Regression Shrinkage and Selection via the Lasso.¶
Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society: Series B (Methodological), 58(1), 267-288.
Cited by¶
4 citations across 4 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Convexity
- Statistics and machine learning. Log-likelihoods of exponential families are concave, so maximum-likelihood is a convex problem; convex relaxations (LP relaxation of integer programs, nuclear-norm for rank, the Lasso for sparsity) make hard tasks tractable, and convex surrogate losses replace the non-convex 0–1 loss.
This sourceThe Lasso — an L1-constrained (convex) reformulation that yields sparse, interpretable models — exemplifying convex relaxation/surrogate machinery in statistics and machine learning.
Supported in partVerified against the publisher's abstract
Tibshirani's abstract supplies the L1-constrained, sparsity-producing Lasso — one instance of convex relaxation — but says nothing about exponential-family concavity, LP relaxation, nuclear norm, or 0–1 surrogate losses.
“The ‘lasso’ minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models.”
- Statistics and machine learning. Log-likelihoods of exponential families are concave, so maximum-likelihood is a convex problem; convex relaxations (LP relaxation of integer programs, nuclear-norm for rank, the Lasso for sparsity) make hard tasks tractable, and convex surrogate losses replace the non-convex 0–1 loss.
- Overfitting
- Listed in the references but not attached to a specific claim.
- Parsimony (Occam's Razor)
- Regularization via LASSO (L1): Tibshirani's LASSO
SupportedVerified against the publisher's abstract
Tibshirani introduces the lasso, which “minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant” and, because of that constraint, “tends to produce some coefficients that are exactly 0 and hence gives interpretable models”. That L1 shrinkage-and-selection method, and its sparsity effect, are what the heading attributes to him. The work states the lasso as a constraint on the sum of absolute coefficients rather than as the lambda-weighted penalty term the surrounding passage writes it as.
“The ‘lasso’ minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models.”
- Regularization via LASSO (L1): Tibshirani's LASSO
- Regularization
- The frame's prior-penalty duality is exact here — ridge is precisely MAP estimation under a Gaussian prior on \(\beta\), so the penalty is a prior — and the choice-of-norm inference is visible by swapping the L2 norm for L1, which yields the lasso and induces sparsity rather than uniform shrinkage.
This sourceIntroduces the lasso (L1 penalty) inducing sparsity; equivalently MAP estimation under a Laplace prior.
Supported in partVerified against the publisher's abstract
Tibshirani's abstract supplies the L1 half of the claim — the lasso constraint drives coefficients to exactly 0, i.e. sparsity — but says nothing about ridge, Gaussian priors, or the penalty-as-prior duality.
“Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models.”
- The frame's prior-penalty duality is exact here — ridge is precisely MAP estimation under a Gaussian prior on \(\beta\), so the penalty is a prior — and the choice-of-norm inference is visible by swapping the L2 norm for L1, which yields the lasso and induces sparsity rather than uniform shrinkage.
Verification¶
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