---
tag: KZ226116HW
kind: concept
name: Lasso
msc: [62J07, 62J05]
sources:
  - url: https://en.wikipedia.org/wiki/Lasso_(statistics)
same_as:
  wikidata: Q20789991
  wikipedia: Lasso (statistics)
---

## Summary

The lasso, short for least absolute shrinkage and selection operator, fits a linear model by penalised least squares.
The penalty is a multiple of the sum of the absolute values of the coefficients.
Because of the corners of the $\ell^1$ ball, the penalty sets some coefficients exactly to zero.
So the lasso performs estimation and variable selection at the same time.
It was introduced in statistics by Robert Tibshirani in 1996 and is a basic tool for regression with many more regressors than observations.
It is a [[shrinkage estimator|GPBSS1V3SS]], and the [[elastic net|TKQPE0Q05Y]] combines it with [[ridge regression|3KZW2930V2]].

## Explanations

### L1 human

The lasso charges a price for every unit of coefficient size, whatever the coefficient.
A regressor whose contribution to the fit is smaller than that price is dropped completely: its coefficient becomes zero.
The result is a model that uses only a few regressors, which is easier to read.
When the regressors are orthonormal, the lasso takes each least squares coefficient, moves it towards zero by the penalty level, and sets it to zero if it would cross zero.
For example, with penalty level $1$, least squares coefficients $3$, $-0.5$ and $-2$ become $2$, $0$ and $-1$.