Identifying Latent Stochastic Differential Equations

Abstract: We present a method for learning latent stochastic differential equations (SDEs) from high-dimensional time series data. Given a high-dimensional time series generated from a lower dimensional latent unknown It^o process, the proposed method learns the mapping from ambient to latent space, and the underlying SDE coefficients, through a self-supervised learning approach. Using the framework of variational autoencoders, we consider a conditional generative model for the data based on the Euler-Maruyama approximation of SDE solutions. Furthermore, we use recent results on identifiability of latent variable models to show that the proposed model can recover not only the underlying SDE coefficients, but also the original latent variables, up to an isometry, in the limit of infinite data. We validate the method through several simulated video processing tasks, where the underlying SDE is known, and through real world datasets.

• The paper demonstrates a novel VAE-based approach to infer latent SDE dynamics from observable data using rigorous loss derivations.
• It employs Gaussian modeling with Taylor approximations and Cholesky decomposition to efficiently capture latent state transitions.
• It establishes theoretical identifiability proofs under affine transformations, supporting applications in finance, biology, and physics.

Identifying Latent Stochastic Differential Equations

The paper "Identifying Latent Stochastic Differential Equations" presents methodologies for inferring underlying latent stochastic processes from observable data, utilizing the Variational Autoencoder (VAE) framework. It extends the application of Stochastic Differential Equations (SDEs) for modeling temporal dynamics in latent spaces.

Variational Autoencoder Framework for SDEs

Calculating the VAE Loss

The VAE loss is a critical component in the model, where the expectation of the log-likelihood of the latent variable, conditioned on observed data, is computed using several techniques. These techniques include exact calculation, the reparameterization trick, and first-order Taylor approximations. The loss function encompasses terms derived from the likelihood of observed data given latent states, the prior over latent variables, and the transition dynamics encoded through SDEs.

Encoding Dynamics with SDEs

An SDE encodes the evolution of latent variables, characterized by drift and diffusion components. For the paper's implementation, Gaussian distributions are used, where the encoder model provides a mean and covariance matrix (captured via Cholesky decomposition). This enables tractable calculation of the latent dynamics necessary for training the VAE.

The methodology involves modeling the transition of latent states using Gaussian priors, where the covariance includes a noise term modulated by a hyper-parameter $\nu$. Key approximations include estimating expectations using Taylor series and addressing conditional independencies between time steps.

Theoretical Contributions

Identifiability of the Generative Model

The identifiability of the generative process, particularly in the context of latent variable models using SDEs, is crucial. The paper outlines conditions under which the latent space and associated temporal dynamics can be uniquely recovered, subject to affine transformations. These theoretical results determine when a given observation space can be reconstructively mapped to the latent space in a manner invariant to affine transformations.

Furthermore, the exploration includes proofs of identifiability under specific model configurations and conditions, ensuring that the latent variables capture intrinsic temporal dynamics systematically.

Lemmas and Propositions

Explicit proofs explore the properties of the model, such as mutual information decompositions and parametric identifiability related to latent dynamics and variational encodings. These proofs address how the VAE framework can decompose complex observational data into interpretable latent representations.

Implications and Applications

The integration of SDEs with VAEs represents a powerful approach to modeling temporally structured latent variables, with notable applications in fields such as finance, systems biology, and physics, where systems evolve in a stochastic manner. Understanding the latent space dynamics could enhance predictive modeling and provide insights into the underlying processes of observed phenomena.

The framework allows for both theoretical exploration and empirical validation, enabling practitioners to deploy these methodologies in real-world scenarios where system dynamics are uncertain or partially observed.

Conclusion

The paper provides a comprehensive examination of integrating SDEs within the VAE framework, emphasizing the identifiability of latent temporal dynamics and the efficient computation of the VAE loss. The theoretical underpinnings support practical applications across various domains where modeling latent dynamics is essential. Potential future directions involve exploring more complex forms of drift and diffusion processes and extending the methodology for broader classes of SDEs.