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An inverse-error chain obeys Minkowski in $L^{2}$ ; root-sum-square is equivalent to aggregate cross-term cancellation

Statement

Let $(Ω, F, μ)$ be a measure space and $H$ a real or complex inner-product space. Let $d_{1}, \dots, d_{N} : Ω \to H$ be a finite list, with $N \geq 1$ , such that $f_{j} (ω) = ∥ d_{j} (ω) ∥_{H}$ and $g (ω) = ∥ \sum_{j = 1}^{N} d_{j} (ω) ∥_{H}$ are measurable and each $f_{j}$ belongs to $L^{2} (μ)$ . Assume also that the scalar pairings $⟨ d_{i}, d_{j} ⟩_{H}$ are measurable. For any such map $d$ whose norm function is in $L^{2} (μ)$ , write

$∥ d ∥_{2, H} : = {(\int_{Ω} ∥ d (ω) ∥_{H}^{2} d μ)}^{1 / 2} .$

Then the error sum obeys

${∥ \sum_{j = 1}^{N} d_{j} ∥}_{2, H} \leq \sum_{j = 1}^{N} ∥ d_{j} ∥_{2, H} .$

Moreover,

${∥ \sum_{j = 1}^{N} d_{j} ∥}_{2, H}^{2} = \sum_{j = 1}^{N} ∥ d_{j} ∥_{2, H}^{2} + 2 \sum_{1 \leq i < j \leq N} \int_{Ω} Re ⟨ d_{i} (ω), d_{j} (ω) ⟩_{H} d μ .$

Thus root-sum-square equality holds if and only if the aggregate cross-term sum $\sum_{1 \leq i < j \leq N} \int_{Ω} Re ⟨ d_{i} (ω), d_{j} (ω) ⟩_{H} d μ = 0.$ Individual cross integrals may be nonzero and cancel; their separate vanishing is sufficient but is not necessary. Root-sum-square does not follow from the pathwise triangle inequality alone. When Countable Choice is assumed as in Tikhonov error separates bias, data noise, discretization and solve residual, the result applies to its finite bias/noise/discretization/solve chains.

Facts & Assumptions

Given: A measure space, a real or complex inner-product space, and the finite list of maps with the scalar measurability and L2 hypotheses in the Statement.

[F1]

Under Countable Choice, Tikhonov error separates bias, data noise, discretization and solve residual: the realization-wise error is the finite sum of its displayed chain differences.

[A1]

For the inverse-problem-chain application, assume Countable Choice as required by [F1].

[F2]

The function space $L^{p} (μ)$ for $0 < p < \infty$ : A measurable scalar function belongs to $L^{2} (μ)$ exactly when its $L^{2}$ functional is finite.

[F3]

Minkowski's inequality for integrals, including $p = \infty$ : For $f, h \in L^{2} (μ)$ , $∥ f + h ∥_{2} \leq ∥ f ∥_{2} + ∥ h ∥_{2}$ .

[F4]

Monotonicity and nonnegative homogeneity of the nonnegative integral: If $0 \leq f \leq h$ , then $\int f d μ \leq \int h d μ$ .

[F5]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: $∣ ⟨ u, v ⟩_{H} ∣ \leq ∥ u ∥_{H} ∥ v ∥_{H}$ .

[F6]

Integrable real and complex functions, and their integrals: A measurable real or complex function is integrable when its absolute value has finite integral.

[F7]

The Lebesgue integral is linear on $L^{1} (μ)$ : The integral is linear on integrable functions.

[F8]

Real and imaginary parts, complex conjugation, and modulus: $Re z$ is the real coordinate of $z \in C$ ; for real scalars read $Re z = z$ .

[F9]

Real and complex inner product spaces, with the inner product linear in the first argument: The inner product is linear in the first argument and conjugate symmetric.

[F10]

The induced length is a norm: The inner-product norm satisfies $∥ u + v ∥ \leq ∥ u ∥ + ∥ v ∥$ .

Proof

1.1givenF2F3F4F10algebra

Put $F_{k} = \sum_{j = 1}^{k} f_{j}$ . Since $F_{1} = f_{1} \in L^{2}$ , induction using [F3] shows $F_{k} \in L^{2}$ and $∥ F_{k} ∥_{2} \leq \sum_{j = 1}^{k} ∥ f_{j} ∥_{2}$ for every $k \leq N$ . The pointwise triangle inequality [F10] gives $0 \leq g \leq F_{N}$ , hence [F4] applied to their squares gives $∥ g ∥_{2} \leq ∥ F_{N} ∥_{2} \leq \sum_{j} ∥ f_{j} ∥_{2}$ . This is the claimed $L^{2}$ error bound by the definition of $∥ \cdot ∥_{2, H}$ .

1.2F2F4F5F6F7F8F9algebra

At each $ω$ , expanding the squared inner-product norm and pairing the two cross terms gives $g (ω)^{2} = \sum_{j} f_{j} (ω)^{2} + 2 \sum_{i < j} Re ⟨ d_{i} (ω), d_{j} (ω) ⟩_{H}$ . By [F5], $∣ ⟨ d_{i}, d_{j} ⟩_{H} ∣ \leq f_{i} f_{j}$ , and $(f_{i} - f_{j})^{2} \geq 0$ gives $2 f_{i} f_{j} \leq f_{i}^{2} + f_{j}^{2}$ . Thus each cross term is integrable: [F2] makes each $f_{i}^{2}$ integrable, [F7] makes their finite sum integrable, and [F4] bounds the integral of the absolute cross term by that sum. All squared-norm terms are integrable by [F2]. The expansion uses [F9] and [F8]. Integrating the finite identity and using [F6]–[F7] gives the displayed cross-term formula; the formula gives exact root-sum-square if and only if the finite sum of cross integrals is zero. Separate vanishing is sufficient, but nonzero cross integrals may cancel.

2.1A1F1step 1.1step 1.2∎

For the stated inverse-problem application, [F1] supplies the finite chain under [A1]. Whenever its differences satisfy the scalar hypotheses of this corollary, steps 1.1 and 1.2 apply to that chain.

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Sources

Sheldon Axler, Measure, Integration & Real Analysis, Minkowski's Inequality (standard reference, not scraped)
John K. Hunter, Measure Theory, Theorem 7.5 (standard reference, not scraped)
Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)

An inverse-error chain obeys Minkowski in L2L2; root-sum-square is equivalent to aggregate cross-term cancellation