This is a experimental project that build a WaveNet-based vocoder which converts mel-spectrogram to raw wave in parallel, inspired by parallel WaveNet paper.

Thanks to the structure of the inverse autoregressive flow(IAF), one of non-autoregressive models, we're able to generate sequential data in parallel. Because to optimize the IAF model directly in maximum likelihood estimation(MLE) fails to find appropriate optimum so that generalization is hard, the paper introduces an alternative method a.k.a probability density distillation.

It trains the original WaveNet(teacher) beforehand and then optimize the IAF model(student) to model similar output probability by minimizing KL divergence between two probabilities. If we think a bit more, we could conclude that the 'autoregressive' loss is forced to the IAF model, which is a non-autoregressive model by itself. That means the 'autoregressive' constraint is still the key when it comes to training sequence generation models.

To optimize the model, I've only tried the simpler one (MLE) because I was curious about the motivation of to devise the alternative one. In conclusion, to optimize the IAF model without autoregressive constraint was almost not feasible in my case. Please refer to the output samples here.

It consists of 4 IAF layers that transform one probability to other probability in a 'inverse autoregressive' way.

That indicates where we assume that input u=(u_1, ..., u_n) and output x=(x_1, ..., x_n) are multivariate, x_t is computed only by u_1, ..., u_t. This is identical to say that Jacobian matrix dx/du is triangular. In short, this model ensures that output data in a current timestep is only conditioned on latent values of all previous timesteps.

In a timestep of a IAF layer, input that follows logistic distribution linearly transforms to other logistic distribution by scaling and shifting. Mean and variance of each layer are computed in a inverse autoregressive way. To compute these, I appled WaveNet model as used in the paper as well so that the total WaveNet models are 8. (surely, it's possible mean and variable to share weights in each layer so that the number of WaveNet is 4.)

In each layer, mel-spectrogram values are conditioned on. As the timesteps of the mel-spectrogram are shorter than the raw wave, it's necessary to expand it to have the same length as the raw wave's one. As a result, two experiments were conducted : 1. use transposed convolution and 2. repeat mel-spectrum multiple times. In my case, the latter was slightly better in quality.

To optimize the model, I use MLE and this is identical to using L1 loss between prediction and ground truth of raw wave. The following is one of the training loss curves I got.

I kept track of exponential moving averages (EMA) of all weights and used them in generation phase.

The dataset I tested on is a bunch of audio files uttered by a female speaker(slt) in CMU arctic dataset. Please refer to the output samples here.

Why optimizing the IAF model directly harder?

• I think that entropy of the probability P(x_t | z_<=t) (non-autogressive) is much larger than that of P(x_t | x_<t) (autoregressive). Since the modality of the latter probability is more deterministic that the former one, the optimization could be done more effectively.