T-distributed stochastic neighbor embedding

The t-SNE algorithm comprises two main stages. First, t-SNE constructs a probability distribution over pairs of high-dimensional objects in such a way that similar objects are assigned a higher probability while dissimilar points are assigned a lower probability. Second, t-SNE defines a similar probability distribution over the points in the low-dimensional map, and it minimizes the Kullback–Leibler divergence (KL divergence) between the two distributions with respect to the locations of the points in the map. While the original algorithm uses the Euclidean distance between objects as the base of its similarity metric, this can be changed as appropriate. A Riemannian variant is UMAP.

t-SNE has been used for visualization in a wide range of applications, including genomics, computer security research, natural language processing, music analysis, cancer research, bioinformatics, geological domain interpretation, and biomedical signal processing.

For a data set with ${\displaystyle n}$ elements, t-SNE runs in ${\displaystyle O(n^2)}$ time and requires ${\displaystyle O(n^2)}$ space.

Details

Given a set of ${\displaystyle N}$ high-dimensional objects ${\displaystyle {\mathbf{x}}_1,\dots ,{\mathbf{x}}_N}$, t-SNE first computes probabilities ${\displaystyle p_{ij}}$ that are proportional to the similarity of objects ${\displaystyle {\mathbf{x}}_i}$ and ${\displaystyle {\mathbf{x}}_j}$, as follows.

For ${\displaystyle i\ne j}$, define

${\displaystyle p_{j\mid i}=\frac{\exp (-\Vert {\mathbf{x}}_i-{\mathbf{x}}_j{\Vert }^2/2{\sigma }_i^2)}{\sum _{k\ne i}\exp (-\Vert {\mathbf{x}}_i-{\mathbf{x}}_k{\Vert }^2/2{\sigma }_i^2)}}$

and set ${\displaystyle p_{i\mid i}=0}$. Note the above denominator ensures ${\displaystyle \sum _j{p}_{j\mid i}=1}$ for all ${\displaystyle i}$.

As van der Maaten and Hinton explained: "The similarity of datapoint ${\displaystyle x_j}$ to datapoint ${\displaystyle x_i}$ is the conditional probability, ${\displaystyle p_{j|i}}$, that ${\displaystyle x_i}$ would pick ${\displaystyle x_j}$ as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at ${\displaystyle x_i}$."

Now define

${\displaystyle p_{ij}=\frac{p_{j\mid i}+p_{i\mid j}}{2N}}$

This is motivated because ${\displaystyle p_i}$ and ${\displaystyle p_j}$ from the N samples are estimated as 1/N, so the conditional probability can be written as ${\displaystyle p_{i\mid j}=N{p}_{ij}}$ and ${\displaystyle p_{j\mid i}=N{p}_{ji}}$ . Since ${\displaystyle p_{ij}=p_{ji}}$, you can obtain previous formula.

Also note that ${\displaystyle p_{ii}=0}$ and ${\displaystyle \sum _{i,j}{p}_{ij}=1}$.

The bandwidth of the Gaussian kernels ${\displaystyle {\sigma }_i}$ is set in such a way that the entropy of the conditional distribution equals a predefined entropy using the bisection method. As a result, the bandwidth is adapted to the density of the data: smaller values of ${\displaystyle {\sigma }_i}$ are used in denser parts of the data space. The entropy increases with the perplexity of this distribution ${\displaystyle P_i}$; this relation is seen as

${\displaystyle Perp(P_i)=2^{H(P_i)}}$

where ${\displaystyle H(P_i)}$ is the Shannon entropy ${\displaystyle H(P_i)=-\sum _j{p}_{j|i}{\log }_2{p}_{j|i}.}$

The perplexity is a hand-chosen parameter of t-SNE, and as the authors state, "perplexity can be interpreted as a smooth measure of the effective number of neighbors. The performance of SNE is fairly robust to changes in the perplexity, and typical values are between 5 and 50.".

Since the Gaussian kernel uses the Euclidean distance ${\displaystyle \Vert x_i-x_j\Vert }$, it is affected by the curse of dimensionality, and in high dimensional data when distances lose the ability to discriminate, the ${\displaystyle p_{ij}}$ become too similar (asymptotically, they would converge to a constant). It has been proposed to adjust the distances with a power transform, based on the intrinsic dimension of each point, to alleviate this.

t-SNE aims to learn a ${\displaystyle d}$-dimensional map ${\displaystyle {\mathbf{y}}_1,\dots ,{\mathbf{y}}_N}$ (with ${\displaystyle {\mathbf{y}}_i\in {\mathbb{R}}^d}$ and ${\displaystyle d}$ typically chosen as 2 or 3) that reflects the similarities ${\displaystyle p_{ij}}$ as well as possible. To this end, it measures similarities ${\displaystyle q_{ij}}$ between two points in the map ${\displaystyle {\mathbf{y}}_i}$ and ${\displaystyle {\mathbf{y}}_j}$, using a very similar approach. Specifically, for ${\displaystyle i\ne j}$, define ${\displaystyle q_{ij}}$ as

${\displaystyle q_{ij}=\frac{(1+\Vert {\mathbf{y}}_i-{\mathbf{y}}_j{\Vert }^2{)}^{-1}}{\sum _k\sum _{l\ne k}(1+\Vert {\mathbf{y}}_k-{\mathbf{y}}_l{\Vert }^2{)}^{-1}}}$

and set ${\displaystyle q_{ii}=0}$. Herein a heavy-tailed Student t-distribution (with one-degree of freedom, which is the same as a Cauchy distribution) is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map.

The locations of the points ${\displaystyle {\mathbf{y}}_i}$ in the map are determined by minimizing the (non-symmetric) Kullback–Leibler divergence of the distribution ${\displaystyle P}$ from the distribution ${\displaystyle Q}$, that is:

${\displaystyle \mathrm{K}\mathrm{L}\left(P\parallel Q\right)=\sum _{i\ne j}{p}_{ij}\log \frac{p_{ij}}{q_{ij}}}$

The minimization of the Kullback–Leibler divergence with respect to the points ${\displaystyle {\mathbf{y}}_i}$ is performed using gradient descent. The result of this optimization is a map that reflects the similarities between the high-dimensional inputs.