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Coordinate Round-Trip Test

Test protocol — instantiates Independent Generating Set Design

Encodes a known object into coordinates and reconstructs it, checking that decode-of-encode returns the original — an end-to-end proof that the basis represents faithfully and uniquely.

Coordinate Round-Trip Test takes a known object, encodes it into coordinates, reconstructs it, and checks that what comes back matches what went in — that decode ∘ encode ≈ identity. Its defining trait is that it is behavioral: it exercises the entire encode/decode pipeline on real objects rather than inspecting any matrix, so it catches faults that a structural check cannot see — a subtle non-invertibility, a lossy encoder, a boundary case where two distinct objects collapse to the same coordinates. Where Experimental Design-Matrix Rank Check inspects a design's rank before any data exists, the round-trip test runs the machinery and watches what actually happens.

Example

A mapping pipeline converts GPS positions in WGS84 latitude/longitude into UTM grid coordinates for local analysis, then back again for display. The round-trip test feeds it a curated set of known survey points — city centres, but also the hard cases: points straddling a UTM zone boundary, near the poles, on the antimeridian. Each point is projected to UTM, inverted back to lat/long, and compared to the original; a pass requires agreement within a set tolerance (say, under a centimetre). Most points round-trip cleanly, but the zone-boundary cases return several metres off — surfacing that the projection is not cleanly invertible there. The failure is invisible in the projection formulas; only running real coordinates through and back reveals it.

How it works

The protocol assembles a validation set that is representative and adversarial — typical objects plus corners, edges, and degeneracies. Each object is encoded, then decoded, and the reconstruction error is measured against a stated tolerance. Separately, the test probes uniqueness: it checks that no two distinct objects encode to the same coordinates (that the encoding is injective across the case set). A pass demands both — small reconstruction error and no coordinate collisions.

Tuning parameters

  • Validation case set — random, representative, or adversarial. Adversarial cases (edges, boundaries, near-degeneracies) are where representation failures actually live, so their inclusion largely determines what the test can catch.
  • Reconstruction tolerance — how close counts as "recovered." Too loose passes a broken basis; too tight fails on benign floating-point rounding.
  • Error metric — max vs. mean vs. relative error. A max norm exposes the single catastrophic point that a mean would average away.
  • Uniqueness probe — whether, and how aggressively, to search for distinct inputs that share coordinates.

When it helps, and when it misleads

Its strength is empirical honesty: by driving real objects through the real pipeline, it certifies faithful, unique representation in a way no amount of formula-inspection can, and it localizes failures to specific cases you can then examine.

Its intrinsic limit is coverage — it certifies only the cases you test, so a thin or comfortable validation set yields false confidence.[1] The matching misuse is loosening the tolerance until a broken basis passes, converting the test from a check into a rubber stamp. The discipline is to fix the tolerance and lock in adversarial cases before running, and to treat a pass as exactly as strong as the case set's coverage — no stronger.

How it implements the components

Coordinate Round-Trip Test realizes the empirical-validation side of the archetype's machinery — the components that certify a representation actually behaves:

  • coordinate_uniqueness_and_identifiability_invariant — the injectivity probe certifies that distinct objects receive distinct coordinates, i.e. the representation is identifiable.
  • basis_validation_case_set — the curated objects it round-trips are this validation case set, and its quality is the test's reach.

It does not build the encoder (Change-of-Basis Matrix) or the synthesis map (Dual-Basis Transform) — it consumes and exercises those; nor does it check identifiability structurally from rank before data (Experimental Design-Matrix Rank Check).

Notes

A round-trip pass certifies faithful reconstruction on the tested cases — nothing about stability under noise. A basis can round-trip a clean object perfectly and still amplify measurement error catastrophically; that is the conditioning audit's question, not this one. The two are complementary acceptance checks: one asks "does it come back?", the other "does it survive being jostled?".

References

[1] When an encoder has an exact left inverse, decode ∘ encode is the identity and reconstruction is lossless — the property this test probes empirically. In signal processing the same guarantee is called perfect reconstruction. ↩