Coranking matrix

Assessing quality of dimensionality reduction by coranking matrix based method. Reference:

1. Lee, John A., and Michel Verleysen. "Quality assessment of dimensionality reduction: Rank-based criteria." Neurocomputing 72.7-9 (2009): 1431-1443.

2. Mokbel, Bassam, et al. "Visualizing the quality of dimensionality reduction." Neurocomputing 112 (2013): 109-123.

Usage

1. Add file coranking.m into Matlab search path

2. Enter original data X and dimensionality reduced map Y with samples on the rows, and features on the columns. Then function coranking computes:

   • Ranking matrix R of X,
   • Ranking matrix r of Y,
   • Coranking matrix C based on R, r,
   • Quality index matrix Q_NX proposed in Ref. 1,
   • Improved quality index matrix Q_ND proposed in Ref. 2.

   Only need one line in Matlab: [R, r, C, Q_NX, Q_ND] = coranking(X, Y).

Examples

1. Use random matrices for X, Y. In Matlab:

N = 6000;       % sample number
n = 100;        % original data dimension (features)
m = 2;          % reduced data dimension (features)

% use random X, Y
X = rand(N, n); 
Y = rand(N, m);
[R, r, C, Q_NX, Q_ND] = coranking(X, Y);

2. File MNIST_tSNE_N6000.mat provides a real dimensionality map using t-SNE on MNIST with 6,000 handwritten digits.

load('./example2_tpSNE/MNIST_tSNE_N6000.mat')
[R, r, C, Q_NX, Q_ND] = coranking(X, Y);

3. File MNIST_pSNE_N6000.mat provides a real dimensionality map using p-SNE on MNIST with 6,000 handwritten digits.

load('./example2_tpSNE/MNIST_pSNE_N6000.mat')
[R, r, C, Q_NX, Q_ND] = coranking(X, Y);

The above matices Q_NX and Q_ND are plotted by the following command in Matlab:

% plot Q_NX
N = size(X, 1);    % sample size
figure
plot(1:N, Q_NX(1:N));
title('Q_{NX}(K)');
xlabel('K', 'FontSize', 12); ylabel('Q_{NX}(K)', 'FontSize', 12);

% plot Q_ND
figure
imagesc(Q_ND);
colormap('hot'); colorbar;
title('Q_{ND}(K_s, K_t)');
xlabel('K_t', 'FontSize', 12); ylabel('K_s', 'FontSize', 12);