Kernel transformations and bounds for smeared spectral functions

Technical Summary: Kernel Transformations and Bounds for Smeared Spectral Functions

Problem Statement
In quantum field theories, particularly Quantum Chromodynamics (QCD), spectral functions are often singular tempered distributions. Phenomenological applications and comparisons between theory (e.g., Lattice QCD) and experiment frequently require evaluating these functions smeared with specific kernels (e.g., Cauchy, Gaussian, or kinematically motivated kernels). While Lattice QCD provides noisy, discrete samples of Euclidean-time correlation functions (which are themselves Laplace-smeared spectral observables), extracting a specific target smeared observable $\rho[\kappa]$ directly from this data constitutes a notoriously ill-posed inverse problem.

This work addresses a related but distinct challenge: the kernel-transformation problem. Given a family of energy-smeared observables $\rho[\phi_\alpha]$ (computed via one kernel family), how can one rigorously compute the same underlying spectral function smeared with a different target kernel $\kappa$? The goal is to establish maps between these families without returning to the original inverse problem of reconstructing the unsmeared spectral function $\rho(\omega)$, while providing rigorous bounds on the resulting observables that account for both statistical noise and systematic errors introduced by the transformation.

Methodology
The authors develop a framework based on the linearity of the spectral functional $\rho[\kappa] = \int d\omega \rho(\omega)\kappa(\omega)$. The core strategy is to express the target kernel $\kappa(\omega)$ as a linear combination (or integral) of the input kernel family $\{\phi_\alpha\}$ via a "transition kernel" $K_{\kappa \leftarrow \phi}(\alpha)$:
$$\kappa(\omega) = \int_A d\alpha K_{\kappa \leftarrow \phi}(\alpha) \phi_\alpha(\omega)$$
If this representation converges, the target observable is obtained directly from the input data:
$$\rho[\kappa] = \int_A d\alpha K_{\kappa \leftarrow \phi}(\alpha) \rho[\phi_\alpha]$$

The methodology is divided into two regimes:

1. Exact Transformations: The paper establishes analytic conditions under which the transition kernel exists and the integral converges without arbitrary regularization. This relies heavily on the analytic structure of the kernels in the complex plane (specifically the location of singularities relative to the mass gap) and Fourier analysis.

   • Key Examples: The paper derives exact transition kernels for Gaussian-to-Gaussian (widening only), Cauchy-to-Cauchy (widening only), and, most significantly, Cauchy-to-Gaussian (exact for all widths).
   • Obstructions: Transformations that require "deconvolution" (e.g., sharpening a Gaussian to a Cauchy, or narrowing a Cauchy) generally fail to converge in the continuum limit due to the essential singularities of the kernels or the growth of their Fourier transforms.

2. Regulated Transformations: When exact maps do not exist or are numerically unstable, the target kernel is replaced by a regulated approximant $\kappa_T$. The difference $\rho[\kappa] - \rho[\kappa_T]$ is treated as a systematic bias. The authors provide a method to bound this bias directly from the input data using Hölder's inequality, independent of the specific regularization scheme used.

Bound Propagation
A central contribution is the propagation of rigorous pointwise bounds (or statistical error budgets) from the input to the target. Two complementary methods are proposed:

• Riesz–Kantorovich (RK) Bounds: A general, non-iterative method that splits the transition kernel into positive and negative parts. It uses the upper bound of the input where the kernel is positive and the lower bound where it is negative (and vice versa for the lower target bound). This method is rigorous and computationally inexpensive but can be conservative as it ignores correlations between input points.
• Positivity-Optimized Bounds: A method that formulates the bound propagation as a Semi-Infinite Program (SIP). It explicitly enforces the physical constraint that the underlying spectral density $\rho(\omega)$ must be non-negative. This yields tighter bounds than RK but requires numerical optimization and a certification step to ensure the solution satisfies continuum inequalities on a finite grid.

For regulated transformations, the total uncertainty is the sum of the propagated statistical/interval uncertainty and the certified systematic bias from the kernel mismatch. The regulator parameter is optimized to minimize this total certified width.

Key Contributions

1. Formalism for Kernel Transformations: The paper establishes the necessary and sufficient analytic conditions for exact kernel-to-kernel maps, identifying that such maps exist if the target kernel's singularities are sufficiently separated from the real axis relative to the input kernel's width.
2. Explicit Analytic Expressions: The authors provide closed-form transition kernels for several classes, including the primary example of Cauchy-to-Gaussian transformation, which is exact and requires no regularization.
3. Rigorous Bias Certification: A novel method is introduced to bound the systematic error of any approximate kernel transformation using only the known input data and the ratio of the target to input kernels (Eq. 71). This allows for "certified" bounds even when exact transformations are impossible.
4. Optimization Framework: The paper formulates the selection of input smearing widths and regularization parameters as optimization problems to minimize the final uncertainty interval.
5. Levy Mixture Construction: The authors demonstrate that a Cauchy-smeared observable can be reconstructed from a continuum of Gaussian-smeared observables (with varying widths) via a Lévy distribution mixture, bypassing the fixed-width obstruction.

Results
Numerical demonstrations on toy models and lattice-like data confirm the framework:

• Cauchy-to-Gaussian: Exact transformations successfully propagate bounds with no kernel-mismatch bias. The optimal input smearing width $\epsilon^*$ is found to be of the order of the target width $\sigma$.
• Cauchy-to-Cauchy (Sharpening): For the ill-posed case of sharpening a Cauchy kernel, the regulated approach successfully reconstructs the target with certified bounds. The results show a trade-off: as the regulator is removed, the systematic bias vanishes but the propagated statistical error explodes; the optimal regulator balances these to minimize the total certified interval.
• Gaussian-to-Cauchy: Using the Lévy mixture, the paper demonstrates that a Cauchy target can be recovered from a family of Gaussian inputs, with the positivity-optimized bounds (SIP) yielding tighter intervals than the RK method.
• Euclidean-to-Smeared: The framework is applied to noisy Euclidean-time correlators, transforming them into Cauchy- and Gaussian-smeared spectral functions with rigorous bounds, demonstrating the viability of the method for practical Lattice QCD analysis.

Significance
The paper claims that this framework offers a robust alternative to solving the full inverse spectral problem for every new phenomenological observable. Instead of reconstructing the full spectral function $\rho(\omega)$ from Euclidean data for every specific kernel, one can compute a convenient intermediate smeared observable (e.g., Cauchy-smeared) and then transform it to the desired target (e.g., Gaussian or kinematic) with certified error bars.

The significance lies in:

• Modularity: It decouples the extraction of spectral information from the specific phenomenological kernel required.
• Rigor: It provides mathematically rigorous upper and lower bounds that account for both statistical noise and the systematic bias of the transformation itself.
• Positivity: It leverages the physical constraint of spectral positivity to tighten bounds beyond what is possible with generic linear propagation.
• Interoperability: It bridges different theoretical and experimental determinations of spectral densities (e.g., relating Lattice QCD results to experimental cross-sections or muon $g-2$ calculations) by providing a controlled mathematical bridge between different smearing conventions.

The authors emphasize that while the methods can be applied directly to Euclidean-time inputs, the primary focus is on transforming between energy-smeared observables, thereby avoiding the repeated solution of the ill-posed inverse problem for every new observable of interest.