Paper:
Unsupervised feature selection involves finding a small number of highly informative features, in the absence of a specific supervised learning task. Selecting a small number of features is an important problem in many scientific domains with high-dimensional observations. Here, we propose the restricted autoencoder (RAE) framework for selecting features that can accurately reconstruct the rest of the features. We justify our approach through a novel proof that the reconstruction ability of a set of features bounds its performance in downstream supervised learning tasks. Based on this theory, we present a learning algorithm for RAEs that iteratively eliminates features using learned per-feature corruption rates. We apply the RAE framework to two high-dimensional biological datasets-single cell RNA sequencing and microarray gene expression data, which pose important problems in cell biology and precision medicine-and demonstrate that RAEs outperform nine baseline methods, often by a large margin.

Review:
Authors of this paper propose the restricted autoencoder (RAE) framework for selecting features that can accurately reconstruct the rest of features. Authors justify the proposed method via the proof that the reconstruction ability of a set of features bounds its performance in downstream supervised learning tasks. The algorithm that iteratively eliminates features using learned per-feature corruption rates is proposed.
The fundamental of this paper is built on the argument that the optimal approach is to select a set of features that can accurately reconstruct all the remaining features for the settings where they will be used in downstream prediction tasks. Authors studied the performance losses of linear and nonlinear models by using the defined imputation losses.  Some concerns are listed as follows:
1. the theorem based on strong assumptions that all learned models are optimal. The applicability of the theoretical results to general prediction model is still questionable. 
2. only the prediction problems of least square (linear or nonlinear) are studied. It is not equivalent to the downstream supervised learning tasks. It is just a special case study.
3. It is unclear how to get the conclusion from Theorem 1 that the linear imputation loss is equal to the sum of eigenvalues. Please clarify it in details.
The RFE-like algorithm is used to solve (7). However, the sensitivity measures used in Algorithm 1 seems to take the different optimization problems since additional regularization terms are added. This is different from RFE where a single SVM optimization problem is used and the ranking score is solely based on the learned SVM classifier. The discussion on the inconsistency of learning h_{\theta} and the sensitivity measures could be interesting. 
In the experiments, authors did not mention the parameter settings of all compared methods. It is known that the unsupervised feature selection methods incorporate priors with usually various parameters. For fair comparisons, it is better to report the properly tuned results since these parameters are often data-dependent.