Space-Time Local Embeddings
Space-time is a profound concept in physics. This concept was shown to be useful for dimensionality reduction. We present basic definitions with interesting counter-intuitions. We give theoretical propositions to show that space-time is a more powerful representation than Euclidean space. We apply this concept to manifold learning for preserving local information. Empirical results on nonmetric datasets show that more information can be preserved in space-time.
This paper proposes a new embedding structure for the output of nonlinear dimensionality reduction (NLDR). It borrows the space-time concept from physics and shows that the new structure can encode (or represent) more information than conventional Euclidean space.The idea is pioneering and wonderful. It goes beyond all conventional embedding structures and has theoretical guarantee to accommodate more symmetry in the input space. It brings much flexibility to the NLDR problem. This can be an important middle step towards better dimensionality reduction.However, how to visualize the space-time structure seems unsolved. 1) The authors proposes to use colors to show the time dimension. But it is pretty hard to read such display. Information visualization should take advantage of the human eye's broad bandwidth pathway into the mind to allow users to digest large amounts of information at once. But Figs. 1 and 2 are rather messy. There is no immediate and clear macro pattern in these figures. Users have to examine the details piece by piece, which is time-consuming.2) the meaning of the time dimension in data science remains unclear in many aspects. For example, in Fig. 1, what is the different meanings of "very red" and "very green" (e.g. Hinton vs. Bengio)? In Fig. 2, what are the meaning of the red groups and blue groups?The KL-divergence is used as benchmarks to show the advantage of using space-time. This is not convincing because there are many other factors that can affect the results. For example, the benefits may vanish when another input kernel width or another optimization algorithm is used. In addition, the SNE and t-SNE objectives are not scale invariant to Y. That is, multiplying a same constant to all y's may change the objective. A smaller divergence may not bring a better visualization. So I don't agree Table 1 is a solid justification.The title is a bit ambiguous. Readers may think this is to find embeddings of spatial-temporal input data. How about "Local Embeddings using Space-Time"?Line 341: what do you mean "an author or a word A at a different time with the majority is close to them all"?Line 346: what do you mean "dark color"?  After feedback: The author feedback is a bit disappointing. Repeatedly arguing the advantages is not very helpful. I believe the idea is new and interesting. It deserves further exploration. But the visualization part needs improvement. From the feedback, I feel that visualizing the information in a space-time is unsolved. I don't agree with the 3D visualization approach because 1) it is restricted to only R^{2,1} case and 2) it is not provided in the paper or supplemental material. Using colors or levels to show the time dimensional is not satisfied either. The authors admit that the visualization work is still ongoing. The time dimension is claimed to show the importance (or ranking) of data points. But this remains unclear in Figure 2. The authors did not answer my question about the meanings of red words and blue words. Furthermore, it seems only the absolute values of time dimension is important. We still don't know the meaning with the signs. For example, Hinton and LeCun are green (blue in the final version), but Bengio is very red. Why? The feedback cannot explain this. The authors admit that selecting the right evaluation measure for visualization is an open problem. Therefore smaller KL-divergence values only partially demonstrate the advantages. It is not "a solid justification" as claimed. As I said, the reduced KL is not necessarily caused by the use of space-time and it may not correspond to better displays.After seeing the final version: The authors have removed most improper claims in the original version. They also improved the quality of the paper. But most of the above questions are not answered yet. They deserve further studies in future research.