text: We identify a fundamental issue in the popular Stochastic Neighbour Embedding (SNE and t-SNE), i.e., the "learned" similarity of any two points in high-dimensional space is not defined and cannot be computed. It underlines two previously unexplored issues in the algorithm which have undermined the quality of its final visualisation output and its ability to process large datasets. The issues are:(a) the reference probability in high-dimensional space is set based on entropy which has undefined relation with local density; and (b) the use of data independent kernel which leads to the need to determine n bandwidths for a dataset of n points. This paper establishes a principle to set the reference probability via a data-dependent kernel which has a well-defined kernel characteristic that linked directly to local density. A solution based on a recent data-dependent kernel called Isolation Kernel addresses the fundamental issue as well as its two ensuing issues. As a result, it significantly improves the quality of the final visualisation output and removes one obstacle that prevents t-SNE from processing large datasets. The solution is extremely simple, i.e., simply replacing the existing data independent kernel with Isolation Kernel, leaving the rest of the t-SNE procedure unchanged.
  @cite_1: This paper considers the problem of learning an embedding of data based on similarity triplets of the form “A is more similar to B than to C”. This learning setting is of relevance to scenarios in which we wish to model human judgements on the similarity of objects. We argue that in order to obtain a truthful embedding of the underlying data, it is insufficient for the embedding to satisfy the constraints encoded by the similarity triplets. In particular, we introduce a new technique called t-Distributed Stochastic Triplet Embedding (t-STE) that collapses similar points and repels dissimilar points in the embedding — even when all triplet constraints are satisfied. Our experimental evaluation on three data sets shows that as a result, t-STE is much better than existing techniques at revealing the underlying data structure.
  @cite_3: Stochastic Neighbor Embedding (SNE) has shown to be quite promising for data visualization. Currently, the most popular implementation, t-SNE, is restricted to a particular Student t-distribution as its embedding distribution. Moreover, it uses a gradient descent algorithm that may require users to tune parameters such as the learning step size, momentum, etc., in finding its optimum. In this paper, we propose the Heavy-tailed Symmetric Stochastic Neighbor Embedding (HSSNE) method, which is a generalization of the t-SNE to accommodate various heavy-tailed embedding similarity functions. With this generalization, we are presented with two difficulties. The first is how to select the best embedding similarity among all heavy-tailed functions and the second is how to optimize the objective function once the heavy-tailed function has been selected. Our contributions then are: (1) we point out that various heavy-tailed embedding similarities can be characterized by their negative score functions. Based on this finding, we present a parameterized subset of similarity functions for choosing the best tail-heaviness for HSSNE; (2) we present a fixed-point optimization algorithm that can be applied to all heavy-tailed functions and does not require the user to set any parameters; and (3) we present two empirical studies, one for unsupervised visualization showing that our optimization algorithm runs as fast and as good as the best known t-SNE implementation and the other for semi-supervised visualization showing quantitative superiority using the homogeneity measure as well as qualitative advantage in cluster separation over t-SNE.
  @cite_8: A perpetual calendar that has a flat frame with two cavities. When cards are arranged in each cavity of the frame it will display a specific month. To display a particular number of days in a month, select the proper 31 day card and arrange the vertical slits on the card with the horizontal slit on the frame to obtain 28, 29, 30 or 31 days. To display a particular month choose the name of the month card to be viewed and insert it in the position of the frame provided for it. The few cards that are not used will remain in the frame cavities behind the month being displayed. This perpetual calendar can also display three consecutive months at one time.
summary: There are some improvements based on revised Gaussian Kernel functions in order to get better similarity measurements. @cite_8 proposes a symmetrised SNE, @cite_3 enable t-SNE to accommodate various heavy-tailed embedding similarity functions; and @cite_1 propose an algorithm based on similarity triplets of the form A is more similar to B than to C'' to model the local structure of the data more effectively.