Window-Deconvolved Quasi-Optimal Estimators

• Window-deconvolved quasi-optimal estimators are techniques that remove the effects of survey masks to yield unbiased, directly comparable cosmological statistics.
• They combine Fourier deconvolution with quadratic, cubic, or quartic maximum-likelihood constructions to efficiently counteract mode mixing from selection functions and beams.
• These methods improve covariance diagonality and computational efficiency in analyses of galaxy redshift surveys and CMB, balancing near-minimum variance with practical weighting approximations.

Searching arXiv for the specified papers and closely related work to ground the article in current literature. arxiv_search {"^^^^^^^^12query12 OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^^^^^", "^^^^^^^^12max_results12 ^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12^^^^, "^^^^^^^^12sort_by12 "^^^^^^^^12submittedDate12 "^^^^^^^^12sort_order12 "^^^^^^^^12descending12 arxiv_search {"^^^^^^^^12query12 of window effect in galaxy power spectrum analysis^^^^^^^^12\12 OR ^^^^^^^^12\12 Without Windows: Quadratic Estimators for the Galaxy Power Spectrum^^^^^^^^12\12 OR ^^^^^^^^12\12 Without Window Functions: Cubic Estimators for the Galaxy Bispectrum^^^^^^^^12\12 OR ^^^^^^^^12\12 A Suite of Optimal and Efficient Power Spectrum and Bispectrum Estimators for Large-Scale Structure^^^^^^^^12\12 "^^^^^^^^12max_results12 ^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12^^^^, "^^^^^^^^12sort_by12 "relevance", "^^^^^^^^12sort_order12 "^^^^^^^^12descending12 Window-deconvolved quasi-optimal estimators are estimators for cosmological two-, three-, and four-point statistics that remove the effects of survey masks, selection functions, beams, or pixellization from the measurement itself, so that the reported quantity can be compared directly to unconvolved theory. In the literature surveyed here, this is realized in two closely related ways: by explicit deconvolution using the Fourier deconvolution theorem for galaxy power-spectrum multipoles, and by quadratic, cubic, or quartic maximum-likelihood constructions in which a Fisher normalization carries the window response and its inverse de-mixes the estimator output. “Quasi-optimal” denotes the use of computationally tractable approximations to inverse-covariance weighting, most commonly FKP-type weights, which remain unbiased and are close to minimum variance under the relevant Gaussian, weakly non-Gaussian, or distant-observer assumptions (&&&^^^^12query12^^^^&&&, &&&^^^^12sort_by12^^^^&&&, &&&^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^&&&, &&&^^^^12max_results12^^^^&&&, &&&^^^^12 OR \12^^^^&&&).

^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^. Conceptual scope and historical development

The modern literature uses closely related estimator architectures across several domains of large-scale-structure and CMB analysis. In redshift surveys, Sato, Hütsi, and Yamamoto developed a deconvolution method for galaxy multipole power spectra based on the deconvolution theorem and compatible with fast Fourier transforms (&&&^^^^12query12^^^^&&&). Sato, Hütsi, Nakamura, and Yamamoto then analyzed the same problem in both convolved and deconvolved forms, emphasizing multipole mixing kernels and the practical treatment of survey windows in SDSS LRG DR^^^^12query12^^^^ (&&&^^^^12submittedDate12^^^^&&&). Philcox reformulated the power-spectrum problem as a quadratic unwindowed estimator (&&&^^^^12sort_by12^^^^&&&), and later generalized the strategy to the galaxy bispectrum with cubic estimators and Fisher deconvolution (&&&^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^&&&). PolyBin^^^^12sort_by12^^^^D places these power-spectrum and bispectrum constructions in a common maximum-likelihood framework with FFT and Monte Carlo implementations (&&&^^^^12max_results12^^^^&&&), while analogous ideas have been extended to mask-deconvolved quasi-optimal CMB trispectrum estimators (&&&^^^^12 OR \12^^^^&&&).

Domain	Estimator class	Window removal mechanism
Galaxy power spectrum	Fourier deconvolution; quadratic estimators	Configuration-space ratio or Fisher inversion (&&&^^^^12query12^^^^&&&, &&&^^^^12sort_by12^^^^&&&)
Galaxy bispectrum	Cubic estimators	Fisher matrix deconvolution (&&&^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^&&&)
Joint LSS power/bispectrum pipelines	Maximum-likelihood “unwindowed” estimators	Fisher normalization with linear filter PRESERVED_PLACEHOLDER_^^^^12query12^^^^ (&&&^^^^12max_results12^^^^&&&)
CMB trispectrum	Quartic template estimators	PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ weighting and deconvolved Fisher normalization (&&&^^^^12 OR \12^^^^&&&)

A common misconception is that “window deconvolution” refers only to explicit inversion of a precomputed mode-coupling matrix. The literature is broader. In the Sato et al. power-spectrum construction, deconvolution is performed by dividing inverse Fourier transforms in configuration space and then transforming back (&&&^^^^12query12^^^^&&&). In the Philcox and PolyBin^^^^12sort_by12^^^^D formalisms, the deconvolution is encoded in the Fisher matrix: the estimator numerator is measured on masked data, and PRESERVED_PLACEHOLDER_^^^^12max_results12^^^^ removes the window-induced mixing (&&&^^^^12sort_by12^^^^&&&, &&&^^^^12max_results12^^^^&&&).

^^^^12max_results12^^^^. Power-spectrum formulation in galaxy redshift surveys

The redshift-survey formulations begin from the FKP galaxy-minus-random field. In the notation of Sato, Hütsi, and Yamamoto,

PRESERVED_PLACEHOLDER_^^^^12sort_by12^^^^

with PRESERVED_PLACEHOLDER_^^^^12submittedDate12^^^^, PRESERVED_PLACEHOLDER_^^^^12sort_order12^^^^ a real-space weight, and PRESERVED_PLACEHOLDER_^^^^12descending12^^^^ a random catalog tracing the selection function. The Fourier-space field is

PRESERVED_PLACEHOLDER_^^^^12query12^^^^

and the convolved power-spectrum estimator is

PRESERVED_PLACEHOLDER_^^^^12\12^^^^

with

PRESERVED_PLACEHOLDER_^^^^12 OR \12^^^^

The corresponding window estimator is

PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12^^^^

In the ideal FKP limit, the measured spectrum obeys the convolution relation

PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^

This is the fundamental window-smearing relation that the deconvolved estimator seeks to undo (&&&^^^^12query12^^^^&&&).

For anisotropic redshift-space clustering, the power depends on PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12max_results12^^^^, and the deconvolved multipoles are computed after reconstruction of the full three-dimensional PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12sort_by12^^^^. Under the distant-observer approximation used for FFT compatibility,

PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12submittedDate12^^^^

with the note that this normalization differs from the conventional one by a factor PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12sort_order12^^^^ (&&&^^^^12query12^^^^&&&). The companion analysis of the window effect also writes the convolved multipoles in kernel form,

PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12descending12^^^^

where the coupling matrix is determined by the multipole moments of the survey window (&&&^^^^12submittedDate12^^^^&&&).

A central distinction in the literature is therefore methodological rather than conceptual. One may keep the measurement convolved and convolve theory with PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12^^^^, or one may deconvolve the measurement and compare it directly to theory. The former is the standard FKP route; the latter defines the window-deconvolved estimator (&&&^^^^12submittedDate12^^^^&&&).

^^^^12sort_by12^^^^. Deconvolution and quasi-optimality

In the Sato et al. power-spectrum method, the key step is the deconvolution theorem. Starting from the convolution relation, the true spectrum is reconstructed as

PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12\12^^^^

Operationally, the inverse transforms of PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12 OR \12^^^^ and PRESERVED_PLACEHOLDER_^^^^12max_results12query12^^^^ are divided pointwise in configuration space and then Fourier transformed back. This yields a deconvolved PRESERVED_PLACEHOLDER_^^^^12max_results12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ before multipole projection (&&&^^^^12query12^^^^&&&).

The term “quasi-optimal” enters through weighting. In the original FKP derivation, the weight

PRESERVED_PLACEHOLDER_^^^^12max_results12max_results12^^^^

minimizes the variance of spherically averaged band-powers for Gaussian fields and slowly varying PRESERVED_PLACEHOLDER_^^^^12max_results12sort_by12^^^^. Sato et al. allow such weights within the deconvolution pipeline, though their SDSS LRG DR^^^^12query12^^^^ application sets PRESERVED_PLACEHOLDER_^^^^12max_results12submittedDate12^^^^ (&&&^^^^12query12^^^^&&&). Sato et al. also describe the FKP weight as “quasi-optimal” under Gaussianity, slowly varying PRESERVED_PLACEHOLDER_^^^^12max_results12sort_order12^^^^, and distant-observer assumptions (&&&^^^^12submittedDate12^^^^&&&).

Philcox recast this logic in a quadratic-estimator language. For an arbitrary symmetric, positive-definite pixel weight PRESERVED_PLACEHOLDER_^^^^12max_results12descending12^^^^,

PRESERVED_PLACEHOLDER_^^^^12max_results12query12^^^^

with unbiased estimator

PRESERVED_PLACEHOLDER_^^^^12max_results12\12^^^^

For PRESERVED_PLACEHOLDER_^^^^12max_results12 OR \12^^^^, this becomes the maximum-likelihood estimator, which saturates the Cramér–Rao bound in the Gaussian limit when PRESERVED_PLACEHOLDER_^^^^12sort_by12query12^^^^. Replacing PRESERVED_PLACEHOLDER_^^^^12sort_by12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ by an FKP-like diagonal approximation yields the quasi-optimal FKP-based estimator (&&&^^^^12sort_by12^^^^&&&).

The same structure persists at higher order. For the bispectrum, the general quasi-optimal estimator is

PRESERVED_PLACEHOLDER_^^^^12sort_by12max_results12^^^^

with

PRESERVED_PLACEHOLDER_^^^^12sort_by12sort_by12^^^^

With inverse-covariance weights this is minimum variance in the weakly non-Gaussian limit; with FKP weights it is “close-to-optimal and easy to compute” (&&&^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^&&&). PolyBin^^^^12sort_by12^^^^D generalizes this to PRESERVED_PLACEHOLDER_^^^^12sort_by12submittedDate12^^^^-point functions by expressing the estimator as a numerator, bias term, and Fisher normalization for a general linear filter PRESERVED_PLACEHOLDER_^^^^12sort_by12sort_order12^^^^, with the window entering through PRESERVED_PLACEHOLDER_^^^^12sort_by12descending12^^^^ and PRESERVED_PLACEHOLDER_^^^^12sort_by12query12^^^^ inside the Fisher and bias terms (&&&^^^^12max_results12^^^^&&&).

^^^^12submittedDate12^^^^. Numerical realization and algorithmic structure

The original FFT-compatible power-spectrum deconvolution pipeline is explicit. One builds random catalogs with PRESERVED_PLACEHOLDER_^^^^12sort_by12\12^^^^, chooses PRESERVED_PLACEHOLDER_^^^^12sort_by12 OR \12^^^^, places galaxies and randoms on an FFT grid, estimates PRESERVED_PLACEHOLDER_^^^^12submittedDate12query12^^^^ and PRESERVED_PLACEHOLDER_^^^^12submittedDate12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^, computes the inverse FFTs

PRESERVED_PLACEHOLDER_^^^^12submittedDate12max_results12^^^^

forms PRESERVED_PLACEHOLDER_^^^^12submittedDate12sort_by12^^^^, and FFTs back to obtain PRESERVED_PLACEHOLDER_^^^^12submittedDate12submittedDate12^^^^. The complexity is dominated by ^^^^12sort_by12^^^^D FFTs, with each forward or inverse FFT scaling as PRESERVED_PLACEHOLDER_^^^^12submittedDate12sort_order12^^^^ (&&&^^^^12query12^^^^&&&).

The quadratic-estimator implementation replaces explicit deconvolution of PRESERVED_PLACEHOLDER_^^^^12submittedDate12descending12^^^^ by repeated applications of covariance operators. In the maximum-likelihood power-spectrum case, PRESERVED_PLACEHOLDER_^^^^12submittedDate12query12^^^^ is applied with preconditioned conjugate gradient descent, using FFT representations of PRESERVED_PLACEHOLDER_^^^^12submittedDate12\12^^^^ and PRESERVED_PLACEHOLDER_^^^^12submittedDate12 OR \12^^^^. One application of PRESERVED_PLACEHOLDER_^^^^12sort_order12query12^^^^ requires PRESERVED_PLACEHOLDER_^^^^12sort_order12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ FFTs, one application of PRESERVED_PLACEHOLDER_^^^^12sort_order12max_results12^^^^ requires PRESERVED_PLACEHOLDER_^^^^12sort_order12sort_by12^^^^ FFTs, and convergence in PRESERVED_PLACEHOLDER_^^^^12sort_order12submittedDate12^^^^ iterations is typical. Monte Carlo estimation of Fisher matrices and biases adds only a PRESERVED_PLACEHOLDER_^^^^12sort_order12sort_order12^^^^ error-bar penalty, quoted as PRESERVED_PLACEHOLDER_^^^^12sort_order12descending12^^^^ for PRESERVED_PLACEHOLDER_^^^^12sort_order12query12^^^^ (&&&^^^^12sort_by12^^^^&&&).

For the bispectrum, the practical implementation relies on separable templates and FFT-based filtered maps. In the FKP-weighted form, the data bispectrum across PRESERVED_PLACEHOLDER_^^^^12sort_order12\12^^^^ mocks requires PRESERVED_PLACEHOLDER_^^^^12sort_order12 OR \12^^^^ CPU-hours, while the Fisher matrix and Monte Carlo corrections are data-independent and require PRESERVED_PLACEHOLDER_^^^^12descending12query12^^^^ CPU-hours for FKP weighting or PRESERVED_PLACEHOLDER_^^^^12descending12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ CPU-hours for ML weighting, with percent-level accuracy for PRESERVED_PLACEHOLDER_^^^^12descending12max_results12^^^^ (&&&^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^&&&). PolyBin^^^^12sort_by12^^^^D reports Monte Carlo convergence for masked bispectra using only a small number of iterations, “typically PRESERVED_PLACEHOLDER_^^^^12descending12sort_by12^^^^ for bispectra,” and supports GPU acceleration using JAX, with speed-ups of PRESERVED_PLACEHOLDER_^^^^12descending12submittedDate12^^^^ versus ^^^^12descending12submittedDate12^^^^-core CPU nodes for representative power-spectrum numerators (&&&^^^^12max_results12^^^^&&&).

These implementations are unified by the same computational principle: shift the expensive mask treatment into precomputable normalization objects, then use FFT-amenable filtered maps for the data-dependent part.

^^^^12sort_order12^^^^. Covariance structure, empirical performance, and survey demonstrations

The principal empirical motivation for deconvolution is covariance simplification. In the SDSS LRG DR^^^^12query12^^^^ study of Sato, Hütsi, and Yamamoto, ^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12query12query12^^^^ LRG-like mocks show that convolved spectra exhibit strong off-diagonal correlations, especially for narrower subsamples, whereas the correlation matrices of deconvolved spectra are “practically diagonal” for PRESERVED_PLACEHOLDER_^^^^12descending12sort_order12^^^^, PRESERVED_PLACEHOLDER_^^^^12descending12descending12^^^^, and the cross-covariances PRESERVED_PLACEHOLDER_^^^^12descending12query12^^^^. Deconvolution also corrects amplitudes measured from different survey divisions, and the BAO signature in deconvolved PRESERVED_PLACEHOLDER_^^^^12descending12\12^^^^ and PRESERVED_PLACEHOLDER_^^^^12descending12 OR \12^^^^ shows larger peak–trough amplitudes and closely matches the “ideal” catalogs, particularly for subsamples where window smoothing is otherwise significant (&&&^^^^12query12^^^^&&&).

The SDSS application used the LRG DR^^^^12query12^^^^ sample with PRESERVED_PLACEHOLDER_^^^^12query12query12^^^^, sky coverage PRESERVED_PLACEHOLDER_^^^^12query12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^, and PRESERVED_PLACEHOLDER_^^^^12query12max_results12^^^^ LRGs, with distances computed assuming flat PRESERVED_PLACEHOLDER_^^^^12query12sort_by12^^^^ with PRESERVED_PLACEHOLDER_^^^^12query12submittedDate12^^^^, PRESERVED_PLACEHOLDER_^^^^12query12sort_order12^^^^, and angular subdivision into ^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12\12^^^^ subsamples of PRESERVED_PLACEHOLDER_^^^^12query12descending12^^^^ or ^^^^12sort_by12max_results12^^^^ subsamples of mean PRESERVED_PLACEHOLDER_^^^^12query12query12^^^^ so that a single line of sight PRESERVED_PLACEHOLDER_^^^^12query12\12^^^^ could be assigned per subsample (&&&^^^^12query12^^^^&&&). The companion window-analysis paper further shows that the convolved monopole and quadrupole amplitudes decrease with decreasing patch size, and that correction factors derived from the measured window can restore consistent amplitudes across different patch divisions (&&&^^^^12submittedDate12^^^^&&&).

In the BOSS DR^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12max_results12^^^^ power-spectrum study, Philcox found that unwindowed band-powers from ML and FKP estimators are statistically consistent and have similar error bars for a low-density sample with a compact window. The unwindowed covariance is nearly diagonal with slight nearest-bin anti-correlations, whereas windowed estimates have smaller per-bin variance at high PRESERVED_PLACEHOLDER_^^^^12query12 OR \12^^^^ because the window induces bin correlations. Compressed coefficients and cosmological posteriors were likewise statistically consistent with conventional windowed analyses, with shifts PRESERVED_PLACEHOLDER_^^^^12\12query12^^^^ (&&&^^^^12sort_by12^^^^&&&).

PolyBin^^^^12sort_by12^^^^D extends these comparisons. For power spectra, unwindowed estimators recover the ideal unmasked spectrum across multipoles, while windowed FKP estimators show PRESERVED_PLACEHOLDER_^^^^12\12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ distortions on large scales, particularly in the quadrupole. For bispectra, unwindowed estimators recover the injected bispectrum within errors, while windowed estimators exhibit significant large-scale biases and strong bin-to-bin correlations. The use of optimal CGD weights reduces large-scale power-spectrum error bars by PRESERVED_PLACEHOLDER_^^^^12\12max_results12^^^^–PRESERVED_PLACEHOLDER_^^^^12\12sort_by12^^^^ relative to FKP in realistic tests (&&&^^^^12max_results12^^^^&&&).

^^^^12descending12^^^^. Generalization to higher-order statistics and other observables

The bispectrum generalization makes the logic of window-deconvolved quasi-optimal estimation especially transparent. The observed bispectrum of the masked field involves a three-leg window convolution,

PRESERVED_PLACEHOLDER_^^^^12\12submittedDate12^^^^

which is a six-dimensional integral at every triangle configuration and MCMC step. Philcox’s cubic estimator avoids this by constructing an estimator for the unwindowed bispectrum,

PRESERVED_PLACEHOLDER_^^^^12\12sort_order12^^^^

where the cubic data statistic PRESERVED_PLACEHOLDER_^^^^12\12descending12^^^^ is measured directly from the masked survey and the Fisher matrix deconvolves the window. In the limit of weak non-Gaussianity, the inverse-covariance-weighted version is minimum variance; the FKP-weighted variant is close-to-optimal and easy to compute (&&&^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^&&&).

PolyBin^^^^12sort_by12^^^^D recasts both power-spectrum and bispectrum estimation in a single maximum-likelihood formalism. For general PRESERVED_PLACEHOLDER_^^^^12\12query12^^^^-point functions,

PRESERVED_PLACEHOLDER_^^^^12\12\12^^^^

with the window entering only through the pointing operator PRESERVED_PLACEHOLDER_^^^^12\12 OR \12^^^^ and the noise PRESERVED_PLACEHOLDER_^^^^12 OR \12query12^^^^ inside the Fisher and bias terms. Because PRESERVED_PLACEHOLDER_^^^^12 OR \12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ carries the mask dependence via PRESERVED_PLACEHOLDER_^^^^12 OR \12max_results12^^^^, applying PRESERVED_PLACEHOLDER_^^^^12 OR \12sort_by12^^^^ de-mixes the window leakage and returns unwindowed band-powers or bispectrum coefficients directly comparable to theory (&&&^^^^12max_results12^^^^&&&).

An analogous strategy has been developed for the CMB trispectrum. There, observed maps are filtered with

PRESERVED_PLACEHOLDER_^^^^12 OR \12submittedDate12^^^^

where PRESERVED_PLACEHOLDER_^^^^12 OR \12sort_order12^^^^ contains the mask, spherical-harmonic synthesis, and beam, and a quartic Edgeworth-derived estimator measures template amplitudes PRESERVED_PLACEHOLDER_^^^^12 OR \12descending12^^^^. The disconnected terms subtract the cut-sky mean field and Gaussian biases, while the same PRESERVED_PLACEHOLDER_^^^^12 OR \12query12^^^^ factors in the Fisher normalization provide exact window and beam deconvolution. The resulting estimators are described as unbiased, minimum variance, mask-deconvolved, and able to account for correlations between templates, including lensing and point sources (&&&^^^^12 OR \12^^^^&&&).

A plausible implication is that “window-deconvolved quasi-optimal estimator” is best understood as a general estimation paradigm rather than a single algorithm: window treatment is transferred from forward-model convolution of theory to estimator normalization and filtering, with optimality controlled by how closely the adopted weighting approximates PRESERVED_PLACEHOLDER_^^^^12 OR \12\12^^^^.

^^^^12query12^^^^. Limitations, assumptions, and methodological trade-offs

The principal numerical vulnerability of explicit deconvolution is division by a small window transform. In the Sato et al. implementation, PRESERVED_PLACEHOLDER_^^^^12 OR \12 OR \12^^^^ must not vanish; the FFT box should be chosen to match the survey footprint so that PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12query12^^^^ does not approach zero at problematic modes. If PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ is small, the deconvolution can amplify noise, and while the paper does not introduce explicit regularization, it notes that masking low-PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12max_results12^^^^ cells, band-limiting, or imposing a floor would help (&&&^^^^12query12^^^^&&&). The ^^^^12max_results12query12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ follow-up similarly emphasizes that deconvolution can amplify noise where the window suppresses signal strongly and may require regularization or restricted PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12sort_by12^^^^-ranges (&&&^^^^12submittedDate12^^^^&&&).

Approximate optimality also has a domain of validity. FKP weights are justified when the field is Gaussian, PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12submittedDate12^^^^ is slowly varying, and one is content with near-minimum variance rather than exact inverse-covariance weighting (&&&^^^^12submittedDate12^^^^&&&). In the quadratic-estimator framework, unbiasedness does not require Gaussianity, but optimality of the ML form does (&&&^^^^12sort_by12^^^^&&&). For the bispectrum and trispectrum, the Edgeworth expansions and Cramér–Rao statements are explicitly tied to weak non-Gaussianity (&&&^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^&&&, &&&^^^^12 OR \12^^^^&&&).

Geometric approximations matter as well. The redshift-space multipole implementations typically assume a distant-observer or Yamamoto-like line-of-sight prescription. In the SDSS LRG deconvolution study, a small low-PRESERVED_PLACEHOLDER_^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12query12sort_order12^^^^ deviation in the quadrupole is attributed to limitations of the distant-observer approximation in finite-area subsamples (&&&^^^^12query12^^^^&&&). The ^^^^12max_results12query12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^ treatment therefore advocates patching the survey so that the line of sight is approximately constant within each patch (&&&^^^^12submittedDate12^^^^&&&).

Finally, there is a trade-off between robustness and formal optimality. Philcox notes that maximum-likelihood weighting is more exact but computationally heavier, whereas FKP weighting is quasi-optimal and often sufficient for compact, low-density surveys such as BOSS DR^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12max_results12^^^^ (&&&^^^^12sort_by12^^^^&&&). The bispectrum and PolyBin^^^^12sort_by12^^^^D studies echo this balance: ML weighting can be numerically delicate in masked regions or for fine binning, while FKP weighting sacrifices a small amount of optimality for robustness, speed, and simpler implementations (&&&^^^^12id:(Sato et al., 2010) OR id:(Philcox, 2021) OR id:(Philcox et al., 2024) OR id:(Philcox, 2020) OR id:(Sato et al., 2013)12^^^^&&&, &&&^^^^12max_results12^^^^&&&).

Within those assumptions, window-deconvolved quasi-optimal estimators provide a coherent route to unbiased, nearly decorrelated cosmological summary statistics whose normalization absorbs survey geometry rather than forcing it into every theory evaluation.