DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Two-class spherical Gaussian model

Definition

Assume the Axiom of Choice. For an integer $d \geq 1$ , a vector $θ \in R^{d}$ , and $σ > 0$ , the two-class spherical Gaussian model is the Borel probability law $P_{θ, σ}$ on $R^{d} \times {- 1, + 1}$ defined by $P_{θ, σ} (C) = \frac{1}{2} \sum_{y \in {- 1, + 1}} N_{d} (y θ, σ^{2} I_{d}) (C_{y}), C_{y} : = {x \in R^{d} : (x, y) \in C},$ for every Borel set $C \subseteq R^{d} \times {- 1, + 1}$ . Thus $Y$ is uniform on ${- 1, + 1}$ and, conditional on $Y = y$ , $X$ has law $N_{d} (y θ, σ^{2} I_{d})$ . In the normalized model used below, $∥ θ ∥_{2} = \sqrt{d}$ . Independent training and test observations are drawn from the corresponding finite product law.

Definition references

Multivariate normal law, including singular covariance: $N_{d} (m, Σ)$ is the Borel law whose linear projections have the corresponding scalar normal laws; its construction assumes AC.

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function.

Probability measures and probability spaces: a probability measure is a measure of total mass one.

Schmidt et al., Definition 1 and the Gaussian-model setup.

Depends on

Used by

One-sample natural error in the Gaussian model Corollary
Gaussian robust risk across dimensions and radii Example
Natural and robust risk in the Gaussian model Example
Signal alignment of a Gaussian sample mean Lemma
Robust sample lower bound in the Gaussian model Theorem
Robust sample upper bound in the Gaussian model Theorem

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ludwig Schmidt, Shibani Santurkar, Dimitris Tsipras, Kunal Talwar, and Aleksander Madry, Adversarially Robust Generalization Requires More Data, arXiv:1804.11285v2, Definition 1 (standard reference, not scraped)