Timezone: »
Performance metrics for binary classification are designed to capture tradeoffs between four fundamental population quantities: true positives, false positives, true negatives and false negatives. Despite significant interest from theoretical and applied communities, little is known about either optimal classifiers or consistent algorithms for optimizing binary classification performance metrics beyond a few special cases. We consider a fairly large family of performance metrics given by ratios of linear combinations of the four fundamental population quantities. This family includes many well known binary classification metrics such as classification accuracy, AM measure, F-measure and the Jaccard similarity coefficient as special cases. Our analysis identifies the optimal classifiers as the sign of the thresholded conditional probability of the positive class, with a performance metric-dependent threshold. The optimal threshold can be constructed using simple plug-in estimators when the performance metric is a linear combination of the population quantities, but alternative techniques are required for the general case. We propose two algorithms for estimating the optimal classifiers, and prove their statistical consistency. Both algorithms are straightforward modifications of standard approaches to address the key challenge of optimal threshold selection, thus are simple to implement in practice. The first algorithm combines a plug-in estimate of the conditional probability of the positive class with optimal threshold selection. The second algorithm leverages recent work on calibrated asymmetric surrogate losses to construct candidate classifiers. We present empirical comparisons between these algorithms on benchmark datasets.
Author Information
Sanmi Koyejo (Stanford, Google Research)

Sanmi Koyejo an Assistant Professor in the Department of Computer Science at Stanford University. Koyejo also spends time at Google as a part of the Brain team. Koyejo's research interests are in developing the principles and practice of trustworthy machine learning. Additionally, Koyejo focuses on applications to neuroscience and healthcare. Koyejo has been the recipient of several awards, including a best paper award from the conference on uncertainty in artificial intelligence (UAI), a Skip Ellis Early Career Award, and a Sloan Fellowship. Koyejo serves as the president of the Black in AI organization.
Nagarajan Natarajan (Microsoft Research, India)
Pradeep Ravikumar (Carnegie Mellon University)
Inderjit Dhillon (Google & UT Austin)
Related Events (a corresponding poster, oral, or spotlight)
-
2014 Spotlight: Consistent Binary Classification with Generalized Performance Metrics »
Wed. Dec 10th 10:40 -- 11:05 PM Room Level 2, room 210
More from the Same Authors
-
2021 : Probabilistic Performance Metric Elicitation »
Zachary Robertson · Hantao Zhang · Sanmi Koyejo -
2021 : Robust and Personalized Federated Learning with Spurious Features: an Adversarial Approach »
Xiaoyang Wang · Han Zhao · Klara Nahrstedt · Sanmi Koyejo -
2021 : RVFR: Robust Vertical Federated Learning via Feature Subspace Recovery »
Jing Liu · Chulin Xie · Krishnaram Kenthapadi · Sanmi Koyejo · Bo Li -
2021 : Secure Byzantine-Robust Distributed Learning via Clustering »
Raj Kiriti Velicheti · Sanmi Koyejo -
2021 : Exploiting Causal Chains for Domain Generalization »
Olawale Salaudeen · Sanmi Koyejo -
2021 : Distribution Preserving Bayesian Coresets using Set Constraints »
Shovik Guha · Rajiv Khanna · Sanmi Koyejo -
2022 : Differentially Private Federated Learning with Normalized Updates »
Rudrajit Das · Abolfazl Hashemi · Sujay Sanghavi · Inderjit Dhillon -
2022 : Metric Elicitation; Moving from Theory to Practice »
Safinah Ali · Sohini Upadhyay · Gaurush Hiranandani · Elena Glassman · Sanmi Koyejo -
2022 : The Curse of Low Task Diversity: On the Failure of Transfer Learning to Outperform MAML and Their Empirical Equivalence »
Brando Miranda · Patrick Yu · Yu-Xiong Wang · Sanmi Koyejo -
2022 : Batch Active Learning from the Perspective of Sparse Approximation »
Maohao Shen · Yibo Jacky Zhang · Bowen Jiang · Sanmi Koyejo -
2022 Spotlight: Lightning Talks 1A-4 »
Siwei Wang · Jing Liu · Nianqiao Ju · Shiqian Li · Eloïse Berthier · Muhammad Faaiz Taufiq · Arsene Fansi Tchango · Chen Liang · Chulin Xie · Jordan Awan · Jean-Francois Ton · Ziad Kobeissi · Wenguan Wang · Xinwang Liu · Kewen Wu · Rishab Goel · Jiaxu Miao · Suyuan Liu · Julien Martel · Ruobin Gong · Francis Bach · Chi Zhang · Rob Cornish · Sanmi Koyejo · Zhi Wen · Yee Whye Teh · Yi Yang · Jiaqi Jin · Bo Li · Yixin Zhu · Vinayak Rao · Wenxuan Tu · Gaetan Marceau Caron · Arnaud Doucet · Xinzhong Zhu · Joumana Ghosn · En Zhu -
2022 Spotlight: CoPur: Certifiably Robust Collaborative Inference via Feature Purification »
Jing Liu · Chulin Xie · Sanmi Koyejo · Bo Li -
2022 Poster: Diagnosing failures of fairness transfer across distribution shift in real-world medical settings »
Jessica Schrouff · Natalie Harris · Sanmi Koyejo · Ibrahim Alabdulmohsin · Eva Schnider · Krista Opsahl-Ong · Alexander Brown · Subhrajit Roy · Diana Mincu · Christina Chen · Awa Dieng · Yuan Liu · Vivek Natarajan · Alan Karthikesalingam · Katherine Heller · Silvia Chiappa · Alexander D'Amour -
2022 Poster: S3GC: Scalable Self-Supervised Graph Clustering »
Fnu Devvrit · Aditya Sinha · Inderjit Dhillon · Prateek Jain -
2022 Poster: ELIAS: End-to-End Learning to Index and Search in Large Output Spaces »
Nilesh Gupta · Patrick Chen · Hsiang-Fu Yu · Cho-Jui Hsieh · Inderjit Dhillon -
2022 Poster: A Reduction to Binary Approach for Debiasing Multiclass Datasets »
Ibrahim Alabdulmohsin · Jessica Schrouff · Sanmi Koyejo -
2022 Poster: CoPur: Certifiably Robust Collaborative Inference via Feature Purification »
Jing Liu · Chulin Xie · Sanmi Koyejo · Bo Li -
2022 Poster: Fair Wrapping for Black-box Predictions »
Alexander Soen · Ibrahim Alabdulmohsin · Sanmi Koyejo · Yishay Mansour · Nyalleng Moorosi · Richard Nock · Ke Sun · Lexing Xie -
2022 Poster: A Nonconvex Framework for Structured Dynamic Covariance Recovery »
Katherine Tsai · Mladen Kolar · Sanmi Koyejo -
2021 Poster: Fast Multi-Resolution Transformer Fine-tuning for Extreme Multi-label Text Classification »
Jiong Zhang · Wei-Cheng Chang · Hsiang-Fu Yu · Inderjit Dhillon -
2021 Poster: Label Disentanglement in Partition-based Extreme Multilabel Classification »
Xuanqing Liu · Wei-Cheng Chang · Hsiang-Fu Yu · Cho-Jui Hsieh · Inderjit Dhillon -
2021 Poster: DRONE: Data-aware Low-rank Compression for Large NLP Models »
Patrick Chen · Hsiang-Fu Yu · Inderjit Dhillon · Cho-Jui Hsieh -
2020 Poster: CSER: Communication-efficient SGD with Error Reset »
Cong Xie · Shuai Zheng · Sanmi Koyejo · Indranil Gupta · Mu Li · Haibin Lin -
2020 Poster: Fairness with Overlapping Groups; a Probabilistic Perspective »
Forest Yang · Mouhamadou M Cisse · Sanmi Koyejo -
2020 Poster: Fair Performance Metric Elicitation »
Gaurush Hiranandani · Harikrishna Narasimhan · Sanmi Koyejo -
2019 : Lunch Break and Posters »
Xingyou Song · Elad Hoffer · Wei-Cheng Chang · Jeremy Cohen · Jyoti Islam · Yaniv Blumenfeld · Andreas Madsen · Jonathan Frankle · Sebastian Goldt · Satrajit Chatterjee · Abhishek Panigrahi · Alex Renda · Brian Bartoldson · Israel Birhane · Aristide Baratin · Niladri Chatterji · Roman Novak · Jessica Forde · YiDing Jiang · Yilun Du · Linara Adilova · Michael Kamp · Berry Weinstein · Itay Hubara · Tal Ben-Nun · Torsten Hoefler · Daniel Soudry · Hsiang-Fu Yu · Kai Zhong · Yiming Yang · Inderjit Dhillon · Jaime Carbonell · Yanqing Zhang · Dar Gilboa · Johannes Brandstetter · Alexander R Johansen · Gintare Karolina Dziugaite · Raghav Somani · Ari Morcos · Freddie Kalaitzis · Hanie Sedghi · Lechao Xiao · John Zech · Muqiao Yang · Simran Kaur · Qianli Ma · Yao-Hung Hubert Tsai · Ruslan Salakhutdinov · Sho Yaida · Zachary Lipton · Daniel Roy · Michael Carbin · Florent Krzakala · Lenka Zdeborová · Guy Gur-Ari · Ethan Dyer · Dilip Krishnan · Hossein Mobahi · Samy Bengio · Behnam Neyshabur · Praneeth Netrapalli · Kris Sankaran · Julien Cornebise · Yoshua Bengio · Vincent Michalski · Samira Ebrahimi Kahou · Md Rifat Arefin · Jiri Hron · Jaehoon Lee · Jascha Sohl-Dickstein · Samuel Schoenholz · David Schwab · Dongyu Li · Sang Keun Choe · Henning Petzka · Ashish Verma · Zhichao Lin · Cristian Sminchisescu -
2019 Poster: Provable Non-linear Inductive Matrix Completion »
Kai Zhong · Zhao Song · Prateek Jain · Inderjit Dhillon -
2019 Poster: Inverting Deep Generative models, One layer at a time »
Qi Lei · Ajil Jalal · Inderjit Dhillon · Alex Dimakis -
2019 Poster: Think Globally, Act Locally: A Deep Neural Network Approach to High-Dimensional Time Series Forecasting »
Rajat Sen · Hsiang-Fu Yu · Inderjit Dhillon -
2019 Poster: AutoAssist: A Framework to Accelerate Training of Deep Neural Networks »
Jiong Zhang · Hsiang-Fu Yu · Inderjit Dhillon -
2019 Poster: Learning Sparse Distributions using Iterative Hard Thresholding »
Jacky Zhang · Rajiv Khanna · Anastasios Kyrillidis · Sanmi Koyejo -
2019 Poster: Primal-Dual Block Generalized Frank-Wolfe »
Qi Lei · JIACHENG ZHUO · Constantine Caramanis · Inderjit Dhillon · Alex Dimakis -
2019 Poster: Multiclass Performance Metric Elicitation »
Gaurush Hiranandani · Shant Boodaghians · Ruta Mehta · Sanmi Koyejo -
2019 Tutorial: Representation Learning and Fairness »
Moustapha Cisse · Sanmi Koyejo -
2017 Poster: A Greedy Approach for Budgeted Maximum Inner Product Search »
Hsiang-Fu Yu · Cho-Jui Hsieh · Qi Lei · Inderjit Dhillon -
2016 Oral: Examples are not enough, learn to criticize! Criticism for Interpretability »
Been Kim · Sanmi Koyejo · Rajiv Khanna -
2016 Poster: Generalized Correspondence-LDA Models (GC-LDA) for Identifying Functional Regions in the Brain »
Timothy Rubin · Sanmi Koyejo · Michael Jones · Tal Yarkoni -
2016 Poster: Preference Completion from Partial Rankings »
Suriya Gunasekar · Sanmi Koyejo · Joydeep Ghosh -
2016 Poster: Asynchronous Parallel Greedy Coordinate Descent »
Yang You · Xiangru Lian · Ji Liu · Hsiang-Fu Yu · Inderjit Dhillon · James Demmel · Cho-Jui Hsieh -
2016 Poster: Regret Bounds for Non-decomposable Metrics with Missing Labels »
Nagarajan Natarajan · Prateek Jain -
2016 Poster: Coordinate-wise Power Method »
Qi Lei · Kai Zhong · Inderjit Dhillon -
2016 Poster: Structured Sparse Regression via Greedy Hard Thresholding »
Prateek Jain · Nikhil Rao · Inderjit Dhillon -
2016 Poster: Examples are not enough, learn to criticize! Criticism for Interpretability »
Been Kim · Sanmi Koyejo · Rajiv Khanna -
2016 Poster: Mixed Linear Regression with Multiple Components »
Kai Zhong · Prateek Jain · Inderjit Dhillon -
2016 Poster: Temporal Regularized Matrix Factorization for High-dimensional Time Series Prediction »
Hsiang-Fu Yu · Nikhil Rao · Inderjit Dhillon -
2016 Poster: Dual Decomposed Learning with Factorwise Oracle for Structural SVM of Large Output Domain »
Ian En-Hsu Yen · Xiangru Huang · Kai Zhong · Ruohan Zhang · Pradeep Ravikumar · Inderjit Dhillon -
2015 Workshop: Multiresolution methods for large-scale learning »
Inderjit Dhillon · Risi Kondor · Rob Nowak · Michael O'Neil · Nedelina Teneva -
2015 Poster: Predtron: A Family of Online Algorithms for General Prediction Problems »
Prateek Jain · Nagarajan Natarajan · Ambuj Tewari -
2015 Poster: Fast Classification Rates for High-dimensional Gaussian Generative Models »
Tianyang Li · Adarsh Prasad · Pradeep Ravikumar -
2015 Poster: Matrix Completion with Noisy Side Information »
Kai-Yang Chiang · Cho-Jui Hsieh · Inderjit Dhillon -
2015 Poster: Collaborative Filtering with Graph Information: Consistency and Scalable Methods »
Nikhil Rao · Hsiang-Fu Yu · Pradeep Ravikumar · Inderjit Dhillon -
2015 Spotlight: Collaborative Filtering with Graph Information: Consistency and Scalable Methods »
Nikhil Rao · Hsiang-Fu Yu · Pradeep Ravikumar · Inderjit Dhillon -
2015 Spotlight: Matrix Completion with Noisy Side Information »
Kai-Yang Chiang · Cho-Jui Hsieh · Inderjit Dhillon -
2015 Poster: Beyond Sub-Gaussian Measurements: High-Dimensional Structured Estimation with Sub-Exponential Designs »
Vidyashankar Sivakumar · Arindam Banerjee · Pradeep Ravikumar -
2015 Poster: Sparse Linear Programming via Primal and Dual Augmented Coordinate Descent »
Ian En-Hsu Yen · Kai Zhong · Cho-Jui Hsieh · Pradeep Ravikumar · Inderjit Dhillon -
2015 Poster: Fixed-Length Poisson MRF: Adding Dependencies to the Multinomial »
David I Inouye · Pradeep Ravikumar · Inderjit Dhillon -
2015 Poster: Consistent Multilabel Classification »
Oluwasanmi Koyejo · Nagarajan Natarajan · Pradeep Ravikumar · Inderjit Dhillon -
2015 Poster: Closed-form Estimators for High-dimensional Generalized Linear Models »
Eunho Yang · Aurelie Lozano · Pradeep Ravikumar -
2015 Spotlight: Closed-form Estimators for High-dimensional Generalized Linear Models »
Eunho Yang · Aurelie Lozano · Pradeep Ravikumar -
2014 Poster: On Prior Distributions and Approximate Inference for Structured Variables »
Sanmi Koyejo · Rajiv Khanna · Joydeep Ghosh · Russell Poldrack -
2014 Poster: QUIC & DIRTY: A Quadratic Approximation Approach for Dirty Statistical Models »
Cho-Jui Hsieh · Inderjit Dhillon · Pradeep Ravikumar · Stephen Becker · Peder A Olsen -
2014 Poster: Fast Prediction for Large-Scale Kernel Machines »
Cho-Jui Hsieh · Si Si · Inderjit Dhillon -
2014 Poster: Multi-Scale Spectral Decomposition of Massive Graphs »
Si Si · Donghyuk Shin · Inderjit Dhillon · Beresford N Parlett -
2014 Poster: On the Information Theoretic Limits of Learning Ising Models »
Rashish Tandon · Karthikeyan Shanmugam · Pradeep Ravikumar · Alex Dimakis -
2014 Poster: Sparse Random Feature Algorithm as Coordinate Descent in Hilbert Space »
Ian En-Hsu Yen · Ting-Wei Lin · Shou-De Lin · Pradeep Ravikumar · Inderjit Dhillon -
2014 Poster: Proximal Quasi-Newton for Computationally Intensive L1-regularized M-estimators »
Kai Zhong · Ian En-Hsu Yen · Inderjit Dhillon · Pradeep Ravikumar -
2014 Poster: A Representation Theory for Ranking Functions »
Harsh H Pareek · Pradeep Ravikumar -
2014 Poster: Capturing Semantically Meaningful Word Dependencies with an Admixture of Poisson MRFs »
David I Inouye · Pradeep Ravikumar · Inderjit Dhillon -
2014 Poster: Constant Nullspace Strong Convexity and Fast Convergence of Proximal Methods under High-Dimensional Settings »
Ian En-Hsu Yen · Cho-Jui Hsieh · Pradeep Ravikumar · Inderjit Dhillon -
2014 Poster: Elementary Estimators for Graphical Models »
Eunho Yang · Aurelie Lozano · Pradeep Ravikumar -
2014 Poster: Sparse Bayesian structure learning with dependent relevance determination prior »
Anqi Wu · Mijung Park · Sanmi Koyejo · Jonathan W Pillow -
2013 Workshop: Discrete Optimization in Machine Learning: Connecting Theory and Practice »
Stefanie Jegelka · Andreas Krause · Pradeep Ravikumar · Kazuo Murota · Jeffrey A Bilmes · Yisong Yue · Michael Jordan -
2013 Poster: Conditional Random Fields via Univariate Exponential Families »
Eunho Yang · Pradeep Ravikumar · Genevera I Allen · Zhandong Liu -
2013 Poster: On Poisson Graphical Models »
Eunho Yang · Pradeep Ravikumar · Genevera I Allen · Zhandong Liu -
2013 Poster: BIG & QUIC: Sparse Inverse Covariance Estimation for a Million Variables »
Cho-Jui Hsieh · Matyas A Sustik · Inderjit Dhillon · Pradeep Ravikumar · Russell Poldrack -
2013 Oral: BIG & QUIC: Sparse Inverse Covariance Estimation for a Million Variables »
Cho-Jui Hsieh · Matyas A Sustik · Inderjit Dhillon · Pradeep Ravikumar · Russell Poldrack -
2013 Poster: Dirty Statistical Models »
Eunho Yang · Pradeep Ravikumar -
2013 Poster: Large Scale Distributed Sparse Precision Estimation »
Huahua Wang · Arindam Banerjee · Cho-Jui Hsieh · Pradeep Ravikumar · Inderjit Dhillon -
2013 Poster: Learning with Noisy Labels »
Nagarajan Natarajan · Inderjit Dhillon · Pradeep Ravikumar · Ambuj Tewari -
2012 Workshop: Discrete Optimization in Machine Learning (DISCML): Structure and Scalability »
Stefanie Jegelka · Andreas Krause · Jeffrey A Bilmes · Pradeep Ravikumar -
2012 Poster: Graphical Models via Generalized Linear Models »
Eunho Yang · Pradeep Ravikumar · Genevera I Allen · zhandong Liu -
2012 Oral: Graphical Models via Generalized Linear Models »
Eunho Yang · Pradeep Ravikumar · Genevera I Allen · zhandong Liu -
2012 Poster: A Divide-and-Conquer Method for Sparse Inverse Covariance Estimation »
Cho-Jui Hsieh · Inderjit Dhillon · Pradeep Ravikumar · Arindam Banerjee -
2011 Workshop: Discrete Optimization in Machine Learning (DISCML): Uncertainty, Generalization and Feedback »
Andreas Krause · Pradeep Ravikumar · Stefanie S Jegelka · Jeffrey A Bilmes -
2011 Poster: On Learning Discrete Graphical Models using Greedy Methods »
Ali Jalali · Christopher C Johnson · Pradeep Ravikumar -
2011 Spotlight: On Learning Discrete Graphical Models using Greedy Methods »
Ali Jalali · Christopher C Johnson · Pradeep Ravikumar -
2011 Poster: Greedy Algorithms for Structurally Constrained High Dimensional Problems »
Ambuj Tewari · Pradeep Ravikumar · Inderjit Dhillon -
2011 Poster: Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximation »
Cho-Jui Hsieh · Matyas A Sustik · Inderjit Dhillon · Pradeep Ravikumar -
2011 Session: Oral Session 5 »
Pradeep Ravikumar -
2011 Poster: Nearest Neighbor based Greedy Coordinate Descent »
Inderjit Dhillon · Pradeep Ravikumar · Ambuj Tewari -
2011 Poster: Orthogonal Matching Pursuit with Replacement »
Prateek Jain · Ambuj Tewari · Inderjit Dhillon -
2010 Workshop: Discrete Optimization in Machine Learning: Structures, Algorithms and Applications »
Andreas Krause · Pradeep Ravikumar · Jeffrey A Bilmes · Stefanie Jegelka -
2010 Workshop: Robust Statistical Learning »
Pradeep Ravikumar · Constantine Caramanis · Sujay Sanghavi -
2010 Session: Oral Session 14 »
Pradeep Ravikumar -
2010 Spotlight: Guaranteed Rank Minimization via Singular Value Projection »
Prateek Jain · Raghu Meka · Inderjit Dhillon -
2010 Poster: Guaranteed Rank Minimization via Singular Value Projection »
Prateek Jain · Raghu Meka · Inderjit Dhillon -
2010 Spotlight: Inductive Regularized Learning of Kernel Functions »
Prateek Jain · Brian Kulis · Inderjit Dhillon -
2010 Oral: A Dirty Model for Multi-task Learning »
Ali Jalali · Pradeep Ravikumar · Sujay Sanghavi · Chao Ruan -
2010 Poster: Inductive Regularized Learning of Kernel Functions »
Prateek Jain · Brian Kulis · Inderjit Dhillon -
2010 Poster: A Dirty Model for Multi-task Learning »
Ali Jalali · Pradeep Ravikumar · Sujay Sanghavi · Chao Ruan -
2009 Workshop: Discrete Optimization in Machine Learning: Submodularity, Polyhedra and Sparsity »
Andreas Krause · Pradeep Ravikumar · Jeffrey A Bilmes -
2009 Poster: Information-theoretic lower bounds on the oracle complexity of convex optimization »
Alekh Agarwal · Peter Bartlett · Pradeep Ravikumar · Martin J Wainwright -
2009 Spotlight: Information-theoretic lower bounds on the oracle complexity of convex optimization »
Alekh Agarwal · Peter Bartlett · Pradeep Ravikumar · Martin J Wainwright -
2009 Poster: A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers »
Sahand N Negahban · Pradeep Ravikumar · Martin J Wainwright · Bin Yu -
2009 Oral: A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers »
Sahand N Negahban · Pradeep Ravikumar · Martin J Wainwright · Bin Yu -
2009 Poster: Matrix Completion from Power-Law Distributed Samples »
Raghu Meka · Prateek Jain · Inderjit Dhillon -
2008 Poster: Nonparametric sparse hierarchical models describe V1 fMRI responses to natural images »
Pradeep Ravikumar · Vincent Vu · Bin Yu · Thomas Naselaris · Kendrick Kay · Jack Gallant -
2008 Spotlight: Nonparametric sparse hierarchical models describe V1 fMRI responses to natural images »
Pradeep Ravikumar · Vincent Vu · Bin Yu · Thomas Naselaris · Kendrick Kay · Jack Gallant -
2008 Poster: Online Metric Learning and Fast Similarity Search »
Prateek Jain · Brian Kulis · Inderjit Dhillon · Kristen Grauman -
2008 Oral: Online Metric Learning and Fast Similarity Search »
Prateek Jain · Brian Kulis · Inderjit Dhillon · Kristen Grauman -
2008 Poster: Model Selection in Gaussian Graphical Models: High-Dimensional Consistency of \ell_1-regularizedMLE »
Pradeep Ravikumar · Garvesh Raskutti · Martin J Wainwright · Bin Yu -
2007 Poster: SpAM: Sparse Additive Models »
Pradeep Ravikumar · Han Liu · John Lafferty · Larry Wasserman -
2007 Spotlight: SpAM: Sparse Additive Models »
Pradeep Ravikumar · Han Liu · John Lafferty · Larry Wasserman -
2006 Poster: Inferring Graphical Model Structure using $\ell_1$-Regularized Pseudo-Likelihood »
Martin J Wainwright · Pradeep Ravikumar · John Lafferty -
2006 Spotlight: Inferring Graphical Model Structure using $\ell_1$-Regularized Pseudo-Likelihood »
Martin J Wainwright · Pradeep Ravikumar · John Lafferty -
2006 Poster: Differential Entropic Clustering of Multivariate Gaussians »
Jason V Davis · Inderjit Dhillon