Shared And View-Specific Factors

Why Separate Shared From View-Specific Structure?

Different omics assays on the same samples can reflect both common biological variation and modality-specific variation. A sample-level pattern may appear in expression and methylation, for example, while another pattern is visible only in the expression measurements. A useful factor representation distinguishes these possibilities without requiring every factor to contribute to every view.

The key is to inspect both parts of a factorization. A score records how strongly each sample expresses a factor; a loading records how that factor contributes to features in a view. A shared factor uses the same sample scores across views, but it generally has different loadings in each view. A view-specific factor has loadings in one view and structural zeros in the others.

Definition: A shared factor is a latent sample pattern whose score vector is used to represent more than one omics view, with nonzero loadings in each view to which it contributes. A view-specific factor is a latent pattern whose loading vector is nonzero in one view and constrained to zero in the other views. “Shared” and “view-specific” describe the factorization structure; they do not by themselves establish a biological cause or independence among factors.

A Matrix Representation of the Two Types of Factors

Suppose \(M\) views measure the same \(n\) samples. Let \(X_m\) be the \(n \times p_m\) data matrix for view \(m\), with samples in rows and features in columns. Assume the rows have been aligned to the same samples in every view, and that the columns have been suitably centered or scaled for the intended analysis. A factor approximation can be written as

X_m \approx S A_m^T + T_m B_m^T + E_m,\qquad m=1,\ldots,M.

Here \(S\) is an \(n \times K_s\) matrix of shared sample scores. \(A_m\) is a \(p_m \times K_s\) loading matrix for those shared factors in view \(m\). The \(K_m\) columns of \(T_m\), an \(n \times K_m\) matrix, are view-\(m\)-specific sample scores, and \(B_m\) is the corresponding \(p_m \times K_m\) loading matrix. \(E_m\) collects what the displayed factors do not reconstruct. The approximation is an algebraic representation; it does not, on its own, specify a probability distribution for \(E_m\).

The shared scores \(S\) are the same in every view. The loading matrices \(A_m\) need not be the same: a shared sample pattern can affect different features, with different signs and magnitudes, in different assays. Conversely, \(T_m B_m^T\) contributes to \(X_m\), but has no corresponding factor contribution in another view.

This distinction is explicit when the data blocks are concatenated horizontally. Let \(X=[X_1\ \cdots\ X_M]\). The shared loading matrix is \([A_1^T\ \cdots\ A_M^T]\); each shared score column therefore has a loading block in every view. A view-specific factor instead has a loading block in its own view and zero blocks elsewhere. For two views, a compact form is

[X_1\ X_2] \approx [S\ T_1\ T_2] \begin{bmatrix} A_1^T & A_2^T\\ B_1^T & 0\\ 0 & B_2^T \end{bmatrix} + [E_1\ E_2].

The zero blocks encode the restriction, not an estimated absence that has merely been rounded to a small value. A model may omit \(T_2\) if it includes no view-2-specific factor, or omit one of the other factor groups if that structure is not being represented.

Worked Example: Reconstructing Shared and Specific Contributions

Consider two hypothetical views measured on three matched samples. Use one shared factor and one factor specific to view 1. Let the shared scores be \(S=(1,0,-1)^T\), and the view-1-specific scores be \(T_1=(0,1,0)^T\). Suppose the loadings are \(A_1=(2,1)^T\), \(A_2=(-1,2)^T\), and \(B_1=(0.5,-1)^T\). There is no view-2-specific factor in this representation.

For sample 1, the shared contribution to view 1 is \(1(2,1)=(2,1)\), and its specific contribution is \(0(0.5,-1)=(0,0)\). The reconstructed row is therefore \((2,1)\). In view 2, the shared contribution is \(1(-1,2)=(-1,2)\).

For sample 2, the shared score is zero, so its shared contribution is zero in both views. Its view-1-specific contribution is \(1(0.5,-1)=(0.5,-1)\); view 2 has no factor contribution for this sample in the displayed approximation. For sample 3, the shared score is \(-1\), giving \((-2,-1)\) in view 1 and \((1,-2)\) in view 2. The reconstructed matrices are

\widehat X_1 = \begin{bmatrix} 2 & 1\\ 0.5 & -1\\ -2 & -1 \end{bmatrix}, \qquad \widehat X_2 = \begin{bmatrix} -1 & 2\\ 0 & 0\\ 1 & -2 \end{bmatrix}.

The shared score vector is reused across both matrices, while each view has its own shared loadings. The view-1-specific score and loading contribute only to \(\widehat X_1\). These are reconstructions from the specified factors, not a claim that real measurements would equal them exactly; an observed matrix could also contain \(E_m\).

Reading the Block Pattern

A factor’s status can be read by following its column through the score and loading matrices. For a shared factor, one score column is paired with a loading column for each view. For a view-specific factor, one score column is paired with a loading column in its assigned view, and its loading blocks for the other views are zero. This block pattern is more informative than the score column alone: scores indicate which samples vary along a factor, but they do not say which features or views the factor affects.

Worked Example: Identifying Factor Types From a Block Loading Matrix

Consider two hypothetical views, each with two features, two samples, one shared factor, and one view-1-specific factor. Let \(S=(1,-1)^T\), \(T_1=(0,2)^T\), \(A_1=(2,1)^T\), \(A_2=(-1,3)^T\), and \(B_1=(1,-2)^T\). There is no view-2-specific factor. The block loading matrix, with rows ordered by view-1 then view-2 features, is

\begin{bmatrix} A_1^T & B_1^T\\ A_2^T & 0 \end{bmatrix} = \begin{bmatrix} 2 & 1\\ 1 & -2\\ -1 & 0\\ 3 & 0 \end{bmatrix}.

The first column is shared: its view-1 loading block is \((2,1)^T\) and its view-2 block is \((-1,3)^T\), both nonzero. The second column is specific to view 1: it has loading \((1,-2)^T\) there and zeros in the view-2 rows.

The reconstruction checks the interpretation. In view 1, sample 1 has contribution \(1(2,1)+0(1,-2)=(2,1)\); sample 2 has contribution \(-1(2,1)+2(1,-2)=(0,-5)\). In view 2, the shared factor gives sample 1 the contribution \(( -1,3)\) and sample 2 the contribution \((1,-3)\). The view-1-specific factor adds nothing to view 2 because its loading block is zero.

Justification: Why the Zero Blocks Matter

The decomposition follows by expanding the score-loading product for each data block. For two views, partition the score matrix into shared, view-1-specific, and view-2-specific columns. Multiplying by the block loading matrix gives

[S\ T_1\ T_2] \begin{bmatrix} A_1^T & A_2^T\\ B_1^T & 0\\ 0 & B_2^T \end{bmatrix} = [S A_1^T + T_1 B_1^T\quad S A_2^T + T_2 B_2^T].

The equality shows precisely how the structure is imposed: \(S\) appears in both reconstructed blocks, while \(T_1\) and \(T_2\) enter only their designated blocks. If a supposedly view-specific factor had nonzero loading blocks in both views, it would not have the stated view-specific structure. If a purported shared factor had a zero loading block in one view, it would not contribute to that view, even if its score column were included in the common score matrix.

This argument is an algebraic justification of the representation, not a guarantee that a data set uniquely determines the factors. Different score and loading matrices can sometimes yield the same or nearly the same fitted blocks. The labels “shared” and “specific” are meaningful relative to the imposed representation and its constraints.

Worked Example: Cross-View Covariance Is Not Proof of a Shared Factor

Take three hypothetical, centered sample scores \(s=(1,-1,0)^T\). Suppose a single shared factor has loading 2 on one feature in view 1 and loading \(-3\) on one feature in view 2. The reconstructed feature values are \(x_1=(2,-2,0)^T\) and \(x_2=(-3,3,0)^T\). Their sample covariance is

\frac{\sum_{i=1}^3(x_{1i}-\bar{x}_1)(x_{2i}-\bar{x}_2)}{3-1} = \frac{2(-3)+(-2)(3)+0(0)}{2} = \frac{-12}{2}=-6.

The negative covariance is consistent with this shared factor: its two loadings have opposite signs. It is not, however, a unique signature of a shared factor. As a counterexample, let separate view-specific score vectors both equal \(t=(1,-1,0)^T\), with loading 1 in view 1 and loading 4 in view 2. The reconstructed features are \(y_1=(1,-1,0)^T\) and \(y_2=(4,-4,0)^T\), giving sample covariance \((4+4+0)/2=4\). The factors are still view-specific by their loading structure, even though the two score vectors happen to align and their reconstructed features covary. Thus cross-view association alone does not establish that a fitted factor is shared.

Assumptions, Diagnostics, and Alternatives

This representation is useful when the views refer to aligned samples and a low-rank approximation is a reasonable description of the structure of interest. The analyst must choose how many shared and view-specific factors to include and how the data are centered or scaled. Features with very different measurement scales can otherwise dominate a fit. The interpretation also assumes that the chosen factorization has a meaningful structural distinction between shared and view-specific loading blocks; it does not assume that specific scores are statistically independent of shared scores or of scores from other views.

Useful checks include the following:

Verify sample alignment. The \(i\)th row in every view must refer to the same sample before a common score column can be interpreted across views.
Inspect loading blocks. A claimed shared factor should have meaningful loadings in each view it is said to affect. A claimed specific factor should have zeros outside its designated view by construction.
Check residual structure. If \(E_m=X_m-\widehat X_m\) retains a strong sample pattern, the factors may not capture important structure. Systematic residual patterns can also indicate that the selected number or type of factors is inadequate.
Assess sensitivity. Strong changes in the apparent shared/specific pattern under reasonable scaling choices, rank choices, or data perturbations make the interpretation less stable.
Look for confounding. A technical effect affecting several assays can appear shared. A shared factor is a cross-view statistical pattern, not proof of a common biological mechanism.

A common alternative is to fit each view separately. That permits view-specific structure without imposing common scores, but it does not explicitly represent shared sample patterns. At the other extreme, a shared-only factorization makes all represented factors common and can force view-specific variation into residuals or shared factors. A model with both shared and view-specific terms permits both patterns, at the cost of more parameters and a need to distinguish their contributions. Earlier material on joint NMF uses related shared and modality-specific component ideas under non-negativity constraints; the matrix pattern here expresses the distinction without requiring nonnegative scores or loadings.

Key takeaway: Sharedness is encoded by reusing a sample-score column across views and giving it loadings in those views. View specificity is encoded by restricting a factor’s loading support to one view. Neither correlated scores nor cross-view covariance alone determines which structure is appropriate.

Common Mistake

Wrong reasoning: “The same samples have correlated measurements in two omics views, so the factor explaining them must be shared.” Correlation can also arise when two view-specific score patterns are correlated, or through other features of the data.

Correction: Identify the factor structure from the score and loading matrices together. A shared factor uses the same score column across views and has nonzero loading blocks where it contributes. A view-specific factor has its loading block restricted to one view. Treat covariance as a diagnostic clue, not as a definition or proof of factor type.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In a two-view factorization, what distinguishes a shared score column from a view-specific score column in the loading matrix?
A factor has the same score vector used in both views, but its loading block is zero in view 2. Does it contribute to view 2 as a shared factor? Explain.
Why must the sample rows be aligned before interpreting one score column as shared across views?
Can correlated view-specific scores produce cross-view covariance? Give a brief explanation.
Name one diagnostic sign that a chosen shared/view-specific factor representation may be inadequate.

Solutions. (1) A shared column has loading blocks in the views to which it contributes; a view-specific column has nonzero loadings in one view and zero blocks elsewhere. (2) No. The zero loading block makes its reconstructed contribution to view 2 equal to zero, regardless of whether its score is included in a common score matrix. (3) Without row alignment, a score at position \(i\) would not necessarily describe the same sample in each view, so reusing that score would not encode a common sample pattern. (4) Yes. If the view-specific scores covary across samples, their separate reconstructions can covary even though their loading support is restricted to separate views. (5) Examples include strong systematic patterns remaining in the residuals, or factor assignments that change substantially under reasonable scaling or rank choices.