# How to achieve variational autoencoder (VAE) with unrestricted input?

For a normal VAE an input and a reconstruction with values in the range of $$[0, 1]$$ are expected. This is necessary since the log loss only makes sense for this range. If the input is not within $$[0, 1]$$ it is common to normalize it with the help of statistics (for example mean-std normalization + logistic function). That is not what I want. I actually want the VAE to have an unrestricted input and reconstruction.

How to formulate reconstruction loss and kullback-leibler (KL) divergence in this unrestricted setting? The code $$z$$ is sampled from a normal distribution that is based on the mean $$m$$ and standard deviation $$s$$ given by a dense neural network layer.

My current thoughts:

The KL divergence is supposed to push the mean $$m$$ to $$0$$ and the standard deviation $$s$$ to $$1$$:

$$m^2 + s^2 - log(s^2) - 1$$ But that doesn't seem to make much sense if as a result the mean $$m$$ and std $$s$$ of the VAE encoding is not in the range $$[0, 1]$$. In my current setup the range is more like $$[-9, 8]$$ for the mean and $$[-2, -4]$$ for the std. So how to go about the KL divergence?

Will I need to restrict the mean and std values? But what are the effects and implications of that? How much would I restrict the VAE or rather how unrestricted is the VAE with that?

The reconstruction loss in a normal VAE punishes differences between input $$x$$ and reconstruction $$y$$ in the higher and lower end of the $$[0, 1]$$ range:

$$-(x * \log(y) + (1 - x) * \log(1- y))$$ For an unrestricted reconstruction loss the square difference could be used but how to combine / weight it against the KL divergence term?

Or are there other more useful reconstruction loss formulations? Are there loss formulations that follow a similar goal as the log loss but for unrestricted values (centered around 0) - and would that even make sense?

For clarification a simple VAE model that I'm using here:

tfd = tf.contrib.distributions

code_size = 5
input_size = input_tensor.shape[-1].value # input_tensor shape: [batch, values]
net = input_tensor

net = tf.layers.dense(net, units=input_size, activation=tf.nn.leaky_relu, name='Encoding_Layer_1')
net = tf.layers.dense(net, units=code_size * 2, activation=None, name='Encoding_Layer_2')

posterior = tfd.MultivariateNormalDiagWithSoftplusScale(net[:, :code_size], net[:, code_size:])

code = posterior.mean()
net = posterior.sample()

net = tf.layers.dense(net, units=input_size, activation=tf.nn.leaky_relu, name='Decoding_Layer_1')
net = tf.layers.dense(net, units=input_size, activation=None, name='Decoding_Layer_2')

reconstruction = net

image_loss = tf.reduce_mean(tf.square(input_tensor - reconstruction), 1)

prior = tfd.MultivariateNormalDiag(tf.zeros(code_size), tf.ones(code_size))
divergence = tfd.kl_divergence(posterior, prior)

loss = image_loss + 1e-4 * divergence

• May I know why you do this: 1e-4 * divergence? – Jacky Guo Apr 2 '19 at 21:52
• The divergence term has way higher values than the image loss term. The 1e-4 is balancing both terms. – Spenhouet Apr 2 '19 at 22:15

If you care about modeling unrestricted $$x$$, you shouldn't worry about $$z$$'s distribution much, so the KL term is not a problem here.

Instead, just replace your Bernoulli decoder $$p(x|z)$$ with a Gaussian one:

$$p(x|z) = \mathcal{N}(\mu(z), \sigma(z))$$

That is, multivariate normal distribution with mean and scale generated by a neural network from $$z$$.

• I get the idea of the guassian decoder but why shouldn't I worry about $z$'s distribution? Isn't that essential for a VAE? Or do you mean I should remove the KL term? If not, than the actual value range for the KL divergence as a loss definitely is important for the standard KL divergence formula to make sense and with unrestricted $x$ that is not the case as I see it. Would you mind further explaining your position? – Spenhouet Mar 6 '19 at 14:55
• @Spen, z is essential, but it doesn't care about your observations' nature, whether they're bounded or not. It absracts representational features away, leaving only those that are importance to discern different observations. Removing KL would only break the VAE. – Artem Sobolev Mar 6 '19 at 19:53
• But the KL divergence formula doesn't make any sense outside of a [0,1] range. Mathematically that seems very significant to me or do you disagree? – Spenhouet Mar 6 '19 at 20:30
• What? The KL between two Gaussian distributions is valid for any mean and any (positive) scale. Let me restate the claim once again: the KL operates in latent space, and it does not care about the observations space, it only cares about observations' features. – Artem Sobolev Mar 7 '19 at 8:27
• What Artem is trying to say is that your embedding space is not a probability distribution and hence is not to be bound to [0,1], the KL divergence is here to serve as a regularization term for your embedding space, and allow your network to learn without having increasingly large values, destroying any meaningful information. – Arnaud De Broissia Jul 19 '19 at 8:25