$\newcommand{\ones}{\mathbf 1}$
Least-Squares
In practical applications of least-squares polynomial fitting, it is always best to use the highest-order polynomial that is computationally feasible.
True
Incorrect.
False
Correct!
Least-squares approximate solution of overdetermined equations
is a method to find an exact solution of $Ax=b$
✓
✗
This option is incorrect. $Ax=b$ may not have an exact solution
is a method to find $A^{-1}b$
✓
✗
This option is incorrect.
is a method for finding a value of $x$ that minimizes $\|Ax−b\|$
✓
✗
This option is correct.
is a method to find $x$ if there exists $x$ that satisfies $Ax=b$
✓
✗
This option is correct.
Submit
Suppose $b=Ax+v$, where $x \in \mathbf{R}^n$ is some set of parameters you wish to estimate, $b \in \mathbf{R}^m$ is a set of measurements, and $v$ represents a noise. We assume $m>n$. Consider an estimator of the form $\hat{x} = Bb$.
Choosing $B$ to be any left inverse of $A$ yields $\hat{x} = x$, matter what x is, provided $v=0$
✓
✗
This option is correct.
The choice $B=(A^TA)^{−1}A^T$ yields $\hat{x}=x$, provided $v$ is small
✓
✗
This option is incorrect.
The choice $B=(A^TA)^{−1}A^T$ yields $\hat{x}$ that is closest to $x$
✓
✗
This option is incorrect.
The choice $B=(A^TA)^{−1}A^T$ yields $\hat{x}$ that minimizes norm of $Ax - b$
✓
✗
This option is correct.
Submit
Least-Squares Data Fitting
Least-squares function fitting works well for interpolation, but should never be used for extrapolation.
True
Incorrect.
False
Correct!
Regularization
Regularized least-squares, i.e., choosing $x$ to minimize $\|Ax−b\|^2 + \mu\|x\|^2, \text{with } \mu > 0$
can always be done, even when A is not wide
Correct!
fails when A is not skinny
Incorrect.
requires only that A is nonzero
Incorrect.
Suppose that $x$ minimizes $J_1(x) + \mu J_2(x)$, for some value of $\mu > 0$, but you'd like to find a point with a smaller value of $J_2$, if possible. You should
decrease the parameter $\mu$ and minimize $J_1+\mu J_2$
Incorrect.
increase the parameter $\mu$ and minimize $J_1+\mu J_2$
Correct!
minimize $J_1 + (1/\mu)J_2$
Incorrect.
minimize $J_1 − \mu J_2$
Incorrect.