Fast DPP Sampling for Nystrom with Application to Kernel Methods

Chengtao Li; Stefanie Jegelka; Suvrit Sra

Fast DPP Sampling for Nystrom with Application to Kernel Methods

Chengtao Li, Stefanie Jegelka, Suvrit Sra ;

Proceedings of The 33rd International Conference on Machine Learning, PMLR 48:2061-2070, 2016.

Abstract

The Nystrom method has long been popular for scaling up kernel methods. Its theoretical guarantees and empirical performance rely critically on the quality of the landmarks selected. We study landmark selection for Nystrom using Determinantal Point Processes (DPPs), discrete probability models that allow tractable generation of diverse samples. We prove that landmarks selected via DPPs guarantee bounds on approximation errors; subsequently, we analyze implications for kernel ridge regression. Contrary to prior reservations due to cubic complexity of DPP sampling, we show that (under certain conditions) Markov chain DPP sampling requires only linear time in the size of the data. We present several empirical results that support our theoretical analysis, and demonstrate the superior performance of DPP-based landmark selection compared with existing approaches.

Related Material

@InProceedings{pmlr-v48-lih16, title = {Fast DPP Sampling for Nystrom with Application to Kernel Methods}, author = {Chengtao Li and Stefanie Jegelka and Suvrit Sra}, booktitle = {Proceedings of The 33rd International Conference on Machine Learning}, pages = {2061--2070}, year = {2016}, editor = {Maria Florina Balcan and Kilian Q. Weinberger}, volume = {48}, series = {Proceedings of Machine Learning Research}, address = {New York, New York, USA}, month = {20--22 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v48/lih16.pdf}, url = {http://proceedings.mlr.press/v48/lih16.html}, abstract = {The Nystrom method has long been popular for scaling up kernel methods. Its theoretical guarantees and empirical performance rely critically on the quality of the landmarks selected. We study landmark selection for Nystrom using Determinantal Point Processes (DPPs), discrete probability models that allow tractable generation of diverse samples. We prove that landmarks selected via DPPs guarantee bounds on approximation errors; subsequently, we analyze implications for kernel ridge regression. Contrary to prior reservations due to cubic complexity of DPP sampling, we show that (under certain conditions) Markov chain DPP sampling requires only linear time in the size of the data. We present several empirical results that support our theoretical analysis, and demonstrate the superior performance of DPP-based landmark selection compared with existing approaches.} }

%0 Conference Paper %T Fast DPP Sampling for Nystrom with Application to Kernel Methods %A Chengtao Li %A Stefanie Jegelka %A Suvrit Sra %B Proceedings of The 33rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2016 %E Maria Florina Balcan %E Kilian Q. Weinberger %F pmlr-v48-lih16 %I PMLR %J Proceedings of Machine Learning Research %P 2061--2070 %U http://proceedings.mlr.press %V 48 %W PMLR %X The Nystrom method has long been popular for scaling up kernel methods. Its theoretical guarantees and empirical performance rely critically on the quality of the landmarks selected. We study landmark selection for Nystrom using Determinantal Point Processes (DPPs), discrete probability models that allow tractable generation of diverse samples. We prove that landmarks selected via DPPs guarantee bounds on approximation errors; subsequently, we analyze implications for kernel ridge regression. Contrary to prior reservations due to cubic complexity of DPP sampling, we show that (under certain conditions) Markov chain DPP sampling requires only linear time in the size of the data. We present several empirical results that support our theoretical analysis, and demonstrate the superior performance of DPP-based landmark selection compared with existing approaches.

TY - CPAPER TI - Fast DPP Sampling for Nystrom with Application to Kernel Methods AU - Chengtao Li AU - Stefanie Jegelka AU - Suvrit Sra BT - Proceedings of The 33rd International Conference on Machine Learning PY - 2016/06/11 DA - 2016/06/11 ED - Maria Florina Balcan ED - Kilian Q. Weinberger ID - pmlr-v48-lih16 PB - PMLR SP - 2061 DP - PMLR EP - 2070 L1 - http://proceedings.mlr.press/v48/lih16.pdf UR - http://proceedings.mlr.press/v48/lih16.html AB - The Nystrom method has long been popular for scaling up kernel methods. Its theoretical guarantees and empirical performance rely critically on the quality of the landmarks selected. We study landmark selection for Nystrom using Determinantal Point Processes (DPPs), discrete probability models that allow tractable generation of diverse samples. We prove that landmarks selected via DPPs guarantee bounds on approximation errors; subsequently, we analyze implications for kernel ridge regression. Contrary to prior reservations due to cubic complexity of DPP sampling, we show that (under certain conditions) Markov chain DPP sampling requires only linear time in the size of the data. We present several empirical results that support our theoretical analysis, and demonstrate the superior performance of DPP-based landmark selection compared with existing approaches. ER -