Attentional Line Message Passing (ALMP)

• ALMP is an attention-based message passing mechanism that integrates line geometry and connectivity using rotary positional encoding.
• It selectively aggregates features from line-connected neighbors through a residual MLP update to improve matching accuracy.
• Empirical results on ETH3D benchmarks show that ALMP achieves higher average precision and efficiency compared to mean-based approaches.

Attentional Line Message Passing (ALMP) is a neural message passing mechanism designed to enhance the representation and matching of line-like structures in graphs by leveraging attention-based aggregation schemes that exploit line connectivity and spatial relationships. ALMP is particularly impactful in geometric deep learning applications, such as joint point and line matching, where encoding line topology and selectively weighting neighbors is critical for robust performance and computational efficiency (Ubingazhibov et al., 18 Oct 2025).

1. Formal Definition and Core Mechanism

ALMP operates on a graph where nodes represent line endpoints and possibly keypoints, and edges encode connectivity among endpoints as induced by detected line segments. Unlike traditional mean-based line message passing (LMP), ALMP replaces uniform aggregation with attention-weighted summation over a node's neighbors, allowing the network to focus on more reliable or repeatable endpoints.

The update rule for a node $i$, associated with feature $\mathbf{x}_i^I$, is

$$\mathbf{x}_i^I \leftarrow \mathbf{x}_i^I + \mathrm{MLP}\left([\mathbf{x}_i^I \mid \mathbf{m}_i^I] \right)$$

where the aggregated message $\mathbf{m}_i^I$ is computed as

$$\mathbf{m}_i^I = \sum_{j \in \mathcal{N}(i) \cup \{i\}} \mathrm{Softmax}_{k \in \mathcal{N}(i) \cup \{i\}} (a_{ik})_j \cdot \mathbf{v}_j^I$$

with the value vector $\mathbf{v}_j^I$ obtained via a linear transformation of $\mathbf{x}_j^I$, and $a_{ik}$ denoting the attention weights.

The attention score between node $i$ and $j$ at positions $\mathbf{p}_i, \mathbf{p}_j$ is

$$a_{ij}^I = \mathbf{q}_i^\top \cdot \mathcal{R}(\mathbf{p}_j - \mathbf{p}_i) \cdot \mathbf{k}_j$$

where $\mathbf{q}_i, \mathbf{k}_j$ are linear projections of input features, and $\mathcal{R}(\cdot)$ is a rotary positional encoding capturing angular and spatial relationships.

This attentional update explicitly incorporates both node features and line geometry, enabling finer control over message propagation.

2. Architectural Context: Integration in LightGlueStick

In LightGlueStick, ALMP is integrated as the principal line message passing mechanism, with each endpoint node aware of its direct line neighbors as determined by the wireframe structure. This contrasts with earlier GNN-based matchers (e.g., GlueStick), which aggregated endpoint features by simple averaging.

The ALMP mechanism, as adopted, leverages a residual update structure combined with an MLP, with the attention-based aggregation restricted to each node's line-connected neighbors plus itself. The inclusion of rotary positional encodings allows the attention module not only to consider feature similarity but also to encode angular geometry between lines, critical for point-line and line-line correspondence tasks.

A summarizing table distinguishes the main message passing modes:

Message Passing Scheme	Aggregation Weights	Geometric Encoding
LMP (mean-based)	Uniform (mean)	None
ALMP (attention)	Learned attention	Rotary positional

3. Theoretical and Methodological Underpinnings

ALMP advances upon standard attentional message passing (Do et al., 2018) and energy-constrained diffusion perspectives (Wu et al., 13 Sep 2024) by specializing its attention computation to structures derived from line connectivity graphs. The update closely mirrors local self-attention along a line-induced neighborhood, supported by geometric encodings, and respects the underlying spatial topology of line arrangements.

The modular update design in ALMP is informed by insights from message passing attention networks for document graphs (Nikolentzos et al., 2019), which demonstrate that weighting neighborhood aggregation through neural attention mechanisms captures richer dependencies than undifferentiated averaging.

A plausible implication is that the framework of ALMP could be generalized to other "higher-order" graph structures where groupwise or topologically induced connectivity (e.g., hyperedges, polygons, filaments) is critical, provided appropriate geometric encoding is defined.

4. Practical Impact: Matching Performance and Efficiency

Empirical evaluation in LightGlueStick (Ubingazhibov et al., 18 Oct 2025) demonstrates that ALMP achieves a superior balance of accuracy and compute cost in line segment matching tasks. On ETH3D benchmarks:

• No LMP: Average Precision (AP) = 68.4, runtime 39 ms
• Mean-based LMP: AP = 73.3, runtime 54 ms
• ALMP (proposed): AP = 74.6, runtime 47 ms

This evidences both a gain in absolute accuracy and favorable runtime scaling compared to prior approaches. The improvement is attributed to ALMP's ability to focus the update on more reliable endpoints, a property that is particularly beneficial when dealing with fragmented or non-repeatable line detections.

Notably, ALMP's attention mechanism not only boosts matching robustness but also avoids the excessive computational overhead characterizing more generalized transformer or global attention backbones.

5. Geometric Consistency and Line Connectivity

One of the defining benefits of ALMP is its explicit encoding of line connectivity. During message passing, the network "reminds" each endpoint of its relationship to line-connected neighbors, which:

• Captures long-range and angular relationships inherent in the geometric structure,
• Enforces geometric consistency across endpoints, and
• Guides the network toward globally coherent matches in the presence of ambiguous or spurious features.

The rotary positional encoding $\mathcal{R}(\mathbf{p}_j - \mathbf{p}_i)$ further ensures that the relative angular orientation of line endpoints contributes to attention weighting, enhancing the reliability of learned descriptors.

6. Comparative Context, Scalability, and Limitations

Compared to general-purpose attentional message passing frameworks (Do et al., 2018, Wu et al., 13 Sep 2024), ALMP is highly specialized for edge-centric (line-centric) scenarios where connectivity is not purely node adjacency but is derived from domain-specific geometric predicates (e.g., line membership).

In practice, the attention computation restricts neighbors to those determined by the line structure, ensuring that computational complexity grows linearly with the number of endpoints and lines, making the scheme suitable for real-time settings and edge devices.

A plausible implication is that, although ALMP is efficient for line-structured graphs, extension to denser or more complex connectivities may require further optimizations or sparsity-inducing mechanisms to maintain scalability.

7. Applications and Future Directions

ALMP has demonstrated effectiveness in geometric computer vision pipelines, notably in applications requiring joint point and line matching such as SLAM and Structure-from-Motion (Ubingazhibov et al., 18 Oct 2025). Its synergy with transformer-style backbones and ability to condition attention on geometric configuration enables robust feature correspondence under challenging visual conditions.

A plausible trajectory for future developments includes the incorporation of ALMP-like mechanisms into broader categories of higher-order message passing networks, leveraging domain-specific structural or spatial encodings to further enhance performance on structured data beyond lines, such as surfaces or meshes.

In summary, Attentional Line Message Passing provides an efficient, geometry-aware, attention-based approach to message propagation in line-centric graphs, enabling robust, accurate, and explainable solutions for matching tasks in geometric deep learning.