When Embedding Islands Do Not Mean Clusters

Why an Island Is Not Automatically a Cluster

A t-SNE or UMAP plot can show cell groups as islands separated by empty space. That appearance is useful for noticing structure, but it does not by itself show that the cells form discrete groups in the original expression-derived representation. The embedding is a low-dimensional construction that emphasizes selected aspects of neighborhood structure; its gaps are not direct measurements of gaps between cell populations.

A practical diagnostic is to ask whether an apparent separation persists when the neighborhood scale changes, and whether cells on opposite sides of the visible boundary are still neighbors in the representation used to build the embedding. The aim is not to prove or disprove a biological classification from a plot. It is to check whether the visual island interpretation is sensitive to the embedding settings or conflicts with local relationships in the higher-dimensional data.

The introductory material on embedding density explains why dense and sparse regions in a plot need cautious interpretation. Here the focus is specifically on apparent separation: a blank strip between plotted regions can coexist with high-dimensional neighbor links connecting cells across that strip.

Two Complementary Checks

First, vary the method’s neighborhood-scale parameter while keeping the cells, input representation, distance metric, and other analysis choices fixed. In t-SNE, perplexity controls the effective neighborhood scale used to define pairwise similarities. In UMAP, n_neighbors affects the scale of the nearest-neighbor information used to construct the fuzzy representation. These parameters are not interchangeable, and neither should be interpreted as a guaranteed number of visible neighbors in the final plot.

Use values that are valid for the number of cells. In particular, t-SNE perplexity must be smaller than the number of observations, and a UMAP neighborhood cannot include more than the other \(N-1\) cells when there are \(N\) cells. Implementations may cap an oversized UMAP setting; report the effective setting rather than describing it as a larger neighborhood than the data allow.

Second, inspect neighbors in the high-dimensional representation supplied to the embedding. For a proposed division into two visually separated regions, count how often a cell’s nearest neighbors lie on the other side. This makes the proposed visual boundary explicit and checks whether local connections cross it. The count is a descriptive diagnostic, not a universal significance test: there is no single cross-boundary proportion that defines a cluster.

The neighbor sets in this calculation are obtained from the same input representation and metric used for the embedding, not from distances in the two-dimensional plot. If distances are tied, the implementation’s tie-handling rule can affect which neighbors are selected; use a consistent rule across comparisons. Likewise, do not change the input representation or metric at the same time as the embedding parameter, because then a change in the plot has more than one possible explanation.

Worked Example: A t-SNE Gap With Crossing Neighbors

Worked Example: UMAP Settings Must Be Valid for the Data Size

What Parameter Sensitivity Can and Cannot Tell You

A meaningful comparison changes one scale parameter at a time. For t-SNE, examine a small set of valid perplexities rather than searching for a setting that produces a preferred picture. For UMAP, compare valid n_neighbors values spanning reasonable local and broader scales. In each case, keep the cells and input representation fixed, and record the effective settings. A parameter change alters the neighborhood information that the method tries to represent, so altered islands or bridges are a diagnostic signal—not a failure of arithmetic or an automatic classification result.

The cross-boundary fraction gives a second, complementary view. Its justification is direct: the denominator \(Nk\) counts every directed neighbor choice made by \(N\) cells when each contributes \(k\) neighbors. The numerator counts the subset whose endpoints fall on opposite sides of the proposed division. Therefore, their ratio is exactly the proportion of those choices that cross the boundary. If the visual separation were a strong local break at this scale, one would expect relatively few such links; if many links cross, the plot’s gap does not reflect a lack of local links.

This reasoning is descriptive. Neighbor links are not independent observations, so treating the \(Nk\) links as independent Bernoulli trials to calculate a standard error or p-value would generally be unjustified. Links share cells, and nearest-neighbor selection constrains which links can occur. Interpret the fraction as a summary for diagnosis, and compare it across explicitly stated scales rather than applying an arbitrary threshold.

Diagnostics, Alternatives, and Common Pitfalls

Look for agreement between the two checks rather than relying on any one visual impression. If islands merge or acquire bridges as perplexity or n_neighbors changes, and high-dimensional links frequently cross the proposed boundary, describe the islands as parameter-sensitive visual regions. If separation persists across tested scales and cross-boundary links are consistently uncommon, the evidence for local separation is stronger, but it is still not a formal cluster assignment.

A useful alternative to reading the embedding alone is to inspect the high-dimensional neighbor relationships directly, as above. Another is to color the same embedding by known cell annotations or measured features, where those are available, and ask whether the visual division aligns with independent information. Such agreement can support interpretation, but labels and markers can be incomplete or confounded; it does not turn a geometric gap into proof of discrete groups. Conversely, if an embedding changes but its high-dimensional links remain similar, the change may be primarily a consequence of the low-dimensional display.

• Pitfall: “There is empty space, so these must be different cell types.” A plotted gap is not a test of group membership. Check neighborhood-scale sensitivity and cross-boundary neighbors in the input representation.
• Pitfall: “I requested more neighbors than there are cells, so I tested a broad scale.” That setting is impossible as stated. For \(N\) cells, at most \(N-1\) other cells are available; report any implementation cap or choose a valid value.
• Pitfall: changing multiple analysis choices together. If the representation, metric, and neighborhood parameter all change, you cannot attribute a changed island pattern to the parameter alone. Hold other choices fixed for a focused diagnostic.
• Pitfall: treating a cross-link percentage as a universal cutoff. The fraction depends on \(k\), the representation, metric, and proposed boundary. Report the choices and interpret the value comparatively, not as a stand-alone decision rule.
• Pitfall: concluding that parameter sensitivity proves there are no real groups. A plot’s sensitivity weakens a claim based solely on that plot; it does not establish the opposite biological conclusion.