WaveNet

WaveNet is an audio generative model based on the PixelCNN architecture. In order to deal with long-range temporal dependencies needed for raw audio generation, architectures are developed based on dilated causal convolutions, which exhibit very large receptive fields.

The joint probability of a waveform $\vec{x} = { x_1, \dots, x_T }$ is factorised as a product of conditional probabilities as follows:

$$p\left(\vec{x}\right) = \prod_{t=1}^{T} p\left(x_t \mid x_1, \dots ,x_{t-1}\right)$$

Each audio sample $x_t$ is therefore conditioned on the samples at all previous timesteps.