Discovering and decoding latent mean-field structure with variational autoencoders

Technical Summary: Discovering and Decoding Latent Mean-Field Structure with Variational Autoencoders

Problem Statement
Generative models, particularly Variational Autoencoders (VAEs), are increasingly employed to capture correlations in many-body systems across physics and neuroscience. However, the representations learned by these models often remain opaque to physical interpretation. A central challenge in the natural sciences is predicting the behavior of systems with complex correlations among constituents, where the joint probability distribution $p(x)$ is non-factorizable. While machine learning tools offer powerful heuristics for identifying collective variables, there is a lack of established criteria determining when and why these approaches succeed or fail. Specifically, it remains unclear under what conditions a low-dimensional latent representation can faithfully reconstruct the joint distribution of a correlated system.

Methodology
The authors establish a theoretical framework linking VAEs to finite-size mean-field theory in statistical physics. The core of the methodology involves:

1. Structural Equivalence: The paper demonstrates that the conditional independence assumption inherent in VAE decoders—where $p_\theta(x|z) = \prod_i p_\theta^{(i)}(x_i|z)$—is structurally identical to a finite-size mean-field factorization. Unlike standard mean-field approximations that assume a deterministic order parameter (valid only in the thermodynamic limit $N \to \infty$), the VAE formulation retains a stochastic latent variable $z$, allowing it to describe non-zero correlations $\langle x_i x_j \rangle - \langle x_i \rangle \langle x_j \rangle \neq 0$ in finite systems.
2. Capacity Criterion: The authors derive a necessary condition for a VAE to successfully model a system. By comparing the information rate of the latent channel ($R$) to the bipartite mutual information of the data ($I_{bip}(p)$), they establish the bound $R \gtrsim I_{bip}(p)$. If the system's correlations require more information than the latent space can encode (e.g., systems without a mean-field description), the VAE will necessarily fail to reconstruct the joint distribution, regardless of hyperparameter tuning.
3. Success Metric: To quantify reconstruction fidelity, the authors introduce the conditional total correlation $TC|z = D_{KL}[p(x|z) || \prod p(x_i|z)]$. A successful VAE should yield $TC|z \approx 0$, indicating that the latent variable $z$ has absorbed all relevant correlations.
4. Parameter Recovery: The framework posits that if a VAE successfully reconstructs the data, the microscopic parameters of the underlying mean-field theory can be directly read from the trained decoder.

Key Results
The authors validate their theory on a hierarchy of solvable models and experimental data:

• Scalar Order Parameters (Curie-Weiss vs. 2D Ising):
  • Curie-Weiss: A VAE with a 1-dimensional latent space perfectly reconstructs the joint distribution, recovering observables like magnetization, susceptibility, and the Binder cumulant across all temperatures. The conditional total correlation is indistinguishable from zero.
  • 2D Ising: The VAE fails to reconstruct the 2D Ising model (which lacks a mean-field description). While single-point observables may be approximated, the conditional total correlation grows significantly near the critical temperature, and the model fails to capture two-point correlation functions.
• Vector Order Parameters (Hopfield Model):
  • A VAE with a latent dimension matching the number of stored patterns ($P=4$) successfully reconstructs the Hopfield model from equilibrium samples alone.
  • Crucially, the authors demonstrate C3: by analyzing the Jacobian of the decoder's logit space, they recover the exact microscopic pattern matrix $\xi$ with 100% accuracy, effectively solving the inverse-Ising problem without prior knowledge of the couplings.
• Tensor Order Parameters (Maier-Saupe):
  • Applied to the nematic-isotropic phase transition, a VAE with a 5-dimensional latent space (matching the degrees of freedom of the nematic tensor) correctly reproduces the first-order transition and order parameters. The latent variables are shown to map directly to the physical nematic tensor.
• Experimental Application (Retinal Recordings):
  • Applied to salamander retinal recordings ($N=40$ ganglion cells), a 2-latent VAE reproduces population statistics (word rates and overlap distributions) that independent models fail to capture.
  • The trained decoder reveals two effective collective variables (stored patterns) and an external field. The resulting model is a generalized Hopfield model that captures higher-order interactions in the tails of the distribution, suggesting a storage capacity exceeding standard quadratic Hopfield models.

Significance and Claims
The paper claims three primary consequences derived from the equivalence between VAEs and finite-size mean-field theory:

1. Failure Mode Identification: Low-dimensional VAEs with factorizable decoders provably fail on systems that do not admit a mean-field description (e.g., 2D Ising). This provides a diagnostic tool to identify when a dataset lacks a low-dimensional collective variable structure.
2. Evidence for Latent Structure: A successful VAE reconstruction serves as direct evidence that the system admits a latent mean-field theory. The success of the model implies the existence of a low-dimensional manifold organizing the correlations.
3. Decoding Microscopic Parameters: When a successful reconstruction occurs, the microscopic parameters of the mean-field theory (such as coupling patterns or tensor order parameters) can be explicitly recovered from the trained decoder network.

The authors conclude that this work bridges the gap between generative machine learning and statistical physics, transforming VAEs from "black box" tools into interpretable frameworks for discovering and quantifying the latent mean-field structures governing complex many-body systems.