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LMC

About

This library offers a suite of techniques for low-rank matrix completion of longitudinal data. These techniques introduces various regularization and descrepancy terms to impose structural information on the completed matrix.

Low-rank matrix completion for longitudinal data

Low-rank matrix completion (LMC) is a technique used to recover a partially observed matrix by exploiting the assumption that the matrix has a low-rank structure. In LMC for longitudinal data, the data was collected over time, potentially at irregular intervals.

In this context, the data can be organized as a partially observed matrix $\mathbf{X} \in \mathbb{R}^{N \times T}$, with $N$ rows corresponding to time-varying entities, and $T$ columns corresponding to the time points. Each entry $X_{n, t}$ would represent an observation for a particular entity $n$ at a particular time $t$.

Low-rank matrix completion aims to recover the missing entries by exploiting the assumption that the matrix, when viewed as time-dependent for each entity, can be approximated by a low-rank matrix. The basic factorization model decomposes $\mathbf{X}$ into a set of shared time-varying basic profiles $\mathbf{v}_1, \dots, \mathbf{v}_r$, where $r \ll \min ({N,T})$, and profile-specific coefficients in $\mathbf{U}_n$. The linear combination $\mathbf{M}_n = \mathbf{U}_n \mathbf{V}^\top$ of coefficients and basic profiles yields the estimate for the reconstructed profile.

Key features

• Longitudinal matrix completion
• Convolutional longitudinal matrix completion
• Total variation longitudinal matrix completion
• Least-angle regression matrix completion
• Phase-shifted matrix completion

Installation

Install the library with pip:

pip install .

This ensures dependencies listed in pyproject.toml are handled correctly.

Usage

A basic example involves estimating the entries of a matrix $X$, given only the entries indicated by $O_{train}$ and use the entries in $O_{test}$ to evaluate the reconstruction accuracy.

# lmc lib
from lmc import CMC
from utils import train_test_data

# third party
from sklearn.metrics import mean_squared_error

X, O_train, O_test = train_test_data()

X_train = X * O_train
X_test = X * O_test

model = CMC(rank=5, n_iter=100)
model.fit(X_train)

Y_test = model.M * O_test

score = mean_squared_error(X_test, Y_test)