Use svdsketch to compute the SVD factors of a low-rank matrix approximation.

Use gallery to create a 200-by-200 random matrix with geometrically distributed singular values.

A = gallery('randsvd',200);

Use svdsketch to calculate the SVD of a low-rank approximation of A.

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

Check the size of the outputs.

size(S)

ans = 1×2

   120   120

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

Specify a tolerance with svdsketch to compute the SVD factors of a low-rank matrix approximation. svdsketch adaptively determines the appropriate rank of the matrix sketch based on the specified tolerance.

Use gallery to create a 200-by-200 random matrix with geometrically distributed singular values.

A = gallery('randsvd',200);

Use svdsketch to calculate the SVD of a low-rank approximation of A. Specify a tolerance of 1e-2, and find the size of the output S to determine the rank svdsketch uses for the matrix sketch.

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

ans = 1×2

    60    60

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

Examine how well the matrix sketch approximates A by comparing A to U*S*V'.

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

ans = 0.0048

The result indicates that the matrix sketch approximates A within the specified tolerance of 1e-2.

Use svdsketch with the MaxSubspaceDimension option on a matrix that has slowly decaying singular values. You can use this option to force svdsketch to use a subset of the features of A in 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 in A decay slowly, A has many important features and does not lend itself to low-rank approximations. To form a matrix sketch that reasonably approximates A, most or nearly all of the features need to be preserved.

Use svdsketch on the matrix with a tolerance of 1e-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 = 1×2

        5000        5000

The size of S indicates that to satisfy the tolerance, svdsketch needs to preserve all of the features of A. For large sparse input matrices, this can present memory issues since the low-rank approximation of A formed by svdsketch is roughly the same size as A and thus might not fit in memory as a dense matrix.

Check the approximation error of the outputs. Since svdsketch preserves everything in A, the computed answer is accurate, but the calculation was just an expensive way to calculate svd(X).

apxError1(end)

ans = 1.9075e-08

Now, do the same calculation but specify MaxSubspaceDimension as 650 to limit the size of the subspace used to sketch A. This is useful to force svdsketch to use only a subset of the features in A to form the matrix sketch, reducing the size of the outputs.

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

ans = 1×2

   650   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 of A does not meet the specified tolerance.

apxError2(end)

ans = 0.8214