arxiv: 2605.06395 · v1 · submitted 2026-05-07 · 💻 cs.LG · cs.AI· eess.SP

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Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

Kartik Tandon , Julian Gould , Tanishq Bhatia , Francesca Dominici , Alejandro Ribeiro , Claudio Battiloro

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Pith reviewed 2026-05-08 12:48 UTC · model grok-4.3

classification 💻 cs.LG cs.AIeess.SP

keywords Hilbert bundlescellular sheavesconnection LaplacianHilbNetsgeometric deep learningmanifold samplingLaplacian convergenceinfinite-dimensional signals

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The pith

Sampling a manifold with a Hilbert bundle induces a cellular sheaf whose Laplacian converges in probability to the connection Laplacian, enabling consistent discrete networks for infinite-dimensional signals.

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

The paper develops HilbNets, convolutional architectures that define filters using the connection Laplacian of a Hilbert bundle so that signals taking values in possibly infinite-dimensional Hilbert spaces can be processed on a manifold. To implement this, the authors show that sampling the manifold produces a discrete structure called a Hilbert cellular sheaf equipped with Hilbert spaces on vertices and coupling rules on edges. They prove that the Laplacian of this sheaf converges in probability to the continuous connection Laplacian as the number of sample points grows. The discretized networks obtained from the sheaf are then shown to converge to the underlying continuous HilbNets and to produce the same outputs when applied to different samplings of the same bundle.

Core claim

We prove that the sheaf Laplacian of the Hilbert cellular sheaf induced by sampling converges in probability to the connection Laplacian of the Hilbert bundle as sampling density increases. This result generalizes the Belkin-Niyogi convergence of graph Laplacians to the infinite-dimensional bundle setting. We further prove that the discretized HilbNets converge to the continuous architectures and remain transferable across different samplings of the same bundle, yielding a consistent learning procedure for infinite-dimensional signals supported on manifolds.

What carries the argument

The Hilbert cellular sheaf: a graph whose vertices carry Hilbert spaces and whose edges carry coupling rules derived from the bundle so that the discrete Laplacian approximates the continuous connection operator.

If this is right

The discretized convolutional filters and neural networks converge to their continuous counterparts as sampling density increases.
The learned networks remain consistent when applied to new samplings of the same underlying bundle.
Convolution becomes implementable on irregular domains for signals whose values at each point live in a Hilbert space.
Geometric learning frameworks that rely on Laplacian operators can be lifted to the infinite-dimensional bundle case without losing consistency guarantees.

Where Pith is reading between the lines

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

The same sampling construction might allow consistent discretizations for other differential operators on Hilbert bundles beyond the connection Laplacian.
The transferability result suggests that models trained at one sampling density could be deployed directly on data collected at a different density without retraining.
The framework could be tested on concrete infinite-dimensional data such as functional time series or distributions defined over point clouds.

Load-bearing premise

The sampling procedure on the manifold must produce edge-wise coupling rules in the induced Hilbert cellular sheaf that preserve the structure needed for Laplacian convergence in the infinite-dimensional setting.

What would settle it

A specific Hilbert bundle together with a sequence of increasingly dense samplings where the empirical sheaf Laplacian does not converge in probability to the connection Laplacian in the appropriate operator norm.

Figures

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

**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: Visualization of the effect of the choice of underlying connection for generating heat flows of vector fields on the sphere S 2 . Left: Generated by considering the standard Levi-Civita connection on TS 2 , and corresponding 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: 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.

Desk Editor's Note private letter to a colleague

The paper sets up a sheaf-based framework to extend Laplacian geometric learning to infinite-dimensional Hilbert bundle signals and claims consistency via a generalized Belkin-Niyogi result, but the infinite-dimensional convergence step looks fragile without extra fiberwise control.

read the letter

The core move is to replace pointwise scalar features with Hilbert-space valued signals on a manifold, model the geometry with a connection Laplacian on a Hilbert bundle, and discretize via Hilbert cellular sheaves whose Laplacian is shown to converge in probability to the continuous operator. They then build HilbNets from the resulting convolution and prove that the discrete networks converge to the continuous ones and transfer across samplings. That generalization from the classical manifold Laplacian setting is the actual novelty, and the two-stage sampling plus discretization argument is laid out cleanly enough to be usable. The abstract also flags the empirical checks on synthetic and real tasks, which at least shows they tried to close the loop. The weak point is exactly the one the stress-test flags. Convergence in probability for the sheaf Laplacian to the connection Laplacian is asserted, but in infinite-dimensional fibers the passage from pointwise or weak operator convergence to the strong topology needed for the quadratic forms and network consistency is not automatic. The paper would need uniform control over the fibers or some compactness on the connection operators; if those are only implicit or missing, the claim does not go through for general Hilbert bundles. The Belkin-Niyogi citation is handled properly as an external base rather than circular, and no free parameters are hidden in the central statement. This is worth referee time for anyone working on geometric or functional-data networks. The framework is coherent on its own terms and the questions it raises are concrete, so a serious editor should send it out even if the proofs ultimately need tightening or extra assumptions.

Referee Report

3 major / 2 minor

Summary. The paper introduces HilbNets, a convolutional framework for infinite-dimensional signals supported on manifolds, based on the connection Laplacian of a Hilbert bundle. It claims that sampling the manifold induces a Hilbert cellular sheaf whose sheaf Laplacian converges in probability to the underlying connection Laplacian (generalizing the Belkin-Niyogi result to infinite-dimensional bundles), that discretized HilbNets converge to their continuous counterparts and are transferable across samplings, and that the framework is validated on synthetic and real-world tasks.

Significance. If the convergence and consistency results hold rigorously, the work would meaningfully extend geometric deep learning beyond finite-dimensional features to settings with signals in Hilbert spaces (e.g., distributions, operators, or time series), providing a unified theoretical foundation via bundles and sheaves. The explicit generalization of the Belkin-Niyogi theorem and the two-stage sampling/discretization procedure are strengths that could support consistency guarantees for learning on irregular domains.

major comments (3)

The main convergence result (sheaf Laplacian to connection Laplacian in probability) is load-bearing for all subsequent claims on HilbNet consistency and transferability. The proof must establish convergence in the strong operator topology (or norm topology) uniformly over fibers; weak or pointwise convergence alone does not suffice to control the quadratic forms or ensure the discretized operators approximate the continuous bundle Laplacian when fibers are infinite-dimensional Hilbert spaces. The manuscript should explicitly state the topology used and any fiberwise compactness or trace-class assumptions on the connection forms that upgrade the convergence.
The edge-wise coupling rules in the induced Hilbert cellular sheaf (parallel transport or connection operators between fibers) are central to preserving the structure needed for the Laplacian approximation. The paper must verify that these operators produce a kernel whose quadratic form converges to the bundle connection Laplacian without implicit finite-rank reductions; otherwise the generalization from the finite-dimensional Belkin-Niyogi setting fails for general Hilbert bundles.
The discretization step and transferability proof rely on the sampling density increasing and the sheaf Laplacian convergence. Any gap in controlling the approximation error uniformly (e.g., via explicit rates or bounds that hold in infinite dimensions) would undermine the claim that discretized HilbNets are consistent and transferable across different samplings of the same bundle.

minor comments (2)

Notation for the Hilbert bundle, connection form, and sheaf Laplacian should be introduced with explicit definitions and compared to standard references (e.g., the precise relation to the Belkin-Niyogi graph Laplacian) to improve readability for readers outside bundle theory.
The empirical validation section would benefit from clearer statements of the data exclusion rules, hyperparameter choices, and how the infinite-dimensional signals are approximated in practice (e.g., truncation of basis expansions).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The emphasis on rigorous convergence in infinite dimensions is appreciated, and we address each major comment below with clarifications and planned revisions.

read point-by-point responses

Referee: The main convergence result (sheaf Laplacian to connection Laplacian in probability) is load-bearing for all subsequent claims on HilbNet consistency and transferability. The proof must establish convergence in the strong operator topology (or norm topology) uniformly over fibers; weak or pointwise convergence alone does not suffice to control the quadratic forms or ensure the discretized operators approximate the continuous bundle Laplacian when fibers are infinite-dimensional Hilbert spaces. The manuscript should explicitly state the topology used and any fiberwise compactness or trace-class assumptions on the connection forms that upgrade the convergence.

Authors: We agree that the topology must be stated explicitly. Theorem 3.2 and its proof in Appendix A establish convergence in the strong operator topology, uniformly over fibers, under the assumption that the connection 1-forms are trace-class. This controls the quadratic forms in the infinite-dimensional setting. We will revise Section 3 to state the topology and trace-class assumption explicitly. revision: yes
Referee: The edge-wise coupling rules in the induced Hilbert cellular sheaf (parallel transport or connection operators between fibers) are central to preserving the structure needed for the Laplacian approximation. The paper must verify that these operators produce a kernel whose quadratic form converges to the bundle connection Laplacian without implicit finite-rank reductions; otherwise the generalization from the finite-dimensional Belkin-Niyogi setting fails for general Hilbert bundles.

Authors: The couplings are realized by parallel transport operators, which are unitary (hence isometries) between the infinite-dimensional Hilbert fibers. No finite-rank reduction is applied; the quadratic-form convergence follows from the sheaf Laplacian result in Theorem 3.2 while preserving full fiber dimension. We will add a clarifying remark in Section 2.3 confirming the generalization holds without rank reduction. revision: yes
Referee: The discretization step and transferability proof rely on the sampling density increasing and the sheaf Laplacian convergence. Any gap in controlling the approximation error uniformly (e.g., via explicit rates or bounds that hold in infinite dimensions) would undermine the claim that discretized HilbNets are consistent and transferable across different samplings of the same bundle.

Authors: Uniform error bounds for the discretization and transferability are derived in the proof of Theorem 4.3 (and Appendix B) under the trace-class assumption; these bounds hold in infinite dimensions and control consistency across samplings. Explicit rates are sampling-density dependent and manifold-specific. We will expand the discussion of these uniform bounds in Section 4. revision: partial

Circularity Check

0 steps flagged

No significant circularity; central convergence is an independent generalization of an external result

full rationale

The paper's load-bearing claim is the probabilistic convergence of the induced sheaf Laplacian to the connection Laplacian on a Hilbert bundle, presented explicitly as a generalization of the external Belkin & Niyogi theorem (cited as [BELKIN20081289]). No equations reduce this convergence to a self-definition, a fitted parameter renamed as a prediction, or a chain of self-citations whose validity is assumed rather than independently established. The Hilbert cellular sheaf construction and discretization steps are derived from the sampling procedure and the external manifold Laplacian result without circular reduction; the framework therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on generalizing existing Laplacian convergence to Hilbert bundles and introducing sheaf structures for discretization; no free parameters are fitted in the abstract, but several domain assumptions about infinite-dimensional signals and manifold sampling are required.

axioms (2)

domain assumption The connection Laplacian associated with a Hilbert bundle serves as a valid convolutional operator for infinite-dimensional signals supported on a manifold.
Invoked to define the continuous HilbNet architecture.
ad hoc to paper Sampling the manifold induces a Hilbert Cellular Sheaf whose Laplacian converges in probability to the bundle connection Laplacian.
This is the load-bearing generalization of the Belkin-Niyogi result stated in the abstract.

invented entities (2)

Hilbert Cellular Sheaf no independent evidence
purpose: Generalized graph structure with Hilbert feature spaces and edge-wise coupling rules to discretize the continuous bundle.
New construct introduced to bridge continuous and implementable discrete models.
HilbNets no independent evidence
purpose: Neural networks derived from Hilbert bundle convolutions that are consistent under discretization.
New architecture family defined via the two-stage sampling procedure.

pith-pipeline@v0.9.0 · 5608 in / 1584 out tokens · 73948 ms · 2026-05-08T12:48:21.143336+00:00 · methodology

discussion (0)

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