletto, Tebe Nigrelli

原文摘要

Balanced Forman (BF) curvature is used in graph learning as a cheap stand-in for Ollivier-Ricci (OR) curvature, yet it does not lower-bound OR curvature. We show that the BF value of an edge certifies a two-sided interval for its lazy OR curvature without solving any optimal transport problem, that BF coincides with a quadrangle-augmented Jost-Liu (JL) bound with its positive parts removed, and that when the 2-hop neighborhood is available, the interval can be sharpened at a cost of $\mathcal O(d_{\max}^{2.5})$ per edge. On 23 benchmark graphs, our quadrangle-augmented bound is tighter than the bounds of JL and Tian-Lubberts-Weber (TLW) on 13.8-100% of the edges and reduces their mean gap to the exact curvature by a median of 31.2%; intersected with the bounds of TLW, the intervals settle the curvature sign of more than half of the edges on 21 of the 23 graphs. On 15 real networks with up to 1,049,866 edges, a certify-then-solve procedure answers sign queries exactly while solving a median of 5.9 times fewer transport problems, for median speedups of 2$\times$ over exact evaluation, measured against a min-cut solver that is itself about two orders of magnitude faster than the widely used $\mathtt{GraphRicciCurvature}$. The main results are machine-checked in Lean 4.

stat.CO · cs.DM · cs.LG · math.CO · math.DG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2511.13025

Recovering Geodesic Distances from Noisy Observations

Charles Fefferman, Jonathan Marty, Kevin Ren

原文摘要

Let $(M,d,μ)$ be a nice geodesic probability space (such as a manifold) and $X=\{x_i\}_{i\in [N]} \subset M$ a finite $μ$-i.i.d. sample. Suppose we observe noisy distance random variables $d'(x_i,x_j)$. When can the true distances be recovered to small additive error? This question has attracted interest in two disparate fields: latent-space graph models and boundary-value inverse problems. We first show that, taking $N \asymp ε^{-2d-2}|\logε|$ and using runtime $O(N^{2.5+o_d (1)})$, we can recover all distances in $X$ to additive error $O(ε|\logε|)$ with high probability, assuming $\mathbb{E} d'(x_i, x_j) = F(d(x_i,x_j))$ for an unknown, increasing bilipschitz $F$ and with the $d'(x_i,x_j)$ allowed to be arbitrarily heterogeneous in their distribution as long as they are bounded in subgaussian norm. This generalizes existing work, which required $F$ to be known and made strict assumptions on the distribution of $d'(x_i, x_j)$. Achieving this generalization required a novel ``ruler-building'' procedure to learn $F$. We extend our results to a missing data case where we randomly lose access to noisy distances. Our second result uses ideas from the first to tackle a partial observability case: $X = Y \cup Z$ with $Z$ sampled from $M$ and $Y$ sampled from an observer subset satisfying visibility conditions that model the assumptions made on the boundary in boundary-value inverse problems. In this setting, we recover all pairwise distances in $Z$ to additive error $O(ε)$ with $|Z| \asymp ε^{-2d-2}|\logε|$, $|Y| \asymp ε^{-\max\{d_ν,4\}}$ with $d_ν$ determined by the volume-growth properties of the probability measure on the observer set, and runtime $O(|Z|^3)$. This is, to our knowledge, the first quantitative guarantee for this observer/observed split under noise, which complements existing stability results for boundary data.

stat.ML · cs.LG · math.DG · math.PR

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arXiv ↗PDF ↗
版本更新 · v5arXiv:2504.20941

Conformal-DP: A Density-Aware Mechanism for Differential Privacy over Riemannian Manifolds via Conformal Transformation

Peilin He, Liou Tang, M. Amin Rahimian, James Joshi

原文摘要

Differential Privacy (DP) is being increasingly adopted for non-Euclidean data that lie on complex, high-dimensional manifolds. Existing DP mechanisms for manifold data consider geometric properties when calibrating privacy perturbations, but they largely fail to capture variations in data density within datasets, leading to biased perturbations and suboptimal privacy-utility trade-offs due to heterogeneous data distributions. In this paper, we propose a novel density-aware differential privacy mechanism on Riemannian manifolds, referred to as Conformal-DP, that leverages conformal transformations to calibrate perturbations based on local densities and to induce a density-balanced geometry. We prove that our mechanism satisfies $ε$-differential privacy on any complete Riemannian manifold under mild regularity assumptions. In addition, we derive a closed-form expected geodesic error bound that depends only on the underlying data density ratio and is independent of global curvature. Our empirical results on synthetic and real-world datasets demonstrate that the proposed Conformal-DP mechanism substantially improves the privacy-utility trade-off in heterogeneous data distribution settings, with worst-case performance comparable to state-of-the-art manifold DP mechanisms that assume uniformly distributed data.

cs.CR · math.DG · stat.OT

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.10727

Kernel-Gradient Drifting Models

Maria Esteban-Casadevall, Jorge Carrasco-Pollo, Max Welling, Jan-Willem van de Meent, Erik J. Bekkers, Floor Eijkelboom

原文摘要

We propose kernel-gradient drifting, a one-step generative modeling framework that replaces the fixed Euclidean displacement direction in drifting models with directions induced by the kernel itself. Standard drifting is attractive because it enables fast, high-quality generation without distilling a large pretrained diffusion model, but its theory is currently understood mainly for Gaussian kernels, where the drift coincides with smoothed score matching and is identifiable. Our gradient-based reformulation exposes this score-based structure for general kernels: the resulting drift is the score difference between kernel-smoothed data and model distributions, yielding identifiability for characteristic kernels and a smoothed-KL descent interpretation of the smoothed drifting dynamics. Since kernel gradients are intrinsic tangent vectors, the same construction extends naturally to Riemannian manifolds and to discrete data via the Fisher-Rao geometry of the probability simplex. Across spherical geospatial data, promoter DNA and molecule generation, kernel-gradient drifting enables state-of-the-art one-step generation beyond the Euclidean setting without distillation. Code is available \href{https://github.com/jcarrascopollo/kgd_thesis}{here}.

cs.LG · math.DG

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2601.10083

Starfield: Demand-Aware Satellite Topology Design for Low-Earth Orbit Mega Constellations

Shayan Hamidi Dehshali, Tzu-Hsuan Liao, Shaileshh Bojja Venkatakrishnan

原文摘要

Low-Earth orbit (LEO) mega-constellations are emerging as high-capacity backbones for next-generation Internet. Deployment of laser terminals enables high-bandwidth, low-latency inter-satellite links (ISLs); however, their limited number, slow acquisition, and instability make forming a stable satellite topology difficult. Existing patterns like +Grid and Motif ignore regional traffic, ground station placement, and constellation geometry. Given sparse population distribution on Earth and the isolation of rural areas, traffic patterns are inherently non-uniform, providing an opportunity to orient inter-satellite links (ISLs) according to these traffic patterns. In this paper, we propose Starfield, a novel demand-aware satellite topology design heuristic algorithm supported by mathematical analysis. We first formulate a vector field on the constellation's shell according to traffic flows and define a corresponding Riemannian metric on the spherical manifold of the shell. The metric, combined with the spatial geometry, is used to assign a distance to each potential ISL, which we then aggregate over all demand flows to generate a score for each satellite's link selection. Inspired by +Grid, each satellite selects the link with the minimum Riemannian distance along with its corresponding angular links. To evaluate Starfield, we developed a custom, link-aware, and link-configurable packet-level simulator, comparing it against LAMP, +Grid, Motif, and Random topologies. For the Phase 1 Starlink, simulation results show up to a 30% reduction in hop count and a 15% improvement in stretch factor across multiple traffic distributions. Moreover, static Starfield, an inter-orbital link matching modification of Starfield, achieves a 15% improvement in stretch factor under realistic traffic patterns compared to +Grid. Experiments demonstrate Starfield's robustness under traffic demand perturbations.

cs.NI · cs.ET · math.DG

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arXiv ↗PDF ↗
版本更新 · v5arXiv:2607.19305

Riemannian Deep Learning: Modules, Networks, and Geometries

Ziheng Chen

原文摘要

Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations. This thesis develops a unified framework for Riemannian deep learning from three complementary perspectives: reusable neural modules, manifold-specific network architectures, and the design of underlying geometries. It generalizes batch normalization from Euclidean spaces and individual manifolds to broad classes of Lie groups and gyrogroups, and extends multinomial logistic regression from Euclidean space to SPD manifolds and then to general Riemannian manifolds. It further develops neural networks for several important geometric representations, including an unconstrained model of hyperbolic space, Busemann-based hyperbolic learning, and full-rank correlation matrices. Finally, it introduces adaptive and computationally efficient Riemannian metrics on SPD manifolds, including learnable Log-Euclidean geometries and fast, stable Cholesky-based geometries. The proposed methods are supported by theoretical analysis and validated through numerical experiments and applications in vision, signal processing, graph learning, and genomics.

cs.LG · cs.AI · math.DG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.37370

The Geometry of Manifold Learning: Singular foliations via Deep Learning models

Gabriele Novelli, Eliot Tron, Emanuele Latini, Rita Fioresi

原文摘要

Deep learning algorithms have become widely used and successful across a wide range of tasks. A key challenge is determining whether these algorithms truly capture the intrinsic geometry of their training data in order to better understand what they learn from data. In this theoretical work, we explore a new geometric structure defined on the data space by a trained neural network through the Data Information Matrix (DIM), a variation of the Fisher Information Matrix. This new geometric structure is a foliation and the main focus of this work is to explore the non-smooth case of ReLU activation functions, where traditional geometric tools often fail. The foliation is defined through a distribution spanned by the columns of the DIM. We first prove that this distribution is non-smooth in the sense of distributions, rendering the existence of an associated foliation much more complex. To circumvent this issue, we introduce a smoothed version of this distribution that differs from the original only on a set of measure zero. Finally, we prove that this smoothed distribution induces a singular foliation on the data space, which is based on the data the network has been trained with. Our theoretical contributions provide a geometric framework to better understand how neural networks organize their input space in non-smooth settings, bridging differential geometry and deep learning theory.

math.DG · cs.LG

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arXiv ↗PDF ↗
版本更新 · v4arXiv:2608.19584

Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold

Andrew Gracyk

原文摘要

We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned, or lack thereof, landscape. We focus on Calabi-Yau metrics specifically in a non-compact setting with a global potential, so defined geometrically rather than invoking the topological requirements of the Calabi conjecture. In non-compact settings, we can write the metric determinant with respect to a background in terms of a function, pluriharmonic or real. Using Yau's Liouville theorem, we get a partial eigenvalue blow-up effect. In an empirical setting, a Ricci-flat metric will not form, but a blow-up can be a local condition and can hold empirically on open sets even partially in generalized Kähler cases due to lower bounds on the Ricci potential. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist via geometric modality, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.

cs.LG · math.DG · stat.ML

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arXiv ↗PDF ↗