Detailed Description
Truncated Least Squares (TLS) robust error model.
This model has a scalar parameter "c" (threshold).
- Loss \rho(x) = 0.5 x^2 if |x|<=c, 0.5 c^2 otherwise
- Derivative \phi(x) = x if |x|<=c, 0 otherwise
Weight w(x) = \phi(x)/x = 1 if |x|<=c, 0 otherwise
TLS has three graduated forms. All are stated in terms of the normalized \mu; the historical Yang/Peng parameter is \theta = \mu / (1 - \mu), which GncOptimizer schedules under the loss-independent name \lambda.
STANDARD TLS loss is graduated by relaxing the threshold with \lambda = 1 / \mu, i.e. by replacing c² with c²/\mu.
- Loss \rho(x,\mu) = 0.5 x^2 if x^2 <= c^2/\mu, 0.5 c^2/\mu otherwise
Weight w(x,\mu) = 1 if x^2 <= c^2/\mu, 0 otherwise
GNC_LINEAR is the normalized Yang GNC-TLS surrogate LB = \mu c^2 UB = c^2 / \mu
- Loss \rho(x,\mu) = 0.5 x^2 if x^2 < LB = 0.5 (2\sqrt(\mu c^2 x^2) - \mu(c^2 + x^2))/(1-\mu) if LB < x^2 < UB = 0.5 c^2 if UB < x^2
- Weight w(x,\mu) = 1 if x^2 < LB = (\sqrt(\mu c^2 / x^2) - \mu)/(1-\mu) if LB < x^2 < UB = 0 if UB < x^2 Yang's \theta -> 0 surrogate is degenerate (every weight vanishes), so the \mu = 0 endpoint is defined to be the all-inlier least-squares initialization step: w = 1 and \rho = 0.5 x^2. \mu = 1 is exact TLS.
GNC_SUPERLINEAR is the normalized Peng MS-GNC-TLS majorizer LB = c^2 UB = c^2 / \mu^2
- Loss \rho(x, \mu) = (not formally defined)
- Weight w(x, \mu) = 1 if x^2 < LB = (\sqrt(c^2 / x^2) - \mu)/(1-\mu) if LB < x^2 < UB = 0 if UB < x^2 Its \mu = 0 endpoint is the non-degenerate convex weight min(1, c/|x|), so unlike GNC_LINEAR it needs no separate initialization convention. \mu = 1 is exact TLS.
Public Member Functions | |
| TruncatedLeastSquares (double c=1.0, const ReweightScheme reweight=Block) | |
| Construct standard TLS loss. | |
| TruncatedLeastSquares (double c, GradScheme graduation, const ReweightScheme reweight=Block) | |
| Construct with an explicit graduation scheme. | |
| double | weight (double distance) const override |
| This method is responsible for returning the weight function for a given amount of error. | |
| double | loss (double distance) const override |
| This method is responsible for returning the total penalty for a given amount of error. | |
| double | graduatedWeight (double distance, double mu) const override |
| This method is responsible for returning the weight for a given amount of error and the current control parameter \(\mu\). | |
| double | graduatedLoss (double distance, double mu) const override |
| This method is responsible for returning the total penalty for a given amount of error and the current control parameter \(\mu\). | |
| void | print (const std::string &s) const override |
| bool | equals (const Base &expected, double tol=1e-8) const override |
| double | modelParameter () const |
| GradScheme | gradScheme () const |
| Returns the graduation scheme used by this loss. | |
| Public Member Functions inherited from gtsam::noiseModel::mEstimator::Base | |
| Base (const ReweightScheme reweight=Block) | |
| ReweightScheme | reweightScheme () const |
| Returns the reweight scheme, as explained in ReweightScheme. | |
| double | sqrtWeight (double distance) const |
| Vector | weight (const Vector &error) const |
| produce a weight vector according to an error vector and the implemented robust function | |
| Vector | sqrtWeight (const Vector &error) const |
| square root version of the weight function | |
| void | reweight (Vector &error) const |
| reweight block matrices and a vector according to their weight implementation | |
| void | reweight (std::vector< Matrix > &A, Vector &error) const |
| void | reweight (Matrix &A, Vector &error) const |
| void | reweight (Matrix &A1, Matrix &A2, Vector &error) const |
| void | reweight (Matrix &A1, Matrix &A2, Matrix &A3, Vector &error) const |
Static Public Member Functions | |
| static shared_ptr | Create (double c, const ReweightScheme reweight=Block) |
| static shared_ptr | Create (double c, GradScheme graduation, const ReweightScheme reweight=Block) |
| Factory for a loss with an explicit graduation scheme. | |
| static double | Weight (double distance2, double c2) |
| Static implementation of TLS Weight. | |
| static double | Loss (double distance2, double c2) |
| Static implementation of TLS Loss. | |
| static double | GraduatedWeight (double distance2, double c2, double mu, GradScheme graduation) |
| Static implementation of TLS Graduated Weight. | |
| static double | GraduatedLoss (double distance2, double c2, double mu, GradScheme graduation) |
| Static implementation of TLS Graduated Loss. | |
Public Types | |
| enum | GradScheme { STANDARD , GNC_LINEAR , GNC_SUPERLINEAR } |
| typedef std::shared_ptr< TruncatedLeastSquares > | shared_ptr |
| Public Types inherited from gtsam::noiseModel::mEstimator::Base | |
| enum | ReweightScheme { Scalar , Block } |
| the rows can be weighted independently according to the error or uniformly with the norm of the right hand side | |
| typedef std::shared_ptr< Base > | shared_ptr |
Protected Attributes | |
| double | c_ |
| double | csquared_ |
| GradScheme | graduation_ |
| Protected Attributes inherited from gtsam::noiseModel::mEstimator::Base | |
| ReweightScheme | reweight_ |
| Strategy for reweighting. | |
Member Function Documentation
◆ equals()
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overridevirtual |
Implements gtsam::noiseModel::mEstimator::Base.
◆ graduatedLoss()
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overridevirtual |
This method is responsible for returning the total penalty for a given amount of error and the current control parameter \(\mu\).
This returns \(\rho(x, \mu)\) in mEstimator
Implements gtsam::noiseModel::mEstimator::Base.
◆ graduatedWeight()
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overridevirtual |
This method is responsible for returning the weight for a given amount of error and the current control parameter \(\mu\).
This returns \(w(x, \mu)\) in mEstimator
Implements gtsam::noiseModel::mEstimator::Base.
◆ loss()
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overridevirtual |
This method is responsible for returning the total penalty for a given amount of error.
For example, this method is responsible for implementing the quadratic function for an L2 penalty, the absolute value function for an L1 penalty, etc.
TODO(mikebosse): When the loss function has as input the norm of the error vector, then it prevents implementations of asymmeric loss functions. It would be better for this function to accept the vector and internally call the norm if necessary.
This returns \(\rho(x)\) in mEstimator
Reimplemented from gtsam::noiseModel::mEstimator::Base.
◆ print()
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overridevirtual |
Implements gtsam::noiseModel::mEstimator::Base.
◆ weight()
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overridevirtual |
This method is responsible for returning the weight function for a given amount of error.
The weight function is related to the analytic derivative of the loss function. See https://members.loria.fr/MOBerger/Enseignement/Master2/Documents/ZhangIVC-97-01.pdf for details. This method is required when optimizing cost functions with robust penalties using iteratively re-weighted least squares.
This returns w(x) in mEstimator
Implements gtsam::noiseModel::mEstimator::Base.
The documentation for this class was generated from the following files:
- /tmp/gtsam-4.3.0-doxygen.rsXPUS/source/gtsam/linear/LossFunctions.h
- /tmp/gtsam-4.3.0-doxygen.rsXPUS/source/gtsam/linear/LossFunctions.cpp
