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QUESTION-286: Under smooth invertible changes of the parameter coordinates, which first-order frustration diagnoses among a learner's demands are invariant: pairwise Euclidean gradient angles, common strict first-order descent, or budgeted feasibility with the update geometry transported?

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Properties5
StatusAnswered
Tagsinformation-geometry, loss-landscapes, representation-learning
Linelearned-structure
RefinesQUESTION-127 — Which diagnoses of frustration among a learner's demands survive quotienting by function-preserving symmetries, which belong only to an update geometry, and which only to a particular training organization with its optimizer?
Cited as “answers” byCLAIM-366 — Whether several objectives share a strict first-order descent direction, 0 ∉ conv{g_i}, is invariant under smooth invertible reparameterization, while the sign of a Euclidean gradient inner product is not, and a negative inner product does not exclude common descent; a budgeted version is invariant only with the metric pulled back

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The question

A35 §2 of the exchange of 2026-10-17 sorted five diagnostics by their expected invariance. The first three are about charts of one parameter space, and this question asks about those three.

The answer

CLAIM-366, Active, proves all three rows:

  • Common strict first-order descent (0 ∉ conv{g_i}) is invariant under smooth invertible reparameterization. The chart’s Jacobian is invertible, so it maps the descent cone of one chart onto that of the other.
  • The sign of a Euclidean gradient inner product is not invariant. A negative inner product does not exclude common descent.
  • Budgeted feasibility is invariant once the update geometry is pulled back along the change of chart.

The other two rows, function-preserving symmetries and finite-step trajectories, are not changes of chart, and they remain QUESTION-127’s.

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