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Basis Conditioning and Perturbation Audit

Audit — instantiates Independent Generating Set Design

Stress-tests a basis by measuring how much small errors in the data or generators blow up in the coordinates, flagging bases that are complete but numerically fragile.

Basis Conditioning and Perturbation Audit asks a question that completeness and independence checks never reach: not does this basis span and is it independent? but how much does a tiny error amplify? Two generators can be genuinely independent yet nearly parallel, and then the coordinates that resolve them become hypersensitive — a whisper of noise in the input swings the coefficients wildly. The audit measures that amplification directly and maps the region in which the basis can be trusted. Its defining focus is the soft failure: a perfectly valid basis that is nonetheless useless on real, noisy data.

Example

An analyst fits a degree-15 polynomial to sensor readings on the interval [0, 1] using the natural monomial basis {1, x, x², …, x¹⁵}. Every monomial is independent; the set is a legitimate basis. But the audit forms the Vandermonde matrix of the sample points and reports a condition number in the billions — the high-degree monomials are nearly indistinguishable over [0, 1], so a rounding error in one reading flips the top coefficients by orders of magnitude and the fitted curve oscillates wildly. The audit then perturbs the inputs within measurement tolerance and traces the coefficient swing to confirm the fragility is real, not cosmetic. Its recommendation: swap to the near-orthogonal Chebyshev basis for the same space, whose condition number is small and whose coefficients barely move under the same noise.

How it works

The audit computes conditioning diagnostics for the basis as used — the condition number, the smallest singular value (how close to collapse), and the mutual coherence between generators (how nearly parallel any pair is). It then applies perturbations within a stated tolerance, to both the data and the generators, and traces how far the coordinates move in response. The output is an envelope: the range of inputs over which coefficients stay stable, plus a named list of the near-dependencies driving any fragility.

Tuning parameters

  • Perturbation model — random noise vs. worst-case adversarial perturbation, and its magnitude. Worst-case is honest but pessimistic; random reflects typical operation.
  • Conditioning metric — condition number vs. smallest singular value vs. mutual coherence; each catches a different flavour of fragility, and a basis can look fine on one and fail another.
  • Tolerance band — the amplification factor above which a basis is declared too fragile to trust.
  • Norm choice — which norm measures the error; a max norm surfaces the single worst-amplified direction that an average norm hides.

When it helps, and when it misleads

Its strength is catching fragility that structural checks miss entirely — the gap between a basis that works in exact arithmetic and one that survives measurement noise, finite precision, and drift. It is the difference between "this is a basis" and "this is a basis you can compute with."

Its subtlety is that conditioning is relative to scaling and norm: a frightening condition number is sometimes an artifact of un-normalized units (metres mixed with microns), fixable by rescaling rather than by redesigning the basis — so the audit can raise false alarms if run on raw, unscaled generators.[1] The matching misuse is to choose the norm or scaling that makes the conditioning look acceptable, waving a genuinely fragile basis through. The discipline is to normalize first, then audit, and to report the metric and norm alongside the number so the verdict can be checked.

How it implements the components

Basis Conditioning and Perturbation Audit realizes the stability side of the archetype's machinery — the components that govern whether coordinates survive real-world error:

  • conditioning_and_stability_envelope — its primary output: the envelope of inputs over which the coordinates stay stable.
  • basis_sensitivity_trace — the traced response of coordinates to bounded perturbations of data and generators.
  • tolerance_and_precision_policy — it sets and applies the amplification and precision thresholds the verdict is judged against.

It does not check whether the generators span or are independent (that is Basis Extraction from a Spanning Set), nor whether the design is even full rank (that is Experimental Design-Matrix Rank Check); conditioning is a question you ask only once a basis exists.

Notes

There is a hard boundary and a soft boundary. Rank-deficiency is the hard one — a generator is a genuine linear combination of the others, so no basis exists until it is removed. Ill-conditioning is the soft one — the generators are independent but so nearly aligned that the coordinates are untrustworthy. This audit lives entirely in the soft regime; the hard boundary belongs to the rank diagnostics.

References

[1] The condition number of a basis (equivalently, of the matrix whose columns are its generators) bounds how much a relative error in the input can be amplified in the coordinates. It is scale-dependent, which is exactly why normalization must precede any conditioning verdict. ↩