Fig 1.
Examples of frequency-modulated sine wave (FM) synthetic time series.
Top: pure and noisy FM-sine wave decomposition. Gaussian noise and frequency modulation are linearly added to decaying sine wave carrier. Bottom: random selection of noisy-input (blue) and pure-target (black) samples, illustrating the effect of the random latent parameter selection.
Fig 2.
Unified DNN architecture and loss description.
The Encoder produces a reduced representation of the input noisy signals. The Encoder output is passed to the Regressor, which outputs the latent parameters’ prediction. The Encoder and Regressor outputs are passed to the Decoder, which produces a noiseless prediction of the inputs. The Regressor and Decoder outputs are used to compute the regression and denoising losses, MSEreg and MSEdec, respectively. The loss used during backpropagation is a weighted sum of MSEreg and MSEdec using a bias parameter β.
Fig 3.
Comparison of DNN post-training performance to LS-fits with true latent-parameters initial guesses for 1000 random monochromatic, decaying sine waves from the test set (unseen during training).
The denoising (MSEdec, top) and latent-parameters relative regression losses (MSEreg, bottom) are sorted by increasing noise levels. The DNN was trained on monochromatic sine waves samples. MSEreg is the MSE from the true latent parameters to the predicted latent parameters. For the DNN, MSEdec is the MSE from the true noiseless signal to the Decoder noiseless-signal prediction. For the LS-fits, MSEdec is computed similarly, but the noiseless-signal prediction is generated by inputting the predicted latent parameters in the noiseless data-generating process. The LS-fit with true initial guesses vastly outperforms the DNN for low-noise signals but both systems reach similar performance for high-noise.
Fig 4.
DNN latent-parameters predictions used as initial guesses for DNN-assisted fits.
Comparison to LS-fit with true initial guesses. Top: The denoising (MSEdec) and latent-parameters relative regression losses (MSEreg) are sorted by increasing noise levels. See Fig 3 for MSEreg and MSEdec computation methods. Bottom: Phase and carrier frequency predictions for the DNN-assisted fits and LS-fits. Both methods converge to the same losses and predictions for over 99% of the samples. The DNN and data employed here are identical as in Fig 3.
Fig 5.
Performance comparison of the specialized DNN (trained on AM-sine waves, tasked to denoise signals and recover all latent parameters of AM-sine waves) and of the partial DNN (trained on monochromatic, AM- and FM-sine waves, tasked to denoise signals and recover the carrier frequency, phase, coherence time and noise level only).
Top: Randomly selected example of noisy input AM-signal, alongside specialized- and partial-DNN denoised and latent predictions. Bottom: Individual latent parameters and signal denoising root mean squared error (RMSE), averaged over the whole AM-sinewave test set (100′000 samples) for both DNNs.
Fig 6.
FM-sine waves test-data prediction errors: Total weighted loss βMSE, denoising loss MSEdec and regression loss MSEreg for varying values of β.
Setting β = 0 or β = 1 fully biases training toward one of the two tasks, preventing the negatively-biased tasks to reach sufficient performance. Middle-range values enable both tasks to be learned simultaneously.