⚡ electrical engineering

Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

This paper introduces HilbNets, a novel geometric deep learning framework that utilizes connection Laplacians on Hilbert bundles to enable consistent, transferable convolutional learning for infinite-dimensional signals on irregular domains by proving that their discrete cellular sheaf approximations converge to the underlying continuous operators.

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

Published 2026-05-08

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a computer to understand complex data, like a weather map, a stock market trend, or a 3D shape. Usually, computers are good at looking at data arranged in neat grids (like pixels in a photo) or simple lists (like a spreadsheet). But the real world is messy. Data often lives on irregular shapes (like a crumpled piece of paper) or changes over time in ways that are hard to pin down.

This paper introduces a new way to teach computers to handle this messy, complex data. The authors call their new system HilbNets.

Here is the breakdown of how it works, using simple analogies:

1. The Problem: Data That Lives in "Infinite" Spaces

Most current AI models treat data points as simple numbers (like a temperature reading of 72°F). But sometimes, a single data point isn't just a number; it's a whole story.

The Analogy: Imagine a weather station. A simple model sees "72°F." A HilbNet sees the entire history of wind, pressure, and temperature for that station as a single, complex "bundle" of information.
The Challenge: These "bundles" can be infinitely complex (like a continuous time series or a probability distribution). Standard AI tools break down when faced with this because they don't know how to compare one complex bundle to another, especially when they are on a weird, curved surface.

2. The Solution: The "Hilbert Bundle"

The authors use a mathematical concept called a Hilbert Bundle.

The Metaphor: Imagine a giant, curved balloon (the "manifold"). At every single point on the balloon's surface, there is a tiny, invisible suitcase (the "fiber").
- In old AI, every suitcase contained just a single number (like a scalar).
- In HilbNets, every suitcase contains a complex, infinite-dimensional object (like a whole song, a video, or a probability curve).
The goal is to move information between these suitcases as you travel across the balloon without losing the shape or meaning of the data inside.

3. The Engine: "Parallel Transport"

To move data between suitcases on a curved surface, you need a rulebook. In math, this is called a connection.

The Analogy: Imagine you are carrying a tray of drinks (the data) while walking on a winding, hilly path. If you just walk straight, the drinks might spill because the ground tilts. You need to constantly tilt your tray to keep the drinks level.
Parallel Transport is the rulebook that tells the AI exactly how to tilt the tray (rotate or shift the data) as it moves from one point to another so that the data remains consistent. The paper shows how to learn these rules automatically.

4. Making it Real: The "Cellular Sheaf"

The math behind HilbNets is continuous and smooth, which is great for theory but impossible to run on a computer (which only understands discrete steps). The authors had to figure out how to turn this smooth theory into a practical computer program.

The Metaphor: Imagine trying to draw a perfect, smooth circle on a computer screen. You can't draw a true circle; you have to use a grid of tiny squares (pixels).
The authors developed a two-step "sampling" process:
1. Sampling the Map: They pick a bunch of points on the balloon to create a rough map (a graph).
2. Sampling the Suitcases: They take a snapshot of the complex data inside each suitcase and turn it into a manageable list of numbers.
This creates a structure they call a Hilbert Cellular Sheaf. Think of it as a "super-graph" where the nodes are the points on the map, and the connections between them are not just simple lines, but complex rules for how to translate the data from one node to the next.

5. The Big Promise: Consistency and Transferability

The most important part of the paper is the proof that this method works reliably.

The Guarantee: The authors proved that as you add more and more points to your map and take more detailed snapshots of the data, your computer program (the discretized HilbNet) gets closer and closer to the perfect, theoretical version.
The Analogy: It's like taking a photo of a sculpture. If you take a low-resolution photo, it looks blocky. If you take a high-resolution photo, it looks smooth. The paper proves that no matter which specific points you choose to photograph (as long as you take enough), the final picture will always converge to the same true image.
Why it matters: This means the AI is robust. If you train it on one set of sensors and test it on a different set of sensors in the same area, it will still work correctly. It doesn't get confused by the specific layout of the data.

6. Real-World Tests

The team tested this on two things:

Synthetic Math: They created a fake world with known rules and showed their AI could perfectly learn the rules for moving data between points.
Traffic Forecasting: They used real traffic data from Los Angeles and San Francisco. In this case, the "suitcase" at each traffic sensor contained a time-series of speed data.
- Result: Their new method (HilbNet) predicted traffic speeds better than standard methods, especially when they allowed the AI to learn how to "tilt the tray" (align the time data) between sensors. Even a simplified version of their method outperformed older models.

Summary

In short, this paper builds a bridge between advanced, infinite-dimensional math and practical AI. It creates a new type of neural network (HilbNet) that can handle data that is both complex (infinite-dimensional) and irregular (on curved surfaces). It proves that this network is mathematically sound and works better than current tools for tasks like predicting traffic, all by treating data as a collection of interconnected, complex "suitcases" that need to be carefully aligned as they move across a map.

Technical Summary: Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

1. Problem Statement

Modern deep learning increasingly encounters signals that are infinite-dimensional (e.g., time series, probability distributions, operators) and defined over irregular domains (e.g., manifolds, graphs). While Geometric Deep Learning (GDL) has successfully generalized Convolutional Neural Networks (CNNs) to non-Euclidean domains like graphs and manifolds, existing frameworks predominantly assume finite-dimensional fibers (e.g., scalar values or finite vectors at each point).

There is a lack of a unified learning theory for infinite-dimensional signals supported on manifolds. Specifically, current methods cannot natively handle signals where the data at each point $x \in M$ resides in its own infinite-dimensional Hilbert space $E_x$ . Furthermore, there is no rigorous theoretical foundation connecting discrete approximations of such structures to their continuous counterparts, leaving a gap in consistency guarantees for learning on these complex data types.

2. Methodology

The authors propose a framework called HilbNets, which unifies and generalizes existing geometric learning approaches through the lens of Hilbert Bundles and Cellular Sheaves.

2.1 Continuous Framework: Hilbert Bundles and Connection Laplacians

Hilbert Bundles: The authors model the data as a section $S$ of a Hilbert bundle $E$ over a base manifold $M$ . Unlike trivial bundles where fibers are identical copies of $\mathbb{R}^n$ , here each fiber $E_x$ is a (possibly infinite-dimensional) separable Hilbert space.
Connection and Parallel Transport: To compare signals across different fibers (which lack a canonical identification), the framework employs a connection $\nabla$ . This induces parallel transport maps $P_{x \to y}: E_x \to E_y$ along paths on the manifold, allowing for coherent comparison of infinite-dimensional data.
Convolution via Connection Laplacian: The core convolutional operator is the connection Laplacian $\Delta_\nabla = \nabla^* \nabla$ . This operator generalizes the Laplace-Beltrami operator (for scalar fields) and the connection Laplacian for tangent bundles.
Filters: Filters are defined using the Borel functional calculus. A filter $g$ is applied to the self-adjoint operator $\Delta_\nabla$ via $g(\Delta_\nabla)$ , allowing for spectral filtering of infinite-dimensional signals without requiring a discrete spectrum.
HilbNets: A HilbNet is a deep architecture composed of layers of these Hilbert bundle filters followed by pointwise non-linearities $\sigma$ .

2.2 Discretization: Two-Stage Sampling

Since continuous HilbNets are not directly implementable, the authors introduce a two-stage discretization procedure to derive Discretized HilbNets:

Manifold Sampling (Spatial Discretization):
- Sampling $n$ points $X_n$ from the manifold $M$ induces a Hilbert Cellular Sheaf $\mathcal{F}^t_n$ .
- This structure assigns a Hilbert space (the fiber) to each node and edge.
- Edge-wise coupling is defined via restriction maps derived from the parallel transport operators and a Gaussian kernel weight $k^t_{ij} = e^{-d_M(x_i, x_j)^2 / 4t}$ .
- The Hilbert Sheaf Laplacian $\Delta_{\mathcal{F}^t_n}$ is constructed from these restriction maps.
Signal Sampling (Fiber Discretization):
- The infinite-dimensional signals in the fibers are projected onto a finite-dimensional subspace of dimension $d$ using an orthogonal basis (e.g., Fourier basis for time series).
- This yields a Network Sheaf $\mathcal{F}^t_{n,d}$ with finite-dimensional stalks ( $\mathbb{R}^d$ ).
- The resulting Network Sheaf Laplacian $\Delta_{\mathcal{F}^t_{n,d}}$ is a block matrix where blocks represent the discretized parallel transport maps.
- The implementable (n, d)-HilbNet is defined as a Sheaf Neural Network operating on this finite structure.

3. Key Contributions

Hilbert Bundle Convolutional Framework: The paper defines a principled convolution operation for infinite-dimensional signals on manifolds using the connection Laplacian and Borel functional calculus, extending GDL beyond finite-dimensional fibers.
Hilbert Cellular Sheaves: The authors establish the first rigorous link between Hilbert bundles and Hilbert Cellular Sheaves, showing that sampling a manifold induces a sheaf structure that captures both the geometry of the base manifold and the coupling rules of the bundle fibers.
Convergence Guarantees (Theoretical Core):
- Theorem 1 (Sheaf Laplacian Convergence): Proves that as the sampling density $n \to \infty$ , the Hilbert Sheaf Laplacian converges in probability to the underlying Connection Laplacian. This is a generalization of the seminal Belkin & Niyogi [14] result (Graph Laplacian $\to$ Manifold Laplacian) to the setting of infinite-dimensional bundles.
- Theorem 2 (Finite-Rank Convergence): Proves that the fully discretized (implementable) Laplacian converges to the continuous operator as both manifold sampling ( $n$ ) and signal sampling ( $d$ ) densities increase.
Consistency and Transferability:
- Corollary 2.1: Establishes that the output of Discretized HilbNets converges to the continuous HilbNet in Mean Squared Error (MSE) as sampling densities increase.
- Corollary 2.2 (Transferability): Demonstrates that Discretized HilbNets are transferable across different samplings of the same underlying bundle. The architecture's output is consistent regardless of the specific point cloud sampled, providing resolution consistency guarantees.

4. Experimental Results

The framework is validated on two tasks:

Synthetic Transport Recovery:
- Setup: A statistical bundle over centered Gaussians (using the Otto-Wasserstein metric) where the ground-truth parallel transport (Levi-Civita connection) is known.
- Result: The model successfully recovers the true parallel transport operators. The "Free $O(d)$ " parametrization recovers the ground truth to numerical precision ( $\approx 1.6 \times 10^{-7}$ ), while restricted classes (e.g., circulant) converge to their analytical projection limits, confirming that the hypothesis class constrains the learning in a predictable way.
Real-World Traffic Forecasting (METR-LA and PEMS-BAY):
- Setup: Predicting traffic speeds where nodes are sensors and fibers are time-series. The framework learns edge-wise transport maps to align temporal patterns before filtering.
- Baselines: Compared against MLPs (fiber-only), Spatiotemporal Graph baselines, and standard GCNs (frozen identity transport).
- Result: Learning non-trivial transports consistently outperforms frozen identity (GCN) at all forecasting horizons. The Free $O(T)$ variant achieves the best accuracy, while the Circulant variant offers competitive performance with significantly fewer parameters (approx. 1/10th), demonstrating the value of structured, physically motivated inductive biases.

5. Significance and Claims

The paper claims that HilbNets provide a unified and general framework for geometric deep learning that lifts classical Laplacian-based methods to settings with infinite-dimensional signals.

Theoretical Impact: The primary contribution is the extension of the Belkin & Niyogi [14] convergence theorem to Hilbert bundles. This provides the theoretical bedrock for Laplacian-based learning on infinite-dimensional data, justifying the use of discrete sheaf approximations for continuous bundle problems.
Practical Impact: The framework allows for the principled incorporation of domain-specific geometric priors (via the choice of connection and transport) into deep learning architectures. This is particularly beneficial in low-data regimes or when the data has inherent structural constraints (e.g., time-series alignment, distributional geometry).
Generality: The authors note that many existing architectures (CNNs, GNNs, Sheaf NNs, Equivariant CNNs) can be viewed as specific instances of HilbNets with particular choices of bundles and connections. Thus, the convergence and transferability results derived here potentially extend to a broad class of existing geometric learning methods.

The paper concludes that this work broadens the scope of geometric learning by providing a consistent, theoretically grounded approach for learning on manifolds where signals live in their own Hilbert spaces.