Nearly Isometric Embedding by Relaxation
Many manifold learning algorithms aim to create embeddings with low or no distortion (isometric). If the data has intrinsic dimension d, it is often impossible to obtain an isometric embedding in d dimensions, but possible in s > d dimensions. Yet, most geometry preserving algorithms cannot do the latter. This paper proposes an embedding algorithm to overcome this. The algorithm accepts as input, besides the dimension d, an embedding dimension s ≥ d. For any data embedding Y, we compute a Loss(Y), based on the push-forward Riemannian metric associated with Y, which measures deviation of Y from from isometry. Riemannian Relaxation iteratively updates Y in order to decrease Loss(Y). The experiments confirm the superiority of our algorithm in obtaining low distortion embeddings.
This paper presents a new type of data embedding that preserves the geometry of the data as defined by a Reimannian metric (inner product on the tensor space of every point). Then they provide a method for gradient descent to solve this problem as the objective is not convex. 
This paper claims that contrary to the plethora of embeddings it utilizes [3, 4,12,19,20] that the Reimannian metric is best for geometry preservation without motivating this thought or providing the intuition for this claim. The authors claim that unlike previous methods they do not provide a one-shot mapping, but they ignore the existence of methods like tsne which do employ gradient descent in deriving the embedding. Finally they do not test this on sufficiently many datasets of significant size show how their embeddings are better than previously existing datasets or are practical in terms of runtime.   Figure 1: Their initialization is already very similar to the true shape; how does the algorithm perform without so much prior information? Also, the plot of MSE/loss vs sigma is poorly drawn. I'm not sure if they're constrained by runtime, but to connect the dots between their data points I would expect to see A) intervals for MSE & loss drawn from several randomized initializations and B) more points tested for sigma  Figure 2: Poorly formatted; it's not immediately obvious that they have two data sets with the same embeddings. Also, top-left/top-right etc is difficult to follow; there should be sections of the figure labeled A-G with corresponding sections in the legend. Again, the line plots are not publication quality. There need to be more points sampled on the X axis or replicates of each condition to provide an interval; I assume the trends going up then down then up are an artifact of individual runs?  Figure 3: Their analysis here is only half-complete. They present a real dataset, show superficial improvements in its analysis as a result of their method, but provide no physics-based analysis that would suggest their method actually improves the ability of scientists to analyze this kind of data.