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arxiv: 1906.10411 · v1 · pith:FXN5VZCVnew · submitted 2019-06-25 · 📡 eess.IV · cs.CV· cs.LG

Deep Learning of Compressed Sensing Operators with Structural Similarity Loss

Yochai Zur , Amir Adler This is my paper

Pith reviewed 2026-05-25 16:19 UTC · model grok-4.3

classification 📡 eess.IV cs.CVcs.LG
keywords compressed sensingdeep learningstructural similarity indexfully-connected networkimage reconstructionsensing matrixend-to-end trainingSSIM loss
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The pith

A fully-connected network jointly learns the sensing matrix and reconstruction operator for compressed sensing by training on structural similarity loss.

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

The paper establishes that a single fully-connected network can handle both the linear sensing stage and the nonlinear reconstruction stage in compressed sensing, with the sensing matrix and reconstructor optimized together using SSIM loss instead of MSE. This joint training produces reconstructions that score higher than state-of-the-art methods when measured by both SSIM and MSE. A reader would care because compressed sensing aims to recover signals from few measurements, and an integrated network that directly targets structural fidelity could simplify pipelines while improving output quality in imaging tasks.

Core claim

The central claim is that an end-to-end deep learning approach in which a fully-connected network performs both linear sensing and nonlinear reconstruction, with the sensing matrix and reconstruction operator jointly optimized using SSIM as the loss function rather than MSE, yields higher reconstruction quality than state-of-the-art methods under both SSIM and MSE metrics.

What carries the argument

A fully-connected network that executes both the linear sensing matrix multiplication and the subsequent nonlinear reconstruction, trained end-to-end with SSIM loss to optimize the matrix and reconstructor simultaneously.

If this is right

  • The sensing matrix and reconstruction operator are learned jointly rather than designed or trained separately.
  • Reconstruction quality improves under both SSIM and MSE metrics relative to prior compressed sensing techniques.
  • SSIM loss serves as an effective objective even when final evaluation includes MSE.
  • The entire compressed sensing pipeline is realized inside one network without separate stages.

Where Pith is reading between the lines

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

  • The learned sensing matrix could be implemented directly in analog hardware sensors tuned to structural image features.
  • The same joint-optimization idea might apply to other linear inverse problems such as super-resolution or denoising.
  • Gains from SSIM training suggest testing alternative perceptual losses in deep reconstruction networks.
  • Performance on natural images raises the question of how the approach behaves on signals with different statistical structure.

Load-bearing premise

That a single fully-connected network can simultaneously learn an effective linear sensing matrix and a high-quality nonlinear reconstructor for the signals of interest, and that SSIM is an appropriate training objective for this joint optimization task.

What would settle it

Training the described network on standard image datasets used in compressed sensing and measuring that its SSIM and MSE scores on held-out test signals do not exceed those of existing state-of-the-art methods.

Figures

Figures reproduced from arXiv: 1906.10411 by Amir Adler, Yochai Zur.

Figure 1
Figure 1. Figure 1: Scheme of the end-to-end deep neural network for reconstruction architecture, which jointly optimizes the sensing and the non-linear reconstruction [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scheme of the CNN kernels (as described in appendix A) for extracting 2x2 patches from a 4x4 image. The CNN last layer reshapes the output [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstruction of 2 test images. SSIM(1) is Equation (12) as weight function. SSIM(2) is a uniform weight function [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scheme of calculating overall SSIM gradient for whole image. The overall SSIM gradient is calculated per pixel. For each pixel we sum all the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

Compressed sensing (CS) is a signal processing framework for efficiently reconstructing a signal from a small number of measurements, obtained by linear projections of the signal. In this paper we present an end-to-end deep learning approach for CS, in which a fully-connected network performs both the linear sensing and non-linear reconstruction stages. During the training phase, the sensing matrix and the non-linear reconstruction operator are jointly optimized using Structural similarity index (SSIM) as loss rather than the standard Mean Squared Error (MSE) loss. We compare the proposed approach with state-of-the-art in terms of reconstruction quality under both losses, i.e. SSIM score and MSE score.

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

1 major / 0 minor

Summary. The manuscript proposes an end-to-end deep learning framework for compressed sensing in which a single fully-connected network simultaneously learns the linear sensing matrix and the nonlinear reconstruction operator. The sensing and reconstruction stages are jointly optimized during training by minimizing a structural similarity (SSIM) loss rather than the conventional MSE loss, and the authors report that the resulting reconstructions outperform state-of-the-art methods when evaluated under both SSIM and MSE metrics.

Significance. If the performance gains can be isolated to the joint FC architecture rather than the SSIM objective alone, the work would provide evidence that perceptual losses and end-to-end optimization of the sensing operator can improve reconstruction quality in CS. The approach is conceptually straightforward and could be relevant to practical CS systems where measurement design and recovery are co-optimized.

major comments (1)
  1. Abstract: the central claim that the proposed FC network outperforms SOTA under both SSIM and MSE is not isolated from the choice of training loss. The abstract states that SOTA methods are compared after being trained (presumably with MSE), so any SSIM improvement is expected while MSE improvement could arise from SSIM acting as a surrogate loss rather than from the joint linear/nonlinear FC design. Without an explicit statement that SOTA baselines were retrained under identical SSIM loss, data, and hyper-parameters, the load-bearing contribution of the proposed architecture remains unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract: the central claim that the proposed FC network outperforms SOTA under both SSIM and MSE is not isolated from the choice of training loss. The abstract states that SOTA methods are compared after being trained (presumably with MSE), so any SSIM improvement is expected while MSE improvement could arise from SSIM acting as a surrogate loss rather than from the joint linear/nonlinear FC design. Without an explicit statement that SOTA baselines were retrained under identical SSIM loss, data, and hyper-parameters, the load-bearing contribution of the proposed architecture remains unverified.

    Authors: We agree that the manuscript does not state or perform retraining of SOTA baselines under the SSIM loss; comparisons use the originally published results of those methods (typically MSE-trained). Our FC network, jointly optimized end-to-end with SSIM, nevertheless reports superior scores on both SSIM and MSE. We will revise the abstract and methods section to explicitly note that baselines reflect their published (MSE) training regimes. This makes clear that the reported gains arise from the combination of the joint FC architecture and SSIM objective. Additional experiments retraining all baselines with SSIM would further isolate the architecture's contribution but are outside the scope of the current submission. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical training procedure with no reducing equations

full rationale

The paper presents an end-to-end training procedure for a fully-connected network that jointly learns sensing and reconstruction operators using SSIM loss, with experimental comparisons to SOTA methods under SSIM and MSE. No derivation chain, equations, or self-citations are described that reduce any claimed result or metric to a fitted parameter or input by construction. The approach is self-contained as an empirical method; no self-definitional, fitted-prediction, or load-bearing self-citation patterns apply. This is the expected outcome for a non-derivational ML paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No mathematical derivations or explicit assumptions are stated in the abstract; the approach rests on the empirical claim that joint optimization under SSIM improves both metrics.

pith-pipeline@v0.9.0 · 5632 in / 1107 out tokens · 22969 ms · 2026-05-25T16:19:41.953077+00:00 · methodology

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Reference graph

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