In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x {\displaystyle x} and y {\displaystyle y} directions contain all of the necessary information; a 2 2 {\displaystyle 2\times 2} matrix would be necessary to fully characterize the two-dimensional variation. Some statisticians, following the probabilist William Feller in his two-volume book An Introduction to Probability Theory and Its Applications,[2] call the matrix K X X {\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }} the variance of the random vector X {\displaystyle \mathbf {X} }, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the covariance matrix, because it is the matrix of covariances between the scalar components of the vector X {\displaystyle \mathbf {X} } .

An autoencoder is a type of artificial neural network used to learn efficient data codings in an unsupervised manner.[1] The aim of an autoencoder is to learn a representation (encoding) for a set of data, typically for dimensionality reduction, by training the network to ignore signal "noise". Along with the reduction side, a reconstructing side is learnt, where the autoencoder tries to generate from the reduced encoding a representation as close as possible to its original input, hence its name. Several variants exist to the basic model, with the aim of forcing the learned representations of the input to assume useful properties.[2] Examples are the regularized autoencoders (Sparse, Denoising and Contractive autoencoders), proven effective in learning representations for subsequent classification tasks,[3] and Variational autoencoders, with their recent applications as generative models.[4] Autoencoders are effectively used for solving many applied problems, from face recognition[5] to acquiring the semantic meaning of words.[6][7]