Active Deep Probabilistic Subsampling
Authors: Hans van Gorp
Iris A.M. Huijben
Bastiaan S. Veeling
Nicola Pezzotti
Ruud Van Sloun
Venue:  (2021)
Keywords: Compressed Sensing
subsampling
active acquisition
accelerated MRI

Abstract: Subsampling a signal of interest can reduce costly data transfer, battery drain, radiation exposure and acquisition time in a wide range of problems. The recently proposed Deep Probabilistic Subsampling (DPS) method effectively integrates subsampling in an end-to-end deep learning model, but learns a static pattern for all datapoints. We generalize DPS to a sequential method that actively picks the next sample based on the information acquired so far; dubbed Active-DPS (A-DPS). We validate that A-DPS  improves over DPS for MNIST classification at high subsampling rates. We observe that A-DPS learns to actively adapt based on the previously sampled elements, yielding different sampling sequences across the dataset. Moreover, we demonstrate strong performance in active acquisition Magnetic Resonance Image (MRI) reconstruction, outperforming DPS and other deep learning methods.

Under review as a conference paper at ICLR 2021
ACTIVE DEEP PROBABILISTIC SUBSAMPLING
Anonymous authors Paper under double-blind review
ABSTRACT
Subsampling a signal of interest can reduce costly data transfer. battery drain, ra- diation exposure and acquisition time in wide range of problems. The recently proposed Deep Probabilistic Subsampling (DPS) method effectively integrates subsampling in an end-to-end deep learning model, but learns a static pattern for all datapoints. We generalize DPS to a sequential method that actively picks the next sample based on the information acquired so far; dubbed Active-DPS (A- DPS). We validate that A-DPS improves over DPS for MNIST classification at high subsampling rates. We observe that A-DPS learns to actively adapt based on the previously sampled elements, yielding different sampling sequences across the dataset. Moreover, we demonstrate strong performance in active acquisition Magnetic Resonance Image (MRI) reconstruction, outperforming DPS and other deep learning methods.
INTRODUCTION
Present-day technologies produce and consume amounts of data, which is typically acquired ing an analog-to-digital converter (ADC). The amount of data digitized by an ADC is determined not only by the temporal sampling rate, but also by the manner in which spatial acquisitions are taken, e.g. by using specific design of sensor arrays. Reducing the number of sample acquisitions needed, can lead meaningful reductions in scanning time, e.g. in Magnetic Resonance Imaging (MRI), radiation exposure, e.g. in Computed Tomography (CT), battery drain, and bandwidth requirements. While the Nyquist theorem traditionally used to provide theoretical bounds on the sampling rate, in recent years signal reconstruction from sub-Nyquist sampled data has been achieved through framework called Compressive Sensing (CS).
First proposed by Donoho (2006), and later applied for MRT by Lustig et al. (2007), CS leverages structural signal priors, specifically sparsity under some known transform. By taking compressive measurements followed by iterative optimization of a linear system under said sparsity prior, recon- struction of the original signal is possible while sampling at sub-Nyquist rates. Researchers have employed CS with great success in wide variety of applications. such as radar (Baraniuk & Steeghs, 2007; Ender, 2010), seismic surveying (Herrmann et al., 2012), spectroscopy (Sanders et al., 2012), and medical imaging (Han al., 2016; Lai et al., 2016).
However, both the need to know the sparsifying basis of the data, and the iterative nature of the reconstruction algorithms, still hamper practical applicability of CS in many situations. These lim- itations can be overcome by the use of deep learning reconstruction models that make the sparsity assumption implicit, and facilitate non-iterative inference once trained. Moreover, the (typically random) nature of the measurement matrix in CS does, despite adhering to the given assumptions, not necessarily result in an optimal measurement given the underlying data statistics and the down- stream system task. This has recently been tackled by algorithms that learn the sampling scheme from a data distribution.
In general, these data-driven sampling algorithms can be divided into two categories: algorithms that learn sampling schemes which are fixed once learned (Huijben et al., 2020a;b;e; Ravishankar
& Bresler, 2011; Sanchez et al., 2020; Bahadir et al., 2019; Bahadir et al., 2020: Weiss et al., 2019), and algorithms that learn to actively sample (Ji et al., 2008; Zhang et al., 2019; Jin et al., 2019; samples based sequential acquisition
Pineda al., 2020; Bakker et al., 2020); selecting of the information. The former type of algorithms learn a sampling scheme that on average selects informative samples of all instances originating from the training distribution. However,

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when this distribution is multi-modal, using one globally optimized sampling scheme, can easily be sub-optimal on instance-level. Active acquisition algorithms deal with such shifts in underlying data statistics by conditioning sampling behavior previously acquired information from the instance (e.g. the image to be sampled). This results in a sampling sequence that varies across test instances, L.e. sampling is adapted to the data. This adaptation achievable sampling rates, or better downstream task performance for the same rate, compared to sampling schemes that operate equivalently on all data.
result of conditioning, promises lower
In this work, we extend the Deep Probabilistic Subsampling (DPS) framework (Huijben et al., 2020a) to an active acquisition framework by making the sampling procedure iterative and con- ditional on the samples already acquired, see Fig. 1. We refer to our method as Active Deep Prob- abilistic Subsampling (A-DPS). We show how A-DPS clearly exploits the ten different modalities (i.e. the digits) present in the MNIST dataset to adopts instance-adaptive sampling sequences. More- over, we demonstrate both on MNIST (LeCun al., 1998) and the real-world fast MRI knee dataset (Zbontar al., 2018), that A-DPS outperforms other state-of-the-e models for learned sub-Nyquis sampling. We make all code publicly available upon publication, in order to facilitate benchmarking to all provided baselines and A-DPS in future research.
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Figure 1: Architectural overview for acquisition steps of A-DPS, with extensions of DPS shown in red.
2 RELATED WORK
Recently, several techniques for learning fixed sampling pattern have been proposed, especially in the field of MR imaging, in which Ravishankar & Bresler (2011) were one of the firsts. this work, the authors make use of non-overlapping cells in k-space, and move samples between these During training Ravishankar & Bresler (201 1) alternate between reconstruction and relocation of sampling positions. After reconstruction step they sort the cells in terms of reconstructing error and an infinite-p norm. Selected samples from lower scoring cells are relocated to higher scoring cells in a greedy fashion.
Sanchez et al. (2020) also propose a greedy approach, in which samples are not relocated between cells, but greedily chosen optimize a reconstruction loss on batch of examples. Both of the types of greedy optimization do however not allow for joint learning of sampling together with a downstream reconstruction/task model, as the reconstruction has either be parameter-free or pretrained to work well with variety of sampling schemes.
Bahadir et al. (2019) on the other hand propose learn the sampling pattern by thresholding pixel- based i.i.d. samples drawn from a uniform distribution, dubbed Learning-based Optimization of the Under-sampling PattErn (LOUPE). The sample rate of LOUPE indirectly controlled by promoting sparsity through the use of an l penalty the thresholds.
One of the first active sampling schemes was proposed by Ji et al. (2008), who leverage CS recon- struction techniques that also give a measure of uncertainty of the reconstruction using Bayesian modeling. et al. (2008) leveraged this uncertainty in the reconstruction to adaptivly select the next measurement that will reduce this uncertainty by the largest amount. However, this method and other similar works from (Carson et al., 2012: Li et al., 2013) rely on linearly combined measurements, rather than discrete sampling, with which concern ourselves here.
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In the field of MRI, Zhang al. (2019) propose an active acquisition scheme by leveraging recon- struction and adversarial neural network. Whereas the reconstruction network is trained to recon- struct MR images from the subsampled Fourier space (k-space), the adversarial network is trained to distinguish between already sampled, and omitted lines this space. The k-space line that is most believed be 'fake' (i.e. filled in by the reconstruction network) by the adversarial network, is sampled next. However, This framework only works for undersampled Fourier to image recon- struction tasks, as the discriminator requires mappings of the image in k-space. Jin et al. (2019) put forth an active acquisition scheme for MRI by leveraging reinforcement learning (RL). Two neural networks, one for sampling and one for reconstruction are trained jointly using Monte-Carlo tree search, resulting in sampling policy that is dependent on the current reconstruction of the image.
Concurrently with our work, both Pineda et al. (2020) and Bakker et al. (2020) proposed RI.-based active acquisition techniques. Pineda et al. (2020) leverages a Double Deep Q-Network. The model trained using a modified c-greedy policy, in which the best action taken with probability c, and an exploratory action is taken with probability Bakker et al. (2020) compare greedy with non- greedy training, finding that the greedy method leads to a higher degree adaptability, especially for tasks with long horizon (i.e. more samples to be taken). Both of the frameworks proposed by (Pineda et al., 2020) and Bakker et al. (2020) make use of a pretrained reconstruction network, which differs from the proposed A-DPS method that enables joint training of both the reconstruction (task) network and sampling network.
Even though subsampling is extreme form of data compression, differentiate from typical data compression architectures like deep encoder-decoder structures (Theis et al., 2017; Ballé et al., 2017), as these methods do not reduce data rates at the measurement stage. The feedback recurrent autoencoder proposed by Yang et al. (2020) is however related to A-DPS through its use of a recur- rent context. But whereas Yang et al. (2020) learn a recurrent context to inform the encoder stage of the network, A-DPS uses this to inform the sampling pattern.
3 METHOD
3.1 GENERAL FRAMEWORK
Given a prediction task & we are interested in learning to predict an optimal subsampling scheme A E (0, M (with M < N) on an input signal x E R resulting in measurement ý E RRM:
ÿ = Ax.
Each row in A is constrained have to oo-norm of I, while each column in A is constrained to have an fg-norm of either 0 or I, i.c. each of the N candidate samples is selected at most once. In the rest of this paper we will index these candidate samples with n E fI...., NY. and the selected samples with m E (1....,M} M). The percentage of selected samples from the candidate samples is called the sampling ratio = M/N 100%.
We also introduce a non-compressed form of the measurement ÿ. called y E R-N, that contains N M zeros, and M non-zeros at the sampled indices specified by A, i.e. the masked input. This way, the location of samples from æ is preserved, which is especially useful when A changes during training. To acquire y from x, one seeks a subsampling mask that can be applied on x via:
y = - x = AT Ac,
where O denotes element- wise multiplication. From the resulting measurement y then aim at predicting the downstream task s through:
= (g)
(3)
where fo(.) is function that is differentiable with respect to its input and parameters 0, e.g. neural network. Normally, optimization of the task model f.). is achieved through backpropagation of some loss function L(s, 8). However, calculating gradients its combinatorial nature, inhibiting joint training of the task with the sampling operation. The DPS framework provides solution to this problem, on which we will elaborate in the next section.
the sampling matrix is blocked by
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3.2 DPS: DEEP PROBABILISTIC SUBSAMPLING
To enable joint training of the sampling operation with the downstream task model, Huijben et al. (2020a) introduce DPS. Rather than optimizing A directly, they propose to optimize generative sampling model P(A(e) where & are learned unnormalized logits of (possibly multiple) categorical distribution(s). Each distribution expresses the probabilities for sampling any of the elements In from æ through sampling matrix A. More specifically, &m,n is the log-probability for setting am,, 1, and thus sampling nn as m sample.
To generate sampling pattern from these unnormalized logits, i.e. implementation of this con- ditional model, the Gumbel-mas trick is leveraged (Gumbel, 1954). In the Gumbel-max trick the unnormalized logits are perturbed with i..d. Gumbel noise samples cm.n Gumbel 1). By selecting the maximum of this perturbation realization of the sampling mask can be found using:
A = one-hoty argmax (wm-1,, + Óm,n + em.n)
(4)
n
where Am, denotes the m-th row of A and one-hoty creates a one-hot vector of length N, with the one the index specified by the argmax operator. Moreover, the cumulative mask Wm-1.x E {-00,00 0} masks previously selected samples by adding minus infinity to those logits, thereby ensuring sampling without replacement.
During backpropagation, gradients are computed by relaxing this sampling procedure using the Gumbel-softmax trick (Jang et al., 2016; Maddison al., 2017), resulting in:
den Am.: di.. Be.,. [softmax, {1m-11. Um,n + em,n)),
(5)
where denotes the temperature parameter of the softmax operator. Setting > 0 results in smoothed sampling matrix A (i.e. elements can have values between and I as well), allowing gradients to distribute over multiple logits during training. In the limit of operator approaches the one-hot argmax function of equation (4). Although this approach - also known straight-through Gumbel-softmax leads biased gradients, it has been shown work well in practice, and Huijben et al. (2020a) keep r at a fixed value during training.
0 the softmax
Huijben et al. (2020a) propose two regimes of DPS. First, Top-1 sampling, expressive form of DPS where each of the M selected samples are separately conditioned on all V candidate samples, resulting in M X N trainable logits Φm.n. Second, Top-M sampling (called Top-K in their paper), constrained form where all M samples together are conditioned on all N candidate samples, i.e. the logits on are shared between the M rows of A, resulting in only N trainable logits. While Top-1 sampling is more expressive, Huijben et al. (2020a) noticed slightly better results for the Top-M regime, possibly thanks to the smaller number of trainable logits, therefore facilitating optimization. For scaleability reasons, we thus choose to continue with Top-M sampling in this work and refer to this regime as DPS in the rest of this paper. We refer the reader to Huijben et al. (2020a) for more details regarding DPS.
3.3 A-DPS: ACTIVE DEEP PROBABILISTIC SUBSAMPLING
We have seen how DPS enables the learning of a sampling scheme that selects M out of N sam- ples. However, these samples are selected simultaneously. A-DPS selects its samples in an iterative fashion, separating the logits into I acquisition steps, i.e. ø' with i E (1,2..... /] and I = M.
Active acquisition is then achieved by introducing dependency between samples, i... the sampling distribution at acquisition step i should depend on the information acquired in previous acquisition steps. To that end, we introduce a context vector C4, that accumulates information from all previous time steps. We then condition the sampling distribution on this context by learning a transformation Φ = (c), where (.) is a function that is differentiable with respect to its input and parameters k. Thus, instead of optimizing the parameters directly (as DPS does), we optimize gx((), which we will refer to as the sampling model.
The question then arises how to best generate this context from previous samples. Here, follow the analysis-br mmalysss-y--ynnthess principle, and let the analysis (the sampling model) depend on the synthesis (the task model). This way, the task model can inform the sampling model what information it needs
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to achieve its assigned task. The iterative analysis-by-syntteet scheme of A-DPS is formalized as follows:
o'= gm(c("))
(6)
=',c' co folds 3 Ox' c.-1).
We hypothesize that would be beneficial to add memory to the context vector, so that it does not contain only information about the previous state of task model fo(-) but contain information from even earlier acquisition steps. To this end, we propose to leverage a recurrent neural network structure in the form of a Long Short-Term Memory (LSTM) cell (Hochreiter & Schmidhuber, 1997). This LSTM is integrated into the neural networks as separate layer. We visualize the architecture of the A-DPS framework in Fig. and discuss its computational complexity of A-DPS in Appendix A.
EXPERIMENTS
To show the applicability of A-DPS on both classification well as reconstruction tasks we evaluate its performance in two experiments. First, we will compare A-DPS with DPS at different subsam- pling ratios on an MNIST classification example in Section 4.1. Second, we will compare A-DPS with contemporary CS and deep learning methods on a MRI example in Section 4.2, leveraging the fast MRI knee dataset (Zbontar et al., 2018).
4.1 MNIST
Experiment setup Classification performance at different sampling rates was tested on the MNIST database (LeCun et al., 1998), consisting of 70,000 grayscale images of 28 x 28 pixels of handwritten digits between 0 and 9. We keep the original data split of 60,000 training and 10,000 testing examples. We train both DPS top-M and A-DPS to take partial measurements in the pixel- domain at different sampling rates.
Reaching back to Fig.1 1 and equations (6) and (7), DPS top-M sampling only consists of the sampling and task model (-)) parts. All M samples are selected at the same time and used once by f-(.) to predict which digit the network looking at. the case of A-DPS however, only 1 sample taken at time and used as input for f.). Here, fa(.) also creates context that is used by the sampling network (-) to select the next sample. A-DPS iterates through this loop M times in order to select all the samples. We keep f.) the same for both DPS and A-DPS. Resulting in the fact that the last iteration of A-DPS is similar to that of DPS top-M (i.c. A samples are selected and fed through f.).)
Task model In the classification network fo(.), all 784 (28 x 28) zero-masked samples are used as input for fully connected layers. The fully connected layers have 784, 256, 128, and 128 nodes, respectively. Moreover, they are all activated by leaky ReLU activation functions with a negative slope of 0.2. The first three layers also have dropout of 30%.
The output of the last fully connected layer then fed into an LSTM with hidden size of 128. The output of the LSTM is both used as the context for the sampling network, well as the input for one final fully connected layer. This final fully connected layer maps the output of the LSTM into the 10 class labels, and is therefore followed by softmax activation function.
In case of A-DPS the output of the LSTM is also used as the input for the sampling network (.). The sampling network consists of two linear layers with output sizes of 256 and 784, respectively. Moreover, after the first layer a leaky ReLU activation function is used with negative slope of 0.2, and dropout of 30% is applied.
Training details Both sampling strategies were trained to minimize the categorical cross-ente cross-entropy loss. The temperature parameter fixed to 2. We again employ SGD with the Adam solver (Ir 2e 4, B, = 0.9. 82 = 0.999, and e = le - 7) to minimize the loss function. Training was performed on batches of 256 examples for 50 epochs.
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The resulting accuracy the test set is shown in Fig. 2a. A-DPS outperforms DPS especially when the actual number of samples taken is very low. It hypothesized that it is especially important to select those candidate samples that carry lot of information based the previous samples for very low data rates. Two examples of the selected sampling masks at = 4% are displayed in Fig. 2b. Here, it is shown how DPS selects all samples once, while A-DPS selects them in an iterative fashion, resulting in different sampling patterns for the two examples.
Results
hiput
DP5
ADPS
90
80
70
all samples
7
60
21
Sampling ratio (86]
b
Figure 2: Results for the MNIST experiment. Classification accuracy the test set for different sampling ratios. Confidence bounds of one standard deviation are shown. The 'all samples' line indicates the accuracy achieved without subsampling. b) Comparison of the sampling masks for two examples at r 4%. The colors denote (for A-DPS) the order in which the samples were taken.
To study the effect of the LSTM this set-up, we perform an ablation study by replacing with fully connected layer, resulting in direct projection of the context vector from the previous state without any further memory. The results of this ablation study are shown in Fig. 3a. Here we can that for the stricter sampling regimes below 3% the use of an LSTM leads higher accuracy.
95
8
(aigi)
90
6
75
70
65
A-DPS
A-DPS Ablation
Punoia
1
2
10
Sampling ratio
0
b
Figure 3: a) Classification accuracy on the test set for different sampling ratios for A-DPS and an ablated version of A-DPS where the LSTM has been replaced by fully connected layer. Confidence bounds of one standard deviation over 5 runs are shown. b) A t-SNE plot of the sampling patterns colored by digit- shows that A-DPS actively generates different sampling masks for different digits
To analyze the sampling patterns created by A-DPS we create a t-SNE plot (Van Der Maaten & Hinton, 2008), which maps the created sampling pattern (for 4%) per image in the test set to one dot in a 2D space. In this 2D space t-SNE aims to preserve spatial relationships from the higher dimension, i.e. similar high dimensional vectors get mapped close to together, while dissimilar ones are mapped further apart. The resulting plot is shown in Fig. 3b, where each dot is colored with the ground truth label (digit) of the corresponding image. The clustering in this figure indicates how similar selected sampling patterns are. For example, the sampling patterns for digit zero tend to be dissimilar from all the other sampling patterns, while the digits four, six, and nine are sampled more similarly using A-DPS.
To get more insight into the inner workings of A-DPS, we perform a similar t-SNE analysis on the context vector that A-DPS constantly updates after acquisition of new sample. The result is shown Fig. 4. Here we show the same seatter plot with two different colorings, namely the current acquisition step and the ground truth label (digit). It can be seen how the context vectors similar

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for the first few acquisition step, but information comes in the context vectors accumulate useful information and branch of into different regions dependent on the ground truth label.
moouu mith label (digit)
25
20 8883
15-00
15
10
b
Figure 4: A t-SNE plot of the generated context vector by A-DPS, for all images in the test set. This context vector changes every time a new sample is acquired. This change is visualized by labeling each dot with the number of samples acquired so far (a), showing that context vectors become more distinct after acquiring more data. (b): The same plot, but now colored by the ground truth label (digit).
4.2 MRI
Experiment setup To show the applicability of A-DPS, we demonstrate its performance on line- based MRI. We make use of the NYU fastMRI database knee MRI volumes (Zbontar et al., 2018). Only the single-coil measurements resulting data was split into 8,000 training. 2,000 validation, and 3,000 testing MRI slices. All slices cropped to the central 208 208 pixels and normalized between 0 and 1. The subsampling operation on one of these MRI slices is then performed in k-space (Fourier-space):
selected, from which the outer slices were removed. The
Y T=IF"DOFX
where |. is the magnitude operator. Moreover, X E NRNN is the fully sampled ground truth image and Y E NxN the subsampled image, both in the pixel domain. this case N equal to 208. Furthermore, F and FH denote the forward and inverse 2D-Fourier transform, respectively. D (0, INNN denotes the sampling mask in k-space.
Normally Y would be complex, due to the asymmetrical nature of MRI measurements and the in- complete subsampling mask. Here, we choose to take the magnitude to simplify reconstruction. We hypothesize that doing does not significantly change the problem, the imaginary part of fully sampled images in the NYU fastMRI dataset is very small compared to the real part.
Task model To reconstruct an estimate of the original image X from the partial measurement Y a deep unfolded proximal gradient method is used (Mardani al., 2018), which iterations of proximal gradient method are unfolded as feed forward neural network following:
(k+1) p((k) x (k)
(k) a(v) (FF"DOFX") -Y)].
(9)
where po()( (c) (.) is a trainable image-to-image proximal mapping and (( is the step size, parameter- ized by ( and vb, respectively. We implement this proximal gradient method for k steps, with the trainable step size (w) implemented as a 3 x 3 convolutional layer. Each proximal mapping is implemented
(k)
(k)
series of 4 convolutions with 16, 16. 16, and I feature(s) each and kernel size of 3 x 3. All convolutions but the last are followed by ReLU activation functions.
We will compare A-DPS to to several relevant sampling baselines, namely, random uniform, low- pass, variable density (VDS), greedy mask selection (Sanchez et al., 2020), LOUPE (Bahadir et al., 2019; Bahadir et al.. 2020), and DPS. Under random uniform regime all N lines are equally likely to be sampled, while under a low-pass regime the M lines closest to the DC frequency will be selected. VDS the other hand is heuristic regime that employs probability density from which

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the desired amount of samples are drawn. Following (Lustig et al., 2007), we here polynomial probability density function with a decay factor of 6.
For the greedy mask selection we follow the approach by Sanchez et al. (2020) and first optimize the sampling mask using the NESTA solver (Becker et al., 2011). After this, we fix the sampling mask and train our proximal gradient network. Results for both reconstruction algorithms are reported.
To generate a sampling mask D using A-DPS we use a sampling network gr((). This network takes two inputs. Firstly, the output of the proximal network, i.e. the reconstructed image at that iteration. This image is analyzed using 3 convolutional layers with kernels sizes of 3 x 3 followed by ReLU activation functions. The output features are of sizes 16, 16, and 32, respectively. The final feature map is aggregated into a feature vector using global average pooling, Next to this feature vector, the indices of the selected lines the previous iteration(s) are also used as input for the sampling network, encoded as M-hot vector of dimension 208. Both the feature and selected indices vector are concatenated and used as input for an LSTM cell, with output dimension of 208. This is followed by two fully connected layers with 208 neurons each, having a ReLU activation after the first layer.
Training details promote the reconstruction of visually plausible images, we leverage both Mean Squared Error (MSE) and adversarial loss (Ledig et al., 2016). To that end we introduce a discriminator network that is trained to distinguish between real and reconstructed MR images. The discriminator is implemented using three convolutional layers with kernel sizes of 3 x 3, stride 2, and 64 feature maps, each with Leaky ReLU activations. After the last convolutional layer the feature maps are aggregated into feature vector using global average pooling, with a dropout rate of 40%, which is mapped to a single output probability using one fully connected layer followed by sigmoid activation function. Next to the MSE loss and adversarial loss, we add third loss term that penalizes the MSE loss between the discriminator features of real and generated images. The total loss function is a weighted summation of these three losses, with weights 5e 6, and 1e- 7, respectively.
All sampling mask selection strategies were then trained using SGD on batches of images for a total of 10 epochs. We again employ the Adam solver (Ir = 2e 4, B1 0.9, B2 = 0.999, and e = le - 7) to minimize the loss function, and set the temperature parameter to 2. We choose
M = 26, which results in a sampling ratio of r 12.5%, an acceleration factor of 8.
k spacce
Sampling Mask
Reconstruction
Ground Truth
-50-
50-
1-2
100
24
22
20
(traadin
Figure 5: A-DPS MR image reconstruction of test-set image by adaptively sampling 26 k-space lines. From left to right: 1) k-space with the first 3 sampled lines, 2) sequence of line indices selected by A-DPS, 3) reconstructed image, 4) fully sampled MR image (ground truth).
Results We score the different strategies based on metrics: the normalized mean square error (NMSE), the peak signal-to noinoise ratio (PSNR), and the structural similarity index (SSIM) (Wang et al., 2004). Averaged results over 5 runs for an acceleration factor of 8 are shown in Table 1, while those for an acceleration factor of 8 are shown in Table 2. We test for the statistical significance of the gains made by A-DPS over DPS in Appendix B. An example of A-DPS reconstruction shown in Fig. 5, while comprehensive overview of all baselines for this example can be found in Appendix C and D.
To analyse the workings of A-DPS in this MRI setting we first plot the relative occurrence of differ- ent line indices over the test set. This is shown in Fig. 6a. We can see that A-DPS always selects a band of lines around 0, with lower occurence of high frequency lines. We also employ t-SNE on the context vector that A-DPS updates every acquisition step. The result of this is shown in Fig. 6b. It can be seen that untill acquisition step 15 the context vectors very similar for the images per
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acquisition step, while after acquisition step 15 the context vectors start fanning out. It hypothe- sized that at the start of the sampling procedure not a lot of actionable information is available to the system, but this increases more samples are taken over time.
Table 1: Averaged results over five runs on the hold-out test set for an acceleration factor of 8.
Sampling Model
Reference
NMSE
PSNR
SSIM
Random uniform
0.324
16.0
0.47
Low pass
0.043
25.2
0.71
Variable density
(Lustig al., 2007)
0.041
25.4
0.73
Greedy Mask NESTA
(Sanchez et al., 2020)
0.043
24.8
0.71
Greedy Mask Prox. Grad.
0.041
25.4
0.73
LOUPE
(Bahadir et al., 2019)
0.045
25.7
0.74
DPS
(Huijben et al., 2020a)
0.039
25.7
0.74
A-DPS (proposed)
0.036
26.1
0.75
Table 2: Averaged results over five runs the hold-out test for an acceleration factor of 16.
Sampling Model
Reference
NMSE
PSNR
SSIM
Random uniform
0.265
18.9
0.5
Low pass
0.060
23.4
0.65
Variable density
(Lustig et al., 2007)
0.058
23.4
0.66
Greedy Mask NESTA
(Sanchez et al., 2020)
0.068
22.3
0.63
Greedy Mask Prox. Grad.
0.059
23.4
0.66
LOUPE
(Bahadir al., 2019)
0.206
20.2
0.55
DPS
(Huijben et al., 2020a)
0.056
23.8
0.67
A-DPS (proposed)
0.049
24.4
0.69
relebivee occurence
20
15
-50
0
a
b
Figure 6: Analysis of A-DPS for MRI. a) Relative occurrence of the line indices. b) t-SNE analysis on the context Vector that is updated every acquisition step. The coloring corresponds the current acquisition step, which is equal to the number of sampled k-space lines.
CONCLUSION
We proposed a generalization of DPS, which enables active acquisition, called A-DPS. We demon- strated its applicability on both MNIST classification task well an MRI reconstruction task. Moreover, we found that the adaptive nature of A-DPS improves performance over other sampling pattern selection methods on downstream task performance. We find that A-DPS uses qualitatively differing sampling strategies depending the context. On a critical note, the black-box nature of A-DPS comes with the traditional machine learning challenges out-of-distribution generalization and overfitting. This means that in practical application, the sub-sampling regime could obfuscate the information required to recognize failure cases. Future work includes exploration on how to improve conditioning of the sampling scheme earlier acquired information and mnta-informttion (such as resolution, sampling ratio, and weighting). Potential future applications include 3D and dynamic MRI, CT, ultrasound, radar, video, and MIMO systems.

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Under review as a conference paper at ICLR 2021
REFERENCES
Cagla D. Bahadir, Adrian V. Dalca, and Mert R. Sabuncu. Learning- Optimization of the Under-sampling Pattern in MRI. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 11492 LNCS:780-792, 2019.
Cagla D. Bahadir, Alan Q. Wang, Adrian V. Dalca, and Mert R. Sabuncu. Deep-learning-bee -based optimization of the under-sampling pattern in mri. IEEE Transactions Computational Imaging, 6: 6:1139-1152, 2020.
Tim Bakker, Herke van Hoof, and Max Welling. Experimental design for mri by greedy policy search, 2020.
Johannes Ballé, Valero Laparra, and Eero Simoncelli. End-to-end Optimized Image Compres- sion. 5th International Conference on Learning Representations, ICLR 2017 Conference Track Proceedings, 11 2017.
Richard Baraniuk and Philippe Steeghs. Compressive radar imaging. In IEEE National Radar Conference Proceedings, pp. 128 133. Institute of Electrical and Electronies Engineers Inc., 2007. ISBN 1424402840. doi: 10. 20...0//AAARR20...7722222022
Stephen Becker, Jérôme Bobin, and Emmanuel J. Candès. NESTA: A fast and accurate first-order method for sparse recovery. SIAM Journal on Imaging Sciences, 4(1):1-39, 2011. ISSN 19364954. doi: 10.1 13/91977097755655
William R Carson, Minhua Chen, Miguel R D Rodrigues, Robert Calderbank, and Lawrence Carin. aoaamunicaiions-nspiiee Projection Design with Application to Compressive Sensing. SIAM Journal on Imaging Sciences, 5(4): 10 2012. doi: 10.1137/120878380.
David L Donoho. Compressed Sensing. IEEE Transactions on Information Theory, 52(4):1289, 2006. doi: 10.1109/TT.22006.975558
Joachim H.G. Ender. On compressive sensing applied to radar. Signal Processing, 9025):1402-1414, 2010. ISSN 01651684. doi:
Emil Gumbel. Statistical theory of extreme values and some practical applications. NBS Applied Mathematics Series, 33, 1954.
Yo Seob Han, Jaejun Yoo, and Jong Chul Ye. Deep Residual Learning for Compressed Sensing CT Reconstruction via Persistent Homology Analysis. 11 2016.
Felix J. Herrmann, Michael P. Friedlander, and Özgür Yilmaz. Fighting the curse of dimensionality: Compressive sensing in exploration seismology. IEEE Signal Processing Magazine, 29(3):88- 100, 2012. ISSN 10535888. doi: 10.1 10.1109//SP221122188888
Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8): 1735-1780, 1997.
Iris A. M. Huijben, Bastiaan S. Veeling, and Ruud J. G. Van Sloun. Deep Probabilistic Subsampling for Aask-Ada Compressed Sensing. In International Conference Learning Representa- tions, ICLR 2020, Addis Ababa, Ethiopia, 2020a.
Iris A. M. Huijben, Bastiaan S. Veeling, and Rund J. G. Van Sloun. Learning sub-sampling and signal recovery with applications in ultrasound imaging. IEEE Transactions on Medical Imaging, 2020b.
Iris A. M. Huijben, Bastiaan S. Veeling, and Ruud J. G. Van Sloun. Learning Sampling and Model- Based Signal Recovery for Compressed Sensing MRI. In International Conference on Acoustics, Speech and Signal Processing, ICASSP 2020, Barcelona, Spain, 2020c.
Eric Jang, Shixiang Gu, and Ben Poole. Categorical Reparameterization with Gumbel-Softnen 5th International Conference on Learning Representations, ICLR 2017 Conference Track Proceed- ings, 11 2016.
10

---

Under review as a conference paper at ICLR 2021
S. Ji, Y. Xue, and L. Carin. Bayesian compressive sensing. JEEE Transactions Signal Processing, 56(6):2346223555 2008. doi: 10.1109/TSP:0007.114355
Kyong Hwan Jin, Michael Unser, and Kwang Moo Yi. Self-Supervised Deep Active Accelerated MRI. 2019.
Zongying Lai, Xiaobo Qu, Yunsong Liu, Di Guo, Jing Ye, Zhifang Zhan, and Zhong Chen, Im- age reconstruction of compressed sensing MRI using graph-based redundant wavelet transform. Medicall Image Analysis, 27:93-104, 2016. ISSN 13618423. doi: 10. 00.1016f..eddia.0155....3
Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(1 1):227) 2:2278-2323, 1998. ISSN 00189219. doi: 10.1109/5.72679..
Christian Ledig, Lucas Theis, Ferenc Huszar, Jose Caballero, Andrew Cunningham, Alejandro Acosta, Andrew Aitken, Alykhan Tejani, Johannes Totz, Zehan Wang, and Wenzhe Shi. Photo- Realistic Single Image Super-Resolune Using a Generative Adversarial Network. CVPR, 2017- January: 105-114, 9 2016.
G. Li, Z. Zhu, D. Yang. L. Chang, and H. Bai. On projection matrix optimization for compressive sensing systems. IEEE Transactions on Signal Processing, 61(11):28877 2013.
Michael Lustig, David Donoho, and John M. Pauly. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic Resonance in Medicine, 188- 12 2007. ISSN 07403194, doi: 10. m0.1002/mm.21191.
Chris Maddison, Andriy Mnih, and Yee Whye Tch. The concrete distribution: A continuous relaxation of discrete random variables. In International Conference Machine Learning, 2017.
Morteza Mardani, Qingyun Sun, Shreyas Vasawanala, Vardan Papyan, Hatef Monajemi, John Pauly, and David Donoho. Neural Proximal Gradient Descent for Compressive Imaging. 6 2018.
Luis Pineda, Sumana Basu, Adriana Romero, Roberto Calandra, and Michal Drozdzal. Active mr k-space sampling with reinforcement learning. In Medical Image Computing and Computer Assisted Intervention - MICCAI 2020, pp. 23-33, Cham, 2020. Springer International Publishing. ISBN 978-3-030-59713-9,
Saiprasad Ravishankar and Yoram Bresler. Adaptive sampling design for compressed sensing mri. In 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 3751-3755, 101
Thomas Sanchez, Baran Gozcu, Ruud B. van Heeswijk, Armin Eftckhari. Efe llicak, Tolga Cukur, and Volkan Cevher. Scalable Learning-Base Sampling Optimization for Compres- sive Dynamic MRI. In ICASSP 2020 2020 IEEE International Conference Acousties, Speech and Signal Processing (ICASSP), pp. 8584, Barcelona, Spain, 4 2020. doi: 10. 00.1009/ccssp407722200000990
Jacob N. Sanders, Semion K. Saikin, Sarah Mostame, Xavier Andrade, Julia R. Widom, Andrew H. Marcus, and Alán Aspuru-Guziz Guzik. Compressed sensing for multidimensional spectroscopy exper- iments. Journal of Physical Chemistry Letters, 3(18):2697-270, 9 2012. ISSN 19487185. doi: 10/0098/
Lucas Theis, Wenzhe Shi, Andrew Cunningham, and Ferenc Huszár. Lossy Image Compression with Compressive Autoencoders. 5th International Conference on Learning Representations, ICLR 2017 Conference Track Proceedings, 2017.
Laurens Van Der Maaten and Geoffrey Hinton. Visualizing Data using t-SNE. Journal of Machine Learning Research, 9:2579-2605, 2008.
Zhou Wang, Alan C Bovik. Hamid R. Sheikh, and Eero P. Simoncelli. Image Quality Assessment: From Error Visibility to Structural Similarity. Search Results Web results IEEE Transactions on Image Processing, 13(4), 2004.
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Tomer Weiss, Ortal Senouf, Sanketh Vedula, Oleg Michailovich, Michael Zibulevsky, and Alex Bronstein. PILOT: Physics-Informan Learned Optimized Trajectories for Accelerated MRI. 2019.
Y. Yang, G. Sautière, J. Ryu, and T. S. Cohen. Feedback recurrent autoencoder. In ICASSP 2020- 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3347-3351, 2020.
Jure Zbontar, Florian Knoll, Anuroop Sriram, Matthew J. Muckley, Mary Bruno, Aaron Defazio, Mare Parente, Krzysztof J. Geras, Joe Katsnelson, Hersh Chandarana, Zizhao Zhang, Michal Drozdzal, Adriana Romero, Michael Rabbat, Pascal Vincent, James Pinkerton, Duo Wang, Nafissa Yakubova, Erich Owens, C. Lawrence Zitnick, Michael P. Recht, Daniel K. Sodickson, and Yvonne W. Lui. fastmri: An open dataset and benchmarks for accelerated MRI. CoRR, abs/1811.0885 2018.
Zizhao Zhang. Adriana Romero, Matthew J. Muckley, Pascal Vincent, Lin Yang, and Michal Drozdzal. Reducing Uncertainty in Undersampled MRI Reconstruction with Active Acquisi- tion. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2019-223 2 2019.
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A COMPUTATIONAL COMPLEXITY
One of the drawbacks A-DPS compared to learned fixed sampling schemes is it higher amount of computational complexity. The main source of this complexity is the unrolling of iterations, leading to a computational complexity of O(1) = O(M/p). Although we set p equal to in all our experiments, one can in fact seamlessly interpolate between A-DPS and DPS by choosing 1 v p S M. This constitutes a trade-off between computational complexity and adaptation rate. We leave further exploration of this trade-off to future work.
We can also express computational complexity in terms of run-time on machine, in our case GeForce GTX 1080 Ti. comparison of DPS and A-DPS in terms of training time per epoch can be seen in Fig. 7. We can see that the training time for A-DPS increases for higher sampling ratios where it needs to unroll through more iterations. By combining the results from Fig. and Fig. one can make a trade-off between run-time and accuracy. Where A-DPS achieves higher accuracy for stricter sampling regimes, while at the same time not increasing run-times by a lot.
DPS
600
ADPS
400
Line 0
300
200
100
sampling ratle [56]
Figure 7: Comparison between DPS and A-DPS of time taken to train for epoch for the MNIST example on a GeForce GTX 1080 Ti.
For the MRI experiment the training times per epoch are and 52 minutes, for DPS and A-DPS, spectively. Inference is however fast: A-DPS only requires ~ 13 ms of processing time to determine the next-to-acquire K-space line and reconstruct image after each step. This is well below the shortest reported Time of Echo (TE) for this MRI acquisition, being 27 ms (Zbontar et al., 2018).
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B STATISTICAL TESTS ON MRI
To analyse whether the gains made by A-DPS over DPS are statistically significant we perform two- sided paired Student's I-fests, the results of which are shown in Table and Table 4. We perform two different types of tests. Firstly, we look at the average results for each run after training on the hold out test set. This results in a t-test with DPS, given a new random initialization of the trainable network weights, and different stochastic optimization behavior. These results are shown in Table 3. Secondly, we perform t-test over the results each individual image (averaged over the 5 runs) of the hold-out test set, resulting in 3000. This test indicates whether A-DPS' performance significantly higher than that of DPS, given test image. These results are shown in Table 4.It can be seen how in all cases for all metries 0.05 indicating that our findings are statistically significant.
5 that conveys how reliably A-DPS outperforms
Table 3: Student's s t-test performed over the average results of the 5 runs (n 5), indicating how reliably A-DPS outperforms DPS given a new random initialization.
8 times acceleration
16 times acceleration
P value
T-statistic
P value
T-statistic
NMSE
0.0272
-2.82
0.00652
-3.65
PSNR
0.0217
2.90
0.00798
3.57
SSIM
0.0170
3.06
0.00852
3.56
Table 4: Student's is t-test performed over the average results on each test image (n = =3000, indicating how reliably A-DPS outperforms DPS given a new image.
times acceleration
16 times acceleration
P value
T-statistic
P value
T-statistic
NMSE
1.44 e-5
-4.34
3.26 e-12
-6.99
PSNR
6.40 e-16
8.10
2.92 e-41
13.6
SSIM
6.10 e-25
10.4
5.05 e-18
8.70
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Under review as a conference paper at ICLR 2021
C
MRI RECONSTRUCTION EXAMPLES FOR ACCELERATION FACTOR 8
Method
Sampling Mask
Reconstruction
Error Map
Ground Truth
100
-50
Random Uniform
50
100
-100
550
Low Pass
line
50
100
0.20
-50
0.55
Variable Density
0.30
line
0.55
100
0.00
0.20
550
0.55
line indix
Greedy Mask Selection
0.00
0.05
0.00
0.00
-50
0.25
LOUPE
0.00
line
50
0.05
0.00
0.50
-50
0.55
DPS
0.10
line
50
0.55
0.00
-100
0.00
035
A-DPS
0-
0.00
line
50
8.55
8.50
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Under review as a conference paper at ICLR 2021
D
MRI RECONSTRUCTION EXAMPLES FOR ACCELERATION FACTOR 16
Method
Sampling Mask
Reconstruction
Error Map
Ground Truth
100
-50
Random Uniform
50
100
-100
550
Low Pass
line
50
100
0.20
550
0.15
Variable Density
0.10
line
0.55
100
0.00
0.20
500
0.55
line Indix
Greedy Mask Selection
0.00
0.05
0.00
0.00
-50
0.25
LOUPE
0.00
line
50
0.05
0.00
0.50
-50
0.55
DPS
0
0.10
line
50
0.55
0.00
-100
0.00
035
A-DPS
0-
0.10
line
50
8.55
8.50
16


Reviews:
[{"review_id": "gYuamGOTJMa", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "SUMMARY:\nThe paper at hand deals with compressed sensing (CS) and introduces an extension to deep probabilistic subsampling (DPS) called active deep probabilistic subsampling (A-DPS): instead of learning a sampling pattern that is equal for each element of the dataset, A-DPS adaptively selects entries (of each element) based on the information acquired so far. It is shown that this active sampling increases performance for different tasks: a toy example that aims to demonstrate the benefits of active sampling, a classification task (from subsampled inputs) on the MNIST dataset and a reconstuction task on the NYU fastMRI database of knee MRI volumes. \n\nSTRENGTHS:\n1. The paper is very clearly written and very comprehensible. Furthermore, it is very detailed about the experimental setup. I also liked the description of the general framework which thoroughly defines the used notation.\n2. The idea is well motivated and the approach of selecting samples depending on the previously selected ones makes intuitively sense.\n3. The results of the experiments on MNIST and the NYU fastMRI data are promising. A plenthora of (non-active) subsampling schemes are benchmarked as well.\n\nWEAKNESSES:\n1. The greatest weakness of this paper is the missing comparison to other active sub-sampling schemes (Zhang et al., 2019; Jin et al., 2019). It would be nice to see wether the proposed method produces better results than the existing methods. \n2. I found the toy example very constructed. It is not really easy to understand and does in my opinion not improve the quality of the paper.\n\nQUESTIONS:\n- What happens when the MNIST sampling ratio in Figure 3a is further increased? Does A-DPS consistently outperform DPS in low sampling ratio regimes?\n\nDECISION:\nOverall, the paper presents an interesting and novel approach. However, it remains an open question wether the proposed A-DPS scheme performs better than already existing active subsampling schemes. Besides this, the experimental evaluation is solid. I lean towards acceptance.\n\nUPDATE AFTER REBUTTAL:\nI thank the authors for their responses and appreciate the inclusion of some of the requested changes in the paper. However, the paper still misses the comparison to other adaptive methods which is the paper's greatest weakness. Therefore, I decided to keep my score at 6.\n\nMINOR REMARKS:\n- Caption of Table 1 could use some more spacing\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "KI_Z-Ca98_b", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "### Summary\nThis paper develops methods to perform active subsampling. That is, given some downstream task like classification or image reconstruction, it sequentially selects which elements of an image or signal to sample so as to perform said task. It does so by extending the Deep Probabilistic Subsampling (DPS) method developed by Huijben et al. The proposed method is applied to two problems as well as a simplified, low-resolution MRI reconstruction problem.\n\n### Strengths\nMotivation: Active sampling is an interesting idea that has been around for some time, but was often computationally impractical. Thanks to GPUs and deep learning, active sampling is becoming more practical and its interesting to see new work in this direction.\n\n### Weaknesses\nNovelty: The method is a small extension to the DPS method where the network that selects which rows to samples is conditioned on the existing measurements.\n\nValidation: The paper did not compare to any other active sampling strategies. The authors made no effort to replicate existing methods.\n\nClarity: \nThe Markov chain example in section 4.1 was hard to follow and more distracting than informative. The phrase \"the task model gets to sample only one position out of every three\" reads as if the model is sampling one position out of every three in the sequence. It took some time before I realized this meant that at every position in the sequence it was probing one of the three states.\n\nImpact: The results with active sampling were only marginally better than results with a fixed (learned) sampling strategy.\n\nLimitations: The method is applicable only to true subsampling problems, not general sensing. That is, one isn't designing the rows of a measurement matrix on the fly but rather selecting which row from an existing matrix (identity in most of the examples) that one would like to sample from.\n\n\n### Recommendation\nThe paper's presentation could be improved and it is sorely missing comparisons to other active sampling methods. I don't think the papers novelty is enough to overcome these issues and so I do not believe it is ready for publication.\n\n### Comments\nWhile the proposed method was computationally impractical, active sampling was discussed extensively in [A] from a information theoretical perspective.\n[A] Ji, S., Xue, Y., & Carin, L. (2008). Bayesian compressive sensing. IEEE Transactions on signal processing, 56(6), 2346-2356.\n\nBecause of the nonlinearity in the forward model, equation (9) is not actually proximal gradient descent. I believe there's a sign(F^HD\\circFX) term missing from the (sub) gradient.\n\n### Update\n\nI thank the authors for their comprehensive response. While its unfortunate they couldn't compare to any other active methods, the related work and overall clarity of the paper is significantly improved. The t-SNE plots were informative and interesting. While I have reservations about the paper's lack of comparisons, I think its publication still might be a net positive for the research community.\n\nI have updated my score.\n\n##### Other comments\nLet A(X)=F^H D\\circ F X. The expression A^H(Ax-Y\\circ sign(A(x))) is a subgradient of 1/2|| Y - |A(X)|||^2 but A^H(|Ax|-Y) is not. I would avoid calling (9) projected gradient descent as the \"gradient\" isn't really a gradient.\n\n\"We have performed a statistical analysis on the performance gains made by A-DPS over DPS in the MRI reconstruction task, concluding that they are indeed statistically significant.\" It would be nice to see confidence intervals in Tables 1 and 2.\n\n#### Questions/comments that do not effect the review:\nWhy use an LSTM/any network with memory? It seems the next sample depends on the previous samples, but not their order. The ablation study on pg 6 shows that memory helps (at low sampling rates), but I don't understand the intuition why. Could the LSTM just have more capacity?\n\nTypos:\npg 2: \"cells.During\" space\npg 3: \"However, This\" capitalization", "extra_scores": {}, "extra_text": {}}, {"review_id": "Im23lluzCeN", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "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": "In this paper, the authors consider the problem of compressed sensing where the underlying signal of interest is captured and restored based only on sparse measurements: Specifically, this paper focuses on the scenario of Deep Probabilistic Subsampling (DPS) which finds sparse measurements in the way that the models designed to solve specific learning problems based on these measurements are jointly optimized. The authors extend DPS to a sequential framework that iteratively and actively selects the next measurement points: The proposed approach encodes the information accumulated until a time step into a context vector which is updated, and used in selecting the next point, in an LSTM-like framework (see minor comments below). In the experiments with two toy problems (including MNIST) and an MRI reconstruction problem, the authors demonstrated that the proposed Active DPS (ADPS) outperforms DPS (in toy problems) and three other compressed sensing algorithms (for MRI reconstruction).\n\nI think this paper makes a borderline case: DPS provides a framework that combines the compressed sensing part (sparse data acquisition) and the subsequent learning part in an end-to-end manner. This paper contributes by extending DPS into an active/sequential learning framework achieving significant performance gains over DPS (mainly on toy problems. see minor comments below). On the other hand, the proposed approach appears to be incremental: ADPS adds a simple sequential update structure (of a context vector) to DPS, which can be described by only two equations (6 and 7). The simplicity of the changes proposed (over DPS) is not a limitation, but it could be accompanied by an in-depth theoretical analysis, a convincing qualitative discussion or _extensive_ experiments demonstrating the practical relevance of the proposed approach.\n\nMinor comments\n- Apart from the last one paragraph, the Introduction Section focuses on discussing the context and motivation of Deep Probabilistic Subsampling (DPS). Instead, the authors could use this space to describe and characterize the proposed Active DPS in detail. \n- I was not sure why the proposed architecture (Figure 1 and equations 6 and 7) is called LSTM, it has a recurrent network structure but I was not able to find any attention (gating) mechanism that characterizes LSTM. Please advise me if I missed anything. \n- Please test if the improvements gained by ADPS over DPS on MRI reconstruction are statistically significant.\n\nUpdate:\n\nThank the authors for their responses, clarification, and additional experiments. I read through authors’ responses and the comments from the other reviewers. I still think this paper makes a borderline case for 1) its technical contribution on extending DPS and thereby achieving significant performance gain on a toy problem and MRI reconstruction tasks, still 2) with limited novelty and room for a more extensive experimental validation (perhaps, beyond MRI). My other concerns on clarity and significance of experiments have been addressed. I would raise my rating to marginally above acceptance threshold (borderline).\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The review phase was very constructive, where reviewers raised several opportunities for improvements. The authors did a very good job in their rebuttal, which led some reviewers to change their opinion in a positive direction. Overall, reviewers agree that this is the borderline paper with remaining concerns about the weak experimentation.  The paper was again discussed by the Area Chair and Program chairs.  Due to the competitive nature of the conference and the high bar of experimental evaluations expected by empirical papers, the paper was finally rejected.  We hope authors will use the feedback from the reviews and make a stronger submission in near future. 
Decision: Reject