Paper ID: 499 Title: Sparse Parameter Recovery from Aggregated Data Review #1 ===== Summary of the paper (Summarize the main claims/contributions of the paper.): The paper studies to estimate the parameter of a linear model when the data are not collected in one time. The paper then gives some analysis when the empirical moments are computed from a sufficiently large number of samples. Clarity - Justification: For me the paper is a bit hard to follow. Significance - Justification: I did not catch the significance of the paper. Detailed comments. (Explain the basis for your ratings while providing constructive feedback.): The paper studies to estimate the parameter of a linear model when the data are not collected in one time. The paper then gives some analysis when the empirical moments are computed from a sufficiently large number of samples. For me the paper is a bit hard to follow. Followings are my major concerns and I would like the authors to clarify. 1. What is real meaning of aggregated data in this paper? Actually ‘Aggregated data’ is a very broad and general term. It could be lots of the representing forms in problems. From the paper, it seems that the paper uses ‘aggregated data’ to emphasize the importance of the problem, which is also a reason that makes me very hard to judge the significance/contribution of the paper without the clearness of ‘the real problem’ studied in the paper. 2. The paper claims that aggregated data is under semi-supervision scenario and highlights this issue in front of other issues. However, from the main technique part, i do not see why semi-supervision is an important issue and how does the proposed approach solve/avoid this issue. In summary, after reading the paper, to be honestly, I did not catch the significance of the studied problem (because the definition or description is too broad in the paper) and the importance of the contribution (because the key issue of the studied problem sounds not clearly justified from the paper). ===== Review #2 ===== Summary of the paper (Summarize the main claims/contributions of the paper.): The authors investigated the problem of parameter recovery for sparse linear models with aggregated data (in the form of first order moments), and proved that the true parameter can be recovered with high probability with mild assumptions under various settings. They also showed the hardness of recovery using higher order moments. Clarity - Justification: The paper is well-written and the problem is well-motivated. All theoretical results are clearly presented with smooth logic flow. Significance - Justification: Please refer to detailed comments. Detailed comments. (Explain the basis for your ratings while providing constructive feedback.): The authors extended the results of compressed sensing to handle scenarios where additional errors are introduced by the data aggregation process. The extension is nontrivial as the two additional noise terms over the aggregated means of input and target are linearly correlated. The paper is technically sound and the experiment results seem reasonable. I skipped the proofs as I’m not strongly experienced in the associated literature. Minor comments: 1. The authors assumed uniform group sizes (line 296). It would be helpful if the authors can provide intuitions to clarity why their results can be trivially generalized to nonuniform cases. 2. Same sentence appeared twice (line 299-302). ===== Review #3 ===== Summary of the paper (Summarize the main claims/contributions of the paper.): The paper extends the theory of compressed sensing to the case where features and target variables are both aggregations of individual data points, such as in the case of healthcare and aggregating data over different hospitals. The problem is still to recover the true model on individual data points. Theory is developed to show then the true models is recoverable. Experiments are given to justify the theory. Clarity - Justification: The paper is well-written with almost no typos. I did not read the supplementary material, but I assume that it provides more explanation of the theory. Significance - Justification: I have not seen such results on aggregated data before. It builds on existing theory, but I believe it to still be more than just incremental. Detailed comments. (Explain the basis for your ratings while providing constructive feedback.): I think the results will be of interest to a wide audience and open the door to thinking about aggregated data in new interesting ways. A few minor comments: pg 3: Section 2.2 2nd paragraph: last sentence repeated twice. pg 4: equation (7) missing n in subscript, \nu_\epsilon should be \nu_{n,\epsilon} pg 5: change "in which the target mean in estimated" to "in which the target mean is estimated" -The relationship of Theorem 3.3 to 3.2 is not clear. Whereas for Theorem 3.2 the error can be anything greater than 0, it seems there must be a much larger error in Theorem 3.3 since \delta\geq 1. An explanation here would help in understanding. Section 4.1: please provide more details on the distributions and noise. ===== Review #4 ===== Summary of the paper (Summarize the main claims/contributions of the paper.): This paper presents the parameter recovery for sparse linear models from aggregated observations in the form of empirical means computed from different clusters of the data collection. The main result shows that in the case where there are sufficient number of observations; to compute the empirical moments, the standard LASSO can recover the true parameters with high probability provided the collection of true group moments is an incoherent matrix. An extension of the framework to the case of moments computed from histograms is also proposed. Experimental results on synthetic and healthcare applications corroborate the theoretical results. Clarity - Justification: Though I am not a specialist of parameter recovery, but I found that the paper is well written and very easy to follow. Significance - Justification: There are many studies on sparse recovery, but the parameter recovery for aggregated data as presented in the paper is novel Detailed comments. (Explain the basis for your ratings while providing constructive feedback.): I am an educated guess in parameter recovery. The paper is well written and the authors succeed to put in front the novelty of their approach. The mathematical justifications are well drawn so my overall rating is justified in this way. =====