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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 <gtsam/base/Testable.h>
#include <gtsam/base/Manifold.h>
#include <boost/foreach.hpp>
#include <boost/tuple/tuple.hpp>
#include <boost/bind.hpp>
#include <boost/lambda/lambda.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>
#include <boost/mpl/transform.hpp>
#include <boost/mpl/at.hpp>
namespace MPL = boost::mpl::placeholders;
#include <new> // for placement new
class ExpressionFactorBinaryTest;
// Forward declare for testing
namespace gtsam {
template<typename T>
class Expression;
typedef std::pair<Key, Eigen::Block<Matrix> > JacobianPair;
typedef std::vector<JacobianPair> JacobianMap;
//-----------------------------------------------------------------------------
/**
* The CallRecord class stores the Jacobians of applying a function
* with respect to each of its arguments. It also stores an executation trace
* (defined below) for each of its arguments.
*
* It is sub-classed in the function-style ExpressionNode sub-classes below.
*/
template<int COLS>
struct CallRecord {
static size_t const N = 0;
virtual void print(const std::string& indent) const {
}
virtual void startReverseAD(JacobianMap& jacobians) const {
}
virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
}
typedef Eigen::Matrix<double, 2, COLS> Jacobian2T;
virtual void reverseAD2(const Jacobian2T& dFdT,
JacobianMap& jacobians) const {
}
};
//-----------------------------------------------------------------------------
/// Handle Leaf Case: reverseAD ends here, by writing a matrix into Jacobians
template<int ROWS, int COLS>
void handleLeafCase(const Eigen::Matrix<double, ROWS, COLS>& dTdA,
JacobianMap& jacobians, Key key) {
JacobianMap::iterator it = std::find_if(jacobians.begin(), jacobians.end(),
boost::bind(&JacobianPair::first, _1)==key);
assert(it~= jacobians.end());
it->second.block<ROWS, COLS>(0, 0) += dTdA; // block makes HUGE difference
}
/// Handle Leaf Case for Dynamic Matrix type (slower)
template<>
void handleLeafCase(
const Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic>& dTdA,
JacobianMap& jacobians, Key key) {
JacobianMap::iterator it = std::find_if(jacobians.begin(), jacobians.end(),
boost::bind(&JacobianPair::first, _1)==key);
assert(it~= jacobians.end());
it->second += dTdA;
}
//-----------------------------------------------------------------------------
/**
* The ExecutionTrace class records a tree-structured expression's execution.
*
* The class looks a bit complicated but it is so for performance.
* It is a tagged union that obviates the need to create
* a ExecutionTrace subclass for Constants and Leaf Expressions. Instead
* the key for the leaf is stored in the space normally used to store a
* CallRecord*. Nothing is stored for a Constant.
*
* A full execution trace of a Binary(Unary(Binary(Leaf,Constant)),Leaf) would be:
* Trace(Function) ->
* BinaryRecord with two traces in it
* trace1(Function) ->
* UnaryRecord with one trace in it
* trace1(Function) ->
* BinaryRecord with two traces in it
* trace1(Leaf)
* trace2(Constant)
* trace2(Leaf)
* Hence, there are three Record structs, written to memory by traceExecution
*/
template<class T>
class ExecutionTrace {
static const int Dim = traits::dimension<T>::value;
enum {
Constant, Leaf, Function
} kind;
union {
Key key;
CallRecord<Dim>* ptr;
} content;
public:
/// Pointer always starts out as a Constant
ExecutionTrace() :
kind(Constant) {
}
/// Change pointer to a Leaf Record
void setLeaf(Key key) {
kind = Leaf;
content.key = key;
}
/// Take ownership of pointer to a Function Record
void setFunction(CallRecord<Dim>* record) {
kind = Function;
content.ptr = record;
}
/// Print
void print(const std::string& indent = "") const {
if (kind == Constant)
std::cout << indent << "Constant" << std::endl;
else if (kind == Leaf)
std::cout << indent << "Leaf, key = " << content.key << std::endl;
else if (kind == Function) {
std::cout << indent << "Function" << std::endl;
content.ptr->print(indent + " ");
}
}
/// Return record pointer, quite unsafe, used only for testing
template<class Record>
boost::optional<Record*> record() {
if (kind != Function)
return boost::none;
else {
Record* p = dynamic_cast<Record*>(content.ptr);
return p ? boost::optional<Record*>(p) : boost::none;
}
}
/**
* *** This is the main entry point for reverseAD, called from Expression ***
* Called only once, either inserts I into Jacobians (Leaf) or starts AD (Function)
*/
typedef Eigen::Matrix<double, Dim, Dim> JacobianTT;
void startReverseAD(JacobianMap& jacobians) const {
if (kind == Leaf) {
// This branch will only be called on trivial Leaf expressions, i.e. Priors
static const JacobianTT I = JacobianTT::Identity();
handleLeafCase(I, jacobians, content.key);
} else if (kind == Function)
// This is the more typical entry point, starting the AD pipeline
// 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 (kind == Leaf)
handleLeafCase(dTdA, jacobians, content.key);
else if (kind == Function)
content.ptr->reverseAD(dTdA, jacobians);
}
// Either add to Jacobians (Leaf) or propagate (Function)
typedef Eigen::Matrix<double, 2, Dim> Jacobian2T;
void reverseAD2(const Jacobian2T& dTdA, JacobianMap& jacobians) const {
if (kind == Leaf)
handleLeafCase(dTdA, jacobians, content.key);
else if (kind == Function)
content.ptr->reverseAD2(dTdA, jacobians);
}
/// Define type so we can apply it as a meta-function
typedef ExecutionTrace<T> type;
};
/// Primary template calls the generic Matrix reverseAD pipeline
template<size_t ROWS, class A>
struct Select {
typedef Eigen::Matrix<double, ROWS, traits::dimension<A>::value> Jacobian;
static void reverseAD(const ExecutionTrace<A>& 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, traits::dimension<A>::value> Jacobian;
static void reverseAD(const ExecutionTrace<A>& trace, const Jacobian& dTdA,
JacobianMap& jacobians) {
trace.reverseAD2(dTdA, jacobians);
}
};
//-----------------------------------------------------------------------------
/**
* 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:
size_t traceSize_;
/// Constructor, traceSize is size of the execution trace of expression rooted here
ExpressionNode(size_t traceSize = 0) :
traceSize_(traceSize) {
}
public:
/// Destructor
virtual ~ExpressionNode() {
}
/// Return keys that play in this expression as a set
virtual std::set<Key> keys() const {
std::set<Key> keys;
return keys;
}
/// Return dimensions for each argument, as a map
virtual void dims(std::map<Key, size_t>& map) const {
}
// Return size needed for memory buffer in traceExecution
size_t traceSize() const {
return traceSize_;
}
/// Return value
virtual T value(const Values& values) const = 0;
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values, ExecutionTrace<T>& trace,
char* raw) 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 value
virtual T value(const Values& values) const {
return constant_;
}
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values, ExecutionTrace<T>& trace,
char* raw) 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 dimensions for each argument
virtual void dims(std::map<Key, size_t>& map) const {
map[key_] = traits::dimension<T>::value;
}
/// Return value
virtual T value(const Values& values) const {
return values.at<T>(key_);
}
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values, ExecutionTrace<T>& trace,
char* raw) const {
trace.setLeaf(key_);
return values.at<T>(key_);
}
};
//-----------------------------------------------------------------------------
// Below we use the "Class Composition" technique described in the book
// C++ Template Metaprogramming: Concepts, Tools, and Techniques from Boost
// and Beyond. Abrahams, David; Gurtovoy, Aleksey. Pearson Education.
// to recursively generate a class, that will be the base for function nodes.
// The class generated, for three arguments A1, A2, and A3 will be
//
// struct Base1 : Argument<T,A1,1>, FunctionalBase<T> {
// ... storage related to A1 ...
// ... methods that work on A1 ...
// };
//
// struct Base2 : Argument<T,A2,2>, Base1 {
// ... storage related to A2 ...
// ... methods that work on A2 and (recursively) on A1 ...
// };
//
// struct Base3 : Argument<T,A3,3>, Base2 {
// ... storage related to A3 ...
// ... methods that work on A3 and (recursively) on A2 and A1 ...
// };
//
// struct FunctionalNode : Base3 {
// Provides convenience access to storage in hierarchy by using
// static_cast<Argument<T, A, N> &>(*this)
// }
//
// All this magic happens when we generate the Base3 base class of FunctionalNode
// by invoking boost::mpl::fold over the meta-function GenerateFunctionalNode
//-----------------------------------------------------------------------------
/// meta-function to generate fixed-size JacobianTA type
template<class T, class A>
struct Jacobian {
typedef Eigen::Matrix<double, traits::dimension<T>::value,
traits::dimension<A>::value> type;
};
/// meta-function to generate JacobianTA optional reference
template<class T, class A>
struct Optional {
typedef Eigen::Matrix<double, traits::dimension<T>::value,
traits::dimension<A>::value> Jacobian;
typedef boost::optional<Jacobian&> type;
};
/**
* Base case for recursive FunctionalNode class
*/
template<class T>
struct FunctionalBase: ExpressionNode<T> {
static size_t const N = 0; // number of arguments
typedef CallRecord<traits::dimension<T>::value> Record;
/// Construct an execution trace for reverse AD
void trace(const Values& values, Record* record, char*& raw) const {
}
};
/**
* Building block for recursive FunctionalNode class
* The integer argument N is to guarantee a unique type signature,
* so we are guaranteed to be able to extract their values by static cast.
*/
template<class T, class A, size_t N>
struct Argument {
/// Expression that will generate value/derivatives for argument
boost::shared_ptr<ExpressionNode<A> > expression;
};
/**
* Building block for Recursive Record Class
* Records the evaluation of a single argument in a functional expression
*/
template<class T, class A, size_t N>
struct JacobianTrace {
A value;
ExecutionTrace<A> trace;
typename Jacobian<T, A>::type dTdA;
};
/**
* Recursive Definition of Functional ExpressionNode
*/
template<class T, class A, class Base>
struct GenerateFunctionalNode: Argument<T, A, Base::N + 1>, Base {
static size_t const N = Base::N + 1; ///< Number of arguments in hierarchy
typedef Argument<T, A, N> This; ///< The storage we have direct access to
/// Return keys that play in this expression
virtual std::set<Key> keys() const {
std::set<Key> keys = Base::keys();
std::set<Key> myKeys = This::expression->keys();
keys.insert(myKeys.begin(), myKeys.end());
return keys;
}
/// Return dimensions for each argument
virtual void dims(std::map<Key, size_t>& map) const {
Base::dims(map);
This::expression->dims(map);
}
/// Recursive Record Class for Functional Expressions
struct Record: JacobianTrace<T, A, N>, Base::Record {
typedef T return_type;
typedef JacobianTrace<T, A, N> This;
/// Print to std::cout
virtual void print(const std::string& indent) const {
Base::Record::print(indent);
static const Eigen::IOFormat matlab(0, 1, " ", "; ", "", "", "[", "]");
std::cout << This::dTdA.format(matlab) << std::endl;
This::trace.print(indent);
}
/// Start the reverse AD process
virtual void startReverseAD(JacobianMap& jacobians) const {
Base::Record::startReverseAD(jacobians);
Select<traits::dimension<T>::value, A>::reverseAD(This::trace, This::dTdA,
jacobians);
}
/// Given df/dT, multiply in dT/dA and continue reverse AD process
virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
Base::Record::reverseAD(dFdT, jacobians);
This::trace.reverseAD(dFdT * This::dTdA, jacobians);
}
/// Version specialized to 2-dimensional output
typedef Eigen::Matrix<double, 2, traits::dimension<T>::value> Jacobian2T;
virtual void reverseAD2(const Jacobian2T& dFdT,
JacobianMap& jacobians) const {
Base::Record::reverseAD2(dFdT, jacobians);
This::trace.reverseAD2(dFdT * This::dTdA, jacobians);
}
};
/// Construct an execution trace for reverse AD
void trace(const Values& values, Record* record, char*& raw) const {
Base::trace(values, record, raw); // recurse
// Write an Expression<A> execution trace in record->trace
// Iff Constant or Leaf, this will not write to raw, only to trace.
// Iff the expression is functional, write all Records in raw buffer
// Return value of type T is recorded in record->value
record->Record::This::value = This::expression->traceExecution(values,
record->Record::This::trace, raw);
// raw is never modified by traceExecution, but if traceExecution has
// written in the buffer, the next caller expects we advance the pointer
raw += This::expression->traceSize();
}
};
/**
* Recursive GenerateFunctionalNode class Generator
*/
template<class T, class TYPES>
struct FunctionalNode {
typedef typename boost::mpl::fold<TYPES, FunctionalBase<T>,
GenerateFunctionalNode<T, MPL::_2, MPL::_1> >::type Base;
struct type: public Base {
// Argument types and derived, note these are base 0 !
typedef TYPES Arguments;
typedef typename boost::mpl::transform<TYPES, Jacobian<T, MPL::_1> >::type Jacobians;
typedef typename boost::mpl::transform<TYPES, Optional<T, MPL::_1> >::type Optionals;
/// Reset expression shared pointer
template<class A, size_t N>
void reset(const boost::shared_ptr<ExpressionNode<A> >& ptr) {
static_cast<Argument<T, A, N> &>(*this).expression = ptr;
}
/// Access Expression shared pointer
template<class A, size_t N>
boost::shared_ptr<ExpressionNode<A> > expression() const {
return static_cast<Argument<T, A, N> const &>(*this).expression;
}
/// Provide convenience access to Record storage
struct Record: public Base::Record {
/// Access Value
template<class A, size_t N>
const A& value() const {
return static_cast<JacobianTrace<T, A, N> const &>(*this).value;
}
/// Access Jacobian
template<class A, size_t N>
typename Jacobian<T, A>::type& jacobian() {
return static_cast<JacobianTrace<T, A, N>&>(*this).dTdA;
}
};
/// Construct an execution trace for reverse AD
Record* trace(const Values& values, char* raw) const {
// Create the record and advance the pointer
Record* record = new (raw) Record();
raw = (char*) (record + 1);
// Record the traces for all arguments
// After this, the raw pointer is set to after what was written
Base::trace(values, record, raw);
// Return the record for this function evaluation
return record;
}
};
};
//-----------------------------------------------------------------------------
/// Unary Function Expression
template<class T, class A1>
class UnaryExpression: public FunctionalNode<T, boost::mpl::vector<A1> >::type {
public:
typedef boost::function<T(const A1&, typename Optional<T, A1>::type)> Function;
typedef typename FunctionalNode<T, boost::mpl::vector<A1> >::type Base;
typedef typename Base::Record Record;
private:
Function function_;
/// Constructor with a unary function f, and input argument e
UnaryExpression(Function f, const Expression<A1>& e1) :
function_(f) {
this->template reset<A1, 1>(e1.root());
ExpressionNode<T>::traceSize_ = sizeof(Record) + e1.traceSize();
}
friend class Expression<T> ;
public:
/// Return value
virtual T value(const Values& values) const {
return function_(this->template expression<A1, 1>()->value(values), boost::none);
}
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values, ExecutionTrace<T>& trace,
char* raw) const {
Record* record = Base::trace(values, raw);
trace.setFunction(record);
return function_(record->template value<A1, 1>(),
record->template jacobian<A1, 1>());
}
};
//-----------------------------------------------------------------------------
/// Binary Expression
template<class T, class A1, class A2>
class BinaryExpression: public FunctionalNode<T, boost::mpl::vector<A1, A2> >::type {
public:
typedef boost::function<
T(const A1&, const A2&, typename Optional<T, A1>::type,
typename Optional<T, A2>::type)> Function;
typedef typename FunctionalNode<T, boost::mpl::vector<A1, A2> >::type Base;
typedef typename Base::Record Record;
private:
Function function_;
/// Constructor with a ternary function f, and three input arguments
BinaryExpression(Function f, const Expression<A1>& e1,
const Expression<A2>& e2) :
function_(f) {
this->template reset<A1, 1>(e1.root());
this->template reset<A2, 2>(e2.root());
ExpressionNode<T>::traceSize_ = //
sizeof(Record) + e1.traceSize() + e2.traceSize();
}
friend class Expression<T> ;
friend class ::ExpressionFactorBinaryTest;
public:
/// Return value
virtual T value(const Values& values) const {
using boost::none;
return function_(this->template expression<A1, 1>()->value(values),
this->template expression<A2, 2>()->value(values),
none, none);
}
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values, ExecutionTrace<T>& trace,
char* raw) const {
Record* record = Base::trace(values, raw);
trace.setFunction(record);
return function_(record->template value<A1, 1>(),
record->template value<A2,2>(), record->template jacobian<A1, 1>(),
record->template jacobian<A2, 2>());
}
};
//-----------------------------------------------------------------------------
/// Ternary Expression
template<class T, class A1, class A2, class A3>
class TernaryExpression: public FunctionalNode<T, boost::mpl::vector<A1, A2, A3> >::type {
public:
typedef boost::function<
T(const A1&, const A2&, const A3&, typename Optional<T, A1>::type,
typename Optional<T, A2>::type, typename Optional<T, A3>::type)> Function;
typedef typename FunctionalNode<T, boost::mpl::vector<A1, A2, A3> >::type Base;
typedef typename Base::Record Record;
private:
Function function_;
/// 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) {
this->template reset<A1, 1>(e1.root());
this->template reset<A2, 2>(e2.root());
this->template reset<A3, 3>(e3.root());
ExpressionNode<T>::traceSize_ = //
sizeof(Record) + e1.traceSize() + e2.traceSize() + e3.traceSize();
}
friend class Expression<T> ;
public:
/// Return value
virtual T value(const Values& values) const {
using boost::none;
return function_(this->template expression<A1, 1>()->value(values),
this->template expression<A2, 2>()->value(values),
this->template expression<A3, 3>()->value(values),
none, none, none);
}
/// Construct an execution trace for reverse AD
virtual T traceExecution(const Values& values, ExecutionTrace<T>& trace,
char* raw) const {
Record* record = Base::trace(values, raw);
trace.setFunction(record);
return function_(
record->template value<A1, 1>(), record->template value<A2, 2>(),
record->template value<A3, 3>(), record->template jacobian<A1, 1>(),
record->template jacobian<A2, 2>(), record->template jacobian<A3, 3>());
}
};
//-----------------------------------------------------------------------------
}
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