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Existence of a finite q=2 ellipsoid extremal profile for small radius

Statement

Assume the minimax detection-risk definition with $q = 2$ , and let $a_{j}$ and $σ_{j}$ be positive nondecreasing sequences tending to infinity. There is a constant $r_{0} > 0$ such that for every $r \in (0, r_{0})$ there exist $A, z_{0} > 0$ with finite nonempty $S = {j : A a_{j}^{2} < 1}$ for which ${\tilde{η}}_{j}^{2} = z_{0}^{2} σ_{j}^{2} (1 - A a_{j}^{2})_{+}$ satisfies $\sum_{j \geq 1} σ_{j}^{2} {\tilde{η}}_{j}^{2} = r^{2}, \sum_{j \geq 1} a_{j}^{2} σ_{j}^{2} {\tilde{η}}_{j}^{2} = 1.$

Statement references

Ingster, Sapatinas and Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, Remark 4.1

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §6.2.3

Facts & Assumptions

Given: Positive nondecreasing $a_{j}, σ_{j} \to \infty$ in the q=2 Gaussian sequence detection class.

[F1]

Minimax detection risk and separation rate: the two constraints defining $Θ_{2} (r)$ are $\sum_{j} a_{j}^{2} σ_{j}^{2} η_{j}^{2} \leq 1$ and $\sum_{j} σ_{j}^{2} η_{j}^{2} \geq r^{2}$ .

[F2]

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ : if a real-valued function is continuous on a closed bounded interval and a target lies between its endpoint values, it takes that target at some point of the interval; no additional choice assumption is needed.

Proof

1.1given

Put $b_{j} = a_{j}^{2}$ and $c_{j} = σ_{j}^{4}$ . For $0 < A < b_{1}^{- 1}$ define the finite sums $N (A) = \sum_{j \geq 1} c_{j} (1 - A b_{j})_{+}, D (A) = \sum_{j \geq 1} c_{j} b_{j} (1 - A b_{j})_{+},$ and $R (A) = N (A) / D (A)$ . The active set is finite because $b_{j} \to \infty$ , and it is nonempty because $A < b_{1}^{- 1}$ . On every compact subinterval of $(0, b_{1}^{- 1})$ , only finitely many indices can be active; each summand is continuous and vanishes at its activation endpoint. Thus $R$ is continuous and $D (A) > 0$ .

2.1step 1.1algebra

The ratio is a weighted average of $1 / b_{j}$ : with $p_{j} (A) = c_{j} b_{j} (1 - A b_{j})_{+} / D (A)$ , the numbers $p_{j} (A)$ are nonnegative, sum to one, and $R (A) = \sum_{j} p_{j} (A) / b_{j}$ . Fix $K > 0$ and choose $j_{0}$ so $b_{j} \geq K$ for $j \geq j_{0}$ . The head weight $\sum_{j < j_{0}} p_{j} (A)$ tends to zero as $A ↓ 0$ : its numerator is bounded by the fixed finite sum $\sum_{j < j_{0}} c_{j} b_{j}$ , while for any $M \geq j_{0}$ the denominator is at least $\frac{1}{2} \sum_{j = j_{0}}^{M} c_{j} b_{j}$ once $A b_{M} \leq 1 / 2$ . Since $c_{j} b_{j} \to \infty$ , these finite partial sums can be made arbitrarily large. Also $R (A) \leq b_{1}^{- 1} \sum_{j < j_{0}} p_{j} (A) + 1 / K$ , so ${lim sup}_{A ↓ 0} R (A) \leq 1 / K$ . As $K$ is arbitrary, $R (A) \to 0$ .

2.2givenstep 1.1algebra

Let $m_{0}$ be the largest index with $b_{m_{0}} = b_{1}$ ; it is finite because $b_{j} \to \infty$ . For $A$ sufficiently close to $b_{1}^{- 1}$ from below, the active set is exactly ${1, \dots, m_{0}}$ and all its $b_{j}$ equal $b_{1}$ . Hence $R (A) = b_{1}^{- 1}$ there.

3.1F1F2step 1.1step 2.1step 2.2constructalgebra∎

By step 1.1, $R$ is continuous on $(0, b_{1}^{- 1})$ . Given $r^{2} \in (0, b_{1}^{- 1})$ , steps 2.1 and 2.2 let us choose $0 < A_{-} < A_{+} < b_{1}^{- 1}$ with $R (A_{-}) < r^{2}$ and $R (A_{+}) = b_{1}^{- 1}$ . Apply [F2] to the restriction of $R$ to $[A_{-}, A_{+}]$ to obtain $A$ with $R (A) = r^{2}$ . Set $z_{0}^{2} = r^{2} / N (A) > 0$ and define ${\tilde{η}}_{j}^{2} = z_{0}^{2} σ_{j}^{2} (1 - A b_{j})_{+}$ . Its support is the finite active set, and $\sum_{j} σ_{j}^{2} {\tilde{η}}_{j}^{2} = z_{0}^{2} N (A) = r^{2}, \sum_{j} a_{j}^{2} σ_{j}^{2} {\tilde{η}}_{j}^{2} = z_{0}^{2} D (A) = \frac{r^{2} D (A)}{N (A)} = 1.$ Taking $r_{0} = b_{1}^{- 1 / 2}$ proves the assertion and, by [F1], the profile belongs to $Θ_{2} (r)$ .

Depends on

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

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Sources

Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, Remark 4.1 (standard reference, not scraped)
Evarist Giné and Richard Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §6.2.3 (standard reference, not scraped)