Operator Domains in Latent Measurement Models

Conceptual map

V-12.01Latent measurement structure
V-12.02A wrapped-Gaussian witness
V-12.03Three operator-domain failures
V-12.04Stronger repairs and their cost
V-12.05Conclusion boundaries

1. A latent measurement model is an operator between named spaces

Injectivity, range, bounded inverse, and cross-candidate composition are different properties. Each must be checked on its actual domain.

Let $\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$ and work in $L^{2}(\mathbb{T})$ with normalized Lebesgue measure. If a latent circular variable is observed with independent wrapped-Gaussian error of variance $\sigma ^{2}$ , the conditional measurement operator $K_{\sigma}$ convolves a latent density with the wrapped error density. In Fourier coordinates,

\[\widehat{(K_{\sigma }f)}(k)=e^{-\sigma ^{2}k^{2}/2}\hat{f}(k), \quad k\in \mathbb{Z}.\](1)

Definition 1 · Operator-domain ledger

For every composition, inverse, adjoint, or spectral projection, record source space, codomain, algebraic domain, closure, norm, and whether the map is bounded. Candidate-specific objects retain candidate-specific domains until a common-space result is proved.

2. Gaussian convolution is injective and ill-posed

Proposition 1 · Range and inverse of wrapped-Gaussian smoothing

For $\sigma>0$ , $K_{\sigma}:L^{2}(\mathbb{T})\to L^{2}(\mathbb{T})$ is bounded, compact, self-adjoint, positive, and injective. Its range is dense and nonclosed, and its inverse on that range is unbounded.

Proof. The Fourier basis diagonalizes the operator with real positive multipliers $a_{k}=e^{-\sigma ^{2}k^{2}/2}$ . Their supremum is one, giving boundedness; they tend to zero, giving compactness. Positivity and self-adjointness follow from the real nonnegative diagonal. No multiplier is zero, so the kernel is trivial. Every finite Fourier polynomial has a finite preimage and such polynomials are dense, so the range is dense. For unit Fourier mode $e_{k}$ , the inverse norm is $1/a_{k}\to\infty$ ; hence the inverse is unbounded and the range cannot be closed. ∎

At $\sigma=0.5$ and frequency $$k=6$$ , the forward multiplier is $e^{-4.5}\approx0.011109$ and inverse amplification is $e^{4.5}\approx90.0171$ . Small high-frequency record errors can therefore dominate an unregularized reconstruction.

3. Cross-candidate transforms have restricted domains

For candidates $\sigma _{1}<\sigma _{2}$ , the formal transform $K_{\sigma _{1}}K_{\sigma _{2}}^{-1}$ is initially defined only on $\operatorname{range}(K_{\sigma _{2}})$ . Its Fourier multiplier is $\exp[(\sigma _{2}^{2}-\sigma _{1}^{2})k^{2}/2]$ , which is unbounded. At $\sigma _{1}=0.25$ , $\sigma _{2}=0.5$ , and $$k=6$$ , the factor is $e^{3.375}\approx29.2243$ .

Three audit failures follow. Candidate-specific injectivity does not make the inverse defined on all observed $L^{2}$ functions. Algebraic similarity cannot be invoked through an unbounded transform without a common invariant domain. An estimated density perturbed off the exact range may have no square-integrable inverse, and truncation can violate nonnegativity or unit mass unless constrained.

4. Regularization changes the solved problem

Failure case · Injective operator reported as stably invertible

Every Fourier coefficient is identified in the population because multipliers are nonzero. Stability fails because their reciprocals diverge.

At $\sigma=0$ the operator is the identity and loses compact smoothing. As $\sigma$ grows, high-frequency singular values decay faster. Finite Fourier truncation gives a bounded inverse on the retained subspace, with condition number determined by the largest retained frequency. Tikhonov and spectral cutoffs introduce bias; their tuning parameter and target norm belong to the estimand. Zeros in a different error characteristic function destroy injectivity at those frequencies. Discrete grids create aliasing, and density constraints do not automatically survive an unconstrained inverse.

5. Implementation, exercises, and sources

Declare Hilbert spaces and Fourier normalization, compute or bound singular multipliers, and test kernel and range claims candidate by candidate. Before composing inverses, verify range inclusion and boundedness on a common domain. Simulate perturbations by frequency, report amplification, and validate regularized outputs for mass, nonnegativity, residual norm, and sensitivity to cutoff.

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Exercises

Reproduce both amplification factors above.
Prove that finite Fourier polynomials lie in Range( $K_{\sigma}$ ).
Find the condition number of the truncated operator retaining $\lvert k\rvert \le K$ .

Partial solutions

1. Evaluate $\operatorname{exp}(4.5)$ and $\exp[(0.25-0.0625)\times36/2]$ . 2. Divide each of finitely many Fourier coefficients by its positive multiplier; the resulting finite sequence defines an $L^{2}$ preimage. 3. The largest multiplier is one and the smallest retained multiplier is $\operatorname{exp}(-\sigma ^{2}K^{2}/2)$ , so the condition number is $\operatorname{exp}(\sigma ^{2}K^{2}/2)$ .

Rainer Kress, Linear Integral Equations, chapters on compact operators and ill-posed equations.Range closure and unbounded inverses.
Marine Carrasco, Jean-Pierre Florens, and Eric Renault (2007), “Linear Inverse Problems in Structural Econometrics: Estimation Based on Spectral Decomposition and Regularization,” in Handbook of Econometrics, Volume 6B, 5633–5751.Spectral regularization in econometric inverse problems.
Alexander Meister, Deconvolution Problems in Nonparametric Statistics.Fourier multipliers and measurement-error inversion.

6. Audit checkpoint

Source and codomain, measure and Fourier normalization, $\sigma$ support, kernel, multiplier zeros, compactness, range density and closure, inverse algebraic domain, boundedness, candidate range inclusion, similarity domain, invariant subspace, observed perturbation, frequency cutoff, singular values, condition number, regularization bias, tuning rule, grid aliasing, mass and nonnegativity, residual norm, and sensitivity envelope.

7. Scope boundary

The chapter covers wrapped-Gaussian convolution on a circle. Noncompact supports, unknown error laws, nonlinear operators, and adaptive regularization require additional theory.