\newpage
\subsection{Non-Linear Least Squares}
Find $\vec{x}^* = \text{argmax}\; \P(\vec{x} | \vec{z}) = \argmin{}(-\log(\P(\vec{x}|\vec{z})))$

\bi{Gauss-Newton} Need func $F$ and its Jacobian (or derivative) $DF$
\rmvspace[0.3]
% TODO: Error propagation laws
{\small
    \begin{minted}[
    breaklines,
    breakindentnchars=2,
]{python}
def gauss_newton(x: np.ndarray, F, DF, tol=1e-6):
    s = np.linalg.lstsq(DF(x), F(x))[0] # least sq.
    x = x-s; k = 1
    while np.linalg.norm(s) > tol * np.linalg.norm(x):
        s = np.linalg.lstsq(DF(x), F(x))[0]
        x = x-s; k += 1 # k optional for max iter
    return x, k
\end{minted}
}
\rmvspace[0.4]

\bi{Levenberg-Marquardt}
\bi{(1)} Pick start point $\overline{\vec{x}}^0$ and start param $\lambda^0 = \max \text{diag}(\mat{A})$ and $v$ (e.g. $v = 2$).;
\bi{(2)} Modified GN sys: $\mat{A} + \lambda \text{diag}(\mat{A}) \Delta \vec{x} = \vec{b}$;
\bi{(3)} Solve for $\Delta \vec{x}$;
\bi{(4)} Update: $\overline{\vec{x}}^{k + 1} = \overline{\vec{x}}^k + \Delta \vec{x}$ and $\lambda^{k + 1} = \lambda^k \div v$ (if cost reduced),
else: $\overline{\vec{x}}^{k + 1} = \overline{\vec{x}}^k$ and $\lambda^{k + 1} = \lambda^k v$, go to step 3;
\bi{(5)} Check convergence, else go to step 2

\bi{Robust Cost Functions} Account for outliers, by mod. err. terms