Choosing an Integration Feature Space for MNNs

Why the Feature Space Matters

A mutual nearest neighbor (MNN) is defined through distances between cells, so it depends not only on which cells are present but also on which features define those distances. A cell pair can be mutual nearest neighbors in one representation and not in another. Choosing genes and dimensions is therefore part of defining the alignment problem, not a neutral preprocessing detail.

Let \(x_i\) denote the normalized expression vector for cell \(i\), with one coordinate per gene. An integration feature space is specified by a set of genes \(G\), any scaling or transformation applied to those genes, and possibly a projection to fewer dimensions. Write the resulting representation as \(z_i=T_G(x_i)\). Distances such as Euclidean distance between \(z_i\) and \(z_j\) determine the neighbor rankings. For MNNs, the cells must also be reciprocal nearest neighbors across the batches; a nearest candidate in only one direction is not, by itself, an MNN.

A useful space should make distances reflect cell-state differences that are relevant to the intended integration target, while limiting variation that is mostly technical or irrelevant to that target. This is a trade-off: excluding informative genes can erase real distinctions, while including noisy or biologically variable genes can create misleading local geometry. The shared-state assumptions and the intended target matter: integration should not be expected to align states that are genuinely absent from one batch.

How Extra Features Change Distances

For a selected set of \(p\) coordinates, squared Euclidean distance is the sum of the squared coordinate differences. Adding coordinates adds nonnegative terms to every squared distance, but it need not add the same amount for each candidate pair. A pair that is close on a few useful features can become farther than another pair if it differs substantially on added features.

A simple noise model helps explain the risk. Suppose \(q\) added coordinates are irrelevant to the cell-state distinctions of interest, and each cell has independent noise with variance \(\tau^2\) on each such coordinate. For any pair of cells, the expected added squared distance is \(2q\tau^2\). This common expectation does not necessarily change the expected ordering of pairwise distances. However, the realized noise contribution differs from pair to pair, so it can obscure small differences in the informative coordinates and change nearest-neighbor rankings. Indeed, each coordinate's squared difference has variance \(8\tau^4\), so the variance of the total added squared distance is \(8q\tau^4\), under this model.

The calculation assumes independent, mean-zero Gaussian noise in each cell and coordinate, with the same variance. Real expression data need not satisfy those assumptions; the calculation is a justification for concern about added noisy dimensions, not a complete model of expression. In practice, a feature can be noisy in one batch, dominated by dropout, or variable for reasons unrelated to the shared states being aligned.

Biological Variation Can Be Misleading for Alignment

Not all problematic features are technical noise. A gene can be measured reliably yet vary because of a condition, activation program, cell cycle, or other biological axis that is not meant to define the shared-state correspondence. If that axis differs systematically across batches, it can dominate distances and pull neighborhoods toward cells that are similar on that axis but not on the identity of interest. Earlier material on confounding is relevant here: if condition and batch are inseparable in the data, feature selection alone cannot identify which differences are batch effects and which are condition effects.

Highly variable genes are often useful because they can distinguish cell states, but high variability alone is not evidence that a gene is suitable for alignment. A batch-specific technical effect can make a gene variable, as can a genuine biological program whose differences should be preserved. Conversely, a gene with modest overall variability may still be important for a rare but relevant state. Feature selection should therefore be checked against the integration target and the experimental design rather than treated as an automatic rule.

Selected Genes and Reduced-Dimension Representations

Using selected genes directly retains a clear coordinate-to-gene interpretation. A low-dimensional representation, such as a truncated principal-component representation, can reduce the number of coordinates and concentrate broad patterns of variation. But a projection changes the distance geometry: it retains variation in directions included in the projection and discards variation in omitted directions. Principal components are ordered by total variance, not by whether a direction represents shared cell identity, technical variation, or a biological difference that should remain distinct.

For an orthonormal projection onto the first \(r\) component directions, the squared distance in the projected representation is the sum of squared differences along those directions. It is no greater than the full squared distance, because omitted squared coordinate differences are nonnegative. The ranking of pairs, however, can change: different pairs lose different amounts of distance when their differences lie in omitted directions.

Here \(v_k\) are orthonormal projection directions and \(P_r\) retains the first \(r\) directions. The inequality follows by expressing the difference in a complete orthonormal basis: the full squared norm sums the squared coefficients over all directions, while the projection sums only the first \(r\). This justification assumes an orthonormal projection and the same original coordinate scaling on both sides. If genes are rescaled before projection, that scaling defines a different geometry.

Choosing and Checking a Representation

A defensible choice begins with the biological target and then examines how plausible feature choices affect neighborhoods. There is no universally optimal number of genes or components: the answer depends on the scale of shared states, the noise structure, the normalization and scaling, and which differences the analysis is intended to preserve.

Comparing alternatives has practical limits. Selected-gene distances are interpretable but can be sensitive to which genes are included and how they are scaled. Truncated PCA can make neighborhoods less sensitive to individual genes, but may discard low-variance, biologically important directions or retain high-variance technical and condition effects. Increasing the number of dimensions preserves more of the original variation but can also admit more noisy coordinates. No option removes the need to check whether the resulting neighborhoods make sense for the intended integration target.

Warning signs include neighbor rankings that change substantially with small feature-set or dimension-count changes, matches driven by a small number of strongly varying coordinates, and apparent matches that disagree on known state markers. These signs do not prove the matches are wrong, but they call for investigation. In particular, a visually compact integrated representation is not by itself evidence that the chosen feature space produced valid cross-batch correspondences.