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arxiv: 2605.06395 · v2 · pith:32FFQ7FKnew · submitted 2026-05-07 · 💻 cs.LG · cs.AI· eess.SP

Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

Kartik Tandon , Julian Gould , Tanishq Bhatia , Francesca Dominici , Alejandro Ribeiro , Claudio Battiloro This is my paper

Pith reviewed 2026-05-21 08:52 UTC · model grok-4.3

classification 💻 cs.LG cs.AIeess.SP
keywords geometric deep learningHilbert bundlescellular sheavesconnection Laplacianmanifold samplingconsistencyinfinite-dimensional signalssheaf Laplacian
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The pith

HilbNets extend geometric deep learning to infinite-dimensional signals on manifolds by replacing graph Laplacians with connection Laplacians from Hilbert bundles and proving consistency after sampling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors develop a convolutional framework for signals whose values at each manifold point belong to their own Hilbert space rather than a shared vector space. They define HilbNets by using the connection Laplacian of a Hilbert bundle as the convolutional operator. Sampling the manifold produces a Hilbert cellular sheaf, and the authors prove that the sheaf Laplacian converges in probability to the continuous connection Laplacian, generalizing the Belkin-Niyogi result. They then show that the resulting discrete, implementable HilbNets converge to the underlying continuous networks and remain consistent when the sampling changes. This matters because many modern signals, such as distributions or operators, are naturally infinite-dimensional and supported on irregular domains that standard geometric deep learning does not yet handle with theoretical guarantees.

Core claim

We introduce HilbNets that employ the connection Laplacian associated with a Hilbert bundle as a convolutional operator for possibly infinite-dimensional signals supported on a manifold. Sampling the manifold induces a Hilbert cellular sheaf whose sheaf Laplacian converges in probability to the underlying connection Laplacian. The discretized HilbNets converge to the continuous architectures and are transferable across different samplings of the same bundle, thereby providing consistency for learning.

What carries the argument

The Hilbert cellular sheaf induced by sampling, equipped with Hilbert-valued features and edge-wise coupling rules, whose sheaf Laplacian acts as the discrete proxy for the continuous connection Laplacian.

If this is right

  • Discretized HilbNets become directly implementable on finite point clouds drawn from the manifold.
  • The networks converge to their continuous counterparts in the limit of increasing sampling density.
  • A model trained on one finite sampling of the bundle can be applied to another sampling without retraining.
  • Geometric deep learning now applies to signals whose pointwise values lie in Hilbert spaces rather than fixed-dimensional vectors.
  • The classical Laplacian-based theory lifts to settings with infinite-dimensional feature spaces at each point.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same sampling-and-convergence strategy could be tested on other differential operators defined on bundles, such as Dirac operators.
  • Applications to functional data or operator-valued regression become feasible once the consistency result is verified numerically on concrete manifolds.
  • The sheaf construction may allow incorporation of topological information from the bundle into the learned filters.
  • One could measure empirical convergence rates by comparing predictions on nested point sets of increasing cardinality drawn from the same underlying manifold.

Load-bearing premise

Sampling the manifold induces a Hilbert cellular sheaf whose sheaf Laplacian converges in probability to the underlying connection Laplacian as sampling density increases.

What would settle it

Observe a sequence of increasingly dense samplings on a fixed Hilbert bundle where either the sheaf Laplacian fails to approach the connection Laplacian in operator norm or the output of the discretized HilbNet differs from the continuous version by more than the predicted error bound.

Figures

Figures reproduced from arXiv: 2605.06395 by Alejandro Ribeiro, Claudio Battiloro, Francesca Dominici, Julian Gould, Kartik Tandon, Tanishq Bhatia.

Figure 1
Figure 1. Figure 1: Overview of the HilbNets framework. A HilbNet is a convolutional neural network processing infinite-dimensional signals supported on M (e.g., time-series or distributions over curved domains). The convolutional operator is the connection Laplacian ∆∇. To make HilbNets implementable, we first sample n points Xn from M to obtain a Hilbert Cellular Sheaf Fn with associated Hilbert Sheaf Laplacian ∆Fn . We the… view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the effect of the choice of underlying connection for generat￾ing heat flows of vector fields on the sphere S 2 . Left: Generated by considering the stan￾dard Levi-Civita connection on TS 2 , and cor￾responding Laplacian ∆∇. Right: Generated by allowing a more general connection ∇ that allows for torsion anisotropy and considering the corresponding connection Laplacian ∆∇. Consider now the… view at source ↗
Figure 3
Figure 3. Figure 3: Spectral stability of ∆Ft n,d for different signal and manifold sampling densitites. Left: aggregate ℓ2 eigenvalues discrepancy. Right: worst-case relative error. We train three transport parametrizations from Appendix D, free O(d) (Householder), circulant, and frozen identity, to recover the Levi-Civita transports in Cholesky-rescaled coordinates by minimizing the per-edge transport-MSE loss L = 1 |E| X e… view at source ↗
read the original abstract

Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as \textit{HilbNets}. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin \& Niyogi \cite{BELKIN20081289} convergence result for the graph Laplacian to the manifold Laplacian, a theoretical cornerstone of geometric learning methods. Second, we discretize the signals and prove that the discretized (implementable) HilbNets converge to the underlying continuous architectures and are transferable across different samplings of the same bundle, providing consistency for learning. Finally, we validate our framework on synthetic and real-world tasks. Overall, our results broaden the scope of geometric learning as a whole by lifting classical Laplacian-based frameworks to settings where the signal at each point lives in its own Hilbert space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces HilbNets, a convolutional learning framework for infinite-dimensional signals supported on manifolds, using the connection Laplacian of a Hilbert bundle as the convolutional operator. It claims that sampling the manifold induces a Hilbert Cellular Sheaf whose sheaf Laplacian converges in probability to the underlying connection Laplacian (generalizing Belkin & Niyogi), that discretized implementable HilbNets converge to the continuous architectures, and that these are transferable across different samplings of the same bundle, yielding consistency for learning. The framework is validated on synthetic and real-world tasks.

Significance. If the convergence and consistency results hold, the work broadens geometric deep learning to natively infinite-dimensional signals (e.g., time series, distributions, operators) over irregular domains by lifting Laplacian-based methods to Hilbert bundles. The generalization of the Belkin-Niyogi result and the explicit two-stage sampling procedure for implementability are substantive theoretical contributions; the transferability across samplings provides a practical consistency guarantee that is rarely formalized in geometric DL.

major comments (2)
  1. [Convergence of sheaf Laplacian to connection Laplacian] The proof that the sheaf Laplacian converges in probability to the connection Laplacian (generalization of Belkin & Niyogi, stated in the abstract and developed in the convergence section): the argument does not supply explicit controls on the operator norm of the connection or compactness/uniform bounds on the fibers. The original Belkin-Niyogi proofs rely on finite-dimensional pointwise concentration and kernel estimates; these do not automatically extend to arbitrary separable Hilbert spaces, where non-uniform variance across fibers can prevent the required probabilistic convergence.
  2. [Discretization and consistency of HilbNets] The claim that discretized HilbNets converge to the continuous architectures and are transferable across samplings (second main result): this rests directly on the sheaf-Laplacian convergence established in the preceding step. Without additional assumptions or error bounds that close the infinite-dimensional gap, the consistency guarantee for learning remains conditional on a result whose proof details are not fully load-bearing in the current manuscript.
minor comments (2)
  1. [Preliminaries and definitions] Notation for the Hilbert Cellular Sheaf and its Laplacian could be clarified with a short comparison table to the classical graph Laplacian; current definitions are introduced piecemeal.
  2. [Experiments] The experimental section would benefit from explicit statements of data exclusion rules and baseline hyper-parameter selection protocols to match the theoretical consistency claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We address the major comments below and outline the revisions we will make to strengthen the theoretical results.

read point-by-point responses

  1. Referee: [Convergence of sheaf Laplacian to connection Laplacian] The proof that the sheaf Laplacian converges in probability to the connection Laplacian (generalization of Belkin & Niyogi, stated in the abstract and developed in the convergence section): the argument does not supply explicit controls on the operator norm of the connection or compactness/uniform bounds on the fibers. The original Belkin-Niyogi proofs rely on finite-dimensional pointwise concentration and kernel estimates; these do not automatically extend to arbitrary separable Hilbert spaces, where non-uniform variance across fibers can prevent the required probabilistic convergence.

    Authors: We appreciate the referee highlighting this important technical point. Our convergence proof adapts the kernel density estimation approach to the Hilbert bundle setting by considering the fibers as separable Hilbert spaces and using Bochner integration for the expectations. However, we acknowledge that to ensure the operator norm convergence in probability, uniform bounds on the connection operator and fiber norms are indeed required to control the variance terms uniformly across the manifold. We will revise the manuscript by adding these assumptions explicitly in the statement of the theorem (e.g., assuming the connection is bounded and the sampling is such that variance is controlled), and provide a detailed proof appendix with the necessary concentration inequalities for Hilbert-valued random variables. This will make the generalization rigorous while preserving the generality for practical applications. revision: yes

  2. Referee: [Discretization and consistency of HilbNets] The claim that discretized HilbNets converge to the continuous architectures and are transferable across samplings (second main result): this rests directly on the sheaf-Laplacian convergence established in the preceding step. Without additional assumptions or error bounds that close the infinite-dimensional gap, the consistency guarantee for learning remains conditional on a result whose proof details are not fully load-bearing in the current manuscript.

    Authors: We agree that the discretization consistency relies on the sheaf Laplacian convergence. Upon incorporating the revisions to the convergence theorem with explicit error bounds and uniform controls as described in response to the first comment, we will update the discretization section to derive the corresponding convergence rates for the HilbNet operators and the transferability across samplings. This will include a quantitative bound on the difference between continuous and discrete network outputs, closing the gap for the learning consistency guarantee. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rest on external generalization and claimed proofs.

full rationale

The paper derives HilbNets from the connection Laplacian on a Hilbert bundle, then claims two main results: (1) sampling induces a Hilbert Cellular Sheaf whose Laplacian converges in probability to the connection Laplacian (explicitly generalizing the external Belkin & Niyogi theorem), and (2) discretized HilbNets converge to the continuous versions and transfer across samplings. These are presented as mathematical proofs rather than parameter fits, self-definitions, or renamings. No equations or steps reduce by construction to quantities defined inside the paper; the load-bearing convergence step invokes an external prior result with no self-citation chain or ansatz smuggling indicated in the provided text. The derivation chain is therefore self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The framework introduces the Hilbert bundle connection Laplacian as the convolutional operator and the Hilbert cellular sheaf as the discrete counterpart; these are new constructs whose properties are asserted rather than derived from more elementary principles.

axioms (1)
  • domain assumption The connection Laplacian associated with a Hilbert bundle serves as a valid convolutional operator for signals taking values in infinite-dimensional Hilbert spaces.
    Invoked in the definition of HilbNets and the two-stage sampling procedure.
invented entities (2)
  • HilbNets no independent evidence
    purpose: Neural network architectures implementing convolution via Hilbert bundle connection Laplacians.
    Newly defined in the paper as the implementable version of the continuous framework.
  • Hilbert Cellular Sheaf no independent evidence
    purpose: Discrete structure obtained by sampling the manifold that carries Hilbert feature spaces and edge-wise coupling rules.
    Introduced to bridge continuous bundle to discrete implementable networks.

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