2.5.6 · D5Unsupervised Learning
Question bank — DBSCAN density-based clustering
This page hunts the misconceptions, not the arithmetic. Each item is a one-line reveal: read the prompt, answer in your head, then check. Every answer gives you the reasoning, so a wrong guess teaches you something. For the mechanics behind these, keep the parent note open.
True or false — justify
TF — DBSCAN needs you to specify the number of clusters up front, like K-means does.
False. DBSCAN discovers the cluster count from the data via density; you supply ε and MinPts, and the number of clusters falls out of how many separate dense regions exist.
TF — Every point that ends up in a cluster is a core point.
False. Border points sit inside a cluster but have fewer than MinPts neighbours; they belong because they lie within ε of a core, not because they are dense themselves.
TF — A noise point is permanent the instant it is first labelled noise.
False. The label is tentative — a point flagged noise can later be reclaimed as a border point when a nearby core is expanded, which is why the algorithm checks "not yet in any cluster" rather than "marked noise".
TF — If A is directly density-reachable from B, then B is directly density-reachable from A.
False. Direct density-reachability is asymmetric: it requires the source to be a core point. A border A can be reached from a core B, but B may not be reachable from A because A lacks MinPts neighbours.
TF — Density-connectedness is symmetric even though density-reachability is not.
True. Two points are density-connected if some core can reach both; swapping the two points doesn't change that shared witness , so the relation is symmetric — which is exactly what lets it define clusters cleanly.
TF — Increasing MinPts always increases the number of clusters found.
False. Raising MinPts makes it harder to be a core point, so dense regions shrink or vanish and more points become noise — clusters typically merge into noise or disappear, not multiply.
TF — DBSCAN can find clusters that are long and curved, not just blob-shaped.
True. Because membership grows by chaining density-reachable neighbours, a cluster can follow any curve as long as the density holds along the path — that's why it beats K-means on the two-moons dataset.
TF — Two clusters can share a border point.
True (edge case). A border point within ε of cores from two different clusters can legitimately be reachable from both; DBSCAN assigns it to whichever core claims it first, so the result can be order-dependent for such points.
TF — If you scale one feature so its range is 10× larger, DBSCAN gives the same clusters.
False. DBSCAN uses a single ε on the raw distance, so a stretched axis dominates the distance and warps every neighbourhood — you must standardise features before clustering.
TF — Core points are never order-dependent in their labelling.
True. A core point is always density-reachable within its own cluster regardless of visit order; only the ambiguous border points can flip clusters depending on processing sequence.
Spot the error
Someone says: "I set MinPts = 1 so that every isolated point becomes its own cluster." — what's wrong?
With MinPts = 1 every point is a core point (its ε-ball always contains itself), so nothing is ever noise and the whole notion of outliers collapses — MinPts must be ≥ 2 to be meaningful.
Someone says: "F and G are 1 unit apart with ε = 1.5, so they clearly form a cluster." — what's wrong?
Closeness alone is not enough. Each needs MinPts neighbours to be core; with only 2 points and MinPts = 3, neither qualifies, so both are noise — DBSCAN deliberately refuses to name tiny groups clusters.
Someone says: "The k-distance elbow gives me the perfect MinPts." — what's wrong?
The k-distance plot tunes ε, not MinPts. You choose MinPts first (often ~2×dimensionality), then use it as the in the plot to read off the ε elbow.
Someone says: "DBSCAN scales fine to 1000-dimensional data because it's with a KD-tree." — what's wrong?
In high dimensions the tree degrades to a linear scan and distances concentrate (all pairs look equally far), so both the speed and the density contrast collapse — see Curse of Dimensionality.
Someone says: "A point with exactly MinPts − 1 neighbours plus itself is a core point." — what's wrong?
The count includes the point itself, and the rule is . With MinPts − 1 others plus itself you have exactly MinPts, so it is core — the trap is forgetting whether "including itself" is already in your tally.
Someone says: "Since clusters can have different shapes, they can also have wildly different densities and DBSCAN handles it."
False comfort. A single ε imposes one density threshold globally, so a dense cluster and a sparse cluster can't both be captured well — this is exactly why OPTICS Algorithm exists.
Why questions
Why is the neighbourhood defined with (a local radius) rather than looking at all pairwise distances?
A global view can't tell that different regions have different densities; a fixed local radius asks "is this neighbourhood crowded?", which is what "packed together" actually means.
Why does DBSCAN produce a "noise" category at all, unlike K-means?
Because density can genuinely run out — a point far from any dense region has no cluster to belong to, so DBSCAN reports it as an outlier instead of forcing it into the nearest blob (see Outlier Detection).
Why must maximality be part of the cluster definition?
Without it you could stop growing a dense region arbitrarily and split one true cluster in two; maximality forces the cluster to expand as far as density-reachability allows, giving a unique, non-arbitrary result.
Why is the rule of thumb MinPts ≈ 2 × dimensionality rather than a fixed small number?
In higher dimensions you need more points to establish that a neighbourhood is a genuine region and not just a coincidental line or plane, so the threshold scales with dimension.
Why does DBSCAN respect the "manifold structure" of interleaved crescents while K-means doesn't?
DBSCAN grows clusters by chaining along dense paths, so it travels around a curve; K-means measures straight-line distance to a centroid, which cuts straight across the gap and mixes the crescents.
Why do we mark a point noise tentatively rather than committing immediately?
A point sparse in its own neighbourhood may still fall within ε of a core discovered later; deferring the decision lets it be reclaimed as a border point instead of being wrongly discarded.
Edge cases
What happens when ε is set enormously large (bigger than the whole dataset span)?
Every point becomes core with every other point as a neighbour, so DBSCAN returns one giant cluster and zero noise — density contrast is destroyed.
What happens when ε is set extremely small (smaller than the closest pair)?
No point reaches MinPts neighbours, so everything is noise and no cluster forms at all.
What if two distinct dense clusters are joined by a thin "bridge" of medium-density points?
If the bridge points are dense enough to be core, DBSCAN chains through them and merges the two clusters into one — the single-density-threshold weakness known as the "bridging" or "chaining" effect.
What if a border point lies within ε of two cores belonging to different clusters?
It is legally reachable from both, so DBSCAN assigns it to whichever cluster is expanded first — the outcome is order-dependent for that one point, though the cores are unaffected.
What clustering happens on a perfectly uniform grid of points where every ε-ball has the same count?
Either all points are core (if that count ≥ MinPts) giving one connected cluster, or all are noise — there is no density variation for DBSCAN to exploit, so it can't carve out meaningful groups.
What does DBSCAN return if MinPts is larger than the total number of points?
No point can ever reach MinPts neighbours, so every point is noise and there are zero clusters — a sanity check worth remembering.
Recall Quick self-test
The three point types are ::: core (dense enough), border (within ε of a core but not dense), and noise (neither). The parameter tuned by the k-distance elbow is ::: ε (with = MinPts chosen beforehand). The property that makes direct density-reachability one-directional is ::: the source must be a core point, which a border point is not.