learner {learner} | R Documentation |
Latent space-based transfer learning
Description
This function applies the LatEnt spAce-based tRaNsfer lEaRning (LEARNER) method (McGrath et al. 2024) to leverage data from a source population to improve estimation of a low rank matrix in an underrepresented target population.
Usage
learner(Y_source, Y_target, r, lambda_1, lambda_2, step_size, control = list())
Arguments
Y_source |
matrix containing the source population data | ||||||
Y_target |
matrix containing the target population data | ||||||
r |
(optional) integer specifying the rank of the knowledge graphs. By default, ScreeNOT (Donoho et al. 2023) is applied to the source population knowledge graph to select the rank. | ||||||
lambda_1 |
numeric scalar specifying the value of | ||||||
lambda_2 |
numeric scalar specifying the value of | ||||||
step_size |
numeric scalar specifying the step size for the Newton steps in the numerical optimization algorithm | ||||||
control |
a list of parameters for controlling the stopping criteria for the numerical optimization algorithm. The list may include the following components:
|
Details
Data and notation:
The data consists of a matrix in the target population Y_0 \in \mathbb{R}^{p \times q}
and the source population Y_1 \in \mathbb{R}^{p \times q}
.
Let \hat{U}_{k} \hat{\Lambda}_{k} \hat{V}_{k}^{\top}
denote the truncated singular value decomposition (SVD) of Y_k
, k = 0, 1
.
For k = 0, 1
, one can view Y_k
as a noisy version of \Theta_k
, referred to as the knowledge graph. The target of inference is the target population knowledge graph, \Theta_0
.
Estimation:
This method estimates \Theta_0
by \tilde{U}\tilde{V}^{\top}
, where (\tilde{U}, \tilde{V})
is the solution to the following optimization problem
\mathrm{arg\,min}_{U \in \mathbb{R}^{p \times r}, V \in \mathbb{R}^{q \times r}} \big\{ \| U V^{\top} - Y_0 \|_F^2 + \lambda_1\| \mathcal{P}_{\perp}(\hat{U}_{1})U \|_F^2 + \lambda_1\| \mathcal{P}_{\perp}(\hat{V}_{1})V \|_F^2 + \lambda_2 \| U^{\top} U - V^{\top} V \|_F^2 \big\}
where \mathcal{P}_{\perp}(\hat{U}_{1}) = I - \hat{U}_{1}^{\top}\hat{U}_{1}
and \mathcal{P}_{\perp}(\hat{V}_{1}) = I - \hat{V}_{1}^{\top}\hat{V}_{1}
.
This function uses an alternating minimization strategy to solve the optimization problem. That is, this approach updates U
by minimizing the objective function (via a gradient descent step) treating V
as fixed. Then, V
is updated treating U
as fixed. These updates of U
and V
are repeated until convergence.
Value
A list with the following elements:
learner_estimate |
matrix containing the LEARNER estimate of the target population knowledge graph |
objective_values |
numeric vector containing the values of the objective function at each iteration |
convergence_criterion |
integer specifying the criterion that was satisfied for terminating the numerical optimization algorithm. A value of 1 indicates the convergence threshold was satisfied; A value of 2 indicates that the maximum number of iterations was satisfied; A value of 3 indicates that the maximum value of the objective function was satisfied. |
r |
rank value used. |
References
McGrath, S., Zhu, C,. Guo, M. and Duan, R. (2024). LEARNER: A transfer learning method for low-rank matrix estimation. arXiv preprint arXiv:2412.20605.
Donoho, D., Gavish, M. and Romanov, E. (2023). ScreeNOT: Exact MSE-optimal singular value thresholding in correlated noise. The Annals of Statistics, 51(1), pp.122-148.
Examples
res <- learner(Y_source = dat_highsim$Y_source,
Y_target = dat_highsim$Y_target,
lambda_1 = 1, lambda_2 = 1,
step_size = 0.003)