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  "body": "\n## Summary\n\nTikhonov regularization is a method to solve an ill-posed linear equation $K x = y$ in a stable way.\nInstead of solving the equation, it minimizes $\\norm{K x - y}^2 + \\alpha \\norm{x}^2$ for a small parameter $\\alpha \u003e 0$.\nThe minimizer always exists, is unique, and depends continuously on the data $y$.\nIt is the standard tool for [[inverse problems|PCMW7KN5P4]] such as first-kind integral equations, and in statistics it is known as [[ridge regression|3KZW2930V2]].\n\n## Explanations\n\n### L1 human\n\nWhen small errors in the data cause huge errors in the solution, solving the equation exactly is useless.\nTikhonov's idea is to accept a small misfit in the equation in exchange for a solution that is not too large.\nThe parameter $\\alpha$ sets the balance: a larger $\\alpha$ gives a smaller, more stable solution that fits the data less well.\nFor a single equation $k x = y$ with a number $k \\ne 0$, the regularized solution is $x = k y / (k^2 + \\alpha)$.\nIt is close to $y / k$ when $\\alpha$ is much smaller than $k^2$, and it stays moderate when $k$ is tiny.\n",
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