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diff --git a/report/paper.md b/report/paper.md index 68df6fd..de62e4f 100755 --- a/report/paper.md +++ b/report/paper.md @@ -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 |