Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty

The optimal reconstruction of cosmic metric perturbations and other signals requires knowledge of their power spectra and other parameters. If these are not known a priori, they have to be measured simultaneously from the same data used for the signal reconstruction. We formulate the general problem of signal inference in the presence of unknown parameters within the framework of information field theory. We develop a generic parameter uncertainty renormalized estimation (PURE) technique and address the problem of reconstructing Gaussian signals with unknown power-spectrum with five different approaches: (i) separate maximum-a-posteriori power spectrum measurement and subsequent reconstruction, (ii) maximum-a-posteriori power reconstruction with marginalized power-spectrum, (iii) maximizing the joint posterior of signal and spectrum, (iv) guessing the spectrum from the variance in the Wiener filter map, and (v) renormalization flow analysis of the field theoretical problem providing the PURE filter. In all cases, the reconstruction can be described or approximated as Wiener filter operations with assumed signal spectra derived from the data according to the same recipe, but with differing coefficients. All of these filters, except the renormalized one, exhibit a perception threshold in case of a Jeffreys prior for the unknown spectrum. Data modes, with variance below this threshold do not affect the signal reconstruction at all. Filter (iv) seems to be similar to the so called Karhune-Loeve and Feldman-Kaiser-Peacock estimators for galaxy power spectra used in cosmology, which therefore should also exhibit a marginal perception threshold if correctly implemented. We present statistical performance tests and show that the PURE filter is superior to the others.

💡 Research Summary

The paper tackles the fundamental problem of reconstructing continuous signals when their power spectra (or other statistical parameters) are unknown a priori. In many cosmological and astrophysical applications, the same data set must be used both to infer the signal (e.g., metric perturbations, density fields) and to estimate the signal’s statistical properties. The authors formulate this joint inference problem within the framework of Information Field Theory (IFT), which treats fields as infinite‑dimensional random variables and expresses Bayesian inference in a field‑theoretic language.

Within this formalism they develop a generic “Parameter Uncertainty Renormalized Estimation” (PURE) technique. PURE treats the unknown spectrum as a dynamical variable whose uncertainty is incorporated through a renormalization flow, analogous to the way interactions are handled in quantum field theory. The flow gradually integrates out the uncertainty, yielding an effective posterior that can be evaluated with a Wiener‑filter‑like operation but with a data‑dependent, self‑consistent spectrum.

To assess the performance of PURE they compare five distinct reconstruction strategies:

1. Separate MAP spectrum estimation + Wiener reconstruction – the spectrum is first obtained by maximizing its marginal posterior, then held fixed while applying a Wiener filter to the data.
2. MAP reconstruction with marginalized spectrum – the spectrum is integrated out analytically (or numerically) and the resulting marginal posterior for the signal is maximized.
3. Joint MAP of signal and spectrum – both quantities are treated as a combined parameter vector and the joint posterior is maximized simultaneously.
4. Variance‑based spectrum guess – the variance of the Wiener‑filtered map is used as an estimator for the spectrum; this is closely related to Karhunen‑Loève (KL) expansions and the Feldman‑Kaiser‑Peacock (FKP) estimator used in galaxy‑survey power‑spectrum analyses.
5. PURE (renormalized) filter – the full field‑theoretic problem is solved by a renormalization‑group flow that yields a self‑consistent, uncertainty‑aware spectrum and corresponding Wiener‑type filter.

All five approaches can be expressed as Wiener‑filter operations with different prescriptions for the assumed signal spectrum. However, when a Jeffreys prior (log‑uniform, scale‑invariant) is adopted for the unknown spectrum, the first four methods exhibit a “perception threshold”: data modes whose variance falls below a certain level do not influence the reconstruction at all. This threshold leads to overly conservative spectra and loss of information, especially in low‑signal‑to‑noise regimes.

PURE, by contrast, does not suffer from this hard threshold. The renormalization flow continuously adjusts the effective spectrum, allowing even weak modes to contribute proportionally to their information content. Consequently, PURE yields lower mean‑square error, higher signal‑to‑noise recovery, and reduced bias in the estimated power spectrum across a wide range of simulated conditions.

The authors validate these claims with extensive numerical experiments. Synthetic Gaussian fields with known spectra are generated, and noise of varying amplitude is added. They then apply each filter, compute reconstruction error metrics, and compare the recovered spectra to the ground truth. PURE consistently outperforms the alternatives, particularly when the true signal‑to‑noise ratio is low or when the spectrum is highly uncertain. Moreover, the computational cost of PURE is comparable to that of a standard Wiener filter because the renormalization flow can be implemented iteratively with a modest number of steps.

In summary, the paper demonstrates that incorporating parameter uncertainty via a renormalization‑group approach yields a robust, near‑optimal estimator for signals with unknown spectra. The PURE filter supersedes traditional MAP or variance‑based methods, eliminates the perception threshold, and retains computational practicality. These results have immediate relevance for cosmological data analysis (e.g., CMB, large‑scale structure, weak lensing) and broader implications for any field where continuous signals must be reconstructed from noisy data without prior knowledge of their statistical structure. Future work may extend PURE to non‑Gaussian signals, anisotropic spectra, and non‑linear observation models, further broadening its applicability.