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The q=2 ellipsoid extremal profile minimizes the fourth-power objective

Statement

Assume the minimax detection-risk definition, $q = 2$ , and positive nondecreasing sequences $a_{j}$ and $σ_{j}$ with $a_{j} \to \infty$ . Let $A, z_{0} > 0$ and suppose $S = {j : A a_{j}^{2} < 1}$ is finite and nonempty. Define ${\tilde{η}}_{j}^{2} = z_{0}^{2} σ_{j}^{2} (1 - A a_{j}^{2})_{+}$ and assume $\sum_{j \geq 1} σ_{j}^{2} {\tilde{η}}_{j}^{2} = r^{2}, \sum_{j \geq 1} a_{j}^{2} σ_{j}^{2} {\tilde{η}}_{j}^{2} = 1.$ Then $\tilde{η} \in Θ_{2} (r)$ and minimizes $\sum_{j \geq 1} η_{j}^{4}$ over $Θ_{2} (r)$ ; the nonnegative minimizer is unique.

Statement references

Ingster, Sapatinas and Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, equations (4.4)–(4.8)

Marteau and Sapatinas, A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection, §3.1

Facts & Assumptions

Given: The $q = 2$ class $Θ_{2} (r)$ , positive nondecreasing $a_{j}, σ_{j}$ with $a_{j} \to \infty$ , a finite nonempty active set $S = {j : A a_{j}^{2} < 1}$ , positive $A, z_{0}$ , and the profile ${\tilde{η}}_{j}^{2} = z_{0}^{2} σ_{j}^{2} (1 - A a_{j}^{2})_{+}$ satisfying the two displayed constraints.

[F1]

Minimax detection risk and separation rate: $η \in Θ_{2} (r)$ means $\sum_{j} a_{j}^{2} σ_{j}^{2} η_{j}^{2} \leq 1$ and $\sum_{j} σ_{j}^{2} η_{j}^{2} \geq r^{2}$ , with $(σ_{j} η_{j}) \in ℓ^{2}$ .

Proof

1.1givenF1

Since $S$ is finite, $\tilde{η}$ has finite support. The second constraint in the Given states $\sum_{j} a_{j}^{2} σ_{j}^{2} {\tilde{η}}_{j}^{2} = 1$ , and the first states $\sum_{j} σ_{j}^{2} {\tilde{η}}_{j}^{2} = r^{2}$ . Thus [F1] gives $\tilde{η} \in Θ_{2} (r)$ .

1.2F1givenalgebra

Fix any $η \in Θ_{2} (r)$ and put $x_{j} = η_{j}^{2}$ and ${\tilde{x}}_{j} = {\tilde{η}}_{j}^{2}$ . Both series $\sum_{j} σ_{j}^{2} x_{j}$ and $\sum_{j} a_{j}^{2} σ_{j}^{2} x_{j}$ are finite by [F1], so their displayed difference is absolutely convergent. For $j \notin S$ , $1 - A a_{j}^{2} \leq 0$ ; since $x_{j} \geq 0$ , $\sum_{j \in S} σ_{j}^{2} (1 - A a_{j}^{2}) x_{j} = \sum_{j \geq 1} σ_{j}^{2} (1 - A a_{j}^{2}) x_{j} - \sum_{j \notin S} σ_{j}^{2} (1 - A a_{j}^{2}) x_{j} \geq r^{2} - A,$ where the first sum is at least $r^{2} - A$ by [F1]. For the candidate, the same quantity equals $\sum_{j \in S} σ_{j}^{2} (1 - A a_{j}^{2}) {\tilde{x}}_{j} = r^{2} - A$ by its two active constraints. Multiplying the inequality by $z_{0}^{2}$ therefore gives $\sum_{j} {\tilde{x}}_{j} x_{j} \geq \sum_{j} {\tilde{x}}_{j}^{2}$ .

2.1F1step 1.2algebra

Expanding the difference of squared $ℓ^{2}$ norms of the nonnegative sequences $x$ and $\tilde{x}$ gives $\sum_{j} η_{j}^{4} - \sum_{j} {\tilde{η}}_{j}^{4} = \sum_{j} (x_{j} - {\tilde{x}}_{j})^{2} + 2 \sum_{j} {\tilde{x}}_{j} (x_{j} - {\tilde{x}}_{j}) \geq 0$ by step 1.2. The sums are well-defined: $\sum_{j} x_{j}^{2} \leq (\sum_{j} x_{j})^{2}$ and $\sum_{j} x_{j} = \sum_{j} η_{j}^{2} < \infty$ by [F1]. This proves minimality.

3.1step 2.1algebra∎

If equality holds in the objective, step 2.1 forces $\sum_{j} (x_{j} - {\tilde{x}}_{j})^{2} = 0$ , hence $x_{j} = {\tilde{x}}_{j}$ for every $j$ . Thus the nonnegative minimizer is unique.

Depends on

Minimax detection risk and separation rate

Used by

Mildly ill-posed analytic q=2 detection rate Example
Mildly ill-posed Sobolev q=2 detection rate Example
Severely ill-posed analytic q=2 detection rate Example
Severely ill-posed Sobolev q=2 detection rate Example
q=2 Gaussian-sequence ellipsoid detection boundary Theorem

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Sources

Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, equations (4.4)–(4.8) (standard reference, not scraped)
Clément Marteau and Theofanis Sapatinas, A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection, §3.1 (standard reference, not scraped)