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A unit-diagonal covariance can have conditional covariance below the reciprocal global eigenvalue bound

Statement refuted

For every real symmetric positive-definite covariance matrix $Σ$ with unit diagonal, every coordinate set $S$ for which both $S$ and $S^{c}$ are nonempty and $Σ_{S S}$ is invertible, and constants $C_{\min} > 0$ , $D_{\max} < \infty$ , $C_{\max} < \infty$ , and $0 < ϵ \leq 1$ satisfying $λ_{\min} (Σ_{S S}) \geq C_{\min}, λ_{\max} (Σ) \leq C_{\max}, ∥ (Σ_{S S})^{- 1} ∥_{\infty} \leq D_{\max}, ∥ Σ_{S^{c} S} (Σ_{S S})^{- 1} ∥_{\infty} \leq 1 - ϵ,$ the Schur conditional covariance $Σ_{S^{c} ∣ S} : = Σ_{S^{c} S^{c}} - Σ_{S^{c} S} (Σ_{S S})^{- 1} Σ_{S S^{c}}$ satisfies $λ_{\min} (Σ_{S^{c} ∣ S}) \geq 1 / C_{\max}$ . Here $∥ M ∥_{\infty} = \max_{i} \sum_{j} ∣ M_{i j} ∣$ .

Statement references

Wainwright 2006, Appendix D.6

Wainwright 2009, §IV.A, equations (28)–(30)

Facts & Assumptions

Given: The matrix inequality stated above is proposed for all positive-definite unit-diagonal $Σ$ and admissible $S, C_{\max}$ .

[A1]

For a block matrix with invertible leading block, the expression $Σ_{S^{c} S^{c}} - Σ_{S^{c} S} (Σ_{S S})^{- 1} Σ_{S S^{c}}$ is the Schur complement; here it is defined by the displayed formula.

Counterexample

1.1givenconstructalgebra

Fix any integer $s \geq 1$ , set $p = 2 s$ , $S = {1, \dots, s}$ , and take $Σ = (\begin{array}{cc} I_{s} & \frac{3}{4} I_{s} \\ \frac{3}{4} I_{s} & I_{s} \end{array})$ with $C_{\min} = 1$ , $C_{\max} = 7 / 4$ , $D_{\max} = 1$ , and $ϵ = 1 / 4$ . The vectors $(u, u)$ and $(u, - u)$ have eigenvalues $7 / 4$ and $1 / 4$ for every $u \in R^{s}$ , and these subspaces span $R^{2 s}$ ; hence $Σ$ is positive definite with unit diagonal and maximum eigenvalue $C_{\max}$ . Also $Σ_{S S} = I_{s}$ , so its minimum eigenvalue is $C_{\min}$ and its inverse infinity norm is $D_{\max}$ , while $∥ Σ_{S^{c} S} Σ_{S S}^{- 1} ∥_{\infty} = 3 / 4 = 1 - ϵ$ . Thus all stated covariance hypotheses hold.

2.1A1step 1.1algebra∎

Substitution in [A1] gives $Σ_{S^{c} ∣ S} = I_{s} - (3 I_{s} / 4) I_{s}^{- 1} (3 I_{s} / 4) = 7 I_{s} / 16$ , so $λ_{\min} (Σ_{S^{c} ∣ S}) = 7 / 16 < 4 / 7 = 1 / C_{\max}$ since $49 < 64$ . Direct block multiplication gives $Σ^{- 1} = (16 / 7) (\begin{array}{cc} I_{s} & - 3 I_{s} / 4 \\ - 3 I_{s} / 4 & I_{s} \end{array})$ , whose lower-right block is $16 I_{s} / 7$ , the inverse of the conditional covariance rather than the conditional covariance itself. Thus the proposed bound is false.

Source notes

For this block family, the lower-right block of $Σ^{- 1}$ is $16 I_{s} / 7$ , the inverse of $Σ_{S^{c} ∣ S}$ , not $Σ_{S^{c} ∣ S}$ itself.

Used by

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Sources

Wainwright, Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using L1-Constrained Quadratic Programming (Lasso), 2009, §IV.A (standard reference, not scraped)
Wainwright, Sharp Thresholds for High-Dimensional and Noisy Recovery of Sparsity, 2006, Appendix D.6 (standard reference, not scraped)