Navigating the Latent Space Dynamics of Neural Models (2505.22785v2)

Abstract: Neural networks transform high-dimensional data into compact, structured representations, often modeled as elements of a lower dimensional latent space. In this paper, we present an alternative interpretation of neural models as dynamical systems acting on the latent manifold. Specifically, we show that autoencoder models implicitly define a latent vector field on the manifold, derived by iteratively applying the encoding-decoding map, without any additional training. We observe that standard training procedures introduce inductive biases that lead to the emergence of attractor points within this vector field. Drawing on this insight, we propose to leverage the vector field as a representation for the network, providing a novel tool to analyze the properties of the model and the data. This representation enables to: (i) analyze the generalization and memorization regimes of neural models, even throughout training; (ii) extract prior knowledge encoded in the network's parameters from the attractors, without requiring any input data; (iii) identify out-of-distribution samples from their trajectories in the vector field. We further validate our approach on vision foundation models, showcasing the applicability and effectiveness of our method in real-world scenarios.

• The paper introduces the idea that iteratively applying the encoder-decoder composition forms a latent vector field with attractor points representing high-density data modes.
• It demonstrates that standard regularization techniques enforce local contractiveness in AutoEncoders, influencing the balance between memorization and generalized feature representation.
• It shows practical applications by leveraging latent dynamics for data-free weight probing and out-of-distribution detection through trajectory analysis.

The paper "Navigating the Latent Space Dynamics of Neural Models" (2505.22785) introduces a novel perspective on AutoEncoder (AE) models, interpreting them as dynamical systems operating within their latent space. The core idea is that iteratively applying the composition of the decoder and encoder, denoted as $f(\mathbf{z}) = E(D(\mathbf{z}))$, defines a vector field in the latent space. The trajectories in this latent vector field represent the evolution of a latent code under repeated mapping through the AE.

The paper demonstrates that standard training procedures for AEs, which often include regularization techniques like weight decay, bottleneck constraints, sparsity penalties, or denoising objectives, implicitly enforce local contractiveness in the learned mapping $f$. This contractiveness ensures the existence of attractor points within the latent vector field. These attractors are fixed points $\mathbf{z}^*$ such that $f(\mathbf{z}^*) = \mathbf{z}^*$. The authors show both theoretically (under certain assumptions) and empirically that these attractors correspond to modes or regions of high probability density in the latent space distribution of the data.

The practical significance of this latent dynamics perspective lies in its ability to reveal insights about the neural model's behavior and the data distribution it has learned, often without requiring access to the original training data. Key applications explored in the paper include:

1. Analyzing Memorization vs. Generalization: The properties of the attractors and their relationship to training data points can characterize where a model sits on the spectrum between memorization (attractors closely match individual training samples) and generalization (attractors represent broader interpolations or modes of the data distribution). The paper empirically shows this transition by varying AE bottleneck dimensions or observing the evolution of attractors during training. More regularized models or models trained on less data tend to exhibit stronger memorization captured by the attractors.
2. Data-Free Weight Probing: The set of attractors derived from the latent vector field can act as a dictionary of signals encoded within the network's weights. By computing attractors starting from simple initial conditions (e.g., Gaussian noise) without using any training data, one can recover meaningful representations. The paper validates this on large vision foundation models (like the AE component of Stable Diffusion), showing that images from diverse datasets can be effectively reconstructed using sparse combinations of noise-derived attractors, outperforming reconstruction using a random orthogonal basis. This suggests the attractors capture salient features learned by the model.
3. Out-of-Distribution (OOD) Detection: The trajectories traced by samples in the latent vector field towards their attractors can be informative about the source distribution. The paper shows that the paths themselves, not just the final attractor points, carry information that distinguishes in-distribution (ID) data from OOD data. By measuring the distance of a sample's latent trajectory to the set of attractors derived from training data, one can define a score for OOD detection. Experiments on Vision Transformer Masked AEs (ViT-MAEs) demonstrate that this trajectory-based score significantly outperforms a simple K-Nearest Neighbor baseline for OOD detection on various benchmark datasets.

The paper provides theoretical grounding connecting the latent vector field to the score function of the learned distribution under local contractiveness. It also analyzes the convergence of the iterative dynamics $\mathbf{z}_{t+1} = f(\mathbf{z}_t)$ to attractors, showing it behaves like gradient descent on the reconstruction loss $\|f(\mathbf{z}) - \mathbf{z}\|^2$ only in specific cases (near isometric regions or near attractors where the Jacobian vanishes), while generally tracing nonlinear paths due to higher-order terms.

From an implementation perspective, analyzing these dynamics involves:

• Implementing the iterative application of the $E \circ D$ composition in the latent space.
• Computing attractors by iterating this map until convergence or a maximum number of steps. Convergence criteria typically involve checking the change in $\mathbf{z}$ between iterations ($\| \mathbf{z}_{t+1} - \mathbf{z}_t \|$).
• Analyzing the properties of the computed attractors (e.g., decoding them and comparing to data) or the trajectories (e.g., distances to attractor sets).
• For data-free probing, initializing the iterations from random noise samples in the latent space.
• For OOD detection, comparing trajectories of test samples to those of training/known ID samples.

The paper highlights that the latent space of trained AEs is not merely a static embedding but a dynamic space shaped by the network's architecture and training objectives, providing a rich structure for analysis and practical applications. Limitations include the current focus on AE-like models and the need for further research into generalizing this perspective to other network architectures and objectives.