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  "body": "\n## Definition\n\nLet $A \\in \\R^{m \\times n}$, $b \\in \\R^m$ and $L \\in \\R^{p \\times n}$ with $\\ker A \\cap \\ker L = \\set{0}$.\nFor $\\lambda \u003e 0$ let $x_\\lambda$ be the unique minimizer of the [[Tikhonov|M68V0M5A3R]] functional\n$$\\norm{A x - b}_2^2 + \\lambda^2 \\norm{L x}_2^2.$$\nThe *L-curve* is the plane curve\n$$\\lambda \\mapsto \\bigl( \\log \\norm{A x_\\lambda - b}_2, \\ \\log \\norm{L x_\\lambda}_2 \\bigr), \\qquad \\lambda \u003e 0,$$\ndefined where both norms are positive.\nThe *L-curve criterion* chooses $\\lambda$ at the point of maximal curvature of this curve.\n\n## Remarks\n\nThe residual norm $\\norm{A x_\\lambda - b}$ is non-decreasing in $\\lambda$, and the seminorm $\\norm{L x_\\lambda}$ is non-increasing.\nSo the curve runs from upper left (small $\\lambda$, noise-dominated) to lower right (large $\\lambda$, over-smoothed).\nExample: for $A = L = I$ and $b \\ne 0$, one gets $x_\\lambda = b/(1 + \\lambda^2)$.\nThen $\\norm{A x_\\lambda - b} = \\lambda^2 \\norm{b}/(1 + \\lambda^2)$ and $\\norm{x_\\lambda} = \\norm{b}/(1 + \\lambda^2)$, so the first increases and the second decreases with $\\lambda$, as stated.\nNon-example: the curve $\\lambda \\mapsto (\\lambda, \\norm{A x_\\lambda - b})$ is not the L-curve, since it plots the parameter itself instead of the solution seminorm.\nSome authors use the squared norms or linear axes, but the corner is clearest on log-log axes.\nThe criterion is a heuristic: it is not a convergent [[regularization|WTY54BFC46]] parameter choice in the worst-case sense, yet it works well on many practical problems.\n\n## Explanations\n\n### L1 human\n\nThe L-curve shows the trade-off between fitting the data and keeping the solution tame, one point for each value of the regularization parameter.\nIts corner marks a good compromise.\n",
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