How can QR-decomposition be used for low-rank estimation? In a coursebook I'm reading, some algorithms used to calculate the QR-decomposition of a matrix are introduced, namely Gram-Schmidt orthogonalization and the use of Givens rotations with or without column pivoting.

A question that has been posed on previous exams asks how the QR-decomposition can be used to do low-rank approximation of a matrix. This is unclear to me. I understand how this can be done using singular value decomposition, but cannot seem to grasp how the same can be achieved using the QR-decomposition.

Thanks in advance!
You want a "rank revealing" QR decomposition, see https://math.berkeley.edu/~mgu/MA273/Strong_RRQR.pdf or https://people.csail.mit.edu/yujia/assets/pdf/rrqr_slides.pdf or https://www.irisa.fr/sage/wg-statlin/WORKSHOPS/LEMASSOL05/SLIDES/QR/Guyomarch.pdf (or even how rank-revealing QR factorization determine the rank of the matrix)
You want a "rank revealing" QR decomposition, see https://math.berkeley.edu/~mgu/MA273/Strong_RRQR.pdf or https://people.csail.mit.edu/yujia/assets/pdf/rrqr_slides.pdf or https://www.irisa.fr/sage/wg-statlin/WORKSHOPS/LEMASSOL05/SLIDES/QR/Guyomarch.pdf (or even how rank-revealing QR factorization determine the rank of the matrix)