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# Generalized Discriminant Analysis Projection Matrix

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wbadry on 6 Jun 2020
Hello,
I tried to perform a supervised dimensionality reduction using GDA which is also known as Kernel Fisher Discriminant Analysis. The code was written by Laurens van der Maaten . The function perfectly works as the dimensionality is reduced to 2 features and separation is good.
My problem is I am unable to find the projection matrix that I should multiply the new incoming features point (say 100 features by 1 sample) so that the result is a new mapped projected features (2 festures by 1 sample). Tried to analyze the variables but unfortunately nothing cameout to be this matrix.
function mappedX = gda(X, Y, no_dims, varargin)
%GDA Perform Generalized Discriminant Analysis
%
% mappedX = gda(X, Y, no_dims)
% mappedX = gda(X, Y, no_dims, kernel)
% mappedX = gda(X, Y, no_dims, kernel, param1)
% mappedX = gda(X, Y, no_dims, kernel, param1, param2)
%
% Perform Generalized Discriminant Analysis. GDA or Kernel LDA is the
% nonlinear generalization of LDA by means of the kernel trick. X is the
% data on which to perform GDA, Y are the corresponding labels.
% The value of kernel determines the used kernel. Possible values are 'linear',
% 'gauss', 'poly', 'subsets', or 'princ_angles' (default = 'gauss'). For
% more info on setting the parameters of the kernel function, type HELP
% GRAM.
% The function returns the locations of the embedded trainingdata in
% mappedX.
%
%
% This file is part of the Matlab Toolbox for Dimensionality Reduction.
% The toolbox can be obtained from http://homepage.tudelft.nl/19j49
% You are free to use, change, or redistribute this code in any way you
% want for non-commercial purposes. However, it is appreciated if you
% maintain the name of the original author.
%
% (C) Laurens van der Maaten, Delft University of Technology
% Process inputs
if ~exist('no_dims', 'var')
no_dims = 2;
end
kernel = 'gauss';
param1 = 1;
param2 = 0;
if length(varargin) > 0 & strcmp(class(varargin{1}), 'char'), kernel = varargin{1}; end
if length(varargin) > 1 & strcmp(class(varargin{2}), 'double'), param1 = varargin{2}; end
if length(varargin) > 2 & strcmp(class(varargin{3}), 'double'), param2 = varargin{3}; end
% Make sure labels are nice
[foo, bar, Y] = unique(Y, 'rows');
% Get dimensions
[n, dim] = size(X);
nclass = max(Y);
% Sort data according to labels
[foo, ind] = sort(Y);
Y = Y(ind);
X = X(ind,:);
% Compute kernel matrix
disp('Computing kernel matrix...');
K = gram(X, X, kernel, param1, param2);
% Compute centering matrix
ell = size(X, 1);
D = sum(K) / ell;
E = sum(D) / ell;
J = ones(ell, 1) * D;
K = K - J - J' + E * ones(ell, ell);
% Perform eigenvector decomposition of kernel matrix (Kc = P * gamma * P')
disp('Performing eigendecomposition of kernel matrix...');
K(isnan(K)) = 0;
K(isinf(K)) = 0;
[P, gamma] = eig(K)
if size(P, 2) < n
error('Singularities in kernel matrix prevent solution.');
end
% Sort eigenvalues and vectors in descending order
[gamma, ind] = sort(diag(gamma), 'descend');
P = P(:,ind);
% Remove eigenvectors with relatively small value
minEigv = max(gamma) / 1e5;
ind = find(gamma > minEigv);
P = P(:,ind);
gamma = gamma(ind);
rankK = length(ind);
% Recompute kernel matrix
K = P * diag(gamma) * P';
% Construct diagonal block matrix W
W = [];
for i=1:nclass
num_data_class = length(find(Y == i));
W = blkdiag(W, ones(num_data_class) / num_data_class)
end
% Determine target dimensionality of data
old_no_dims = no_dims;
no_dims = min([no_dims rankK nclass]);
if old_no_dims > no_dims
warning(['Target dimensionality reduced to ' num2str(no_dims) '.']);
end
% Perform eigendecomposition of matrix (P' * W * P)
disp('Performing GDA eigenanalysis...');
[Beta, lambda] = eig(P' * W * P)
lambda = diag(lambda)
% Sort eigenvalues and eigenvectors in descending order
[lambda, ind] = sort(lambda, 'descend')
Beta = Beta(:,ind(1:no_dims))
% Compute final embedding mappedX
mappedX = P * diag(1 ./ gamma) * Beta;
% Normalize embedding
for i=1:no_dims
mappedX(:,i) = mappedX(:,i) / sqrt(mappedX(:,i)' * K * mappedX(:,i));
end

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