In this thesis, we explore the power of non-convex penalty in signal denoising problems. Both 1-d and 2-d signals are considered. We focus on the case where the original signal is piecewise constant. Friedman et al. (2007) proposed the Fused Lasso Signal Approximator (FLSA) to denoise this kind of signals specifically. It penalizes the differences between adjacent signal points. However, this method is imperfect. We propose a new method by combining MCP with the fusion type of penalty, named fused-MCP. MCP is a non-convex penalty presented by Zhang (2010). Since it ameliorates the over-selection and bias problem of LASSO, MCP improves the performance of FLSA in both capturing the profile of the original signal and detecting the change points. We show that, in certain cases, fused-MCP po...[ Read more ]

In this thesis, we explore the power of non-convex penalty in signal denoising problems. Both 1-d and 2-d signals are considered. We focus on the case where the original signal is piecewise constant. Friedman et al. (2007) proposed the Fused Lasso Signal Approximator (FLSA) to denoise this kind of signals specifically. It penalizes the differences between adjacent signal points. However, this method is imperfect. We propose a new method by combining MCP with the fusion type of penalty, named fused-MCP. MCP is a non-convex penalty presented by Zhang (2010). Since it ameliorates the over-selection and bias problem of LASSO, MCP improves the performance of FLSA in both capturing the profile of the original signal and detecting the change points. We show that, in certain cases, fused-MCP possesses an oracle property. Namely, with large probability, it selects the right change points and obtains oracle LSE of the signals. This property enables fused-MCP to keep the change size of the signals while FLSA usually shrinks it too much. We derive algorithms to solve the optimization problems formulated in our method. They are either transformed to the original form of MCP penalized regression problem or solved by an adjusted majorization-minimization algorithm, depending on their specific form.