This paper can be thought as proposing a variational autoencoder applied to a form of meta-learning, i.e. where the input is not a single input but a dataset of inputs. For this, in addition to having to learn an approximate inference network over the latent variable $z_i$ for each input $x_i$ in an input dataset $D$, approximate inference is also learned over a latent variable $c$ that is global to the dataset $D$. By using Gaussian distributions for $z_i$ and $c$, the reparametrization trick can be used to train the variational autoencoder.

The generative model factorizes as

$p(D=(x_1,\dots,x_N), (z_1,\dots,z_N), c) = p(c) \prod_i p(z_i|c) p(x_i|z_i,c)$

and learning is based on the following variational posterior decomposition:

$q((z_1,\dots,z_N), c|D=(x_1,\dots,x_N)) = q(c|D) \prod_i q(z_i|x_i,c)$.

Moreover, latent variable $z_i$ is decomposed into multiple ($L$) layers $z_i = (z_{i,1}, \dots, z_{i,L})$. Each layer in the generative model is directly connected to the input. The layers are generated from $z_{i,L}$ to $z_{i,1}$, each layer being conditioned on the previous (see Figure 1 *Right* for the graphical model), with the approximate posterior following a similar decomposition.

The architecture for the approximate inference network $q(c|D)$ first maps all inputs $x_i\in D$ into a vector representation, then performs mean pooling of these representations to obtain a single vector, followed by a few more layers to produce the parameters of the Gaussian distribution over $c$. 

Training is performed by stochastic gradient descent, over minibatches of datasets (i.e. multiple sets $D$).

The model has multiple applications, explored in the experiments. One is of summarizing a dataset $D$ into a smaller subset $S\in D$. This is done by initializing $S\leftarrow D$ and greedily removing elements of $S$, each time minimizing the KL divergence between $q(c|D)$ and $q(c|S)$ (see the experiments on a synthetic Spatial MNIST problem of section 5.3).

Another application is few-shot classification, where very few examples of a number of classes are given, and a new test example $x'$ must be assigned to one of these classes. Classification is performed by treating the small set of examples of each class $k$ as its own dataset $D_k$. Then, test example $x$ is classified into class $k$ for which the KL divergence between $q(c|x')$ and $q(c|D_k)$ is smallest. Positive results are reported when training on OMNIGLOT classes and testing on either the MNIST classes or unseen OMNIGLOT datasets, when compared to a 1-nearest neighbor classifier based on the raw input or on a representation learned by a regular autoencoder.

Finally, another application is that of generating new samples from an input dataset of examples. The approximate posterior is used to compute $q(c|D)$. Then, $c$ is assigned to its posterior mean, from which a value for the hidden layers $z$ and finally a sample $x$ can be generated. It is shown that this procedure produces convincing samples that are visually similar from those in the input set $D$.

**My two cents**

Another really nice example of deep learning applied to a form of meta-learning, i.e. learning a model that is trained to take *new* datasets as input and generalize even if confronted to datasets coming from an unseen data distribution. I'm particularly impressed by the many tasks explored successfully with the same approach: few-shot classification and generative sampling, as well as a form of summarization (though this last probably isn't really meta-learning). Overall, the approach is quite elegant and appealing.

The very simple, synthetic experiments of section 5.1 and 5.2 are also interesting. Section 5.2 presents the notion of a *prior-interpolation layer*, which is well motivated but seems to be used only in that section. I wonder how important it is, outside of the specific case of section 5.2.

Overall, very excited by this work, which further explores the theme of meta-learning in an interesting way.