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/* ----------------------------------------------------------------------------
* GTSAM Copyright 2010, Georgia Tech Research Corporation,
* Atlanta, Georgia 30332-0415
* All Rights Reserved
* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
* See LICENSE for the license information
* -------------------------------------------------------------------------- */
/**
* @file Expression-inl.h
* @date September 18, 2014
* @author Frank Dellaert
* @author Paul Furgale
* @brief Internals for Expression.h, not for general consumption
*/
#pragma once
#include <gtsam/nonlinear/Values.h>
#include <gtsam/base/Matrix.h>
#include <boost/foreach.hpp>
#include <boost/tuple/tuple.hpp>
// template meta-programming headers
#include <boost/mpl/vector.hpp>
#include <boost/mpl/plus.hpp>
#include <boost/mpl/front.hpp>
#include <boost/mpl/pop_front.hpp>
#include <boost/mpl/fold.hpp>
#include<boost/mpl/empty_base.hpp>
#include<boost/mpl/placeholders.hpp>
namespace MPL = boost::mpl::placeholders;
namespace gtsam {
template<typename T>
class Expression;
typedef std::map<Key, Matrix> JacobianMap;
//-----------------------------------------------------------------------------
/// The JacobinaTrace class records a tree-structured expression's execution
template<class T>
struct JacobianTrace {
// Some fixed matrix sizes we need
typedef Eigen::Matrix<double, 2, T::dimension> Jacobian2T;
/**
* The Pointer class is a tagged union that obviates the need to create
* a JacobianTrace subclass for Constants and Leaf Expressions. Instead
* the key for the leaf is stored in the space normally used to store a
* JacobianTrace*. Nothing is stored for a Constant.
*/
///
class Pointer {
enum {
Constant, Leaf, Function
} type;
union {
Key key;
JacobianTrace* ptr;
} content;
public:
/// Pointer always starts out as a Constant
Pointer() :
type(Constant) {
}
/// Destructor cleans up pointer if Function
~Pointer() {
if (type == Function)
delete content.ptr;
}
/// Change pointer to a Leaf Trace
void setLeaf(Key key) {
type = Leaf;
content.key = key;
}
/// Take ownership of pointer to a Function Trace
void setFunction(JacobianTrace* trace) {
type = Function;
content.ptr = trace;
}
// *** This is the main entry point for reverseAD, called from Expression::augmented ***
// Called only once, either inserts identity into Jacobians (Leaf) or starts AD (Function)
void startReverseAD(JacobianMap& jacobians) const {
if (type == Leaf) {
// This branch will only be called on trivial Leaf expressions, i.e. Priors
size_t n = T::Dim();
jacobians[content.key] = Eigen::MatrixXd::Identity(n, n);
} else if (type == Function)
// This is the more typical entry point, starting the AD pipeline
// It is inside the startReverseAD that the correctly dimensioned pipeline is chosen.
content.ptr->startReverseAD(jacobians);
}
// Either add to Jacobians (Leaf) or propagate (Function)
void reverseAD(const Matrix& dTdA, JacobianMap& jacobians) const {
if (type == Leaf) {
JacobianMap::iterator it = jacobians.find(content.key);
if (it != jacobians.end())
it->second += dTdA;
else
jacobians[content.key] = dTdA;
} else if (type == Function)
content.ptr->reverseAD(dTdA, jacobians);
}
// Either add to Jacobians (Leaf) or propagate (Function)
void reverseAD2(const Jacobian2T& dTdA, JacobianMap& jacobians) const {
if (type == Leaf) {
JacobianMap::iterator it = jacobians.find(content.key);
if (it != jacobians.end())
it->second += dTdA;
else
jacobians[content.key] = dTdA;
} else if (type == Function)
content.ptr->reverseAD2(dTdA, jacobians);
}
};
/// Make sure destructor is virtual
virtual ~JacobianTrace() {
}
virtual void startReverseAD(JacobianMap& jacobians) const {
}
virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
}
virtual void reverseAD2(const Jacobian2T& dFdT,
JacobianMap& jacobians) const {
}
};
/// Primary template calls the generic Matrix reverseAD pipeline
template<size_t M, class A>
struct Select {
typedef Eigen::Matrix<double, M, A::dimension> Jacobian;
static void reverseAD(const typename JacobianTrace<A>::Pointer& trace,
const Jacobian& dTdA, JacobianMap& jacobians) {
trace.reverseAD(dTdA, jacobians);
}
};
/// Partially specialized template calls the 2-dimensional output version
template<class A>
struct Select<2, A> {
typedef Eigen::Matrix<double, 2, A::dimension> Jacobian;
static void reverseAD(const typename JacobianTrace<A>::Pointer& trace,
const Jacobian& dTdA, JacobianMap& jacobians) {
trace.reverseAD2(dTdA, jacobians);
}
};
/**
* Recursive Trace Class for Functional Expressions
* Abrahams, David; Gurtovoy, Aleksey (2004-12-10).
* C++ Template Metaprogramming: Concepts, Tools, and Techniques from Boost
* and Beyond (Kindle Location 1244). Pearson Education.
*/
template<class T, class A, class More>
struct Trace: More {
// define dimensions
static const size_t m = T::dimension;
static const size_t n = A::dimension;
// define fixed size Jacobian matrix types
typedef Eigen::Matrix<double, m, n> JacobianTA;
typedef Eigen::Matrix<double, 2, m> Jacobian2T;
// declare trace that produces value A, and corresponding Jacobian
typename JacobianTrace<A>::Pointer trace;
JacobianTA dTdA;
/// Start the reverse AD process
virtual void startReverseAD(JacobianMap& jacobians) const {
More::startReverseAD(jacobians);
Select<T::dimension, A>::reverseAD(trace, dTdA, jacobians);
}
/// Given df/dT, multiply in dT/dA and continue reverse AD process
virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
More::reverseAD(dFdT, jacobians);
trace.reverseAD(dFdT * dTdA, jacobians);
}
/// Version specialized to 2-dimensional output
virtual void reverseAD2(const Jacobian2T& dFdT,
JacobianMap& jacobians) const {
More::reverseAD2(dFdT, jacobians);
trace.reverseAD2(dFdT * dTdA, jacobians);
}
};
/// Recursive Trace Class Generator
template<class T, class TYPES>
struct GenerateTrace {
typedef typename boost::mpl::fold<TYPES, JacobianTrace<T>,
Trace<T, MPL::_2, MPL::_1> >::type type;
};
//-----------------------------------------------------------------------------
/**
* Value and Jacobians
*/
template<class T>
class Augmented {
private:
T value_;
JacobianMap jacobians_;
typedef std::pair<Key, Matrix> Pair;
/// Insert terms into jacobians_, adding if already exists
void add(const JacobianMap& terms) {
BOOST_FOREACH(const Pair& term, terms) {
JacobianMap::iterator it = jacobians_.find(term.first);
if (it != jacobians_.end())
it->second += term.second;
else
jacobians_[term.first] = term.second;
}
}
/// Insert terms into jacobians_, premultiplying by H, adding if already exists
void add(const Matrix& H, const JacobianMap& terms) {
BOOST_FOREACH(const Pair& term, terms) {
JacobianMap::iterator it = jacobians_.find(term.first);
if (it != jacobians_.end())
it->second += H * term.second;
else
jacobians_[term.first] = H * term.second;
}
}
public:
/// Construct value that does not depend on anything
Augmented(const T& t) :
value_(t) {
}
/// Construct value dependent on a single key
Augmented(const T& t, Key key) :
value_(t) {
size_t n = t.dim();
jacobians_[key] = Eigen::MatrixXd::Identity(n, n);
}
/// Construct value, pre-multiply jacobians by dTdA
Augmented(const T& t, const Matrix& dTdA, const JacobianMap& jacobians) :
value_(t) {
add(dTdA, jacobians);
}
/// Construct value, pre-multiply jacobians
Augmented(const T& t, const Matrix& dTdA1, const JacobianMap& jacobians1,
const Matrix& dTdA2, const JacobianMap& jacobians2) :
value_(t) {
add(dTdA1, jacobians1);
add(dTdA2, jacobians2);
}
/// Construct value, pre-multiply jacobians
Augmented(const T& t, const Matrix& dTdA1, const JacobianMap& jacobians1,
const Matrix& dTdA2, const JacobianMap& jacobians2, const Matrix& dTdA3,
const JacobianMap& jacobians3) :
value_(t) {
add(dTdA1, jacobians1);
add(dTdA2, jacobians2);
add(dTdA3, jacobians3);
}
/// Return value
const T& value() const {
return value_;
}
/// Return jacobians
const JacobianMap& jacobians() const {
return jacobians_;
}
/// Return jacobians
JacobianMap& jacobians() {
return jacobians_;
}
/// Not dependent on any key
bool constant() const {
return jacobians_.empty();
}
/// debugging
void print(const KeyFormatter& keyFormatter = DefaultKeyFormatter) {
BOOST_FOREACH(const Pair& term, jacobians_)
std::cout << "(" << keyFormatter(term.first) << ", " << term.second.rows()
<< "x" << term.second.cols() << ") ";
std::cout << std::endl;
}
/// Move terms to array, destroys content
void move(std::vector<Matrix>& H) {
assert(H.size()==jacobains.size());
size_t j = 0;
JacobianMap::iterator it = jacobians_.begin();
for (; it != jacobians_.end(); ++it)
it->second.swap(H[j++]);
}
};
//-----------------------------------------------------------------------------
/**
* Expression node. The superclass for objects that do the heavy lifting
* An Expression<T> has a pointer to an ExpressionNode<T> underneath
* allowing Expressions to have polymorphic behaviour even though they
* are passed by value. This is the same way boost::function works.
* http://loki-lib.sourceforge.net/html/a00652.html
*/
template<class T>
class ExpressionNode {
protected:
ExpressionNode() {
}
public:
/// Destructor
virtual ~ExpressionNode() {
}
/// Return keys that play in this expression as a set
virtual std::set<Key> keys() const = 0;
/// Return value
virtual T value(const Values& values) const = 0;
/// Return value and derivatives
virtual Augmented<T> forward(const Values& values) const = 0;
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values,
typename JacobianTrace<T>::Pointer& p) const = 0;
};
//-----------------------------------------------------------------------------
/// Constant Expression
template<class T>
class ConstantExpression: public ExpressionNode<T> {
/// The constant value
T constant_;
/// Constructor with a value, yielding a constant
ConstantExpression(const T& value) :
constant_(value) {
}
friend class Expression<T> ;
public:
/// Return keys that play in this expression, i.e., the empty set
virtual std::set<Key> keys() const {
std::set<Key> keys;
return keys;
}
/// Return value
virtual T value(const Values& values) const {
return constant_;
}
/// Return value and derivatives
virtual Augmented<T> forward(const Values& values) const {
return Augmented<T>(constant_);
}
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values,
typename JacobianTrace<T>::Pointer& p) const {
return constant_;
}
};
//-----------------------------------------------------------------------------
/// Leaf Expression
template<class T>
class LeafExpression: public ExpressionNode<T> {
/// The key into values
Key key_;
/// Constructor with a single key
LeafExpression(Key key) :
key_(key) {
}
friend class Expression<T> ;
public:
/// Return keys that play in this expression
virtual std::set<Key> keys() const {
std::set<Key> keys;
keys.insert(key_);
return keys;
}
/// Return value
virtual T value(const Values& values) const {
return values.at<T>(key_);
}
/// Return value and derivatives
virtual Augmented<T> forward(const Values& values) const {
return Augmented<T>(values.at<T>(key_), key_);
}
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values,
typename JacobianTrace<T>::Pointer& p) const {
p.setLeaf(key_);
return values.at<T>(key_);
}
};
//-----------------------------------------------------------------------------
/// Unary Function Expression
template<class T, class A1>
class UnaryExpression: public ExpressionNode<T> {
public:
typedef Eigen::Matrix<double, T::dimension, A1::dimension> JacobianTA;
typedef boost::function<T(const A1&, boost::optional<JacobianTA&>)> Function;
private:
Function function_;
boost::shared_ptr<ExpressionNode<A1> > expressionA1_;
/// Constructor with a unary function f, and input argument e
UnaryExpression(Function f, const Expression<A1>& e) :
function_(f), expressionA1_(e.root()) {
}
friend class Expression<T> ;
public:
/// Return keys that play in this expression
virtual std::set<Key> keys() const {
return expressionA1_->keys();
}
/// Return value
virtual T value(const Values& values) const {
return function_(this->expressionA1_->value(values), boost::none);
}
/// Return value and derivatives
virtual Augmented<T> forward(const Values& values) const {
using boost::none;
Augmented<A1> argument = this->expressionA1_->forward(values);
JacobianTA dTdA;
T t = function_(argument.value(),
argument.constant() ? none : boost::optional<JacobianTA&>(dTdA));
return Augmented<T>(t, dTdA, argument.jacobians());
}
/// Trace structure for reverse AD
typedef boost::mpl::vector<A1> Arguments;
typedef typename GenerateTrace<T, Arguments>::type Trace;
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values,
typename JacobianTrace<T>::Pointer& p) const {
Trace* trace = new Trace();
p.setFunction(trace);
A1 a = this->expressionA1_->traceExecution(values, trace->trace);
return function_(a, trace->dTdA);
}
};
//-----------------------------------------------------------------------------
/// Binary Expression
template<class T, class A1, class A2>
class BinaryExpression: public ExpressionNode<T> {
public:
typedef Eigen::Matrix<double, T::dimension, A1::dimension> JacobianTA1;
typedef Eigen::Matrix<double, T::dimension, A2::dimension> JacobianTA2;
typedef boost::function<
T(const A1&, const A2&, boost::optional<JacobianTA1&>,
boost::optional<JacobianTA2&>)> Function;
private:
Function function_;
boost::shared_ptr<ExpressionNode<A1> > expressionA1_;
boost::shared_ptr<ExpressionNode<A2> > expressionA2_;
/// Constructor with a binary function f, and two input arguments
BinaryExpression(Function f, //
const Expression<A1>& e1, const Expression<A2>& e2) :
function_(f), expressionA1_(e1.root()), expressionA2_(e2.root()) {
}
friend class Expression<T> ;
public:
/// Return keys that play in this expression
virtual std::set<Key> keys() const {
std::set<Key> keys1 = expressionA1_->keys();
std::set<Key> keys2 = expressionA2_->keys();
keys1.insert(keys2.begin(), keys2.end());
return keys1;
}
/// Return value
virtual T value(const Values& values) const {
using boost::none;
return function_(this->expressionA1_->value(values),
this->expressionA2_->value(values), none, none);
}
/// Return value and derivatives
virtual Augmented<T> forward(const Values& values) const {
using boost::none;
Augmented<A1> a1 = this->expressionA1_->forward(values);
Augmented<A2> a2 = this->expressionA2_->forward(values);
JacobianTA1 dTdA1;
JacobianTA2 dTdA2;
T t = function_(a1.value(), a2.value(),
a1.constant() ? none : boost::optional<JacobianTA1&>(dTdA1),
a2.constant() ? none : boost::optional<JacobianTA2&>(dTdA2));
return Augmented<T>(t, dTdA1, a1.jacobians(), dTdA2, a2.jacobians());
}
/// Trace structure for reverse AD
struct Trace: public JacobianTrace<T> {
typename JacobianTrace<A1>::Pointer trace1;
typename JacobianTrace<A2>::Pointer trace2;
JacobianTA1 dTdA1;
JacobianTA2 dTdA2;
/// Start the reverse AD process
virtual void startReverseAD(JacobianMap& jacobians) const {
Select<T::dimension, A1>::reverseAD(trace1, dTdA1, jacobians);
Select<T::dimension, A2>::reverseAD(trace2, dTdA2, jacobians);
}
/// Given df/dT, multiply in dT/dA and continue reverse AD process
virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
trace1.reverseAD(dFdT * dTdA1, jacobians);
trace2.reverseAD(dFdT * dTdA2, jacobians);
}
/// Version specialized to 2-dimensional output
typedef Eigen::Matrix<double, 2, T::dimension> Jacobian2T;
virtual void reverseAD2(const Jacobian2T& dFdT,
JacobianMap& jacobians) const {
trace1.reverseAD2(dFdT * dTdA1, jacobians);
trace2.reverseAD2(dFdT * dTdA2, jacobians);
}
};
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values,
typename JacobianTrace<T>::Pointer& p) const {
Trace* trace = new Trace();
p.setFunction(trace);
A1 a1 = this->expressionA1_->traceExecution(values, trace->trace1);
A2 a2 = this->expressionA2_->traceExecution(values, trace->trace2);
return function_(a1, a2, trace->dTdA1, trace->dTdA2);
}
};
//-----------------------------------------------------------------------------
/// Ternary Expression
template<class T, class A1, class A2, class A3>
class TernaryExpression: public ExpressionNode<T> {
public:
typedef Eigen::Matrix<double, T::dimension, A1::dimension> JacobianTA1;
typedef Eigen::Matrix<double, T::dimension, A2::dimension> JacobianTA2;
typedef Eigen::Matrix<double, T::dimension, A3::dimension> JacobianTA3;
typedef boost::function<
T(const A1&, const A2&, const A3&, boost::optional<JacobianTA1&>,
boost::optional<JacobianTA2&>, boost::optional<JacobianTA3&>)> Function;
private:
Function function_;
boost::shared_ptr<ExpressionNode<A1> > expressionA1_;
boost::shared_ptr<ExpressionNode<A2> > expressionA2_;
boost::shared_ptr<ExpressionNode<A3> > expressionA3_;
/// Constructor with a ternary function f, and three input arguments
TernaryExpression(
Function f, //
const Expression<A1>& e1, const Expression<A2>& e2,
const Expression<A3>& e3) :
function_(f), expressionA1_(e1.root()), expressionA2_(e2.root()), expressionA3_(
e3.root()) {
}
friend class Expression<T> ;
public:
/// Return keys that play in this expression
virtual std::set<Key> keys() const {
std::set<Key> keys1 = expressionA1_->keys();
std::set<Key> keys2 = expressionA2_->keys();
std::set<Key> keys3 = expressionA3_->keys();
keys2.insert(keys3.begin(), keys3.end());
keys1.insert(keys2.begin(), keys2.end());
return keys1;
}
/// Return value
virtual T value(const Values& values) const {
using boost::none;
return function_(this->expressionA1_->value(values),
this->expressionA2_->value(values), this->expressionA3_->value(values),
none, none, none);
}
/// Return value and derivatives
virtual Augmented<T> forward(const Values& values) const {
using boost::none;
Augmented<A1> a1 = this->expressionA1_->forward(values);
Augmented<A2> a2 = this->expressionA2_->forward(values);
Augmented<A3> a3 = this->expressionA3_->forward(values);
JacobianTA1 dTdA1;
JacobianTA2 dTdA2;
JacobianTA3 dTdA3;
T t = function_(a1.value(), a2.value(), a3.value(),
a1.constant() ? none : boost::optional<JacobianTA1&>(dTdA1),
a2.constant() ? none : boost::optional<JacobianTA2&>(dTdA2),
a3.constant() ? none : boost::optional<JacobianTA3&>(dTdA3));
return Augmented<T>(t, dTdA1, a1.jacobians(), dTdA2, a2.jacobians(), dTdA3,
a3.jacobians());
}
/// Trace structure for reverse AD
struct Trace: public JacobianTrace<T> {
typename JacobianTrace<A1>::Pointer trace1;
typename JacobianTrace<A2>::Pointer trace2;
typename JacobianTrace<A3>::Pointer trace3;
JacobianTA1 dTdA1;
JacobianTA2 dTdA2;
JacobianTA3 dTdA3;
/// Start the reverse AD process
virtual void startReverseAD(JacobianMap& jacobians) const {
Select<T::dimension, A1>::reverseAD(trace1, dTdA1, jacobians);
Select<T::dimension, A2>::reverseAD(trace2, dTdA2, jacobians);
Select<T::dimension, A3>::reverseAD(trace3, dTdA3, jacobians);
}
/// Given df/dT, multiply in dT/dA and continue reverse AD process
virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
trace1.reverseAD(dFdT * dTdA1, jacobians);
trace2.reverseAD(dFdT * dTdA2, jacobians);
trace3.reverseAD(dFdT * dTdA3, jacobians);
}
/// Version specialized to 2-dimensional output
typedef Eigen::Matrix<double, 2, T::dimension> Jacobian2T;
virtual void reverseAD2(const Jacobian2T& dFdT,
JacobianMap& jacobians) const {
trace1.reverseAD2(dFdT * dTdA1, jacobians);
trace2.reverseAD2(dFdT * dTdA2, jacobians);
trace3.reverseAD2(dFdT * dTdA3, jacobians);
}
};
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values,
typename JacobianTrace<T>::Pointer& p) const {
Trace* trace = new Trace();
p.setFunction(trace);
A1 a1 = this->expressionA1_->traceExecution(values, trace->trace1);
A2 a2 = this->expressionA2_->traceExecution(values, trace->trace2);
A3 a3 = this->expressionA3_->traceExecution(values, trace->trace3);
return function_(a1, a2, a3, trace->dTdA1, trace->dTdA2, trace->dTdA3);
}
};
//-----------------------------------------------------------------------------
}
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