Nonnegative Matrix Factorization as a Data Model

Introduction

Multi-omics measurements often contain nonnegative quantities: counts, intensities, concentrations, or nonnegative scores. Such data can be represented as combinations of a smaller number of nonnegative patterns. Nonnegative matrix factorization (NMF) expresses this idea by approximating a data matrix with the product of two smaller matrices. The product links two kinds of quantities: which features characterize each component, and how strongly each component contributes to each sample.

The interpretation depends on how the data matrix is arranged and measured. A clear convention is therefore essential: rows will represent features and columns will represent samples. With that convention, the factor matrices have direct, distinct roles.

Dimensions and the Entry-Wise Model

The dimensions must agree for the product \(WH\) to be defined. The \(r\) columns of \(W\) correspond, in order, to the \(r\) rows of \(H\). The resulting product has \(m\) rows and \(n\) columns, just like \(X\). Thus each reconstructed entry corresponds to a particular feature in a particular sample.

Matrix multiplication gives a precise expression for the model. For feature \(i\) and sample \(j\), the fitted or reconstructed value is the sum of the component-specific contributions for that feature and sample.

Here, \(W_{ik}\) describes the contribution of feature \(i\) to the profile of component \(k\), while \(H_{kj}\) describes the abundance of component \(k\) in sample \(j\). Their product is that component's contribution to the reconstructed measurement for feature \(i\), sample \(j\). Adding those contributions across components gives \(\widehat{x}_{ij}\). This is an additive data model: a feature value is represented as a sum, not as a choice of just one component.

The dimensions also indicate how to read the factors. A column of \(W\) has one entry per feature, so it describes a pattern across features. A column of \(H\) has one entry per component, so it describes a sample across components. Confusing these orientations can produce a plausible-looking product with the wrong scientific interpretation.

What Nonnegativity Says About the Model

The constraints \(X \geq 0\), \(W \geq 0\), and \(H \geq 0\) apply entry by entry. If \(W\) and \(H\) are nonnegative, then every component contribution \(W_{ik}H_{kj}\) is nonnegative, and their sum \(\widehat{x}_{ij}\) is nonnegative. This fits a model in which components add to a measured quantity without subtracting one component from another.

That statement does not imply that the residuals are nonnegative. Define the residual matrix \(E=X-WH\); its entries can be positive, negative, or zero. A positive residual means the observed measurement exceeds its reconstruction, and a negative residual means the reconstruction exceeds the observation. Nonnegative factors restrict the representation, not the direction of its errors.

The data domain is an assumption to check, not a cosmetic preference. A measurement matrix with negative entries cannot be represented by a product of nonnegative factors, because such a product is nonnegative entry by entry. Centering features by subtracting their means commonly creates negative values, so the centered matrix is no longer directly within the basic NMF data domain. Similarly, preprocessing should not be selected merely because it is familiar from another analysis.

Assumptions, Units, and Preprocessing

The basic model is most interpretable when features and samples have been arranged consistently, measurements are nonnegative, and additive combinations of feature profiles are scientifically meaningful as an approximation. These conditions do not guarantee that NMF will recover uniquely defined biological entities, or that each component corresponds to a single biological mechanism. The factors summarize structure under the chosen data representation; interpretation still requires scientific judgment.

Units deserve attention. If \(X\) contains counts, then its reconstructed values are on the count scale. If it contains intensities, the reconstruction is on that intensity scale. In the product \(W_{ik}H_{kj}\), the units are determined jointly by the factor conventions: one factor's scale and the other's reciprocal scale can describe the same product. Therefore, numerical entries of a factor should not be given standalone physical units or interpreted as absolute abundance without specifying how the factors were scaled.

A transformation must be checked against the nonnegative constraint. The ordinary logarithm does not generally preserve nonnegativity: for example, \(\ln(0.5)\approx-0.6931\), and \(\ln(1)=0\). It produces nonnegative values only for inputs at least 1. A transformation such as \(\ln(1+x)\) is positive when \(x>0\) and equals zero at \(x=0\), but it changes the scale and the meaning of additive combinations. A model for transformed values is not automatically a model for the original measurements.

Diagnostics and Alternative Representations

Several checks help reveal when the data model is being used inappropriately. First, verify the dimensions and orientation before fitting: rows should correspond to the defined features and columns to the defined samples, and the dimensions of \(W\), \(H\), and \(WH\) should match that convention. Second, inspect the input for negative values, missing values, and zeros. A missing measurement is not automatically a measured zero; treating it as zero changes the data being represented. Third, compare observed entries with reconstructed entries and inspect residuals by feature and sample. Large or systematic residual patterns indicate that the chosen representation may not capture important structure.

Also check whether component profiles are interpretable in context and whether their sample abundances vary in a way that is credible. A nonnegative factorization can be mathematically valid while its components are unstable or scientifically unhelpful. Examine whether conclusions depend on preprocessing choices, feature selection, or the chosen number of components. These checks diagnose the adequacy and interpretability of the representation; they do not by themselves prove that a component is a real biological source.

NMF differs from centered PCA-based representations, including methods discussed in the earlier material on Multiple Factor Analysis and joint PCA. PCA represents deviations around feature means, and its loadings and scores can be negative. NMF instead uses nonnegative profiles and abundances to represent nonnegative measurements as additive contributions. Centering before NMF can therefore defeat its basic domain constraint. Neither representation is universally preferable: the appropriate choice depends on the data scale and the scientific meaning of the desired components.

Common Mistakes and Limitations

• Reversing the factor interpretation. With features in rows of \(X\), the columns of \(W\), not \(H\), are feature profiles. The columns of \(H\) describe components within samples.
• Assuming the factors must reproduce every entry exactly. The model is generally approximate. Compare \(X\) with \(WH\), and distinguish observations from their reconstructions.
• Assuming all residuals are nonnegative. The factors and reconstruction are nonnegative, but observed-minus-reconstructed residuals can have either sign.
• Logging nonnegative data without checking the result. The ordinary logarithm of values between 0 and 1 is negative, and zero has no ordinary logarithm. Check the transformed values and the interpretation of the new scale.
• Treating any fitted component as a biological entity. A component is a pattern in a chosen representation. Its biological interpretation requires evidence beyond the factorization itself.

The basic model also has limits: its additive, nonnegative representation may not suit centered data, measurements with meaningful negative values, or relationships that are not well described by additive component contributions. A good fit alone does not establish a causal mechanism or uniquely identify biological sources.