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arxiv: 2605.20770 · v1 · pith:VCH4XQGEnew · submitted 2026-05-20 · 🧮 math.NA · cs.NA

Data-informed posterior approximation for Bayesian linear inverse problems

Haibo Li This is my paper

Pith reviewed 2026-05-21 02:49 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Bayesian inverse problemsposterior approximationdata-informed subspaceisometric embeddingGolub-Kahan bidiagonalizationempirical Bayesian inferenceKrylov subspacesmatrix-free methods
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The pith

In Bayesian linear inverse problems the prior-to-posterior update is confined to an isometric embedding of a low-dimensional data space.

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

The paper develops a data-informed framework that moves the computational burden in large-scale Bayesian linear inverse problems from the high-dimensional parameter space to a much smaller data space. It rigorously shows that this data space is intrinsically low-dimensional, that it embeds isometrically into the parameter space, and that the entire prior-to-posterior update occurs inside the image of that embedding. Because the update is confined to this subspace, posterior inference can be performed in a reduced-dimensional setting rather than the full parameter space. The authors then build a matrix-free iterative procedure that constructs data-informed Krylov subspaces via quotient-space Golub-Kahan bidiagonalization while simultaneously estimating hyperparameters with empirical Bayes.

Core claim

We rigorously characterize an intrinsically low-dimensional data space, establish its isometric embedding into the parameter space, and show that the prior-to-posterior update is confined to a data-informed subspace. This perspective allows posterior inference to be carried out in a reduced data-informed subspace. Based on this formulation, we propose a quotient-space Golub-Kahan bidiagonalization method to construct data-informed Krylov subspaces, and integrate empirical Bayesian inference into the iterative framework, enabling simultaneous hyperparameter estimation and posterior approximation in a matrix-free manner.

What carries the argument

isometric embedding of the intrinsically low-dimensional data space into the parameter space that confines the prior-to-posterior update

Load-bearing premise

The data-informed subspace remains low-dimensional and the isometric embedding preserves the essential posterior information without significant truncation error.

What would settle it

A concrete numerical experiment in which the posterior mean or covariance obtained from the reduced data-informed subspace differs substantially from a reference posterior computed in the full parameter space would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.20770 by Haibo Li.

Figure 5.1
Figure 5.1. Figure 5.1: Illustration of the true solution and noisy observation data. [PITH_FULL_IMAGE:figures/full_fig_p020_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Eigenvalue decay of (M, Γ), reconstructed solution, and convergence behavior of the Q￾GKB method for the first example. relationship σ 2 = 1/λ, the Q-GKB based empirical Bayesian inference (QGKB-EB) method iteratively estimates the optimal hyperparameter λ, thereby obtaining an accurate estimate of σ, which is then be used to compute the posterior distribution. In [PITH_FULL_IMAGE:figures/full_fig_p021_… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: The F¨orstner distance and the KL divergence between the exact and approximate posterior, [PITH_FULL_IMAGE:figures/full_fig_p022_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Contour plots of the marginal distributions of six randomly selected [PITH_FULL_IMAGE:figures/full_fig_p022_5_4.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Illustration of the true image and noisy blurred image. [PITH_FULL_IMAGE:figures/full_fig_p023_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Eigenvalue decay of (M, Γ) and convergence behavior of the Q-GKB method for the first example [PITH_FULL_IMAGE:figures/full_fig_p024_5_6.png] view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Illustration of the prior image, deblurred image at [PITH_FULL_IMAGE:figures/full_fig_p024_5_7.png] view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: The relative error of the approximate posterior mean and the corresponding mean standard [PITH_FULL_IMAGE:figures/full_fig_p025_5_8.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: Comparison of the running time for the Q-GKB method and LIS method as the iteration [PITH_FULL_IMAGE:figures/full_fig_p026_5_9.png] view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: Illustration of the true image and noisy observation data. [PITH_FULL_IMAGE:figures/full_fig_p027_5_10.png] view at source ↗
Figure 5.11
Figure 5.11. Figure 5.11: Evolution of the approximate posterior mean and variance during the Q-GKB iteration. [PITH_FULL_IMAGE:figures/full_fig_p028_5_11.png] view at source ↗
Figure 5.12
Figure 5.12. Figure 5.12: Illustration of the prior image, reconstructed image, and corresponding pixel-wise variance [PITH_FULL_IMAGE:figures/full_fig_p028_5_12.png] view at source ↗
read the original abstract

Computing posterior distributions in large-scale Bayesian linear inverse problems is challenging due to the high dimensionality of the parameter space. In this work, we develop a data-informed framework that shifts the computational focus from the parameter space to the data space. We rigorously characterize an intrinsically low-dimensional data space, establish its isometric embedding into the parameter space, and show that the prior-to-posterior update is confined to a data-informed subspace. This perspective allows posterior inference to be carried out in a reduced data-informed subspace. Based on this formulation, we propose a quotient-space Golub--Kahan bidiagonalization method to construct data-informed Krylov subspaces, and integrate empirical Bayesian inference into the iterative framework, enabling simultaneous hyperparameter estimation and posterior approximation in a matrix-free manner. Numerical experiments on representative problems support the theoretical framework and demonstrate the effectiveness of the resulting method.

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 / 2 minor

Summary. The paper develops a data-informed framework for posterior approximation in large-scale Bayesian linear inverse problems. It rigorously characterizes an intrinsically low-dimensional data space, establishes its isometric embedding into the parameter space, and shows that the prior-to-posterior update is confined to this data-informed subspace. Based on this, the authors propose a quotient-space Golub-Kahan bidiagonalization method to build data-informed Krylov subspaces and integrate empirical Bayesian inference for simultaneous hyperparameter estimation and matrix-free posterior approximation. Numerical experiments on representative problems are included to support the claims.

Significance. If the theoretical characterization and isometric embedding hold under the paper's assumptions, the work offers a promising shift from high-dimensional parameter space to lower-dimensional data space for Bayesian inverse problems, potentially improving scalability in applications such as imaging or parameter estimation. The matrix-free iterative approach combined with empirical Bayes is a practical strength, and the numerical results indicate effectiveness. Explicit error bounds or operator conditions would further strengthen the significance.

major comments (2)
  1. [Theoretical framework] Theoretical framework section: The central claim that the data-informed subspace remains intrinsically low-dimensional and that the isometric embedding preserves essential posterior information without significant truncation error lacks explicit bounds, singular-value decay assumptions on the forward map, or discretization conditions. This is load-bearing for the applicability to the target class of inverse problems, as the abstract and framework rely on it for the reduced-space inference.
  2. [Quotient-space Golub-Kahan bidiagonalization] Section on the quotient-space Golub-Kahan bidiagonalization: The construction of data-informed Krylov subspaces via the quotient-space method requires clarification on how the isometry is maintained during iteration and whether truncation in the bidiagonalization introduces errors that propagate to the posterior approximation; this directly affects the matrix-free claim.
minor comments (2)
  1. [Abstract] Abstract: The description of numerical experiments could briefly note the specific inverse problems considered (e.g., dimensions or forward operators) to better contextualize the support for the theoretical framework.
  2. [Notation and definitions] Notation: Ensure consistent use of symbols for the data-informed subspace and embedding operator across sections to avoid potential confusion in the reduced-space formulation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment point by point below, indicating the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Theoretical framework] Theoretical framework section: The central claim that the data-informed subspace remains intrinsically low-dimensional and that the isometric embedding preserves essential posterior information without significant truncation error lacks explicit bounds, singular-value decay assumptions on the forward map, or discretization conditions. This is load-bearing for the applicability to the target class of inverse problems, as the abstract and framework rely on it for the reduced-space inference.

    Authors: We agree that the manuscript would benefit from more explicit quantitative statements. The current theoretical framework characterizes the data-informed subspace as the range of the adjoint of the forward operator composed with the prior covariance and establishes the isometric embedding via the inner-product structure induced by the prior. However, we acknowledge that explicit a priori error bounds in terms of singular-value decay and discretization error estimates are not stated. In the revised manuscript we will add a new subsection containing such bounds under standard assumptions on the decay of the singular values of the discretized forward map and on the mesh size. This will make the load-bearing claims fully quantitative while preserving the matrix-free character of the method. revision: yes

  2. Referee: [Quotient-space Golub-Kahan bidiagonalization] Section on the quotient-space Golub-Kahan bidiagonalization: The construction of data-informed Krylov subspaces via the quotient-space method requires clarification on how the isometry is maintained during iteration and whether truncation in the bidiagonalization introduces errors that propagate to the posterior approximation; this directly affects the matrix-free claim.

    Authors: We appreciate the request for clarification. The quotient-space formulation performs the bidiagonalization entirely in the data space; the isometry is maintained at every step by construction because the inner products are taken with respect to the data-space metric induced by the noise covariance and the prior is used only to map the resulting basis vectors back to the parameter space via the adjoint operator. Truncation after k steps produces a rank-k approximation whose error is controlled by the residual of the bidiagonalization process. In the revised version we will insert a short paragraph (and, if space permits, a brief lemma) that explicitly states how the isometry is preserved iteration by iteration and that bounds the propagation of the truncation error into the posterior covariance and mean. These additions will also reinforce the matrix-free nature of the overall algorithm. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework builds on standard linear algebra and Bayesian tools

full rationale

The paper's central claims involve characterizing a low-dimensional data space and its isometric embedding into parameter space for Bayesian linear inverse problems, using quotient-space Golub-Kahan bidiagonalization and empirical Bayesian inference. These steps rely on established operator theory, Krylov subspace methods, and standard Bayesian updating without reducing any prediction or result to a fitted parameter or self-citation by construction. The abstract and framework present the low-dimensionality as a rigorous characterization under typical assumptions on the forward map, not as a tautology or renamed input. No load-bearing step equates outputs to inputs via definition or self-referential fitting. Minor self-citation risk exists in any iterative method paper but is not central here. Derivation remains self-contained against external benchmarks like standard SVD-based dimension reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of an intrinsically low-dimensional data space whose isometric embedding confines the prior-to-posterior update; these are presented as rigorously established but their precise mathematical assumptions are not visible in the abstract.

axioms (2)
  • domain assumption The data space is intrinsically low-dimensional for the class of linear inverse problems considered
    Abstract states 'rigorously characterize an intrinsically low-dimensional data space'
  • domain assumption An isometric embedding of the data space into the parameter space exists and preserves the update
    Abstract claims 'establish its isometric embedding into the parameter space'

pith-pipeline@v0.9.0 · 5661 in / 1388 out tokens · 30491 ms · 2026-05-21T02:49:49.826194+00:00 · methodology

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