Module: tfg.math.optimizer.levenberg_marquardt
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This module implements a Levenberg-Marquardt optimizer.
Minimizes \(\min_{\mathbf{x} } \sum_i \|\mathbf{r}_i(\mathbf{x})\|^2_2\) where
\(\mathbf{r}_i(\mathbf{x})\)
are the residuals. This function implements Levenberg-Marquardt, an iterative
process that linearizes the residuals and iteratively finds a displacement
\(\Delta \mathbf{x}\) such that at iteration \(t\) an update
\(\mathbf{x}_{t+1} = \mathbf{x}_{t} + \Delta \mathbf{x}\) improving the
loss can be computed. The displacement is computed by solving an optimization
problem
\(\min_{\Delta \mathbf{x} } \sum_i
\|\mathbf{J}_i(\mathbf{x}_{t})\Delta\mathbf{x} +
\mathbf{r}_i(\mathbf{x}_t)\|^2_2 + \lambda\|\Delta \mathbf{x} \|_2^2\) where
\(\mathbf{J}_i(\mathbf{x}_{t})\) is the Jacobian of \(\mathbf{r}_i\)
computed at \(\mathbf{x}_t\), and \(\lambda\) is a scalar weight.
More details on Levenberg-Marquardt can be found on this page.
Functions
minimize(...): Minimizes a set of residuals in the least-squares sense.
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Last updated 2022-08-26 UTC.
[null,null,["Last updated 2022-08-26 UTC."],[],[],null,["# Module: tfg.math.optimizer.levenberg_marquardt\n\n|---------------------------------------------------------------------------------------------------------------------------------------|\n| [View source on GitHub](https://github.com/tensorflow/graphics/blob/master/tensorflow_graphics/math/optimizer/levenberg_marquardt.py) |\n\nThis module implements a Levenberg-Marquardt optimizer.\n\nMinimizes \\\\(\\\\min_{\\\\mathbf{x} } \\\\sum_i \\\\\\|\\\\mathbf{r}_i(\\\\mathbf{x})\\\\\\|\\^2_2\\\\) where\n\\\\(\\\\mathbf{r}_i(\\\\mathbf{x})\\\\)\nare the residuals. This function implements Levenberg-Marquardt, an iterative\nprocess that linearizes the residuals and iteratively finds a displacement\n\\\\(\\\\Delta \\\\mathbf{x}\\\\) such that at iteration \\\\(t\\\\) an update\n\\\\(\\\\mathbf{x}_{t+1} = \\\\mathbf{x}_{t} + \\\\Delta \\\\mathbf{x}\\\\) improving the\nloss can be computed. The displacement is computed by solving an optimization\nproblem\n\\\\(\\\\min_{\\\\Delta \\\\mathbf{x} } \\\\sum_i\n\\\\\\|\\\\mathbf{J}_i(\\\\mathbf{x}_{t})\\\\Delta\\\\mathbf{x} +\n\\\\mathbf{r}_i(\\\\mathbf{x}_t)\\\\\\|\\^2_2 + \\\\lambda\\\\\\|\\\\Delta \\\\mathbf{x} \\\\\\|_2\\^2\\\\) where\n\\\\(\\\\mathbf{J}_i(\\\\mathbf{x}_{t})\\\\) is the Jacobian of \\\\(\\\\mathbf{r}_i\\\\)\ncomputed at \\\\(\\\\mathbf{x}_t\\\\), and \\\\(\\\\lambda\\\\) is a scalar weight.\n\nMore details on Levenberg-Marquardt can be found on [this page.](https://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm)\n\nFunctions\n---------\n\n[`minimize(...)`](../../../tfg/math/optimizer/levenberg_marquardt/minimize): Minimizes a set of residuals in the least-squares sense."]]
View source on GitHub