Info-Geometric Graph Sequential Growth

• The paper introduces a fully-corrective coordinate descent strategy that leverages KL divergence reductions to optimally select edges in sparse graphical models.
• Approximate heuristics like Best Block Improvement and Gauss–Southwell–Lipschitz accelerate edge evaluation, ensuring scalability in high-dimensional settings.
• Empirical evaluations on Gaussian models and lane topology reconstruction demonstrate state-of-the-art performance with robust activation ranks and efficient model recovery.

Information-geometry-driven graph sequential growth encompasses regularisation-free methods for sparse graphical model inference, linking information geometry and coordinate-wise optimization on the space of precision matrices, as well as sequential graph construction for topological learning. It includes both theoretical and empirical results demonstrating the efficacy of information-geometric formulations for graph growth, information-optimal coordinate selection, and scalable approximations in both statistical inference and topological prediction tasks (Bond et al., 29 Jan 2026, Xie et al., 7 Jul 2025).

1. Information-Geometric Frameworks for Graph Growth

Let $x \sim \mathcal{N}(0, \Sigma)$ be a $d$-dimensional Gaussian with precision matrix $Q=\Sigma^{-1} \in S_+^d$. The negative log-likelihood for $Q$ with sample covariance $S$ is $f_S(Q) = \text{tr}(S Q) - \log\det Q$, which, up to an additive constant, is twice the Kullback–Leibler divergence between $\mathcal{N}(0, S)$ and $\mathcal{N}(0, Q^{-1})$. The manifold $S_+^d$ is endowed with the Fisher–Rao metric, $g_Q(H_1, H_2) = \text{tr}(Q^{-1} H_1 Q^{-1} H_2)$, where $H_1, H_2$ are tangent directions. Gradients, Hessians, and geodesic flows on this manifold admit closed-form expressions, facilitating coordinate descent procedures in graphical model learning (Bond et al., 29 Jan 2026).

In topological learning (e.g., lane graph expansion for autonomous driving), a directed graph $G=(V,E)$ models intersections ($V$) and lanes ($E$). Adjacency matrix $A$ and geometry matrix $M$ (with quadratic Bézier control points) describe connectivity and shape. The graph is grown sequentially, node-by-node, mirroring incremental expansion on matrix manifolds (Xie et al., 7 Jul 2025).

2. Exact and Approximate Sequential Graph Growth Algorithms

The fully-corrective (information-optimal) strategy seeks, at each step, the edge whose inclusion yields maximal reduction in $f_S$, subject to graphical constraints. Formally, for edge set $E$, the constrained loss $\widetilde{f}_S(G)$ is minimized over symmetric positive-definite $Q$ with sparsity pattern $D \cup E$. Optimal updates align with the activation of off-diagonal coordinates in $Q$’s basis (Bond et al., 29 Jan 2026):

• Fully-corrective coordinate descent (BFCI): At each iteration, select the edge $e$ maximizing the KL-divergence decrease $$I_S^{\mathrm{FC}}(Q, e) = f_S(Q) - \widetilde{f}_S(G \cup \{e\})$$.

This process constructs a sparse graph corresponding to the precision matrix’s zero pattern while being statistically optimal at each activation. Complexity is $O(M^2 \cdot C_{\mathrm{full}})$ per outer step due to repeated completions of partial precision matrices.

To scale, efficient approximations are introduced:

• Best Block Improvement (BBI): For $Q$ and candidate $(i,j)$, evaluate $I_S^{B}(Q, (i,j))$ using 2×2 blocks of $S$ and $Q^{-1}$.
• Gauss–Southwell–Lipschitz (GSL): For each basis direction, maximize $[Df_S(Q)[B(i,j)]]^2 / D^2f_S(Q)[B(i,j), B(i,j)]$.

Both approximate selection heuristics require only $O(M)$ block evaluations per iteration. Update steps involve restricted coordinate descent, using either cyclic, Gauss–Southwell, or block-wise updates.

In lane topology reconstruction, incremental graph expansion is cast into a chain of adjacency and geometry matrix updates with each addition of a node. This process is deterministically serialized (e.g., DFS order) and encoded as a sequence for autoregressive modeling (Xie et al., 7 Jul 2025).

3. Serialization and Autoregressive Modeling

Sequential graph growth enables serialization of the graph-expansion process for deep learning architectures:

• Sequence construction: Each node addition is described by its coordinates, incoming/outgoing edges, and associated Bézier control points. Ordered token sequences encode the full structure within a fixed vocabulary.
• Autoregressive transformer: A BEV-encoded feature map feeds a cross-attentive transformer decoder, which outputs the sequence auto-regressively: $$p(S|\mathrm{FEAT}) = \prod_{t=1}^T p(x_t | x_{<t},\mathrm{FEAT})$$, where each block corresponds to the addition of one node and incident edges.

The sequence order (DFS is most effective) and tokenization scheme (quantized bins for coordinates and control points) establish the transformer input/output for lane graph learning (Xie et al., 7 Jul 2025).

4. Activation Ranks and Edge Relevance

In information-geometry-driven sequential graph growth, the concept of activation rank indexes the iteration $k(e)$ at which edge $e$ is added to the growing graph. Edges with smaller ranks correspond to larger immediate decreases in $f_S$ and are thus more “informationally relevant.” Distributions of activation ranks, visualized via boxplots over bootstrap or subsample replicates, serve as a stability diagnostic akin to classical stability selection. This supports model selection by indicating which edges reliably have high informational value and discriminates true versus spurious connections (Bond et al., 29 Jan 2026).

5. Theoretical Guarantees and Recovery

Within fixed support, cyclic or Gauss–Southwell coordinate descent on $f_S(Q)$ converges linearly (Theorem 4.1) as:

$$f_S(Q^{(t)}) - f_S(Q_{S,E}) \leq (1-\mu/(mL))^t \left[f_S(Q^{(0)})-f_S(Q_{S,E})\right]$$

where $m = |D \cup E|$ and $L$, $\mu$ are determined by the smallest/largest eigenvalues of a compact sublevel set $K$.

While explicit irrepresentability or incoherence bounds are not provided, information-optimal sequential growth heuristically recovers the correct graph under standard high-dimensional assumptions about restricted eigenvalues and minimum signal strength—for the first $|E_{\text{true}}|$ activations. Exact rates remain an open research question (Bond et al., 29 Jan 2026).

6. Empirical Evaluation and Applications

Empirical studies encompass both statistical graphical model selection and topological reconstruction in computer vision contexts:

• Gaussian Graphical Model Benchmarks: On synthetic, block, and genomics data, BFCI (exact), BBI, and GSL methods outperform classical penalized approaches (Glasso) in recovering true edges with fewer false positives and demonstrate linear scaling and tuning-free operation. Activation-rank diagnostics provide insight for edge inclusion (Bond et al., 29 Jan 2026).
• Lane Topology Reconstruction: SeqGrowGraph autoregressively expands lane graphs in nuScenes and Argoverse 2 datasets, achieving superior performance on landmark F1 and reachability F1 scores compared to prior baselines (Toponet, LaneGAP, RNTR):

Method	L-F (nuScenes)	R-F (nuScenes)
TopoNet	49.6	17.5
LaneGAP	53.2	47.5
RNTR	48.9	52.8
LaneGraph2Seq	45.2	46.2
SeqGrowGraph	56.4	67.8

SeqGrowGraph also achieves state-of-the-art centerline matching and connectivity F1 metrics, robustly outperforming alternatives (Xie et al., 7 Jul 2025).

7. Significance and Outlook

Information-geometry-driven sequential graph growth establishes a unified geometric and optimization-theoretic basis for both graphical statistical inference and sequential topological prediction. It offers parameter-free, computationally tractable selection rules, interpretable edge relevance diagnostics, and compatibility with gradient/coordinate-based, as well as deep autoregressive, learning paradigms. The approach is validated across statistical and computer vision domains, with theoretical and empirical guarantees in sparse regime performance and model interpretability. Further investigation into recovery theory and scaling to larger graphs remains an active direction (Bond et al., 29 Jan 2026, Xie et al., 7 Jul 2025).