%% ============================================================
%   LEVERAGE SCORE DEMONSTRATION (MATLAB ONLY)
%   Shows WHY leverage uses U_k, not A.
%   Produces plots + heatmaps for direct comparison.
%% ============================================================

close all; clear; clc;

%% ---- Load 256x256 grayscale image ----
A = double(imread('cameraman.bmp'));
A = A(:,:,1);           % ensure grayscale
[m,n] = size(A);

%% ---- Choose rank k ----
k = 50;                 % good test value

%% ---- Compute rank-k SVD ----
[U,S,V] = svds(A, k);

%% ---- TRUE leverage scores (from U_k) ----
true_row_lev = sum(U.^2, 2);     % m×1
true_col_lev = sum(V.^2, 2);     % n×1

%% ---- FAKE leverage scores (wrong: directly from A) ----
fake_row_lev = sum(A.^2, 2);     % sum of row squares
fake_col_lev = sum(A.^2, 1)';    % sum of col squares

%% ---- Normalize both for comparison ----
true_row_lev = true_row_lev / sum(true_row_lev);
fake_row_lev = fake_row_lev / sum(fake_row_lev);

true_col_lev = true_col_lev / sum(true_col_lev);
fake_col_lev = fake_col_lev / sum(fake_col_lev);

%% ============================================================
%                   VISUALIZATIONS 
% ============================================================

figure('Position',[200 200 1200 700]);

%% ---- Plot row leverage scores ----
subplot(2,2,1);
plot(true_row_lev, 'LineWidth', 2); hold on;
plot(fake_row_lev, '--', 'LineWidth', 2);
title('Row Leverage Scores');
legend('True leverage (U_k)', 'Fake leverage (A)', 'Location','Best');
xlabel('Row index'); ylabel('Score');

%% ---- Plot column leverage scores ----
subplot(2,2,2);
plot(true_col_lev, 'LineWidth', 2); hold on;
plot(fake_col_lev, '--', 'LineWidth', 2);
title('Column Leverage Scores');
legend('True leverage (V_k)', 'Fake leverage (A)', 'Location','Best');
xlabel('Column index'); ylabel('Score');

%% ---- Heatmap of true leverage (rows) ----
subplot(2,2,3);
imagesc(repmat(true_row_lev, 1, n)); 
colormap turbo; colorbar;
title('TRUE Row Leverage Heatmap (from U_k)');
xlabel('Column'); ylabel('Row');

%% ---- Heatmap of fake leverage (rows) ----
subplot(2,2,4);
imagesc(repmat(fake_row_lev, 1, n)); 
colormap turbo; colorbar;
title('FAKE Row Leverage Heatmap (from A)');
xlabel('Column'); ylabel('Row');

sgtitle('Why CUR Uses U_k for Leverage Scores, Not A', 'FontSize', 16);