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Convolutional Sparse Support Estimator Network (CSEN) From energy efficient support estimation to learning-aided Compressive Sensing

Support estimation (SE) of a sparse signal refers to finding the location indices of the non-zero elements in a sparse representation. Most of the traditional approaches dealing with SE problem are iterative algorithms based on greedy methods or optimization techniques. Indeed, a vast majority of them use sparse signal recovery techniques to obtain support sets instead of directly mapping the non-zero locations from denser measurements (e.g., Compressively Sensed Measurements). This study proposes a novel approach for learning such a mapping from a training set. To accomplish this objective, the Convolutional Support Estimator Networks (CSENs), each with a compact configuration, are designed. The proposed CSEN can be a crucial tool for the following scenarios: (i) Real-time and low-cost support estimation can be applied in any mobile and low-power edge device for anomaly localization, simultaneous face recognition, etc. (ii) CSEN’s output can directly be used as ”prior information” which improves the performance of sparse signal recovery algorithms. The results over the benchmark datasets show that state-of-the-art performance levels can be achieved by the proposed approach with a significantly reduced computational complexity.


I. INTRODUCTION
S PARSE Representation or Sparse Coding (SC) denotes representing a signal as a linear combination of only a small subset of a pre-defined set of waveforms. Compressive Sensing (CS) [1], [2] can be seen as a special form of SC while a signal, s ∈ R d which has a sparse representation, x ∈ R n in a dictionary or basis Φ ∈ R d×n , can be acquired in a compressed manner using a linear dimensional reductional matrix, A ∈ R m×d . Therefore, this signal can also be represented in a sparse manner in the dictionary, D ∈ R m×n , (that can be called equivalent dictionary [3], where m << n, and typically assumed to be full-row rank), which is the matrix multiplication of the measurement matrix, A and predefined dictionary, Φ, i.e., D = AΦ. In SC literature, signal synthesis refers to producing a signal, y = Dx ∈ R m , using a sparse code, x ∈ R n and a pre-specified dictionary, D. On the other hand, signal analysis deals with finding the sparse codes, x from the given measurements, y, with respect to the dictionary D [4]. Sparse Support Estimation or simply Support Estimation (SE) [5], [6], [7], refers to finding the location indices of non-zero elements in SCs. In other words, it is the localization of the smallest subset of the atoms, which are the basis waveforms in the dictionary, whose linear combination represents the given signal well enough. On the other hand, sparse Signal Recovery (SR) refers to finding the values of these non-zero elements of SCs. SE and SR are intimately linked in such a way that the SE of a sparse signal is first performed, then an SR will be trivial using the ordinary Least Squares optimization. Actually, this is the main principle of most greedy algorithms [8], [9] The literature that purely targets SE is relatively short compared to extensive studies on sparse signal recovery [10]. Many existing works, first apply a coarse SR using existing SR methods, and then SE can be easily performed if SE is the main objective. Indeed, there are many applications where computing the support set is more important than computing the magnitudes of SCs. For instance, in an SR based classification (SRC) [11], such as face recognition [12], the training samples are stacked in the dictionary in such a way that a subset of the columns consists of the samples of a specific class. As another example, in cognitive radio systems, only a small ratio of all spectrum is occupied for a given time interval. Therefore, finding the occupied spectrum (i.e., the support set) is the primary concern [13], [14]. Similarly, in a ground-penetrating radar imaging system, finding the location of the target is more important than predicting the actual signal magnitudes [15].
In this study, a novel Convolutional Support Estimator Network (CSEN) is proposed with two primary objectives as in Figure 1. First, this approach enables learning-based noniterative support estimation with minimal computational complexity. To accomplish this, we use two compact Convolutional Neural Network (CNN) configurations, both of which are designed without the dense layers [16]. The proposed CSENs are trained to optimize the support estimations. To our knowledge, this is the first study that proposes a learning-based approach for non-iterative support estimation. Hence, in order to perform comparative evaluations we train the following state-of-the- art CS signal reconstruction deep neural networks as the support estimators: 1) ReconNet [17] that originally works on the spatial domain and, 2) the Learned AMP (LAMP) [18], that is the deep version of AMP [19], which is the state-ofthe-art optimization scheme working on sparse domain. An extensive set of experiments over three benchmark datasets has demonstrated that the proposed CSEN approach outperforms both deep counterparts, especially dealing with a structural sparse signal. In the first experimental setup, we simulate a CS system making data acquisition from the MNIST data set in different measurement rates. Moreover, the proposed SE system is shown to improve the SE performance when compared to its deep counterparts, especially in low measurement rates and imperfect sparsity (in the case of CS of approximate sparse signal or noisy environment). Furthermore, CSEN is tested on a well-known support recovery problem, where face recognition is performed based on sparse codes [11]. We use two benchmark datasets, Yale-B [20], and CelebA [21], in our experiments. Comparative evaluations performed against the two state-of-the-art dictionary-based (representation-based) face recognition methods in the literature, SR based face recognition [11], and collaborative learning [22], have demonstrated that the proposed CSEN approach outperformed both methods.
As for the second objective, we focus on an alternative usage of CSENs. Instead of using them as support estimators, which naturally requires the hard-thresholding of the network outputs, these outputs can be directly used as prior information about the sparse signals. It is a well-known fact that having prior information about the non-zero locations such as the probability map, p(x) (or simply p), on the support set, could improve the conventional signal recovery algorithms [23]. However, in many cases, it is not clear how to obtain such prior information in advance. The most common usage of such a system appears in dynamical sparse recovery [24], where previous support estimations can be used as priors for the next estimation. In this study, we have demonstrated that CSEN outputs can be a better alternative for the prior information of the non-zero locations. Therefore, CSEN is now used as a learning-aided CS reconstruction scheme, where the prior information comes directly from the CSEN outputs. A wide range of experiments shows that this approach has great potential to improve SR performance of traditional approaches for sparse signal recovery problems. As mentioned above, we used CS imaging simulation, but this time signal reconstruction error is compared with state-of-the-art conventional SR approaches. Figure 1 illustrates a representative graph of two different applications of CSENs; a) performing SE from CS measurement vector, y, and b) the output of CSEN is used as the side information, p, which gives the estimated probability of being non-zero for each index. In this simple illustration we assume that the hand-writing signal '2' is sparse in spatial domain such that Φ = I; therefore, D = AI = A and B is a denoiser matrix such as D T , or D T D + λI −1 D T where λ is the regularization parameter.

II. NOTATIONS
In this work, we define the p norm of any vector x ∈ R n as The 0 -norm of the vector x ∈ R n is given as x n 0 = lim p→0 n i=1 |x i | p = #{j : x j = 0} and the ∞ is defined as x n ∞ = max i=1,...,n (|x i |). A signal s can be defined as a strictly k-sparse signal if it can be represented with less than k + 1 non-zero coefficients in a proper basis Φ, i.e., x 0 ≤ k, where s = Φ x. We also define a sparse support set or simply support set, Λ ⊂ {1, 2, 3, ..., n}, as the set of indices that represents the non-zero coefficients, i.e., Λ := {i : x i = 0}. The complement of support set, Λ, w.r.t {1, 2, 3, ..., n} is given as Λ c = {1, 2, 3, ..., n} \ Λ. In this manner, x Λ ∈ R |Λ| is a vector consisting of non-zero elements of x ∈ R n , where |Λ| refers to the number of the non-zero coefficients . Similarly, M Λ ∈ R m×|Λ| denotes a matrix that consists of the columns of a matrix M ∈ R m×n indexed by support Λ.

III. RELATED WORK
CS theory claims that a signal s can be sensed using far fewer linear measurements m than Nyquist/Shannon based traditional methods use, d, i.e., where A ∈ R m×d is the measurement matrix and D ∈ R m×n is called the equivalent dictionary. It can be demonstrated that sparse representation, is unique if m ≥ 2k [25] and x 0 ≤ k. In brief, the uniqueness of the sparse representation in Eq. (2) shows that any k-sparse signal pair can still be distinguished in the equivalent dictionary, D. However, the problem in Eq. (2) is that this is a non-convex problem and known to be NP-hard. One of the most common approaches is the relaxation of the 0 norm as follows: which is known as Lasso [26] formulation, and also known to produce stable solutions in noisy cases and exact solutions in noise free cases [27].

A. Generic Sparse Support Estimation (SE)
In many application scenarios, detecting the indices of the non-zero coefficients' location, Λ, is more important than computing these coefficients such as sparse representation based classifications [11], [12] and radar imaging [15], [28], CS-based active user detection in the downlink of a CDMA system [29], and for the uplink of a NOMA [30], [31] system. Mathematically speaking, for the linear measurement model given in Eq. (1) and with additive noise, y = Dx+z, we define the following support estimator E(., .), whereΛ is the estimated support. For the noise-free case and x is exactly k-sparse, the exact Λ recovery performance of an algorithm coincides with the sparse signal recovery performance. This an expected outcome since the unique representation is satisfied when m > 2k. In the noisy case, even if the exact signal recovery is not possible, it is still possible to recover the support set exactly. In the literature, several studies have proposed to provide information-theoretical (i.e., the optimal decoder, E's performance) guarantee conditions for exact [32], [5], [33], [10], and partial support estimation [7], [34], [10]. However, in most of the practical applications, a tractable signal recovery method is applied first to find an estimationx of the sparse signal x, then a component-wise thresholding is applied tox to compute the estimated support as illustrated in Figure 2. A common approach is to follow an iterative sparse signal recovery method from the CS literature. For instance, it is proven in [35] that if min i∈Λ |x i | > 8σ 2 * log(n), then one can recover the support set exactly using Lasso with λ = 2 2 * log(n), where σ 2 is variance of the measurement noise. This theorem is valid in the case that the equivalent dictionary satisfies the mutual coherence property defined in [35]. One may clearly deduce from their results that accurate support estimation is possible via Lasso if the non-zero coefficients' magnitudes are above a certain level determined by the noise. Similarly, the conditions of exact support recovery under noise using OMP are given in [36], and partial support recovery performance bounds of AMP are in [37]. Along with these SR algorithms in CS literature, which are iterative methods, traditional linear decoders such as Maximum Correlation (MC) [38],x M C = D T y and LMSEE [37],x LM M SE = D T D + σ 2 z I n×n −1 D T y are also used in many applications. The theoretical performance bounds of these methods are also given in [37].

B. Case study of SE: Representation based classification
Consider an image from a particular class is queried: It can be expected from the estimated SCs,x to have significant (non-zero) entries which are located in a specific location so that the corresponding columns in the dictionary matrix, D, are the samples from the actual class of the image. This problem is also known as the representation based classification, which is a typical example where the support set location is the main information we are seeking.
In [11], 1 -minimization is used to obtain such a sparse code to determine the identity of face images. However, in reality such an ideal decomposition is not accomplished in general because face images show a high correlation among different classes. This is why, instead of using the estimated sparse codes,x obtained by an SR technique such as Eq. (3), the authors propose a four steps solution: (i) Normalization: Normalize all the atoms in D and y to have unit 2 -norm, (ii) SR:x = arg min x x 1 s.t y − Dx 2 , (iii) Residual finding: e i = y − D ixi 2 , wherex i is the estimated coefficients corresponding the class i, (iv) Class determination: Class (y) = arg min (e i ).
This technique and its similar variants have been reported to perform well not only in face recognition but many other classification problems [39], [40]. Later, the authors of [22] propose to change the second step, from , which has a closed-form This collaborative representation based classification (CRC) was reported to achieve a comparable classification performance for different classification problems. For face recognition problems, in particular, the authors reported that high classification accuracies were obtained especially for high measurement rates (MRs). C. Sparse signal reconstruction with side information of support set Consider the case where SE is not the main concern but SR is. In case side information is available about the support set, an improvement to 1 -minimization can be achieved in sparse signal recovery as follows: where is element-wise multiplication operator and w is the predefined cost that imposes the prior information about each element's values. In the concept of modified CS [41] and CS with prior information literature, the cost function, w generally appears in the form of w i = 1 pi+ , where > 0 is a predefined constant and p i is the i th element of the vector p which is a type of a measure such as prior likelihood [23] of the support set, which could represent the probability of (i) th element being non-zero.

D. Limitations of existing Support Estimators
Both SE and SR algorithms guarantee to perform well if the equivalent dictionary D satisfies certain properties such as mutual incoherence [42]. However, in many practical scenarios, D fails to satisfy these properties, e.g., in face recognition problem, the atoms of D, vectorized faces, are highly correlated. The second limitation of traditional sparse recovery algorithms is that they are iterative methods and computationally costly. Therefore, the support estimators relying on these sparse recovery algorithms may not be feasible, especially in real-time applications. The third limitation of state-of-the-art SR techniques such as 1 -minimization is that there is a lower limit for MR (see phase transition [43]); below this limit, the SR algorithms start to fail completely. This limit generally depends on the wellness of D (defined by properties such as mutual incoherence [42] ). Therefore, SE techniques that build upon an SR algorithm tend to fail if D does not satisfy the required properties, e.g., if the atoms of D are highly correlated.
On the other hand, when it comes to SR techniques leveraging SE as prior information, despite the fact that a good improvement can be achieved using such prior information, most of the works assume that the information is available in advance; however, they do not mention how to obtain such a p.

IV. CONVOLUTIONAL SUPPORT ESTIMATOR NETWORK
Recent advance in deep neural networks [44], [18] enables a non-iterative solution for the sparse signal recovery. It is often reported that they produce a solutionx, which is closer to x than the ones obtained by an iterative approach. They can still work under those measurement rates where classical CS recovery algorithms fail. Nevertheless, a crucial disadvantage is that they require a large number of training samples to achieve a high generalization capability. Second, their complex configuration with millions of parameters causes certain computational complexity issues such as speed and memory problems, especially when they are used in edge devices with limited power, speed and memory.
If one may wish to find only support Λ instead of the sign and amplitude of x, a traditional Machine Learning approach would be sufficient. In this study, we propose a support estimator, E(.), that can be performed by a compact CSEN network. Another crucial objective is to have the ability to learn from a minimal training set with a limited number of labeled data. A typical application where this approach can benefit from is face recognition via sparse representations, where only a few samples of each identity are available.
To accomplish this objective, first, the CSEN network with input and output, P (y, D ) : R n → [0, 1] n , produces a vector p that gives the information about the probability of each index to be in support set such that p i ∈ [0, 1]. Then, the final support estimator, E(y, D) will produce a support estimation such that Λ = { i ∈ { 1, 2, .., n} : p i > τ }, by thresholding p with τ where τ is a fixed threshold.
As shown in Figure 3, the proposed SE approach is different from the conventional SR based methods, which directly thresholdx for support estimation. Moreover, the input-output pair is different. The proposed CSEN learns over y train , v train to compute p while the conventional SR methods work with y train , x train to first make the sparse signal estimation, and then compute support estimation by thresholding it. As evident in Figure 1, the application of direct signal recovery may cause noisy estimation of the support codes while the proposed CSEN has the advantage of learning the pattern of the support codes and, therefore, can predict their most-likely location with proper training.
In this study, the proposed CSEN models consist of only convolutional layers in the type of fully convolutional networks [16] that are trained by optimizing the support estimations. Since the SE problem involves one-to-one mapping, other network types such as Multi-Layer Perceptrons (MLPs) can also be used as in [18]. However, this brings two limitations compared to CSENs: high computational complexity and over-fitting due to the limited training data and number of parameters in the network. In Section V, it will be shown that such an approach yields a poor generalization and is not robust to noise.
When a CSEN is trained, it learns the following transformation:v ← P (x), wherev is the estimation of binary mask representing the estimated support for the signal x, and the proxyx = Bx with B = D T , or D T D + λI −1 D T , i.e., the Maximum Correlation and LMMSE formula in [37]; and hence, x,x ∈ R N . First, the proxyx is reshaped to 2-D plane (e.g., original size of the image or pre-defined search grid). Correspondingly, the proxyx is convolved with W 1 , the weight kernels connecting the input layer to the next layer  with N filters to form the input of the next layer with the summation of weight biases b 1 as follows: where S(.) is the down-or up-sampling operation and ReLu(x) = max(0, x). In more general form, the k th feature map of layer l can be expressed as, The trainable parameters of the network would be: In the proposed approach, the Mean-Square Error (MSE) of a sparse code x, E(x), is computed between its binary mask, v, and CSEN's actual output, P Θ (x) p as follows: where v p is the p th pixel of v. The CSEN network is trained using samples in the train data, D train = (x (1) , v (1) ), (x (2) , v (2) ), ...(x (s) , v (s) ) .

V. RESULTS
In order to evaluate the effect of different network configurations, in this study, we use two different CSEN configurations and perform a comprehensive analysis over each of them. Generally, each convolutional layer has a dimension reduction capability with pooling functions. However, the first proposed network architecture consists of only convolutional layers with ReLu activation functions to preserve the sparse signal (e.g., image) dimensions at the output layer. In this configuration (CSEN1), we propose to use three convolutional layers with 48 and 24 hidden neurons and 3×3 filter size as given in Figure  4. CSEN2 is a slight modification of CSEN1 configuration, as shown in Figure 5 by using up-and down-sampling layers. Although this modification increases the number of parameters, in return, it yields substantial performance improvement over MNIST. While the SE performance analysis over MNIST has done using CSEN1 and CSEN2, only CSEN1 results are reported since CSEN2 produces similar recognition rates (∼ 0.001 difference) for face recognition. In any case, both network configurations are compact compared to the deep CNNs that have been proposed recently. For example, the study in [17] proposes ReconNet for SR, which consists of six convolutional layers with 32 neurons or more in each layer.
Since there is no competing method for SE that is similar to the proposed method, we use the ReconNet [17] in this study on the SE problem by directly givingx as the input, and removing the denoiser block at the end for comparative evaluations. Finally, we apply thresholding over the output of ReconNet to generate SE i.e.,Λ R = { i ∈ { 1, 2, .., n} : P R (x) > τ }, where P R (.) is ReconNet with fully convolutional layers. ReconNet is originally a CS recovery algorithm working directly on spatial domain, i.e.,ŝ ← P (y) instead of solving them in the sparsifying dictionary, i.e.,ŝ = Φx wherex ← P (y). Therefore, ReconNet serves as a deep CSEN approach against which the performance of the two compact CSENs will be compared. Moreover, we also train the state-of-the-art deep SR solution, LAMP, in order to use it over SE problem. For LAMP method, it is possible to predefine the number of layers in advance. For a fair comparison, we have tested the algorithm for three different setups: 2, 3, and 4 layers design using their provided implementation. Next, in the experiments of face recognition based on SR, we consider both speed and recognition accuracy of the algorithms as it is performed only for 1 -minimization toolbox in [45]. Thus, in order to perform comparative evaluations, the proposed CSEN approach is evaluated against most of the conventional state-of-the-art SR techniques along with ReconNet. Finally, CSEN2 is applied as a pre-processing step for the CS-recovery to obtain w in the cost function as illustrated in Figure 1. The experiments in this study have been carried out on a workstation that has four Nvidia R TITAN-X GPU cards and Intel R Xeon(R) CPU E5-2637 v4 at 3.50GHz with 128 GB memory. Tensorflow library [46] is used with Python. ADAM optimizer [47] is utilized during the training with the proposed default values of the learning parameters: learning rate, α = 0.001, and moment updates β 1 = 0.9, β 2 = 0.999 with only 100 and 30 Back-Propagation iterations for MNIST and face recognition experiments, respectively.

A. Experiment I: Support Estimation from CS measurements
For the experiments in this section, MNIST dataset is used. This dataset contains 70000 samples (50K/10K/10K as the sizes of the train/validation/test sets) of the handwritten digits (0 to 9). Each image in the dataset is a 28×28 pixel resolution with intensity values ranging from 0 (black, background) to 1 (white, foreground). Since the background covers more area than the foreground, each image can be considered as a sparse signal. Mathematically speaking, we may assume that the i th vectorized, x i ∈ R n=784 can be considered as the k i -sparse signal. The sparsity rates of each sample is calculated as ρ i = ki n , and its histogram is given in Figure 6. We have designed an experimental setup where these sparse signals (sparse in canonical basis) x i 's are compressively sensed, where D = A ∈ R m×n since Φ = I. We calculate the measurement rate as MR = m n . Therefore, the problem is SE from each CS measurement, i.e., findingΛ i from each y i in the test dataset.  For this dataset, the measurement rate, (MR) is varied from 0.05 to 0.25 in order to investigate the effect of MR on the SE performance. The measurement matrix is then chosen as the "Gaussian", the matrix whose elements A i,j are i.i.d. drawn from N 0, 1 m . It is worth mentioning that the approximate message passing (AMP) algorithm is a welloptimized method for the Gaussian measurement matrix, and LAMP is a learned version of this algorithm. Therefore, they are reported to be state-of-the-art if the measurement matrix is Gaussian but they do not even guarantee the converge for other types of measurement matrices. On the other hand, the comparative performance evaluations against LAMP and deep CS-sparse methods are presented in Table I, and the results clearly indicate that the proposed method achieves the best SE performance in terms of F1 measure for MR = 0.25 and 0.05 and comparable for MR = 0.1. The results presented in Table I indicate that despite its deep and complex configuration, compact CSENs achieve superior performance levels compared to ReconNet.
Furthermore, comparative evaluations are performed when the measurements are exposed to noise in the test set, i.e., The results presented in Figure 7 show that SE performances of the LAMP method are adversely affected by increased measurement noise. Their performance gets even worse when the number of layers is increased (i.e., see results for LAMP (2) to LAMP (4)). CSEN2, on the other hand, achieves the highest F1 measure for all noise levels.
B. Experiment II: Face Recognition based on Sparse Representation As explained in Section III-B, the dictionary-based (representation based) classification could be seen as a SE problem. Therefore, CSEN presents an alternative and better approach to both CRC and SRC solutions. In this manner, the proposed CSEN approach is evaluated against both CRC and the stateof-the-art SRC techniques recently proposed. The algorithms are chosen by considering both their speed and performance on the SR problem, since the speed-accuracy performance of SRC directly depends on the performance of the sparse signal recovery algorithm [45], and there is no unique winner to achieve the top performance level for all databases. The proposed method is, of course, not limited to face recognition but can be applied in any other representation based classification problem. For the sake of brevity, in this study, we focus only on the face recognition problem. In dictionary-based classification designs, the samples of a specific class are stacked in the dictionary as atoms with pre-defined indices, e.g., the atoms belonging to a particular class can be located in concatenate manner. Consequently, in sparse representation based classification, instead of using 1 -minimization in Eq. (3), group 1 -minimization can be introduced as follows, where x Gi is the group of coefficients corresponds to class i. Hence, the MSE cost function in Eq. (9) can be modified accordingly: To approximate this, a simple average pooling can be applied after the last layer of CSEN which is then followed by SoftMax function to produce class probabilities. Therefore, the modified cost function with Cross-Entropy loss at the output would be: where t i and P Θ (x) are the real and the predicted values by CSEN, respectively, for class i ∈ C. By this way, the modified network can directly produce the predicted class labels as the output. In the experiments, we have used Yale-B [20] and CelebA [21] databases. In Yale-B dataset, there are 2414 face images with 38 identities; and a sub-set of CelebA is chosen with 5600 images and 200 identities. The face recognition experiments are repeated 5 times with samples randomly selected to build the dictionary, train, and test sets with 32, 16, 16, and 8, 12, 8 samples each for Yale-B and CelebA, respectively, and 25% of training data is separated as validation set. The selected subset of CelebA dataset is also different between each repeated run. For Yale-B database, we use vectorized images in the dictionary. Earlier studies reported that both SRC and CRC techniques achieve a high recognition accuracy such as 97 -98%, especially for high MR rate scenarios (m/d > 0.25 for A ∈ R m×d ). On the other hand, for CalebA both CRC and SRC solution tends to fail when we use raw atoms in the dictionary without extracting descriptive features. This is why in this study, we propose to use a more representative dictionary. Instead of using raw images, the atoms consist of more descriptive features extracted by a neural networkbased face feature extractor in the library [48]. The proposed method is compared against CRC and SRC techniques with the following 7 state-of -art SR solver: ADDM [49], Dalm [45], OMP [45], Homotopy [50], GPSR [51], L1LS [52], 1 -magic [53], and Palm [45].
Overall, when we perform experiments in two facial image databases, Yale-B and CalebA for different MRs, the CSEN based classification proves to be very stable; and in all MRs, it gives the highest or comparable recognition accuracy to the highest ones for all experiments as presented in Figure  8 and 9. Furthermore, it is significantly superior in terms of computational speed when compared with SRC solutions.

C. Experiment III: Learning-aided Compressive Sensing
As the experimental setup, we randomly choose, sparse signals, x in the MNIST database and use the Gaussian measurement matrix, A to simulate the CS, i.e., y = Ax. Then, we recover the sparse signal from y by using the aforementioned state-of-the-art SR tools and the proposed weighted 1 -minimization Eq. (5), where the weights w are obtained using CSEN output such that w = 1 p+ . The two examples where signals are compressively sensed with M R = 0.25 and their estimated versions by different SR methods are shown in Figure 10. It is clear that the proposed approach recovers the sparse signal with the best quality while the other state-of-art SR techniques perform poorly. Figure 11 shows an illustration of how proposed compressive sensing reconstruction scheme differs from traditional compressive sensing recovery setup. Using the output of CSEN as prior information not only provide more accurate signal recovery, but also faster convergence of iterative sparse signal recovery such as 1 -minimization. Furthermore, we draw the estimated phase transition of the algorithms in Figure 12 using an experimental setup whose procedure is explained in [19]. Briefly summarizing the procedure, a grid of (MR, ρ) is generated for each algorithm, with 20 independent realization of the problem: according to their sparsity ratios, ρ, randomly chosen sparse signal x, among 10000 MNIST test images, are compressively sensed with independent realization of measurement matrices. Then, they are recovered using the competing algorithms, and each realization is considered a success for the specific algorithm if x −x 2 x ≤ tol (13) where tol is a predefined parameter, we choose tol = 10 −1 in our experiments. For a specific algorithm, we draw the phase transition in the border where a 50% success rate is achieved. The procedure is similar to [19], with the exception that they repeated the experiment only once, while we repeat it 100 times for each method, except L1LS due to its infeasibly high computational cost (it took almost two weeks with an ordinary computer). With an accurate SR algorithm, we expect the transition border to be close to the left-top corner in the phase transition graph because it is a good indicator that the algorithm performs well in low MRs and with high sparsity ratio, ρ. From the Figure, one can easily deduce that the proposed CS-reconstruction approach clearly outperforms all competing state-of-the-art SR reconstruction methods.

VI. CONCLUSIONS
Sparse support estimators that work based on traditional sparse signal recovery techniques suffer from computational complexity and noise. Moreover, they tend to fail at low MRs completely. The proposed CSENs can be considered as reconstruction-free and non-iterative support estimators. Of course, despite their high computational complexity, recent state-of-the-art deep signal reconstruction algorithms may be a cure to sparse recovery methods. However, they are still redundant if SR is not the main concern. In addition, such deep networks often require a large amount of training data that is not available in many practical applications. To address these drawbacks and limitations, in this study, we introduce a novel learning-based support estimators, which have compact network designs. The highlights of the proposed system are as follows: i) Signal reconstruction-free support estimation where sparse estimation can be done in a feed-forward manner, noniteratively at a low cost. ii) Compact network designs enabling efficient learning even from a small-size training set. iii) The proposed solution is generic; it could be used in any support estimation task such as SE based classification.