Rank r matrix approximation via svd in julia

When working with matrices in Julia, it is often useful to approximate a given matrix using its singular value decomposition (SVD). The SVD allows us to decompose a matrix into three separate matrices, which can then be used to approximate the original matrix with a lower rank. In this article, we will explore three different ways to perform rank r matrix approximation via SVD in Julia.

Option 1: Using the `svd` function

The first option is to use the built-in `svd` function in Julia. This function computes the singular value decomposition of a given matrix. To perform rank r matrix approximation, we can simply compute the SVD of the original matrix and then truncate the resulting matrices to keep only the first r singular values and corresponding singular vectors.

using LinearAlgebra

function rank_r_approximation(matrix, r)
    U, Σ, V = svd(matrix)
    U_r = U[:, 1:r]
    Σ_r = Diagonal(Σ[1:r])
    V_r = V[:, 1:r]
    return U_r * Σ_r * V_r'
end

In this code snippet, we first compute the SVD of the input matrix `matrix` using the `svd` function. We then extract the first r columns of the left singular matrix U, the first r singular values as a diagonal matrix Σ, and the first r columns of the right singular matrix V. Finally, we multiply these truncated matrices to obtain the rank r approximation of the original matrix.

Option 2: Using the `econsvd` function

The second option is to use the `econsvd` function from the `LinearAlgebra` package. This function computes the “economy-sized” SVD, which only returns the first r singular values and corresponding singular vectors. This can be more efficient than computing the full SVD and then truncating the matrices.

using LinearAlgebra

function rank_r_approximation(matrix, r)
    U, Σ, V = econsvd(matrix)
    return U[:, 1:r] * Diagonal(Σ[1:r]) * V[:, 1:r]'
end

In this code snippet, we use the `econsvd` function to directly compute the economy-sized SVD of the input matrix. We then extract the first r columns of the left singular matrix U, the first r singular values as a diagonal matrix Σ, and the first r columns of the right singular matrix V. Finally, we multiply these truncated matrices to obtain the rank r approximation of the original matrix.

Option 3: Using the `LowRankApprox` package

The third option is to use the `LowRankApprox` package, which provides a convenient interface for performing low-rank matrix approximation. This package uses randomized algorithms to efficiently compute the rank r approximation of a given matrix.

using LowRankApprox

function rank_r_approximation(matrix, r)
    return approx_svd(matrix, r)
end

In this code snippet, we simply call the `approx_svd` function from the `LowRankApprox` package, passing in the input matrix and the desired rank r. This function internally uses randomized algorithms to compute the rank r approximation of the matrix.

After exploring these three options, it is clear that the second option, using the `econsvd` function, is the most efficient and concise way to perform rank r matrix approximation via SVD in Julia. This option directly computes the economy-sized SVD, avoiding the need to truncate the matrices after computing the full SVD. Therefore, it is recommended to use the `econsvd` function for rank r matrix approximation in Julia.