Usesvdsketchto compute the SVD factors of a low-rank matrix approximation.

Usegalleryto create a 200-by-200 random matrix with geometrically distributed singular values.

A = gallery('randsvd',200);

Usesvdsketchto calculate the SVD of a low-rank approximation ofA.

[U,S,V] = svdsketch(A);

Check the size of the outputs.

size(S)

ans =1×2120 120

The results indicate that the low-rank matrix approximation ofAhas a rank of 120.

Specify a tolerance withsvdsketchto compute the SVD factors of a low-rank matrix approximation.svdsketchadaptively determines the appropriate rank of the matrix sketch based on the specified tolerance.

Usegalleryto create a 200-by-200 random matrix with geometrically distributed singular values.

A = gallery('randsvd',200);

Usesvdsketchto calculate the SVD of a low-rank approximation of A. Specify a tolerance of1e-2, and find the size of the outputSto determine the ranksvdsketchuses for the matrix sketch.

tol = 1e-2; [U,S,V] = svdsketch(A,tol); size(S)

ans =1×260 60

The results indicate that the low-rank matrix approximation ofAhas a rank of 60.

Examine how well the matrix sketch approximatesAby comparingAtoU*S*V'.

norm(U*S*V' - A,'fro')/norm(A,'fro')

ans = 0.0048

The result indicates that the matrix sketch approximatesAwithin the specified tolerance of1e-2.

Usesvdsketchwith theMaxSubspaceDimensionoption on a matrix that has slowly decaying singular values. You can use this option to forcesvdsketchto use a subset of the features ofAin the matrix sketch.

Create a 5000-by-5000 matrix with values drawn from the standard normal distribution. View the distribution of matrix singular values.

A = randn(5000); semilogy(svd(A),'-o')

Since the singular values inAdecay slowly,Ahas many important features and does not lend itself to low-rank approximations. To form a matrix sketch that reasonably approximatesA, most or nearly all of the features need to be preserved.

Usesvdsketchon the matrix with a tolerance of1e-5. Specify four outputs to return the SVD factors as well as the relative approximation error in each iteration.

tol = 1e-5; [U1,S1,V1,apxError1] = svdsketch(A,tol); size(S1)

ans =1×25000 5000

The size ofSindicates that to satisfy the tolerance,svdsketchneeds to preserve all of the features ofA. For large sparse input matrices, this can present memory issues since the low-rank approximation ofAformed bysvdsketchis roughly the same size asAand thus might not fit in memory as a dense matrix.

Check the approximation error of the outputs. Sincesvdsketchpreserves everything inA, the computed answer is accurate, but the calculation was just an expensive way to calculatesvd(X).

apxError1(end)

ans = 1.9075e-08

Now, do the same calculation but specifyMaxSubspaceDimensionas 650 to limit the size of the subspace used to sketchA. This is useful to forcesvdsketchto use only a subset of the features inAto form the matrix sketch, reducing the size of the outputs.

[U2,S2,V2,apxError2] = svdsketch(A,tol,'MaxSubspaceDimension',650); size(S2)

ans =1×2650 650

The outputs now have a smaller size.

Check the approximation error of the new outputs. The tradeoff for forcing the outputs to be smaller is that many important features of A need to be omitted in the matrix sketch, and the resulting rank 650 approximation ofAdoes not meet the specified tolerance.

apxError2(end)

ans = 0.8214