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QUESTION-245: For linear decisions at the population level, if a kind's admissible transformations are given as a distribution over pairs of its realizations, is the top eigenspace of the pair operator a stable, decision-sufficient span, and what span takes that role when the pairs are directed?

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StatusAnswered
Tagsrepresentation-learning, information-theory, mathematics
Linepragmatic-transport
RefinesQUESTION-052 — Beyond linear decisions at the population level, if a kind's admissible transformations are given as a distribution over pairs of its realizations, is the top eigenspace of the pair operator, or the top left singular space when the transformations are directed, its stable, decision-sufficient signature, and what makes one span the signature when other spans tie?
Answered byTHEORY-007 — Kernel PCA under the positive-pair density ratio recovers the eigenfunctions of the positive-pair Markov chain, and their top span is minimax-optimal for linear prediction of approximately view-invariant targets, THEORY-228 — On a finite augmentation space, at the population optimum of the spectral contrastive loss with k ≥ 2r, linear-probe error is bounded, up to constant and logarithmic factors, by the error of recovering the label from an augmentation divided by the squared sparsest-partition conductance of the augmentation graph, THEORY-239 — For any joint law of an input and a context, symmetric or not, the top-d left singular functions of the conditional-expectation operator attain the minimum, over d-dimensional spans, of worst-case linear-probe approximation error over tasks of compatibility at least 1 − ε, namely (s₁² − (1 − ε)²)/(s₁² − s²_{d+1}) when s_{d+1} < 1 − ε ≤ s₁, though other spans can tie, THEORY-319 — When two views are redundant for a label, the predictor that infers one view from the other is within εX + 2√(εXεZ) + εZ of the Bayes predictor in mean squared error, and it is linear in the contrastive density ratio, so for a bounded label and a square-integrable ratio a linear readout of m random evaluations of the exact ratio approaches it at rate 1/m, with no symmetry between the views required

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Answer

Yes for symmetric pairs, and the top left singular span for directed ones. Each result is at the population level, for linear probes, and for a stated class of targets. This is the part of QUESTION-052 that the Active readings settle, split off on 2026-10-21 (ADR-049).

  • Symmetric, stability and optimality. THEORY-007 (Active), Johnson, El Hanchi & Maddison. On a finite augmentation space, the top eigenfunctions of the positive-pair chain are the most view-invariant directions (Prop. 3.1). Their span is minimax-optimal for linear prediction of approximately view-invariant targets, given d ≥ d*, the number of communicating classes (Thm 4.1).
  • Symmetric, classification. THEORY-228 (Active), HaoChen et al. On a finite augmentation space, every population minimizer of the spectral contrastive loss spans the top eigenvectors. Its linear-probe error is Õ(α/ρ²), the error of recovering the label from an augmentation over the squared sparsest-partition conductance (Thm 3.8).
  • Directed. THEORY-239 (Active), Zhai. For any joint law, symmetric or not, the top-d left singular functions of the conditional-expectation operator attain the minimum worst-case linear-probe error over tasks of compatibility at least 1 − ε. Its note of 2026-10-19 re-derives this without symmetry and works a non-reversible three-state chain on one space: the left and right singular functions differ, and the guarantee sits on the left ones.
  • Directed, the whole kernel. THEORY-319 (Active), Tosh, Krishnamurthy & Hsu. When two views are redundant for a label, a linear readout of random evaluations of the density ratio approaches the near-Bayes predictor at rate 1/m, with no symmetry between the views.

So the answer is “a minimax-optimal span”, not “the signature”. THEORY-239’s note shows other spans can tie, even with a gap in the singular values.

What it leaves to QUESTION-052

Whether any span is the kind’s signature for nonlinear decisions, and what picks one out when other spans tie. None of the four says what a trained network reaches. THEORY-007 and THEORY-228 hold on finite spaces only. The task classes differ: view invariance, label recoverability with conductance, compatibility, and redundancy. CLAIM-210 holds that kind-sharing is symmetric, so whether a directed pair distribution describes a kind at all, or a transport, also stays with the parent.

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