The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer

Juan Xie; Jinwang Zha; Jie Ren; Hui Zhang; Yadong Li; Xing Hu; Li Xiao; Gan Wang; Jian Lan; Caihong Cao

doi:doi:10.11648/j.mlr.20261101.11

Research Article |

| Peer-Reviewed

The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer

Juan Xie

, Jinwang Zha

, Jie Ren

, Hui Zhang

, Yadong Li

, Xing Hu^*

, Li Xiao

, Gan Wang

, Jian Lan

, Caihong Cao

Published in Machine Learning Research (Volume 11, Issue 1)

Received: 11 December 2025 Accepted: 29 December 2025 Published: 20 February 2026

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Abstract

The compression sensing reconstruction for the 1-d signal can contribute to the communication of autonomous driving, intelligent robots, and fire exploration robots. To address the issue that fully connected layers in the LISTA method lack the ability to extract non-local features, this paper primarily designs a non-local fully connection layer and proposes a novel compressed sensing reconstruction method for audio signals. This paper designs a compression sensing reconstruction method for the 1-d signal. To reconstruct 1-d signal, the deep learning method LISTA is used. Then, the linear full connection layer in LISTA is improved by combining the output of three full connection layer to capture the non-local information. The computing regions of improved non-local full connection layer contain: 1) the full connection before; 2) the current full connection; and 3) the full connection after. Experimental results show the reconstruction results of LISTA and LISTA_nf are both close to the real signal. The MSE of LISTA_nf is reduced by 0.1 than the MSE of ISTA under the same experimental settings. The non-local full connection layer in the LISTA_nf consumes longer computing time. The LISTA_nf increase the computing time by 0.07s than the computing time of the ISTA. Experimental results show the effectiveness of the proposed method.

Published in	Machine Learning Research (Volume 11, Issue 1)
DOI	10.11648/j.mlr.20261101.11
Page(s)	1-7
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Compression Sensing Reconstruction, 1-d Signal, LISTA, Non-local Full Connection

1. Introduction

Compression sensing reconstruction for the 1-d signal can contribute to the communication of autonomous driving, intelligent robots, and fire exploration robots.

The traditional compression sensing reconstruction methods include OMP

[1]

, SAMP

[2]

, ROMP

[3]

, StOMP

[4]

, CoSaMP

[5]

. With the developments of machine learning, the finite state machine (FSM) and conditional random field (CRF) are introduced to traditional compression sensing reconstruction methods. Then, with the developments of deep learning, the deep neural networks are also introduced to the compression sensing reconstruction. However, the methods based on deep learning need larger training datasets compared with traditional methods.

This paper designs a compression sensing reconstruction method for the 1-d signal. To reconstruct 1-d signal, the deep learning method LISTA is used. Then, the linear full connection layer in LISTA is improved by non-local full connection layer. The computing regions of improved non-local full connection layer contain: 1) the full connection before; 2) the current full connection; and 3) the full connection after. Experimental results show the reconstruction results of LISTA and LISTA_nf are both close to the real signal. The MSE of LISTA_nf is better than the MSE of ISTA and LISTA. The non-local full connection layer in the LISTA_nf consumes longer computing time.

2. Related Works

(1) The Compression Sensing Reconstruction of 1-d Signal

1) Traditional Methods

Defraene et al.

[7]

applied compressive sensing for clipping processing of one-dimensional signals. M. Kasem and EI-Sabrouty

[8]

proposed a comparative study on audio compression based on compressive sensing and sparse fast Fourier transform (SFFT): performance and challenges. Lin et al.

[9]

developed a multi-frame prediction M-LVC method for video compression. Grrggin et al.

[10]

proposed using compressive sensing for encoding the sinusoidal model of audio signals. A. Verburg and Fernandez-Grande

[11]

suggested using compressive sensing to reconstruct sound fields inside buildings. Griffin et al.

[12]

presented compressive sensing-based single-channel and multi-channel sinusoidal sound encoding. Valenzise et al.

[13]

proposed using distributed source coding and compressive sensing techniques to identify sparse audio tampering. Zahedi et al.

[14]

proposed audio coding for wireless acoustic sensor networks. Griffin and Tsakalides

[15]

presented compressive sensing reconstruction of audio signals using multiple sensors. Moreno-Alvarado

[16]

designed research on simultaneous audio encryption and compression using compressive sensing technology. Katzberg

[17]

developed a compressive sensing framework for dynamic sound field measurement. Yin et al.

[18]

designed a compressive sensing method based on sampling and reconstruction, applied to wireless sensor matrix networks. Mads Græsbøl

[19]

discussed compressive sensing methods and their applications in speech and audio signal processing. Wahab Abood

[20]

designed provably secure and efficient audio compression based on compressive sensing. Albiges

[21]

used compressive sensing data for audio signal reconstruction to achieve intelligent classification of chronic respiratory diseases. Upadhyaya

[22]

designed quality parameter estimation for sparse audio signal reconstruction based on compressive sensing.

2) Machine Learning Methods and Deep Learning Methods

Russell and Wang

[23]

proposed the application of physics-informed deep learning in big data signal compression and reconstruction for industrial condition monitoring. Palangi designed a deep learning-based distributed compressed sensing method. Bao

[24]

developed a machine learning approach for compressive sensing data reconstruction in structural health monitoring. L. Machidon and Pejović

[25]

designed applications of deep learning in compressive sensing from a ubiquitous systems perspective. Lu and Bo

[26]

proposed WDLReconNet: deep learning compressive sensing reconstruction for wireless fading channels. Bright

[27]

developed a compressive sensing-based machine learning strategy for characterizing flow around a cylinder with limited pressure measurements. Wang

[28]

designed time-domain signal reconstruction of vehicle interior noise based on deep learning and compressed sensing techniques. L. Machidon and Pejović

[29]

presented applications of deep learning in compressive sensing from a ubiquitous systems perspective. Du

[30]

proposed a deep learning-based fast reconstruction method for EEG signal compressive sensing. Wu

[31]

developed a deep compressive sensing approach.

(2) The Methods for Deformable Convolution

Dai et al.

[32]

proposed deformable convolutional networks. Liu et al.

[33]

designed attention-injective deformable convolutional network for crowd understanding. Thomas et al.

[34]

improved the flexible and deformable convolution for point clouds. Shi et al.

[35]

proposed the video frame interpolation method via generalized deformable convolution. Wang et al.

[36]

designed a video restoration method with enhanced deformable convolutional networks. Wang et al.

[37]

explored large-scale vision foundation Models with deformable convolutions. Zhu et al.

[38]

designed deformable convNets v2: more deformable, better results method. Chen et al.

[39]

designed temporal deformable convolutional encoder-decoder networks for video captioning. Deng et al.

[40]

improved the spatio-temporal deformable convolution for compressed video quality enhancement. Park et al.

[41]

designed deformable graph convolutional networks.

3. Methods

In LISTA

[6]

method, the convolution adopts the full connection layer. However, the full connection layer adopted in LISTA can only capture the local region features, and the non-local region features can not be captured. Therefore, the non-local full connection layer is proposed. The full connection layer in LISTA is improved by non-local full connection layer. The computing regions of improved non-local full connection contain: 1) the full connection before; 2) the current full connection; and 3) the full connection after.

Non-local Full Connection

The normal convolution computing is

s (i) = \sum_{c = 1}^{c_{0}} \sum_{k = 0}^{k_s - 1} I (i + k) K_{1} (k)

(1)

The computing regions of improved non-local full connection layer contain: 1) the full connection before; 2) the current full connection; and 3) the full connection after.

In the before the full connection before, the computing is

s_{1} (i) = \sum_{c = 1}^{c_{0}} \sum_{k = 0}^{k_s - 1} I (i - k) K_{2} (k)

(2)

where

s_{1} (i)

is the output feature, the channel of the input feature is

c_{0}

, the kernel size is

k_s

, the input feature or data is

I

, and the convolution kernel is

K_{2}

In the current full connection, the computing is

s_{2} (i) = \sum_{c = 1}^{c_{0}} \sum_{k = 0}^{k_s - 1} I (i + k) K_{3} (k)

(3)

where

s_{2} (i)

is the output feature, the channel of the input feature is

c_{0}

, the kernel size is

k_s

, the input feature or data is

I

, and the convolution kernel is

K_{3}

In the full connection after, the computing is

s_{3} (i) = \sum_{c = 1}^{c_{0}} \sum_{k = 0}^{k_s - 1} I (i + k_s + k) K_{4} (k)

(4)

where

s_{3} (i)

is the output feature, the channel of the input feature is

c_{0}

, the kernel size is

k_s

, the input feature or data is

I

, and the convolution kernel is also

K_{4}

The output feature is

{s (i) = (s}_{1} (i)

s_{2} (i)

s_{3} (i)) / 3.0

(5)

and the

s (i)

is the output feature for the convolution

i

-point. The different weights are set across the “before,” “current,” and “after” full connections layer to capture more non-local information.

4. Experimental Results

(1) The Experiments Setting

The dimensions of the sparse signal and measurement are 128, 64, respectively. The training and testing signal are no-sparsity signal. The training signals are simulated by random signal. The eps threshold is 0.00001, the maximum iteration time is 1000, and the threshold for the shrinkage computing is 0.05. The training parameters of LISTA and LISTA_nf are: the batch size is 128, the training epoch is 100, the training loss functions are the MSE loss function combined with the L1 loss function, the maximum iteration time is 32, the optimizer is SGD. The ISTA, LISTA, and LISTA_nf are tuned using identical hyperparameters for a fair comparison.

(2) The Ablation and Comparison Experiments

Download: Download full-size image

Figure 1. The comparison results of the left above subfigure a) the sparse coefficient, the right above subfigure b) the real signal, the left down subfigure c) the reconstruction result of LISTA for the real signal, and the right down subfigure d) the reconstruction result of ISTA for the real signal.

The comparison results of the comparison results of the left above subfigure a) the sparse coefficient, the right above subfigure b) the real signal, the left down subfigure c) the reconstruction result of LISTA for the real signal, and the right down subfigure d) the reconstruction result of ISTA for the real signal are as shown in Figure 1. From Figure 1, the reconstruction results of ISTA and LISTA are both close to the real signal.

Download: Download full-size image

Figure 2. The comparison results of the left above subfigure a) the sparse coefficient, the right above subfigure b) the real signal, the left down subfigure c) the reconstruction result of LISTA for the real signal, and the right down subfigure d) the reconstruction result of ISTA for the real signal.

The comparison results of the comparison results of the left above subfigure a) the sparse coefficient, the right above subfigure b) the real signal, the left down subfigure c) the reconstruction result of LISTA for the real signal, and the right down subfigure d) the reconstruction result of ISTA for the real signal are as shown in Figure 2. From Figure 2, the reconstruction results of ISTA and LISTA are both close to the real signal.

Download: Download full-size image

Figure 3. The comparison results of the left above subfigure a) the sparse coefficient, the right above subfigure b) the real signal, the left down subfigure c) the reconstruction result of LISTA for the real signal, and the right down subfigure d) the reconstruction result of LISTA_nf for the real signal.

The comparison results of the comparison results of the left above subfigure a) the sparse coefficient, the right above subfigure b) the real signal, the left down subfigure c) the reconstruction result of LISTA for the real signal, and the right down subfigure d) the reconstruction result of LISTA-nf for the real signal are as shown in Figure 3. From Figure 3, the reconstruction results of LISTA and LISTA_nf are both close to the real signal.

Table 1. The Reconstruction MSE.

Different Methods	MSE
ISTA	0.35
LISTA	0.33
LISTA_nf	0.25

The reconstruction mean-square error (MSE) is shown as Table 1. From Table 1, the MSE of ISTA, LISTA, and LISTA_nf are 0.35, 0.33, and 0.25, respectively. The MSE of LISTA_nf is better than the MSE of ISTA and LISTA.

(3) The Reconstruction Time

Table 2. The Reconstruction Time.

Different Methods	Time (s)
ISTA	0.1
LISTA	0.12
LISTA_nf	0.17

The reconstruction time of ISTA, LISTA, and LISTA_nf are 0.1, 0.12, and 0.17, respectively, which is shown in Table 2. The reconstruction time of LISTA_nf is longer than the reconstruction time of ISTA and LISTA. The non-local full connection layer in the LISTA_nf consumes longer computing time.

5. Conclusions

Abbreviations

ISTA	Iterative Shrinkage-Thresholding Algorithm
LISTA	Learned Iterative Shrinkage and Thresholding Algorithm
LISTA-nf	Learned Iterative Shrinkage and Thresholding Algorithm-Non-local Full Connection
OMP	Orthogonal Matching Pursuit
SAMP	Sparsity Adaptive Matching Pursuit
ROMP	Regularized Orthogonal Matching Pursuit
StOMP	Stagewise Orthogonal Matching Pursuit
CoSaMP	Compressive Sampling Matching Pursuit

Acknowledgments

Thanks for my good friend Na Li, my good friends Shunan Zhao, Lin Wu, and Zhixin Zhang.

Data Availability Statement

The code, data for this paper, and datasets are opened in https://download.csdn.net/download/accdgh/91668667

Conflicts of Interest

The authors declare no conflicts of interest.

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APA Style

Xie, J., Zha, J., Ren, J., Zhang, H., Li, Y., et al. (2026). The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer. Machine Learning Research, 11(1), 1-7. https://doi.org/10.11648/j.mlr.20261101.11

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Xie, J.; Zha, J.; Ren, J.; Zhang, H.; Li, Y., et al. The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer. Mach. Learn. Res. 2026, 11(1), 1-7. doi: 10.11648/j.mlr.20261101.11

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Xie J, Zha J, Ren J, Zhang H, Li Y, et al. The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer. Mach Learn Res. 2026;11(1):1-7. doi: 10.11648/j.mlr.20261101.11

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@article{10.11648/j.mlr.20261101.11,
  author = {Juan Xie and Jinwang Zha and Jie Ren and Hui Zhang and Yadong Li and Xing Hu and Li Xiao and Gan Wang and Jian Lan and Caihong Cao},
  title = {The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer},
  journal = {Machine Learning Research},
  volume = {11},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.mlr.20261101.11},
  url = {https://doi.org/10.11648/j.mlr.20261101.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mlr.20261101.11},
  abstract = {The compression sensing reconstruction for the 1-d signal can contribute to the communication of autonomous driving, intelligent robots, and fire exploration robots. To address the issue that fully connected layers in the LISTA method lack the ability to extract non-local features, this paper primarily designs a non-local fully connection layer and proposes a novel compressed sensing reconstruction method for audio signals. This paper designs a compression sensing reconstruction method for the 1-d signal. To reconstruct 1-d signal, the deep learning method LISTA is used. Then, the linear full connection layer in LISTA is improved by combining the output of three full connection layer to capture the non-local information. The computing regions of improved non-local full connection layer contain: 1) the full connection before; 2) the current full connection; and 3) the full connection after. Experimental results show the reconstruction results of LISTA and LISTA_nf are both close to the real signal. The MSE of LISTA_nf is reduced by 0.1 than the MSE of ISTA under the same experimental settings. The non-local full connection layer in the LISTA_nf consumes longer computing time. The LISTA_nf increase the computing time by 0.07s than the computing time of the ISTA. Experimental results show the effectiveness of the proposed method.},
 year = {2026}
}

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TY - JOUR
T1 - The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer
AU - Juan Xie
AU - Jinwang Zha
AU - Jie Ren
AU - Hui Zhang
AU - Yadong Li
AU - Xing Hu
AU - Li Xiao
AU - Gan Wang
AU - Jian Lan
AU - Caihong Cao
Y1 - 2026/02/20
PY - 2026
N1 - https://doi.org/10.11648/j.mlr.20261101.11
DO - 10.11648/j.mlr.20261101.11
T2 - Machine Learning Research
JF - Machine Learning Research
JO - Machine Learning Research
SP - 1
EP - 7
PB - Science Publishing Group
SN - 2637-5680
UR - https://doi.org/10.11648/j.mlr.20261101.11
AB - The compression sensing reconstruction for the 1-d signal can contribute to the communication of autonomous driving, intelligent robots, and fire exploration robots. To address the issue that fully connected layers in the LISTA method lack the ability to extract non-local features, this paper primarily designs a non-local fully connection layer and proposes a novel compressed sensing reconstruction method for audio signals. This paper designs a compression sensing reconstruction method for the 1-d signal. To reconstruct 1-d signal, the deep learning method LISTA is used. Then, the linear full connection layer in LISTA is improved by combining the output of three full connection layer to capture the non-local information. The computing regions of improved non-local full connection layer contain: 1) the full connection before; 2) the current full connection; and 3) the full connection after. Experimental results show the reconstruction results of LISTA and LISTA_nf are both close to the real signal. The MSE of LISTA_nf is reduced by 0.1 than the MSE of ISTA under the same experimental settings. The non-local full connection layer in the LISTA_nf consumes longer computing time. The LISTA_nf increase the computing time by 0.07s than the computing time of the ISTA. Experimental results show the effectiveness of the proposed method.
VL - 11
IS - 1
ER -

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Author Information

Juan Xie

College of Information Technology, Nanchang Vocational University, Nanchang, China

http://orcid.org/0009-0001-6819-3270
Jinwang Zha

College of Information Technology, Nanchang Vocational University, Nanchang, China

http://orcid.org/0009-0005-4573-0482
Jie Ren

Big Data Technology Teaching and Research Office, Nanchang Vocational University, Nanchang, China

http://orcid.org/0009-0002-3351-5812
Hui Zhang

College of Information Technology, Nanchang Vocational University, Nanchang, China

http://orcid.org/0009-0006-1802-3122
Yadong Li

Beijing Qianfeng Internet Technology Co., Ltd. (Professional Co-construction Department), Beijing, China

http://orcid.org/0009-0002-7180-9846
Xing Hu

College of Information Technology, Nanchang Vocational University, Nanchang, China;School of Control Science and Engineering, Dalian University of Technology, Dalian, China

Contact Email

http://orcid.org/0000-0002-3410-8322
Li Xiao

College of Information Technology, Nanchang Vocational University, Nanchang, China

http://orcid.org/0009-0003-9222-078X
Gan Wang

Huanggang Education Valley Investment Holding Co., Ltd., Huanggang City, China

http://orcid.org/0009-0006-9843-4983
Jian Lan

Huanggang Education Valley Investment Holding Co., Ltd., Huanggang City, China

http://orcid.org/0009-0005-0615-6297
Caihong Cao

Nanchang Vocational University Library, Nanchang Vocational University, Nanchang, China

http://orcid.org/0009-0006-9214-0140

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APA Style

Xie, J., Zha, J., Ren, J., Zhang, H., Li, Y., et al. (2026). The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer. Machine Learning Research, 11(1), 1-7. https://doi.org/10.11648/j.mlr.20261101.11

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ACS Style

Xie, J.; Zha, J.; Ren, J.; Zhang, H.; Li, Y., et al. The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer. Mach. Learn. Res. 2026, 11(1), 1-7. doi: 10.11648/j.mlr.20261101.11

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AMA Style

Xie J, Zha J, Ren J, Zhang H, Li Y, et al. The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer. Mach Learn Res. 2026;11(1):1-7. doi: 10.11648/j.mlr.20261101.11

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@article{10.11648/j.mlr.20261101.11,
  author = {Juan Xie and Jinwang Zha and Jie Ren and Hui Zhang and Yadong Li and Xing Hu and Li Xiao and Gan Wang and Jian Lan and Caihong Cao},
  title = {The Compression Sensing Reconstruction for the 1-d Signal Based on Non-local Full Connection Layer},
  journal = {Machine Learning Research},
  volume = {11},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.mlr.20261101.11},
  url = {https://doi.org/10.11648/j.mlr.20261101.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mlr.20261101.11},
  abstract = {The compression sensing reconstruction for the 1-d signal can contribute to the communication of autonomous driving, intelligent robots, and fire exploration robots. To address the issue that fully connected layers in the LISTA method lack the ability to extract non-local features, this paper primarily designs a non-local fully connection layer and proposes a novel compressed sensing reconstruction method for audio signals. This paper designs a compression sensing reconstruction method for the 1-d signal. To reconstruct 1-d signal, the deep learning method LISTA is used. Then, the linear full connection layer in LISTA is improved by combining the output of three full connection layer to capture the non-local information. The computing regions of improved non-local full connection layer contain: 1) the full connection before; 2) the current full connection; and 3) the full connection after. Experimental results show the reconstruction results of LISTA and LISTA_nf are both close to the real signal. The MSE of LISTA_nf is reduced by 0.1 than the MSE of ISTA under the same experimental settings. The non-local full connection layer in the LISTA_nf consumes longer computing time. The LISTA_nf increase the computing time by 0.07s than the computing time of the ISTA. Experimental results show the effectiveness of the proposed method.},
 year = {2026}
}

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