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+math/fsvd.m

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+function [U,S,V] = fsvd(A, k, i, usePowerMethod)
+% FSVD Fast Singular Value Decomposition 
+% 
+%   [U,S,V] = FSVD(A,k,i,usePowerMethod) computes the truncated singular
+%   value decomposition of the input matrix A upto rank k using i levels of
+%   Krylov method as given in [1], p. 3.
+% 
+%   If usePowerMethod is given as true, then only exponent i is used (i.e.
+%   as power method). See [2] p.9, Randomized PCA algorithm for details.
+% 
+%   [1] Halko, N., Martinsson, P. G., Shkolnisky, Y., & Tygert, M. (2010).
+%   An algorithm for the principal component analysis of large data sets.
+%   Arxiv preprint arXiv:1007.5510, 0526. Retrieved April 1, 2011, from
+%   http://arxiv.org/abs/1007.5510. 
+%   
+%   [2] Halko, N., Martinsson, P. G., & Tropp, J. A. (2009). Finding
+%   structure with randomness: Probabilistic algorithms for constructing
+%   approximate matrix decompositions. Arxiv preprint arXiv:0909.4061.
+%   Retrieved April 1, 2011, from http://arxiv.org/abs/0909.4061.
+% 
+%   See also SVD.
+% 
+%   Copyright 2011 Ismail Ari, http://ismailari.com.
+
+    
+    isSparse =  issparse(A);
+
+
+    if nargin < 3
+        i = 1;
+    end
+
+    % Take (conjugate) transpose if necessary. It makes H smaller thus
+    % leading the computations to be faster
+    if size(A,1) < size(A,2)
+        A = A';
+        isTransposed = true;
+    else
+        isTransposed = false;
+    end
+
+    n = size(A,2);
+    extra_margin = 3;  % slighly improve precision 
+    l = k + extra_margin;
+
+    % Form a real n×l matrix G whose entries are iid Gaussian r.v.s of zero
+    % mean and unit variance
+    G = randn(n,l, 'single');
+
+
+    if nargin >= 4 && usePowerMethod
+        % Use only the given exponent
+        H = A*G;
+        for j = 2:i+1
+            H = A * (A'*H);
+        end
+    else
+        % Compute the m×l matrices H^{(0)}, ..., H^{(i)}
+        % Note that this is done implicitly in each iteration below.
+        
+            if isSparse
+                H = sparse(size(A,1), l * (i+1) );
+            else
+                H = zeros(size(A,1), l * (i+1), 'like', A);
+            end
+                        
+            H(:,1:l) = A*G;
+
+            for j = 2:i+1
+               H(:,(j-1)*l +  (1:l)) = A * (A'*H(: , (j-2)*l +  (1:l)));
+            end
+
+        % Form the m×((i+1)l) matrix H
+       
+    end
+
+    % Using the pivoted QR-decomposiion, form a real m×((i+1)l) matrix Q
+    % whose columns are orthonormal, s.t. there exists a real
+    % ((i+1)l)×((i+1)l) matrix R for which H = QR.  
+    % XXX: Buradaki column pivoting ile yapılmayan hali.
+    [Q,~] = qr(H,0);
+
+   
+    % Compute the n×((i+1)l) product matrix T = A^T Q
+    T = A'*Q;
+
+    % Form an SVD of T
+    [Vt, St, W] = svd(T,'econ');
+
+    % Compute the m×((i+1)l) product matrix
+    Ut = Q*W;
+
+    % Retrieve the leftmost m×k block U of Ut, the leftmost n×k block V of
+    % Vt, and the leftmost uppermost k×k block S of St. The product U S V^T
+    % then approxiamtes A. 
+
+    
+    if isTransposed
+        V = Ut(:,1:k);
+        U = Vt(:,1:k);     
+    else
+        U = Ut(:,1:k);
+        V = Vt(:,1:k);
+    end
+    S = single(St(1:k,1:k));
+end
+
+
+