An explanation for the source code of finding the alignment path in GlowTTS?

[alifarrokh]

Hi. I'm reading the source code of GlowTTS model for educational purposes. One of the sections that I can't really understand is where we try to find the alignment path using Monotonic Alignment Search in the training phase. Could anyone please explain me the following lines of code?

with torch.no_grad():
    x_s_sq_r = torch.exp(-2 * x_logs)
    logp1 = torch.sum(-0.5 * math.log(2 * math.pi) - x_logs, [1]).unsqueeze(-1) # [b, t, 1]
    logp2 = torch.matmul(x_s_sq_r.transpose(1,2), -0.5 * (z ** 2)) # [b, t, d] x [b, d, t'] = [b, t, t']
    logp3 = torch.matmul((x_m * x_s_sq_r).transpose(1,2), z) # [b, t, d] x [b, d, t'] = [b, t, t']
    logp4 = torch.sum(-0.5 * (x_m ** 2) * x_s_sq_r, [1]).unsqueeze(-1) # [b, t, 1]
    logp = logp1 + logp2 + logp3 + logp4 # [b, t, t']
attn = monotonic_align.maximum_path(logp, attn_mask.squeeze(1)).unsqueeze(1).detach()

Thanks in advance.

[shivammehta25]

Hi! I am assuming this late you might not need this! But I am still writing it for the future in case someone else also encounters this!

    x_s_sq_r = torch.exp(-2 * x_logs)
    logp1 = torch.sum(-0.5 * math.log(2 * math.pi) - x_logs, [1]).unsqueeze(-1) # [b, t, 1]
    logp2 = torch.matmul(x_s_sq_r.transpose(1,2), -0.5 * (z ** 2)) # [b, t, d] x [b, d, t'] = [b, t, t']
    logp3 = torch.matmul((x_m * x_s_sq_r).transpose(1,2), z) # [b, t, d] x [b, d, t'] = [b, t, t']
    logp4 = torch.sum(-0.5 * (x_m ** 2) * x_s_sq_r, [1]).unsqueeze(-1) # [b, t, 1]
    logp = logp1 + logp2 + logp3 + logp4 # [b, t, t']

It is the log-likelihood computation from a gaussian centred at (x_m, x_logs).

And in

attn = monotonic_align.maximum_path(logp, attn_mask.squeeze(1)).unsqueeze(1).detach()

They find a Viterbi approximation (using dynamic programming) over the data likelihood to maximise it further.

Hope this helps!

[xiaozhah]

For a Gaussian distribution $\mathcal{N}(\mu, \sigma^2)$, the probability density function is:

$$f(x; \mu, \sigma) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$

Taking the logarithm, we get:

$$\log f(x; \mu, \sigma) = -\frac{1}{2}\log(2\pi\sigma^2) - \frac{(x-\mu)^2}{2\sigma^2}$$

Now, let's see how each term in the code corresponds to the above formula:

0. x_logs corresponds to $\log\sigma$, thus x_s_sq_r = torch.exp(-2 * x_logs) corresponds to $\frac{1}{\sigma^2}$.

1. logp1 = torch.sum(-0.5 * math.log(2 * math.pi) - x_logs, [1]).unsqueeze(-1) # [b, t, 1]:
   This term corresponds to $-\frac{1}{2}\log(2\pi\sigma^2)$, which is the logarithm of the normalization constant of the Gaussian distribution.

2. logp2 = torch.matmul(x_s_sq_r.transpose(1,2), -0.5 * (z ** 2)) # [b, t, d] x [b, d, t'] = [b, t, t']:
   This term corresponds to $-\frac{x^2}{2\sigma^2}$ in $-\frac{(x-\mu)^2}{2\sigma^2}$.

3. logp3 = torch.matmul((x_m * x_s_sq_r).transpose(1,2), z) # [b, t, d] x [b, d, t'] = [b, t, t']:
   This term corresponds to $\frac{x\mu}{\sigma^2}$ in $-\frac{(x-\mu)^2}{2\sigma^2}$.

4. logp4 = torch.sum(-0.5 * (x_m ** 2) * x_s_sq_r, [1]).unsqueeze(-1) # [b, t, 1]:
   This term corresponds to $-\frac{\mu^2}{2\sigma^2}$ in $-\frac{(x-\mu)^2}{2\sigma^2}$.

Adding these four terms together, we get:
logp = logp1 + logp2 + logp3 + logp4 # [b, t, t'] corresponds to $\log f(x; \mu, \sigma)$.