Cached Parameterization Concept#

This concept provides the interface expected by the MonotoneComponent class for the definition of the generally non-monotone function $f(x):{\mathbb{R}}^d\to \mathbb{R}$. This concept defines the ExpansionType template argument in the MonotoneComponent definition.

The MonotoneComponent class will evaluate $f(x)$ at many points $x^{(k)}$ that differ only in the $d^{th}$ component. That is, for any $k_1$ and $k_2$, $x_j^{(k_1)}=x_j^{(k_2)}$ for $j<d$. In many cases, it is possible to take advantage of this fact to precompute some quantities related to $x_{1:d-1}$ that we know will not change as the MonotoneComponent class varies $x_d$.

To enable this, ExpansionType needs to be able to interact with a preallocated block of memory called the cache below. The cache will be allocated outside the ExpansionType class, and then passed to ExpansionType::FillCache1 to do any necessary precomputations.

When the value of the last component $x_d$ is altered, the ExpansionType::FillCache2 function will then be called, before calling ExpansionType::Evaluate to actually evaluate $f(x)$. The ExpansionType::Evaluate function only has access to the memory in the cache, so its important to perform all necessary computations in either the ExpansionType::FillCache1 or ExpansionType::FillCache2 functions.

The details of these functions, as well as all the functions that MonotoneComponent expects ExpansionType to have, are provided in the Specific Requirements section below.

ExpansionType expansion;
// ... Define the expansion

// Evaluate the
std::vector<double> cache( expansion.CacheSize() );

Eigen::VectorXd coeffs = Eigen::VectorXd::Random( expansion.NumCoeffs() );

unsigned int dim = expansion.InputSize();
Eigen::VectorXd evalPt = Eigen::VectorXd::Random( dim );

expansion.FillCache1(&cache[0], evalPt, DerivativeFlags::None);
expansion.FillCache2(&cache[0], evalPt, evalPt(dim-1), DerivativeFlags::None);

double f = expansion.Evaluate(&cache[0], coeffs);

// Now perturb x_d and re-evaluate
double xd = evalPt(dim-1) + 1e-2;
expansion.FillCache2(&cache[0], evalPt, xd, DerivativeFlags::None);

double f2 = expansion.Evaluate(&cache[0], coeffs);