Project matrix to rank r

proj_rank_r() implements a best (i.e. closest) rank-r approximation of an input matrix.

This is computed via a simple truncated singular value decomposition (SVD), retaining the first r leading singular values/vectors of D. This is equivalent to solving the following optimization problem: \(min ||X-D||_F s.t. rank(X) <= r\), where X is the approximated solution and D is the input matrix.

proj_rank_r() is used to iteratively model the low-rank L matrix in the non-convex PCP function rrmc(), providing a non-convex replacement for the prox_nuclear() method used in the convex PCP function root_pcp().

Intuitively, proj_rank_r() can also be thought of as providing a PCA estimate of a rank-r matrix L from observed data D.

Usage

proj_rank_r(D, r)

Arguments

D

The input data matrix (cannot have NA values).

r

The rank that D should be projected/truncated to.

Value

The best rank-r approximation to D via a truncated SVD.

See also

rrmc()

Examples

# Simulating a simple dataset D with the sim_data() function.
# The dataset will be a 10x5 matrix comprised of:
# 1. A rank-1 component as the ground truth L matrix; and
# 2. A dense Gaussian noise component corrupting L, making L full-rank
data <- sim_data(10, 5, 1, numeric(), 0.01)
# The observed matrix D is full-rank, while L is rank-1:
data.frame("D_rank" = matrix_rank(data$D), "L_rank" = matrix_rank(data$L))
#>   D_rank L_rank
#> 1      5      1
before_proj_err <- norm(data$D - data$L, "F") / norm(data$L, "F")
# Projecting D onto the nearest rank-1 approximation, X, via proj_rank_r()
X <- proj_rank_r(data$D, r = 1)
after_proj_err <- norm(X - data$L, "F") / norm(data$L, "F")
proj_v_obs_err <- norm(X - data$D, "F") / norm(data$D, "F")
data.frame(
  "Observed_error" = before_proj_err,
  "Projected_error" = after_proj_err,
  "Projected_vs_observed_error" = proj_v_obs_err
)
#>   Observed_error Projected_error Projected_vs_observed_error
#> 1      0.0283898      0.01435178                  0.02436504