
Submitted by Assigned_Reviewer_1
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
The paper is interesting and the authors do a good job explaining the idea and the material. Your models seem to be most closely related to the one in [11], where you instead have used the Coulomb repulsive process for the manifold points. Nevertheless, [11] is not included in your overview of related work in the introduction, only its prior work in [6]. Could you more carefully explain how you work relates to [11]?
In the experimental section your onedimensional manifold corresponds to time, since the video frames constitute a time series. This opens up to comparisons with dynamical models and especially hidden Markov models to describe such data. Could you relate your model to HMMs? By using the sequential order of the video frames, the prediction performance should be increased. Such models would also automatically solve the "gap" issue that you encounter. Further, in the experimental section you state on row 310/311 that "As can be seen the right panel, optimizing x's without any constraint results in "holes" on [0, 1]". To me this is not clear. Please clarify how to see this from the figures. Do also give a clearer fig reference. In addition, you alternate between comparing with GPLVM and Bayesian GPLVM in the results sections. Please be consistent or give a motivation for your choices. Finally, are there any situations that the proposed model does not handle? It would be fair to give such comments as well to increase the objectivity of the paper.
I think the model is interesting and it seems to solve problems present in previous related models. The results are promising and worth to be published.
Minors:  Fig 3. the caption should be improved since it does not fully explain the content in the figures. It would be helpful to mark the regions in figure 2 where you have zoomed in.  In fig 4, I think you should exchange rows and columns in the right panel. Now it is a bit confusing and not consistent with the left panel.
Q2: Please summarize your review in 12 sentences
The model is interesting and it seems to solve problems present in previous related models. The results are promising and worth to be published.
Submitted by Assigned_Reviewer_2
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
=Summary=
This paper proposes a maximum penalized likelihood version of the GPLVM (equation after eq. 4). The penalty term added to the GPLVM log likelihood is the log joint probability density function of the inputs under a Coulomb repulsive process.
=Evaluation=
Quality: medium because of the missing references. Clarity: good Originality: good, to the best of my knowledge Significance: medium (early days, still far from an easy to reimplement approach)
=Details=
The optimization depends critically on initialization, and the authors propose a heuristic that relies on using a similarity preserving traditional embedding, as well as a way to initialize the GPs hyperparameters.
Obtaining posterior uncertainty is pretty tedious and there isn't a good solution. The authors propose a heuristic.
Under the GPLVM model, two latent points that are nearby must be associated with nearby data points. However, there is no guarantee that two nearby data points will be mapped to two latent points that are near one another. This causes many of the ailments of the GPLVM mentioned in the current paper. The authors should consider citing the constrained likelihood approach, that imposes that the inputs be a smooth map from data to latent [1, 3]. These formulations do not correspond to a proper Bayesian generative model.
There exists however a Bayesian formulation of the GPLVM [2] the authors should relate their method to.
Minor:
 figure 2 is really hard to read.
Refs:
[1] "Local Distance Preservation in the GPLVM Through Back Constraints" Neil Lawrence and Joaquin Quinonero Candela Proceedings of the 23rd International Conference on Machine Learning 2006, pages 512520, Pittsburgh, PA
[2]
"Bayesian Gaussian process latent variable model"
M. K. Titsias and N. D. Lawrence. (2010) in Y. W. Teh and D. M. Titterington (eds) Proceedings of the Thirteenth International Workshop on Artificial Intelligence and Statistics, JMLR W&CP 9, Chia Laguna Resort, Sardinia, Italy, pp 844851.
[3] "Topologicallyconstrained latent variable models"
R. Urtasun, D. J. Fleet, A. Geiger, J. Popovi, T. J. Darrell and N. D. Lawrence. (2008) in S. Roweis and A. Mccallum (eds) Proceedings of the International Conference in Machine Learning, Omnipress, , pp 10801087.
Q2: Please summarize your review in 12 sentences
An interesting penalized maximum likelihood version of the GPLVM to force better distributed embeddings. The paper suffers from not having referenced more modern versions of the GPLVM which in different ways attempt to improve the properties of the learnt embeddings, and to a smaller extent from a somewhat tedious heuristic for computing posterior embedding uncertainty.
Submitted by Assigned_Reviewer_3
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
This paper proposes a new method to learn a mapping function from a ddimensional data point, y, to onedimensional embedding, x.
The paper extends the standard GP to a hierarchical model by incorporating what they call Coulomb repulsive process (Corp). It first generates samples of the onedimensional x and then generates observable ddim data points through a GP, whose covariance is constrained to be diagonal.
Intuitively, Corp is used to ensure reasonable interpoint distance in the embedding space through the repulsive force in [0,1] to make the distance of 0.5 is the least stressful state. The equidistance prior in [0,1] looks like a natural choice when the original distribution for y forms a compact single continuous manifold in the original space.
To find the embeddings, they propose an alternating algorithm, where GP parameters and the embeddings are solved alternatingly.
To the best of my knowledge, the use of Corp in the context of GPLVM is new. The proposed approach looks interesting enough to accept the paper.
I personally think that the paper can be improved by describing theoretical aspects of the model more carefully. It looks obvious to me that the model still has major limitations such as the strong dependency on the initial guess (x_0). I feel that the authors are overselling the model by making too strong statements on the practical utility of the model.
For example, in practice, we all know that careful preprocess is required to make "your favorite distancepreserving manifold" work. What if the distribution looks multimodal? What if x_0 misses to capture the major features of the original distribution? Your claim in Sec. 5 is scientifically sound?
I know recently more and more reviewers tend to demand extensive experimental comparison. However, for papers like this, the most important thing is to build a common understanding with the audience on the major features of the model and wide applicability of the model. I hope the cameraready version gets more understandable and sophisticated by addressing it.
Minor comments.  Do not mix between bold and nonbold notation to represent vectors. Use boldface for e.g. \mu in Introduction.
 Introduction of Corp in Sec.2 can be improved. This is basically to introduce the repulsive force. Don't make it overly mathematical.
 If you are to use r=1, you don't have to show r=20. You can show illustrative figures instead.
 Dependency on x_0 should be discussed in depth. If you cannot, do not overclaim the practical utility.
 Also, describe practical limitations in addition to practical advantages.
 Sec. 5 sounds like a frivolous sales talk. This is a good paper as is. You don't have to worry about the "more experiment!" criticism. I'd like you to make more sophisticated academic comments.
 Redraw Fig.2. Visibility is extremely bad.
Q2: Please summarize your review in 12 sentences
The paper proposes a new prior to GPLVM to address wellknown shortcomings of GPLVM to realize a natural distribution of embedded coordinates. The paper seems to be rather overclaiming the practical utility of the proposed method,
but the approach looks novel enough to be accepted.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 5000 characters. Note
however, that reviewers and area chairs are busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We thank the reviewers for the time and expertise
they have invested in providing their feedback.
To Reviewer
1: We chose [6] as a direct comparison with electroGP since they both
try to find the MAP of the latent x's. We agree that the relation to [11]
should be discussed and will include it in the introduction of the final
version.
By "hole" we meant, in the right panel of Figure 3, the
vacant space between the leftmost and the second leftmost x's
(triangle).
The proposed model is currently constrained to the
simplest scenario, which is to learn a compact smooth curve. We hope to
explore its ability of handling more complex situation in the future. We
also want to thank the reviewer for the comments on the clearness of the
figures and will improve it in our final version.
To Reviewer
2: We agree with the reviewer and will explore and include the
mentioned references in the final version. And we will redraw Figure
2.
We also agree that the optimization relies heavily on initial
ordering of x's. Fortunately, in our experience, the traditional manifold
learning methods provide good enough initialization. This is probably due
to that we are only solving the simplest onedimensional curve learning
task. We admit that the model is still in its early days. Generalizing it
to multidimensional manifold problems and relaxing the dependency on
initialization are both on our todo list.
To Reviewer 3: We
thank the reviewer for pointing out the limitations of our paper.
Exploring the theoretical properties of the model and relaxing the
dependency on the initialization are both on our todo list.
The
reviewer also raised a good point of the potential multimodality in the
data, which is not concerned in this paper since we only consider a simple
scenario where the data live on a smooth compact curve. This is certainly
a limitation and might be solved by a mixture of electroGP. We hope to
explore this possibility in the future. We also thank the reviewer for
the helpful comments and will try our best to address them in the final
version.
To Reviewer 4, 6 and 7: We thank the reviewers for
their accurate summaries of this paper. 
