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();
.......

Inheritance diagram for gtsam::HessianFactor:

Public Member Functions
	HessianFactor ()
	default constructor for I/O

	HessianFactor (Key j, const Matrix &G, const Vector &g, double f)
	Construct a unary factor. More...

	HessianFactor (Key j, const Vector &mu, const Matrix &Sigma)
	Construct a unary factor, given a mean and covariance matrix. More...

	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. More...

	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. More...

	HessianFactor (const KeyVector &js, const std::vector< Matrix > &Gs, const std::vector< Vector > &gs, double f)
	Construct an n-way factor. More...

template<typename KEYS >
	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 (const JacobianFactor &cg)
	Construct from a JacobianFactor (or from a GaussianConditional since it derives from it)

	HessianFactor (const GaussianFactor &factor)
	Attempt to construct from any GaussianFactor - currently supports JacobianFactor, HessianFactor, GaussianConditional, or any derived classes.

	HessianFactor (const GaussianFactorGraph &factors, const Scatter &scatter)
	Combine a set of factors into a single dense HessianFactor.

	HessianFactor (const GaussianFactorGraph &factors)
	Combine a set of factors into a single dense HessianFactor.

	~HessianFactor () override
	Destructor.

GaussianFactor::shared_ptr	clone () const override
	Clone this HessianFactor. More...

void	print (const std::string &s="", const KeyFormatter &formatter=DefaultKeyFormatter) const override
	Print the factor for debugging and testing (implementing Testable) More...

bool	equals (const GaussianFactor &lf, double tol=1e-9) const override
	Compare to another factor for testing (implementing Testable) More...

double	error (const VectorValues &c) const override
	Evaluate the factor error f(x). More...

DenseIndex	getDim (const_iterator variable) const override
	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? More...

size_t	rows () const
	Return the number of columns and rows of the Hessian matrix, including the information vector.

GaussianFactor::shared_ptr	negate () const override
	Construct the corresponding anti-factor to negate information stored stored in this factor. More...

bool	empty () const override
	Check if the factor is empty. More...

double	constantTerm () const
	Return the constant term \( f \) as described above. More...

double &	constantTerm ()
	Return the constant term \( f \) as described above. More...

SymmetricBlockMatrix::constBlock	linearTerm (const_iterator j) const
	Return the part of linear term \( g \) as described above corresponding to the requested variable. More...

SymmetricBlockMatrix::constBlock	linearTerm () const
	Return the complete linear term \( g \) as described above. More...

SymmetricBlockMatrix::Block	linearTerm ()
	Return the complete linear term \( g \) as described above. More...

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

SymmetricBlockMatrix &	info ()
	Return non-const information matrix. More...

Matrix	augmentedInformation () const override
	Return the augmented information matrix represented by this GaussianFactor. More...

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

Matrix	information () const override
	Return the non-augmented information matrix represented by this GaussianFactor. More...

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

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

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

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

Matrix	augmentedJacobian () const override
	Return (dense) matrix associated with factor The returned system is an augmented matrix: [A b]. More...

void	updateHessian (const KeyVector &keys, SymmetricBlockMatrix *info) const override
	Update an information matrix by adding the information corresponding to this factor (used internally during elimination). More...

void	updateHessian (HessianFactor *other) const
	Update another Hessian factor. More...

void	multiplyHessianAdd (double alpha, const VectorValues &x, VectorValues &y) const override
	y += alpha * A'Ax More...

VectorValues	gradientAtZero () const override
	eta for Hessian More...

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

Vector	gradient (Key key, const VectorValues &x) const override
	Compute the gradient at a key: \grad f(x_i) = \sum_j G_ij*x_j - g_i. More...

boost::shared_ptr< GaussianConditional >	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. More...

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

Public Member Functions inherited from gtsam::GaussianFactor
	GaussianFactor ()
	Default constructor creates empty factor.

template<typename CONTAINER >
	GaussianFactor (const CONTAINER &keys)
	Construct from container of keys. More...

virtual	~GaussianFactor ()
	Destructor.

void	print (const std::string &s="", const KeyFormatter &formatter=DefaultKeyFormatter) const override=0
	print More...

virtual bool	equals (const GaussianFactor &lf, double tol=1e-9) const =0
	Equals for testable. More...

virtual double	error (const VectorValues &c) const =0
	Print for testable. More...

virtual DenseIndex	getDim (const_iterator variable) const =0
	0.5(Ax-b)'D(A*x-b) More...

virtual Matrix	augmentedJacobian () const =0
	Return a dense \( [ \;A\;b\; ] \in \mathbb{R}^{m \times n+1} \) Jacobian matrix, augmented with b with the noise models baked into A and b. More...

virtual std::pair< Matrix, Vector >	jacobian () const =0
	Return the dense Jacobian \( A \) and right-hand-side \( b \), with the noise models baked into A and b. More...

virtual Matrix	augmentedInformation () const =0
	Return the augmented information matrix represented by this GaussianFactor. More...

virtual Matrix	information () const =0
	Return the non-augmented information matrix represented by this GaussianFactor. More...

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

virtual void	hessianDiagonalAdd (VectorValues &d) const =0
	Add the current diagonal to a VectorValues instance. More...

virtual void	hessianDiagonal (double *d) const =0
	Raw memory access version of hessianDiagonal. More...

virtual std::map< Key, Matrix >	hessianBlockDiagonal () const =0
	Return the block diagonal of the Hessian for this factor. More...

virtual GaussianFactor::shared_ptr	clone () const =0
	Clone a factor (make a deep copy) More...

virtual bool	empty () const =0
	Test whether the factor is empty. More...

virtual GaussianFactor::shared_ptr	negate () const =0
	Construct the corresponding anti-factor to negate information stored stored in this factor. More...

virtual void	updateHessian (const KeyVector &keys, SymmetricBlockMatrix *info) const =0
	Update an information matrix by adding the information corresponding to this factor (used internally during elimination). More...

virtual void	multiplyHessianAdd (double alpha, const VectorValues &x, VectorValues &y) const =0
	y += alpha * A'Ax More...

virtual VectorValues	gradientAtZero () const =0
	A'*b for Jacobian, eta for Hessian. More...

virtual void	gradientAtZero (double *d) const =0
	Raw memory access version of gradientAtZero. More...

virtual Vector	gradient (Key key, const VectorValues &x) const =0
	Gradient wrt a key at any values. More...

Public Member Functions inherited from gtsam::Factor
virtual	~Factor ()=default
	Default destructor.

KeyVector &	keys ()

iterator	begin ()
	Iterator at beginning of involved variable keys.

iterator	end ()
	Iterator at end of involved variable keys.

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

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
	Iterator at beginning of involved variable keys.

const_iterator	end () const
	Iterator at end of involved variable keys.

size_t	size () const

Public Types
typedef GaussianFactor	Base
	Typedef to base class.

typedef HessianFactor	This
	Typedef to this class.

typedef boost::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)

Public Types inherited from gtsam::GaussianFactor
typedef GaussianFactor	This
	This class.

typedef boost::shared_ptr< This >	shared_ptr
	shared_ptr to this class

typedef Factor	Base
	Our base class.

Public Types inherited from gtsam::Factor
typedef KeyVector::iterator	iterator
	Iterator over keys.

typedef KeyVector::const_iterator	const_iterator
	Const iterator over keys.

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

Protected Attributes inherited from gtsam::Factor
KeyVector	keys_
	The keys involved in this factor.

Friends
class	NonlinearFactorGraph

class	NonlinearClusterTree

class	boost::serialization::access
	Serialization function.

Additional Inherited Members
Static Public Member Functions inherited from gtsam::GaussianFactor
template<typename CONTAINER >
static DenseIndex	Slot (const CONTAINER &keys, Key key)

Protected Member Functions inherited from gtsam::Factor
	Factor ()
	Default constructor for I/O.

template<typename CONTAINER >
	Factor (const CONTAINER &keys)
	Construct factor from container of keys. More...

template<typename ITERATOR >
	Factor (ITERATOR first, ITERATOR last)
	Construct factor from iterator keys. More...

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

Static Protected Member Functions inherited from gtsam::Factor
template<typename CONTAINER >
static Factor	FromKeys (const CONTAINER &keys)
	Construct factor from container of keys. More...

template<typename ITERATOR >
static Factor	FromIterators (ITERATOR first, ITERATOR last)
	Construct factor from iterator keys. More...

Constructor & Destructor Documentation

◆ HessianFactor() [1/5]

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() [2/5]

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() [3/5]

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() [4/5]

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() [5/5]

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.

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.

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

Detailed Description

Public Member Functions

Public Types

Protected Attributes

Friends

Additional Inherited Members

Constructor & Destructor Documentation

◆ HessianFactor() [1/5]

◆ HessianFactor() [2/5]

◆ HessianFactor() [3/5]

◆ HessianFactor() [4/5]

◆ HessianFactor() [5/5]

Member Function Documentation

◆ augmentedInformation()

◆ augmentedJacobian()

◆ clone()

◆ constantTerm() [1/2]

◆ constantTerm() [2/2]

◆ eliminateCholesky()

◆ empty()

◆ equals()

◆ error()

◆ getDim()

◆ gradient()

◆ gradientAtZero() [1/2]

◆ gradientAtZero() [2/2]

◆ hessianBlockDiagonal()

◆ hessianDiagonal()

◆ hessianDiagonalAdd()

◆ info()

◆ information()

◆ jacobian()

◆ linearTerm() [1/3]

◆ linearTerm() [2/3]

◆ linearTerm() [3/3]

◆ multiplyHessianAdd()

◆ negate()

◆ print()

◆ updateHessian() [1/2]

◆ updateHessian() [2/2]