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QUESTION-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?

4 min read

Properties7
StatusOpen
Version2
Tagsrepresentation-learning, information-theory, mathematics
Linepragmatic-transport
RefinesQUESTION-027 — Is there a relational signature of a communicative kind that is stable under admissible transformations, sufficient for a family of decisions and discriminating on designated contrasts, and can it be identified from the available probes, for decisions outside them, when the full latent model cannot?
Refined byQUESTION-141 — For an asymmetric self-supervised objective, such as BYOL or SimSiam with a predictor and stop-gradient, does the eigenfunction characterization of spectral methods fail, or is it replaced by the singular functions of a non-self-adjoint operator?, QUESTION-182 — Does the landmark readout of a learned contrastive density ratio keep its 1/m rate under a finite second moment of the ratio alone, without a uniform bound on it?, QUESTION-183 — Does the spectral contrastive guarantee hold off a finite augmentation space, and is it non-vacuous on a real augmentation graph?, 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?
BearingTHEORY-565 — From a symmetric, positivity-preserving, positive semi-definite affinity, the normalized Markov operator is conjugate to a symmetric one, so its leading eigenfunctions scaled by powers of their eigenvalues embed the data with Euclidean distance equal to diffusion distance, and a density-corrected normalization recovers the Laplace–Beltrami operator independently of how the data were sampled, THEORY-320 — In a spectral contrastive graph that joins labelled known classes to unlabelled data, a novel labelling's linear-probe residual is exactly its part outside the top eigenspace that the labelled rows of the discarded eigenvectors cannot absorb, so which partition of the unlabelled data is discovered depends on the known classes: ones sharing the target's attribute can remove the residual, and ones sharing a rival attribute can raise it, THEORY-005 — The positive-pair density ratio is the kernel of the conditional-expectation operator on L²(p), so spectral representations are that operator's eigenfunctions, well defined when the positive-pair χ²-divergence is finite, THEORY-001 — On a finite augmentation space, InfoNCE, logistic and spectral contrastive losses share one population optimum: the positive-pair density ratio, THEORY-009 — In the positive-pair spectral self-supervised papers, self-adjointness comes from the symmetry of the positive-pair distribution, not from the Riesz representation theorem; elsewhere in the family it comes from a symmetrized loss, a precondition on the kernel, or composing the conditional-expectation operator with its adjoint, THEORY-006 — InfoNCE is a lower bound on mutual information for every critic and can never exceed the log of the batch size, THEORY-003 — Maximizing HSIC between representations and image identity maximizes the average squared MMD between the images' view distributions, whose kernel mean embeddings exist by the Riesz representation theorem

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The question

Treat “these two realizations are of the same kind” as a joint distribution over pairs. That distribution defines a conditional-expectation operator on functions of realizations. The question is whether its top eigenspace is the signature QUESTION-027 asks for: stable under the transformations, sufficient for a stated family of decisions, and discriminating between kinds. It also asks what replaces the eigenspace when the pairing is not symmetric.

Why the record needs it

QUESTION-027 asks for a signature with three properties and names no construction. This is one candidate. The mechanism transfers because the operator is defined for any symmetric pair distribution on any space. Nothing in it is specific to networks. CLAIM-098 holds that translation is directed and need not be invertible, and directedness is exactly where the construction breaks. CLAIM-164 says the definite object is the eigenspace, not a single vector. CLAIM-193 says linear probes see only the kernel, and here the kernel is the density ratio.

What the record holds

  • THEORY-007: the top eigenfunctions are “the most view-invariant directions”, and their span is minimax-optimal for linear prediction of approximately invariant targets. Limit: finite spaces, population level, linear decisions only.
  • THEORY-005: the operator is compact when “the positive-pair χ²-divergence is finite”. Limit: the condition fails when realizations determine each other almost exactly.
  • THEORY-009: self-adjointness comes “from the symmetry of the positive-pair distribution”. For directed pairs “no such account was found”.
  • THEORY-001: three losses “share one population optimum”, so the candidate signature does not depend on the objective used. Limit: low-rank truncations differ.
  • THEORY-003: HSIC maximizes the “average squared MMD” between kinds, which gives discrimination. Limit: it is injective only for a characteristic kernel.
  • THEORY-006: the information bound “can never exceed the log of the batch size” and does not track quality, so the signature is not an amount of information (compare CLAIM-043).
  • THEORY-228: at the population optimum of the spectral contrastive loss, linear-probe error is bounded by the probability that augmentation changes the label over the squared sparsest-partition conductance of the augmentation graph. So the top eigenvectors are decision-sufficient for linear classification of any labelling that rarely crosses the graph’s sparse cuts. Limit: finite augmentation spaces, symmetric pairs, the optimum of one loss; nothing about what training reaches.
  • 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 worst-case linear-probe error over tasks of high compatibility, though other spans can tie. This is the singular-function result asked for below, for linear decisions, between two different spaces. Limit: population level and linear probes only, and the span is not unique: other spans can tie. The thesis never works out a directed kernel on a single space. THEORY-239’s note of 2026-10-19 does, with a non-reversible three-state chain, and the guarantee sits on the left singular functions.
  • THEORY-319: when two views are redundant for a label, a linear readout of random evaluations of the density ratio approaches the near-Bayes predictor, with no symmetry between the views. Limit: the full kernel through landmarks, not a low-dimensional eigenspace, and the exact ratio at the population level.

What would answer it

A singular-function result for a non-symmetric pair operator, with its own sufficiency guarantee. Or a showing that directed transports admit no such signature.

Note of 2026-10-12: the thesis this candidate would instantiate

PR #121 filed CLAIM-209 as a proposed answer to QUESTION-027, and it says “No signature is yet named”. The top eigenspace asked about here is a named candidate for the signature that thesis needs. The directed case left open here is where that thesis has no construction, so a showing that directed transports admit no such signature would bear on its first defeat disjunct. CLAIM-210, Active, holds that “kind-sharing is symmetric”, while fidelity belongs to a transport from a given source. On that reading a distribution over same-kind pairs is symmetric and directedness enters with transport; whether that removes this question’s second half or moves it to decisions about transported targets is not settled here.

Split, 2026-10-21 (ADR-049). For linear decisions at the population level the question is answered, and that part is QUESTION-245. For symmetric pairs the top eigenspace is a stable, minimax-optimal span (THEORY-007, THEORY-228). For directed pairs, on two spaces or on one, the top left singular span takes that role (THEORY-239), and the full density ratio suffices for redundant labels (THEORY-319). Other spans can tie. This question keeps what they leave: whether any span is the kind’s signature for nonlinear decisions, and what makes one span the signature when others tie. Whether a directed pair distribution describes a kind or a transport (CLAIM-210) also stays here.

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