Inquiry about reconstruction loss

[ken-take-it-so-so]

Hello.
I have a question about below formula. How did you derive this?
https://github.com/yoyololicon/diffwave-sr/blob/cab5c4e330c8b6d8b329a6c85812a7328fe3431c/loss.py#L20
In this research, audio data is used and is it continuous?
I would appreciate your cooperation.

[yoyolicoris]

Hi @ken-take-it-so-so , good question.

Here, ll means the term $\mathbb{E}_{q(\mathbf{z}_1|\mathbf{x})}[\log p(\mathbf{x}|\mathbf{z}_1)]$.
We estimate it by sampling $\mathbf{z}_1 \sim \mathcal{N}(\alpha_1 \mathbf{x}, \sigma_1^2 \mathbf{I})$ and parameterize $p(\mathbf{x}|\mathbf{z}_1)$ as $\mathcal{N}(\frac{\mathbf{z}_1}{ \alpha_1}, \frac{\sigma_1^2}{\alpha_1^2} \mathbf{I})$.
Thus, ll $= - \frac{N}{2} \log (\frac{\sigma_1^2}{\alpha_1^2}) - \frac{\alpha_1^2}{2 \sigma_1^2} (\mathbf{x} - \frac{\mathbf{z}_1}{ \alpha_1})^T(\mathbf{x} - \frac{\mathbf{z}_1}{ \alpha_1}) + \mathcal{C}$ where $N$ is the signal length and $\mathcal{C}$ is just a constant.
Using the fact that $\frac{\mathbf{z}_1}{ \alpha_1} \sim \mathcal{N}(\mathbf{x}, \frac{\sigma_1^2}{\alpha_1^2} \mathbf{I})$, we approximate ll as $- \frac{N}{2} \log (\frac{\sigma_1^2}{\alpha_1^2}) - \frac{N}{2} + \mathcal{C} = \frac{N}{2} \delta_{max}- \frac{N}{2} + \mathcal{C}$ assuming $N$ is large enough.

In this research, audio data is used and is it continuous?

Yes, we treat audio as continuous data.