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A Case for using Trend Filtering over Splines

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This talk will be about fast optimization algorithms (specifically specialized ADMM ) for a common practical problem – estimating piecewise constant/linear/quadratic fits to time series data. I will first introduce Trend Filtering, a recently proposed tool for this problem by Kim, Koh, Boyd and Gorinevsky (2009), and compare it to the popular smoothing splines and locally adaptive regression splines. Tibshirani (2014) showed that trend filtering estimates converge at the minimax optimal if the true underlying function (or its derivatives) has bounded total variation. Hence, the only roadblock to using it in practice is having robust and efficient algorithms. We take a major step in overcoming this problem, by providing a more efficient and robust solution than the current interior point methods in use. Furthermore, the proposed ADMM implementation is very simple, and importantly, it is flexible enough to extend to many interesting related problems, such as sparse trend filtering and isotonic trend filtering. Software for our method will be made freely available, written in C++, and also in R (see the {\tt trendfilter} function in the R package {\tt genlasso}).

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