Elastic Net Regularization and Gabor Dictionary for Classification of Heart Sound Signals using Deep Learning

Mahmoud Fakhry; Ascensión Gallardo-Antolín

cs.SDApr 14, 2026

Elastic Net Regularization and Gabor Dictionary for Classification of Heart Sound Signals using Deep Learning

Authors: Mahmoud Fakhry, Ascensión Gallardo-Antolín

Organizations: Department of Signal Theory and Communications, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, Leganés (Madrid), 28911, Spain · Department of Electrical Engineering, Faculty of Engineering, Aswan University, Aswan, 81542, Egypt

Abstract

In this article, we propose the optimization of the resolution of time-frequency atoms and the regularization of fitting models to obtain better representations of heart sound signals. This is done by evaluating the classification performance of deep learning (DL) networks in discriminating five heart valvular conditions based on a new class of time-frequency feature matrices derived from the fitting models. We inspect several combinations of resolution and regularization, and the optimal one is that provides the highest performance. To this end, a fitting model is obtained based on a heart sound signal and an overcomplete dictionary of Gabor atoms using elastic net regularization of linear models. We consider two different DL architectures, the first mainly consisting of a 1D convolutional neural network (CNN) layer and a long short-term memory (LSTM) layer, while the second is composed of 1D and 2D CNN layers followed by an LSTM layer. The networks are trained with two algorithms, namely stochastic gradient descent with momentum (SGDM) and adaptive moment (ADAM). Extensive experimentation has been conducted using a database containing heart sound signals of five heart valvular conditions. The best classification accuracy of $98.95\%$ is achieved with the second architecture when trained with ADAM and feature matrices derived from optimal models obtained with a Gabor dictionary consisting of atoms with high-time low-frequency resolution and imposing sparsity on the models.

Figures & tables

Figure 1: Plots of heart sound signals and their corresponding Fourier transforms and spectrograms at a sampling frequency of $f_{s}=1000$ Hz, for a healthy person and four patients with CVDs. The signal length is $2$ seconds and the frequency is between $0$ and $\pi$ .

Figure 2: Flowchart of the proposed method.

$j$	$\beta$	$\omega$	$\tau$
$1$	$2^{1}$	$0:\frac{\pi}{4}:\frac{3\pi}{4}$	$0:2^{0}:(2^{11}-2^{0})$
$2$	$2^{2}$	$0:\frac{\pi}{8}:\frac{7\pi}{8}$	$0:2^{1}:(2^{11}-2^{1})$
$3$	$2^{3}$	$0:\frac{\pi}{16}:\frac{15\pi}{16}$	$0:2^{2}:(2^{11}-2^{2})$
$4$	$2^{4}$	$0:\frac{\pi}{32}:\frac{31\pi}{32}$	$0:2^{3}:(2^{11}-2^{3})$
$5$	$2^{5}$	$0:\frac{\pi}{64}:\frac{63\pi}{64}$	$0:2^{4}:(2^{11}-2^{4})$
$6$	$2^{6}$	$0:\frac{\pi}{128}:\frac{127\pi}{128}$	$0:2^{5}:(2^{11}-2^{5})$

Table 1: Variables of the

j

th Gabor dictionary

\mathbf{D}_{j}

for all values of

j

N=11

and

\theta=0

. A discrete interval of

\omega

\tau

defined as

z_{0}:\delta z:z_{\infty}

begins at

z_{0}

, ends at

z_{\infty}

and is discretized with a step of

\delta z

Figure 3: Time and frequency representations of atoms for $3$ values for $j$ at $N=7$ , $\omega=\pi/2$ and $\tau=2^{6}$ . Increasing $\beta$ improves spectral resolution at the cost of temporal resolution.

Figure 4: Approximation error measured as the average energy of all residuals

\|\mathbf{x}-\mathbf{D}_{j}\mathbf{a}_{j,\alpha}\|_{2}^{2}

, calculated for all the recordings of the dataset.

Figure 5: Average energy

\|\mathbf{a}_{j,\alpha}\|_{2}^{2}

, calculated for all the recordings of the dataset.

Figure 6: Average number of atoms corresponding to the number of non-zero entries in

\mathbf{a}_{j,\alpha}

. These numbers and their standard deviations are computed using

200

PCG signals for each heart condition. Signals are truncated to the length of

2^{14}

samples, downsampled by a factor of

8

, and then approximated with a Gabor dictionary of size

2^{11}\times 2^{13}

Figure 7: Three different transformation functions for the calculation of features.

Figure 8: Time-frequency feature matrices for PCG signals of five conditions. Each matrix is obtained by reshaping the weighted logarithmic values applied to the coefficient vector of the corresponding signal. Vectors are calculated using ridge regression ( $\alpha=0$ ) at different time-frequency resolutions for the atoms.

Figure 9: CNN-LSTM network consisting of 1D and 2D CNN, and LSTM layers.

Parameters	Value
No. classes	$5$
Truncated signal length (samples)	$2^{14}$
Downsampling factor	$2^{3}$
Downsampled signal length ( $L=2^{N}$ )	$2^{11}$
$j$	$1,2,...,10$
Size of input feature matrices ( $\mathbf{B}_{j,\alpha}$ )	$2^{j+1}\times 2^{N-j+1}$

Table 2: Implementation parameters of the CNN-LSTM classification model

Panel of Figure 10 — Figure 10: Average classification accuracy. Feature matrices used to train and evaluate the model comprise the weighted logarithmic of $\mathbf{a}_{j,\alpha}$ .

$\alpha$	Metric	$\beta$
$\alpha$	Metric	$2^{1}$	$2^{2}$	$2^{4}$	$2^{7}$	$2^{10}$
$0$	PRE	$97.90(0.88)$	$97.12(1.00)$	$93.20(1.40)$	$92.06(1.67)$	$94.46(0.96)$
	REC	$97.86(0.90)$	$97.07(1.03)$	$93.07(1.45)$	$91.77(1.75)$	$94.25(0.99)$
	SPE	$99.46(0.22)$	$99.27(0.26)$	$98.27(0.36)$	$97.94(0.44)$	$98.56(0.25)$
	F1 score	$97.85(0.90)$	$97.07(1.03)$	$93.05(1.45)$	$91.76(1.76)$	$94.26(0.98)$
$0.10$	PRE	$97.99(0.81)$	$96.92(0.99)$	$90.80(1.77)$	$91.55(1.40)$	$92.98(1.28)$

Table 3: Classification metrics and standard deviations (in parenthesis) for the two network architectures when trained with ADAM.

$\alpha$	Metric	$\beta$
$\alpha$	Metric	$2^{1}$	$2^{2}$	$2^{4}$	$2^{7}$	$2^{10}$
$0$	PRE	$98.39(0.66)$	$97.59(0.95)$	$94.13(1.12)$	$92.19(1.49)$	$95.59(1.12)$
	REC	$98.37(0.68)$	$97.55(0.96)$	$94.05(1.14)$	$91.99(1.84)$	$95.49(1.16)$
	SPE	$99.59(0.17)$	$99.39(0.24)$	$98.51(0.29)$	$98.00(0.46)$	$98.87(0.29)$
	F1 score	$98.37(0.68)$	$97.55(0.96)$	$94.03(1.15)$	$91.93(1.86)$	$95.49(1.15)$
$0.10$	PRE	$97.85(0.79)$	$96.93(0.90)$	$92.04(1.26)$	$91.56(1.25)$	$94.49(1.40)$

Table 4: Classification metrics and standard deviations (in parenthesis) for the two network architectures when trained with SGDM.

Figure 11: The confusion matrix of the five heart valvular conditions for $100$ experiments. It is obtained using the Gabor dictionary constituted with $\beta=2^{1}$ and using the elastic net regularization with $\alpha=0.1$ . The classification model consists of 1D and 2D convolutional layers and an LSTM layer and is trained using ADAM.

Metric	Heart valvular condition
Metric	N	MS	MVP	AS	MR
PRE	$99.83$	$98.74$	$98.10$	$99.22$	$98.85$
REC	$99.58$	$98.75$	$97.75$	$99.74$	$98.91$
SPE	$99.96$	$99.68$	$99.53$	$99.80$	$99.71$
F1 score	$99.71$	$98.75$	$97.93$	$99.48$	$98.88$

Table 5: Classification metrics of the confusion matrix in Figure 11 .

Ref.	Features	Classifier	Training	Accu.
[ 3 ]	Raw PCG signals	Deep 2D CNN-LSTM	ADAM	$98.48$
[ 3 ]	Transformed PCG signals	Deep 2D CNN-LSTM	ADAM	$95.40$
[ 15 ]	VMD + weighted logarithmic	Light 1D CNN-LSTM	SGDM	$98.65$
Prop.	Gabor dictionary ( $\beta=2^{1}$ )+	1D CNN-LSTM	ADAM	$97.97$
Prop.	elastic net ( $\alpha=0.1$ )	1D CNN-LSTM	ADAM	$97.97$
Prop.	Gabor dictionary ( $\beta=2^{1}$ )+	1D CNN-LSTM	SGDM	$98.39$

Table 6: Comparison of the proposed method and baseline methods.

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