Temporal Geometry of Deep Networks: Hyperbolic Representations of Training Dynamics for Intrinsic Explainability

Ambarish Moharil

cs.LGOct 2, 2026

Temporal Geometry of Deep Networks: Hyperbolic Representations of Training Dynamics for Intrinsic Explainability

Authors: Ambarish Moharil

Organizations: Data Governance Department Jheronimus Academy of Data Science Eindhoven University of Technology

Abstract

Intrinsic explainability remains a challenging problem, particularly in contexts where multilayer perceptrons (MLPs) require dynamic re-training within an optimization environment. This paper investigates how MLPs and their training dynamics can be represented and studied in non-Euclidean spaces; our representation features the Poincaré model of hyperbolic geometry. We aim to capture the geometric evolution of their weighted topology and self-organization over time. Instead of restricting the analysis to single checkpoints---as per established measure-based explainability methods---we construct temporal parameter graphs, i.e., snapshots over time $T$ steps of the optimization/training process for MLPs. This reflects the view that neural networks encode information not only in their weights but also in the trajectory traced during training. Drawing on the idea that many complex networks admit embeddings in hidden metric spaces where distances correspond to connection likelihood, we present a geometric and temporal graph-based metalearning framework for obtaining dynamic hyperbolic representations of the underlying neural parameter graphs. Our model embeds temporal parameter graphs in the Poincaré model ball, and learns from them while maintaining equivariance to within-snapshot neuron permutations and invariance to permutations of past snapshots. In doing so, the approach preserves functional equivalence over time and recovers the latent evolving geometry of the network. Experiments on regression and classification tasks with trained MLPs show strong meta-network performance, accompanied by hyperbolic temporal representations. This reveals how the network structure emerges over time under specific training environments, thus providing insights into the network's self-organization.

Figures & tables

Figure 1: Architecture of the Geometric Temporal Hyperbolic Graph Meta-Network (GTH-GMN).

Model	MNIST INR	Fashion-MNIST INR
MLP	17.55 $\pm$ 0.01	19.91 $\pm$ 0.47
MLP + Perm. aug	29.26 $\pm$ 0.18	22.76 $\pm$ 0.13
MLP + Alignment	58.98 $\pm$ 0.52	47.79 $\pm$ 1.03
INR2Vec (Arch.)	23.69 $\pm$ 0.10	22.33 $\pm$ 0.41
Transformer	26.57 $\pm$ 0.18	26.97 $\pm$ 0.33
DWSNets	85.71 $\pm$ 0.57	67.06 $\pm$ 0.29

Table 1: INR classification in static vs. temporal settings. Average test accuracies (

\pm

standard error) on MNIST and Fashion-MNIST INRs. Baselines are reproduced from DWSNets ( Navon et al., 2024 ) and NFNs ( Zhou et al., 2023a ) . Our spatio-temporal variant uses pooled representations across

K

traces.

Table 3

Panel of Figure 2 — Figure 2: Hyperbolic embeddings of parameter graphs from a 13-layer MLP trained as an INR on CIFAR-10, shown at checkpoints $t=11$ (left) and $t=97$ (right). Embeddings are learned in $d=10$ dimensions with 4 HGAT layers and 4 EvolveGCN–H kernels (tangent dim. 128) and projected to the Poincaré disk via PCA. Nodes are colored by layer index, with edges in gray, illustrating the progressive reorganization and separation of layers over training.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter Graphs & Features
$G_{t}$	Parameter-graph snapshot at epoch $t$
$V,\ E_{t}$	Node set; edge set at epoch $t$
$w^{\star}_{uv}$	Signed edge label on $\{u,v\}$
$e_{uv}$	Edge context (bias/kernel, interlayer flag)
$L,\ d,\ c$	#layers; embedding dim; curvature parameter
$\ell(u)$	Layer label of node $u$

Appendix

Table 5

Figure 3: Predicted versus actual generalization performance on CIFAR-10 checkpoint traces. Each point corresponds to a model trial at a given epoch, where predictions are obtained from the learned GTH-GMN embeddings. The strong monotonic trend (Kendall’s $\tau=0.846\pm 0.004$ ) indicates that hyperbolic embeddings capture meaningful structure in the parameter dynamics, even if absolute correlation lags slightly behind NFN baselines.

Panel of Figure 4 — Figure 4: Relationship between hyperbolic distance $d$ and edge weights $w$ on CIFAR-10 INR traces, shown as log-count hexbins. (Top Left, Bottom Left) Distance versus absolute weight $|w|$ (at $t\in T$ ), highlighting the concentration of large weights at small distances and the rapid decay at larger distances. (Top Right, Bottom Right) Distance versus log-magnitude $\log|w|$ (at $t^{\prime}\in T$ ), which reveals an approximately linear inverse trend, consistent with a power–law dependence of weight magnitudes on hyperbolic distance. Together, these plots illustrate that the hyperbolic geometry learned by GTH-GMN aligns distances with functional weight strengths, capturing both edge existence and graded magnitude in a geometrically coherent manner.

Component	Hyperparameter	Range / Value
Manifold	Curvature $\|c\|$	fixed to $1.0$
	Embedding dimension $d$	$80\;-\;125$
HGAT (per layer)	# layers $L$	$4$
	# heads $H$	$4\;-\;8$ (tuned)
	Head temperature	learned, init $0.5\;-\;1.0$
	Residual gate	learned scalar $\in(0,1)$

Appendix

Table 5: Architecture and data settings for our Experiments.

Component	Hyperparameter	Range / Value
Euclidean params	Optimizer / LR	Adam, $\text{lr}=1\!\times\!10^{-3}\;-\;3\!\times\!10^{-3}$
(kernels, heads)	Weight decay	$10^{-6}\;-\;10^{-4}$
	LR schedule	StepLR ( $\text{step}=10\;-\;20$ , $\gamma=0.6\;-\;0.8$ )
	Epochs	$30\;-\;50$
Manifold params	Optimizer	RAMSGrad (Riemannian)
(node positions $z$ )	LR / trust radius	$\text{lr}=1\!\times\!10^{-3}\;-\;3\!\times\!10^{-3}$ , $\tau=0.05\;-\;0.15$

Appendix

Table 6: Optimization and loss settings for INR-trace image classification. Ranges correspond to grid or random search over hyperparameters.

\nabla^{\mathrm{Riem}}_{z^{(t)}_{i}}\mathcal{L}=\lambda_{z^{(t)}_{i}}^{-2}\,\nabla^{\mathrm{Euc}}_{z^{(t)}_{i}}\mathcal{L}

Appendix

Algorithm 1 Intrinsic Riemannian Optimization of Node Positions on the Poincaré Ball

Panel of Figure 5 — Figure 5: Hyperbolic embeddings of parameter graphs from a 25-layer MLP trained as an INR on CIFAR-10, shown at checkpoints. Embeddings are learned in $d=10,25$ dimensions with 4 HGAT layers and 4 EvolveGCN–H kernels (tangent dim. 128) and projected to the Poincaré disk via PCA. Nodes are colored by layer index, with edges in gray, illustrating the progressive reorganization and separation of layers over training.

Variant	Question	$L_{\text{FD}}$	$L_{\text{wrec}}$	$L_{\text{sign}}$	$L_{\alpha}$	$L_{\text{rank}}$	$L_{\text{smooth}}$
Full objective	–	✓	✓	✓	✓	✓	✓
No ranking loss	(1)	✓	✓	✓	✓	✗	✓
No ranking & no smoothness	(2)	✓	✓	✓	✓	✗	✗
No signed regression head	(3)	✓	✗	✗	✗	✓	✓

Appendix

Table 7: Ablation scenarios and their relation to the three guiding questions. Each row indicates which loss terms are active in the corresponding variant.

Condition	Metric	MNIST Value	Fashion-MNIST Value
Full Objective	$\tau$	$0.946$	$0.935$
Full Objective	Slope ( $b$ )	${-0.617}$	${-0.322}$
No Ranking Loss	$\tau$	$0.811$	$0.797$
No Ranking Loss	Slope ( $b$ )	${-0.473}$	${-0.189}$
No Rank/Smooth	$\tau$	$0.765$	$0.764$
No Rank/Smooth	Slope ( $b$ )	${-0.365}$	${-0.061}$

Appendix

Table 8: Ablation Study Table

Panel of Figure 6 — Figure 6: Kendall $\tau$ over the full objective for the INR classification task on MNIST and Fashion-MNIST datasets. GTH-GMN with full objective achieves a Kendall’s $\tau$ of ${0.946}$ on MNIST and ${0.935}$ on Fashion–MNIST datasets.

Panel of Figure 8 — Figure 8: Kendall $\tau$ over the ”No ranking loss” condition for the INR classification task on MNIST and Fashion-MNIST datasets. GTH-GMN without $\mathcal{L}_{\text{rank}}$ achieves a Kendall’s $\tau$ of $0.811$ on MNIST and $0.797$ on Fashion–MNIST datasets, indicating a noticeable degradation in the monotonicity of the predictive signal compared to the full objective.

Panel of Figure 9 — Figure 9: Analysis of the relationship between hyperbolic distance ( $d$ ) and edge weights for the ”No ranking loss” condition. The top row displays results for MNIST, while the bottom row shows Fashion-MNIST. For each dataset, the left image shows the hexbin plot of distance vs. $\log|w|$ with a fitted linear regression line (dashed orange). The right image shows the hexbin plot of distance vs. the raw weight magnitude $|w|$ . The fitted slope for MNIST in this condition ( $-0.473d$ ) indicates a moderate decay of edge weights, which is substantially shallower than the full objective model, while the decay for Fashion-MNIST ( $-0.189d$ ) is notably weaker. This indicates that removing the ranking loss leads to a loss of the crisp, monotone geometric ordering.

Panel of Figure 10 — Figure 10: Kendall $\tau$ over the ”No ranking and no temporal smoothness” condition for the INR classification task on MNIST and Fashion-MNIST datasets. GTH-GMN without both $\mathcal{L}_{\text{rank}}$ and $\mathcal{L}_{\text{smooth}}$ achieves a Kendall’s $\tau$ of $0.765$ on MNIST and $0.764$ on Fashion–MNIST datasets, indicating a further decline in the ordering coherence of the predictive signal.

Panel of Figure 11 — Figure 11: Kendall $\tau$ over the ”No signed regression head” condition for the INR classification task on MNIST and Fashion-MNIST datasets. GTH-GMN without the signed head achieves a Kendall’s $\tau$ of $0.738$ on MNIST and $0.726$ on Fashion–MNIST datasets, representing the strongest degradation in the geometric ordering of the predictive signal.

Panel of Figure 12 — Figure 12: Analysis of the relationship between hyperbolic distance ( $d$ ) and edge weights for the ”No ranking and no temporal smoothness loss” condition. The top row displays results for MNIST, while the bottom row shows Fashion-MNIST. For each dataset, the left image shows the hexbin plot of distance vs. $\log|w|$ with a fitted linear regression line (dashed orange). The right image shows the hexbin plot of distance vs. the raw weight magnitude $|w|$ . The fitted slope for MNIST in this condition ( $-0.365d$ ) indicates a substantially shallower decay of edge weights compared to the full model, while the decay for Fashion-MNIST is minimal ( $-0.061d$ ), reflecting a loss of coherent geometric structure when both regularization terms are removed.

Panel of Figure 13 — Figure 13: Analysis of the relationship between hyperbolic distance ( $d$ ) and edge weights for the ”No signed regression head” condition. The top row displays results for MNIST, while the bottom row shows Fashion-MNIST. For each dataset, the left image shows the hexbin plot of distance vs. $\log|w|$ with a fitted linear regression line (dashed orange). The right image shows the hexbin plot of distance vs. the raw weight magnitude $|w|$ . The fitted slope for MNIST in this condition ( $-0.211d$ ) is near-zero, and the slope for Fashion-MNIST ( $+0.125d$ ) is positive , indicating that the expected decay of weights with distance has collapsed or reversed. This confirms that the signed head is the primary mechanism binding the hyperbolic geometry to the functional magnitude of the weights.

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