DimensionalityReduction.tex

%\begin{frame}
%\frametitle{Curse of dimensionality}
%\begin{center}
%{\Huge Large $p$, small $n$.\par}
%\end{center}
%\end{frame}

%\begin{frame}
%\frametitle{Nearest-neighbour classification}
%\begin{columns}[c]
%\column{0.7\textwidth}
%\includegraphics[width=\textwidth]{voronoi}
%\column{0.3\textwidth}
%\begin{itemize}
%\item Not nice smooth separations.
%\item Lots of sharp corners.
%\item May be improved with \emph{K-nearest neighbours}.
%\end{itemize}
%\end{columns}
%\end{frame}

%\begin{frame}
%\frametitle{Rule-based approaches}
%\begin{columns}[c]
%\column{0.7\textwidth}
%\includegraphics[width=\textwidth]{rule_based}
%\column{0.3\textwidth}
%\begin{itemize}
%\item Not nice smooth separations.
%\item Lots of sharp corners.
%\end{itemize}
%\end{columns}
%\end{frame}


%\begin{frame}
%\frametitle{Unintuitive behaviour in high-dimensions}
%\begin{columns}[c]
%\column{0.4\textwidth}
%\includegraphics[width=\textwidth]{circle}
%\column{0.6\textwidth}
%\includegraphics[width=\textwidth]{sphere}
%\end{columns}
%\end{frame}




%\begin{frame}
%\frametitle{Unintuitive behaviour in high-dimensions}
%\begin{columns}[c]
%\column{0.2\textwidth}
%\includegraphics[width=.8\textwidth]{circle}\par
%\includegraphics[width=\textwidth]{sphere}\par
%\column{0.8\textwidth}
%\includegraphics[width=\textwidth]{corners}
%\end{columns}
%\end{frame}




\begin{frame}
\frametitle{Dimensionality $\ne$ number of voxels}
\begin{itemize}
\item Lots of effort on data-driven feature selection methods.
\begin{itemize}
\item Involves estimating\\
      $\boldsymbol\Sigma_0 = diag({\bf s}), s_i \in \{0,w\}$, where $w \in \mathcal{R}^+$.
\item Lots of parameters needed to achieve this.
\end{itemize}
\item Many papers claim excellent results.
\item Little evidence to suggest that most voxel-based feature selection methods help.
\begin{itemize}
\item Little or no increase in predictive accuracy.
\item Commonly perceived as being more ``interpretable''.
\end{itemize}
%\item Prior knowledge derived from independent data is the most reliable way to improve accuracy.
%\begin{itemize}
%\item e.g. search the literature for clues about which regions to weight more heavily.
%\end{itemize}
\end{itemize}
\end{frame}

%\begin{frame}
%\frametitle{``Data-driven feature selection''}
%\begin{itemize}
%\item Lots of effort on data-driven feature selection methods.
%\item Many papers claim excellent results.
%\item Not much evidence that it really improves accuracy.
%\end{itemize}
%\end{frame}



\begin{frame}
\frametitle{``Data-driven feature selection''}
%The main objectives of the feature selection step are to keep only informative features and to reduce the dimensionality of the feature space.
\begin{quote}
``In our evaluation, two methods included a feature selection step: Voxel-STAND and Voxel-COMPARE.
Overall, these methods did not perform substantially better than simpler ones... ...
%In particular, their results might be more sensitive to the training set.
%Indeed, feature selection can be regarded as a learning step.
%In such a case, the feature selection step increases the class of all possible classification functions, which could lead to overfitting the data.
A more robust way to decrease the dimensionality of the features way would be to use more prior knowledge of the disease.''

%Besides features selection can be time consuming as it adds new hyperparameters and thus makes the grid search less tractable.
%Compared to Voxel-Direct and Voxel-Atlas, Voxel-STAND and Voxel-COMPARE are time consuming (up to weeks), mostly because of the number of hyperparameters to be tuned.

%Nevertheless, feature selection proved useful in two specific cases.
%First, these methods proved less sensitive when increasing the dimensionality of the feature space by adding WM and CSF maps.
%They also tended to be more accurate for the MCIc vs MCInc experiment, where only a few brain regions are informative.
\end{quote}

\begin{center}
\begin{tiny}
Cuingnet et al. ``Automatic classification of patients with Alzheimer's disease from structural MRI: A comparison of ten methods using the ADNI database''.  NeuroImage 56(2):766--781 (2011).\par
\end{tiny}
\end{center}
\end{frame}




\begin{frame}
\frametitle{``Data-driven feature selection''}
\begin{center}
\includegraphics[height=0.55\textwidth]{data_driven_feature_selection.png}\par
\begin{tiny}
Chu et al. ``Does feature selection improve classification accuracy? Impact of sample size and feature selection on classification using anatomical magnetic resonance images''.  NeuroImage 60:59--70 (2012).\par
\end{tiny}
\end{center}
\end{frame}




\begin{frame}
\frametitle{``Data-driven feature selection''}
...did not help the winning entry.
\begin{center}
\includegraphics[height=0.5\textwidth]{pbaic2007}\par
\url{http://www.lrdc.pitt.edu/ebc/2007/2007.html}\par
\end{center}
\end{frame}




\begin{frame}
\frametitle{``Data-driven feature selection''}
\begin{center}
\includegraphics[height=0.5\textwidth]{pbaic2007_2}\par
\url{http://www.lrdc.pitt.edu/ebc/2007/2007.html}\par
\end{center}
\end{frame}




\begin{frame}
\frametitle{Modelling the nonlinearities}
Instead of using nonlinear pattern recognition methods...
\begin{itemize}
\item Capture nonlinearities by appropriate preprocessing.
\item Allows nonlinear effects to be modelled by a linear classifier.
\item Gives more interpretable characterisations of differences.
\item May lead to more accurate predictions.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Transformed images fall on manifolds}
Rotating an image leads to points on a manifold.\par
\includegraphics[width=\textwidth]{manifold_rotation}

Rigid-body motion leads to a 6-dimensional manifold (not shown).

\end{frame}

%\begin{frame}
%\frametitle{Local linearisation through smoothing}
%\begin{columns}[c]
%\column{0.7\textwidth}
%\includegraphics[width=\textwidth]{manifold_unsmoothed}\par
%\includegraphics[width=\textwidth]{manifold_smoothed}
%\column{0.3\textwidth}
%Spatial smoothing can make the manifolds more linear with respect to small misregistrations.
%
%Some information is inevitably lost.
%\end{columns}
%\end{frame}

\begin{frame}
\frametitle{One mode of geometric variability}
\begin{columns}[c]
\column{0.5\textwidth}
Simulated images\par
\includegraphics[height=0.9\textwidth]{circles}
\column{0.5\textwidth}
Principal components\par
\includegraphics[height=0.9\textwidth]{circles_pca}
\end{columns}
A suitable model would reduce this variability to a single dimension.
\end{frame}




\begin{frame}
\frametitle{Two modes of geometric variability}
\begin{columns}[c]
\column{0.5\textwidth}
Simulated images\par
\includegraphics[height=0.9\textwidth]{things}
\column{0.5\textwidth}
Principal components\par
\includegraphics[height=0.9\textwidth]{things_pca}
\end{columns}
A suitable model would reduce this variability to two dimensions.
\end{frame}

%\begin{frame}
%\frametitle{Simulated shapes}
%\includegraphics[width=0.9\textwidth]{original}
%\end{frame}

%\begin{frame}
%\frametitle{Simulated nonlinear variability}
%\includegraphics[width=0.9\textwidth]{simulated}
%\end{frame}

%\begin{frame}
%\frametitle{Simulated linear variability}
%\includegraphics[width=0.9\textwidth]{simulated_lin}
%\end{frame}