My professor is teaching us about a clustering algorithm called K-means clustering. He used the term projection in a way I'm unfamiliar with.
Assume all vectors in this example are in $\mathbb{R}^d$. Suppose we are given a collection of vectors $C = \lbrace c_1,...,c_K\rbrace$. Now consider a vector $x$ and let $d$ be the Euclidean metric. Then, suppose I want to find the $c_k \in C$ such that
$$d(x,c_k)\leq d(x,c_j), \forall j\ne k$$
My professor chose to denote this
$$\Pi_C(x)=\underset{c_k\in C}{\text{argmin}}\lbrace \lVert x-c_k\rVert_2 \rbrace$$
He called this the "projection" of $x$ onto $C$.
In what sense is this a projection? I thought a projection was a linear transformation $T:V\rightarrow W$ where $V,W$ are both vector spaces. What sense of the term projection is he using?
Wikipedia's page on projections implicitly covers the type of map your professor describes, since under "Definition" it says:
and having a right inverse is the same thing as being surjective. Your arg min map is obviously surjective as a map from $\mathbb R^d$ to $C \subset \mathbb R^d$.
For a more geometric perspective, note that $\lVert x - c_k \rVert_2$ is the distance between $x$ and $c_k$, as induced by the norm $\lVert \cdot \rVert_2$. So your professor is asking for the point in $C$ that is closest to $x$. This is very much consistent with what it intuitively means geometrically to project a point $y$ onto a set $A$, like the $x$-$y$ plane for example.