Details view: t-Distributed Stochastic Neighbor Embedding (t-SNE)

t-Distributed Stochastic Neighbor Embedding (t-SNE) Τμήμα Άποψης 1 #307498 t-Distributed Stochastic Neighbor Embedding (t-SNE) is a machine learning algorithm for dimensionality reduction developed by Laurens van der Maaten and Geoffrey Hinton.[1] It is a nonlinear dimensionality reduction technique that is particularly well suited for embedding high-dimensional data into a space of two or three dimensions, which can then be visualized in a scatter plot.
t-Distributed Stochastic Neighbor Embedding (t-SNE) is a machine learning algorithm for dimensionality reduction developed by Laurens van der Maaten and Geoffrey Hinton.^[1] It is a nonlinear dimensionality reduction technique that is particularly well suited for embedding high-dimensional data into a space of two or three dimensions, which can then be visualized in a scatter plot. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points. The t-SNE algorithms comprises two main stages. First, t-SNE constructs a probability distribution over pairs of high-dimensional objects in such a way that similar objects have a high probability of being picked, whilst dissimilar points have an infinitesimalprobability of being picked. Second, t-SNE defines a similar probability distribution over the points in the low-dimensional map, and it minimizes the Kullback-Leibler divergence between the two distributions with respect to the locations of the points in the map. t-SNE has been used in a wide range of applications, including computer security research,^[2] music analysis,^[3] cancer research,^[4] and bio-informatics.^[5] Details[edit] Given a set of $N$ high-dimensional objects $\mathbf{x}_1, \dots, \mathbf{x}_N$ , t-SNE first computes probabilities $p_{ij}$ that are proportional to the similarity of objects $\mathbf{x}_i$ and $\mathbf{x}_j$ , as follows: $p_{j\|i} = \frac{\exp(-\lVert\mathbf{x}_i, \mathbf{x}_j\rVert^2 / 2\sigma_i^2)}{\sum_{k \neq i} \exp(-\lVert\mathbf{x}_i, \mathbf{x}_k\rVert^2 / 2\sigma_i^2)},$ $p_{ij} = \frac{p_{j\|i} + p_{i\|j}}{2N}$ The bandwidth of the Gaussian kernels $\sigma_i$ , is set in such a way that the perplexity of the conditional distribution equals a predefined perplexity using a binary search. As a result, the bandwidth is adapted to the density of the data: smaller values of $\sigma_i$ are used in denser parts of the data space. t-SNE aims to learn a $d$ -dimensional map $\mathbf{y}_1, \dots, \mathbf{y}_N$ (with $\mathbf{y}_i \in \mathbb{R}^d$ ) that reflects the similarities $p_{ij}$ as good as possible. To this end, it measures similarities $q_{ij}$ between two points in the map $\mathbf{y}_i$ and $\mathbf{y}_j$ , using a very similar approach. Specifically, $q_{ij}$ is defined as: $q_{ij} = \frac{(1 + \lVert \mathbf{y}_i - \mathbf{y}_j\rVert^2)^{-1}}{\sum_{k \neq l} (1 + \lVert \mathbf{y}_k - \mathbf{y}_l\rVert^2)^{-1}}$ Herein a heavy-tailed Student-t distribution is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map CITATION. The locations of the points $\mathbf{y}_i$ in the map are determined by minimizing the Kullback-Leibler divergence between the two distributions $P$ and $Q$ : $KL(P\|\|Q) = \sum_{i \neq j} p_{ij} \log \frac{p_{ij}}{q_{ij}}$ The minimization of the Kullback-Leibler divergence with respect to the points $\mathbf{y}_i$ is performed using gradient descent. The result of this optimization is a map that reflects the similarities between the high-dimensional inputs well. References[edit] Jump up^ van der Maaten, L.J.P.; Hinton, G.E. (Nov 2008). "Visualizing High-Dimensional Data Using t-SNE". Journal of Machine Learning Research 9: 2579–2605. Jump up^ Gashi, I.; Stankovic, V., Leita, C., Thonnard, O. (2009). "An Experimental Study of Diversity with Off-the-shelf AntiVirus Engines". Proceedings of the IEEE International Symposium on Network Computing and Applications: 4–11. Jump up^ Hamel, P.; Eck, D. (2010). "Learning Features from Music Audio with Deep Belief Networks". Proceedings of the International Society for Music Information Retrieval Conference: 339–344. Jump up^ Jamieson, A.R.; Giger, M.L., Drukker, K., Lui, H., Yuan, Y., Bhooshan, N. (2010). "Exploring Nonlinear Feature Space Dimension Reduction and Data Representation in Breast CADx with Laplacian Eigenmaps and t-SNE". Medical Physics 37(1) 37: 339–351.doi:10.1118/1.3267037. Jump up^ Wallach, I.; Liliean, R. (2009). "The Protein-Small-Molecule Database, A Non-Redundant Structural Resource for the Analysis of Protein-Ligand Binding". Bioinformatics 25(5) 25 (5): 615–620. doi:10.1093/bioinformatics/btp035.

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