LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Class-collision loss is bounded by within-class covariance deviation

Statement

In the one-negative latent-class model, for every bounded representation $f$ and either source loss, the population contrastive loss decomposes as

$L_{u n} (f) = τ L_{u n}^{=} (f) + (1 - τ) L_{u n}^{\neq} (f) .$

If $τ > 0$ , the same-class conditional loss obeys

$1 \leq L_{u n}^{=} (f) \leq 1 + c_{0} s (f), c_{0} : = \sqrt{2} L_{0},$

where $L_{0}$ is the common Lipschitz constant of the hinge and logistic losses. If $τ = 0$ , then $L_{u n} (f) = L_{u n}^{\neq} (f)$ and the model's convention is $s (f) = 0$ .

Statement references

Latent-class contrastive representation learning: defines $τ$ , the conditional collision/distinct losses, the covariance matrix, and $s (f)$ .

Convexity, Lipschitz bounds, and oscillation of the source contrastive losses: both losses are convex, $L_{0}$ -Lipschitz, and have value $1$ at zero.

Moments, variance, and covariance on a probability space

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: supply the covariance and operator-norm conventions.

Saunshi et al., Equations (8)–(9) and Lemma 4.4.

Saunshi et al., Supplementary Lemma A.1 and complete proof.

Facts & Assumptions

Given: The one-negative latent-class model, $f \in F$ , $τ < 1$ , and $ℓ \in {ℓ_{h}, ℓ_{l o g}}$ .

[F1]

$L_{u n}$ is the risk under independent class draws; conditioning on equality gives $L_{u n}^{=}$ and conditioning on inequality gives $L_{u n}^{\neq}$ . For $τ > 0$ , $s (f)$ averages the within-class covariance deviation under the collision-conditioned class law $ν$ . Latent-class contrastive representation learning

[F2]

The source losses are convex, $L_{0}$ -Lipschitz, and satisfy $ℓ (0) = 1$ . Convexity, Lipschitz bounds, and oscillation of the source contrastive losses

[F3]

For $Y = f (X)$ with $X \sim D_{c}$ , $Σ_{f} (c) = E [(Y - μ_{c}) (Y - μ_{c})^{⊤}]$ and $u^{⊤} Σ_{f} (c) u = E [(u^{⊤} (Y - μ_{c}))^{2}] \geq 0$ ; its operator norm is the induced Euclidean norm. Moments, variance, and covariance on a probability space, The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||

[F4]

Conditional on a collision class $c$ , the positive and negative examples are independent with common law $D_{c}$ , so their joint law is $D_{c} \otimes D_{c}$ . Independent random elements have product joint law

[F5]

Bounded measurable functions on probability product spaces are integrable, and iterated integrals agree with the product integral. Fubini's theorem for L^1 functions on a sigma-finite product

[F6]

The integral is linear on integrable functions. The Lebesgue integral is linear on $L^{1} (μ)$

[F7]

For vectors $u, v$ , $∣ u^{⊤} v ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2}$ . Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors

[F8]

For an integrable real random variable $Z$ with integrable square, $(E ∣ Z ∣)^{2} \leq E [∣ Z ∣^{2}]$ . Jensen's inequality for expectation

[F9]

For a bounded real random variable $Z$ , the Lipschitz source loss is integrable and Jensen's inequality gives $ℓ (E Z) \leq E [ℓ (Z)]$ . Jensen's inequality for expectation

Proof

Given: The objects and hypotheses in the Statement.

1.1F1givenalgebra

Partition the independent class-pair law into the collision event, of probability $τ$ , and its complement, of probability $1 - τ$ . The conditional loss on these two events is respectively $L_{u n}^{=} (f)$ and $L_{u n}^{\neq} (f)$ , giving the displayed decomposition when $τ > 0$ . If $τ = 0$ , the collision event is null and $L_{u n} (f) = L_{u n}^{\neq} (f)$ .

1.2F1F2F4F6F9given

Suppose $τ > 0$ . Fix a collision class $c$ and anchor $x$ . Conditional on $x, c$ , let $Y_{+} = f (X^{+})$ and $Y_{-} = f (X^{-})$ be independent copies with mean $μ_{c}$ . Then $E [f (x)^{⊤} (Y_{-} - Y_{+})] = 0$ by [F4] and [F6]. Jensen's inequality [F9], using convexity [F2], gives $E [ℓ (f (x)^{⊤} (Y_{-} - Y_{+}))] \geq ℓ (0) = 1$ . Averaging over $x \sim D_{c}$ and $c \sim ν$ proves $L_{u n}^{=} (f) \geq 1$ .

1.3F3F4F5F6F7givenalgebra

With $Z_{+} : = f (X^{+}) - μ_{c}$ and $Z_{-} : = f (X^{-}) - μ_{c}$ , the conditional vectors are independent, centered, and have covariance $Σ_{f} (c)$ . For each matrix entry, Fubini and product-law independence factor the cross expectation as $E [(Z_{-})_{r} (Z_{+})_{s}] = E [(Z_{-})_{r}] E [(Z_{+})_{s}] = 0$ , and likewise with $+$ and $-$ exchanged. Linearity therefore gives $E [(Z_{-} - Z_{+}) (Z_{-} - Z_{+})^{⊤}] = 2 Σ_{f} (c)$ . By [F3], $Σ_{f} (c)$ is positive semidefinite; by [F7] and the operator-norm definition, $f (x)^{⊤} Σ_{f} (c) f (x) \leq ∥ f (x) ∥_{2} ∥ Σ_{f} (c) f (x) ∥_{2} \leq ∥ Σ_{f} (c) ∥_{o p} ∥ f (x) ∥_{2}^{2}$ . Thus $E [(f (x)^{⊤} (Y_{-} - Y_{+}))^{2}] = 2 f (x)^{⊤} Σ_{f} (c) f (x) \leq 2 ∥ Σ_{f} (c) ∥_{o p} ∥ f (x) ∥_{2}^{2} .$

2.1F1F2F3F8step 1.3givenalgebra

Apply [F8] to the bounded score in step 1.3 to get $E ∣ f (x)^{⊤} (Y_{-} - Y_{+}) ∣ \leq \sqrt{2} \sqrt{∥ Σ_{f} (c) ∥_{o p}} ∥ f (x) ∥_{2}$ . By [F2], $E [ℓ (f (x)^{⊤} (Y_{-} - Y_{+}))] - 1 \leq L_{0} E ∣ f (x)^{⊤} (Y_{-} - Y_{+}) ∣$ . Average over $x \sim D_{c}$ and $c \sim ν$ to obtain $L_{u n}^{=} (f) - 1 \leq \sqrt{2} L_{0} s (f)$ .

3.1step 1.1step 1.2step 2.1∎

Steps 1.2 and 2.1 establish both bounds on $L_{u n}^{=} (f)$ , while step 1.1 gives the unconditional decomposition and the zero-collision case.

Depends on

Used by

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Sources

Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Equation (8), Equation (9), and Lemma 4.4 (standard reference, not scraped)
Nikunj Saunshi et al., Contrastive Unsupervised Representation Learning, Supplementary Lemma A.1 and proof (standard reference, not scraped)