Weighted Nearest Neighbors for Multimodal Cells

Why Combine Neighborhoods Rather Than Measurements?

A cell can resemble one set of cells in its RNA profile and a different set in its protein measurements. This is not necessarily a contradiction: the modalities record different molecular features, and their usefulness can vary across cells. Weighted nearest neighbors (WNN) combines the evidence from modality-specific neighborhoods to build a joint neighborhood graph. The central operation is local weighting: each cell’s RNA and protein similarities contribute according to modality weights for that cell.

WNN operates on representations of the modalities, such as low-dimensional RNA and protein embeddings, rather than requiring raw counts from different assays to be added together. The earlier material on scaling modalities and canonical correlation provides relevant context: representation and preprocessing affect distances, but WNN’s task is to combine neighborhood evidence after those representations have been chosen.

From Modality-Specific Neighborhoods to a Joint Graph

First, represent every paired cell in each modality. For example, RNA may be represented by selected principal components, while antibody-derived protein measurements may be represented by a normalized and reduced protein embedding. For each modality separately, calculate distances between cells and identify nearby cells. Convert distances or neighbor relations into affinities: a close cell can receive a larger affinity than a more distant cell. Affinity construction may use a distance kernel or a locally scaled distance, and the exact transformation depends on the implementation.

The modalities need not use the same coordinate system or have comparable raw distances. The key requirement is that each modality’s affinity values carry a meaningful within-modality interpretation. A similarity of 0.8 in RNA is not automatically equivalent to a raw distance of 0.8 in protein; the joint calculation uses affinities and weights, not unstandardized cross-modality distances.

In a WNN procedure, local modality weights are estimated from how informative each modality is for each cell. A common principle is to compare how well a cell’s observations in one modality can be predicted using neighbors identified in that modality versus neighbors identified through another modality. A modality that provides more useful local predictive information receives more weight for that cell. Implementations can differ in the prediction-error calculation, distance kernels, and transformations used to obtain normalized weights; those details should be reported when results need to be reproduced.

Here \(S_{ij}\) is the combined score of cell \(j\) as a neighbor of cell \(i\), \(M\) is the number of modalities, \(w_i^{(m)}\) is cell \(i\)'s weight for modality \(m\), and \(A_{ij}^{(m)}\) is the affinity between \(i\) and \(j\) in that modality. A high score means that the combined, locally weighted evidence supports the pair as neighbors. The WNN neighborhood for \(i\) is formed by selecting cells with high joint scores, subject to the graph-construction rule used by the analysis.

This weighted-sum form has a simple justification. Imagine selecting one modality at random for evaluating the pair \((i,j)\), with probability \(w_i^{(m)}\) of selecting modality \(m\). The expected affinity under that selection is

Thus the combined score is the average affinity supported by the modalities, weighted by their local contributions. This does not claim that the cell’s biological state is literally generated by randomly choosing one assay; it explains the arithmetic of the aggregation.

Worked Calculations

Assumptions, Diagnostics, and Alternatives

The construction is meaningful only if its inputs are suitable. Cells must be correctly paired across modalities, and the measurements used to build each representation must be sufficiently reliable for local distances to be informative. The neighborhood size and affinity transformation should not make nearly all cells equally similar or isolate most cells as unconnected. Local weights must be nonnegative and normalized if the stated weighted-average interpretation is used. Before interpreting a joint graph, check that modality-specific neighborhoods are not dominated by technical artifacts, and that the resulting graph is not determined solely by an unintended scale or preprocessing choice.

Useful checks include examining the distribution of modality-specific neighbor counts and affinities, checking whether the joint graph contains plausible connectivity, and comparing joint neighbor rankings with the separate modality rankings for selected cells. If a small preprocessing change radically rearranges the graph, or if many cells have nearly identical affinities to almost every candidate, the neighborhood structure may be unstable or uninformative. These checks identify warning signs; they do not by themselves prove that a graph is biologically valid.

WNN is one choice among several. Separate modality graphs preserve each assay’s neighborhood structure but do not provide one combined graph. Concatenating features or embeddings before calculating neighbors gives modalities influence through the concatenated representation and its scaling; it does not directly use cell-specific neighborhood weights. Canonical correlation can identify paired directions of shared variation, but it is a different operation from weighting modality-specific affinities to construct a graph. The choice depends on the analysis goal, the quality of each modality, and whether a single joint neighborhood is needed.

WNN’s main advantage is flexibility: different modalities can contribute differently from cell to cell, while retaining their own neighborhood calculations. Its limitations follow from the same design. A poor representation or unreliable modality can distort affinities; an unsuitable weight-estimation procedure can then amplify or suppress that distortion. The graph also depends on implementation choices such as neighborhood size, affinity kernel, and graph symmetrization. Report these choices and treat the joint graph as an analysis representation whose sensitivity should be checked, rather than as a uniquely determined account of cellular similarity.