Geometry-Aware Latent Representation

• Geometry-aware latent representation is a framework that endows latent spaces with a learned Riemannian metric, reflecting both local curvature and global topology.
• Decoder ensembles estimate an expected pull-back metric that robustly quantifies uncertainty, ensuring reliable, semantically meaningful interpolations across the manifold.
• Geodesic computations based on this expected metric yield stable latent trajectories and visualizations that respect the true data topology in generative models.

A geometry-aware latent representation encodes the latent variables of a model within a space intrinsically endowed with meaningful geometric structure, typically via a learned Riemannian metric that captures both local data-manifold curvature and global topological features. Such representations are foundational for differential geometric analysis, semantically faithful interpolation, and robust generative modeling in deep latent variable models.

1. Mathematical Foundation: The Pull-Back Riemannian Metric

In the context of deep generative models, let the data manifold $M \subset X$ be parametrized by latent variables $z \in Z \subset \mathbb{R}^d$, with a (potentially nonlinear) decoder $D: Z \to X$, $z \mapsto D(z)$. Geometry-awareness is achieved by endowing $Z$ with a Riemannian metric via the pull-back construction: $g(z) = J_D(z)^{T} J_D(z),$ where $$J_D(z) = \frac{\partial D}{\partial z}|_{z} \in \mathbb{R}^{n \times d}$$ is the Jacobian of the decoder. The pull-back metric measures infinitesimal latent-space distances that match Euclidean displacements on the data manifold: $$\langle \dot z, \dot z \rangle_{g(z)} = \dot z^{T} g(z) \dot z = \| \mathrm{D}D(z)[\dot z] \|^2.$$ This geometric structure enables the definition of geodesics—curves of minimal length with respect to $g(z)$—providing a rigorous notion of interpolation aligned with data semantics.

2. Topological Mismatch and Uncertainty: Motivating Decoder Ensembles

In practice, the data manifold $M$ is often compact and may exhibit nontrivial topology (holes, disconnected components) that Euclidean latent spaces cannot faithfully capture. This creates a structural mismatch: the pull-back metric $g(z)$ may poorly reflect true uncertainty in regions unsupported by data, leading to unreliable geodesics and interpolations.

To address this, let uncertainty act as a proxy for topological features. However, single-decoder models lack a principled notion of uncertainty. Previous solutions employed heuristic variance inflation (e.g., RBF kernels), which are fragile and do not scale to high latent dimensions.

By constructing ensembles of decoders $\{D_i\}_{i=1}^K$, with each $D_i \sim q(\theta)$ sampled from the posterior over parameters, model uncertainty in the decoder is captured robustly. The expected pull-back metric over the ensemble,

$$\bar g(z) = \frac{1}{K}\sum_{i=1}^K J_{D_i}(z)^{T}J_{D_i}(z),$$

inflates in data-sparse regions (where decoders disagree), naturally revealing the latent space's topological complexity and providing an empirically meaningful measure of "geometric uncertainty."

3. Geodesic Computation under the Expected Metric

The Riemannian geodesic $\gamma: [0,1] \to Z$, minimizing

$$\mathrm{Length}[\gamma] = \int_0^1 \sqrt{\dot\gamma(t)^\top \bar g(\gamma(t))\,\dot\gamma(t)}\,dt,$$

encodes the shortest latent-space path reflecting the true manifold geometry. The geodesic ODE, in local coordinates, is

$$\ddot z^a + \Gamma^a_{bc}(z)\,\dot z^b\,\dot z^c = 0,$$

with Christoffel symbols

$$\Gamma^a_{bc}(z) = \tfrac12\,\bar g^{a e}(z)\left( \partial_b \bar g_{e c}(z) +\partial_c \bar g_{e b}(z) -\partial_e \bar g_{b c}(z) \right).$$

Numerical implementation:

• The geodesic path is discretized by $T+1$ spline control points $\{\gamma_t\}_{t=0}^T$ with endpoints $\gamma_0 = z_0, \gamma_T = z_1$.
• The discrete energy is

$$\mathcal{E}[\gamma] \approx \sum_{t=0}^{T-1} \mathbb{E}_{i,i'}\bigl\|D_i(\gamma_{t+1})-D_{i'}(\gamma_t)\bigr\|^2,$$

where the ensemble expectation penalizes regions of high decoder disagreement.

• Smoothness is ensured via cubic-spline parameterization; optimization is conducted by gradient descent on the discretized energy, dropping cross-covariance terms for stability.

This procedure yields geodesics that "hug" the data manifold and avoid navigating holes or disconnected components—an ability absent in geometry-unaware baselines.

4. Empirical Evaluation and Comparison: Robustness and Fidelity

Experimental protocol spans MNIST and FashionMNIST for $d=2$ and $d=50$, comparing decoder ensemble geometry to an RBF-heuristic variance baseline [Arvanitidis et al., 2021].

Interpolation Quality:

• Geodesic interpolation between test latent codes produces decodings with semantically smooth transitions; ensemble geodesics reliably avoid "empty" latent regions where single-decoders may cut through, leading to more realistic outputs.

Stability Across Seeds:

• Repeated training with 30 random seeds shows that geodesic distances under the ensemble geometry exhibit substantially lower coefficient of variation (CV) than the RBF baseline (paired one-sided Student’s $t$-test yields $p \approx 1.0$), indicating higher geometric robustness.

Visualization:

• Latent spaces colored by predictive uncertainty show ensemble geometry produces uncertainty "halos" around clusters.
• Geodesic grids in 2D reflect true data topology (holes, cluster boundaries).

5. Applications and Practical Considerations

Interpolation and Sampling:

• Ensemble-based geodesics enable meaningful latent interpolations for data augmentation, generative design, and visualization.
• Sampling along geodesics produces diverse, realistic samples traversing the learned manifold, respecting topological features and avoiding collapse into high-uncertainty/low-density regions.

Visualization:

• Manifold-respecting grids and geodesics in visualization tasks grant insight into cluster adjacency, topology, and separation.

Limitations and Scalability:

• Ensemble methods (size $K$) incur additional computational and memory overhead, scaling linearly with $K$.
• Monte Carlo (MC) variance in estimating $\bar g(z)$ may necessitate large $K$ or advanced variance reduction.
• Neglecting cross-covariances in the energy computation is a heuristic; a more complete treatment using full covariance would enhance geometric accuracy.

Potential extensions:

• Use of diverse ensembling (bootstrap, Bayesian NNs, tempered posteriors).
• Extension to higher-dimensional latent spaces via low-rank metric approximations or graph-based geodesic solvers.
• Integration with topological priors or non-Euclidean latent constructions (e.g., multi-chart, toroidal) for manifolds with global structure irreducible to Euclidean charts.

6. Theoretical and Methodological Context

The decoder-ensemble approach generalizes previous Riemannian latent geometry [e.g., "Pulling back information geometry," (Arvanitidis et al., 2021)], replacing manual uncertainty heuristics with a robust, learned geometric structure that scales to deep neural models. The ensemble's expected metric is directly motivated by posterior uncertainty, bridging geometry and epistemics in the latent space. Geometric quantities—distance, volume element, curvature—become well-defined, supporting rigorous analysis, visualization, and semantically meaningful computations.

7. Summary and Outlook

The geometry-aware latent representation induced by decoder ensembles realizes a Riemannian metric on latent space that simultaneously reflects local data-curvature and topological uncertainty. Unlike heuristically designed geometric proxies, this framework is reliable and robust, offering scalable and semantically aligned geodesic computations applicable across deep generative models. The trade-off between computational resources and geometric fidelity can be calibrated via ensemble size and MC estimation strategies, with further extensions possible through Bayesian deep learning, topological regularization, and high-dimensional latent approximations. The empirical evidence confirms that this method yields stable, interpretable, and topology-respecting latent geometries, advancing the practical utility of geometric deep learning in generative modeling (Syrota et al., 14 Aug 2024).