Alphabeta MathHow 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.
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- 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.
Single-block sparse covariance class and restricted directions
Definition
For , let and, for an integer , let Here is the diagonal matrix with the same diagonal entries as . Thus all nonzero off-diagonal entries lie in one common coordinate block of size at most ; no row-sparsity condition is implied.
For , define When , the restricted direction set is
Depends on
Used by
- Matrix depth and covariance estimators Definition
- A covariance matrix with one correlated block Example
- A matrix-depth maximizer exists on the bounded covariance classes Lemma
- Clean block-sparse Gaussian covariance has squared operator lower rate s log(ep/s) over n Lemma
- Huber covariance modulus lower bound for unrestricted matrix-valued actions Lemma
- Measurable selection of a matrix-depth maximizer Lemma
- Operator error bound for the Gaussian Huber matrix-depth estimator Lemma
- The Huber covariance contamination modulus has a quadratic lower bound Lemma
- Single-block sparse Gaussian Huber covariance: high-probability upper and low-ratio lower bounds Theorem
Dependency tree · two levels
7 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
- Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, §3.4 (standard reference, not scraped)