QR Decomp for Linear Least Squares

This question focuses on implementing linear least squares regression when you cannot call a library routine directly. The setup is a standard multiple linear regression with more observations than features, and the candidate is asked to design a method from first principles using QR decomposition. It is common in quantitative developer and quant research roles, especially where candidates are expected to understand the numerical linear algebra underlying regression rather than treating it as a black-box function call.

The discussion leans on ideas from matrix factorization, orthogonal transformations, and solving triangular systems. It requires recognizing how to re-express the fitting problem using a QR factorization of the design matrix and then translating that into a concrete computational procedure. Interviewers are looking for clear reasoning about why this approach is preferred over directly solving normal equations, an awareness of numerical stability issues, and the ability to articulate an algorithm step by step, including how the final coefficients are recovered and what assumptions are needed for the method to work.

What it tests

The core structure underlying least squares regression is the minimization of the squared distance between observed data and the linear model's predictions, which algebraically translates to solving an overdetermined system $X\beta = Y$ in the least-squares sense. The key insight is that orthogonal transformations, such as those provided by QR decomposition, preserve distances and angles, allowing us to transform the regression problem into a simpler equivalent one without changing its solution. By decomposing $X$ into $Q$ (orthogonal) and $R$ (upper-triangular), we exploit the fact that multiplying by an orthogonal matrix does not alter the least squares objective, so we can reduce the problem to solving a much simpler triangular system. This approach is numerically stable and avoids the pitfalls of directly inverting $X^TX$, which can be ill-conditioned and amplify errors. The pattern is that orthogonality lets us re-express complex fitting problems as tractable triangular systems, making the solution both efficient and robust.