What do row and column marginals reveal about your dataset?
Numerous datasets ranging from group memberships within social networks to purchase histories on e-commerce sites are represented by binary matrices. While this data is often either proprietary or sensitive, aggregated data, notably row and column marginals, is often viewed as much less sensitive, and may be furnished for analysis. Here, we investigate how these data can be exploited to make inferences about the underlying matrix H. Instead of assuming a generative model for H, we view the input marginals as constraints on the dataspace of possible realizations of H and compute the probability density function of particular entries H(i, j) of interest. We do this for all the cells of H simultaneously, without generating realizations, but rather via implicitly sampling the datasets that satisfy the input marginals. The end result is an efficient algorithm with asymptotic running time the same as that required by standard sampling techniques to generate a single dataset from the same dataspace. Our experimental evaluation demonstrates the efficiency and the efficacy of our framework in multiple settings.
The paper concerns extracting reliable information 
      about binary matrices, given only their column and row sums. These sums 
      are considered as constraints on the space of possible matrices. The goal 
      is to estimate the entry-wise probability density functions and to do this 
      simultaneously via implicit sampling, making the proposed algorithm much 
      more efficient than existing techniques. The paper is well written and 
      easy to follow. -- For sparse matrices, which is the case in the 
      considered DBLP and NOLA datasets, the probability of having 1 is low, and 
      the proposed approach computes low values as well. However, how useful is 
      this information? It would be interesting to see how the algorithm 
      performs on real or simulated matrices with density around 50%. Would we 
      have many interpretable entries with probabilities close to 0 or close to 
      1, or would we get probabilities around 0.5? -- Minor remarks: 
      1. Is there a reference for the considered LS approach with \bar h 
      being the average entry value (Section 4.1)? If \bar h is close to 0.5, 
      wouldn't this approach select an approximation with entries as close to 
      0.5 as possible, making the approximation less interpretable? 2. The 
      legend of Figure 1(b) contains undefined labels. -- Further 
      questions: 1. Is the approach generalizable to integer matrices? 
      2. Is the proposed approach generalizable to the noisy case, i.e. how 
      much would the entry-wise PDFs change if a given row or column sum is 
      increased or decreased by one? 3. To speed up the proposed algorithm, 
      columns having the same sum are considered to be the same and duplicates 
      are removed from the computational step. Is it possible to extend the 
      algorithm to be able to incorporate other types of prior knowledge? For 
      example, a common assumption about data matrices is that they have low 
      rank. (Redundant columns/rows lead to low rank matrices, but the low rank 
      assumption is more general than that.)