Paper:
discovering-order-unordered
The assumption that data samples are independently identically distributed is the backbone of many learning algorithms. Nevertheless, datasets often exhibit rich structures in practice, and we argue that there exist some unknown orders within the data instances. Aiming to find such orders, we introduce a novel Generative Markov Network (GMN) which we use to extract the order of data instances automatically. Specifically, we assume that the instances are sampled from a Markov chain. Our goal is to learn the transitional operator of the chain as well as the generation order by maximizing the generation probability under all possible data permutations. One of our key ideas is to use neural networks as a soft lookup table for approximating the possibly huge, but discrete transition matrix. This strategy allows us to amortize the space complexity with a single model and make the transitional operator generalizable to unseen instances. To ensure the learned Markov chain is ergodic, we propose a greedy batch-wise permutation scheme that allows fast training. Empirically, we evaluate the learned Markov chain by showing that GMNs are able to discover orders among data instances and also perform comparably well to state-of-the-art methods on the one-shot recognition benchmark task.

Review:
The authors deal with the problem of implicit ordering in a dataset and the challenge of recovering it, i.e. when given a random dataset with no explicit ordering in the samples, the model is able to recover an ordering. They propose to learn a distance-metric-free model that assumes a Markov chain as the generative mechanism of the data and learns not only the transition matrix but also the optimal ordering of the observations.
> Abstract
“Aiming to find such orders, we introduce a novel Generative Markov Network (GMN) which we use to extract the order of data instances automatically. ”
I am not sure what automatically refers here to. Do the authors mean that the GMN model does not explicitly assume any ordering in the observed dataset? This needs to be better stated here. 
“Aiming to find such orders, we introduce a novel Generative Markov Network (GMN) which we use to extract the order of data instances automatically; given an unordered dataset, it outputs the best -most possible- ordering.”
Most of the models assume an explicit ordering in the dataset and use it as an integral modelling assumption. Contrary to that they propose a model where no ordering assumption is made explicitly, but the model itself will recover it if any.
> Introduction
The introduction is fairly well structured and the example of the joint locations in different days helps the reader.  
In the last paragraph of page 1, “we argue that … a temporal model can generate it.”, the authors present very good examples where ordered observations (ballerina poses, video frames) can be shuffled and then the proposed model can recover a temporal ordering out of them. What I would like to think also here is about an example where the recovered ordering will also be useful as such. An example where the recovered ordering will increase the importance of the inferred solution would be more interesting..
2. Related work
This whole section is not clear how it relates to the proposed model GMN. Rewriting is strongly suggested. 
The authors mention Deep Generative models and One-shot learning methods as related work but the way this section is constructed makes it hard for the reader to see the relation. It is important that first the authors discuss the characteristics of GMN that makes it similar to Deep generative models and the one-shot learning models. They should briefly explain the characteristics of DGN and one-shot learning so that the readers see the relationship. 
Also, the authors never mention that the architecture they propose is deep.
 
Regarding the last paragraph of page 2, “Our approach can be categorised … can be computed efficiently.”:
Not sure why the authors assume that the samples can be sampled from an unmixed chain. An unmixed chain can also result in observing data that do not exhibit the real underlying relationships. Also the authors mention couple of characteristics of the GMN but without really explaining them.  What are the explicit and implicit models [1] … this needs more details. 
[1] P. J. Diggle and R. J. Gratton. Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society. Series B (Methodological), pages 193–227, 1984. 
“Second, prior approaches were proposed based on the notion of denoising models. In other words, their goal was generating high-quality images; on the other hand, we aim at discovering orders in datasets.” —>this bit is confusing. Do the authors mean that prior approaches were considering the observed ordering as part of the model assumptions and were just focusing on the denoising? 
3. Generative Markov models
First, I would like to draw the attention of the authors on the terminology they use. The states here are not the latent states usually referred in the literature of Markov chains. The states here are observed and should not be confused with the emissions also usually stated in the corresponding literature. There are as many states as the number of observations and not differentiation is made for ties. All these are based on my understanding of the model.
In  the Equation just before equation (1),  on the left hand side, shouldn’t \pi be after the `;’. It’s an average over the possible \pi.  We cannot  consider the average over \pi when we also want to find the optimal \pi.  The sum doesn’t need to be there. Shouldn’t it just be  max_{\theta, \pi} log P({s_i}^{n}_{i=1}; \pi, \theta) ?
Equation (1), same. The summation over the possible \pi is confusing. It’s an optimisation problem…
page 4, section 3.1: The discussion about the use of Neural Net for the construction of the transition matrix needs expansion. It is unclear how the matrix is constructed. Please add more details. E.g. use of soft-max non-linear transformation so that the output of the Neural Net can be interpreted as the probabilities of jumping to one of the possible states. In this fashion, we map the input (current state) and transform it to the probability gf occupying states at the next time step.
Why this needs expansion: The construction of the transition matrix is the one that actually plays the role of the distance metric in the related models. More specifically, the choice of the non-linear function that outputs the transition probability is crucial; e.g. a smooth function will output comparable transition probabilities to similar inputs (i.e. similar states). 
section 3.2: 
My concern about averaging over \pi applies on the equations here too. 
“However, without further assumption on the structure of the transitional operator..”—> I think the choice of the nonlinear function in the output node of the NN is actually related to the transition matrix and defines the probabilities. It is a confusing statement to make and authors need to discuss more about it. After all, what is the driving force of the inference? This is a problem/task where the observations are considered in a number of different permutations. As such, the ordering is not fixed and the main driving force regarding the best choice of ordering should come from the architecture of the transition matrix; what kind of transitions does the Neural Net architecture favour? Distance free metric but still assumptions are made that favour specific transitions over others. 
“At first, Alg. 1 enumerates all the possible states appearing in the first time step. For each of the following steps, it finds the next state by maximizing the transition probability at the current step, i.e., a local search to find the next state. ” —>  local search in the sense that the algorithm chooses as the next state the state with the biggest transition probability (to it) as defined in the Neural Net (transition operator) output? This is a deterministic step, right? 
4.1 DISCOVERING ORDERS IN DATASETS 
Nice description of the datasets. In the <MSR_SenseCam> the choice of one of the classes needs to be supported.  Why? What do the authors expect to happen if a number of instances from different classes are chosen? 
4.1.1 IMPLICIT ORDERS IN DATASETS 
The explanation of the inferred orderings for the GMN and Nearest Neighbour model is not clear. In figure 2, what forces the GMN to make distinguishable transitions as opposed to the Nearest neighbour approach that prefers to get stuck to similar states? Is it the transition matrix architecture as defined by the neural network? 
>> Figure 10: why use of X here? Why not keep being consistent by using s?
*** DO the authors test the model performance on a ordered dataset (after shuffling it…) ?  Is the model able of recovering the order? **