Learning Sub-Sampling and Signal Recovery with Applications in Ultrasound Imaging

Bastiaan S. Veeling; Iris A.M. Huijben; Kees Janse; Massimo Mischi; Ruud J.G. van Sloun

REVIEW 3 major objections 4 minor 60 references

Learning Sub-Sampling and Signal Recovery with Applications in Ultrasound Imaging

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A single end-to-end training procedure, Deep Probabilistic Sub-sampling, learns fixed sub-sampling masks that retain task-relevant information at least as well as random sampling and better than uniform sampling for ultrasound B-mode and…

desk verdict A genuinely useful and honestly reported learned-subsampling paper; the one real gap is that the 'fixed pattern' claim rests on a single draw from distributions that Appendix A shows never converged for channel Doppler. read the letter →

arxiv 1908.05764 v5 pith:H6DEDCO5 submitted 2019-08-15 eess.IV cs.LG

Iris A.M. Huijben , Bastiaan S. Veeling , Kees Janse , Massimo Mischi , Ruud J.G. van Sloun This is my paper · ORCID

classification eess.IVcs.LG

keywords deepprobabilisticsub-samplingtask-adaptivesamplingcompressedsensingGumbel-SoftmaxultrasoundimagingDopplersparsearrayslearnedreconstruction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the sampling pattern itself, which elements of a signal get measured, can be learned from data at the same time as the network that turns those measurements into a useful output. The proposed method, Deep Probabilistic Sub-sampling (DPS), models each of the $M$ measurements as a categorical draw over the $N$ candidate sample locations, uses the Gumbel-Softmax relaxation to backpropagate through the discrete choice, and after training fixes a mask drawn from the learned distributions. The authors show that DPS produces task-specific patterns: near-uniform for B-mode ultrasound, clustered slow-time ensembles for Doppler, and center-weighted channel selection for sparse arrays. The payoff, if correct, is that imaging systems can cut data rates and power consumption using fixed, hardware-friendly non-uniform sampling that retains task-relevant information at least as well as random or hand-designed sampling.

What carries the argument

The load-bearing object is the DPS generative sampling model: for each measurement $m \in \{1,\dots,M\}$, a categorical random variable $r_m \sim \text{Cat}(N,\pi_m)$ over the $N$ candidate positions, with probabilities $\pi_{m,n} = \exp(\phi_{m,n}) / \sum_i \exp(\phi_{m,i})$ derived from trainable logits $\phi_{m,n}$. A one-hot sample is drawn through the Gumbel-max trick, and sampling without replacement is enforced by adding $-\infty$ to previously chosen positions via a mask $w_{m-1}$. Training uses the straight-through Gumbel estimator: the forward pass commits to the hard $\arg\max$ sample, while the gradient is computed through $\mathrm{softmax}_\tau(w_{m-1} + \phi_m + e_m)$, with temperature annealed from $5.0$ to $0.5$ and an entropy penalty pushing the distributions toward one-hot masks. This machinery is what makes joint, differentiable optimization of a discrete sampling pattern and a downstream network possible.

What would settle it

Train DPS multiple times from different random seeds on the same in-vivo ultrasound data, fix one mask per run, and evaluate each on the hold-out test set; if the test MSEs across realizations vary widely, or if a fixed random mask at the same sub-sampling factor matches or beats the learned mask, then the claim that learned patterns retain task-relevant information better than standard sampling is not supported.

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Extended reading notes

Core claim

The paper's central claim is that a task-driven sub-sampling mask can be optimized end-to-end without relaxing the constraint that measurements are direct selections of input elements. DPS represents the mask as a matrix of trainable logits $\Phi \in R^{M \times N}$, draws rows as one-hot samples without replacement via the Gumbel-max trick, and trains $\Phi$ jointly with a neural task model $g_\theta$ by minimizing mean squared error plus an entropy penalty. Because the forward pass uses hard one-hot sampling while the backward pass uses the softmax temperature surrogate, the learned pattern converges to a fixed, discrete mask. Across partial Fourier recovery, slow-time ultrasound, and channel sub-sampling, the resulting masks match or outperform fixed uniform and random sampling when paired with the jointly trained network, and in Doppler imaging the learned ensemble pattern substantially reduces aliasing compared with naive interpolation plus the Kasai estimator.

Load-bearing premise

The claim rests on the assumption that the straight-through Gumbel-Softmax surrogate gradient, with the chosen temperature annealing and entropy penalty, actually steers the hard sampling mask toward a near-optimal pattern; this is supported empirically but not by any formal guarantee, and the evaluated masks are from a single stochastic realization.

Editorial extensions

If this is right

Once training is done, the learned mask is a fixed set of sample locations, so it can be implemented by non-uniform analog-to-digital conversion, sparse array design, or slow-time pulsing schemes without any per-acquisition optimization.
The same training procedure adapts the pattern to the task: B-mode recovery drives near-uniform slow-time sampling, Doppler recovery drives ensemble-like clustered sampling, and channel sub-sampling drives center-weighted arrays.
Jointly trained sampling plus task model outperforms fixed sampling with a trained model, and for Doppler it avoids the strong aliasing seen when interpolating frames before applying the Kasai auto-correlator.
The unfolded learned-ISTA task model recovers sparse signals from partial Fourier measurements faster than 300-iteration ISTA by a factor over 1000, with better MSE at high sub-sampling factors.
Learned sparse arrays can match a hand-designed full-sum-coarray array in B-mode MSE, suggesting DPS discovers known good array geometries from data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's claims, the same sub-sampling formalism would transfer naturally to MRI k-space or sparse-view CT, since those modalities already acquire partial Fourier or projection measurements; the partial-Fourier experiment is the closest evidence.
Because the reported masks are single stochastic realizations of trained distributions, a seed-variability study would tell whether the learned pattern itself is stable; the paper does not report this, so the realized mask should be read as one sample, not as the unique optimum.
One could test the method's sensitivity to the straight-through surrogate by training with several temperature schedules or entropy penalty weights; if performance is strongly schedule-dependent, the relaxation's behavior, not the sampling concept, may be the limiting factor.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about arXiv:1908.05764. First, the core idea is real and was genuinely new at the time: instead of learning weighted linear measurements like most learned-CS work, DPS learns categorical distributions over which signal elements to keep, using Gumbel-Softmax plus a masking trick for sampling without replacement. That closes a practical gap—weighted combinations are hard to build into hardware, discrete sub-sampling masks are not. Second, the paper is honest in ways that make it better than average. It reports an essentially negative result for slow-time B-mode (the learned pattern is nearly uniform, performance matches uniform sampling), and Appendix A states openly that for channel sub-sampling with color Doppler the trained distributions were far from one-hot.

The math in Section III is coherent. The row-wise one-hot constraint on the sampling matrix, the w_{m-1} masking for sampling without replacement, the straight-through Gumbel gradient with temperature annealing, and the entropy penalty are all set up correctly, and the pseudocode is clear enough to implement. The ultrasound targets (B-mode envelope, Kasai Doppler) are computed from fully sampled data, so there is no circularity. Comparing the learned sparse array against a fixed full-sum-coarray design is the right kind of check. The Fourier experiment is a useful sanity check: DPS matches random sampling, and both beat uniform, which is what CS theory would suggest.

The soft spot is the one the authors themselves flag. The central claim that DPS yields a fixed, hardware-implementable pattern is not supported for exactly the channel-Doppler case where the distributions are least concentrated. The reported result is a single stochastic draw from distributions that still assign substantial probability across many channels, and a different draw could give a different sparse array with different grating-lobe behavior. There are no error bars or repeated-realization statistics anywhere in the results, and the ultrasound validation uses one train/test split from a two-animal cohort. That limits the strength of the channel-Doppler comparison in Fig. 8b, though the Fourier and slow-time results are more robust. The Gumbel-Softmax surrogate being heuristic is a common field-wide issue, not a paper-specific flaw. Minor point: the method overlaps with the authors' own ICLR 2020 paper, and the journal version should state clearly what is added beyond it (the ultrasound applications).

This paper is for anyone working on learned sensing, compressed ultrasound, or hardware-aware acquisition. I would send it to peer review and would cite the DPS method. The main things I'd ask a revision to add: repeated realizations and error bars for the non-converged distributions, and a concrete statement of how the pattern is selected at deployment time.

Referee Report

3 major / 4 minor

Summary. The paper proposes Deep Probabilistic Sub-sampling (DPS), a framework that jointly learns a categorical distribution over candidate measurement indices and a neural task model in an end-to-end fashion. Sampling is performed without replacement via a Gumbel-max trick with masking, and gradients are obtained through a straight-through Gumbel-Softmax surrogate with temperature annealing. The method is validated on three settings: sparse signal recovery from partial Fourier measurements, slow-time sub-sampling for ultrasound B-mode and color Doppler imaging, and channel sub-sampling for sparse array design. The authors report that DPS yields task-specific patterns, that learned patterns are comparable or superior to uniform and random fixed sampling when combined with the learned task model, and that the resulting fixed patterns are hardware-implementable.

Significance. If the reported results are robust, the work is significant: it offers a practical way to learn hard, task-driven sub-sampling masks with a trainable downstream network, avoiding the hardware challenges of randomly weighted linear measurement schemes. The method is clearly described with pseudocode and the masking mechanism for sampling without replacement is coherent. The use of in-vivo porcine ultrasound data and the comparison with a hand-designed full-sum-coarray sparse array are strong points, as is the interpretable finding that B-mode and Doppler tasks induce qualitatively different learned sampling patterns. The main limitations are the reliance on single stochastic realizations for some reported patterns and the absence of error bars or repeated-run statistics in the quantitative comparisons.

major comments (3)

[Section V-C / Fig. 8 / Appendix A] The central claim that DPS produces a fixed, hardware-implementable pattern that outperforms uniform sampling is not established for channel sub-sampling with color Doppler. Appendix A explicitly reports that the learned probability distributions for this task did not converge to near-one-hot (Fig. 10d), yet the quantitative DPS results in Fig. 8b and the qualitative images in Fig. 7 are based on a single stochastic realization of those distributions. Different realizations can yield different sparse arrays with different grating-lobe behavior and different Doppler MSE, so the reported advantage over uniform sampling may be realization-specific. Please report the mean and spread of test MSE over multiple draws from the trained distributions, and specify the procedure by which a single deployment pattern would be selected from a non-concentrated distribution.
[Section III-D, Eq. (11)] The entropy penalty LS is computed from the unmasked probabilities πm defined in Eq. (5), while the actual forward sampling in Eq. (7) uses masked logits wm−1 + φm to enforce sampling without replacement. Penalizing the unmasked entropy does not guarantee that the conditional distribution used at row m is concentrated after earlier rows have removed candidates. This mismatch plausibly contributes to the non-convergence reported in Appendix A for channel Doppler and should be fixed or analyzed; otherwise the training objective does not directly promote the one-hot patterns on which the 'fixed pattern' claim rests.
[Figs. 4, 6, 8] All quantitative comparisons are reported as point estimates without error bars, confidence intervals, or significance tests. Because DPS training is stochastic (Gumbel noise, mini-batch sampling) and the final evaluation itself uses a random realization, the observed differences between DPS and Random+LISTA or between DPS and Uniform+task-model could be within run-to-run variability. Please report results over at least three independent training runs and multiple test realizations for the main comparisons.

minor comments (4)

[Section IV-B1] The transducer system is referred to as 'Verasonics Vantrage'; this appears to be a typo for 'Verasonics Vantage'.
[Fig. 9 caption] The caption contains a duplicated MSE value ('MSE: 2.1e-62.1e-6') and the typo 'task model.s'; both should be corrected.
[Section III-B, Eq. (8)] The notation ∇φm am := ∇φm E[softmaxτ(...)] is nonstandard because the left-hand side denotes a gradient of a hard one-hot sample. Please clarify that this is the surrogate gradient used in backpropagation, not the true gradient of am.
[Section V-A] The statement that the learned sensing matrix 'showed to be RIP-compliant' should specify how the check was performed, for example whether all M×K submatrices were tested and what numerical criterion was used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: DPS optimizes a sampling distribution against fully-sampled task targets, and all comparisons are external.

full rationale

The paper's derivation chain is self-contained. DPS defines the measurement vector as y = A_Phi x (Eq. 1), the task target as z = f(x) (Eq. 2), and the prediction as z_hat = g_theta(y) (Eq. 3). The training objective (Eqs. 12-14) minimizes the MSE between z_hat and the fully-sampled target z, plus an entropy penalty on the sampling distribution. The targets are computed independently of the learned pattern: for Doppler they come from the Kasai auto-correlator applied to fully sampled data, and for B-mode from the envelope of fully sampled beamformed data. The learned sub-sampling matrix is therefore optimized against external task labels, not against a quantity derived from the pattern itself. The paper also evaluates against untrained uniform sampling, random sampling, and a hand-designed full-sum-coarray sparse array, so the claimed improvements are externally anchored. The only self-citations (Refs. [28], [49]) are prior methodological or architectural work and are not load-bearing: the paper provides its own equations, algorithm, and training details. The Appendix A caveat that channel-Doppler distributions did not converge to near-one-hot is a robustness/limitation issue, not circularity. There is no fitted parameter renamed as a prediction, no uniqueness theorem imported from the authors, and no ansatz smuggled in via citation. Accordingly, the circularity score is 0.

Assumptions & free parameters 7 free parameters · 5 assumptions · 1 invented entities

The central claim rests on a new training procedure rather than new physics. The main free parameters are hyperparameters of the optimization procedure, and the main axioms are domain assumptions about the surrogate gradient, model capacity, and representativeness of the ultrasound data. The paper introduces DPS as an architectural entity, not a new natural phenomenon.

free parameters (7)

temperature schedule (tau) = 5.0 to 0.5
The annealing schedule for the Gumbel-Softmax temperature is chosen by hand and affects the variance and convergence of the learned sampling patterns.
entropy penalty multiplier mu = 1e-7 to 1e-8, increasing per epoch
The entropy penalty is used to push the learned distributions toward one-hot patterns; its value is chosen by hand.
L2 penalty multiplier lambda = 0, 1e-5, 1e-5
The weight decay on the task model parameters differs across experiments; the value is chosen by hand.
learning rates eta_Phi and eta_theta = 5e-3/1e-3, 2e-3/1e-4
The learning rates for the logits matrix and task model are chosen by hand and are reported in Section IV.
logits initialization quartic coefficients (alpha, beta) = alpha = -2.73e-7, beta = -2.73e-3
The initialization in Eq. (15) biases the sampling matrix toward a diagonal pattern; the constants appear hand-chosen and affect the final learned pattern.
Gaussian noise scale on logits initialization = sigma = 0.01
Small random perturbation added to the logits initialization, chosen by hand.
number of training iterations and mini-batch sizes = 96000, 640000, 192000
Training budget is chosen by hand; the paper notes that fine-tuning was out of scope.

assumptions (5)

domain assumption The Gumbel-Softmax relaxation, Eq. (8), provides a differentiable surrogate that, combined with the annealing schedule, yields optimal or near-optimal sub-sampling patterns for the hard, non-differentiable forward operation in Eq. (7).
The paper uses the straight-through Gumbel estimator and temperature annealing without a formal convergence or optimality guarantee. The load-bearing validity of the learned patterns rests on this empiricism.
domain assumption The convolutional task model architecture is sufficiently expressive and generalizes to the test distribution.
The task model is a feedforward CNN; its ability to reconstruct from the sub-sampled data is validated only by MSE on the held-out test set, without an analysis of distribution shift or architecture sensitivity. The paper acknowledges this limitation in Section VI.
domain assumption The in-vivo porcine data, acquired with a single ultrasound setup, is representative enough for the learned patterns to be useful in the target deployment scenario.
The training set is only 1340 frames from one porcine model and the test set is 335 frames from another model with a different cardiac condition. The learned patterns are therefore only validated for that narrow acquisition context.
domain assumption The non-learned baselines (ISTA, cubic interpolation, DAS, Kasai) are implemented sufficiently well to provide a fair comparison.
The paper compares against fixed uniform/random patterns with the same task model and against naive processing, but it does not provide details on hyperparameters or implementations of the baselines, so the strength of the baseline is uncertain.
domain assumption The sparse signal model for the synthetic partial Fourier experiment is representative of real compressed sensing scenarios.
Only random K-sparse signals in the frequency domain are tested; the signal model is synthetic and does not include noise, extension to non-sparse signals, or the structured signals found in real imaging.

invented entities (1)

Deep Probabilistic Sub-sampling (DPS) layer
purpose: Learns a categorical distribution over possible sampling indices, enabling joint optimization of the sampling mask and the task model.
DPS is a new architectural component, not a new physical entity; its evidence is the paper's own experiments. There is no external falsifiable handle outside the framework.

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Cite this review

Pith. "Pith review of Learning Sub-Sampling and Signal Recovery with Applications in Ultrasound Imaging." pith.science (2026). https://pith.science/paper/H6DEDCO5

@misc{pith2026190805764,
  author       = {Pith},
  title        = {Pith review of: Learning Sub-Sampling and Signal Recovery with Applications in Ultrasound Imaging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6DEDCO5}},
  note         = {Machine review of arXiv:1908.05764}
}

read the original abstract

Limitations on bandwidth and power consumption impose strict bounds on data rates of diagnostic imaging systems. Consequently, the design of suitable (i.e. task- and data-aware) compression and reconstruction techniques has attracted considerable attention in recent years. Compressed sensing emerged as a popular framework for sparse signal reconstruction from a small set of compressed measurements. However, typical compressed sensing designs measure a (non)linearly weighted combination of all input signal elements, which poses practical challenges. These designs are also not necessarily task-optimal. In addition, real-time recovery is hampered by the iterative and time-consuming nature of sparse recovery algorithms. Recently, deep learning methods have shown promise for fast recovery from compressed measurements, but the design of adequate and practical sensing strategies remains a challenge. Here, we propose a deep learning solution termed Deep Probabilistic Sub-sampling (DPS), that learns a task-driven sub-sampling pattern, while jointly training a subsequent task model. Once learned, the task-based sub-sampling patterns are fixed and straightforwardly implementable, e.g. by non-uniform analog-to-digital conversion, sparse array design, or slow-time ultrasound pulsing schemes. The effectiveness of our framework is demonstrated in-silico for sparse signal recovery from partial Fourier measurements, and in-vivo for both anatomical image and tissue-motion (Doppler) reconstruction from sub-sampled medical ultrasound imaging data.

Figures

Figures reproduced from arXiv: 1908.05764 by the authors.

**Figure 2.** An illustration of the sensing pipeline for the considered US [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Averaged inference results across all signals from the hold-out test [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (7 more)

**Figure 3.** Figure 3: Left column: Fixed uniform (a), fixed random (b) and learned (using N [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]

**Figure 5.** Figure 5: Reconstructed anatomical B- and M-mode (a-e) and color Doppler images (f-j) after sub-sampling by a factor 4 across slow-time frames, either [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: MSE for B-mode (a) and Doppler (b) recovery from a sub-set of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Reconstructed anatomical B-mode images, M-mode images, and line profiles (a-e), as well as Color-Doppler images (f-j) using either uniform (a, b, [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: MSE for B-mode (a) and Doppler (b) reconstruction after channel [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: Comparison of B-mode recovery for half of the channels selected, [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 10.** Figure 10: Learned probability distributions in DPS across ultrasound tasks [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

Discussion (0). Continue with ORCID to comment.

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