Mirror-Based Graph Attention Network

• Mirror-GAT is a graph attention framework enhanced by coupling a mirrored process (via dual–primal or iterative feedback) with traditional node-level attention.
• It improves feature contextualization by allowing edge-level information and incident-edge interactions to modulate node aggregation, surpassing standard GAT limitations.
• The architecture achieves efficiency gains with fewer parameters and lower signaling overhead, making it effective in ISAC and dynamic graph optimization scenarios.

Mirror-Based Graph Attention Network (Mirror-GAT) denotes, in the literature considered here, a family of graph-attention architectures in which a graph-attention process is coupled to a mirrored companion process rather than being applied only once on a single graph. One usage is a dual–primal construction in which node-level attention on a graph is mirrored by edge-level attention on its dual or line graph, so that context among incident edges modulates node aggregation (Monti et al., 2018). A second, explicitly named usage is a lightweight, adjacency-shared, bilevel heterogeneous GAT for cooperative cell-free integrated sensing and communication (ISAC), where two heterogeneous attention modules mirror one another across precoding and association/mode-selection subproblems and are coupled through echo feedback from a frozen 3D-CNN estimator (Jiang et al., 9 Jul 2025). Both constructions are rooted in the masked self-attention mechanism of Graph Attention Networks (GAT), which applies learned, neighborhood-normalized coefficients to graph-structured data without spectral decomposition (Veličković et al., 2017).

1. Foundations in graph attention

Standard GAT provides the immediate substrate for Mirror-GAT. Given a graph $G=(V,E)$ with node features $H \in \mathbb{R}^{N \times F}$, a single attention head first applies a shared linear projection $W \in \mathbb{R}^{F' \times F}$ and then computes masked neighborhood scores only for $j \in \mathcal N_i$, where $\mathcal N_i$ is the $1$-hop neighborhood including the node itself. The core equations are

$$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$

$$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$

$$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$

The attention mechanism is shared across all edges, LeakyReLU uses negative slope $\alpha=0.2$, hidden layers use ELU after aggregation, and multi-head attention concatenates head outputs in hidden layers while averaging them in output layers (Veličković et al., 2017).

Several properties of this baseline are directly inherited or explicitly modified by mirror-based variants. GAT avoids Laplacian eigendecomposition and graph-dependent spectral filters; its attention is local, permutation-invariant over neighbor ordering, and parallelizable across edges. A single head has per-layer time complexity

$H \in \mathbb{R}^{N \times F}$0

with the first term coming from linear projection and the second from attention-score computation and weighted aggregation. The original GAT formulation also supports both transductive and inductive regimes because the shared edge-level mechanism does not depend on a global eigenbasis (Veličković et al., 2017).

Mirror-GAT constructions depart from standard GAT at the point where $H \in \mathbb{R}^{N \times F}$1 is formed. In vanilla GAT, $H \in \mathbb{R}^{N \times F}$2 is a pairwise function of $H \in \mathbb{R}^{N \times F}$3 after projection. In mirror-based formulations, $H \in \mathbb{R}^{N \times F}$4 becomes coupled either to an auxiliary attention process on edges or to a second GAT solving a linked subproblem. This shift from isolated pairwise edge scoring to mirrored contextual scoring is the defining architectural change.

2. Dual–primal mirroring on the line graph

A broader mirror-based interpretation appears in Dual–Primal Graph Convolutional Networks, which can be read as a Mirror-GAT in which the “mirror” is the dual or line graph $H \in \mathbb{R}^{N \times F}$5 (Monti et al., 2018). For a primal graph $H \in \mathbb{R}^{N \times F}$6, the dual graph treats directed primal edges as dual nodes. With source-incidence matrix $H \in \mathbb{R}^{N \times F}$7, a natural dual adjacency is

$H \in \mathbb{R}^{N \times F}$8

which connects edges that share the same source node. For a primal node $H \in \mathbb{R}^{N \times F}$9, the corresponding dual neighborhood is the out-star $W \in \mathbb{R}^{F' \times F}$0.

Within a dual–primal block, node features are first projected, and initial edge features are constructed from endpoint features, for example

$W \in \mathbb{R}^{F' \times F}$1

Attention is then performed on the dual graph. For head $W \in \mathbb{R}^{F' \times F}$2,

$W \in \mathbb{R}^{F' \times F}$3

$W \in \mathbb{R}^{F' \times F}$4

$W \in \mathbb{R}^{F' \times F}$5

The updated edge embedding $W \in \mathbb{R}^{F' \times F}$6 is then converted into the primal attention score

$W \in \mathbb{R}^{F' \times F}$7

Primal node aggregation proceeds with these context-aware $W \in \mathbb{R}^{F' \times F}$8 values rather than the standard GAT score computed directly from $W \in \mathbb{R}^{F' \times F}$9 (Monti et al., 2018).

This construction strictly generalizes GAT in the sense stated in the source summary: if the dual update is removed and $j \in \mathcal N_i$0 is replaced by a fixed function of the endpoint projections, the formulation collapses to vanilla GAT. The conceptual significance is that attention on a single edge $j \in \mathcal N_i$1 becomes a function of the entire star $j \in \mathcal N_i$2, allowing competitive or cooperative interactions among incident edges to shape node aggregation. That richer attention family is obtained at nontrivial computational cost: dual attention scales as $j \in \mathcal N_i$3 per head, so high-degree nodes create the main overhead beyond standard GAT (Monti et al., 2018).

3. Heterogeneous graph formulation in cooperative cell-free ISAC

In the explicit Mirror-GAT of cooperative cell-free ISAC, the architecture is anchored in a heterogeneous network model with $j \in \mathcal N_i$4 distributed dual-function access points (APs), each equipped with $j \in \mathcal N_i$5 antennas in a uniform linear array, serving $j \in \mathcal N_i$6 single-antenna communication users (CUs) and sensing $j \in \mathcal N_i$7 targets (Jiang et al., 9 Jul 2025). Each AP operates per snapshot either as a transmit AP or a receive AP. The mode-selection vector is

$j \in \mathcal N_i$8

with $j \in \mathcal N_i$9 for Tx, $\mathcal N_i$0 for Rx, and $\mathcal N_i$1, where $\mathcal N_i$2.

The physical layer is modeled with OFDM using $\mathcal N_i$3 subcarriers and $\mathcal N_i$4 OFDM symbols. At Tx-AP $\mathcal N_i$5, the discrete frequency-time transmit signal is

$\mathcal N_i$6

where $\mathcal N_i$7 encodes user association and satisfies $\mathcal N_i$8. The communication objective is summarized through the achievable sum-rate

$\mathcal N_i$9

while the sensing side stacks received echoes into

$1$0

from which target positions and velocities are estimated (Jiang et al., 9 Jul 2025).

Mirror-GAT represents this system as two mirror graphs with a shared heterogeneous topology. In the precoding-oriented graph, node types are Tx-AP, Rx-AP, and CU. In the association-oriented graph, node types are AP and CU. Communication edges $1$1 encode AP–CU links, while sensing edges $1$2 encode line-of-sight AP–AP and radar illumination/interference pathways. Communication edge features are constructed from real and imaginary parts of stacked channels,

$1$3

and sensing edge features from AP–AP channel matrices,

$1$4

The two mirror graphs share a relation-composed adjacency

$1$5

where $1$6. According to the source, this shared adjacency reduces parameter duplication, backhaul signaling, and repeated neighborhood discovery (Jiang et al., 9 Jul 2025).

4. Bi-level mirror mechanism and heterogeneous attention

The defining architectural move in the ISAC formulation is a bi-level iterative structure that alternates between two interdependent subproblems (Jiang et al., 9 Jul 2025). The lower level, implemented by a precoding-oriented graph $1$7, takes the current mode vector $1$8 and association matrix $1$9 as given and produces AP embeddings that are read out into precoders $$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$0. The upper level, implemented by an association-oriented graph $$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$1, takes locally optimal precoders as given and updates AP mode selection and user association. The two heterogeneous GATs use the same $$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$2 but differ in message-passing direction and readout: $$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$3 is edge-to-node and precoding-oriented, while $$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$4 is node-to-edge or edge-centric and scheduling-oriented.

For a node $$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$5 of type $$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$6 with current representation $$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$7, Mirror-GAT first computes a type-specific projection

$$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$8

For a neighbor $$e_{ij} = \mathrm{LeakyReLU}\!\left(\mathbf a^\top [W h_i \Vert W h_j]\right),$$9 connected via relation $$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$0, attention incorporates both node states and relation-specific edge features. The summary gives the node-centric update as

$$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$1

followed by

$$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$2

Residual updating is written as $$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$3, and the architecture uses multi-head attention with either concatenation or averaging, residual connections, and layer normalization to stabilize training (Jiang et al., 9 Jul 2025).

The mirror mechanism is not merely shared topology. After each pair of forward passes through $$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$4 and $$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$5, reconstructed echo features from a fixed 3D-CNN estimator are fed back to refresh node and edge features for the next iteration. The paper describes this as closing an optimization-to-estimation loop. At the association level, the model outputs soft association weights $$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$6 subject to $$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$7, and soft Tx-mode scores $$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$8 subject to $$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal N_i} \exp(e_{ik})},$$9. Training uses these soft variables in the loss, while inference hardens them by

$$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$0

The paper notes that the implementation uses Top$$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$1 at selection steps while maintaining the differentiable path through attention weights in backpropagation (Jiang et al., 9 Jul 2025).

5. Optimization, training procedure, and computational profile

The ISAC Mirror-GAT is formulated as a bilevel optimization problem unified with supervised sensing estimation (Jiang et al., 9 Jul 2025). At the lower level, with fixed $$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$2 and $$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$3, the model optimizes precoders by

$$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$4

subject to $$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$5 and $$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$6 for all transmitting APs. At the upper level, with fixed $$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$7, the model optimizes $$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$8 and $$h_i' = \sigma\!\left(\sum_{j \in \mathcal N_i} \alpha_{ij} W h_j\right).$$9 under the same sensing objective, the same rate constraint, and cardinality constraints on AP modes and user associations.

The unified training loss is

$\alpha=0.2$0

The source adds that one may in practice use $\alpha=0.2$1 to avoid negative penalties, although the paper writes the linear penalty term. Ground-truth positions and velocities supervise the sensing losses, whereas AP modes and associations are not labeled and are learned end-to-end from the joint sensing-plus-rate objective (Jiang et al., 9 Jul 2025).

Training is staged. First, the 3D-CNN estimator $\alpha=0.2$2 is pretrained on supervised position and velocity targets using the sensing loss alone, with random $\alpha=0.2$3 and without rate constraints. Second, the pretrained 3D-CNN is frozen and Mirror-GAT is trained end-to-end on $\alpha=0.2$4, alternating $\alpha=0.2$5 and $\alpha=0.2$6 for $\alpha=0.2$7 mirror iterations per batch. The pseudo-code in the source specifies, for each mirror iteration, shared-adjacency construction, a $\alpha=0.2$8-layer pass through $\alpha=0.2$9, a $H \in \mathbb{R}^{N \times F}$00-layer pass through $H \in \mathbb{R}^{N \times F}$01, echo reconstruction, frozen-estimator inference, backpropagation through both GATs, and feedback augmentation of the next iteration’s initial features (Jiang et al., 9 Jul 2025).

The computational profile is explicitly lightweight relative to the paper’s dynamic graph learning framework. With $H \in \mathbb{R}^{N \times F}$02 nodes, $H \in \mathbb{R}^{N \times F}$03 edges, $H \in \mathbb{R}^{N \times F}$04 attention heads, and head dimension $H \in \mathbb{R}^{N \times F}$05, one heterogeneous GAT layer costs $H \in \mathbb{R}^{N \times F}$06 for attention coefficients and $H \in \mathbb{R}^{N \times F}$07 for aggregation and projections, so a stack of $H \in \mathbb{R}^{N \times F}$08 layers costs $H \in \mathbb{R}^{N \times F}$09. The full Mirror-GAT iteration therefore scales as

$H \in \mathbb{R}^{N \times F}$10

The paper contrasts this with a dynamic framework whose cost scales roughly as $H \in \mathbb{R}^{N \times F}$11 plus the 3D-CNN. Backhaul complexity is also given explicitly:

$H \in \mathbb{R}^{N \times F}$12

for the dynamic framework, versus

$H \in \mathbb{R}^{N \times F}$13

for Mirror-GAT, measured in doubles. The reduction follows from exchanging only valid local CSI for selected Tx-APs and their $H \in \mathbb{R}^{N \times F}$14 users, together with compressed features of dimension $H \in \mathbb{R}^{N \times F}$15 (Jiang et al., 9 Jul 2025).

6. Empirical behavior, comparative position, and limitations

The reported ISAC evaluation uses carrier frequency $H \in \mathbb{R}^{N \times F}$16 GHz, bandwidth $H \in \mathbb{R}^{N \times F}$17 MHz, $H \in \mathbb{R}^{N \times F}$18 subcarriers, and $H \in \mathbb{R}^{N \times F}$19 OFDM symbols; $H \in \mathbb{R}^{N \times F}$20 APs with $H \in \mathbb{R}^{N \times F}$21 Rx-APs and $H \in \mathbb{R}^{N \times F}$22 Tx-APs per snapshot; $H \in \mathbb{R}^{N \times F}$23 users with $H \in \mathbb{R}^{N \times F}$24 associations per AP; $H \in \mathbb{R}^{N \times F}$25 targets and $H \in \mathbb{R}^{N \times F}$26 clutter scatterers; $H \in \mathbb{R}^{N \times F}$27 antennas per AP; per-AP transmit power $H \in \mathbb{R}^{N \times F}$28 dBm; noise powers $H \in \mathbb{R}^{N \times F}$29 dBm; total hidden layers $H \in \mathbb{R}^{N \times F}$30; mirror iterations $H \in \mathbb{R}^{N \times F}$31; feature compression $H \in \mathbb{R}^{N \times F}$32; dataset size $H \in \mathbb{R}^{N \times F}$33; and learning rate $H \in \mathbb{R}^{N \times F}$34 (Jiang et al., 9 Jul 2025). Under this setup, both proposed graph-learning frameworks outperform heuristic, random, and optimization-classical baselines. With moderate $H \in \mathbb{R}^{N \times F}$35, both achieve position RMSE below $H \in \mathbb{R}^{N \times F}$36 m and velocity RMSE below $H \in \mathbb{R}^{N \times F}$37 m/s. As the number of antennas $H \in \mathbb{R}^{N \times F}$38 increases, position RMSE decreases because of better angular resolution; as the number of OFDM symbols $H \in \mathbb{R}^{N \times F}$39 increases, velocity RMSE decreases because of finer Doppler resolution. Increasing the minimum-rate threshold $H \in \mathbb{R}^{N \times F}$40 increases both position and velocity RMSE, reflecting the communication–sensing trade-off (Jiang et al., 9 Jul 2025).

The comparative position of Mirror-GAT is explicitly framed as an efficiency-oriented alternative to the paper’s dynamic graph learning framework. The dynamic framework, with structural and temporal attention plus 3D-CNN processing, reaches the highest accuracy because of richer temporal modeling. Mirror-GAT, however, achieves comparable accuracy with fewer parameters, lower inference time, and significantly reduced signaling through shared adjacency and local-CSI exchange (Jiang et al., 9 Jul 2025). Relative to standard GAT, the distinction is sharper: standard GAT does not incorporate heterogeneous relations, shared adjacency across mirrored subproblems, or iterative echo feedback. Relative to the dual–primal mirror interpretation of DP-GCNN, the difference is that the mirror is not a line graph over edges but a paired optimization structure in which two heterogeneous GATs solve coupled subproblems on the same sparse topology (Monti et al., 2018).

Several limitations are stated directly. The ISAC formulation assumes OFDM frame-level synchronization across APs, a shared clock and Doppler model, line-of-sight AP–AP channels, and quasi-orthogonal Zadoff–Chu sensing sequences. Adjacency is shared and static during an iteration, with topology refined only across mirror iterations via echo feedback; ultra-fast dynamics may therefore require more iterations or hybrid temporal modeling. CSI is assumed known for both training and inference, and robustness to CSI errors is not explicitly evaluated. Scaling to extremely large $H \in \mathbb{R}^{N \times F}$41 may require further sparsification, such as $H \in \mathbb{R}^{N \times F}$42NN neighborhoods, and distributed training. The paper does not provide explicit numeric ablations, although it indicates qualitatively that removing mirror feedback degrades sensing accuracy, not sharing adjacency increases parameters and runtime without accuracy gains, increasing heads or layers helps until saturation, and sparser graphs reduce overhead but can hurt rate and sensing unless compensated by better mode selection (Jiang et al., 9 Jul 2025).

In the broader mirror-based lineage, the main limitation shifts from signaling to dual-graph cost. Dual–primal mirroring enriches attention by making $H \in \mathbb{R}^{N \times F}$43 a function of the entire local edge context, but the dual step scales with $H \in \mathbb{R}^{N \times F}$44, making hubs expensive and motivating localized processing rather than explicit materialization of the dual adjacency (Monti et al., 2018). Taken together, these formulations place Mirror-GAT at the intersection of two ideas: contextualization of attention through a mirrored companion domain, and architectural reuse of shared sparse structure to couple graph learning with a linked optimization or estimation process.