{ "id": "epxts1rfkd", "slug": "sparse-estimation-statistics", "title": "Sparse estimation (statistics)", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" }, { "id": "physical.math.statistics.stats_statistical_inference.estimation", "label": "Estimation theory and estimator families", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference.estimation" }, { "id": "physical.math.statistics.stats_statistical_inference.estimation_overview", "label": "Estimation: overview", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference.estimation_overview" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation", "label": "Non-geographic · 1946 to 2000: Estimation theory and estimator families", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation", "label": "Estimation theory and estimator families", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation" } ] } ], "excerpt": "Sparse estimation is a class of statistical methods that fit high-dimensional models by penalizing coefficients so many are exactly zero, as in Robert Tibshirani's 1996 lasso.", "snippet": "Sparse estimation is a class of statistical methods that fit high-dimensional models by penalizing coefficients so many are exactly zero, as in Robert Tibshirani's 1996 lasso.", "node": "physical.math.statistics.stats_statistical_inference.estimation_overview", "markdown": "# Sparse estimation (statistics)\n\nSparse estimation is a class of statistical methods that fit high-dimensional models by penalizing or constraining the coefficients so that many are estimated as exactly zero, leaving a small selected subset. The leading example, the lasso, minimizes the residual sum of squares subject to the sum of absolute coefficient values being less than a constant, which tends to produce some coefficients that are exactly 0 and hence interpretable models.[1](https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1996.tb02080.x) Equivalently it minimizes \\( \\tfrac{1}{2}\\|y - X\\beta\\|_2^2 + \\lambda\\|\\beta\\|_1 \\) for a tuning parameter \\( \\lambda \\ge 0 \\), and larger \\( \\lambda \\) gives sparser solutions.[2](https://www.stat.berkeley.edu/~ryantibs/statlearn-s23/lectures/lasso.pdf) Sparsity serves two purposes: it performs variable selection, aiding interpretation, and it can predict better when the true regression function is well approximated by a sparse linear model.[2](https://www.stat.berkeley.edu/~ryantibs/statlearn-s23/lectures/lasso.pdf)\n\n| Key fact | Statement |\n|---|---|\n| Output | A fitted model in which many coefficients are exactly zero, giving automatic variable selection while remaining convex and efficiently solvable.[1](https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1996.tb02080.x) |\n| Objective | \\( \\tfrac{1}{2}\\|y - X\\beta\\|_2^2 + \\lambda\\|\\beta\\|_1 \\); larger \\( \\lambda \\) means sparser solutions.[2](https://www.stat.berkeley.edu/~ryantibs/statlearn-s23/lectures/lasso.pdf) |\n| Mechanism | With orthonormal designs the lasso is a soft-thresholding rule \\( S_{\\lambda}(x) = \\mathrm{sign}(x)(\\lvert x\\rvert - \\lambda)_+ \\), which sets small coefficients to zero.[3](https://www.ime.unicamp.br/~dias/SLS.pdf) |\n| Saturation | The lasso solution has at most \\( \\min(N, p) \\) nonzero coefficients; with \\( p = 40{,}000 \\) and \\( N = 100 \\), at most 100 are nonzero.[4](https://web.stanford.edu/group/SOL/papers/fused-lasso-JRSSB.pdf) |\n| Tuning \\( \\lambda \\) | Cross-validation with \\( K = 5 \\) or 10 folds is standard for prediction; theory suggests \\( \\lambda \\) of order \\( \\sigma\\sqrt{\\log(p)/n} \\), but CV-tuned lasso is inconsistent for model selection.[3](https://www.ime.unicamp.br/~dias/SLS.pdf) • [5](https://ar5iv.labs.arxiv.org/html/2012.11470) |\n| Algorithms | Pathwise coordinate descent has surpassed LARS in high dimension; FISTA is a proximal-gradient alternative.[6](https://ar5iv.labs.arxiv.org/html/1902.06547) |\n| Oracle property | The adaptive lasso, SCAD, and MCP have it; the plain lasso does not.[7](https://pages.stat.wisc.edu/~wahba/stat860/talks/allpapers/860talks/Stat860_Wenzhi_Cao/zou2006.pdf) • [6](https://ar5iv.labs.arxiv.org/html/1902.06547) |\n\n## How it works\n\nThe lasso replaces the cardinality of the coefficient support, an \\( \\ell_0 \\) count whose minimization is NP-hard in general, with the \\( \\ell_1 \\) norm, the traditional convex approximation for sparse selection.[8](https://www.di.ens.fr/~fbach/bach_jenatton_mairal_obozinski_FOT.pdf) • [9](https://people.eecs.berkeley.edu/~jordan/sail/readings/Wainwright_SharpThreshold.pdf) Geometry explains the exact zeros: the \\( \\ell_1 \\) constraint region \\( \\lvert\\beta_1\\rvert + \\lvert\\beta_2\\rvert \\le t \\) has sharp corners, and in high dimensions sparsity arises from corners and edges of the constraint region, whereas ridge regression's smooth constraint region does not produce them.[10](https://tibshirani.su.domains/ftp/tibs-copss.pdf) Analytically, a vector w is a lasso solution if and only if for every j, \\( \\lvert X_j^{\\top}(y - Xw)\\rvert \\le n\\lambda \\) when \\( w_j = 0 \\) and \\( X_j^{\\top}(y - Xw) = n\\lambda\\,\\mathrm{sign}(w_j) \\) when \\( w_j \\ne 0 \\); coefficients whose correlation with the residual falls below the threshold stay at exactly zero.[8](https://www.di.ens.fr/~fbach/bach_jenatton_mairal_obozinski_FOT.pdf)\n\n## How it is done\n\nPathwise coordinate descent starts at \\( \\lambda_{\\max} = \\max_j \\lvert \\tfrac{1}{N}\\langle x_j, y\\rangle \\rvert \\), the smallest \\( \\lambda \\) giving the all-zero solution, then decreases \\( \\lambda \\) over a grid, running coordinate descent to convergence at each step and using the previous solution as a warm start.[3](https://www.ime.unicamp.br/~dias/SLS.pdf) [Coordinate descent](https://www.edgepedia.org/coordinate-descent) for penalized regression was reported by Wenjiang J. Fu in 1998 in the Journal of Computational and Graphical Statistics,[11](https://doi.org/10.1080/10618600.1998.10474784) and pathwise coordinate descent in the glmnet R package exploits sparsity, active-set convergence, and strong rules to compute the entire path rapidly.[12](https://mail.hastie.su.domains/public/TALKS/hastieJSM2017.pdf) Tuning is by [K-fold cross-validation](https://www.edgepedia.org/k-fold-cross-validation), typically \\( K = 5 \\) or 10, fitting on \\( K - 1 \\) groups and averaging mean-squared prediction errors over the test folds.[3](https://www.ime.unicamp.br/~dias/SLS.pdf)\n\n## Origin\n\n[Robert Tibshirani](https://www.edgepedia.org/robert-tibshirani) introduced the lasso in 1996 in the [Journal of the Royal Statistical Society](https://www.edgepedia.org/journal-of-the-royal-statistical-society): Series B (Methodological).[1](https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1996.tb02080.x) The paper was motivated by an earlier two-stage proposal, the nonnegative garrote, which starts with ordinary least squares estimates and shrinks them by nonnegative factors whose sum is constrained; a drawback is its dependence on the sign and magnitude of the OLS estimates, which behave poorly in overfit or highly correlated settings.[1](https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1996.tb02080.x) A second precursor was bridge regression, a bound on the \\( L_q \\)-norm of the parameters in which the lasso corresponds to \\( q = 1 \\), subset selection to \\( q = 0 \\), and ridge regression to \\( q = 2 \\), with \\( q = 1 \\) the smallest value giving a convex region.[1](https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1996.tb02080.x) In signal processing, basis pursuit, reported by Scott Shaobing Chen, David L. Donoho, and Michael A. Saunders in 2001 in SIAM Review, uses the same \\( \\ell_1 \\) idea.[13](https://doi.org/10.1137/s003614450037906x)\n\n## Variants\n\n**Elastic net.** Hui Zou and [Trevor Hastie](https://www.edgepedia.org/trevor-hastie) proposed the elastic net in 2005 in the Journal of the Royal Statistical Society Series B.[14](https://doi.org/10.1111/j.1467-9868.2005.00503.x) It adds an \\( \\ell_2 \\) term, \\( \\lambda\\|\\beta\\|_1 + \\delta\\|\\beta\\|_2^2 \\), combining ridge's predictive properties with the lasso's sparsity and guaranteeing uniqueness.[15](https://www.stat.cmu.edu/~ryantibs/statml/lectures/sparsity.pdf) It encourages a grouping effect in which strongly correlated predictors enter or leave the model together, and is particularly useful when p is much bigger than n, where the lasso is unsatisfactory.[14](https://doi.org/10.1111/j.1467-9868.2005.00503.x)\n\n**Adaptive lasso.** Hui Zou proposed the adaptive lasso in 2006 in the [Journal of the American Statistical Association](https://www.edgepedia.org/journal-of-the-american-statistical-association).[16](https://doi.org/10.1198/016214506000000735) It uses adaptive weights in the \\( \\ell_1 \\) penalty, reducing shrinkage for strong signals and increasing it for weaker ones, and enjoys the oracle property under mild conditions: it correctly identifies the true model with probability tending to one and produces asymptotically unbiased estimates for nonzero coefficients.[17](https://pmc.ncbi.nlm.nih.gov/articles/PMC12888550/)\n\n**Fused lasso.** Robert Tibshirani, Michael Saunders, Saharon Rosset, Ji Zhu, and Keith Knight's fused lasso (2005) penalizes the \\( \\ell_1 \\)-norm of both the coefficients and their successive differences, encouraging sparsity and local constancy for ordered features.[4](https://web.stanford.edu/group/SOL/papers/fused-lasso-JRSSB.pdf)\n\n**Group and graphical lasso.** The mixed \\( \\ell_1/\\ell_2 \\)-norm yields the group lasso, which sets whole groups of coefficients to zero; structured extensions include overlapping groups, hierarchical penalties, and graph-structured penalties.[8](https://www.di.ens.fr/~fbach/bach_jenatton_mairal_obozinski_FOT.pdf) M. Yuan and Y. Lin's 2007 [Biometrika](https://www.edgepedia.org/biometrika) paper covers model selection and estimation in the [Gaussian graphical model](https://www.edgepedia.org/gaussian-graphical-model).[18](https://doi.org/10.1093/biomet/asm018)\n\n**Non-convex penalties.** The SCAD (smoothly clipped absolute deviation) penalty was introduced to ameliorate the \\( \\ell_1 \\) penalty's properties, leaving large coefficients not excessively penalized while keeping the solution continuous.[19](https://escholarship.org/content/qt8j29393d/qt8j29393d.pdf)\n\n**Other estimators.** Emmanuel Candes and [Terence Tao](https://www.edgepedia.org/terence-tao)'s Dantzig selector (2007, The Annals of Statistics) addresses estimation when p is much larger than n.[20](https://doi.org/10.1214/009053606000001523) Stability Selection, by Nicolai Meinshausen and [Peter Bühlmann](https://www.edgepedia.org/peter-buhlmann) (2010, JRSS-B), subsamples a sparse estimator to control selection error.[21](https://doi.org/10.1111/j.1467-9868.2010.00740.x) SLOPE (Sorted L-One Penalized Estimation), by Małgorzata Bogdan, Ewout van den Berg, Chiara Sabatti, Weijie Su, and Emmanuel J. Candès (2015, The Annals of Applied Statistics), adapts the penalty across sorted coefficients.[22](https://doi.org/10.1214/15-aoas842)\n\n## Applications\n\nStructured sparsity penalties built on the group lasso are applied in computer vision, text processing, bioinformatics, and audio processing.[8](https://www.di.ens.fr/~fbach/bach_jenatton_mairal_obozinski_FOT.pdf) In genomics, benchmarking studies evaluate lasso, elastic net, and adaptive lasso for sparse variable selection on high-dimensional molecular data.[17](https://pmc.ncbi.nlm.nih.gov/articles/PMC12888550/) The same \\( \\ell_1 \\) machinery underlies compressed sensing and basis pursuit for signal recovery from undersampled measurements,[9](https://people.eecs.berkeley.edu/~jordan/sail/readings/Wainwright_SharpThreshold.pdf) and extends to matrix completion, where Emmanuel J. Candès and [Benjamin Recht](https://www.edgepedia.org/benjamin-recht) (2009, Foundations of Computational Mathematics) showed exact recovery of low-rank matrices via convex optimization.[23](https://doi.org/10.1007/s10208-009-9045-5)\n\n## Limitations and alternatives\n\n**Correlated predictors.** When several highly correlated covariates relate to the response, the lasso tends to pick randomly only one or a few of them and shrink the rest to 0, losing information under strong dependence.[5](https://ar5iv.labs.arxiv.org/html/2012.11470) The elastic net, which has no limitation on the number of selected features, is the standard remedy.[14](https://doi.org/10.1111/j.1467-9868.2005.00503.x)\n\n**Saturation and shrinkage bias.** The lasso can select at most \\( \\min(n, p) \\) variables before saturating.[4](https://web.stanford.edu/group/SOL/papers/fused-lasso-JRSSB.pdf) The penalty shrinks small and large coefficients equally, causing bias for large coefficients; when \\( \\lambda \\) must be large for proper variable selection, the estimator is seriously biased downward, and shrinkage noise in residuals can dwarf strong signals and select null variables.[24](https://www.mdpi.com/2076-3417/15/13/7443) • [5](https://ar5iv.labs.arxiv.org/html/2012.11470) Two-stage remedies include the relaxed lasso and thresholded lasso, and reweighting via the adaptive lasso.[5](https://ar5iv.labs.arxiv.org/html/2012.11470)\n\n**False positives.** [Lasso](https://www.edgepedia.org/lasso-statistics) tuned by cross-validation for prediction often selects many variables with a high false-positive rate,[24](https://www.mdpi.com/2076-3417/15/13/7443) and in feature-selection experiments lasso-based estimators returned at least 80% of non-significant features, while MCP and SCAD had false detection rates around 15–30%.[6](https://ar5iv.labs.arxiv.org/html/1902.06547)\n\n**Selection consistency.** [Peng Zhao](https://www.edgepedia.org/peng-zhao) and [Bin Yu](https://www.edgepedia.org/bin-yu) showed that a single condition on the covariance of the predictors, the Irrepresentable Condition, is almost necessary and sufficient for the lasso to select the true model consistently in both fixed-p and large-p settings; when the strong version holds, the selection probability approaches 1 at an exponential rate assuming only a finite second moment of the noise.[25](https://jmlr.org/papers/volume7/zhao06a/zhao06a.pdf) The plain lasso fails to meet both oracle properties simultaneously, because the double-exponential prior's tails are too light; the adaptive lasso remedies this when the initial estimate is \\( \\sqrt{n} \\)-consistent.[15](https://www.stat.cmu.edu/~ryantibs/statml/lectures/sparsity.pdf) • [26](https://tibshirani.su.domains/ftp/lasso-retro.pdf)\n\n**Empirical comparisons.** A large benchmark spanning more than 2300 data-generating scenarios found no unambiguous winner among lasso, adaptive lasso, elastic net, ridge, SCAD, Dantzig selector, and stability selection across prediction, selection, and ranking goals.[27](https://www.repository.cam.ac.uk/items/4957bc68-5490-42d2-9e79-4a6e13b9c667)\n\n**Inference after selection.** A line of 2014 work proposed and analyzed the debiased (desparsified) lasso, a procedure to fix the bias introduced by the \\( \\ell_1 \\) penalty; its error decomposes into a Gaussian component plus a remainder term that vanishes asymptotically with high probability.[28](https://arxiv.org/html/2407.13666) For path-based inference, a covariance test statistic for predictors entering the lasso path converges under the null to an exponential random variable with unit mean as \\( p \\to \\infty \\), providing p-values along the path.[10](https://tibshirani.su.domains/ftp/tibs-copss.pdf)\n\n## References\n\n1. [Regression Shrinkage and Selection Via the Lasso (Tibshirani, 1996)](https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1996.tb02080.x)\n2. [High-Dimensional Regression: Lasso (lecture notes, Berkeley StatLearn S23)](https://www.stat.berkeley.edu/~ryantibs/statlearn-s23/lectures/lasso.pdf)\n3. [Statistical Learning with Sparsity (Hastie, Tibshirani, Wainwright), Chapter 2: The Lasso for Linear Models](https://www.ime.unicamp.br/~dias/SLS.pdf)\n4. [Sparsity and smoothness via the fused lasso (Tibshirani, Saunders et al.)](https://web.stanford.edu/group/SOL/papers/fused-lasso-JRSSB.pdf)\n5. [A critical review of LASSO and its derivatives for variable selection under dependence among covariates (arXiv 2012.11470)](https://ar5iv.labs.arxiv.org/html/2012.11470)\n6. [Sparse regression: Scalable algorithms and empirical performance (arXiv 1902.06547)](https://ar5iv.labs.arxiv.org/html/1902.06547)\n7. [The Adaptive Lasso and Its Oracle Properties (Zou, 2006)](https://pages.stat.wisc.edu/~wahba/stat860/talks/allpapers/860talks/Stat860_Wenzhi_Cao/zou2006.pdf)\n8. [Optimization with Sparsity-Inducing Penalties (Bach, Jenatton, Mairal, Obozinski)](https://www.di.ens.fr/~fbach/bach_jenatton_mairal_obozinski_FOT.pdf)\n9. [Sharp thresholds for high-dimensional and noisy recovery of sparsity using ℓ1-constrained quadratic programming (Wainwright)](https://people.eecs.berkeley.edu/~jordan/sail/readings/Wainwright_SharpThreshold.pdf)\n10. [In praise of sparsity and convexity (Tibshirani COPSS piece)](https://tibshirani.su.domains/ftp/tibs-copss.pdf)\n11. [Wenjiang J. Fu (1998). Penalized Regressions: The Bridge versus the Lasso. Journal of Computational and Graphical Statistics.](https://doi.org/10.1080/10618600.1998.10474784)\n12. [Variable Selection at Scale (Hastie JSM 2017 talk)](https://mail.hastie.su.domains/public/TALKS/hastieJSM2017.pdf)\n13. [Scott Shaobing Chen, David L. Donoho, Michael A. Saunders (2001). Atomic Decomposition by Basis Pursuit. SIAM Review.](https://doi.org/10.1137/s003614450037906x)\n14. [Hui Zou, Trevor Hastie (2005). Regularization and Variable Selection Via the Elastic Net. Journal of the Royal Statistical Society Series B (Statistical Methodology).](https://doi.org/10.1111/j.1467-9868.2005.00503.x)\n15. [Sparsity, the Lasso, and Friends (R. Tibshirani lecture notes, CMU)](https://www.stat.cmu.edu/~ryantibs/statml/lectures/sparsity.pdf)\n16. [Hui Zou (2006). The Adaptive Lasso and Its Oracle Properties. Journal of the American Statistical Association.](https://doi.org/10.1198/016214506000000735)\n17. [Benchmarking Sparse Variable Selection Methods for Genomic Data Analyses (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC12888550/)\n18. [M. Yuan, Y. Lin (2007). Model selection and estimation in the Gaussian graphical model. Biometrika.](https://doi.org/10.1093/biomet/asm018)\n19. [Variable Selection via Penalized Likelihood (Fan & Li, SCAD)](https://escholarship.org/content/qt8j29393d/qt8j29393d.pdf)\n20. [Emmanuel Candes, Terence Tao (2007). The Dantzig selector: Statistical estimation when p is much larger than n. The Annals of Statistics.](https://doi.org/10.1214/009053606000001523)\n21. [Nicolai Meinshausen, Peter Bühlmann (2010). Stability Selection. Journal of the Royal Statistical Society Series B (Statistical Methodology).](https://doi.org/10.1111/j.1467-9868.2010.00740.x)\n22. [Małgorzata Bogdan and colleagues (2015). SLOPE, Adaptive variable selection via convex optimization. The Annals of Applied Statistics.](https://doi.org/10.1214/15-aoas842)\n23. [Emmanuel J. Candès, Benjamin Recht (2009). Exact Matrix Completion via Convex Optimization. Foundations of Computational Mathematics.](https://doi.org/10.1007/s10208-009-9045-5)\n24. [Evaluating Prediction Performance: A Simulation Study Comparing Penalized and Classical Variable Selection Methods in Low-Dimensional Data (Applied Sciences, 2025)](https://www.mdpi.com/2076-3417/15/13/7443)\n25. [On Model Selection Consistency of Lasso (Zhao & Yu, JMLR 7)](https://jmlr.org/papers/volume7/zhao06a/zhao06a.pdf)\n26. [Regression Shrinkage and Selection via the Lasso: a Retrospective (JRSSB 2011)](https://tibshirani.su.domains/ftp/lasso-retro.pdf)\n27. [High-dimensional regression in practice (Wang, Mukherjee, Richardson, Hill; Statistics and Computing)](https://www.repository.cam.ac.uk/items/4957bc68-5490-42d2-9e79-4a6e13b9c667)\n28. [Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning (arXiv 2407.13666, 2024)](https://arxiv.org/html/2407.13666)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing › Estimation theory and estimator families › Estimation: overview*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://www.stat.berkeley.edu/~ryantibs/statlearn-s23/lectures/lasso.pdf", "https://pages.stat.wisc.edu/~wahba/stat860/talks/allpapers/860talks/Stat860_Wenzhi_Cao/zou2006.pdf", "https://people.eecs.berkeley.edu/~jordan/sail/readings/Wainwright_SharpThreshold.pdf", "https://www.stat.cmu.edu/~ryantibs/statml/lectures/sparsity.pdf" ], "url": "https://www.edgepedia.org/sparse-estimation-statistics", "markdown_url": "https://www.edgepedia.org/sparse-estimation-statistics.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. 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Edgepedia Community License 1.0.", "speakable": "Sparse estimation is a class of statistical methods that fit high-dimensional models by penalizing coefficients so many are exactly zero, as in Robert Tibshirani's 1996 lasso." }