A Deep Dive into Linear vs. Non-Linear Methods for Multi-Omics Data Integration

In recent years, the analysis of multi-omics data has become increasingly vital for understanding complex biological systems and predicting disease outcomes. Multi-omics refers to the integration of diverse biological datasets—such as genomics, transcriptomics, proteomics, and metabolomics—collected from the same sample. While these rich datasets hold immense promise, the challenge lies in effectively integrating them to extract meaningful biological insights. Joint embedding methods have emerged as a solution to this problem by reducing the dimensionality of multi-omics data and projecting different modalities into a shared, lower-dimensional space.

In this blog post, we will explore the results of a recent study that compares linear and non-linear joint embedding methods for both bulk and single-cell multi-omics data. The study benchmarks various techniques for tasks like missing data imputation, downstream prediction (e.g., survival analysis and cell type classification), and understanding latent space coherence.

The Challenge of Multi-Omics Integration

As multi-omics data continues to grow in both volume and complexity, analyzing these datasets together presents both opportunities and challenges. The primary challenge is how to integrate these multiple data types into a unified model that reveals the underlying biological signals while accounting for each data modality’s unique characteristics. Joint embedding methods aim to reduce the complexity of these data types by mapping them into a shared latent space, where relationships between different modalities can be explored and analyzed.

Linear vs. Non-Linear Methods for Data Integration

Traditional methods for multi-omics integration primarily use linear techniques. These methods are grounded in well-established approaches like Principal Component Analysis (PCA), which reduces data dimensionality by projecting it into a lower-dimensional space while retaining the maximum variance. For multi-omics, linear techniques like Multi-Omics Factor Analysis (MOFA+) and Multiple Co-Inertia Analysis (MCIA) have been commonly used to analyze the covariance between different omics data modalities.

However, the rise of machine learning and deep learning, particularly neural networks, has led to the development of non-linear methods. These methods are capable of capturing more complex relationships between data modalities. For example, Variational Autoencoders (VAEs), Product of Experts (PoE), and Mixture of Experts (MoE) are non-linear models that have gained traction, especially in single-cell transcriptomics, where non-linear relationships are more likely to occur.

Key Findings from the Study

The study compared several linear and non-linear joint embedding methods, evaluating their performance on key tasks such as imputation of missing data, latent space coherence, and downstream predictive tasks like survival analysis and cell type classification. Here are some of the main takeaways:

1. Imputation: Non-Linear Methods Excel

Non-linear methods consistently outperformed linear methods when it came to imputing missing data. Methods like PoE, MoE, and Cross-Generating VAE (CGVAE) demonstrated superior imputation performance compared to linear methods such as MCIA and MOFA+. Imputation is a critical task in multi-omics analysis, as missing data are common in real-world datasets. Non-linear models were particularly successful at predicting missing modalities, making them ideal for scenarios where data completeness is not guaranteed.

2. Latent Space Coherence: A Trade-Off with Imputation

While non-linear methods excelled at imputation, they struggled with latent space coherence. Latent space coherence refers to the degree to which different data modalities align in the latent space, with similar biological entities grouped together. Non-linear methods like PoE and MoE produced less coherent latent spaces compared to simpler non-linear models like ccVAE and CGVAE. This suggests a trade-off between imputation performance and latent space coherence, where non-linear models may sacrifice coherence in favor of better handling missing data.

3. Downstream Tasks: Survival Analysis and Cell Type Classification

When evaluating the performance of these methods on downstream tasks such as survival analysis and cell type classification, the results varied depending on the availability of data modalities:

• Survival Analysis: In tasks like predicting patient survival using bulk tumor data, simply concatenating the principal components of each modality (a linear approach) proved to be a strong baseline. However, when only one modality was available during testing, non-linear methods such as PoE showed an improvement in predictive performance.
• Cell Type Classification: Similar results were observed for single-cell data analysis. When all modalities were available at the test time, linear methods performed well. However, when only one modality was available, non-linear methods such as PoE and MoE showed improved performance over the baseline.

4. PoE: A Consistently Strong Performer

Among the non-linear methods, PoE emerged as a standout performer, excelling in most tasks while maintaining a reasonable speed. This makes PoE an excellent option for many scenarios where multi-omics data integration is required.

5. UniPort: A Versatile Method for Bulk and Single-Cell Data

UniPort, originally designed for single-cell data, also performed well when applied to bulk datasets, outperforming both MOFA+ and MCIA in all tasks related to The Cancer Genome Atlas (TCGA) data. This highlights the versatility of UniPort in different types of multi-omics data analysis.

6. Early Integration Methods: Not Always the Best

Early integration methods, such as concatenating modalities before encoding (as done in ccVAE), performed poorly, particularly in imputation tasks. This suggests that more sophisticated approaches, which better account for the unique features of each modality, are necessary for successful multi-omics data integration.

Understanding the Methods: A Closer Look

The study evaluated a variety of linear and non-linear methods for multi-omics data integration. Here’s a brief overview of some of the key techniques:

• MOFA+: A linear method that generalizes PCA to multiple modalities using variational inference. It seeks to minimize reconstruction error across modalities while promoting a sparse latent space.
• MCIA: Another linear method that maximizes the covariance between input profiles and their latent representation.
• VAE: A deep learning-based model that uses an encoder-decoder architecture to map data to a latent space and reconstruct it back to the original space.
• PoE: A non-linear method that uses a separate VAE for each modality and combines the latent representations to generate a joint posterior distribution.
• MoE: Similar to PoE but combines the modality-specific VAEs using a mixture model instead of a product of experts.

Practical Implications of the Findings

The findings of this study have significant practical implications for researchers working with multi-omics data:

• Non-linear methods are crucial for imputing missing data, especially when working with incomplete datasets.
• When all modalities are available, linear methods like principal component concatenation provide a competitive baseline for comparison.
• For tasks with limited data, non-linear methods like PoE and MoE can enhance performance, even when only one modality is available.
• Artificial -omics profiles generated by non-linear methods can be used for supervised tasks with only a slight drop in performance, suggesting that these models can generate realistic biological data.

Limitations and Future Directions

The authors acknowledge several limitations in their study. For example, although they conducted an extensive hyperparameter search, the validation loss did not always predict performance in downstream tasks. They also observed that some joint embedding methods tend to learn modality-specific factors rather than capturing a common biological signal. To address this, they developed a method to classify factors as either joint or modality-specific using mutual information.

Future research in this area could focus on developing semi-supervised training schemes, improving the coherence of non-linear methods, and leveraging prior knowledge of cell types for feature pre-selection.

Conclusion

This study provides a comprehensive comparison of linear and non-linear joint embedding methods for multi-omics data integration. It highlights the advantages of non-linear methods in imputing missing data and their potential to improve performance in specific downstream tasks. As multi-omics data becomes more complex and ubiquitous, the findings of this study will be critical for researchers seeking to integrate and analyze diverse biological datasets effectively. With continued advancements in machine learning and deep learning, the future of multi-omics integration looks promising, and new methods will likely emerge to further enhance our ability to interpret and predict biological phenomena.

Frequently Asked Questions About Multi-Omic Data Integration

1. What are multi-omic analyses and why are they important?

Multi-omic analyses involve measuring multiple data modalities (e.g., gene expression, DNA methylation, protein levels) from the same biological sample. This approach is crucial for understanding complex biological processes at both the tissue and cellular levels. By integrating different types of molecular information, researchers can uncover intricate relationships between various biological components and enhance the accuracy of disease prediction and outcome analysis. For example, joint profiling of genetic and transcriptomic data helps identify expression quantitative trait loci (eQTLs), revealing how genetic variations impact gene expression.

2. What is joint embedding and how does it help with multi-omic data analysis?

Joint embedding, also known as joint dimensionality reduction, is a computational technique that projects data from multiple modalities into a single, lower-dimensional space. This “joint space” is designed to capture the information shared across all modalities while filtering out modality-specific noise. The benefits of joint embedding include reducing the effects of experimental noise, revealing relationships between different data types, and facilitating downstream analyses. Common methods for joint embedding include extensions of single-modality techniques like factor analysis or principal component analysis (PCA).

3. What are some examples of linear and non-linear joint embedding methods and how do they compare?

Linear joint embedding methods, such as Multi-Omics Factor Analysis (MOFA+) and Multiple Co-Inertia Analysis (MCIA), are extensions of methods like PCA which seek linear relationships between features. These methods are computationally efficient and well-suited for identifying basic relationships. In contrast, non-linear methods, often based on neural networks and deep learning (e.g., Variational Autoencoders or VAEs like Product of Experts (PoE) and Mixture of Experts (MoE)), can capture more complex, non-linear patterns that are commonly observed in biological data. The study found that non-linear methods, like PoE, generally outperformed linear methods in tasks like missing modality imputation. However, they also found that, for some downstream classification tasks, even a simple baseline like concatenating the principal components of each modality could be hard to beat.

4. What is missing modality imputation and why is it important in multi-omics?

Missing modality imputation involves predicting the values of one modality using the data from another modality. In real-world multi-omic studies, it’s common for some data types to be missing or incomplete for certain samples. Accurate imputation techniques can fill these gaps, enabling more comprehensive downstream analyses. The study found that non-linear methods like PoE showed significant advantages in imputing missing modalities compared to linear methods or a simple regression baseline. For example, PoE performed well in imputing DNA methylation from gene expression and vice versa.

5. What is ‘generation coherence’ and why is it a relevant evaluation metric for joint embedding methods?

“Generation coherence” refers to the ability of a joint embedding model to generate realistic data profiles from the learned latent space. It’s evaluated by checking if the data generated from a single latent space point, when decoded back to each modality, share common characteristics – for example, belonging to the same cancer or cell type. This is an important metric because a coherent latent space is expected to capture shared biological signals. Interestingly, the study found that while non-linear methods like PoE and MoE excelled at imputation, they showed lower coherence compared to models like CGVAE and ccVAE which directly learn a joint latent space for all modalities. This highlights the trade-off between imputation accuracy and latent space coherence.

6. How do joint embedding methods perform in downstream supervised tasks, such as survival analysis and cell type classification?

The study evaluated the utility of joint embedding for downstream supervised tasks like survival analysis (predicting patient survival using bulk tumor data) and cell type classification (predicting cell identity using single-cell data). It was found that while joint embedding can lead to improved performance when only one modality is measured at test time, it does not necessarily provide a significant advantage when data from all modalities are available at test time. In fact, concatenating the principal components (PCs) of each modality often served as a surprisingly competitive baseline. For cell type classification, joint embedding methods could only slightly outperform PCA in cases when trained on a single modality. This means that for many supervised downstream tasks, simple approaches like PCs from all available modalities may be sufficient for good performance.

7. Can artificially generated profiles from joint embedding models be used in downstream analysis?

Yes, the study showed that artificial -omic profiles (e.g., imputed gene expression or protein levels) generated by non-linear joint embedding methods are realistic enough to be used for supervised tasks with limited performance drops. For example, imputed profiles could be used in a classifier trained on real data and still achieve a reasonable performance. While this approach of imputing one modality and then using a multimodal classifier was generally worse than using a joint space classifier of a single modality directly, it did still outperform models trained only on the one directly measured modality. This highlights the power of these methods to capture underlying biological structures.

8. What are some key considerations and recommendations for using joint embedding methods?

The study highlighted several key considerations when applying joint embedding methods:

• Non-linear modeling is essential for imputation: Linear methods are often inadequate for tasks like missing modality imputation.
• Compare against simple baselines: Performance of joint embedding models should always be compared against a baseline that concatenates the principal components of each modality, as it can be surprisingly hard to beat.
• Joint space training improves single-modality downstream tasks: If only one modality will be measured at test time, a classifier trained in the joint space of that modality will perform better than simply using the PCs of that single modality.
• Realistic profiles from imputation: Non-linear joint embedding methods can generate realistic -omic profiles suitable for downstream supervized analysis.
• Hyperparameter tuning is important: It is important to do hyperparameter optimization.
• Semi-supervised approaches might be better for supervised downstream tasks: If the final aim is to excel on a specific downstream task, joint embedding methods can benefit from simultaneously learning the joint space and optimizing for the supervised task.
• Consider cell type markers over variable genes: Cell type specific marker genes may improve model performance compared to simply selecting the most variable genes.

Additionally, computational efficiency and scalability are important for large datasets; VAEs are particularly good at handling very large datasets that may be too big for techniques involving eigenvalue decomposition. The choice of joint embedding method should align with specific research questions and data characteristics.

Glossary of Key Terms

• Multi-omics: The study of biological systems by measuring and integrating multiple “omics” datasets (e.g., genomics, transcriptomics, proteomics, metabolomics).
• Joint Embedding: A method that projects data from multiple modalities into a shared, lower-dimensional space to capture common information.
• Dimensionality Reduction: The process of reducing the number of variables or features in a dataset while preserving the most important information.
• Linear Methods: Mathematical methods that assume a linear relationship between the input data and the output, like MOFA+ and MCIA.
• Non-Linear Methods: Methods, such as neural networks, that can capture non-linear relationships, often more suitable for biological data.
• Neural Network: A computational model inspired by the structure and function of the human brain, composed of interconnected nodes or “neurons” arranged in layers.
• Variational Autoencoder (VAE): A type of neural network used for dimensionality reduction, which consists of an encoder and a decoder.
• Product of Experts (PoE): A neural network architecture that combines multiple encoders for different modalities and combines their outputs into a joint latent space.
• Mixture of Experts (MoE): A neural network architecture similar to PoE that combines outputs using a mixture instead of a product of posteriors.
• Imputation: The process of predicting or filling in missing data based on the available data, often in a related modality.
• Latent Space: The lower-dimensional space to which data are projected using dimensionality reduction techniques.
• Generation Coherence: The degree to which decodings from the same point in a joint latent space generate similar data samples.
• Downstream Supervised Tasks: Tasks such as survival analysis or cell type classification, where a model is trained on labeled data to make predictions.
• TCGA (The Cancer Genome Atlas): A large-scale project that cataloged the genetic and molecular changes in different types of cancer.
• CITE-Seq: A single-cell technology that measures gene expression and protein expression simultaneously.
• RNA-Seq: A sequencing technique used to determine the presence and quantity of RNA molecules in a sample.
• ATAC-Seq: A sequencing technique used to assess the accessibility of chromatin regions in the genome.
• Principal Component Analysis (PCA): A linear dimensionality reduction technique used to project high-dimensional data into a space with reduced dimensions while retaining maximal variance.
• Akaike Information Criterion (AIC): A metric used to evaluate the trade-off between model fit and model complexity.
• Matthews Correlation Coefficient (MCC): A metric used to measure the quality of a classification model, especially for imbalanced datasets.
• Generalized Linear Model (GLM): A flexible generalization of ordinary linear regression that allows for response variables with error distributions other than a normal distribution.
• Early Integration: A technique for combining data from different modalities, in which the modalities are simply concatenated prior to processing, as in ccVAE.
• Out-of-Sample Extension: The ability of a model to make predictions on new, unseen data samples.
• Batch Normalization: A neural network technique that stabilizes training by normalizing the output of a layer.
• Dropout: A regularization technique for neural networks where randomly selected neurons are ignored during training to avoid overfitting.
• Mutual Information: A measure of the amount of information that can be obtained about one random variable by observing another.

Multi-Omics Joint Embedding Study Guide

Short Answer Quiz

1. What is the primary focus of the research described in the paper? The research focuses on comparing linear and non-linear joint embedding methods for multi-omics data, both bulk and single-cell, evaluating their performance in tasks like missing modality imputation, latent space coherence, and downstream supervised tasks. The paper seeks to provide insights into which joint embedding methods are best for various multi-omic applications.
2. Why is multi-omic analysis important in biological research? Multi-omic analysis is important because it allows researchers to gain a deeper understanding of complex biological processes by simultaneously measuring different data modalities from the same sample, such as gene expression and DNA methylation, enabling the discovery of relationships between these different types of information.
3. What is the purpose of joint dimensionality reduction or joint embedding? Joint dimensionality reduction, or joint embedding, projects different modalities of data into the same lower-dimensional space, attempting to encode shared information while filtering out modality-specific signals and noise; this helps reveal relationships between modalities and can improve the performance of downstream tasks.
4. What are some examples of linear methods used for joint embedding, as mentioned in the text? The linear methods discussed in the paper include Multi-Omics Factor Analysis (MOFA+) and Multiple Co-Inertia Analysis (MCIA); these methods are extensions of single modality dimensionality reduction, like principal components analysis.
5. What are the main advantages of using neural networks for dimensionality reduction in -omics data? Neural networks can identify non-linear patterns in the data that are expected to be present in -omics data, something that linear methods cannot do. Additionally, they can handle large datasets via stochastic or batch gradient descent.
6. Name two neural network architectures used for joint embedding in the study. Two neural network architectures used are Variational Autoencoders (VAEs) and Product of Experts (PoE) autoencoders. Other models mentioned include Mixture of Experts (MoE) autoencoders and models that use concatenated VAE (ccVAE).
7. What does the term “imputation” mean in the context of this paper? In this context, “imputation” refers to the ability of a joint embedding method to predict or reconstruct one data modality from another. In essence, it refers to estimating missing data using information from other related modalities.
8. What does “generation coherence” measure in this study? Generation coherence assesses whether decodings of random points in the latent space from a joint embedding method generate instances of each modality that have similar characteristics. It measures if the decodings are classified as the same cell or cancer type.
9. What was a key finding regarding the use of joint embeddings for supervised tasks when both modalities are available at test time? The key finding was that if both modalities are available at test time, using a joint embedding method did not provide a significant advantage for downstream supervised tasks when compared to the concatenation of principal components of each modality.
10. What is the main advantage of VAEs over linear methods for analyzing large datasets? VAEs are designed to handle large training datasets through the use of stochastic or batch gradient descent, while many linear methods such as MCIA require eigendecomposition or singular value decomposition steps, which can become computationally expensive for large sample and feature sizes.

Answer Key

1. The research focuses on comparing linear and non-linear joint embedding methods for multi-omics data, both bulk and single-cell, evaluating their performance in tasks like missing modality imputation, latent space coherence, and downstream supervised tasks. The paper seeks to provide insights into which joint embedding methods are best for various multi-omic applications.
2. Multi-omic analysis is important because it allows researchers to gain a deeper understanding of complex biological processes by simultaneously measuring different data modalities from the same sample, such as gene expression and DNA methylation, enabling the discovery of relationships between these different types of information.
3. Joint dimensionality reduction, or joint embedding, projects different modalities of data into the same lower-dimensional space, attempting to encode shared information while filtering out modality-specific signals and noise; this helps reveal relationships between modalities and can improve the performance of downstream tasks.
4. The linear methods discussed in the paper include Multi-Omics Factor Analysis (MOFA+) and Multiple Co-Inertia Analysis (MCIA); these methods are extensions of single modality dimensionality reduction, like principal components analysis.
5. Neural networks can identify non-linear patterns in the data that are expected to be present in -omics data, something that linear methods cannot do. Additionally, they can handle large datasets via stochastic or batch gradient descent.
6. Two neural network architectures used are Variational Autoencoders (VAEs) and Product of Experts (PoE) autoencoders. Other models mentioned include Mixture of Experts (MoE) autoencoders and models that use concatenated VAE (ccVAE).
7. In this context, “imputation” refers to the ability of a joint embedding method to predict or reconstruct one data modality from another. In essence, it refers to estimating missing data using information from other related modalities.
8. Generation coherence assesses whether decodings of random points in the latent space from a joint embedding method generate instances of each modality that have similar characteristics. It measures if the decodings are classified as the same cell or cancer type.
9. The key finding was that if both modalities are available at test time, using a joint embedding method did not provide a significant advantage for downstream supervised tasks when compared to the concatenation of principal components of each modality.
10. VAEs are designed to handle large training datasets through the use of stochastic or batch gradient descent, while many linear methods such as MCIA require eigendecomposition or singular value decomposition steps, which can become computationally expensive for large sample and feature sizes.

Essay Questions

1. Discuss the differences between linear and non-linear joint embedding methods, including their strengths and weaknesses, based on the findings of this research. In what specific scenarios are each approach preferable, and what are the potential drawbacks of each?
2. Analyze the challenges and benefits of applying joint embedding methods to multi-omics data, and explain how the findings in this paper address these challenges. Specifically, consider the issues of scalability, imputation, and coherence of latent spaces when working with large biological datasets.
3. Evaluate the utility of joint embedding methods for downstream supervised tasks based on the experiments described in the paper. Focus on the use cases for missing modality imputation and the performance differences when one or both modalities are available during testing.
4. Examine the role of baseline models in this study. Why is it important to include simple baseline models when evaluating sophisticated joint embedding techniques? What conclusions can be drawn from the results of baseline comparisons, and how do they influence the overall interpretation of the study’s findings?
5. Based on this research, what are some potential future directions for the development and application of joint embedding methods in multi-omics analysis? Consider both technical improvements to the existing methods and novel use cases for applying these methods to real-world biological problems.

Reference

Makrodimitris, S., Pronk, B., Abdelaal, T., & Reinders, M. (2024). An in-depth comparison of linear and non-linear joint embedding methods for bulk and single-cell multi-omics. Briefings in Bioinformatics, 25(1), bbad416.