What is the goal of sufficient dimension reduction? Under what circumstances can it be achieved? I have recently heard the term "sufficient dimension reduction" tossed around, although I have struggled to find material on the concept that I fully understand or that clearly explains why this specialized variant of dimension reduction is needed to begin with.
What is the goal of sufficient dimension reduction techniques? Why can't it already be accomplished by non-sufficient dimension reduction techniques? When can this goal be achieved and when is is impossible?
I've heard that "sufficient dimension reduction" simply denotes a reduction of dimensions without a loss of information, but I struggle to understand when this could occur unless there are linear dependencies in the data (and in such a situation I don't see why a new theoretical framework of data reduction would be necessary to eliminate the dependency).
See Adragni and Cook (2009) "Sufficient dimension reduction and prediction in regression," which contains a definition of "sufficient dimension reduction" a few paragraphs into the article. This article may be accessed here for those with access to Royal Society Publishing, or check the Wikipedia article here for a brief overview.
The few researchers who discuss this concept tend to present it as a new "paradigm," but the concept is over a decade old and does not seem to have caught on beyond very niche groups mainly in theoretical research.
EDIT: It may be helpful if someone could find an example of an applied use of sufficient dimension reduction. Is there a study that has found and used a sufficient dimension reduction on a large real world dataset?
 A: I will give my interpretation of the Wikipedia article, demonstrated through an R simulation.
library(ggplot2)
set.seed(2021)
N <- 1000
x0 <- x1 <- seq(-1, 1, 0.01)
y0 <- sqrt(1 - x0^2)
y1 <- -sqrt(1 - x1^2)
x <- c(x0, x1)
y <- c(y0, y1)
g <- as.factor(c(rep(0, length(x0)), rep(1, length(x0))))
d <- data.frame(x, y, g)
ggplot(d, aes(x = x, y = y, col = g)) +
    geom_point() +
    theme_bw()


If the $(x,y)$ coordinates are your features, you might think you have a two-dimensional feature space. After all, one dimension is for $x$, and another is for $y$. However, you can, without losing any information, reduce your feature space to one dimension, by only considering the angle.
Thus, your model would look at the angle $\theta$ and assign $P(\text{red}\vert\theta)$ based on the angle...and only the angle, which would be equivalent to $P(\text{red}\vert x, y)$. Since all points are distance $1$ from the origin, the distance from the point does not matter, just the angle.
That would be sufficient dimension reduction from $2$ to $1$.
A: 
What is the goal of sufficient dimension reduction techniques? Why can't it already be accomplished by non-sufficient dimension reduction techniques? When can this goal be achieved and when is is impossible?

The goal of Sufficient Dimension Reduction is to find a lower-dimensional transformation $R(X)$ of the predictor vector $X$ in a regression/classification scenario in such a way that no predictive power for the response $Y$ is lost. It is a supervised approach to dimension reduction.
Historically, only linear transformations were permitted but even these can found under quite general relationships between $X$ and $Y$. Consider, for example, the regression model $Y = f(\beta^{T}X) + \epsilon$. Here $R(X) = \beta^{T}X$ is a one-dimensional reduction which contains all the predictive power that $X$ has for $Y$. More recent work allows for nonlinear reductions.

EDIT: It may be helpful if someone could find an example of an applied use of sufficient dimension reduction. Is there a study that has found and used a sufficient dimension reduction on a large real world dataset?

See https://www.sciencedirect.com/science/article/abs/pii/S0025556401001067 for an application in Genetics.
The main reference book is "Sufficient Dimension Reduction: Methods and Applications with R" by Bing Li.
