Exercise · ZW-LINEAR-LEAST-SQUARES-001 · ★☆☆ Basic
What is the gradient of \(f(x)=\|Ax-b\|_2^2\) with respect to \(x\)?
Choose one option.
Show answer
- C. \(2A^\top(Ax-b)\)
Explanation
The gradient of the squared norm is \(2A^\top(Ax-b)\), with the transpose of \(A\) multiplying the residual. The option using \(A\) without transposing is dimensionally incompatible when \(A\) is not square.
Review the concept: Linear least squares
Optimization problem that minimizes the sum of squared residuals \(\|Ax-b\|_2^2\) to approximate a vector \(b\) by linear combinations of the columns of \(A\).
\[\min_{x}\ \|Ax-b\|_2^2 \quad \Rightarrow \quad A^\top A\,x = A^\top b\]
More practice problems: linear least squares
- ZW-LINEAR-LEAST-SQUARES-002 What are the normal equations of the least squares problem?
- ZW-LINEAR-LEAST-SQUARES-003 If A has linearly independent columns, how many solutions does the least squares problem have?
- ZW-LINEAR-LEAST-SQUARES-004 What does Ax^* represent geometrically, where x^* is the least squares solution?
- ZW-LINEAR-LEAST-SQUARES-005 If A is a 3× 2 matrix, what is the dimension of A^top A?
- ZW-LINEAR-LEAST-SQUARES-006 Given A=[1 0; 0 1; 1 1] and b=(1,2,4)^top, what is A^top b?
- ZW-LINEAR-LEAST-SQUARES-007 With A^top A=[2 1; 1 2] and A^top b=(3,3)^top, what is the least squares solution?
- ZW-LINEAR-LEAST-SQUARES-008 To fit y=c t (no intercept) to the points (1,2) and (2,3), what is c?
- ZW-LINEAR-LEAST-SQUARES-009 Why is it not advisable to compute the least squares solution as x=(A^top A)^-1A^top b when A^top A…