R2RILS

This repository contains Python and Matlab implementations for R2RILS as described in J. Bauch and B. Nadler (2020), available in preprint link, as well as simple demos demonstrating the usage of the Matlab and Python implementations.

Usage

Python

The entry point to run R2RILS is a function with the same name which expects the following parameters:

• X: input matrix to complete. This should be a numpy array of dimensions m x n.
• omega: a mask matrix. 1 on observed entries, 0 otherwise.
• rank: the target rank.
• t_max: maximal number of iterations.
• early_stopping_epsilon: early stopping parameter as described in the paper.
• smart_tol: should increase LSQR's accuracy according to the current quality of the objective. Defaults to True.

This method returns X_hat - R2RILS estimate for X0.

Matlab

The entry point for running R2RILS in Matlab is again a function bearing the same name
function [X_hat U_hat V_hat] = R2RILS(X, omega, rank, t_max)
and expects the following parameters:

• X: matrix with observed entries in the set omega.
• omega: array of pairs (i,j) indicating which entries are observed.
• rank: the target rank.
• t_max: maximal number of iterations.

This method returns [X_hat U_hat lambda_hat V_hat, observed_RMSE] where:

• X_hat: rank 2r approximation of X0 (note that if R2RILS converges than the limiting point is in fact rank r).
• U_hat: matrix of left singular vectors of the best rank r approximation of X_hat.
• lambda_hat: singular values of the best rank r approximation of X0.
• V_hat: matrix of right singular vectors of the best rank r approximation of X_hat.