Deep probabilistic subsampling for task-adaptive compressed sensing
Authors: Iris A.M. Huijben
Bastiaan S. Veeling
Ruud J.G. van Sloun
Venue:  (2020)

Abstract: The field of deep learning is commonly concerned with optimizing predictive models using large pre-acquired datasets of densely sampled datapoints or signals. In this work, we demonstrate that the deep learning paradigm can be extended to incorporate a subsampling scheme that is jointly optimized under a desired minimum sample rate. We present Deep Probabilistic Subsampling (DPS), a widely applicable framework for task-adaptive compressed sensing that enables end-to end optimization of an optimal subset of signal samples with a subsequent model that performs a required task. We demonstrate strong performance on reconstruction and classification tasks of a toy dataset, MNIST, and CIFAR10 under stringent subsampling rates in both the pixel and the spatial frequency domain. Due to the task-agnostic nature of the framework, DPS is directly applicable to all real-world domains that benefit from sample rate reduction.

Published as a conference paper at ICLR 2020
DEEP PROBABILISTIC SUBSAMPLING FOR TASK-ADAPTIV COMPRESSED SENSING
Iris A.M. Huijben"
Bastiaan S. Veeling'
Department of Electrical Engineering Eindhoven University of Technology Eindhoven, The Netherlands
Department of Computer Science, University of Amsterdam Amsterdam, The Netherlands basveell ing i 1, com
i.a. m. hui jben wereentuuu..e. tue.
Ruud J.G. van Sloun
Department of Electrical Engineering Eindhoven University of Technology Eindhoven, The Netherlands r.j.g.v wlo.nntue.nn
ABSTRACT
The field of deep learning is commonly concerned with optimizing predictive models using large pre-acquired datasets of densely sampled datapoints signals. In this work, we demonstrate that the deep learning paradigm can extended to incorporate subsampling scheme that is jointly optimized under desired sam- pling rate. We present Deep Probabilistic Subsampling (DPS), widely appli- cable framework for task-adaptive compressed sensing that enables end-to-end optimization of an optimal subset of signal samples with subsequent model that performs required task. We demonstrate strong performance on reconstruction and classification tasks of a toy dataset, MNIST, and CIFAR10 under stringent subsampling rates both the pixel and the spatial frequency domain. Thanks the data-driven nature of the framework, DPS is directly applicable to all real- world domains that benefit from sample rate reduction. The code used for this paper is made publicly available
INTRODUCTION
In many real-world prediction problems, acquiring data is expensive and often bandwidth- constrained. Such is the case in regimes medical imaging (Lustig et al. 20077: Choi et al. 2010 Chernyakova & Eldar 2014), radar (Baraniuk] 2007). and seismic surveying (Herrmann et al. 2012).
By carefully reducing the number of samples acquired over time, in pixel-coordinate space k- space, efficient subsampling schemes lead to meaningful reductions in acquisition time, radiation exposure, battery drain, and data transfer.
Subsampling is traditionally approached by exploiting expert knowledge on the signal of interest. Famously, the Nyquist theorem states that when the maximum frequency of continuous signal is known, perfect reconstruction is possible when sampled at twice this frequency. More recently, it has been shown that if the signal is sparse in certain domain, sub-Nyquist rate sampling can be achieved through compressive measurements and subsequent optimization of linear system under
said sparsity prior: a framework known compressed sensing (CS) (Donoho et al. 2006 Eldar & Baraniuk (007).
Kutyniok 2012:
CS methods however lack in the sense that they do not (under a given data distribution) focus solely on the information required to solve the downstream task of interest, such as disease prediction or semantic segmentation. Formalizing such knowledge challenging in its own right and would require careful analysis for each modality and downstream task. In this work, we propose to explore
"Equal contribution.
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the deep learning hypothesis promising alternative: reducing the need for expert knowledge in lieu large datasets and end-to-end optimization of neural networks.
As subsampling is non-differentiable, its integration into end-to-end optimized deep learning model is non-trivial. Here we take a probabilistic approach: rather than lcarning a subsampling scheme directly, we pose probability distribution that expresses belief over effective subsampling patterns and optimize the distribution'ss 's parameters instead. To enable differentiable sampling from this distribution, leverage recent advancements in a continuous relaxation of this sampling pro- cess, known as Gumbel-softmax sampling or sampling from a concrete distribution (Jang et al., 2017 Maddison al., 2016). This enables end-to-end training of both the subsampling scheme and the downstream model.
Naively, the number of parameters of a distribution over an n-choose-k problem scales factorially, which is intractable for all practical purposes. We propose novel, expressive yet tractable, parame- terization for the subsampling distribution that conditions on the output sample index. We hypothe- size that such conditioning prevents redundant sampling: it enables modeling the scenario in which multiple candidate samples can be equally good, yet redundant combination. Furthermore, we investigate rertmeter-resttriced approach with single parameter per candidate sample, balancing tractability and exploration (Kool et al. 2019). In this case, we adopt a continuous relaxation of top-K K sampling guarantee differentiability (Pöötz & Roth 2018).
Our main contributions are as follows:
DPS: A new regime for task-adaptive subsampling using novel probabilistic deep learning framework for incorporating sub-Nyquist sampling scheme into an end-to-end network.
DPS enables joint optimization of subsampling pattern with a predictive downstream model, without the need for explicit knowledge on a sparsifying basis.
We demonstrate improved performance over strong subsampling baselines in image classi- fication and reconstruction, while sampling both in Fourier and pixel space.
2 RELATED WORK
Recent works have proposed Seep-learning-baat based subsampling methods for fast MRI, historically being one of the most prominent applications of CS. Weiss et al. (2019) exploit gradient back- propagation to a fixed set of real-valued coordinates, enabled by their subsequent (limteed-support) interpolation on the discrete k-space grid, and Bahadir et al. (2019) formulate the sampling problem by learning pixel-based thresholding i.i.d. samples drawn from a uniform distribution, referred to as LOUPE. Where the former suffers from limited exploratory capabilities (likely due to its compact support), with learned sampling schemes typically not deviating far from their initialization, the lat- ter controls the sample rate only indirectly, through the addition of sparsity-promoting penalty on the sampling mask.
Related our methodology, yet different in purpose, Plötz. & Roth (2018); Xie & Ermon (019); Kool et al. (2019) leverage an extension to the Gumbel-max trick (Gumbel 1954) for (ordered) subset selection using a categorical distribution with N free parameters. They rely on top-K sampling, as first proposed by Vieira (014)). Based upon the continuous relaxation of the cate- gorical distribution, known in the deep learning community
the Gumbel-softmax trick or the Plötz & Roth (2018) show that such relaxation also exists for top-K sampling. In this work, we investigate the use of Gumbel-based relaxations for task-adaptive compressive sampling, and further propose a novel parametrization where we condition the sampling distribution on the output sample index.
concrete distribution (Jang et al. 2017] Maddison et al.
We differentiate our contribution from deep encoder-decoder methods for data compression (Baldi & Hornik 1989: Hinton Zemel 1993 Blier & Ollivier 2018 Habibian et al. 2019), which do not aim at reducing data rates already at the sensing and digitization stage. Related work by Mousavi et al. (2019) and WW et al. (2019), focusses on the problem of learning compressive linear encoders/iilters. rather than discrete subsampling addressed here. The authors of Baln et al. (2019) use a Gumbel-mai max- maabbeddd auto-encoder for discrete feature selection, which is however not task-adaptive.
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a.
Generative sampling model
b.
Input
Sampling
fa
Task
Φ m
E Vm
2
A
digit
m
Gumbel(0,1)
C.
Input
Sompling
fo
Task
e 0://
T - T
2
E/en//
am
x
y
$
A
fv
Sampling
Task model
[s,
Forward
Backpropagation
Desired task
S
Figure 1: (a) System-level overview of the proposed framework, in which probabilistic generative sampling model (DPS) and a subsequent task model are jointly trained to fulfill a desired system task. (b,c) Two illustrative task-based sampling paradigms: image classification from partial set of pixels (b), and image reconstruction from partial Fourier measurements (c), respectively.
Through the lens of contemporary deep learning, subsampling can be interpreted as a form of tention (Bahdanau al.) 2014, Kim et al. 2017 Parikh et al. 2016; Vaswani et al., (2117). Rather than attending on intermediate representations, our model "attends" directly on the input signal. For subsampling to be effective, sparse weights are essential. In the space of attention, this is known as hard attention (Xu et al.) 2015), and is typically optimized using the REINFORCE gradient estima- tor (Williams 1992). In contrast the method of attention applied in these works, method aims for a fixed, reduced subsampling rate.
3 METHOD
3.1 TASK-ADAPTIVE SYSTEM MODEL
We consider the problem of predicting some downstream task 8, through a learned subsampling scheme A on samples signal x E RR resulting measurements y R with M << N:
Ax,
where A E {0, 10,1 is the subsampling measurement matrix. We here concern ourselves specifically with scenarios in which the rows of A are constrained to having cardinality one, i.e. |amlllo = 1, Vm E
J. In such cases, A serves as subset selector, sampling M out of N elements in x. From the resulting low-rate measurements y, we aim at predicting task through:
fo(y):
with fe(-) being function that is differentiable with respect its input and parameters 0, e.g. neural network.
Given downstream task and dataset, we are interested in learning both the optimal processing pa- rameters 0, and the sampling scheme as described in eq. ( To circumvent the non-differential nature of discrete sampling, will introduce novel fully probabilistic sampling strategy that al- lows for gradient- based learning through error backpropagation, on which we detail in the following section.

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3.2 DPS: DEEP PROBABILISTIC SUBSAMPLING
Since direct optimization of the elements in A is intractable due to its combinatorial nature, we here instead propose to leverage a tractable generative sampling model that is governed by learned subsampling distribution, parameterized by d:
( ~ 2)
(3)
Thus, rather than optimizing ( we optimize the distribution parameters Φ. To warrant sufficient expressiveness while maintaining tractability, we learn the parameters Φm IR of M independent categorical distributions (rather than the joint distribution, which scales factorially), being the rows of & E IR M
Formally, we define each m measurement row am of an independent categorical random variable I' Cat(N, mm) We define n. TNN [Tm.II..... Tm.N N. being vector containing N class probabilities, and parameterize it in terms of its unnormalized logits Φm such that:
{0,1) in A(a) one-hot encoding
exp Pm.n
Mm.
(4)
exp Φm,i
We sample from Cat(N, N, mm) by leveraging the Gumbel-max trick (Gumbel 1954). reparameteri- zation of the sampling procedure. It perturbs the unnormalized logits Om.n with i.idd Gumbel noise samples em,n Gumbel 1). The argmax of these perturbed logits, results in a sample from the original categorical distribution Cat(NV mm) realization i'm then defined as:
Fi = argmax (mm-1,n + Omn Cm,)): m M].
(5)
in which Wm-1,n { to 0} for adding -oo to the logit of the previously selected category, enforcing sampling without replacement among the M distributions. Introducing the function one one not as the operator that returns one-hot vector of length N, we finally obtain:
[I...N) mask previous realizations Fi,-.. Fm--1 1. by
am = one hot m one one hot argmax (m 1m-11,n + Pm. + em
(6)
To permit error backpropagation for efficient optimization of + we require V an to exist vm E (1...., M). Since argmax(1 (.) is mon-differentiabee operator, we adopt the Etraight-Through
Gumbel Estimator (Jang et al. 2017 Maddison al. 2016) as a surrogate for V tt ami
db Ee,,, [softmaxx (wm-1 + Φm + +mm)]
am
exp((w + Φm + em)/T}
Eeem Φm
+ om,i + cm,
with (row operator) softmax,() (.) as a continuous differentiable approximation of the one-hot en- coded argmaxx-) operation. Appendix A provides the full derivation of V dt., ams and fig. shows schematic overview of the proposed framework.
We refer to sampling using the softmax, (-) function as soft sampling. Its temperature parameter serves as gradient distributor over multiple entries (i.c. logits) in Φm. Using a relatively high value enables updating of multiple logits during training, even though hard sample was taken in the forward pass. In the limit of r - O. soft sampling approaches the one-hot encoded argmax(-) operator in eq. (6) (Jang al., 2017, Maddison et al. 2016). While lowering reduces the gradient estimator's is bias, it comes at the cost of higher variance. We find however that using fixed (tuned) value of 7 worked well for our experiments, and not explore methods to reduce the introduced gradient bias (Grathwohl al. 2017 Tucker et al. 2017
The Gumbel distribution is typically used model the maximum probability density function is of the form: f(x) ------=* with 2 =i" where μ and are the mean and standard deviation, respectively. The standard Gumbel distribution, with μ 0 and B 1 is adopted in the Gumhel-max trick.
independent samples. Its
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To study the trade-off between parameter efficiency and convergence, further study more parameter-restricted version of DPS that shares the weights Φm across all M distributions. This allows for et al. 2019) (rather than Gumbel-max), and its corresponding relaxation (Plötz & Roth 2018). We refer this particular model as DPS-topK, and adopt DPS-topl to indicate the sampling model pre- sented in equations [ and 7 where weights are not shared across distributions, and one sample is drawn from each distribution. Algorithm in appendix B describes the DPS algorithm.
meeorr--effcieenl implementation, by leveraging the Gumbel top-K trick (Kool
4 EXPERIMENTS
We test the applicability of the proposed task-adaptive DPS framework for three datasets and two distinct tasks: image classification and image reconstruction. We explore subsampling in pixel- coordinate space as well as k-space. The latter is relevant for scenarios in which data acquired in the frequency domain, such as (but not limited to) magnetic resonance imaging (MRI).
4.1 MNIST CLASSIFICATION
Experiment setup Classification performance was tested on the MNIST database (LeCun al. 1998), comprising 70,000 28 28 grayscale images of handwritten digits O to 9. We split the dataset into 50,000 training images, 5,000 validation, and 5,000 test images. We train DPS-topl and DPS-topK to take partial measurements in either the image or Fourier domain, and process them through the task model to yield a classification outcome. Results are compared to those obtained using uniformly distributed pixel/Fourier samples, sampled disk/low pass filter, and the data-driven LOUPE method (Bahadir et al. 2019).
Besides, show the benefit of task-adaptive sampling, i.e. jointly training sampling with the task, we add a case in which we disjointly learn the classification task from the sampling pattern. As such, train DPS-topK for MNIST reconstruction, after which the learned sampling pattern is frozen, and the classifier is trained.
Task model After sampling M elements, all N zero-masked samples (or 2N in the case of complex Fourier samples) are passed through series of 5 fully-connected layers, having N. 256, 128, 128 and 10 output nodes, respectively. The activations for all but the last layer were leaky Re- LUs, and 20% dropout was applied after the first three layers. The 10 outputs were normalized by softmax function to yield the respective classification probabilities. Table T in appendix C shows a detailed overview of the complete architecture of the task model. Zero-filling and connecting all pos- sible samples, rather than only connecting the M selected samples, facilitated faster co-adaptation of the network to different sampling patterns during training.
We train the network to maximize the log-likelihood of the observations D {(x,, s;) i O,.... [} through minimization of the categorical cross-entropy between the predictions and the labels, denoted by E,. We moreover promote training towards one-hot distribu- tions Cat(N, (N, nm) by penalizing high entropy, adopting:
Training details
M
N
L.
Mm. log Tm,n:
(8)
22ml
with Tm, defined as in eq. 44. The total optimization problem is thus:
Φ 0
argmin E ((x,)) C. HCCo
(9)
4,0
where Pp is the data generating distribution. Penalty multiplier μ was set to linearly increase 1e-5 per epoch, starting from 0.0. Increasing this entropy penalty enforces one-hot distributions, and therefore less variations in the realizations, i.e. sampling patterns, when training evolves. The tem- perature parameter 7 in eq. 7 was set to 2.0, and the sampling distribution parameters & were initialized randomly, following zero-mean Gaussian distribution with standard deviation 0.25. Equation X was optimized using stochastic gradient descent on batches of 32 examples, approxi- mating the expectation by mean across the train dataset. To that end, used the ADAM solver
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(a)
Image-domain sampling
(b)
Fourler-domain sampling
Example imputs
2
C
6
7
Plaels removed: 87.5%
Pixels removed: 96.8%
Fourler coeff. removed: 87.5%
Fourier coeff. removed: 96.8%
DPS-topk
DPS-top1
Perc. error:
Perc. error:
Perc. error:
Perc. error:
DPS-topK
2.1%
6.6%
1.3%
1.3%
DPS-top1
1.2%
LOUPE
2.1%
19.9%
6.2%
32.0%
Uniform 4.2%
28. 6%
9.1%
54.5%
Disk
6.3%
27.3%
1.3%
1.7%
Figure 2: MNIST classification for (a) image-domain, and (b) Fourier-domain subsampling. (top) Several example images in the respective sampling domains, (2nd and 3rd row) learned task-adaptive (DPS-topK and DPS-topl, respectively) subsampling patterns, with their relative sample incidence across 1000 such realizations (inset), and (bottom) hold-out classification results of the pro- posed DPS methods, compared the LOUPE baseline, and two non-learned baseline sampling approaches.
(3 0.9, 32 0.999, and 1e-7) (Kingma & Ba 2014), and trained until the vali- dation loss plateaued. We adopted different learning rates for the sampling parameters and the parameters of the task model 6, being 2e and 2e 4, respectively.
Results The results presented fig. 2h show that image-domain sampling using DPS sig- nificantly outperforms the fixed sampling baselines (uniform and disk). The data-driven LOUPE method outperformed in case of a strong subsampling factor. The resulting patterns qualita- tively demonstrate how, for this task, a sensible selection of pixels that are made (slightly slanted, and capturing discriminative areas). Notably, partial Fourier measurements (fig. (2)) allowed for much greater reduction of the number samples, with DPS sampling out- performing uniform sampling and LOUPE, and showing similar performance fixed disk sam- pling. Interestingly, the DC and very-low frequency components were consistently not selected.
informative was
Figure shows the learned sampling pattern in pixel space (96.8% pixels removed), when training DPS-topK the reconstruction task. The adopted reconstruction network architecture is the same as the one used for CIFARIO reconstruction (see section 4.3). The subse- quently trained classifier resulted in classification ror of 9.3%, compared to 6.6% for task-adaptive learning with DPS-topK. This case clearly illustrates the benefit of lask-adaptive learning of a sampling pattern.
Figure 3: A realization of the learned distribution (inset) for reconstruction.
4.2 'LINES AND CIRCLES' IMAGE RECONSTRUCTION
Experiment setup To evaluate reconstruction of structured images from highly undersampled partial Fourier (k-space) measurements (keeping 3.1% of the coefficients), we generated synthetic toy data comprising images that each contain up horizontal lines and randomly-sized circles. Lines and circles were placed at random positions and their pixel intensity was drawn from a uniform distribution between and 10. Examples generated in on-line fashion during training. A

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Proposed
partial k-space sampling 3.15
Uniform
Random
Low
LOUPE
DPS-top1
DPS-topK
Fourier transform
Iuspace
Target
PSNR: 16.7d8 (b)
PSNI:: 27.4 dB
PSNR: 28.7 dA
PSNR: 35.5 dB (e)
PSNR: 37.6 dB
PSNR: 39.6 de
(a)
(c)
(d)
(f)
(g)
Figure 4: Image reconstruction performance from partial k-space (Fourier) measurements on cus- tom toy dataset consisting of lines and circles with random locations and sizes. Illustrative examples of the k-space and target images are given in (a). The sampling patterns, reconstructed images and PSNR value (across the entire test set) for the different sampling strategies are displayed in: (b)
uniform, (c) random, (d) low pass, (e) LOUPE, (f) DPS-topl, and (g) DPS-topK, respectively. In all cases, only 3.1% the Fourier coefficients have been selected.
pre-generated hold-out test 1000 randomly generated examples was used for all cases. Two illustrative test examples, along with their Fourier-domain representations, are given in (a We compare the results to those obtained using three fixed partial Fourier measurement baselines, following uniform, random, and low pass subsampling patterns, respectively, and the data-driven LOUPE method.
Task model Image reconstruction from partial Fourier measurements was performed by fol- lowing methodology in [Zhu et al. (2018). We use a deep neural network consisting of two subsequent fulle-connected layers with tanh activations that map the 2.V (zero-filled Fourier coef- ficients (stacked real and imaginary values) to a vector of length N. This vector was subsequently reshaped into a VN x VN image, and processed by 3 convolutional layers. Table in appendix C provides the layer parameters of this task network.
Training details
The optimization problem is the same as in eq. 9. however with the loss C, now defined the negative log-likelihood:
C. = E (x.)) [Ifo(A s||2-
(10)
The learning rates for + and 0 were 1e - and 1e 4, respectively, and μ and 7 were respectively to 2e 4 and 5.0. The ADAM optimizer (with settings as provided in section 4.1) was used to train in mini-batches of 128 examples. Training was stopped when the validation loss plateaued.
overview of the results is given in fig. As expected, uniform subsampling leads to strong spatial aliasing that can not be recovered by the task model due to violation of the Nyquist cri- terion. In turn, random subsampling introduces an incoherent aliasing pattern, that can only partly be recovered. Although not suffering from aliasing, low pass sampling deteriorates resolution, of which the effect is particularly evident for the broadband/sharp horizontal lines. contrast, data-driven sampling methods (LOUPE, DPS-topl, and DPS-topK) prevent aliasing by learning appropriate sampling pattern. DPS sampling specifically, yields high-resolution accurate reconstructions with an improved PSNR evaluated the entire test set compared the baselines. Note how the learned
Results

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Proposed
partial k-space
Uniform
Random
Low
LOUPE
DPS-top1
DPS-topK
sampling 12.9%
Fourier transform
hupace
PSNR: 14.8 dB (b)
PSNR: 12.7 dB
PSNR: 25.3 (d)
PSNR: d5 (e)
PSNR: 27.1 65 0.80
PSNR: 28.3 dB
(a)
(c)
(f)
(g)
Figure 5: Image reconstruction performance from partial k-space (Fourier) measurements on the CIFARIO 10 database. Illustrative examples of the k-space and target images are given in (a). The sampling patterns, images and statistical quality metrics for uniform, random, low pass, LOUPE, DPS-topl, and DPS-tope given in (b-g), respectively. In all cases, 12.5% of the Fourier coeffi- cients have been selected.
sampling patterns (top row) have evolved from random initialization, similar that of (c), the ultimate task-adaptive scheme (e-g).
4.3 CIFARIO IMAGE RECONSTRUCTION
Experiment setup The CIFAR10 database (Krizhevsky al., 2009) contains 60,000 images of 32 32 pixels in 10 different classes. We converted all images to grayscale, and subsequently split them into 50,000 training images, 5,000 validation and 5,000 test images. We again learn partial Fourier sampling and image reconstruction, keeping only 12.5% of the Fourier coefficients, and compare ourselves to the three fixed partial Fourier measurement baselines and LOUPE, as described in section 42
Task model The challenging reconstruction task for CIFARIO motivates the adoption of struc- tured model to enable strong reconstruction. We draw inspiration from iterative proximal-gradien schemes (Parikh et al. 2014). which are dedicated to solving the ill-posed linear measurement prob- lem in eq. (1) To that end, unfold K such iterations, learning adequate image-domain proximal mapping (kk) and stepsize (k) at each fold:
g (k+1) p(()) (s
F ATo) (A(o) A(o)x))
(11)
where E NN is discrete Fourier transform (DFT) matrix, and (.) denotes the Hermitian (conjugate transpose). In the above formulation, each fold a step is taken towards the sampling- consistent subspace that adequately represents the physical measurement of by A F. The trained
proximal operator (*()) a 7-layer convolutional network, then projects this onto the manifold of visually plausible images (Mardani al., 2018). removing noise, aliasing, or blurring artifacts. Table in appendix C provides details on the layers in the task model.
Training details Optimization settings similar to those in section 4.2 leveraging mean-
squared-error (negative log-likellin likeliioohhod) reconstruction cost and a distribution entropy penalty on n. To promote visually plausible reconstructions, added an adversarial (Ledig et al. 2017 cost by
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adopting discriminator network (-) that aims to discriminate between images reconstructed from partial Fourier measurements (8) and actual images s. The discriminator comprised convolu- tional layers with leaky ReLU activations, followed by global average pooling, 40% dropout, and logistic binary classification model (parameters are provided in table appendix The sampling- and task model parameters were then trained to both minimize the negative log- likelihood and dis- criminator loss LD.. in addition to the entropy penalty Le:
LD. Ess (log(Dx(s)) + log(1 - D& (8)),,
(12)
$,0, 1b argmin argmax B(x.s)- [IJo(AA - s|1]] + uhe + SLro..
(13)
4.0
where \ weighs adherence to the data-driven MSE loss and the discriminator loss. was empirically set to 0.004. The learning rates for { } and 0 were Ie 3 and 2e - 4, respectively, and was empirically set to le 6, balancing sharpness and high frequency hallucinations. Temperature parameter T set at constant value of 2.0. Training
performed using the ADAM optimizer (with settings as given in section 441 in batches of 8 images, until the validation loss plateaued.
Results Figure 5 shows how task-adaptive sampling significantly outperforms all baselines. Both the uniform and random subsampling schemes suffer from severe aliasing which the task model is not able adequately restore. While low pass sampling does not lead to aliasing, the absence of real frequecyy information causes the task model to 'hallucinate' such (super-resolution) content. Interestingly. LOUPE (e) learned a rather compact, i.e. around DC frequencies, sampling scheme, resulting in blurry images, but DPS learned a more wide-spread pattern, also capturing higher frequencies. The result of this particularly notable for the example displayed in the bottom row of fig. (5) being less structured (or predictable) than that in the row above it. DPS thus enabled high-quality, high-resolution reconstructions that go beyond those obtained with the other methods, reaching a PSNR of 27.1 dB and 28.3 dB, and structural similarity index (SSIM) of 0.88 and 0.90 across the 5000 test examples, for topK and topl respectively. For the most competitive methods, i.e. fixed sampling low pass and data-driven LOUPE, these statistics were 25.3 dB and 0.82, and 25.3 dB and 0.86, respectively.
5 CONCLUSIONS
We have introduced Deep Probabilistic Subsampling (DPS), framework that enables jointly learn- ing a data- and task-driven sampling pattern, with a subsequent task-performing model. The frame- work is generic and can be combined with any network architecture that performs the required task using the subsampled set of signal elements. Empirically we find the method to perform strongly on toy datasets and canonical deep learning problems. Further work can explore the effectiveness of DPS in real-world problems such as MRI scanning. Even though LOUPE (Bahadir et al.] 2019) learned similar subsampling patterns in all experiments, compared to DPS, its performance downstreak task showed to be lower. We argue that LOUPE probably suffers from the fact that samples in the training phase are relaxed. As such, the task network optimized on a relaxed sub- sampled pattern, whereas hard thresholded pattern is used for inference on the test set to make fair comparisons to DPS.
the
In comparing the restrretrre DPS-topK versus the more expressive DPS-topl, we find the former to perform better in our experiments. Studying the sampling distributions after training, we find that D exhibits more uncertainty. The insets in fig. 2 show this by the higher variance present among the 1000 realizations for DPS-topl compared to DPS-topK. We tentatively attribute this due to difficulties with convergence. Further work will explore if the DPS-topl's more pressive parametrization can improve over DPS-topK in more complex problems that might benefit from broader hypothesis space during training, well under more carefully tuned softmax temperature annealing schemes.
Our data-driven method does not explicitly need knowledge regarding the, often unknown, spar- sifying basis, whereas conventional CS algorithms do require this. Like all data-driven optimized methods, learned sampling scheme is at risk of overfitting small training dataset. A Although

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did not observe this issue in our experiments, careful regularization might be required to ensure that this effect minimal in such high-risk tasks. The fully-differenniable DPS framework allows for flexibility, and interesting extensions be explored in future work. Rather than learning fixed set of parameters for the subsampling distribution, a neural network can be used to predict the pa- rameters instead, conditioned contextual data the samples acquired so far. Finally, our method currently requires the desired sampling rate to be predetermined as a hyperparameter. Future work can explore if this rate can be jointly optimized to incorporate optimization of the subsampling rate.
ACKNOWLEDGMENTS
This research supported in part by Philips Research. It is also part of research program Ru- bicon ENW 2018-3 with project number 119.183.EN0014 which is financed by the Dutch Research Council (NWO). We thank Wouter Kool and the anonymous reviewers for their valuable feedback.
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Published as a conference paper at ICLR 2020
A GRADIENT OF GUMBEL-SOFTMAN SAMPLING
For any mhth pair of rows (am, Pm), any n element Om.n in am can be differentiated towards all elements in om through:
en Om,
om Bem
softmax, (Wm-1 + om em
Eom
V om
softmmx, (Wm-l + Φm + em
exp((wm-1 + pm + em)/T)
Eem
V @m
(14)
Li=1 exp(((mm1))) + Φm,i + Cm, /
Gumbel noise vector can be reparametrized as function of a uniform noise vector i..d., through:
U(0,1)
6m. - log( log((m)))
(15)
This allows rewriting eq.
14
into:
E expl (Wm + Dm,i - log( - log (ogg(<m.))))// I.]
exp((wm- + Pm log( (eg(-log(Emm)))) log(<m)))//T}
Ven amn.
Een
V
Vm
expí(wm. Φm log( log log(k))) dkN ... dki
P(em = [ki,
kNl] vam N El=1 e + 2m,t log( (kg((k)))//]
-00
-00
/ P Pmmm11 = ki P.] = 12] ... PlEm,N==1
exp <xp((wm-11 + Φm log( - log log((loo(k)))/////lloogk ( dkN ... dki
mm
exp((Wm-ix) + 2m, log( log log(k:)))/T} n
exp((wm- Φm log()))g(k)))//
dkN ... dkg.
m
N Di=1 xpp(Wmm11, + Φm.i log( (k))))
(16)
13

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Published as a conference paper at ICLR 2020
B ALGORITHM DESCRIPTION
Algorithm Deep Probabilistic Subsampling (DPS)
Require:
Training dataset D, Number of iterations Piter- temperature parameter initialized train-
able parameters & and 0.
Ensure:
Trained logits matrix & and task network parameters 0.
for i = 1 to Titee do
Draw mini-batches (x,,S,): random subset of D
Compute As :an using:
if DPS-topl then
Initialize mask: Wo 0 for m = 1 to do
Draw i.i.d. Gumbel noise samples em E RN
Sample from the distribution fm = argmax 1m-1, + Pm. + Cm,n
Create one-hot vector: am. oww_hot.vvee (fm)
Take current mask: wm Wm-1
-ppdate mask: Wm.F.m -0
end for
else if DPS-topK then
Draw i.i.d. Gumbel noise samples e₁ E N
Sample M samples from the distribution and create an M-hot vector:
M-hot = Topk + )
Expaad the M-hot vector in M one-hot vectors am, with m E (1....,M M} am, Vm (1, (1,..., M)
Create mask per sample: Wm.
Crrette the cumulative masks wm = Ei= w,, Vm E {1,...,,, M).
end if'
Subsample the signal: y; = A
Compute the task: s; = fy()
Compute loss using L, = [s $ 2 + Cpen
Vm E {l...., .M) }. redefine
am &m Ben |sottmxx + om + +
where Φm Φi for DPS-topK
Use Adam optimizer to update and 0
end for
14

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Published as a conference paper at ICLR 2020
C HYPERPARAMETERS OF ADOPTED TASK MODELS
This appendix contains tables that provide details on the adopted task model architectures for the different experiments. We use the abbreviation FC, which stands for 'fully-connected layer'.
Table 1: The layer parameters corresponding the task model in MNIST classification.
Layer
Output size
Init
Activation
FC
Glorot uniform
LeakyReLU (0.2)
Dropout (20%)
FC
256
Glorot uniform
LeakyRel.U (0.2)
Dropout (20%)
256
FC
128
Glorot uniform
LeakyReLU (0.2)
Dropout 3 (20%)
128
FC
128
Glorot uniform
LeakyReLU (0.2)
FC 5
10
Glorot uniform
Softmax
Table The layer parameters corresponding to task model in 'lines and circles' reconstruction.
Layer
Output size
Init
Activation
Kernel size
Strides
FC
N
Glorot uniform
tanh
2
Glorot uniform
tanh
Reshape
V
Conv2D
VN X
N
64
Glorot uniform
ReLU
5x5
1x 1
Conv2D 2
V V 64
Glorot uniform
ReLU
5x
1x1 1
3 3
V X1
Glorot uniform
None
7x7
1x1
Glorot & Bengio 2010
15

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Published as a conference paper at ICLR 2020
Table 3: The layer parameters corresponding the task model in CIFARIO reconstruction.
Network type
Layer
Output size
Init
Activation
Kernel size
Strides
Generator
Proximal operator (k)
6x Conv2D
N X N x 64
Glorot uniform
ReLU ReLU
3x3
I x1 x1
1x Conv2D
N X X
Glorot uniform
3x3
Stepsize (8)
1 Conv2D
N X V x1
Glorot uniform
ReLU
3x3
x1
Discriminator D
Conv2D
128
Glorot uniform
LeakyReLU (0.2)
3x3
x2
2
2
Conv2D
VN
N
128
Glorot uniform
LeakyReLU (0.2)
3x3
2 x2
Conv2D
VN
N
128
Glorot uniform
LeakyReLU (0.2)
3x3
2 x2
Giobal//eengeePoolnn22))
128
Dropout 40%
128
FC 1
1
Sigmoid
D EXAMPLE TRAINING GRAPH FOR TRAINING OF 'LINES AND CIRCLES' RECONSTRUCTION USING DPS-TOPK
(b)
Training and validation loss
Training and validation PSNR
20.0
0.8
17.5
0.6
PSMA
0.4
30.0
7.5
0.2
5.0
0.0
750 1000
1500
1750
2000
Figure 6: Training graphs for the 'lines and circles' reconstruction problem from section 4.2 (a) The graph shows that even though neural network loss functions typically non-convex, the op- timization trajectory of the loss value still followed a smooth path. The train and validation loss exactly behave the way, suggesting no overfitting. (b) The corresponding PSNR value during training.
16


Reviews:
[{"review_id": "r1xxdbB9KH", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a learning-based adaptive compressed sensing framework in which both the sampling and the task functions (e.g., classification) are learned jointly end-to-end. The main contribution includes using the Gumbel-softmax trick to relax categorical distributions and use back-propagation to estimate the gradient jointly with the tas neural network. The proposed solution has the flexibility of able to be used in several different tasks, such as inverse problems ( super-resolution or image completion) or classification tasks.  The paper is very well written. \n\nThe paper locates itself well in current baselines and explains Experiments mostly well. However, there are significant limitations in demonstrating the effectiveness/impact of the proposed technique: \n1) The only comparison to another non-fixed sampling baseline is Kool et al. 2019. The visualization and a thorough comparison were missing in MNIST classification. This baseline was also missing in image reconstruction.  \n2) Compressive Sensing incorporates vast literature of algorithms focusing on different aspects of improvements; algorithms focused on classification and inverse problems. Even if done disjointly, how does the proposed joint learning is compared to those algorithms in these domains? \n3) Top row of Figure 3 nicely explains how the learned sampling paradigm performs compared to other mechanisms (such as uniform, random, low-pass). But there is no comparision against other non-fixed techniques.  ", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}, {"review_id": "r1lbykR6YB", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors propose a new approach of deep probabilistic subsampling for compressed sensing, based on Gumbel-softmax, which is interesting.\n\nA few points should be clarified:\n\n- in compressed sensing one has e.g. the restricted isometry property (RIP) related to recovery. How does the new method relate to such theoretical results? Are the results and findings along similar lines as (classical) compressed sensing theory? \n\n- Methods in compressed sensing are typically convex, e.g. using l1-regularization. What are the drawbacks of using deep learning in this context, e.g. related to non-convexity? What is the role of initialization?\n\n- Does the method both work for underdetermined and overdetermined problems (number of data versus number of unknowns)?\n\n- What is the influence of the hyper-parameters mu and lambda in eq (14)? How should the model selection be done (currently lambda is set to 0.004 without further motivation)? \n\n- MNIST: 60,000 instead of 70,000?\n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "Syerno7kqH", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper introduces  a novel DPS(Deep Probabilistic Subsampling) framework for the task-adaptive  subsampling case, which attempts to resolve the issue of end-to-end optimization of an optimal subset of signal with jointly learning a sub-Nyquist sampling scheme and a predictive model for downstream tasks. The parameterization is used to simplify the subsampling distribution and ensure an expressive yet tractable distribution.  The new approach contribution is applied to  both reconstruction and classification tasks and demonstrated with a suite of experiments in a toy dataset, MINIST, and COFAR10.\n\n\nOverall, the paper requires significant improvement. \n\n1. The approach is not well justified either by theory or practice. There is no experiment clearly shows convincing evidence of the correctness of the proposed approach or its utility compared to existing approaches (Xie & Ermon (2019); Kool et al. (2019); PlÂšotz & Roth (2018) ).\n\n2. The paper never clearly demonstrates the problem they are trying to solve (nor well differentiates it from the compressed sensing problem  or sample selection problem)\n\n   The method is difficult to understand, missing many details and essential explanation, and generally does not support a significant contribution. \n\n3. The paper is not nicely written or rather easy to follow. The model is not well motivated and the optimization algorithm is also not well described.\n\n4. A theoretical analysis of the convergence of the optimization algorithm could be needed.\n\n5. The paper is imprecise and unpolished and the presentation needs improvement.\n\n**There are so many missing details or questions to answer**\n\n1. What is the Gumbel-max trick? \n2. How to tune the parameters discussed in training details in the experiments?\n3. Why to use experience replay for the linear experiments?\n4. Are there evaluations on the utility of proposed compared to existing approaches?\n5. Does the proposed approach work in real-world problems?\n6. Was there any concrete theoretical guarantee to ensure the convergence of the algorithm.\n\n[Post Review after discussion]\u0010: The uploaded version has significantly improved over the first submission. It is now acceptable. \n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper introduces a probabilistic data subsampling scheme that can be optimized end-to-end.  The experimental evaluation is a bit weak, focusing mostly on toy-scale problems, and I would have liked to see a discussion of bias in the Gumbel-max gradient estimator.  

It's also not clear how the free hyperparameters for this method were chosen, which makes me suspect they were tuned on the test set.

However, the overall idea is sensible, and the area seems under-explored.
Decision: Accept (Poster)