In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors X = (X₁, ..., X_n) and Y = (Y₁, ..., Y_m) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of X and Y that have a maximum correlation with each other. T. R. Knapp notes that "virtually all of the commonly encountered parametric tests of significance can be treated as special cases of canonical-correlation analysis, which is the general procedure for investigating the relationships between two sets of variables." The method was first introduced by Harold Hotelling in 1936, although in the context of angles between flats the mathematical concept was published by Camille Jordan in 1875.

CCA is now a cornerstone of multivariate statistics and multi-view learning, and a great number of interpretations and extensions have been proposed, such as probabilistic CCA, sparse CCA, multi-view CCA, deep CCA, and DeepGeoCCA. Unfortunately, perhaps because of its popularity, the literature can be inconsistent with notation, we attempt to highlight such inconsistencies in this article to help the reader make best use of the existing literature and techniques available.

Like its sister method PCA, CCA can be viewed in population form (corresponding to random vectors and their covariance matrices) or in sample form (corresponding to datasets and their sample covariance matrices). These two forms are almost exact analogues of each other, which is why their distinction is often overlooked, but they can behave very differently in high dimensional settings. We next give explicit mathematical definitions for the population problem and highlight the different objects in the so-called canonical decomposition - understanding the differences between these objects is crucial for interpretation of the technique.

Given two column vectors ${\displaystyle X=(x_{1},\dots ,x_{n})^{T}}$ and ${\displaystyle Y=(y_{1},\dots ,y_{m})^{T}}$ of random variables with finite second moments, one may define the cross-covariance ${\displaystyle \Sigma _{XY}=\operatorname {cov} (X,Y)}$ to be the ${\displaystyle n\times m}$ matrix whose ${\displaystyle (i,j)}$ entry is the covariance ${\displaystyle \operatorname {cov} (x_{i},y_{j})}$. In practice, we would estimate the covariance matrix based on sampled data from ${\displaystyle X}$ and ${\displaystyle Y}$ (i.e. from a pair of data matrices).

Canonical-correlation analysis seeks a sequence of vectors ${\displaystyle a_{k}}$ (${\displaystyle a_{k}\in \mathbb {R} ^{n}}$) and ${\displaystyle b_{k}}$ (${\displaystyle b_{k}\in \mathbb {R} ^{m}}$) such that the random variables ${\displaystyle a_{k}^{T}X}$ and ${\displaystyle b_{k}^{T}Y}$ maximize the correlation ${\displaystyle \rho =\operatorname {corr} (a_{k}^{T}X,b_{k}^{T}Y)}$. The (scalar) random variables ${\displaystyle U=a_{1}^{T}X}$ and ${\displaystyle V=b_{1}^{T}Y}$ are the first pair of canonical variables. Then one seeks vectors maximizing the same correlation subject to the constraint that they are to be uncorrelated with the first pair of canonical variables; this gives the second pair of canonical variables. This procedure may be continued up to ${\displaystyle \min\{m,n\}}$ times.

The sets of vectors ${\displaystyle a_{k},b_{k}}$ are called canonical directions or weight vectors or simply weights. The 'dual' sets of vectors ${\displaystyle \Sigma _{XX}a_{k},\Sigma _{YY}b_{k}}$ are called canonical loading vectors or simply loadings; these are often more straightforward to interpret than the weights.

Let ${\displaystyle \Sigma _{XY}}$ be the cross-covariance matrix for any pair of (vector-shaped) random variables ${\displaystyle X}$ and ${\displaystyle Y}$. The target function to maximize is

The first step is to define a change of basis and define