Atomic Decomposition by Basis Pursuit
SIAM Review
Convex Optimization
An Interior-Point Method for Large-Scale l1-Regularized Logistic Regression
The Journal of Machine Learning Research
Relaxed maximum a posteriori fault identification
Signal Processing
Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and Convergence
SIAM Journal on Optimization
IEEE Transactions on Information Theory
Just relax: convex programming methods for identifying sparse signals in noise
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
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We consider a mixed linear system model, with both continuous and discrete inputs and outputs, described by a coefficient matrix and a set of noise variances. When the discrete inputs and outputs are absent, the model reduces to the usual noise-corrupted linear system. With discrete inputs only, the model has been used in fault estimation, and with discrete outputs only, the system reduces to a probit model. We consider two fundamental problems: estimating the model input, given the model parameters and the model output; and identifying the model parameters, given a training set of input-output pairs. The estimation problem leads to a mixed Boolean-convex optimization problem, which can be solved exactly when the number of discrete variables is small enough. In other cases, when the number of discrete variables is large, the estimation problem can be solved approximately, by solving a convex relaxation, rounding, and possibly, carrying out a local optimization step. The identification problem is convex and so can be exactly solved. Adding @?"1 regularization to the identification problem allows us to trade off model fit and model parsimony. We illustrate the identification and estimation methods with a numerical example.