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Documentation bug fixes.

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Thanks Vladimir Chalupecky

Change-Id: I52a11d75adbf7adb1233c4d5ec2bc599448ab240
This commit is contained in:
Sameer Agarwal
2012-11-01 10:22:11 -07:00
parent 3dadfb78b7
commit c811d010c0
1 changed files with 10 additions and 8 deletions
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docs/modeling.tex
+10 -8
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@@ -2,10 +2,12 @@
\chapter{Modeling}
\label{chapter:api}
\section{\texttt{CostFunction}}
Given parameter blocks $\left[x_{i_1}, \hdots , x_{i_k}\right]$, a \texttt{CostFunction} is responsible for computing
a vector of residuals and if asked a vector of Jacobian matrices, i.e., given $\left[x_{i_1}, \hdots , x_{i_k}\right]$, compute the vector $f_i\left(x_{i_1},\hdots,x_{k_i}\right)$ and the matrices
Given parameter blocks $\left[x_{i_1}, \hdots , x_{i_k}\right]$, a
\texttt{CostFunction} is responsible for computing
a vector of residuals and if asked a vector of Jacobian matrices, i.e., given $\left[x_{i_1}, \hdots , x_{i_k}\right]$, compute the vector $f_i\left(x_{i_1},\hdots,x_{i_k}\right)$ and the matrices
\begin{equation}
J_{ij} = \frac{\partial}{\partial x_{i_j}}f_i\left(x_{i_1},\hdots,x_{k_i}\right),\quad \forall j = i_1,\hdots, i_k
J_{ij} = \frac{\partial}{\partial x_{j}}f_i\left(x_{i_1},\hdots,x_{i_k}\right),\quad \forall j \in \{i_1,\hdots, i_k\}
\end{equation}
\begin{minted}{c++}
class CostFunction {
@@ -90,8 +92,8 @@ class MyScalarCostFunction {
MyScalarCostFunction(double k): k_(k) {}
template <typename T>
bool operator()(const T* const x , const T* const y, T* e) const {
e[0] = T(k_) - x[0] * y[0] + x[1] * y[1]
return true;
e[0] = T(k_) - x[0] * y[0] - x[1] * y[1];
return true;
}
private:
@@ -265,12 +267,12 @@ block.
Then, the robustified gradient and the Gauss-Newton Hessian are
\begin{align}
g(x) &= \rho'J^\top(x)f(x)\\
H(x) &= J^\top(x)\left(\rho' + 2 \rho''f(x)f^\top(x)\right)J(x)
H(x) &= J^\top(x)\left(\rho' + 2 \rho''f(x)f^\top(x)\right)J(x)
\end{align}
where the terms involving the second derivatives of $f(x)$ have been ignored. Note that $H(x)$ is indefinite if $\rho''f(x)^\top f(x) + \frac{1}{2}\rho' < 0$. If this is not the case, then its possible to re-weight the residual and the Jacobian matrix such that the corresponding linear least squares problem for the robustified Gauss-Newton step.
where the terms involving the second derivatives of $f(x)$ have been ignored. Note that $H(x)$ is indefinite if $\rho''f(x)^\top f(x) + \frac{1}{2}\rho' < 0$. If this is not the case, then its possible to re-weight the residual and the Jacobian matrix such that the corresponding linear least squares problem for the robustified Gauss-Newton step.
Let $\alpha$ be a root of
Let $\alpha$ be a root of
\begin{equation}
\frac{1}{2}\alpha^2 - \alpha - \frac{\rho''}{\rho'}\|f(x)\|^2 = 0.
\end{equation}