Multi-Omics Factor Analysis As A Shared Latent Representation

Why Use a Shared Latent Representation?

A multi-omics study may measure gene expression, protein abundance, and metabolite levels on the same participants. Each assay has its own features, units, and number of measurements, but the samples may exhibit coordinated patterns across assays. A latent factor representation describes those patterns using a small number of numerical scores for the samples and a separate set of feature weights for each assay.

The purpose here is to define that representation and make its dimensions and interpretation concrete. It is a low-rank approximation of the measured data, not by itself a complete probability model, a causal explanation, or evidence that a latent dimension corresponds to one biological mechanism.

Data Matrices, Scores, and Loadings

Suppose there are \(M\) assays measured on the same \(n\) samples. For assay \(m\), let \(p_m\) be its number of features and let \(X^{(m)}\) be its \(p_m \times n\) data matrix. Rows correspond to features, columns correspond to samples, and \(X^{(m)}_{ij}\) is the value for feature \(i\) in sample \(j\). The assays may have different numbers of features, so their matrices need not have the same number of rows.

Let \(K\) be the number of latent dimensions retained. The score matrix \(Z\) has size \(K \times n\): its \(j\)th column contains the \(K\) scores for sample \(j\). For assay \(m\), the loading matrix \(W^{(m)}\) has size \(p_m \times K\). Its \(i\)th row describes how feature \(i\) in that assay is represented by the \(K\) dimensions.

Definition: A multi-omics factor representation with \(K\) dimensions writes each assay matrix as \(X^{(m)} \approx W^{(m)}Z\), for \(m=1,\ldots,M\). The columns of \(Z\) are sample-level latent score profiles used across the assays; \(W^{(m)}\) contains the feature loadings specific to assay \(m\). The difference \(E^{(m)}=X^{(m)}-W^{(m)}Z\) is the reconstruction residual for that assay.

The multiplication is dimensionally valid: \(W^{(m)}\) is \(p_m \times K\), \(Z\) is \(K \times n\), and their product is \(p_m \times n\), matching \(X^{(m)}\). For a single feature \(i\) and sample \(j\), the approximation says

X^{(m)}_{ij} \approx \sum_{k=1}^{K} W^{(m)}_{ik}Z_{kj}.

Thus, the score \(Z_{kj}\) describes the position of sample \(j\) along dimension \(k\), while \(W^{(m)}_{ik}\) describes how that dimension contributes to feature \(i\) in assay \(m\). The same sample scores are reused across assays, but the loadings map those scores to different feature sets. Scores and loadings are numerical components of a representation; neither is automatically a directly observed biological quantity.

Worked Example: Checking Dimensions Across Two Assays

Suppose a hypothetical study measures 120 RNA features and 35 protein features on the same 18 samples, and the selected representation has \(K=2\) dimensions. The RNA matrix has size \(120 \times 18\), and the protein matrix has size \(35 \times 18\). The shared score matrix has size \(2 \times 18\). The RNA loading matrix is \(120 \times 2\), while the protein loading matrix is \(35 \times 2\).

The products have the required shapes:

(120 \times 2)(2 \times 18)=120 \times 18,\qquad (35 \times 2)(2 \times 18)=35 \times 18.

Each assay can therefore be reconstructed using the same two scores for each sample, while having a different loading for every feature. This does not mean the assays have the same features or values; it means their sample patterns are expressed through a common two-dimensional score matrix.

How a Few Dimensions Represent Coordinated Variation

The key algebraic property is that every feature row in every assay is represented using the same \(K\) sample-score vectors. For dimension \(k\), the vector \((Z_{k1},\ldots,Z_{kn})\) gives its score across the \(n\) samples. In assay \(m\), each feature can use that vector with a feature-specific weight \(W^{(m)}_{ik}\). A different assay can assign different weights to the same score pattern.

This reuse is what makes the representation multi-omics: a sample pattern need not be independently rediscovered in every assay. If samples with high scores on one dimension also show high values for positively loaded features in several assays, the fitted representation summarizes that coordinated pattern. Whether it is scientifically meaningful depends on the data, preprocessing, and diagnostics; the algebra alone does not establish a biological explanation.

Justification: For any assay \(m\), the \(i\)th reconstructed feature row is \(\sum_{k=1}^{K}W^{(m)}_{ik}Z_{k\cdot}\), where \(Z_{k\cdot}\) denotes row \(k\) of \(Z\). Consequently, every reconstructed row lies in the span of the same \(K\) sample-score rows. The reconstructed matrix has rank at most \(K\). Because those score rows are reused for every assay, a low-dimensional sample pattern can contribute to feature reconstructions in multiple assays.

This argument establishes what the representation can do, not that it will fit well. A small \(K\) yields a compact summary but can omit important patterns. A large \(K\) can fit more variation while making the summary less compact and potentially capturing noise. The useful choice depends on the data and the purpose of the analysis.

Worked Example: Reconstructing Assay Values

Consider one hypothetical dimension and three samples. Let the score row be \(Z_{1\cdot}=(1,-1,0.5)\). In an RNA assay, a feature has loading \(2\); in a protein assay, a feature has loading \(-1.5\). With one dimension, each reconstructed feature row is its loading multiplied by the score row.

For the RNA feature, the reconstructed values are

$$ 2(1,-1,0.5)=(2,-2,1). $$

For the protein feature, they are

$$ -1.5(1,-1,0.5)=(-1.5,1.5,-0.75). $$

Both features use the same sample scores. The positive RNA loading gives a reconstructed feature that increases with the score, while the negative protein loading gives one that decreases with it. These are reconstructed values in the representation's units; if the assay values were transformed or standardized, that scale must be considered when interpreting them.

Assumptions and Conditions for Interpretation

The matrix representation requires correctly paired samples: column \(j\) of every \(X^{(m)}\) must refer to the same sample \(j\). If one assay's columns are in a different order, multiplying its loading matrix by the shared \(Z\) assigns scores to the wrong samples. Assays may have different feature counts, but the sample columns must correspond.

The product \(W^{(m)}Z\) also presumes that the values being represented can be meaningfully approximated by weighted sums of latent scores on their current scales. This is a modeling choice, not a universal property of omics data. Transformations, centering, and scaling affect what patterns dominate a fit. For example, an assay with much larger numerical variation can have a disproportionate influence if the fitting objective treats raw squared errors equally across all entries.

This formulation is deliberately an algebraic representation. It does not specify a distribution for measurements or residuals, nor does it state how a fitting algorithm estimates \(W^{(m)}\) and \(Z\). Those choices depend on the observation types and model being used. As written, values and residuals can have either sign; it does not impose the non-negativity constraints used in NMF.

Worked Example: Detecting a Sample-Alignment Error

Suppose two assays each have four samples, labelled A, B, C, and D. The score row for one dimension, in that order, is \((0,1,2,3)\). A protein loading is \(2\), so the intended reconstructed protein values are \((0,2,4,6)\), corresponding to A, B, C, and D.

Now suppose the protein matrix columns were accidentally stored in the order C, A, D, B, but the shared score row was used as if the order were A, B, C, D. The multiplication pairs the protein columns with scores \(0,1,2,3\) in the wrong order. Even if the fitted arithmetic is carried out correctly, the reconstructed values are attached to incorrect sample labels.

The check is not a numerical diagnostic alone: verify sample identifiers and column order before fitting and when joining results. A low reconstruction error cannot correct a mislabelled alignment; it may instead produce a misleading cross-assay summary.

Diagnostics, Alternatives, and Limitations

Diagnostics should assess whether the low-dimensional representation captures relevant structure in each assay and whether the shared score patterns are being used sensibly.

Inspect residuals by assay and feature. Large or structured residuals indicate that the chosen dimensions or linear representation may not capture important patterns. A residual pattern that varies systematically with sample groups or feature subsets is more concerning than isolated small discrepancies.
Compare reconstruction quality across assays. A strong aggregate fit can hide poor representation of one assay, especially when assays differ greatly in feature count or scale. Report assay-specific summaries as well as any combined summary.
Check preprocessing and scale. If one assay's units or variability dominate the fitting criterion, shared dimensions may mainly describe that assay. Confirm that the preprocessing matches the scientific question and examine whether conclusions change under reasonable alternatives.
Check sample correspondence. Confirm matching identifiers and ordering across all assay matrices. Misalignment undermines the intended interpretation of shared sample scores.
Look for under- or over-compression. If residuals retain broad patterns, \(K\) may be too small or the linear representation may be unsuitable. If a large \(K\) is needed to fit idiosyncratic fluctuations, the representation may no longer be a useful concise summary.

Separate PCA fits each assay on its own and can provide assay-specific low-dimensional summaries, but its components are not automatically expressed on a common sample-score matrix. The representation defined here uses common sample scores with different assay loadings to summarize patterns jointly. Joint NMF is another joint representation, with non-negativity restrictions and its own objective choices; it is not interchangeable with this unconstrained linear form. The choice among these approaches depends on the data scale, required constraints, and scientific objective.

A loading pattern or a sample score pattern is descriptive, not causal. A latent dimension may combine multiple sources of variation, including technical effects, biological variation, or both. A concise representation can aid exploration and downstream analysis, but it does not on its own establish factor meaning, prove that all assays contribute usefully, or guarantee that a low-rank approximation is adequate.

Common pitfall: Treating a shared score as though it were an identical measured effect in every assay. The score is reused across assays, but each assay has its own feature loadings and residuals. A high score alone does not imply that every feature in every assay is high; the feature's loading determines how the score contributes to its reconstruction.

Check Your Understanding

Answer the questions, then compare with the solutions.

An assay has 80 features measured on 25 aligned samples, and \(K=3\). What are the dimensions of its data matrix, loading matrix, and the shared score matrix?
For one feature, suppose its two loadings are \(1.2\) and \(-0.5\), and a sample has scores \(2\) and \(1\). What is the reconstructed feature value?
Why can different assays use the same sample scores while having different reconstructed feature values?
Name two checks that can reveal a poor or misleading representation.

Solutions. (1) The data matrix is \(80 \times 25\), the loading matrix is \(80 \times 3\), and the shared score matrix is \(3 \times 25\). (2) The reconstruction is \(1.2(2)+(-0.5)(1)=2.4-0.5=1.9\), in the scale of that feature's data. (3) Each assay has its own feature loadings, so the same sample-score vector is mapped differently to each assay's features. (4) Examples include inspecting assay-specific residual patterns, checking sample identifiers and column order, and examining whether preprocessing or scale causes one assay to dominate.