ℓ 0 Trend Filtering

Canhong Wen (), Xueqin Wang () and Aijun Zhang ()
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Canhong Wen: International Institute of Finance, School of Management, University of Science and Technology of China, Hefei 230000, China
Xueqin Wang: International Institute of Finance, School of Management, University of Science and Technology of China, Hefei 230000, China
Aijun Zhang: Department of Statistics and Actuarial Science, The University of Hong Kong, 999077 Hong Kong

INFORMS Journal on Computing, 2023, vol. 35, issue 6, 1491-1510

Abstract: The ℓ 0 trend filtering ( ℓ 0 -TF) is a new effective tool for nonparametric regression with the power of automatic knot detection in function values or derivatives. It overcomes the drawback of ℓ 1 -TF that is known to have bias issues. To solve the ℓ 0 -TF problem, we propose an alternating minimization induced active set (AMIAS) search method based on the necessary optimality conditions derived from an augmented Lagrangian framework. The proposed method takes full advantage of the primal and dual variables with complementary supports, and decouples the high-dimensional problem into two subsystems on the active and inactive sets, respectively. A sequential AMIAS algorithm with warm start initialization is developed for efficient determination of the cardinality parameter, along with the output of solution paths. Theoretically, the oracle estimator of ℓ 0 -TF is justified to behave like regression splines under the continuous time setting with mild conditions. Our numerical experiments include simulation studies for comparing ℓ 0 -TF to ℓ 1 -TF and free-knot splines on several synthetic examples, and a real data application of time series segmentation on Hong Kong PM2.5 indexes.

Keywords: active set search; knot detection; primal-dual formulation; hard thresholding; trend filtering (search for similar items in EconPapers)
Date: 2023
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