In statistics, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of dimension reduction with the concept of sufficiency.

Dimension reduction has long been a primary goal of regression analysis. Given a response variable y and a p-dimensional predictor vector ${\displaystyle {\textbf {x}}}$, regression analysis aims to study the distribution of ${\displaystyle y\mid {\textbf {x}}}$, the conditional distribution of ${\displaystyle y}$ given ${\displaystyle {\textbf {x}}}$. A dimension reduction is a function ${\displaystyle R({\textbf {x}})}$ that maps ${\displaystyle {\textbf {x}}}$ to a subset of ${\displaystyle \mathbb {R} ^{k}}$, k < p, thereby reducing the dimension of ${\displaystyle {\textbf {x}}}$. For example, ${\displaystyle R({\textbf {x}})}$ may be one or more linear combinations of ${\displaystyle {\textbf {x}}}$.

A dimension reduction ${\displaystyle R({\textbf {x}})}$ is said to be sufficient if the distribution of ${\displaystyle y\mid R({\textbf {x}})}$ is the same as that of ${\displaystyle y\mid {\textbf {x}}}$. In other words, no information about the regression is lost in reducing the dimension of ${\displaystyle {\textbf {x}}}$ if the reduction is sufficient.

In a regression setting, it is often useful to summarize the distribution of ${\displaystyle y\mid {\textbf {x}}}$ graphically. For instance, one may consider a scatterplot of ${\displaystyle y}$ versus one or more of the predictors or a linear combination of the predictors. A scatterplot that contains all available regression information is called a sufficient summary plot.

When ${\displaystyle {\textbf {x}}}$ is high-dimensional, particularly when ${\displaystyle p\geq 3}$, it becomes increasingly challenging to construct and visually interpret sufficiency summary plots without reducing the data. Even three-dimensional scatter plots must be viewed via a computer program, and the third dimension can only be visualized by rotating the coordinate axes. However, if there exists a sufficient dimension reduction ${\displaystyle R({\textbf {x}})}$ with small enough dimension, a sufficient summary plot of ${\displaystyle y}$ versus ${\displaystyle R({\textbf {x}})}$ may be constructed and visually interpreted with relative ease.

Hence sufficient dimension reduction allows for graphical intuition about the distribution of ${\displaystyle y\mid {\textbf {x}}}$, which might not have otherwise been available for high-dimensional data.

Most graphical methodology focuses primarily on dimension reduction involving linear combinations of ${\displaystyle {\textbf {x}}}$. The rest of this article deals only with such reductions.

Suppose ${\displaystyle R({\textbf {x}})=A^{T}{\textbf {x}}}$ is a sufficient dimension reduction, where ${\displaystyle A}$ is a ${\displaystyle p\times k}$ matrix with rank ${\displaystyle k\leq p}$. Then the regression information for ${\displaystyle y\mid {\textbf {x}}}$ can be inferred by studying the distribution of ${\displaystyle y\mid A^{T}{\textbf {x}}}$, and the plot of ${\displaystyle y}$ versus ${\displaystyle A^{T}{\textbf {x}}}$ is a sufficient summary plot.