Solves
min over L, a, b of (1/2) * squaredFrobenius(O o (Y - L - a1' - 1b')) + lambda * nuclearNorm(L)
by alternating a back-fitting update of the fixed effects with one
soft-thresholded SVD step for L (Mazumder, Hastie & Tibshirani 2010).
Usage
soft_impute_fe_cpp(
Y,
O,
lambda,
estimate_fe = TRUE,
max_iter = 1000L,
tol = 1e-05,
fe_sweeps = 5L
)Arguments
- Y
Observed outcome matrix (T x N). Unobserved entries should be 0.
- O
Binary mask matrix (T x N): 1 = observed, 0 = missing (treated post).
- lambda
Nuclear norm penalty (soft-threshold on singular values).
- estimate_fe
Estimate unpenalised row and column effects.
- max_iter
Maximum iterations.
- tol
Convergence tolerance (relative Frobenius norm change).
- fe_sweeps
Back-fitting sweeps per iteration. The mask makes the two effects non-orthogonal, so one closed-form pass does not solve for both.
Value
A list with L (the penalised low-rank part), a (row effects),
b (column effects), and M, the completed matrix L + a 1' + 1 b'.
Details
The row and column effects are the unpenalised fixed effects of Athey et
al. (2021, S.8.1): "we regularize the estimates of L*, but do not wish to
regularize the estimates of the fixed effects". Leaving them inside L
shrinks the panel's level towards zero along with its factor structure,
which biases the imputed counterfactual whenever the level is far from
zero. estimate_fe = false reproduces the S.4 baseline model Y = L + e.
Both effects absorb an additive constant, so a and b are identified
only up to a shift between them; their sum, which is what enters the
completed matrix, is unaffected.