Matching Factors Under Sign And Permutation Ambiguity

Why Factor Matches Need Alignment

Two fits of a latent factor model can describe similar structure while listing factors in different orders or with opposite signs. Neither change necessarily reflects a biological difference: permuting factor columns and applying a corresponding permutation to scores leaves the model representation unchanged, while reversing a factor’s sign in both its scores and loadings also leaves the fitted signal unchanged. Directly comparing “factor 1” from one fit with “factor 1” from another can therefore be misleading.

The task here is to match factors across two fitted solutions using their loading profiles. A loading profile is the vector of feature loadings for one factor. The comparison requires the same features, in the same order, in both profiles. This tutorial focuses on sign reversals and permutations; it does not treat general rotations that mix several factors together.

Before calculating similarities, confirm that the two fits use comparable loading scales and the same feature definitions. If features were transformed or standardized differently, apparent differences may reflect preprocessing rather than factors. Restrict both profiles to the same shared features, aligned in the same order. A factor with a zero loading vector cannot be compared by cosine similarity.

Measuring Similarity While Allowing Sign Reversal

A useful scale-invariant comparison is cosine similarity. It is the dot product of two loading vectors divided by the product of their Euclidean lengths. Cosine similarity is \(1\) for identical directions, \(-1\) for opposite directions, and \(0\) for orthogonal vectors. Since opposite signs can represent the same factor, use its absolute value when assessing a potential match.

Here \(c_{ab}\) is the signed cosine and \(s_{ab}\), between 0 and 1, is the sign-invariant similarity. When a pair is matched, choose the sign that makes its signed cosine nonnegative: retain \(l^{(B)}_b\) if \(c_{ab}\geq 0\), and replace it by \(-l^{(B)}_b\) if \(c_{ab}<0\). This convention aligns loading directions; the corresponding factor scores must also have their signs reversed if the fitted representation is to remain unchanged.

Cosine similarity compares directions and ignores overall magnitude. That is helpful when fits have different factor scales, but it also means that a small loading vector can appear highly similar to a large one. If magnitudes are scientifically meaningful and the fits use comparable scales, record them separately rather than expecting cosine similarity to assess them. Correlation between loading columns is another option when similarity of centered feature patterns is the target. It centers each vector before comparison, so it answers a different question and can be unstable when a centered vector has very little variation.

One-to-One Assignment

After computing all pairwise similarities, choose the matching that maximizes their total. A one-to-one assignment means each factor is used at most once: one factor in fit B cannot be assigned to two factors in fit A. For equal factor counts \(K\), let \(\pi\) be a permutation of \(1,\ldots,K\), where A factor \(a\) is matched to B factor \(\pi(a)\). Choose the permutation maximizing the sum of absolute cosines.

This is a linear assignment problem and can be solved by the Hungarian algorithm. The objective is a global sum, not a sequence of independent choices of the largest entry in each row. For unequal numbers of factors, a rectangular assignment can match every factor on the smaller side to a distinct factor on the larger side. If the scientific goal is to permit unmatched factors on either side, add dummy assignments with a prespecified score or use an explicit match threshold; report that choice because it changes which pairings are allowed.

Interpreting Match Quality and Its Limits

The assignment returns a best pairing under the chosen similarity measure; it does not certify that the pairing is strong or uniquely interpretable. Inspect the signed and absolute similarity matrices, the selected similarities, and plausible competing assignments. A selected similarity close to 1 indicates similar loading directions on the shared features. A low selected value means the best available match is still weak. Similar totals for two different assignments indicate ambiguity even if every selected pair looks moderately similar.

Do not treat a universal cutoff as a definition of a valid factor match. What counts as adequate similarity depends on the number and type of shared features, loading sparsity, noise, and the intended interpretation. A factor may also look different because its signal is distributed across several columns in one fit. Sign-and-permutation matching cannot resolve that case: it only tests whether one column resembles another up to sign.

The main alternative to a one-to-one assignment is manual or independent pairwise matching. It can be useful for small problems and for inspecting biologically meaningful loading patterns, but independent comparisons can assign the same factor twice and need not maximize overall agreement. A correlation-based similarity may better suit centered pattern comparison, while a signed dot product on a common scale retains both direction and magnitude; each changes the target of matching. Whatever measure is chosen, use it consistently and state it.

General rotations are a more substantial limitation. If two solutions differ by a rotation that mixes factors, no permutation and sign changes alone need align their columns. That situation calls for a method designed to compare subspaces or to align rotated solutions, not a claim that a poor one-to-one match demonstrates absence of shared structure. Matching also does not establish biological identity or causal meaning; it is a descriptive comparison of fitted loading profiles.

Practical Alignment and Diagnostics

Common warning signs include inconsistent feature ordering, incomparable transformations, near-tied assignments, low similarities, zero or nearly zero loading columns, and matches driven by a small number of features. If several fits are being compared, a reference fit and consistent preprocessing can make results easier to describe, but the chosen reference can affect the labels. Matching is an alignment step, not a substitute for examining the loading patterns or the fitted model.