Enter the Matrix: Factorization Uncovers Knowledge from Omics
Authors: Genevieve L. Stein-O’Brien, Raman Arora, Aedin C. Culhane, Alexander V. Favorov, Lana X. Garmire, Casey S. Greene, Loyal A. Goff, Yifeng Li, Aloune Ngom, Michael F. Ochs, Yanxun Xu, Elana J. Fertig
Journal: Trends in Genetics (2018)

MFs techniques infer low-dimensional structure from high-dimensional omics data to enable visualization and inference of complex biological processes (CBPs). Different MFs applied to the same data will learn different factors. Exploratory data analysis should employ multiple MFs, whereas a specific biological question should employ a specific MF tailored to that problem. MFs learn two sets of low-dimensional representations (in each matrix factor) from high-dimensional data: one defining molecular relationships (amplitude) and another defining sample-level relationships (pattern). Data-driven functional pathways, biomarkers, and epistatic interactions can be learned from the amplitude matrix. Clustering, subtype discovery, in silico microdissection, and timecourse analysis are all enabled by analysis of the pattern matrix. MF enables both multi-omics analyses and analyses of single-cell data. Omics data contain signals from the molecular, physical, and kinetic inter- and intracellular interactions that control biological systems. Matrix factorization (MF) techniques can reveal low-dimensional structure from high-dimensional data that reflect these interactions. These techniques can uncover new biological knowledge from diverse high-throughput omics data in applications ranging from pathway discovery to timecourse analysis. We review exemplary applications of MF for systems-level analyses. We discuss appropriate applications of these methods, their limitations, and focus on the analysis of results to facilitate optimal biological interpretation. The inference of biologically relevant features with MF enables discovery from high-throughput data beyond the limits of current biological knowledge – answering questions from high-dimensional data that we have not yet thought to ask. Omics data contain signals from the molecular, physical, and kinetic inter- and intracellular interactions that control biological systems. Matrix factorization (MF) techniques can reveal low-dimensional structure from high-dimensional data that reflect these interactions. These techniques can uncover new biological knowledge from diverse high-throughput omics data in applications ranging from pathway discovery to timecourse analysis. We review exemplary applications of MF for systems-level analyses. We discuss appropriate applications of these methods, their limitations, and focus on the analysis of results to facilitate optimal biological interpretation. The inference of biologically relevant features with MF enables discovery from high-throughput data beyond the limits of current biological knowledge – answering questions from high-dimensional data that we have not yet thought to ask. High-throughput technologies have ushered in an era of big data in biology [1Bell G. et al.Beyond the data deluge.Science. 2009; 323: 1297-1298Crossref PubMed Scopus (0) Google Scholar, 2Sagoff M. Data deluge and the human microbiome project.Issues Sci. Technol. 2012; 28http://issues.org/28-4/sagoff-3/Google Scholar] and empowered in silico experimentation which is poised to characterize complex biological processes (CBPs; see Glossary) [3Alter O. Discovery of principles of nature from mathematical modeling of DNA microarray data.Proc. Natl. Acad. Sci. U. S. A. 2006; 103: 16063-16064Crossref PubMed Scopus (18) Google Scholar]. The natural representation of high-dimensional biological data is a matrix of the measured values (expression counts, methylation levels, protein concentrations, etc.) in rows and individual samples in columns (Figure 1). Columns corresponding to experimental replicates, or samples with similar phenotypes, will have values from the same distribution of biological variation. Related structures in the data are observed because they share one or more CBP. The activity of CBPs need not be identical in each sample. In these cases, the values of all molecular components that are associated with a CBP will change proportionally to the relative activity of that CBP. These phenotypes and CBP activities are often unknown a priori, requiring computational techniques for unsupervised learning to discover CBPs directly from the biological data. The relationships between CBPs and similarities between samples constrain high-dimensional datasets to have low-dimensional structure. The number of genes, proteins, and pathways that are concurrently active within any cell is constrained by its energy and free-molecule limitations [4Heyn P. et al.Introns and gene expression: cellular constraints, transcriptional regulation, and evolutionary consequences.Bioessays. 2015; 37: 148-154Crossref PubMed Scopus (25) Google Scholar]. Only a characteristic subset of CBPs will be active in any cell at a given time. Thus, for a dataset where columns share CBP, a low-dimensional structure can be extracted which is smaller than either the number of rows or the number of columns. Matrix factorization (MF) is a class of unsupervised techniques that provide a set of principled approaches to parsimoniously reveal the low-dimensional structure while preserving as much information as possible from the original data. MF is also referred to as matrix decomposition, and the corresponding inference problem as deconvolution. Other reviews discuss the mathematical and technical details of MF techniques [5Ochs M.F. Fertig E.J. Matrix factorization for transcriptional regulatory network inference.IEEE Symp. Comput. Intell. Bioinforma. Comput. Biol. Proc. 2012; 2012: 387-396PubMed Google Scholar, 6Abdi H. et al.Multiple factor analysis: principal component analysis for multitable and multiblock data sets.WIREs Comp. Stat. 2013; 5: 149-179Crossref Scopus (82) Google Scholar, 7Meng C. et al.Dimension reduction techniques for the integrative analysis of multi-omics data.Brief. Bioinform. 2016; 17: 628-641Crossref PubMed Scopus (5) Google Scholar, 8Li Y. et al.A review on machine learning principles for multi-view biological data integration.Brief. Bioinform. 2016; 19: 325-340Google Scholar] and their applications to microarray data [9Devarajan K. Nonnegative matrix factorization: an analytical and interpretive tool in computational biology.PLoS Comput. Biol. 2008; 4e1000029Crossref PubMed Scopus (0) Google Scholar]. We focus here on the biological applications of MF techniques and the interpretation of their results since the advent of sequencing technologies. We describe a variety of MF techniques applied to high-throughput data analysis, and compare and contrast their use for biological inference from bulk and single-cell data. After data preprocessing, most high-throughput molecular datasets can be represented as a matrix in which each element contains the measurement of a single molecule in a single experimental condition. In the example of RNA sequencing (RNA-seq), the number of short reads from each gene are summarized into gene level counts. The resulting high-dimensional dataset is formulated by representing these gene level counts for each sample as a column in the data matrix (Figure 1). MF methods can then be applied to these count matrices to learn CBPs from the data. Many MF techniques described are for RNA-seq data that are preprocessed using log transformation [10Ritchie M.E. et al.limma powers differential expression analyses for RNA-sequencing and microarray studies.Nucleic Acids Res. 2015; 43: e47Crossref PubMed Scopus (2443) Google Scholar] or models of sequencing depth [11Anders S. Huber W. Differential expression analysis for sequence count data.Genome Biol. 2010; 11: R106Crossref PubMed Scopus (5331) Google Scholar], while others directly model read counts [12Xie F. et al.BayCount: a Bayesian decomposition method for inferring tumor heterogeneity using RNA-Seq counts.bioRxiv. 2017; (Published online November 13, 2017)https://doi.org/10.1101/218511Crossref Google Scholar]. Most examples featured in this review are based on such preprocessed RNA-seq analysis of gene counts or log-transformed gene counts. We note that applications of MF are not limited to this data modality or this preprocessing pipeline. For example, MF has also been applied to define mutational signatures in cancer [13Alexandrov L.B. et al.Signatures of mutational processes in human cancer.Nature. 2013; 500: 415-421Crossref PubMed Scopus (2340) Google Scholar, 14Alexandrov L.B. et al.Mutational signatures associated with tobacco smoking in human cancer.Science. 2016; 354: 618-622Crossref PubMed Scopus (120) Google Scholar], allele combinations in phenotypes [15Favorov A.V. et al.A Markov chain Monte Carlo technique for identification of combinations of allelic variants underlying complex diseases in humans.Genetics. 2005; 171: 2113-2121Crossref PubMed Scopus (0) Google Scholar], transcript regulation of genes [16Zakeri M. et al.Improved data-driven likelihood factorizations for transcript abundance estimation.Bioinformatics. 2017; 33: i142-i151Crossref PubMed Scopus (0) Google Scholar], and distributions of transcript lengths [17Bertagnolli N.M. et al.SVD identifies transcript length distribution functions from DNA microarray data and reveals evolutionary forces globally affecting GBM metabolism.PLoS One. 2013; 8e78913Crossref PubMed Scopus (5) Google Scholar], as well as to discriminate peptides in mass spectrometry proteomics [18Peckner R. et al.Specter: linear deconvolution as a new paradigm for targeted analysis of data-independent acquisition mass spectrometry proteomics.bioRxiv. 2017; (Published online September 8, 2017)https://doi.org/10.1101/152744Crossref Google Scholar]. To apply MF to other data modalities, they must also be properly preprocessed into a data matrix with a distribution appropriate to the MF analysis method. When applied to high-throughput omics data, MF techniques learn two matrices: one describes the structure between features (e.g., genes) and another describes the structure between samples (Figure 2, Key Figure). We call the former feature-level matrix the amplitude matrix and the latter sample-level matrix the pattern matrix. Additional terms have been coined for the amplitude and pattern matrices based upon the MF problem applied and on the specific application to high-throughput biological data (Box 1). The values in each column of the amplitude matrix are continuous weights describing the relative contribution of a molecule in each inferred factor. In cases where factors distinguish between CBPs, the relative weights of these molecules can be associated with functional pathways. The same molecule may have high values in multiple columns of the amplitude matrix. Thus, MF techniques are able to account for the cumulative effect of genes that participate in multiple pathways.Box 1Common Terminology in the LiteratureHistorically, the independent discovery of MF in multiple fields including mathematics, computer science, and statistics created distinct terminologies that are often used interchangeably as analytical orthologs in genomics. For example, the term 'factorization' is often used interchangeably with 'decomposition'. Other terms, such as 'features', 'components', 'latent variable', or 'latent factors' are used to refer to relationships between either molecular measurements or samples, depending on the context.The specific terminology for the amplitude and pattern matrices also varies according to the method, and are preferably labeled with different variable names. In PCA, the amplitude matrix is often called the score or rotation matrix, and pattern matrix called the loadings. In ICA, the amplitude matrix is called the unmixing matrix and the pattern matrix is called the source matrix. In NMF, the amplitude matrix is commonly called the weights matrix and the pattern matrix is called the features matrix.Throughout this paper we use 'amplitude matrix' to refer to the matrix that contains vectors which represent relationships between molecular measurements. Other terms used in the literature for these molecular relationships include modules, meta-pathways, and signatures. Similarly, we use 'pattern matrix' to refer to the matrix that contains vectors which represent relationships between samples. Other literature terms for these sample-level relationships include patterns, metagenes, eigengenes, sources, and controlling factors. Historically, the independent discovery of MF in multiple fields including mathematics, computer science, and statistics created distinct terminologies that are often used interchangeably as analytical orthologs in genomics. For example, the term 'factorization' is often used interchangeably with 'decomposition'. Other terms, such as 'features', 'components', 'latent variable', or 'latent factors' are used to refer to relationships between either molecular measurements or samples, depending on the context. The specific terminology for the amplitude and pattern matrices also varies according to the method, and are preferably labeled with different variable names. In PCA, the amplitude matrix is often called the score or rotation matrix, and pattern matrix called the loadings. In ICA, the amplitude matrix is called the unmixing matrix and the pattern matrix is called the source matrix. In NMF, the amplitude matrix is commonly called the weights matrix and the pattern matrix is called the features matrix. Throughout this paper we use 'amplitude matrix' to refer to the matrix that contains vectors which represent relationships between molecular measurements. Other terms used in the literature for these molecular relationships include modules, meta-pathways, and signatures. Similarly, we use 'pattern matrix' to refer to the matrix that contains vectors which represent relationships between samples. Other literature terms for these sample-level relationships include patterns, metagenes, eigengenes, sources, and controlling factors. Whereas each column of the amplitude matrix describes the relative contributions of molecules to a factor, each row of the pattern matrix describes the relative contributions of samples to a factor (Figure 2). Sample groups can be learned by comparing the relative weights in each row of the pattern matrix. The pattern matrix from MF can also be binarized to perform clustering [19Yeung K.Y. et al.Model-based clustering and data transformations for gene expression data.Bioinformatics. 2001; 17: 977-987Crossref PubMed Google Scholar, 20Jiang D. et al.Cluster analysis for gene expression data: a survey.IEEE Trans. Knowl. Data Eng. 2004; 16: 1370-1386Crossref Scopus (0) Google Scholar] or kept as continuous values to define relationships between samples [21Venet D. et al.Separation of samples into their constituents using gene expression data.Bioinformatics. 2001; 17: S279-S287Crossref PubMed Google Scholar, 22Abbas A.R. et al.Deconvolution of blood microarray data identifies cellular activation patterns in systemic lupus erythematosus.PLoS One. 2009; 4e6098Crossref PubMed Scopus (145) Google Scholar, 23Erkkilä T. et al.Probabilistic analysis of gene expression measurements from heterogeneous tissues.Bioinformatics. 2010; 26: 2571-2577Crossref PubMed Scopus (37) Google Scholar]. In the same way as molecules with high weights within a column of the amplitude matrix are associated with a common pathway, samples with high weights within a row of the pattern matrix can be assumed to share a common phenotype or CBP. The optimal number of columns of the amplitude matrix and rows of the pattern matrix is often referred to as the dimensionality of the MF problem, and learning this value remains an open problem for the MF research community [24Leek J.T. Asymptotic conditional singular value decomposition for high-dimensional genomic data.Biometrics. 2011; 67: 344-352Crossref PubMed Scopus (19) Google Scholar, 25Kelton C.J. et al.The estimation of dimensionality in gene expression data using nonnegative matrix factorization.2015 IEEE International Conference on Bioinformatics and Biomedicine (BIBM). 2015; : 1642-1649Crossref Scopus (0) Google Scholar]. We also note that MF is not a single computational method. Instead, there is a wide body of literature on numerous MF techniques that have been applied throughout computational biology. The properties of both the amplitude and pattern matrices, and subsequently the interpretation of their values, depend crucially on the specific MF problem and the algorithm selected for analysis. There are numerous approaches to MF. The three most prominent MF approaches are principal component analysis (PCA), independent component analysis (ICA), and non-negative matrix factorization (NMF). Each of these techniques has a distinct mathematical formulation of a distinct MF problem (as described in the supplemental information online and in other reviews [5Ochs M.F. Fertig E.J. Matrix factorization for transcriptional regulatory network inference.IEEE Symp. Comput. Intell. Bioinforma. Comput. Biol. Proc. 2012; 2012: 387-396PubMed Google Scholar, 8Li Y. et al.A review on machine learning principles for multi-view biological data integration.Brief. Bioinform. 2016; 19: 325-340Google Scholar, 26Meng C. et al.Dimension reduction techniques for the integrative analysis of multi-omics data.Brief. Bioinform. 2016; 17: 628-641Crossref PubMed Scopus (5) Google Scholar, 27Berry M.W. et al.Algorithms and applications for approximate nonnegative matrix factorization.Comput. Stat. Data Anal. 2007; 52: 155-173Crossref Scopus (670) Google Scholar, 28Wang Y.-X. Zhang Y.-J. Nonnegative matrix factorization: a comprehensive review.IEEE Trans. Knowl. Data Eng. 2013; 25: 1336-1353Crossref Scopus (209) Google Scholar, 29Zhou G. et al.Nonnegative matrix and tensor factorizations: an algorithmic perspective.IEEE Signal Process. Mag. 2014; 31: 54-65Crossref Scopus (41) Google Scholar]). Briefly, PCA finds dominant sources of variation in high-dimensional datasets, inferring genes that distinguish between samples. Maximizing the variability captured in specific factors, as opposed to spreading relatively evenly among factors, may mix the signal from multiple CBPs in a single component. Therefore, PCA may conflate processes that sometimes occur and complicate interpretation of the amplitude matrix for defining data-driven gene sets or the inference of specific CBPs. ICA and NMF learn distinct processes from an input data matrix using different techniques. ICA learns factors that are statistically independent, resulting in more accurate association with literature-derived gene sets [30Lee S.-I. Batzoglou S. Application of independent component analysis to microarrays.Genome Biol. 2003; 4: R76Crossref PubMed Scopus (0) Google Scholar, 31Engreitz J.M. et al.Independent component analysis: mining microarray data for fundamental human gene expression modules.J. Biomed. Bioinf. 2010; 43: 932-944Google Scholar, 32Teschendorff A.E. et al.Elucidating the altered transcriptional programs in breast cancer using independent component analysis.PLoS Comput. Biol. 2007; 3: e161Crossref PubMed Scopus (71) Google Scholar]. NMF methods constrain all elements of the amplitude and pattern matrices to be greater than or equal to zero [33Ochs M.F. et al.A new method for spectral decomposition using a bilinear Bayesian approach.J. Magn. Reson. 1999; 137: 161-176Crossref PubMed Scopus (54) Google Scholar, 34Lee D.D. Seung H.S. Learning the parts of objects by non-negative matrix factorization.Nature. 1999; 401: 788-791Crossref PubMed Scopus (5898) Google Scholar]. Whereas the features in PCA can be ranked by the extent to which they explain the variation in the data, the features in both ICA and NMF are assumed to have equal weight. NMF is well suited to transcriptional data, which is typically non-negative itself, and semi-NMF is also applicable to data that can have negative values. The assumptions of NMF model both the additive nature of CBPs and parsimony, generating solutions that are biologically intuitive to interpret [35Moloshok T.D. et al.Application of Bayesian decomposition for analysing microarray data.Bioinformatics. 2002; 18: 566-575Crossref PubMed Google Scholar]. The solutions from both ICA and NMF may vary depending upon the initialization of the algorithm, leading to disparate amplitude and pattern matrices. Therefore, it is crucial to ensure that particular solution used for analysis provides an optimal and robust solution before using the results of the factorization to interpret CBPs. We previously found that Bayesian techniques to solve NMF have more robust amplitude matrices than gradient-based techniques, and thus generate more accurate associations between the values in the amplitude matrix and functional pathways [5Ochs M.F. Fertig E.J. Matrix factorization for transcriptional regulatory network inference.IEEE Symp. Comput. Intell. Bioinforma. Comput. Biol. Proc. 2012; 2012: 387-396PubMed Google Scholar, 36Kossenkov A.V. et al.Determining transcription factor activity from microarray data using Bayesian Markov chain Monte Carlo sampling.Stud. Health Technol. Inform. 2007; 129: 1250-1254PubMed Google Scholar]. Other studies have found that gradient-based approaches have similar computational performance to their Bayesian counterparts [37Mairal J. et al.Online learning for matrix factorization and sparse coding.J. Mach. Learn. Res. 2010; 11: 19-60Google Scholar], and new techniques are being developed to enhance the stability of the factorization [38Wu S. et al.Stability-driven nonnegative matrix factorization to interpret spatial gene expression and build local gene networks.Proc. Natl. Acad. Sci. U. S. A. 2016; 113: 4290-4295Crossref PubMed Scopus (9) Google Scholar]. Further integrated computational/experimental investigation is necessary to assess the biological relevance of solutions from both classes of techniques and their robustness. These associations also depend crucially on the input data. Therefore, to learn CBPs from data, MF must be applied to datasets with sufficient measurements of the experimental perturbations or conditions relevant to the specific biological problem being addressed. Applying multiple types of MF techniques to the same dataset or a single MF to distinct subsets of a dataset can also find distinct sources of variability. Selecting which MF method to use to learn relevant CBPs from a dataset is then crucial, and developing standardized metrics to choose between them is an active area of research in computational biology. To illustrate the differences between MF methods, we apply PCA, ICA (CRAN package fastICA [39Hyvärinen A. Oja E. Independent component analysis: algorithms and applications.Neural Netw. 2000; 13: 411-430Crossref PubMed Scopus (4350) Google Scholar]), and NMF (R/Bioconductor package CoGAPS [40Fertig E.J. et al.CoGAPS: an R/C++ package to identify patterns and biological process activity in transcriptomic data.Bioinformatics. 2010; 26: 2792-2793Crossref PubMed Scopus (20) Google Scholar, 41Stein-O'Brien G.L. et al.PatternMarkers & GWCoGAPS for novel data-driven biomarkers via whole transcriptome NMF.Bioinformatics. 2017; 33: 1892-1894Crossref PubMed Scopus (1) Google Scholar]) to a single dataset from postmortem samples in the GTEx project [42Dey K.K. et al.Visualizing the structure of RNA-seq expression data using grade of membership models.PLoS Genet. 2017; 13e1006599Crossref PubMed Scopus (1) Google Scholar] (Figure 3). Specifically, we select a subset of GTEx data containing 12 brain tissues in the GTEx data for seven individuals for which we had previously performed only NMF analysis [41Stein-O'Brien G.L. et al.PatternMarkers & GWCoGAPS for novel data-driven biomarkers via whole transcriptome NMF.Bioinformatics. 2017; 33: 1892-1894Crossref PubMed Scopus (1) Google Scholar] (codes are provided in the supplemental information online). We select this problem because the CBPs (tissue and individual) are readily separated and are known a priori, providing a known ground truth to facilitate comparison of methods. First we apply PCA to this sample GTEx dataset (Figure 3A). The components learned from PCA can be ranked by the amount of variation that they explain in the data, with the first two components explaining 89.6% of the variation in this dataset. The amount of variation explained by these components can be used to determine the optimal dimensionality for PCA [24Leek J.T. Asymptotic conditional singular value decomposition for high-dimensional genomic data.Biometrics. 2011; 67: 344-352Crossref PubMed Scopus (19) Google Scholar]. A typical PCA analysis explores the association between these components and biological covariates in low-dimensional plots in which each axis is defined by the weights in one row of the pattern matrix (Figure 3A). Similarly to clustering analyses, these plots can be used to determine which biological features can be separated from the data. In this application, we observe that the cerebellum (light blue) and first cervical spinal cord (yellow) cluster separately from all other brain tissues (PC1 and PC2, or rows 1 and 2 of the pattern matrix, respectively). No separation between individuals is observed in this PCA analysis. In contrast to PCA, the components of ICA and NMF cannot be ranked by percent variation explained. Instead, each row of the pattern matrix is an equally important CBP in the data. Therefore, these patterns are plotted independently relative to biological covariates, and not relative to one another as in PCA analyses. When applied to the sample GTEx dataset, rows of the pattern matrix from both ICA (Figure 3B) and NMF (Figure 3C) also distinguish the cerebellum, similarly to PCA. Whereas PC1 has large positive values for the cerebellum and large negative values for the other brain tissues, both NMF and ICA have large absolute value only for the tissue of interest and are near zero for other tissues. As a result, both ICA and NMF provide tissue-specific patterns and PCA provides tissue-segregating patterns. We note that the magnitude of all these patterns is unit-less, reflecting the relative weights of each sample in the pattern and not a measurable quantity. Because gene weights can be either positive or negative in ICA, the patterns weights can also. By contrast, by construction the NMF values are all non-negative. As a result, the pattern corresponding to the cerebellum samples from ICA is still specific to that tissue, with genes overexpressed in this tissue having negative weights in the corresponding column of the amplitude matrix, and genes underexpressed in that tissue having positive weights. Indeed, the gene weights in the column of the amplitude matrix corresponding to the NMF cerebellum pattern are significantly anti-correlated with the gene weights in the column of the amplitude matrix corresponding to the ICA cerebellum pattern (R = −0.72, P < 2 × 10−16). In contrast to PCA, both ICA and NMF also infer patterns that distinguish individuals with common weights across all tissues. In the NMF analysis, each individual has a separate pattern. The ICA analysis has a single pattern that has large positive values for one of the individuals distinguished in one NMF pattern, and large negative values for another individual distinguished in different NMF pattern. This discrepancy highlights the difference between inferring independent sources of variation with ICA and NMF. Specifically, ICA may combine multiple CBPs using common genes whose sign changes by experimental condition, whereas NMF will find CBPs that are additive, corresponding only to overexpression of genes in that condition. PCA, ICA, and NMF are equally valid, and their distinct formulation gives rise to the distinct features observed in the data. Applications of multiple types of MF techniques, or even the same MF algorithm with different parameters, may infer several CBPs or phenotypes within a single dataset, in essence providing answers to different questions. We further note that the specific techniques for PCA, ICA, and NMF selected for analysis in this example are only one of a multitude of variants of techniques which have been developed for computational biology. Applying multiple types of MF techniques to the same dataset, or a single MF to distinct subsets of a dataset, can also find distinct sources of variability. However, the general properties of these methods will remain and be consistent with our example. Briefly, we observe in this example that PCA finds sources of separation in the data, whereas both ICA and NMF find independent sources of variation. ICA can find both over- and underexpression of genes in a single CBP, whereas NMF can find only overexpressed genes in a single CBP. As a result, ICA may better model both repression and activation than NMF, but as a side effect may have greater mixture of CBPs than NMF. Regardless of the technique selected, the results will also be sensitive to the input data. For example, a different NMF-class algorithm called 'grade of membership' (GOM) was also applied to a larger set of postmortem samples in GTEx. This algorithm found a pattern that combined all samples from brain regions when applied to all tissue samples in GTEx, but separated the distinct brain regions when applied only to tissue samples from the brain. Thus, applying multiple types of MF techniques to the same dataset, or a single MF to distinct subsets of a dataset, can also find distinct sources of variability that are essential for exploratory data analysis. Exactly as we observed in the GTEx example, a single factorization of complex datasets can find multiple distinct sources of variation. For example, the power of MF to identify multiple sources of variation was seen when multiple technical factors from sample processing and biological factors were discovered in an ICA of gene expression profiles of 198 bladder cancer samples [43Biton A. et al.Independent component analysis uncovers the landscape of the bladder tumor transcriptome and r