TY - JOUR
T1 - Epigraphical Relaxation for Minimizing Layered Mixed Norms
AU - Kyochi, Seisuke
AU - Ono, Shunsuke
AU - Selesnick, Ivan
N1 - Funding Information:
Manuscript received July 14, 2020; revised December 23, 2020; accepted January 17, 2021. Date of publication February 3, 2021; date of current version May 28, 2021. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ketan Rajawat. This work was supported by in part by JSPS KAKENHI Grants 17K14683, 18H05413, 19H04140, and 20H02145 and in part by JST CREST under Grants JPMJCR1662 and JPMJCR1666. (Corresponding author: Seisuke Kyochi.) Seisuke Kyochi is with the Department of Information and Systems Engineering, The University of Kitakyushu, Kitakyushu, Fukuoka 808-0135, Japan (e-mail: kyochi@ieee.org).
Publisher Copyright:
© 1991-2012 IEEE.
PY - 2021
Y1 - 2021
N2 - This paper proposes an epigraphical relaxation (ERx) technique for non-proximable mixed norm minimization. Mixed norm regularization methods play a central role in signal reconstruction and processing, where their optimization relies on the fact that the proximity operators of the mixed norms can be computed efficiently. To bring out the power of regularization, sophisticated layered modeling of mixed norms that can capture inherent signal structure is a key ingredient, but the proximity operator of such a mixed norm is often unavailable (non-proximable). Our ERx decouples a layered non-proximable mixed norm into a norm and multiple epigraphical constraints. This enables us to handle a wide range of non-proximable mixed norms in optimization, as long as both the proximal operator of the outermost norm and the projection onto each epigraphical constraint are efficiently computable. Moreover, under mild conditions, we prove that ERx does not change the minimizer of the original problem despite relaxing equality constraints into inequality ones. We also develop new regularizers based on ERx:one is decorrelated structure-tensor total variation for color image restoration, and the other is amplitude-spectrum nuclear norm for low-rank amplitude recovery. We examine the power of these regularizers through experiments, which illustrates the utility of ERx.
AB - This paper proposes an epigraphical relaxation (ERx) technique for non-proximable mixed norm minimization. Mixed norm regularization methods play a central role in signal reconstruction and processing, where their optimization relies on the fact that the proximity operators of the mixed norms can be computed efficiently. To bring out the power of regularization, sophisticated layered modeling of mixed norms that can capture inherent signal structure is a key ingredient, but the proximity operator of such a mixed norm is often unavailable (non-proximable). Our ERx decouples a layered non-proximable mixed norm into a norm and multiple epigraphical constraints. This enables us to handle a wide range of non-proximable mixed norms in optimization, as long as both the proximal operator of the outermost norm and the projection onto each epigraphical constraint are efficiently computable. Moreover, under mild conditions, we prove that ERx does not change the minimizer of the original problem despite relaxing equality constraints into inequality ones. We also develop new regularizers based on ERx:one is decorrelated structure-tensor total variation for color image restoration, and the other is amplitude-spectrum nuclear norm for low-rank amplitude recovery. We examine the power of these regularizers through experiments, which illustrates the utility of ERx.
KW - Convex optimization
KW - epigraph
KW - epigraphical projection
KW - image recovery
KW - structure tensor total variation
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U2 - 10.1109/TSP.2021.3056598
DO - 10.1109/TSP.2021.3056598
M3 - Article
AN - SCOPUS:85100802010
VL - 69
SP - 2923
EP - 2938
JO - IRE Transactions on Audio
JF - IRE Transactions on Audio
SN - 1053-587X
M1 - 9346035
ER -