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

Mahmoud Fakhry, Ascensión Gallardo-Antolín

arXiv:2604.12483v1cs.SDcs.AI

TL;DR

The paper addresses how to improve heart-sound representations for classifying five valvular conditions. It combines optimized Gabor dictionaries with elastic-net fitting and deep learning, achieving 98.95% accuracy with a 1D+2D CNN-LSTM model under the reported optimal setting.

  • Problem

    The paper investigates how time-frequency resolution and model regularization can be optimized to obtain more discriminative representations of heart sound signals for cardiovascular-disease classification.

  • Method

    The method fits heart sounds with elastic-net linear models over an overcomplete Gabor dictionary and converts the coefficient vectors into sparse time-frequency feature matrices for CNN-LSTM classifiers.

  • Results

    98.95% classification accuracy is achieved by the 1D+2D CNN-LSTM architecture using high-time and low-frequency resolution feature matrices.

  • Takeaways & Limitations

    The reported optimal configuration uses sparse features from high-time-resolution Gabor atoms with elastic-net parameter α = 0.1, ADAM training, and the more complex network architecture.

  • Takeaways & Limitations

    The authors identify replacing the real Gabor dictionary with a complex one and evaluating transfer learning as future work.

Abstract

from arXiv · show

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.

1. Introduction

The paper develops a heart-sound classification approach by optimizing Gabor time-frequency resolution and elastic-net regularization, then using the resulting coefficient vectors as inputs to deep neural networks.

  • Motivation: PCG signals provide acoustic information for automated cardiovascular-disease examination, while their nonstationary structure motivates joint time-frequency analysis.Heart sounds contain transient heartbeats and murmurs whose timing and frequency characteristics can be quantified.
  • Dictionary design: Gabor dictionaries are optimized by varying time-frequency atom resolution to represent PCG signals efficiently with low modeling error and cost.The paper focuses on the tradeoff between dictionary completeness and unnecessary redundancy.
  • Regularization: Elastic-net fitting combines l1 sparsity with squared l2 regularization to select atoms while handling multicollinearity.The regularization parameter is examined from 0, corresponding to ridge regression, to 1, corresponding to LASSO.
  • Deep learning: The study evaluates deep classifiers combining CNN and LSTM components for cardiovascular-disease classification.CNN layers extract representative information, while LSTM layers model recurrent dependencies.
  • Contributions: The proposed feature matrices are derived from discriminative coefficient vectors obtained after optimizing dictionary resolution and regularization.The stated contribution is to use these matrices to build a deep neural network classifier for CVD diagnosis.

2. Heart sound signals

Heart sound recordings contain normal cardiac sounds and murmurs associated with distinct valvular abnormalities. The study uses a five-category dataset and standardized, downsampled PCG signals to represent these conditions for analysis.

  • Valvular conditions: Heart valves can malfunction through regurgitation, which permits backward blood flow, or stenosis, which narrows or damages the valve and restricts flow.These mechanisms produce different cardiac sound patterns and murmurs.
  • Dataset: The dataset contains 1,000 recordings across five categories: normal, mitral valve prolapse, mitral stenosis, mitral regurgitation, and aortic stenosis.Each category contains 200 audio files, with recordings averaging nearly 20,000 samples or 2.5 seconds at 8,000 Hz.
  • Heart sounds: Normal phonocardiograms contain S1 and S2 heartbeats separated by silent systolic and diastolic intervals.S1 occurs at the beginning of systole, while S2 occurs at the beginning of diastole.
  • Valvular conditions: The four abnormal classes have characteristic sound findings, including clicks or late-systolic murmurs in mitral valve prolapse, diastolic murmurs in mitral stenosis, and high-pitched diamond-shaped systolic murmurs in aortic stenosis.Mitral regurgitation is associated with murmurs beginning after S1 and increasing toward S2.
  • Preprocessing: Preprocessing clips or zero-pads recordings to 214 samples, downsamples them by a factor of 23 to L = 211 at fs = 1000 Hz, and standardizes each signal.Standardization sets the mean to 0 and the standard deviation to 1.
  • Signal representations: Figure 1 compares time-domain signals, Fourier transforms, and spectrograms for a healthy recording and four cardiovascular-disease recordings.The plotted signals use a sampling frequency of 1000 Hz and a 2-second duration.

3. Related works

Related work develops time-frequency representations and sparse atom-based modeling for PCG analysis, while this paper combines Gabor features, elastic net regularization, and CNN-LSTM classification for five valvular conditions.

  • Time-frequency analysis: PCG signals are nonstationary, so their diagnostic features include time-frequency changes across heartbeats, murmurs, and cardiac intervals.These changes motivate time-frequency analysis for heart-sound classification.
  • Time-frequency analysis: STFT analysis has a fixed-window resolution trade-off: shorter windows improve temporal tracking but reduce frequency resolution, whereas longer windows do the reverse.Wavelet transforms address this trade-off with frequency-dependent time and frequency resolution.
  • Atom-based representations: Gabor dictionaries represent signals with localized Gaussian atoms formed through scaling, translation, and frequency modulation, supporting analysis of both transient beats and longer murmurs.Earlier atom designs included damped sinusoids and Gaussian functions with varying duration.
  • Prior classification methods: Matching pursuit with Gabor dictionaries has extracted PCG features for valvular-disease classification using conventional MLP and SVM classifiers.These approaches represent prior dictionary-based classification work.
  • Position of this work: The paper integrates Gabor dictionaries with elastic net and replaces cycle segmentation and conventional classifiers with whole-signal CNN-LSTM modeling.The proposed framework estimates features from the entire PCG signal rather than requiring an additional cardiac-cycle segmentation step.
  • Position of this work: The proposed contribution combines dictionary-derived features with CNN and LSTM architectures, a combination the authors report had not previously been proposed for PCG classification.The coefficient vector is reshaped into a time-frequency matrix for deep-learning input.

4. Method

The method models whole PCG signals with overcomplete Gabor dictionaries and elastic net regularization, then reshapes estimated coefficients into time-frequency matrices for CNN-LSTM classification. Dictionary resolution is varied because increasing scale improves spectral resolution while reducing temporal resolution.

  • System overview: The system has feature-computation and classification stages, modeling each PCG signal as a projection onto an overcomplete dictionary of normalized atoms.The dictionary D has signal length L, more atoms than signal dimensions M, and L < M.
  • Regularized modeling: Elastic net replaces matching pursuit to control coefficient sparsity and handle multicollinearity through a regularization parameter.The parameter is varied to estimate coefficient vectors for dictionaries with different time-frequency resolutions.
  • Deep classification: The classification stage uses CNN and LSTM layers with time-frequency matrices obtained by reshaping coefficient vectors and applying mathematical functions to their entries.This matrix representation is used to discriminate five heart valvular conditions.
  • Gabor dictionary: The Gabor dictionary uses Gaussian-window atoms that are scaled, translated, and frequency-modulated.The Gaussian window length matches the signal length, and the construction maintains 50% atom overlap.
  • Gabor dictionary: For N = 11, each dictionary Dj contains 2^N+2 atoms arranged in a 2^N × 2^N+2 matrix, with j controlling dictionary variables and atom resolution.The phase shift is fixed at θ = 0, and the dictionary is constructed for each permissible j.
  • Resolution trade-off: Increasing β improves spectral resolution at the cost of temporal resolution because broader Gaussian atoms have narrower frequency bandwidth.Figure 3 illustrates this trade-off for three j values at N = 7, ω = π/2, and τ = 26.

4.2. Elastic net regularization of linear models

The paper uses elastic net regularization to balance sparsity and multicollinearity when fitting PCG signals with Gabor dictionaries. Very high-time, low-frequency resolutions provide the strongest approximation and sparsity properties across regularization settings.

  • Regularization: Elastic net combines l1 sparsity with squared l2 regularization to select atoms while handling multicollinearity.The parameter α controls the balance between these two penalties; α = 0 gives ridge regression and α = 1 gives LASSO.
  • Model assessment: The fitting problem is optimized with ADMM, and approximation error, coefficient energy, and nonzero-atom counts are used to assess dictionary and regularization choices.These quantities correspond to Figures 4, 5, and 6.
  • Approximation behavior: Very small approximation error occurs at β = 2^1 and β = 2^8, 2^9, and 2^10 for all α values.For β between 2^3 and 2^7, LASSO produces smaller error than ridge regression, with error decreasing as α increases.
  • Approximation behavior: Coefficient-vector energy follows the approximation-error trend across β but changes inversely with α.The energy is evaluated over all 1000 recordings from the five heart conditions.
  • Sparsity: Very high-time, low-frequency dictionaries at β = 2^1 yield highly sparse coefficient vectors because only a few atoms are needed to approximate PCG signals.Healthy-heart signals use fewer atoms than valvular-condition signals, and the number decreases as α increases.

4.3. Feature computation and representation

The feature pipeline transforms regularized coefficient vectors and reshapes them into time-frequency matrices for CNN-based classification. The weighted logarithmic transformation is identified as preferable to linear and magnitude-squared alternatives.

  • Feature representation: Coefficient vectors are standardized, normalized to [−1, 1], transformed into feature vectors, and reshaped into two-dimensional time-frequency matrices.The matrix B_j,α has size 2^(j+1) × 2^(N−j+1).
  • Feature transformation: The weighted logarithmic function is used to construct b_j,α from the coefficient vector a_j,α.The paper contrasts this option with raw coefficients and magnitude-squared features.
  • Feature transformation: Weighted logarithmic features provide better CNN-LSTM classification performance than linear and magnitude-squared feature calculations.Figure 7 compares the three transformation functions over inputs between −1 and 1.
  • Interpretation: The weighted logarithmic transformation can be interpreted using the absolute coefficient values as a probability distribution and the resulting quantity as weighted entropy over temporal neighbors.This interpretation connects the feature calculation to locally weighted coefficient structure.
  • Feature representation: The feature matrix reshaping enables CNN models to process neighboring coefficient entries as two-dimensional imagery-like structure.The original vector has length 2^(N+2), whereas the reshaped matrix preserves the time-frequency arrangement.

4.4. CNN-LSTM classification network

The proposed classification network combines one-dimensional and two-dimensional CNN processing with an LSTM layer. It is trained using either SGDM or ADAM, with convolutional dimensions adapted to the feature-matrix resolution.

  • Architecture: The CNN-LSTM architecture contains 1D CNN, 2D CNN, and LSTM layers followed by dense and softmax classification layers.The LSTM receives flattened structures extracted by the 2D CNN.
  • Architecture: The network uses convolutional layers to extract hierarchical patterns before recurrent processing by the LSTM.The combined CNN-LSTM design is motivated by the complementary roles of convolutional and recurrent layers.
  • Optimization: Training experiments compare stochastic gradient descent with momentum (SGDM) and adaptive moment optimization (ADAM).SGDM uses current and previous gradients to accelerate movement toward a loss minimum, whereas ADAM uses adaptive updates.
  • Resolution handling: Convolutional filter and stride sizes are adjusted to match feature matrices whose shapes vary with time-frequency resolution.Small j produces wide, short matrices, while large j produces narrow, tall matrices.

4.5. Experiments

The experiments evaluate the proposed classification models on 1000 feature matrices using repeated disjoint training and testing splits. The implementation parameters are summarized in Table 2.

  • Protocol: The experiments split 675 feature matrices for training and 325 for testing, repeating each model evaluation 100 times with random selections.Training and testing sets are disjoint in every experiment.
  • Implementation: Table 2 reports the implementation parameters of the CNN-LSTM classification model.The table accompanies the resolution-dependent architecture design.

4.6. Performance metrics

The paper evaluates classification using accuracy, precision, recall, specificity, and F1 score derived from confusion-matrix counts. These metrics quantify correct predictions and class-specific errors for the five heart-valvular conditions.

  • Precision measures correctly classified positive samples among all samples classified as positive.
  • Recall measures correctly classified positive samples among all actual positive samples.
  • Accuracy is the ratio of correct predictions to all predictions.
  • Specificity measures correctly classified negative samples among all actual negative samples.
  • F1 score is the harmonic mean of precision and recall, combining both metrics into one measure.

4.7. Classification results

Classification performance depends on dictionary resolution, elastic-net regularization, architecture, and optimizer. The best result is obtained with the 1D+2D CNN-LSTM architecture trained using ADAM and the β = 21, α = 0.1 configuration.

  • β = 21 generally provides the best accuracy, corresponding to Gabor atoms with high-time and low-frequency resolution.
  • α = 0 often performs best, while α = 0.1 provides comparable or better accuracy at some resolutions, including β = 21 with the second architecture and ADAM.
  • Better classification accuracy is associated with low-energy coefficient vectors and low-energy approximation residuals.
  • 98.95% accuracy is achieved by the 1D+2D CNN-LSTM architecture with β = 21 and α = 0.1.This configuration uses the Gabor dictionary and ADAM training.
  • The four additional classification metrics follow accuracy trends, and their standard deviations decrease as the metric values increase.
  • The best model discriminates healthy hearts most accurately, followed by aortic stenosis, across 100 experiments.

4.8. Discussion

The proposed feature and network configurations outperform the compared baselines on five-class PCG classification. Results favor high-time-resolution Gabor representations, selected sparsity, ADAM for larger networks, and SGDM for smaller networks.

  • High-time-resolution Gabor dictionaries and α = 0.1 improve classification accuracy in the reported optimal combinations.
  • Larger networks perform better with ADAM, whereas SGDM is suitable for smaller networks.
  • 30.92% relative classification error reduction is achieved versus the raw-PCG baseline.
  • 22.22% relative classification error reduction is achieved versus the VMD-based baseline.The authors state that Gabor-dictionary and elastic-net features are more suitable for modeling PCG signals than VMD features.

5. Conclusion and future work

The paper models heart sounds with Gabor-atom fitting and elastic-net regularization, then derives sparse time-frequency matrices for deep-learning classification. Across two architectures and two optimizers, the more complex architecture reaches 98.95% accuracy, while future work proposes complex dictionaries and transfer learning.

  • Elastic-net fitting with an overcomplete Gabor dictionary produces sparse time-frequency feature matrices for heart sound signals.The matrices are intended to capture time-frequency structure while preserving sparsity.
  • The study evaluates 1D CNN-LSTM and 1D+2D CNN-LSTM architectures trained with SGDM and ADAM on a public heart-sound database.
  • 98.95% accuracy is obtained with the second architecture using high-time and low-frequency resolution matrices.The second architecture is more complex but provides better performance.
  • The authors suggest that deeper architectures can better capture complex structures in time-frequency matrices and improve classification accuracy.
  • Future work will test complex Gabor dictionaries and pre-trained deep models with transfer learning.The authors expect complex atoms to produce sparser approximations and plan comparisons with STFT-based representations.
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