Distance-Based Network Recovery under Feature Correlation
We present an inference method for Gaussian graphical models when only pairwise distances of n objects are observed. Formally, this is a problem of estimating an n × n covariance matrix from the Mahalanobis distances d MH (x i , x j ), where object x i lives in a latent feature space. We solve the problem in fully Bayesian fashion by integrating over the Matrix-Normal likelihood and a MatrixGamma prior; the resulting Matrix-T posterior enables network recovery even under strongly correlated features. Hereby, we generalize TiWnet [19], which assumes Euclidean distances with strict feature independence. In spite of the greatly increased flexibility, our model neither loses statistical power nor entails more computational cost. We argue that the extension is highly relevant as it yields significantly better results in both synthetic and real-world experiments, which is successfully demonstrated for a network of biological pathways in cancer patients.
Summary: This work proposes a new method for learning the structure ofa Gaussian graphical model.  It seeks to account for hidden featurecorrelations when determining the structure by modeling the featurecovariance matrix using a Matrix-Gamma prior.  In contrast, prior worksuch as TiWnet [19] assumes feature independence, which can lead to anoverly-dense graph.  Synthetic experiments, as well as experiments inwhich networks of genetic pathways are learned, clearly demonstratethe superiority of the proposed method over TiWnet [19].Clarity, quality, originality, significance: The exposition is fairlyclear.  There seem to be many similarities to the TiWnet work, so,while the background is a little sparse, most readers' questions canbe answered by referring to [19].  Figure 2 also provides a very nice,clean comparison to related work, and section 2.1.1's explanation ofwhy Sigma can't simply be removed by setting the derivative equal tozero is helpful.  While it's not a large theoretical delta over theTiWnet work, the experiments indicate the practical edge the use ofthe Matrix-Gamma prior gives, which testifies to the work'ssignificance.More detailed comments:It seems that there are many similarities between this work and theTiWnet work that aren't exactly explicit in the paper.  For instance,Algorithm 1 is nearly identical to the algorithm for TiWnet, exceptfor the beta sampling step.  It would be helpful to the reader ifthese sort of similarities and differences were more clearly marked.Please explain in more detail exactly what is done experimentally to"obtain the final W via annealing".  How is the temperature d adjustedto converge on a final W?In the experiments, how is lambda for the prior p_2 chosen?  Sincethis prior is meant to induce sparsity, why not just turn up lambdafor TiWnet?  This seems like it might at least partially solve themain problem observed in the experiments, where TiWnet learns far toodense of a graph.It would be interesting to see some discussion of future work and thedeficiencies of the proposed method.  Does TiMT have any obviousweaknesses like TiWnet has?  What sort of connections does it fail topredict and why?  Is the fact that the W matrix is only allowed threelevels (+1, -1, 0) something that could be easily improved upon?  Itseems like most of the analysis of Algorithm 1's efficiency dependsupon the fact that the update to W can be expressed in terms of asmall number of rank-1 matrices.  Perhaps this could be furtherleveraged to increase the number of levels in W without significantlyincreasing the computational cost.Thanks very much for including code with this submission.  One smallcomment --- while the demo_script is easy to parse, it would behelpful if the timt script had some comments in the code.Minor notes:0) "Gaussian Graphical Models" -> "Gaussian graphical models"1) "our approach is capable to infer" -> "our approach is capable ofinferring"2) "can be removed to receive" -> "can be removed to recover"3) "it is possibly to measure" -> "it is possible to measure"4) "levels of hundred thousands of genes" -> "levels of hundreds ofthousands of genes"5) "’a complete success’" -> "'a complete success'"6) "’a spectacular failure’" -> "'a spectacular failure'"7) Copy "Black/red edges refers to +/− edge weight." from the captionof Figure 5 up to the caption of Figure 4.8) "Bhattacharyya distance[11] are computed as" -> "Bhattacharyyadistances [11] are computed as"9) "and to which degree" -> "and to what degree"10) "cannot be verified, however, there" -> "cannot be verified;however, there"11) "information loss about underlying features" -> "information lostabout underlying features"12) Supplement: "we are interested to find" -> "we are interested infinding"