(Seed for Discussion) What would be an example of a non-invertible transform? What would be the shape of a square (in 2D) or a cube (in 3D) after applying such transformations?

I suppose any Matrix with a 0 determinant would be non-inversible, and therefore be a non-invertible transform, but its harder to think of geometric examples. Maybe a very simple geometric example would be scaling something down to 0? There would be a mathematical loss of data there, so it wouldn't be possible to invert this.

Invertible matrix is rank-deficient, which means one or more column will make no use in the tranform process, and there exists some other vectors that will give us same results if we replace the column with it. Therefore, we can't find a inverse of the transform as there should also be many we can choose to trasform the point back. One simple example in 2D space is the transform matrix $\begin{matrix} 2 & 2 \\ 0 & 0 \end{matrix}$. If we want to do the inverse transform, we can choose $\begin{matrix} 0.5 & 0.5 \\ 0 & 0 \end{matrix}$ or $\begin{matrix} 0 & 0 \\ 0.5 & 0.5 \end{matrix}$ and many others. So this transform is invertible.
Hope this makes sense.

Using this picture as a example, a non-invertible transform could be one that "compress" the square into a line segment. In this way, different points in the square are mapped to the same point after transformation, which makes inverting this transformation impossible.

I believe a non-invertible operation would be an operation that would lead to loss of information. For example, as @Sicheng-Pan mentioned compression, or something simpler like masking half of the shape.