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Topographic Error Measure

Topology-quality metric — instantiates Neighborhood-Preserving Substrate Mapping

Reports the fraction of inputs whose best and second-best units are not neighbours on the grid — a single number for how often the map's local topology is broken.

Topographic Error Measure reduces "is the map's topology locally intact?" to one number: the fraction of source samples whose best and second-best matching units are not neighbours on the substrate grid. When an input's two closest representations sit side by side, local topology held there; when they land in different parts of the grid, the map has a local tear at that point. Its defining idea is that it reads the adjacency-preservation invariant directly off the substrate grid's neighbour structure, as a discrete violation rate — not a continuous distortion, not a source-side rank comparison, but a plain count of "were the top two units adjacent?" That makes it a single, comparable scalar, ideal for gating a retrain or A/B-ing two maps.

Example

A speech team trains a self-organizing map of vowel sounds so that acoustically similar vowels occupy neighbouring cells — the backbone of a pronunciation trainer that shows a learner where their vowel "landed." Before shipping, they need the topology to be sound. Topographic error checks it: for each of thousands of recorded vowels, find the best-matching cell and the second-best, and ask whether those two cells are adjacent on the grid. Suppose ≈3% of vowels have their top two cells far apart — the map has occasional local tears, points where it folded, and a vowel whose two best matches sit in opposite corners is exactly such a point. A low error certifies that "neighbouring cells = similar vowels" holds almost everywhere; a high one warns that the feedback would sometimes point learners at a misleading region of the map.

How it works

  • For each sample, locate its BMU and 2nd-BMU — the closest and second-closest prototype units.
  • Test adjacency — are those two units neighbours in the substrate grid, using the grid's own connectivity?
  • Score the violation — 1 if not adjacent, 0 if adjacent.
  • Average across all samples → the topographic error rate.

It reads the adjacency invariant against the substrate geometry; it does not build the map, does not weight by a task-relevant neighbourhood scale, and does not say where each fold sits — only how often topology breaks.

Tuning parameters

  • Grid-neighbour definition — 4- vs. 8-connectivity, or a radius; a looser neighbour rule forgives near-misses and lowers the reported error.
  • Which units are compared — 1st-vs-2nd (standard) or 1st-vs-k-th; a larger k probes topology at a slightly coarser scale.
  • Per-region vs. global rate — one scalar for the whole map, or an error field showing which regions tear the most.
  • Sample weighting — weight by frequency or by importance so tears in high-traffic regions count for more.
  • Pass/fail threshold — the error rate above which the map is rejected and retrained.

When it helps, and when it misleads

Its strength is a single interpretable, comparable scalar for topology preservation — exactly what you want to compare two trainings or gate a retrain automatically.

Its failure mode is that it is blind to magnitude and location: a 2% error could be two catastrophic folds or a scatter of harmless near-misses, and the number alone cannot tell you which; it also says nothing about how well the units fit the data. The classic misuse is loosening the grid-neighbour definition until the error looks acceptable, or reporting topographic error alone as if it certified the whole map. The discipline is to pair it with a fold scan (for where) and a quantization measure (for fit), and to freeze the neighbour rule before comparing maps.[1]

How it implements the components

  • adjacency_preservation_invariant — it is a direct violation-rate reading of exactly this invariant: how often adjacent-in-source fails to stay adjacent-in-substrate.
  • substrate_geometry_and_capacity — the test is defined by the grid's neighbour structure, so it measures against the substrate geometry that decides which units count as adjacent.

It does not compare source-side K-nearest-neighbour sets or weight by task scale — that is Neighborhood Trustworthiness and Continuity Metric — does not locate individual folds (Map Fold and Collision Scan), and does not build the map (Self-Organizing Map Training).

Notes

Topographic error trades off against quantization error — how tightly units fit the data. Shrinking the neighbourhood to lower quantization error tends to raise topographic error (units fit better but the map folds), and forcing topology to be perfect tends to raise quantization error. The pair should always be reported together; either number alone can be driven down while the map as a whole gets worse.

References

[1] Topographic error is the standard scalar diagnostic for a self-organizing map's topology preservation, and it is deliberately distinct from quantization error, which measures fit rather than ordering. A trustworthy map needs both to be low; this measure only speaks to the first. ↩