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A compact diagonal map has a closure-only feature

Example

Assume Countable Choice. Set $I = {j \in N : j \geq 1}$ . On the real Hilbert space $X = H = ℓ^{2} (I, R)$ with standard basis $(e_{j})_{j \in I}$ , define $K x = (x_{j} / j)_{j \in I}$ and the feature vector $h = (1 / j)_{j \in I}$ . Fix $ϵ > 0$ , let $Z$ be a jointly Gaussian centered family indexed by $H$ , linear into $L^{2}$ with covariance $E [Z (g) Z (k)] = ⟨ g, k ⟩_{H}$ , and define $Y_{x} (g) = ⟨ K x, g ⟩_{H} + ϵ Z (g)$ . The map $K$ is compact and self-adjoint. Its adjoint range contains every finitely supported sequence, so $\overline{R (K^{*})} = ℓ^{2} (I, R)$ . The vector $h$ belongs to $ℓ^{2} (I, R)$ but not to $R (K^{*})$ , since any preimage would have all coordinates equal to $1$ . Thus this feature lies in the closure of the adjoint range but is not exactly estimable by one probe.

Example references

Exact linear feature estimability through the adjoint range

Compact linear inverse model between Hilbert spaces

Convergence and divergence of real p-series

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §1.2.2

Facts & Assumptions

Given: The real square-summable sequence Hilbert space, its standard basis, the diagonal map and feature in the Example.

[F1]

The real square-summable sequences form a Hilbert space with inner product $⟨ x, y ⟩ = \sum_{j} x_{j} y_{j}$ and norm squared $\sum_{j} x_{j}^{2}$ . This holds for $I$ as well as $N$ : the bijective linear reindexing $(x_{j})_{j \in I} \mapsto (x_{n + 1})_{n \in N}$ preserves inner products and norms, hence transfers completeness from $ℓ^{2} (N, R)$ . The real square-summable sequence space is complete, Hilbert space

[F2]

For a square-summable sequence indexed by $I$ , its coordinate truncations converge in norm because $\sum_{j > N} x_{j}^{2} \to 0$ . Thus the standard coordinate vectors are orthonormal with dense span, hence form a complete orthonormal basis to which Fourier expansion applies. Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ , Fourier expansion in a Hilbert space

[F3]

The series $\sum_{j \geq 1} j^{- 2}$ converges. The p-series for a real exponent p converges exactly when p is greater than one

[F4]

Every bounded finite-rank operator is compact, and under Countable Choice an operator-norm limit of compact operators into a Banach space is compact. Bounded finite rank operators are compact, Norm limit of compact operators is compact, The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F5]

The adjoint identity is $⟨ K x, y ⟩ = ⟨ x, K^{*} y ⟩$ . The Hilbert-space adjoint of a bounded operator

[F6]

A feature vector is exactly estimable by one probe iff it belongs to $R (K^{*})$ . Exact linear feature estimability through the adjoint range

[F7]

The compact inverse model has Hilbert parameter and data spaces and a compact forward map. Compact linear inverse model between Hilbert spaces

[A1]

Countable Choice is inherited through the compactness of norm limits in [F4], the Hilbert adjoint in [F5], Fourier expansion in [F2], and the exact-estimability interface in [F6]. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Proof

1.1givenF1F2F4F5F7A1algebraconstruct

The map is linear and $∥ K x ∥_{2}^{2} = \sum_{j} x_{j}^{2} / j^{2} \leq ∥ x ∥_{2}^{2}$ , so it is bounded. For $N \geq 1$ , let $K_{N} x$ retain only the first $N$ output coordinates. Its range is spanned by $e_{1}, \dots, e_{N}$ , and $∥ K_{N} x ∥_{2} \leq ∥ x ∥_{2}$ makes it bounded, so [F4] makes $K_{N}$ compact. Also $∥ (K - K_{N}) x ∥_{2}^{2} = \sum_{j > N} x_{j}^{2} / j^{2} \leq (N + 1)^{- 2} ∥ x ∥_{2}^{2}$ , with equality in the operator norm on $e_{N + 1}$ . Thus $∥ K - K_{N} ∥ = 1 / (N + 1) \to 0$ , and [F4] makes $K$ compact. The diagonal real entries show $⟨ K x, y ⟩ = \sum_{j} x_{j} y_{j} / j = ⟨ x, K y ⟩$ ; by [F5], $K^{*} = K$ . For every finitely supported $z$ , the sequence $x_{j} = j z_{j}$ is finitely supported and $K x = z$ , so $c_{00} \subseteq R (K^{*})$ . Truncations are dense in $ℓ^{2}$ by [F2], proving $\overline{R (K^{*})} = ℓ^{2}$ .

2.1givenF1F3F6step 1.1algebra∎

By [F3], $h = (1 / j)$ is square-summable. If $K^{*} x = h$ , self-adjointness and the coordinate equations give $x_{j} / j = 1 / j$ , hence $x_{j} = 1$ for every $j$ . But the first $N$ squared coordinates of such an $x$ sum to $N$ for every $N$ , so no finite bound can contain all partial sums; this contradicts $x \in ℓ^{2}$ . Therefore $h \notin R (K^{*})$ . Since $h \in \overline{R (K^{*})}$ by step 1.1, [F6] shows that the feature is not exactly estimable by a single probe despite lying in the closure of the adjoint range.

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Sources

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §1.2.2 (standard reference, not scraped)
Kretschmann and Werner, Maximum a posteriori testing in statistical inverse problems, §1.1 (standard reference, not scraped)