bernoulli_tangent_sampling_concentration_formula_bound_dense_positive_samples

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by Minghui · Jun 24, 2026 · Mathlib c5ea003 (Lean v4.30.0)

candes-rechtconcentrationexact-matrix-completionformal-bridgepositive-samplestangent-sampling

This is a formal bridge for the dense Bernoulli tangent-sampling concentration route with the Lean-required positive-samples guard made explicit.

Source: Candes--Recht, Exact Matrix Completion via Convex Optimization, PDF p. 18, Section 4.2, Theorem 4.1, equation (4.5), for the tangent sampling concentration target; PDF p. 19, Section 4.2, Theorem 4.2, equation (4.9), for the Rudelson expectation estimate behind the deviation child; PDF p. 24, Section 6.1, Lemma 6.1 and equations (6.5)--(6.7), for the fixed-matrix/Rudelson estimate used in that expectation branch; and Appendix 9.1, PDF p. 46, Theorem 9.1 / equation (9.2), for the Talagrand product-space tail feeding the dense deviation bridge.

Mathematical statement: let $M$ be an $n_1\times n_2$ rank- $r$ matrix with SVD data $S$ , let $n=\max(n_1,n_2)$ , and let $p=m/(n_1n_2)$ be the Bernoulli sampling rate for the random sample set $\Omega$ . Assume $0<n_1$ , $0<n_2$ , $0<r$ , $0<m\le n_1n_2$ , $1\le\mu_0$ , $1\le\mu_1$ , and the incoherence conditions $A0(S,\mu_0)$ and $A1(S,\mu_1)$ . If $\beta>2$ and

m\ge \beta\mu_0 n r\log n,

then there are universal constants $C,c>0$ such that

\mathbb P_p\{\operatorname{TangentSamplingConcentration}(\Omega,S,p,\mathrm{scale}(C,\beta,\mu_0,n,r,m))\} \ge 1-c n^{-\beta}.

Here TangentSamplingConcentration is the pointwise tangent-operator concentration event.

Formalization note: this is a formal bridge, not a new analytic concentration theorem. It composes the source-backed child bernoulli_tangent_sampling_deviation_formula_bound_dense_positive_samples with the already-proved source-backed child bernoulli_tangent_sampling_concentration_from_positive_deviation_bound, using the local arithmetic fact that $0<m$ and $0<n_1,n_2$ imply $0<p\le1$ . The explicit $0<m$ hypothesis is deliberate and this node should not be read as closing the older dense concentration theorem without a positive-sample guard.

Preamble

import Definitions.Def_matrix_completion_tangent

open MatrixCompletion

Formal statement

theorem bernoulli_tangent_sampling_concentration_formula_bound_dense_positive_samples :
    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
      ∀ (β : ℝ), 2 < β →
      ∀ (n₁ n₂ r m : ℕ) (M : Matrix (Fin n₁) (Fin n₂) ℝ)
        (μ₀ μ₁ : ℝ) (S : SVD M r),
        0 < n₁ → 0 < n₂ → 0 < r → 0 < m → m ≤ n₁ * n₂ →
        1 ≤ μ₀ → 1 ≤ μ₁ →
        A0 S μ₀ → A1 S μ₁ →
        (m : ℝ) ≥ β * μ₀ * (↑(max n₁ n₂)) * (r : ℝ) *
          Real.log (↑(max n₁ n₂)) →
        bernoulliEventProb ((m : ℝ) / ((n₁ : ℝ) * (n₂ : ℝ)))
            (fun Omega =>
              TangentSamplingConcentration Omega S
                ((m : ℝ) / ((n₁ : ℝ) * (n₂ : ℝ)))
                (tangentSamplingDeviationScale C β μ₀ (max n₁ n₂) r m)) ≥
          1 - c * Real.rpow (↑(max n₁ n₂)) (-β) := by sorry

Source

Candes, Emmanuel, and Benjamin Recht. Exact Matrix Completion via Convex Optimization. arXiv:0805.4471 / Foundations of Computational Mathematics 9 (2009). Exact locations: PDF p. 18, Section 4.2, Theorem 4.1, equation (4.5); PDF p. 19, Section 4.2, Theorem 4.2, equation (4.9); PDF p. 24, Section 6.1, Lemma 6.1, equations (6.5)--(6.7); Appendix 9.1, PDF p. 46, Theorem 9.1 / equation (9.2).

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