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arxiv: 2512.23405 · v4 · submitted 2025-12-29 · 💻 cs.LG · stat.ML

On the Sample Complexity of Learning for Blind Inverse Problems

Nathan Buskulic , Luca Calatroni , Lorenzo Rosasco , Silvia Villa This is my paper

Pith reviewed 2026-05-16 19:35 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords blind inverse problemssample complexitylinear minimum mean square estimatorTikhonov regularizationfinite-sample boundssource conditionoperator randomness
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The pith

Blind inverse problems admit optimal linear estimators equivalent to Tikhonov regularization whose structure depends on the distributions of signal, noise, and random operator.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper analyzes learning for blind inverse problems where both the unknown signal and the forward operator are only partially known. It works in the linear minimum mean square estimator framework to derive closed-form optimal estimators and shows these coincide with specially chosen Tikhonov-regularized reconstructions. The regularization term is explicitly built from the covariance structures of the signal, the noise, and the random operator. Under a source condition the reconstruction error is shown to vanish as noise level and operator randomness go to zero, and finite-sample error bounds are given that track the dependence on sample size, conditioning, and the randomness factors.

Core claim

In the blind setting the optimal estimator is the linear minimum mean square estimator whose closed form is equivalent to a Tikhonov-regularized inverse problem whose regularization operator is assembled from the second-order statistics of the signal, noise, and random forward operator. Under a source condition the mean-squared reconstruction error converges to zero when both the noise variance and the randomness of the operator tend to zero. Finite-sample bounds are derived that make the dependence on the number of training samples, the conditioning of the problem, and the variance of the operator explicit.

What carries the argument

Linear Minimum Mean Square Estimator (LMMSE), which yields the optimal linear estimator by minimizing expected squared error using the joint second-order statistics of signal, noise, and random operator.

If this is right

  • Optimal blind estimators can be realized by solving a Tikhonov problem whose regularization matrix is built directly from estimated covariances of signal, noise, and operator.
  • Reconstruction error vanishes under the source condition once noise variance and operator variability are driven toward zero.
  • The derived bounds give explicit rates that worsen with stronger operator randomness and improve with more samples.
  • The same bounds quantify how problem conditioning interacts with sample size to control the error.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The distribution-dependent regularization may offer a principled way to set parameters in blind imaging calibration tasks when covariances can be estimated from calibration data.
  • The finite-sample analysis could be used to decide how many measurements are needed before operator randomness dominates the error budget.
  • Similar closed-form equivalences might be sought for other statistical estimators beyond the linear case when the operator is random.

Load-bearing premise

A source condition on the unknown signal is needed for the convergence statement when noise and operator randomness decrease, and the relevant distributions must be known or estimable to obtain the closed-form expressions.

What would settle it

Measure whether the empirical reconstruction error stops converging to zero when the source condition is deliberately violated while noise level and operator randomness are reduced; or compare observed errors against the predicted finite-sample bounds across increasing sample sizes and varying operator randomness.

read the original abstract

Blind inverse problems arise in many experimental settings where both the signal of interest and the forward operator are (partially) unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner due to identifiability issues and symmetric solutions inherent to the blind setting. Recently, data-driven approaches have been proposed to address such problems, demonstrating strong empirical performance and adaptability. However, these methods often lack interpretability and are not supported by theoretical guarantees, limiting their reliability in domains such as applied imaging where a blind approach often relates to a calibration of the acquisition device. In this work, we shed light on learning in blind inverse problems within the insightful framework of Linear Minimum Mean Square Estimators (LMMSEs). We provide a theoretical analysis, deriving closed-form expressions for optimal estimators and extending classical recovery results to the blind setting. In particular, we establish equivalences with tailored Tikhonov-regularized formulations, where the regularization structure depends explicitly on the distributions of the unknown signal, of the noise, and of the random forward operator. We also show how the reconstruction error converges as the noise and the randomness of the operator diminish when we use a source condition assumption. Furthermore, we derive finite-sample error bounds that characterize the performance of the learned estimators as a function of the noise level, problem conditioning, and number of available samples. These bounds explicitly quantify the impact of operator randomness and show explicitly the dependence of the associated convergence rates to this randomness factors. Finally, we validate our theoretical findings through illustrative exemplar numerical experiments that confirm the predicted convergence behavior.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper analyzes learning for blind inverse problems in the LMMSE framework. It derives closed-form expressions for optimal estimators and establishes equivalences to Tikhonov-regularized formulations whose regularization structure depends explicitly on the joint distributions of the unknown signal, noise, and random forward operator. Under a source condition assumption, the reconstruction error is shown to converge to zero as noise level and operator randomness diminish. Finite-sample error bounds are derived that characterize estimator performance in terms of noise level, problem conditioning, number of samples, and randomness factors; these bounds are illustrated via numerical experiments.

Significance. If the derivations and extensions hold, the work is significant for supplying theoretical guarantees and interpretability to data-driven methods for blind inverse problems, which arise in imaging and calibration tasks. The explicit dependence of regularization and error bounds on operator randomness, together with the finite-sample analysis, extends classical LMMSE results in a manner that can guide estimator design and quantify the impact of randomness.

major comments (2)
  1. [Convergence section] The convergence result (abstract and the section establishing error convergence under source condition): the source condition is invoked to obtain convergence as noise and operator randomness vanish, but the manuscript does not clearly verify or adapt the condition to the random-operator distribution model. If the condition is stated only for fixed A, the claimed limit does not automatically transfer and is load-bearing for both the convergence statement and the subsequent finite-sample bounds.
  2. [Finite-sample bounds section] Finite-sample bounds section: the bounds are presented as functions of noise level, conditioning, samples, and randomness factors, yet it is unclear whether they assume the distributions of signal, noise, and operator are known exactly or whether they incorporate the additional sample complexity required to estimate those distributions from data. This distinction affects the tightness and applicability of the bounds.
minor comments (1)
  1. [Abstract] The abstract refers to 'illustrative exemplar numerical experiments'; adding a brief description of the specific inverse problem (e.g., deconvolution with random kernel) would improve context without lengthening the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments help clarify key aspects of the convergence analysis and the interpretation of the finite-sample bounds. We address each major comment below and have revised the manuscript to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Convergence section] The convergence result (abstract and the section establishing error convergence under source condition): the source condition is invoked to obtain convergence as noise and operator randomness vanish, but the manuscript does not clearly verify or adapt the condition to the random-operator distribution model. If the condition is stated only for fixed A, the claimed limit does not automatically transfer and is load-bearing for both the convergence statement and the subsequent finite-sample bounds.

    Authors: We appreciate the referee's careful reading. The source condition in the manuscript is formulated with respect to the joint distribution of the signal and the random operator A (specifically, it takes the form of an expectation E_{x,A}[|| (A^* A + lambda I)^{-1/2} x ||^2] bounded by a constant). This adaptation ensures that the convergence of the reconstruction error to zero holds as both the noise variance and the randomness measure of A (e.g., its variance) tend to zero. However, we agree that the presentation could be more explicit. In the revised manuscript we have added a dedicated paragraph in the convergence section that (i) restates the source condition under the random-operator model and (ii) sketches why the classical deterministic proof carries over after taking the expectation over A. This change does not alter the mathematical result but removes any ambiguity. revision: yes

  2. Referee: [Finite-sample bounds section] Finite-sample bounds section: the bounds are presented as functions of noise level, conditioning, samples, and randomness factors, yet it is unclear whether they assume the distributions of signal, noise, and operator are known exactly or whether they incorporate the additional sample complexity required to estimate those distributions from data. This distinction affects the tightness and applicability of the bounds.

    Authors: The finite-sample bounds are derived for the learned (empirical) LMMSE estimator, i.e., they explicitly incorporate the sample complexity of estimating the required second-order statistics (covariances of the signal, noise, and random operator) from n training pairs. The bounds therefore contain two additive terms: one that vanishes with the noise level and operator randomness, and a second term that decays with n (via standard covariance concentration inequalities). We acknowledge that this distinction was not stated with sufficient clarity in the original text. The revised version adds an explicit sentence at the beginning of the finite-sample bounds section and a short remark after the main theorem that separates the approximation error from the estimation error and references the sample-size dependence. No change to the mathematical statements is required. revision: yes

Circularity Check

0 steps flagged

Derivations self-contained under explicit distributional assumptions

full rationale

The paper derives closed-form LMMSE estimators for blind inverse problems, establishes equivalences to tailored Tikhonov forms whose regularization depends on the joint distributions of signal, noise, and random operator, proves convergence of reconstruction error to zero under a source condition as noise and operator randomness vanish, and supplies finite-sample bounds in terms of noise level, conditioning, and sample count. All steps are presented as direct consequences of the stated assumptions and standard LMMSE theory; no step reduces a claimed prediction or first-principles result to a fitted parameter or self-citation by construction. The source condition is invoked as an external hypothesis rather than derived from the target bounds, and no self-citation load-bearing chains appear in the provided derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard LMMSE assumptions and a source condition typical in inverse-problems theory; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Distributions of the unknown signal, noise, and random forward operator are available or estimable
    Required to obtain the closed-form optimal estimators and the explicit regularization structure.
  • domain assumption Source condition holds to establish convergence
    Invoked to prove that reconstruction error decreases as noise and operator randomness diminish.

pith-pipeline@v0.9.0 · 5589 in / 1389 out tokens · 51445 ms · 2026-05-16T19:35:49.872113+00:00 · methodology

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