Multivariate tests of association based on univariate tests
For testing two vector random variables for independence, we propose testing whether the distance of one vector from an arbitrary center point is independent from the distance of the other vector from another arbitrary center point by a univariate test. We prove that under minimal assumptions, it is enough to have a consistent univariate independence test on the distances, to guarantee that the power to detect dependence between the random vectors increases to one with sample size. If the univariate test is distribution-free, the multivariate test will also be distribution-free. If we consider multiple center points and aggregate the center-specific univariate tests, the power may be further improved, and the resulting multivariate test may have a distribution-free critical value for specific aggregation methods (if the univariate test is distribution free). We show that certain multivariate tests recently proposed in the literature can be viewed as instances of this general approach. Moreover, we show in experiments that novel tests constructed using our approach can have better power and computational time than competing approaches.
The authors propose a modular framework for designing omnibus tests of independence between multivariate distributions. Given (X_i,Y_i) pairs, the authors propose designating a center point z_x and z_y for each, reducing to a bivariate problem by computing (||X_i-z_x||, ||Y_i-z_y||) for each i, and testing whether the resulting distances are independent using some off-the-shelf test of independence for the reduced problem. A similar construction is given for reducing to a univariate K-sample test if X is categorical with K categories. More generally, we can repeat the same process for several different choices of central points and adjust appropriately for multiplicity. This approach is appealing both for its generality and its computational simplicity, since using off-the-shelf bivariate independence tests lets us look up critical values in a table rather than having to compute them by, e.g., permutations.  The authors show under mild conditions that their tests are consistent for almost every choice of central points z_x and z_y. They evaluate their methods in simulation, showing favorable power as compared to competing approaches. 
The paper is well-written and well-structured. The organization is clear in the introduction and executed well throughout. The statements of the theorems are clear and there are few typos.  The technical level is solid. The theorems are interesting, though I would have liked to see more refined results. I am by no means an expert in this area but your theorems did not, in my mind, establish that anyone would want to use the test. Power --> 1 against a fixed alternative, with an infinite sample, is a very low bar that would be cleared by many truly terrible tests (to name a terrible idea off the top of my head: Pearson's chi^2 test after discretizing the data into a multidimensional grid, where the discretization gets finer and finer as the sample size grows).  The paper gave a nice, general framework encompassing a plethora of tests we could try but stopped short of giving specific recommendations, citing the need for further empirical and theoretical work to determine what tests will work well for various data types and against various alternatives. But one has certain alternatives in mind (as I suspect one would in most scientific applications) then why use an omnibus test?  My biggest reservation has to do with practical utility, which is, in my view, questionable. In what concrete applications will your approach actually represent an improvement on already-available methods? And what do we really learn when an omnibus test of independence rejects the null? Formally rejecting independence, unless we can make a concrete statement about how the distributions differ, doesn't seem very useful -- for example, was there ever any doubt whether men's and women's microbiomes were identically distributed, and what do we learn by declaring that they are not?  All but one of the simulation scenarios were for two-sample tests in dimension 2. The other example was for dimension 5, but there you were comparing a very heavy-tailed distribution to a very light-tailed one, so it is unsurprising that a procedure based on the distribution of inter-point distances, or distance from the centroid, would do well. These are very low-dimensional examples, leaving the reader to wonder whether your tests will have any appreciable power in intermediate or high dimensions.  On this point, I did not feel that the AGP experiment added anything to the paper; it sounds like the null was very obviously not true, and every test was able to pick that up very easily. It would be more interesting to compare the power of different approaches by subsampling the data.  Bottom line: there are several valid ways to make the case for your method over already-existing methods: rigorous theory, heuristic arguments, simulations, or on a real data set that represents an important use case. But in my judgement you haven't succeeded on any one of these counts. Your theoretical results can't make the case for your method over other consistent methods; you don't really give heuristic arguments about why or when your method should be more powerful; your simulations are unconvincing because they don't really go beyond dimension 2, and in your real data experiment all methods apparently have good power.  Minor comments:  Table 1: The numbers in this table should be reported with most two significant digits. For example, 250 rejections on 500 tries should give estimate 0.5 and standard error .022. 100 rejections on 1000 tries in the first column gives estimate 0.1 and standard error 0.013. More generally, please report standard errors (if only by reporting a maximum standard error of 0.022 for all values given in the table).  Prop 2.1: this is stated much more generally than what you seem to actually need; taking the time to parse the definitions involved briefly derailed my otherwise-smooth progress through the paper. Can you state it in a simpler way?  - Please state somewhere that you mean Euclidean distance (I assume you do) - L.77: R^p formatted differently than elsewhere - Labels on Fig. 1 are very small - You define the term "distribution-free" as "the null distribution of the test statistic does not depend on the marginal distributions of X and Y." In your proposal to combine p-values via e.g. Bonferroni, the above definition would not apply: the minimum p-value's null distribution does, in fact, depend on the dependence of the p-values (of course, the test still works). Please resolve this issue, e.g. by modifying your definition to say "the critical value of the test statistic ..." - L.150: "two sample" -> "two-sample" - L.219: the multivariate t-distribution is usually defined not as a product of univariate t-distributions, but rather as a multivariate normal divided by an independent chi-squared distribution. I would call this a "product t-distribution" or something. - L.281: "hypohtesis"