Integrating Random Effects in Variational Autoencoders for Dimensionality Reduction of Correlated Data (2412.16899v2)

Abstract: Variational Autoencoders (VAE) are widely used for dimensionality reduction of large-scale tabular and image datasets, under the assumption of independence between data observations. In practice, however, datasets are often correlated, with typical sources of correlation including spatial, temporal and clustering structures. Inspired by the literature on linear mixed models (LMM), we propose LMMVAE -- a novel model which separates the classic VAE latent model into fixed and random parts. While the fixed part assumes the latent variables are independent as usual, the random part consists of latent variables which are correlated between similar clusters in the data such as nearby locations or successive measurements. The classic VAE architecture and loss are modified accordingly. LMMVAE is shown to improve squared reconstruction error and negative likelihood loss significantly on unseen data, with simulated as well as real datasets from various applications and correlation scenarios. It also shows improvement in the performance of downstream tasks such as supervised classification on the learned representations.

Integrating Random Effects in Variational Autoencoders for Dimensionality Reduction of Correlated Data

The paper discusses LMMVAE, a novel approach that integrates the concept of random effects into the framework of Variational Autoencoders (VAEs) to address the dimensionality reduction of correlated datasets. This integration is inspired by linear mixed models (LMM), which traditionally address correlation in data through fixed and random effects. The primary innovation in LMMVAE is the separation of the latent space into fixed and random parts to better capture data dependencies, such as those arising due to spatial, temporal, or clustering factors.

Overview

Standard VAEs assume data independence, which is often violated in real-world datasets displaying significant correlations due to shared environments or sequential measurements. LMMVAE aims to decompose the latent space typical to VAEs by associating the fixed component with independent latent variables while introducing a correlated random component for datasets exhibiting any systematic correlation structure.

The architecture of LMMVAE allows the random effects to be modeled using matrix-normal distributions, where separate encoders handle the fixed and random components, thus accommodating the dependencies between observations. The framework defines an evidence lower bound (ELBO) adjusted for both fixed and random effects, which aids in achieving more nuanced dimensional reduction in the presence of inherent data correlations.

Experimental Results

The experiments demonstrate robust performance improvements in terms of reconstruction error and negative log-likelihood (NLL) when compared to several other methods, including traditional PCA, VAE, and even models like VRAE and SVGPVAE. Notably, LMMVAE shows significant superiority in handling high-cardinality categorical data, longitudinal data, and spatially dependent data. Moreover, LMMVAE effectively separates latent variable contributions from random effects, as evidenced by simulated datasets and diverse real-world data such as the UK Biobank and CelebA.

Another highlight comes from a downstream analysis showing that the reduced-dimensional latent variables from LMMVAE also outperform those from other methods in classification accuracy, suggesting potential utility beyond just data reconstruction.

Implications and Future Directions

The incorporation of random effects into VAEs offers a significant methodological advancement for machine learning, particularly in fields like biostatistics, finance, and geospatial analytics, where correlated data is ubiquitous. The method provides improved model interpretability and robustness in capturing the essential data structure, which traditional approaches might overlook due to simplifications assuming data independence.

Future research could extend this model to higher-dimensional image data with additional external features, further enhancing LMMVAE's applicability. Additionally, exploring different forms of random effects beyond matrix-normal distributions could broaden LMMVAE's adaptability across various domains of machine learning applications.

In conclusion, LMMVAE sets a benchmark for handling correlation in dimensionality reduction, bridging the gap between statistical modeling and modern deep learning paradigms. The results suggest it could redefine best practices in scenarios where traditional models fall short due to their limiting assumptions of data independence.