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-rwxr-xr-xreport/paper.md33
1 files changed, 25 insertions, 8 deletions
diff --git a/report/paper.md b/report/paper.md
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@@ -143,11 +143,13 @@ use effectively 97% of the information from our initial training data for recons
\end{center}
\end{figure}
-The analysed classification methods used for face recognition are **Nearest Neighbor** and
-**alternative method** through reconstruction error.
-EXPLAIN THE METHODS
+The analysed classification methods used for face recognition are Nearest Neighbor and
+alternative method through reconstruction error.
-REFER TO ACCURACY GRAPH 1 FOR NN. MAYBE WE CAN ALSO ADD SAME GRAPH WITH DIFFERENT K
+Nearest Neighbor projects the test data onto the generated subspace and find the closest
+element to the projected test image, assigning the same class as the neighbor found.
+
+Recognition accuracy of NN classification can be observed in Figure 4 (CHANGE TO ALWAYS POINT AT THE GRAPH, DUNNO HOW).
A confusion matrix showing success and failure cases for Nearest Neighbor classfication
can be observed below:
@@ -179,6 +181,11 @@ classification.
\end{center}
\end{figure}
+The process for alternative method is somewhat similar to LDA. One different
+subspace is generated for each class. These subspaces are then used for reconstruction
+of the test image and the class of the subspace that generated the minimum reconstruction
+error is assigned.
+
The alternative method shows overall a better performance, with peak accuracy of 69%
for M=5. The maximum M non zero eigenvectors that can be used will in this case be at most
the amount of training samples per class minus one, since the same amount of eigenvectors
@@ -225,7 +232,15 @@ The pictures on the right show the reconstructed images.
# Question 2, Generative and Discriminative Subspace Learning
-Maximize function J(W) (Fisher's Criterion):
+As mentioned in the introduction, PCA is a generative method that allows to perform dimensionality
+reduction while keeping most of the information from the initial training data. It is a very good method for
+reconstruction and allows very fast computation. LDA is instead a discriminative method that uses a high
+dimensional space for computation. It comes with a very high classification accuracy, with the tradeoff of
+being slightly slower than PCA, and not as good for face reconstruction.
+
+To combine both method it is possible to perform LDA in a generative subspace created by PCA. In order to
+maximize class separation and minimize the distance between elements of the same class it is necessary to
+maximize the function J(W) (Fisher's Criterion):
$$ J(W) = \frac{W\textsuperscript{T}S\textsubscript{B}W}{W\textsuperscript{T}S\textsubscript{W}W}\textrm{ or } J(W) = \frac{W\textsuperscript{T}S\textsubscript{B}W}{W\textsuperscript{T}S\textsubscript{t}W} $$
@@ -263,12 +278,14 @@ $$ P\textsuperscript{T}S\textsubscript{B}P = \widetilde{S}\textsubscript{B} \tex
$$ J(W) = \widetilde{J}(W) = \frac{X\textsuperscript{T}\widetilde{S}\textsubscript{B}X}{X\textsuperscript{T}\widetilde{S}\textsubscript{t}X} $$
$\widetilde{S}\textsubscript{B} \textrm{ and } \widetilde{S}\textsubscript{t}$
-are respectively semi-positive definite and positive definite. So $\widetilde{J}(X)$
-acts like Fisher's criterion but in PCA transformed space. This method
-does not result in any loss of data.
+are respectively semi-positive definite and positive definite. $\widetilde{J}(X)$
+similarly to the original J(X), applies Fisher's criterion in a PCA generated subspace.
+This enables to perform LDA minimizing loss of data.
*Proof:*
+REWRITE FROM HERE
+
The set of optimal discriminant vectors can be found in R\textsuperscript{n}
for LDA. But, this is a difficult computation because the dimension is very
high. Besides, S\textsubscript{t} is always singular. Fortunately, it is possible