Adiabatic Quantum Feature Selection for Sparse Linear Regression

Linear regression is a popular machine learning approach to learn and predict real valued outputs or dependent variables from independent variables or features. In many real world problems, its beneficial to perform sparse linear regression to identify important features helpful in predicting the dependent variable. It not only helps in getting interpretable results but also avoids overfitting when the number of features is large, and the amount of data is small. The most natural way to achieve this is by using `best subset selection' which penalizes non-zero model parameters by adding $\ell_0$ norm over parameters to the least squares loss. However, this makes the objective function non-convex and intractable even for a small number of features. This paper aims to address the intractability of sparse linear regression with $\ell_0$ norm using adiabatic quantum computing, a quantum computing paradigm that is particularly useful for solving optimization problems faster. We formulate the $\ell_0$ optimization problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem and solve it using the D-Wave adiabatic quantum computer. We study and compare the quality of QUBO solution on synthetic and real world datasets. The results demonstrate the effectiveness of the proposed adiabatic quantum computing approach in finding the optimal solution. The QUBO solution matches the optimal solution for a wide range of sparsity penalty values across the datasets.