Why Distances Between Embedding Islands Are Unreliable

Why the gap between islands is tempting—and risky

A two-dimensional embedding can display several compact islands with visible gaps between them. It is natural to read a larger gap as greater dissimilarity, or to interpret the direction in which one island lies as meaningful. For t-SNE and UMAP, those readings are generally not justified by the plot alone. These methods are designed primarily to represent local relationships; the arrangement of separated groups is not a calibrated map of their original-space distances.

An island here means a separated region of displayed points, not necessarily a biological cluster. The earlier material on t-SNE pairwise similarities and UMAP’s nearest-neighbor graph established the local relationships each method tries to represent. This tutorial focuses on a different question: what can the displayed geometry between separated regions tell us?

The distinction is important. If cells remain close to the same neighbors within each island, that local pattern may be informative about the embedding. But the gap between the islands, their left-to-right order, and their relative orientation can be much less constrained. In particular, a visual gap does not supply a unit of biological dissimilarity: twice the gap on the plot does not mean twice the difference in expression profiles.

A geometric justification: preserve local shape, change separation

Consider embedded points \(y_i\in\mathbb{R}^2\), and suppose the points are divided into two separated islands. Translate every point in one island by the same vector \(t\), so that \(y_i'=y_i+t\) for points in that island. For any two points \(i,j\) in that island,

Thus every within-island pairwise distance is unchanged. A common translation of the whole island also preserves its internal nearest-neighbor ordering. Yet distances from its points to the other island can change because the other island was not translated. If the translation moves the island farther away, those cross-island distances increase while its internal geometry stays fixed.

The same reasoning applies to rotation or reflection within an island. An orthogonal transformation \(R\), satisfying \(R^\mathsf{T}R=I\), preserves distances because \(\|R(y_i-y_j)\|^2=(y_i-y_j)^\mathsf{T}R^\mathsf{T}R(y_i-y_j)=\|y_i-y_j\|^2\). Rotating one island around its own center can therefore preserve its shape and local neighborhoods while changing how its points line up with points in another island. By contrast, translating or rotating the entire embedding preserves every pairwise distance; it only changes the plot’s position or orientation on the page.

This geometric argument explains why a local pattern alone cannot determine a between-island arrangement. It does not show that an embedding algorithm’s full objective is exactly unchanged by independently moving islands. Both methods include interactions beyond a list of within-island neighbors. The argument establishes that the displayed local geometry does not, by itself, fix the gap or orientation.

What the t-SNE objective does—and does not—guarantee

In t-SNE, \(p_{ij}\) represents the normalized high-dimensional similarity assigned to a pair of observations, and \(q_{ij}\) is the corresponding low-dimensional similarity. The embedding is optimized to reduce the Kullback–Leibler divergence

The low-dimensional similarities depend on distances between points, typically through a heavy-tailed Student distribution and a normalization across pairs. Consequently, moving an island can affect \(q_{ij}\) for pairs in different islands and can change the objective. It would be incorrect to say that t-SNE is indifferent to every change in between-island distance.

However, this objective does not make the plotted gap a direct readout of high-dimensional dissimilarity. The original similarities are principally local, and low-dimensional similarities are jointly normalized. A cross-island pair’s displayed distance is affected by the whole embedding and its optimization, not simply set equal to an original-space distance. The objective can favor separating points that should have low similarity, but that does not calibrate the distance between island centers in the input space. Thus, a gap may reflect optimization and the need to arrange many points, rather than a quantitative measure of how different two cell groups are.

UMAP: graph connections and nonedge repulsion

UMAP constructs a weighted neighborhood graph in the input space and seeks an embedding that represents its fuzzy connections. In a common description of its low-dimensional membership curve,

where \(a\) and \(b\) control the curve’s shape. If \(w_{ij}\) is the input graph’s fuzzy membership, the cross-entropy objective has attractive terms for connected pairs and repulsive terms for pairs with little or no input membership. For a pair, these contributions have the form

In practice, UMAP commonly uses negative sampling to approximate repulsion rather than evaluating every nonedge in full. When two islands have no graph edges between them, there is no attractive graph connection directly specifying their relative placement. Repulsion still influences their arrangement, but it does not turn the resulting gap into an estimate of an input-space distance. Optimization details and finite computation also matter.

Orientation, diagnostics, and comparisons the plot cannot support

A shared rotation of a whole embedding has no effect on any pairwise distance. More subtly, the internal orientation of one island relative to another need not encode a biological relationship. If the local shape of each island is preserved, changing one island’s orientation can alter which points appear to face or align with points in another island without changing its internal neighborhoods.

Useful diagnostics begin by asking what claim is being made. A claim about whether points remain local neighbors concerns a different property from a claim about the distance between island centers. Inspect neighborhood relationships and the original representation when the question concerns local similarity; do not substitute a gap measurement for an input-space comparison. Also check whether the apparent separation is sensitive to reasonable analysis choices. A gap that changes while internal neighborhoods appear similar is a warning against interpreting it as a stable between-group quantity.

The central comparisons that the plot alone cannot justify include ranking islands by how dissimilar they are, treating ratios of displayed gaps as ratios of biological differences, and interpreting left-right order or relative orientation as meaningful. Nor does a visible gap establish that a transition is absent in the underlying data. These conclusions require evidence beyond the displayed arrangement.

Alternatives depend on the scientific question. If the question concerns distances in a specified input representation, calculate those distances using the chosen metric. If it concerns whether groups are linked by the input neighborhood structure, examine the graph construction and its connections. If the question concerns a display method that more directly preserves large-scale linear distances, principal component analysis may be relevant, while still requiring attention to which components are displayed and what information they omit. Each choice answers a different question; none makes a t-SNE or UMAP gap a calibrated distance after the fact.

Common mistake and practical rule

• Wrong reasoning: “Island B is twice as far from island A in the plot, so the cell groups are twice as different.”
• Correction: The plotted gap is measured in embedding coordinates, not a calibrated scale of input-space dissimilarity. Compare the relevant input-space quantities directly if the scientific question is about between-group distance.
• Wrong reasoning: “The islands face each other, so their facing edges must be biologically related.”
• Correction: Relative orientation can change while each island retains its internal geometry. Orientation alone does not establish correspondence or biological relationship.