---
tag: M68V0M5A3R
kind: concept
name: Tikhonov regularization
msc: [47A52, 45Q05, 65R30]
sources:
  - url: https://en.wikipedia.org/wiki/Ridge_regression
---

## Summary

Tikhonov regularization is a method to solve an ill-posed linear equation $K x = y$ in a stable way.
Instead of solving the equation, it minimizes $\norm{K x - y}^2 + \alpha \norm{x}^2$ for a small parameter $\alpha > 0$.
The minimizer always exists, is unique, and depends continuously on the data $y$.
It is the standard tool for [[inverse problems|PCMW7KN5P4]] such as first-kind integral equations, and in statistics it is known as [[ridge regression|3KZW2930V2]].

## Explanations

### L1 human

When small errors in the data cause huge errors in the solution, solving the equation exactly is useless.
Tikhonov's idea is to accept a small misfit in the equation in exchange for a solution that is not too large.
The parameter $\alpha$ sets the balance: a larger $\alpha$ gives a smaller, more stable solution that fits the data less well.
For a single equation $k x = y$ with a number $k \ne 0$, the regularized solution is $x = k y / (k^2 + \alpha)$.
It is close to $y / k$ when $\alpha$ is much smaller than $k^2$, and it stays moderate when $k$ is tiny.