Deep Subspace Clustering Networks
We present a novel deep neural network architecture for unsupervised subspace clustering. This architecture is built upon deep auto-encoders, which non-linearly map the input data into a latent space. Our key idea is to introduce a novel self-expressive layer between the encoder and the decoder to mimic the "selfexpressiveness" property that has proven effective in traditional subspace clustering. Being differentiable, our new self-expressive layer provides a simple but effective way to learn pairwise affinities between all data points through a standard backpropagation procedure. Being nonlinear, our neural-network based method is able to cluster data points having complex (often nonlinear) structures. We further propose pre-training and fine-tuning strategies that let us effectively learn the parameters of our subspace clustering networks. Our experiments show that our method significantly outperforms the state-of-the-art unsupervised subspace clustering techniques.
The paper addresses the problem of subspace clustering, i.e., separating a collection of data points lying in a union of subspaces according to underlying subspaces, using deep neural networks. To do so, the paper builds on the sparse subspace clustering idea: among all possible representations of a data point as a combination of other points in the dataset, the representation that uses the minimum number of points, corresponds to points from the same subspace. In other words, SSC uses the idea that for a data matrix $X$, a sparse solution of $X = X C$ (subject to $diag(C) = 0$) represents each point as a combination of a few other points from the same subspace. The paper proposes a deep neural network to transform the data into a new representation $Z = f_W(X)$ for which one searches for a sparse representation of $Z = Z C$, with the hope to learn more effective representations of data for clustering. To achieve this, the paper uses an auto-encoder scheme, where the middle hidden layer outputs are used as $Z$. To enforce $Z - Z C = 0$, the paper repeats the middle layer, with the hope that these two consecutive layers obtain same activations. If so, the weights  between the two layers would correspond to coefficient matrix $C$ which will be used then, similar to existing methods, to build an affinity matrix on data and to obtain clustering. The paper demonstrates experiments on Extended YaleB  and ORL face datasets as well as COIL20/100.
The reviewer believes that the paper is generally well-written and easy to read. The paper also takes a sufficiently interesting approach to connect subspace clustering with deep learning. On the other hand, there are several issues with the technical approach and the experiments that limit the novelty and correctness of the paper.
1) The reviewer believes that the idea of the paper is similar to the one in [36], which uses an auto-encoder scheme, taking the middle layer as $Z$. While [36] fixes the $C$ and solves for $Z$, the proposed method repeats the middle layer to also solve for $C$. Despite this similarity, these is only a sentence in the paper (line 82) referring to this work and the experiments do not compare against [36]. Thus there is a need to clearly discuss advantages and disadvantages with respect to [36] and compare against it in the experiments.
2) It is not clear to the reviewer how the paper enforces that the activations of the middle and the middle plus one layer must be the same. While, clearly one can initialize $C$ so that the consecutive activations will be the same, one the back propagation is done and the weights of the auto-encoder change, the activations of middle layer and the one after it will not be the same (unless the $\lambda_2$ is very large and the network achieves it global minima, which is generally not the case). As a result, in the next iteration, the network will minimize $Z - Y C$ for different $Y$ and $Z$, which is not desired. 
3) One of the drawbacks of the proposed approach is the fact that the number of weights between the middle layers of the network is $N^2$, given $N$ data points. Thus, it seems that the method will not be scalable to very large datasets.
4) In the experiments (line 243), the paper discusses training using mini-batches. Notice that while using mini-batches makes sense in the context of recognition, where each sample can be applied separately to the network, using mini-batches does not make sense in the context of clustering discussed in the paper. Using a mini-batch of size $M < N$, how does the paper train a neural network between two layers, each with $N$ nodes? In other words, it is not clear how the mini-batch instances and in what order will be input the the network.