A Gaussian factor using the canonical parameters (information form) More...

#include <HessianFactor.h>

Inheritance diagram for gtsam::HessianFactor:

Collaboration diagram for gtsam::HessianFactor:

Public Types
typedef GaussianFactor	Base
	Typedef to base class.

typedef HessianFactor	This
	Typedef to this class.

typedef std::shared_ptr< This >	shared_ptr
	A shared_ptr to this class.

typedef SymmetricBlockMatrix::Block	Block
	A block from the Hessian matrix.

typedef SymmetricBlockMatrix::constBlock	constBlock
	A block from the Hessian matrix (const version)

typedef KeyVector::iterator	iterator
	Iterator over keys.

typedef KeyVector::const_iterator	const_iterator
	Const iterator over keys.

Public Member Functions
	HessianFactor ()

	HessianFactor (Key j, const Matrix &G, const Vector &g, double f)

	HessianFactor (Key j, const Vector &mu, const Matrix &Sigma)

	HessianFactor (Key j1, Key j2, const Matrix &G11, const Matrix &G12, const Vector &g1, const Matrix &G22, const Vector &g2, double f)

	HessianFactor (Key j1, Key j2, Key j3, const Matrix &G11, const Matrix &G12, const Matrix &G13, const Vector &g1, const Matrix &G22, const Matrix &G23, const Vector &g2, const Matrix &G33, const Vector &g3, double f)

	HessianFactor (const KeyVector &js, const std::vector< Matrix > &Gs, const std::vector< Vector > &gs, double f)

template<typename KEYS >
	HessianFactor (const KEYS &keys, const SymmetricBlockMatrix &augmentedInformation)

	HessianFactor (const JacobianFactor &cg)

	HessianFactor (const GaussianFactor &factor)

	HessianFactor (const GaussianFactorGraph &factors, const Scatter &scatter)

	HessianFactor (const GaussianFactorGraph &factors)

	~HessianFactor () override

GaussianFactor::shared_ptr	clone () const override

void	print (const std::string &s="", const KeyFormatter &formatter=DefaultKeyFormatter) const override

bool	equals (const GaussianFactor &lf, double tol=1e-9) const override

double	error (const VectorValues &c) const override

DenseIndex	getDim (const_iterator variable) const override

size_t	rows () const

GaussianFactor::shared_ptr	negate () const override

double	constantTerm () const

double &	constantTerm ()

SymmetricBlockMatrix::constBlock	linearTerm (const_iterator j) const

SymmetricBlockMatrix::constBlock	linearTerm () const

SymmetricBlockMatrix::Block	linearTerm ()

const SymmetricBlockMatrix &	info () const
	Return underlying information matrix.

SymmetricBlockMatrix &	info ()

Matrix	augmentedInformation () const override

Eigen::SelfAdjointView< SymmetricBlockMatrix::constBlock, Eigen::Upper >	informationView () const
	Return self-adjoint view onto the information matrix (NOT augmented).

Matrix	information () const override

void	hessianDiagonalAdd (VectorValues &d) const override
	Add the current diagonal to a VectorValues instance.

void	hessianDiagonal (double *d) const override
	Raw memory access version of hessianDiagonal.

std::map< Key, Matrix >	hessianBlockDiagonal () const override
	Return the block diagonal of the Hessian for this factor.

std::pair< Matrix, Vector >	jacobian () const override
	Return (dense) matrix associated with factor.

Matrix	augmentedJacobian () const override

void	updateHessian (const KeyVector &keys, SymmetricBlockMatrix *info) const override

void	updateHessian (HessianFactor *other) const

void	multiplyHessianAdd (double alpha, const VectorValues &x, VectorValues &y) const override

VectorValues	gradientAtZero () const override
	eta for Hessian

void	gradientAtZero (double *d) const override
	Raw memory access version of gradientAtZero.

Vector	gradient (Key key, const VectorValues &x) const override

std::shared_ptr< GaussianConditional >	eliminateCholesky (const Ordering &keys)

VectorValues	solve ()
	Solve the system A'A delta = A'b in-place, return delta as VectorValues.

Testable
bool	equals (const This &other, double tol=1e-9) const
	check equality

virtual void	printKeys (const std::string &s="Factor", const KeyFormatter &formatter=DefaultKeyFormatter) const
	print only keys

Standard Interface
double	error (const HybridValues &c) const override

VectorValues	hessianDiagonal () const
	Return the diagonal of the Hessian for this factor.

Standard Interface
bool	empty () const
	Whether the factor is empty (involves zero variables).

Key	front () const
	First key.

Key	back () const
	Last key.

const_iterator	find (Key key) const
	find

const KeyVector &	keys () const
	Access the factor's involved variable keys.

const_iterator	begin () const

const_iterator	end () const

size_t	size () const

Advanced Interface
KeyVector &	keys ()

iterator	begin ()

iterator	end ()

Static Public Member Functions
Advanced Interface
template<typename CONTAINER >
static DenseIndex	Slot (const CONTAINER &keys, Key key)

Static Protected Member Functions
Standard Constructors
template<typename CONTAINER >
static Factor	FromKeys (const CONTAINER &keys)

template<typename ITERATOR >
static Factor	FromIterators (ITERATOR first, ITERATOR last)

Protected Attributes
SymmetricBlockMatrix	info_
	The full augmented information matrix, s.t. the quadratic error is 0.5*[x -1]'H[x -1].

KeyVector	keys_
	The keys involved in this factor.

Friends
class	NonlinearFactorGraph

class	NonlinearClusterTree

Detailed Description

A Gaussian factor using the canonical parameters (information form)

HessianFactor implements a general quadratic factor of the form

\[ E(x) = 0.5 x^T G x - x^T g + 0.5 f \]

that stores the matrix \( G \), the vector \( g \), and the constant term \( f \).

When \( G \) is positive semidefinite, this factor represents a Gaussian, in which case \( G \) is the information matrix \( \Lambda \), \( g \) is the information vector \( \eta \), and \( f \) is the residual sum-square-error at the mean, when \( x = \mu \).

Indeed, the negative log-likelihood of a Gaussian is (up to a constant) \( E(x) = 0.5(x-\mu)^T P^{-1} (x-\mu) \) with \( \mu \) the mean and \( P \) the covariance matrix. Expanding the product we get

\[ E(x) = 0.5 x^T P^{-1} x - x^T P^{-1} \mu + 0.5 \mu^T P^{-1} \mu \]

We define the Information matrix (or Hessian) \( \Lambda = P^{-1} \) and the information vector \( \eta = P^{-1} \mu = \Lambda \mu \) to arrive at the canonical form of the Gaussian:

\[ E(x) = 0.5 x^T \Lambda x - x^T \eta + 0.5 \mu^T \Lambda \mu \]

This factor is one of the factors that can be in a GaussianFactorGraph. It may be returned from NonlinearFactor::linearize(), but is also used internally to store the Hessian during Cholesky elimination.

This can represent a quadratic factor with characteristics that cannot be represented using a JacobianFactor (which has the form \( E(x) = \Vert Ax - b \Vert^2 \) and stores the Jacobian \( A \) and error vector \( b \), i.e. is a sum-of-squares factor). For example, a HessianFactor need not be positive semidefinite, it can be indefinite or even negative semidefinite.

If a HessianFactor is indefinite or negative semi-definite, then in order for solving the linear system to be possible, the Hessian of the full system must be positive definite (i.e. when all small Hessians are combined, the result must be positive definite). If this is not the case, an error will occur during elimination.

This class stores G, g, and f as an augmented matrix HessianFactor::matrix_. The upper-left n x n blocks of HessianFactor::matrix_ store the upper-right triangle of G, the upper-right-most column of length n of HessianFactor::matrix_ stores g, and the lower-right entry of HessianFactor::matrix_ stores f, i.e.

HessianFactor::matrix_ = [ G11 G12 G13 ... g1
                             0 G22 G23 ... g2
                             0   0 G33 ... g3
                             :   :   :      :
                             0   0   0 ...  f ]

Blocks can be accessed as follows:

G11 = info(begin(), begin());
G12 = info(begin(), begin()+1);
G23 = info(begin()+1, begin()+2);
g2 = linearTerm(begin()+1);
f = constantTerm();
.......

Constructor & Destructor Documentation

◆ HessianFactor() [1/11]

gtsam::HessianFactor::HessianFactor ( )

default constructor for I/O

◆ HessianFactor() [2/11]

gtsam::HessianFactor::HessianFactor	(	Key	j,
		const Matrix &	G,
		const Vector &	g,
		double	f
	)

Construct a unary factor. G is the quadratic term (Hessian matrix), g the linear term (a vector), and f the constant term. The quadratic error is: 0.5*(f - 2*x'*g + x'*G*x)

◆ HessianFactor() [3/11]

gtsam::HessianFactor::HessianFactor	(	Key	j,
		const Vector &	mu,
		const Matrix &	Sigma
	)

Construct a unary factor, given a mean and covariance matrix. error is 0.5*(x-mu)'inv(Sigma)(x-mu)

◆ HessianFactor() [4/11]

gtsam::HessianFactor::HessianFactor	(	Key	j1,
		Key	j2,
		const Matrix &	G11,
		const Matrix &	G12,
		const Vector &	g1,
		const Matrix &	G22,
		const Vector &	g2,
		double	f
	)

Construct a binary factor. Gxx are the upper-triangle blocks of the quadratic term (the Hessian matrix), gx the pieces of the linear vector term, and f the constant term. JacobianFactor error is

\[ 0.5* (Ax-b)' M (Ax-b) = 0.5*x'A'MAx - x'A'Mb + 0.5*b'Mb \]

HessianFactor error is

\[ 0.5*(x'Gx - 2x'g + f) = 0.5*x'Gx - x'*g + 0.5*f \]

So, with \( A = [A1 A2] \) and \( G=A*'M*A = [A1';A2']*M*[A1 A2] \) we have

n1*n1 G11 = A1'*M*A1
n1*n2 G12 = A1'*M*A2
n2*n2 G22 = A2'*M*A2
n1*1   g1 = A1'*M*b
n2*1   g2 = A2'*M*b
 1*1    f =  b'*M*b

◆ HessianFactor() [5/11]

gtsam::HessianFactor::HessianFactor	(	Key	j1,
		Key	j2,
		Key	j3,
		const Matrix &	G11,
		const Matrix &	G12,
		const Matrix &	G13,
		const Vector &	g1,
		const Matrix &	G22,
		const Matrix &	G23,
		const Vector &	g2,
		const Matrix &	G33,
		const Vector &	g3,
		double	f
	)

Construct a ternary factor. Gxx are the upper-triangle blocks of the quadratic term (the Hessian matrix), gx the pieces of the linear vector term, and f the constant term.

◆ HessianFactor() [6/11]

gtsam::HessianFactor::HessianFactor	(	const KeyVector &	js,
		const std::vector< Matrix > &	Gs,
		const std::vector< Vector > &	gs,
		double	f
	)

Construct an n-way factor. Gs contains the upper-triangle blocks of the quadratic term (the Hessian matrix) provided in row-order, gs the pieces of the linear vector term, and f the constant term.

◆ HessianFactor() [7/11]

template<typename KEYS >

gtsam::HessianFactor::HessianFactor	(	const KEYS &	keys,
		const SymmetricBlockMatrix &	augmentedInformation
	)

Constructor with an arbitrary number of keys and with the augmented information matrix specified as a block matrix.

◆ HessianFactor() [8/11]

gtsam::HessianFactor::HessianFactor ( const JacobianFactor & cg )

explicit

Construct from a JacobianFactor (or from a GaussianConditional since it derives from it)

◆ HessianFactor() [9/11]

gtsam::HessianFactor::HessianFactor ( const GaussianFactor & factor )

explicit

Attempt to construct from any GaussianFactor - currently supports JacobianFactor, HessianFactor, GaussianConditional, or any derived classes.

◆ HessianFactor() [10/11]

gtsam::HessianFactor::HessianFactor	(	const GaussianFactorGraph &	factors,
		const Scatter &	scatter
	)

explicit

Combine a set of factors into a single dense HessianFactor

◆ HessianFactor() [11/11]

gtsam::HessianFactor::HessianFactor ( const GaussianFactorGraph & factors )

inlineexplicit

Combine a set of factors into a single dense HessianFactor

◆ ~HessianFactor()

gtsam::HessianFactor::~HessianFactor ( )

inlineoverride

Destructor

Member Function Documentation

◆ augmentedInformation()

Matrix gtsam::HessianFactor::augmentedInformation ( ) const

overridevirtual

Return the augmented information matrix represented by this GaussianFactor. The augmented information matrix contains the information matrix with an additional column holding the information vector, and an additional row holding the transpose of the information vector. The lower-right entry contains the constant error term (when \( \delta x = 0 \)). The augmented information matrix is described in more detail in HessianFactor, which in fact stores an augmented information matrix.

For HessianFactor, this is the same as info() except that this function returns a complete symmetric matrix whereas info() returns a matrix where only the upper triangle is valid, but should be interpreted as symmetric. This is because info() returns only a reference to the internal representation of the augmented information matrix, which stores only the upper triangle.

Implements gtsam::GaussianFactor.

◆ augmentedJacobian()

Matrix gtsam::HessianFactor::augmentedJacobian ( ) const

overridevirtual

Return (dense) matrix associated with factor The returned system is an augmented matrix: [A b]

Parameters

set	weight to use whitening to bake in weights

Implements gtsam::GaussianFactor.

◆ begin() [1/2]

const_iterator gtsam::Factor::begin ( ) const

inlineinherited

Iterator at beginning of involved variable keys

◆ begin() [2/2]

iterator gtsam::Factor::begin ( )

inlineinherited

Iterator at beginning of involved variable keys

◆ clone()

GaussianFactor::shared_ptr gtsam::HessianFactor::clone ( ) const

inlineoverridevirtual

Clone this HessianFactor

Implements gtsam::GaussianFactor.

◆ constantTerm() [1/2]

double gtsam::HessianFactor::constantTerm ( ) const

inline

Return the constant term \( f \) as described above

Returns: The constant term \( f \)

◆ constantTerm() [2/2]

double& gtsam::HessianFactor::constantTerm ( )

inline

Return the constant term \( f \) as described above

Returns: The constant term \( f \)

◆ eliminateCholesky()

std::shared_ptr<GaussianConditional> gtsam::HessianFactor::eliminateCholesky ( const Ordering & keys )

In-place elimination that returns a conditional on (ordered) keys specified, and leaves this factor to be on the remaining keys (separator) only. Does dense partial Cholesky.

◆ end() [1/2]

const_iterator gtsam::Factor::end ( ) const

inlineinherited

Iterator at end of involved variable keys

◆ end() [2/2]

iterator gtsam::Factor::end ( )

inlineinherited

Iterator at end of involved variable keys

◆ equals()

bool gtsam::HessianFactor::equals	(	const GaussianFactor &	lf,
		double	tol = `1e-9`
	)		const

overridevirtual

Compare to another factor for testing (implementing Testable)

Implements gtsam::GaussianFactor.

◆ error() [1/2]

double gtsam::GaussianFactor::error ( const HybridValues & c ) const

overridevirtualinherited

All factor types need to implement an error function. In factor graphs, this is the negative log-likelihood.

Reimplemented from gtsam::Factor.

◆ error() [2/2]

double gtsam::HessianFactor::error ( const VectorValues & c ) const

overridevirtual

Evaluate the factor error f(x). returns 0.5*[x -1]'H[x -1] (also see constructor documentation)

Reimplemented from gtsam::GaussianFactor.

◆ FromIterators()

template<typename ITERATOR >

static Factor gtsam::Factor::FromIterators	(	ITERATOR	first,
		ITERATOR	last
	)

inlinestaticprotectedinherited

Construct factor from iterator keys. This is called internally from derived factor static factor methods, as a workaround for not being able to call the protected constructors above.

◆ FromKeys()

template<typename CONTAINER >

static Factor gtsam::Factor::FromKeys ( const CONTAINER & keys )

inlinestaticprotectedinherited

Construct factor from container of keys. This is called internally from derived factor static factor methods, as a workaround for not being able to call the protected constructors above.

◆ getDim()

DenseIndex gtsam::HessianFactor::getDim ( const_iterator variable ) const

inlineoverridevirtual

Return the dimension of the variable pointed to by the given key iterator todo: Remove this in favor of keeping track of dimensions with variables?

Parameters

variable An iterator pointing to the slot in this factor. You can use, for example, begin() + 2 to get the 3rd variable in this factor.

Implements gtsam::GaussianFactor.

◆ gradient()

Vector gtsam::HessianFactor::gradient	(	Key	key,
		const VectorValues &	x
	)		const

overridevirtual

Compute the gradient at a key: \( \grad f(x_i) = \sum_j G_ij*x_j - g_i \)

Implements gtsam::GaussianFactor.

◆ info()

SymmetricBlockMatrix& gtsam::HessianFactor::info ( )

inline

Return non-const information matrix. TODO(gareth): Review the sanity of having non-const access to this.

◆ information()

Matrix gtsam::HessianFactor::information ( ) const

overridevirtual

Return the non-augmented information matrix represented by this GaussianFactor.

Implements gtsam::GaussianFactor.

◆ keys()

KeyVector& gtsam::Factor::keys ( )

inlineinherited

Returns: keys involved in this factor

◆ linearTerm() [1/3]

SymmetricBlockMatrix::constBlock gtsam::HessianFactor::linearTerm ( const_iterator j ) const

inline

Return the part of linear term \( g \) as described above corresponding to the requested variable.

Parameters

j	Which block row to get, as an iterator pointing to the slot in this factor. You can use, for example, begin() + 2 to get the 3rd variable in this factor.

Returns: The linear term \( g \)

◆ linearTerm() [2/3]

SymmetricBlockMatrix::constBlock gtsam::HessianFactor::linearTerm ( ) const

inline

Return the complete linear term \( g \) as described above.

Returns: The linear term \( g \)

◆ linearTerm() [3/3]

SymmetricBlockMatrix::Block gtsam::HessianFactor::linearTerm ( )

inline

Return the complete linear term \( g \) as described above.

Returns: The linear term \( g \)

◆ multiplyHessianAdd()

void gtsam::HessianFactor::multiplyHessianAdd	(	double	alpha,
		const VectorValues &	x,
		VectorValues &	y
	)		const

overridevirtual

y += alpha * A'*A*x

Implements gtsam::GaussianFactor.

Reimplemented in gtsam::RegularHessianFactor< D >.

◆ negate()

GaussianFactor::shared_ptr gtsam::HessianFactor::negate ( ) const

overridevirtual

Construct the corresponding anti-factor to negate information stored stored in this factor.

Returns: a HessianFactor with negated Hessian matrices

Implements gtsam::GaussianFactor.

◆ print()

void gtsam::HessianFactor::print	(	const std::string &	s = `""`,
		const KeyFormatter &	formatter = `DefaultKeyFormatter`
	)		const

overridevirtual

Print the factor for debugging and testing (implementing Testable)

Implements gtsam::GaussianFactor.

◆ rows()

size_t gtsam::HessianFactor::rows ( ) const

inline

Return the number of columns and rows of the Hessian matrix, including the information vector.

◆ size()

size_t gtsam::Factor::size ( ) const

inlineinherited

Returns: the number of variables involved in this factor

◆ updateHessian() [1/2]

void gtsam::HessianFactor::updateHessian	(	const KeyVector &	keys,
		SymmetricBlockMatrix *	info
	)		const

overridevirtual

Update an information matrix by adding the information corresponding to this factor (used internally during elimination).

Parameters

keys	THe ordered vector of keys for the information matrix to be updated
info	The information matrix to be updated

Implements gtsam::GaussianFactor.

◆ updateHessian() [2/2]

void gtsam::HessianFactor::updateHessian ( HessianFactor * other ) const

inline

Update another Hessian factor

Parameters

other the HessianFactor to be updated

The documentation for this class was generated from the following files:

/home/docs/checkouts/readthedocs.org/user_builds/gtsam-jlblanco-docs/checkouts/latest/gtsam/linear/HessianFactor.h
/home/docs/checkouts/readthedocs.org/user_builds/gtsam-jlblanco-docs/checkouts/latest/gtsam/linear/HessianFactor-inl.h

Public Types

Public Member Functions

Static Public Member Functions

Static Protected Member Functions

Protected Attributes

Friends

Detailed Description

Constructor & Destructor Documentation

◆ HessianFactor() [1/11]

◆ HessianFactor() [2/11]

◆ HessianFactor() [3/11]

◆ HessianFactor() [4/11]

◆ HessianFactor() [5/11]

◆ HessianFactor() [6/11]

◆ HessianFactor() [7/11]

◆ HessianFactor() [8/11]

◆ HessianFactor() [9/11]

◆ HessianFactor() [10/11]

◆ HessianFactor() [11/11]

◆ ~HessianFactor()

Member Function Documentation

◆ augmentedInformation()

◆ augmentedJacobian()

◆ begin() [1/2]

◆ begin() [2/2]

◆ clone()

◆ constantTerm() [1/2]

◆ constantTerm() [2/2]

◆ eliminateCholesky()

◆ end() [1/2]

◆ end() [2/2]

◆ equals()

◆ error() [1/2]

◆ error() [2/2]

◆ FromIterators()

◆ FromKeys()

◆ getDim()

◆ gradient()

◆ info()

◆ information()

◆ keys()

◆ linearTerm() [1/3]

◆ linearTerm() [2/3]

◆ linearTerm() [3/3]

◆ multiplyHessianAdd()

◆ negate()

◆ print()

◆ rows()

◆ size()

◆ updateHessian() [1/2]

◆ updateHessian() [2/2]