Nonlinear multi-study factor analysis

TL;DR

A nonlinear multi-study factor model, which allows for both shared and specific factors, is proposed, and a multi-study sparse variational autoencoder is proposed, which helps separate the shared factors from the group-specific factors.

Abstract

High-dimensional data often exhibit variation that can be captured by lower dimensional factors. For high-dimensional data from multiple studies or environments, one goal is to understand which underlying factors are common to all studies, and which factors are study or environment-specific. As a particular example, we consider platelet gene expression data from patients in different disease groups. In this data, factors correspond to clusters of genes which are co-expressed; we may expect some clusters (or biological pathways) to be active for all diseases, while some clusters are only active for a specific disease. To learn these factors, we consider a nonlinear multi-study factor model, which allows for both shared and specific factors. To fit this model, we propose a multi-study sparse variational autoencoder. The underlying model is sparse in that each observed feature (i.e. each dimension of the data) depends on a small subset of the latent factors. In the genomics example, this means each gene is active in only a few biological processes. Further, the model implicitly induces a penalty on the number of latent factors, which helps separate the shared factors from the group-specific factors. We prove that the latent factors are identified, and demonstrate our method recovers meaningful factors in the platelet gene expression data.

Figures (6)

• Figure 1: Example of a two-study sparse model: both study 1 and 2 share factor $z_1$, whereas factors $z_2,z_3$ are specific to study 1, and factors $z_4, z_5$ are specific to study 2.
• Figure 2: Gaussian nonlinear factor analysis. (a) Results over 30 replications. (b-e) Result of one replication as an example. MSSVAE successfully estimates $\bm{W}$, including factor dimension (after removing columns of all zeros).
• Figure 3: Synthetic gene expression data. (a) Results over 30 replications. (b-e) Result of one replication as an example.
• Figure 4: Summary of gene over-representation results for shared latent dimensions. For each latent dimension, we plot the top 3 biological processes by fold enrichment, after filtering for biological processes occurring in more than 10% of latent dimensions.
• Figure 5: Summary of gene over-representation results for disease-state-specific latent dimensions. Displayed are the top 5 biological processes for each latent dimension by fold enrichment, after filtering for biological processes occurring in more than 10% of latent dimensions.

Theorems & Definitions (22)

• Remark 1
• Theorem 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 2
• Remark 5
• Remark 6
• Lemma 1
• proof