Blind Solving: Inferring Hidden Operators

Updated 9 July 2026

Blind Solving is a problem class where missing operative information—such as unknown operators or state trajectories—is inferred jointly with the target signal or structure.
It employs alternating block-structured inference techniques, using methods like plug-and-play, Gibbs sampling, and diffusion priors to update both primary targets and hidden parameters.
The approach relies on strong priors and structured reparameterizations but faces challenges like ambiguity, high computational cost, and reliance on low-dimensional operator models.

Blind solving denotes problem settings in which a solver must recover a target state, signal, structure, or action sequence while some operative information is unavailable at inference time. In the cited literature, this includes blind inverse problems in which both the target signal and the measurement operator are unknown and one seeks $(x,\theta)$ from $y=A(\theta)x+e$ or $y=\mathcal A_\phi(x)+n$ ; maze routing in which a robot executes a finite or infinite instruction sequence without feedback about whether a move succeeded; and blindfolded or token-only puzzle regimes in which solution proceeds without direct visual access to the state (Gan et al., 2023, Dontas et al., 2024, David et al., 2018, Krakauer et al., 23 Nov 2025, Elkn et al., 9 Nov 2025). The term therefore does not identify a single algorithmic family. Rather, it names a class of problems in which hidden operators, hidden correspondences, hidden environments, or hidden state trajectories must be inferred jointly with the primary object of interest.

1. Core meaning and formal structures

Across the literature, blindness is defined operationally by what is missing during solution. In imaging and inverse problems, the missing quantity is often the forward operator or its parameters. A standard formulation is

$y = A(\theta)x + e,$

with unknown image $x$ and unknown operator parameters $\theta$ (Gan et al., 2023). Closely related formulations write

$y = \mathcal{A}_\phi(x) + n,$

with unknown clean image $x$ and unknown operator parameters $\phi$ (Dontas et al., 2024). In blind deconvolution, the operator becomes an unknown kernel, as in

$\mathbf{y}=\mathbf{x}\ast \mathbf{k}+\mathbf{n}$

$y=A(\theta)x+e$ 0

(Tofighi et al., 2017, Li et al., 28 May 2025).

In non-imaging settings, the missing structure takes different forms. In blind maze routing, a maze is a quadruple $y=A(\theta)x+e$ 1 with a countable strongly connected digraph, a proper colouring of edges, an origin, and a destination; the robot follows an instruction sequence but receives no feedback about whether a move occurred (David et al., 2018). In blind Perspective-n-Point, the unknown quantity is not the 3D geometry or the image points themselves, but the 2D-3D correspondences that must be inferred jointly with camera pose (Campbell et al., 2020). In determined blind source separation, local frequency-bin separation may be correct while source order across larger frequency regions is inconsistent, producing the block permutation problem (Matsumoto et al., 2024). In blindfolded Rubik’s Cube solving, the scramble must first be encoded and maintained in memory, so the task is constrained not only by expert knowledge but also by the skill improvements required to overcome short-term memory bottlenecks (Krakauer et al., 23 Nov 2025).

Setting	Hidden quantity	Representative formulation
Blind inverse problems	Signal and operator parameters	$y=A(\theta)x+e$ 2, $y=A(\theta)x+e$ 3
Blind maze solving	Maze geometry and move success feedback	Finite or infinite instruction sequence
Blind source separation	Sources, permutations, or source identities	$y=A(\theta)x+e$ 4, $y=A(\theta)x+e$ 5
Blind camera pose	2D-3D correspondences	Joint optimization over pose and correspondence matrix
Blindfold or token-only puzzles	Visual state during solving	Memorization-and-execution or Seq2Seq over token sequences

This diversity suggests that blind solving is best understood as a structural property of a problem rather than as a domain-specific label. The solver must compensate for missing operative information by exploiting priors, repeated structure, combinatorial regularity, geometric constraints, or learned generative models.

The most developed contemporary use of blind solving appears in imaging inverse problems. A central pattern is joint estimation of a latent image and a latent operator. In "Block Coordinate Plug-and-Play Methods for Blind Inverse Problems" (Gan et al., 2023), the blind setting is defined by a parametric forward model $y=A(\theta)x+e$ 6 with unknown parameters $y=A(\theta)x+e$ 7, and the method seeks $y=A(\theta)x+e$ 8 jointly. The paper emphasizes that this is the standard blind-inverse-problem structure relevant to automatic coil sensitivity estimation in magnetic resonance imaging and blind image deblurring. Its conceptual contribution is to treat operator parameters not as nuisance variables to be calibrated separately, but as objects that can also carry learned denoiser priors.

Subsequent work broadens this formulation from point estimation to posterior inference. "Plug-and-Play Posterior Sampling for Blind Inverse Problems" (Li et al., 28 May 2025) frames the problem as joint Bayesian inference over image and operator, then performs Gibbs sampling with two diffusion priors, one for the image and one for the operator parameters. In blind deblurring, the operator is a convolution kernel $y=A(\theta)x+e$ 9, and the paper states that blind solving becomes possible because the image and the operator constrain each other through alternating Gaussian-denoising subproblems. This replaces explicit hand-crafted priors with learned diffusion-based priors on both latent blocks.

Several papers address how the operator prior itself can be learned. "Double Blind Imaging with Generative Modeling" (Levac et al., 27 Mar 2025) studies a setting with clean images and corrupted measurements that are unpaired, independent, and disjoint, and uses an AmbientGAN-based method to learn the distribution of imaging-system parameters from those unpaired datasets. The learned operator prior is then inserted into Blind Diffusion Posterior Sampling for blind deconvolution. "Training Adaptive Reconstruction Networks for Blind Inverse Problems" (Gossard et al., 2022) takes a different route: train a reconstruction network on a family of operators and then infer the unknown operator parameter at test time by minimizing a consistency objective over $y=\mathcal A_\phi(x)+n$ 0.

The class extends beyond classical 2D deblurring. "Blind Inversion using Latent Diffusion Priors" (Bai et al., 2024) formulates blind inversion with latent diffusion priors inside an EM framework, with 2D blind deblurring and pose-free sparse-view 3D reconstruction as representative linear and nonlinear cases. "Blind Inverse Problem Solving Made Easy by Text-to-Image Latent Diffusion" (Dontas et al., 2024) uses a pretrained text-to-image latent diffusion model, positive and negative prompts, and iterative operator refinement to address both linear and nonlinear blind restoration, including JPEG decompression without task-specific JPEG metadata. "Solving a Nonlinear Blind Inverse Problem for Tagged MRI with Physics and Deep Generative Priors" (Bian et al., 1 Mar 2026) makes the blind structure explicitly nonlinear: tagged MRI measurements depend jointly on unknown anatomy, unknown blur, unknown tag formation and fading, and unknown diffeomorphic motion.

Other sensing problems instantiate the same logic with different hidden operators. In blind pansharpening, the blur kernel, spatial displacement, and spectral response are not known and are estimated directly from low-resolution multispectral and high-resolution panchromatic data (Yu et al., 2021). In blind ptychography, both the sample and the illuminating mask are unknown and are recovered by recasting the measurement model as a structured blind deconvolution problem (Roach, 2023). In blind free deconvolution, neither component spectral measure is known; the problem is made tractable by assuming one-parameter sparse families and linearizing free addition or free product with the $y=\mathcal A_\phi(x)+n$ 1- or $y=\mathcal A_\phi(x)+n$ 2-transform (Ying, 18 Jan 2025).

3. Recurring algorithmic paradigms

A striking regularity across modern blind-solving methods is the move from monolithic optimization to alternating or block-structured inference. BC-PnP updates one block at a time with block-specific denoisers, and for $y=\mathcal A_\phi(x)+n$ 3 reduces to ordinary PnP-ISTA,

$y=\mathcal A_\phi(x)+n$ 4

(Gan et al., 2023). Blind-PnPDM uses split Gibbs sampling, alternating between image and operator variables, with each conditional decomposed into a likelihood step and a prior step implemented by a diffusion model (Li et al., 28 May 2025). GibbsDDRM similarly alternates between diffusion-state updates and operator-parameter Langevin updates inside a partially collapsed Gibbs sampler for blind linear inverse problems (Murata et al., 2023). LatentDEM adopts an EM structure in which the E-step recovers clean images with a latent diffusion prior and the M-step estimates forward-operator parameters (Bai et al., 2024). Tagged MRI recovery uses Coordinate Descent with Diffusion Prior, alternating anatomy sampling with maximum-likelihood updates of blur, tags, fading, and motion (Bian et al., 1 Mar 2026).

A second recurring paradigm is that blind methods are usually prior-rich rather than prior-free. BC-PnP assumes approximate MMSE denoisers on both the image block and the operator-parameter block (Gan et al., 2023). Blind-PnPDM uses two diffusion models, one for natural images and one for valid blur kernels (Li et al., 28 May 2025). Double blind imaging first learns a generator over operator parameters and then samples from that learned distribution to train a kernel diffusion model (Levac et al., 27 Mar 2025). LADiBI uses text prompts with classifier-free guidance to approximate a joint prior over image and operator (Dontas et al., 2024). GibbsDDRM is an exception only in a restricted sense: it avoids a learned operator prior, but still relies on a strong pre-trained diffusion model for the signal and a simple generic prior on operator parameters (Murata et al., 2023).

A third paradigm is reduction by structured reparameterization. BD-RCS lifts blind deblurring to recovery of a rank-one matrix $y=\mathcal A_\phi(x)+n$ 5 and adds row and column sparsity to infer blur-kernel and image supports (Tofighi et al., 2017). Blind ptychography converts ptychographic magnitude data into a blind deconvolutional form and then into a rank-one recovery problem solved by spectral initialization and regularized Wirtinger descent (Roach, 2023). Blind free deconvolution converts nonlinear free inverse problems into linear sparse recovery by using $y=\mathcal A_\phi(x)+n$ 6 for additive cases and $y=\mathcal A_\phi(x)+n$ 7 for multiplicative cases, followed by the eigenmatrix method (Ying, 18 Jan 2025). These constructions do not eliminate the blind structure; they move it into a form with better algebraic or variational control.

This suggests that blind solving progresses when the hidden structure is partitioned into blocks that admit compatible priors and compatible update rules. The blind problem is then attacked not as an opaque joint search, but as a sequence of constrained subproblems that exchange information.

Blind solving is also prominent in combinatorial control, source separation, and geometry. In the maze model of "Solvability of Mazes by Blind Robots" (David et al., 2018), a robot receives only a colour instruction sequence and cannot tell whether a move succeeded. The paper proves that there exists an infinite sequence of instructions for a robot to visit all accessible vertices in any maze whose board is $y=\mathcal A_\phi(x)+n$ 8 with arbitrarily many horizontal edges removed but only finitely many vertical edges removed in consecutive columns. The construction relies on oscillating eastward and westward sweeps, controlled latitude probes, and compactness-based gluing of solvable classes.

In blind source separation, blindness can mean unknown sources, unknown mixture structure, or unknown cross-frequency identity alignment. "Subband Splitting: Simple, Efficient and Effective Technique for Solving Block Permutation Problem in Determined Blind Source Separation" (Matsumoto et al., 2024) addresses the block permutation problem by splitting the frequency axis into overlapping subbands, solving each with an existing BSS method, and using the overlap as a label anchor so that permutations align across bands. "Towards Unsupervised Single-Channel Blind Source Separation using Adversarial Pair Unmix-and-Remix" (Hoshen, 2018) tackles the single-channel setting with no clean supervision by exploiting source independence: estimated sources from two mixtures are swapped and remixed, and adversarial training enforces that the resulting mixtures remain in the real mixture distribution. "Blind source separation using Fast-ICA with a novel nonlinear function" (Xu et al., 2019) remains in the classical instantaneous linear ICA model but proposes $y=\mathcal A_\phi(x)+n$ 9 as the Fast-ICA nonlinearity, reporting improved separation accuracy and faster convergence in the provided simulations.

In geometric vision, "Solving the Blind Perspective-n-Point Problem End-To-End With Robust Differentiable Geometric Optimization" (Campbell et al., 2020) treats blind PnP as simultaneous estimation of camera pose and unknown 2D-3D correspondences. The method combines learned features, Sinkhorn-based soft matching, RANSAC initialization, and weighted nonlinear PnP refinement, with gradients propagated through solution mappings by implicit differentiation.

Puzzle and expertise studies widen the notion further. "Seq2Seq Models Reconstruct Visual Jigsaw Puzzles without Seeing Them" (Elkn et al., 9 Nov 2025) defines blind solving as reconstruction from discrete token sequences rather than raw visual input; on ImageNet 3×3, the reported absolute accuracy is $y = A(\theta)x + e,$ 0 and perfect accuracy is $y = A(\theta)x + e,$ 1. "Universality in Collective Intelligence on the Rubik's Cube" (Krakauer et al., 23 Nov 2025) argues that blindfolded cube solving is a distinct problem class from sighted solving because the solver must first encode and store the scramble state using mnemonic systems such as memory palaces, then execute algorithms without visual refresh.

5. Guarantees, identifiability, and ambiguities

Blind solving is usually more ill-posed than its non-blind counterpart, so the literature repeatedly introduces explicit identifiability assumptions, stationarity notions, or ambiguity classes. BC-PnP develops convergence theory compatible with blind inverse problems by allowing nonconvex data fidelity and expansive denoisers, and shows stationary-point convergence under approximate MMSE denoisers. Under sequential block updates and a stepsize $y = A(\theta)x + e,$ 2, it proves

$y = A(\theta)x + e,$ 3

and if the denoiser errors are square-summable, then $y = A(\theta)x + e,$ 4 (Gan et al., 2023). For random block selection, the analogous stationarity measure $y = A(\theta)x + e,$ 5 converges to zero almost surely under square-summable errors.

In blind maze solving, solvability is constructive rather than variational. Proposition 3.3 in (David et al., 2018) uses compactness on families of subgraphs of $y = A(\theta)x + e,$ 6 to combine algorithms for solvable subfamilies and exploits the fact that if a possibly infinite algorithm solves a compact family of mazes, then some finite initial segment already solves it. This gives a different kind of blind-solving guarantee: universal success over a whole maze class rather than convergence of iterates.

Blind deconvolutional models expose standard ambiguity classes. Blind ptychography has a global phase ambiguity and a scaling ambiguity in the deconvolution reformulation, so recovery is unique only up to unavoidable transformations (Roach, 2023). BD-RCS notes the scale ambiguity of blind deconvolution and recovers kernel and image coefficients from the dominant singular vectors of an approximately rank-one matrix (Tofighi et al., 2017). BPnPNet addresses the combinatorial ambiguity of unknown correspondences with one-to-one matching constraints, Sinkhorn transport, and robust initialization (Campbell et al., 2020). Blind free deconvolution avoids a full identifiability theory by restricting the unknown factors to one-parameter families and requiring analytic dependence and numerical linear independence in the eigenmatrix construction (Ying, 18 Jan 2025).

A common consequence is that guarantees are rarely unconditional. They depend on low-dimensional operator parameterizations, compact support or subspace models, bounded noise, approximate-MMSE denoisers, or structured priors over admissible operators. Blind solving is therefore typically principled under assumptions rather than universally identified in the absence of structure.

6. Misconceptions, limitations, and open problems

A common misconception is that blind solving means prior-free solving. The cited literature points in the opposite direction. Blind inverse methods use learned denoisers, diffusion models, operator priors, family-trained reconstruction networks, text prompts, physical motion models, or simple generic priors on hidden operators (Gan et al., 2023, Li et al., 28 May 2025, Gossard et al., 2022, Dontas et al., 2024). Even when the method is described as training-free, as in LADiBI, it still depends on a strong pretrained text-to-image latent diffusion model and task-specific prompts (Dontas et al., 2024).

Another misconception is that blind solving always seeks a single estimate. Several recent methods explicitly target posterior sampling rather than point recovery, including Blind-PnPDM, GibbsDDRM, and diffusion posterior sampling pipelines used after operator-prior learning (Li et al., 28 May 2025, Murata et al., 2023, Levac et al., 27 Mar 2025). This suggests that, in highly ambiguous blind settings, uncertainty quantification is part of the problem definition rather than an optional add-on.

The limitations are equally recurrent. Many methods assume that the unknown operator can be parameterized by a manageable low-dimensional variable or by a differentiable family (Li et al., 28 May 2025, Gossard et al., 2022). Performance can degrade under prior mismatch, poor prompts, weak initialization, or unstable operator optimization (Dontas et al., 2024, Bai et al., 2024). Blind deblurring remains particularly difficult: in the family-trained reconstruction-network framework, the optimization for Fresnel diffraction kernels tends to return the same smallest-support kernel, close to an Airy pattern, so the true kernel is not identified reliably (Gossard et al., 2022). Computational cost is often high; GibbsDDRM depends on an SVD of the operator, and LADiBI reports about 5 minutes on a single NVIDIA A6000 for the general operator initialization pipeline (Murata et al., 2023, Dontas et al., 2024).

Outside imaging, the open questions are more foundational. The Leader–Spink question on whether there exists a single algorithm solving all relevant blind-robot mazes in $y = A(\theta)x + e,$ 7 remains open beyond the subclass handled in (David et al., 2018). Blindfolded cube solving remains constrained by mnemonic architecture as well as algorithmic expertise, and the paper on collective intelligence treats this as a distinct developmental regime rather than as a trivial variant of sighted solving (Krakauer et al., 23 Nov 2025). The jigsaw Seq2Seq approach is limited to square puzzles and depends on a hand-designed tokenizer with hyperparameters such as $y = A(\theta)x + e,$ 8, $y = A(\theta)x + e,$ 9, and $x$ 0 (Elkn et al., 9 Nov 2025).

Taken together, these works indicate that blind solving is not a marginal special case of ordinary solving. It is a broad technical regime in which hidden operators, hidden correspondences, hidden environments, or hidden state representations must be inferred jointly with the target solution. The principal methodological themes are blockwise inference, learned or structured priors on both primary and nuisance variables, and careful management of ambiguity through geometry, algebra, compactness, or posterior sampling.