ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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An augmentation that erases the label-relevant class coordinate

Example

Let the raw input be $(c, n) \in {- 1, + 1}^{2}$ , where the latent class is $c$ and, conditional on either class, the nuisance coordinate $n$ is uniform on ${- 1, + 1}$ . Apply the augmentation $T (c, n) = n$ before forming positive pairs and negative samples. The resulting input law $D_{c}$ is the same uniform law for both classes, so $D_{s i m} = D_{n e g} \otimes D_{n e g}$ and the observed triplets carry no class-dependent distributional signal. Consequently, every deterministic representation of the augmented input is independent of the binary task label, and its optimal average hinge risk for a uniform-label two-class task is exactly $1$ ; the erased class coordinate is unavailable to both the contrastive learner and the downstream linear classifier.

Example references

Latent-class contrastive representation learning: defines the class-conditional post-augmentation laws and binary tasks.

Convexity, Lipschitz bounds, and oscillation of the source contrastive losses: defines the hinge loss used to compute the downstream risk.

Chen et al., Section 3: explains how augmentation choices define the predictive tasks in contrastive learning.

Facts & Assumptions

Given: The two latent classes, the nuisance law, the deterministic augmentation $T (c, n) = n$ , and a deterministic representation $f$ of the augmented input.

[F1]

After augmentation, $D_{+ 1} = D_{- 1}$ is uniform on ${- 1, + 1}$ , so the observed input has the same law under both labels. Latent-class contrastive representation learning

[F2]

For a fixed score $z = w^{⊤} f (x)$ , the average hinge loss over uniform labels is $\frac{1}{2} [(1 - z)_{+} + (1 + z)_{+}]$ , which is at least $1$ for every real $z$ ; choosing $w = 0$ gives risk $1$ . Convexity, Lipschitz bounds, and oscillation of the source contrastive losses

Proof

Given: The finite model in the Example.

1.1F1given

Since $T (c, n) = n$ and $n$ is uniform independently of $c$ , each post-augmentation class law is the same uniform law $U$ . The mixture formula gives $D_{n e g} = U$ and $D_{s i m} (A \times B) = U (A) U (B)$ for all subsets $A, B$ of the finite input space, hence $D_{s i m} = U \otimes U$ . Thus the observed triplet law is $U \otimes U \otimes U$ , independent of the latent task labels, and every score $z = w^{⊤} f (n)$ has the same distribution under both labels.

2.1F1F2step 1.1givenalgebra∎

For each fixed score $z$ , if $∣ z ∣ \leq 1$ then $(1 - z)_{+} + (1 + z)_{+} = 2$ . If $z \geq 1$ , the sum is $0 + (1 + z) \geq 2$ ; if $z \leq - 1$ , it is $(1 - z) + 0 \geq 2$ . Therefore [F2] gives average risk at least $1$ after integrating over the common input law, and $w = 0$ attains $1$ . The class coordinate erased by $T$ cannot be recovered by this downstream classifier.

Depends on

Used by

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Sources

Ting Chen et al., A Simple Framework for Contrastive Learning of Visual Representations, Section 3 (standard reference, not scraped)