Hessian Sparsity Pattern: Reverse Mode

Processing math: 0%

$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$ This is cppad-20221105 documentation. Here is a link to its current documentation . Hessian Sparsity Pattern: Reverse Mode
Syntax

h = f.RevSparseHes(q, s)

h = f.RevSparseHes(q, s, transpose)

Purpose
We use

$F : \B{R}^n \rightarrow \B{R}^m$ to denote the AD function corresponding to f . For a fixed matrix

$R \in \B{R}^{n \times q}$ and a fixed vector

$S \in \B{R}^{1 \times m}$ , we define

$\begin{array}{rcl} H(x) & = & \partial_x \left[ \partial_u S * F[ x + R * u ] \right]_{u=0} \\ & = & R^\R{T} * (S * F)^{(2)} ( x ) \\ H(x)^\R{T} & = & (S * F)^{(2)} ( x ) * R \end{array}$ Given a sparsity pattern for the matrix

$R$ and the vector

$S$ , RevSparseHes returns a sparsity pattern for the

$H(x)$ .

f
The object f has prototype



    const ADFun<Base> f

x
If the operation sequence in f is independent of the independent variables in

$x \in \B{R}^n$ , the sparsity pattern is valid for all values of (even if it has CondExp or VecAD operations).

q
The argument q has prototype



    size_t q

It specifies the number of columns in

$R \in \B{R}^{n \times q}$ and the number of rows in

$H(x) \in \B{R}^{q \times n}$ . It must be the same value as in the previous ForSparseJac call



    f.ForSparseJac(q, r, r_transpose)

Note that if r_transpose is true, r in the call above corresponding to

$R^\R{T} \in \B{R}^{q \times n}$

transpose
The argument transpose has prototype



    bool transpose

The default value false is used when transpose is not present.

r
The matrix

$R$ is specified by the previous call



    f.ForSparseJac(q, r, transpose)

see r . The type of the elements of SetVector must be the same as the type of the elements of r .

s
The argument s has prototype



    const SetVector& s

(see SetVector below) If it has elements of type bool, its size is

$m$ . If it has elements of type std::set<size_t>, its size is one and all the elements of s[0] are between zero and

$m - 1$ . It specifies a sparsity pattern for the vector S .

h
The result h has prototype



    SetVector& h

(see SetVector below).

transpose false
If h has elements of type bool, its size is

$q * n$ . If it has elements of type std::set<size_t>, its size is

$q$ and all the set elements are between zero and n-1 inclusive. It specifies a sparsity pattern for the matrix

$H(x)$ .

transpose true
If h has elements of type bool, its size is

$n * q$ . If it has elements of type std::set<size_t>, its size is

$n$ and all the set elements are between zero and q-1 inclusive. It specifies a sparsity pattern for the matrix

$H(x)^\R{T}$ .

SetVector
The type SetVector must be a SimpleVector class with elements of type bool or std::set<size_t>; see sparsity pattern for a discussion of the difference. The type of the elements of SetVector must be the same as the type of the elements of r .

Entire Sparsity Pattern
Suppose that

$q = n$ and

$R \in \B{R}^{n \times n}$ is the

$n \times n$ identity matrix. Further suppose that the

$S$ is the k-th elementary vector ; i.e.

$S_j = \left\{ \begin{array}{ll} 1 & {\rm if} \; j = k \\ 0 & {\rm otherwise} \end{array} \right.$ In this case, the corresponding value h is a sparsity pattern for the Hessian matrix

$F_k^{(2)} (x) \in \B{R}^{n \times n}$ .

Example
The file rev_sparse_hes.cpp contains an example and test of this operation. The file sparsity_sub.cpp contains an example and test of using RevSparseHes to compute the sparsity pattern for a subset of the Hessian.

Input File: include/cppad/core/rev_sparse_hes.hpp