t-distributed stochastic neighbor embedding (t-SNE) is a statistical method for visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. It is based on Stochastic Neighbor Embedding originally developed by Geoffrey Hinton and Sam Roweis, where Laurens van der Maaten and Hinton proposed the t-distributed variant. It is a nonlinear dimensionality reduction technique for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. 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 with high probability.

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 n elements, t-SNE runs in O(n²) time and requires O(n²) space.

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 ${\displaystyle i\neq j}$, define

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 ${\displaystyle x_{i}}$."