LatBuilder::Norm::PAlphaTilde Class Reference

Bound on the weighted \(\mathcal \tilde{P}_\alpha\) discrepancy. More...

#include <PAlphaTilde.h>

Inherits LatBuilder::Norm::NormAlphaBase< PAlphaTilde >.

Public Member Functions
	PAlphaTilde (unsigned int alpha, const LatticeTester::Weights &weights, Real normType=2)
	Constructor.
template<LatticeType LR, EmbeddingType L>
Real	value (Real lambda, const SizeParam< LR, L > &sizeParam, Dimension dimension, Real norm=1.0) const
std::string	name () const
Public Member Functions inherited from LatBuilder::Norm::NormAlphaBase< PAlphaTilde >
	NormAlphaBase (unsigned int alpha, Real normType)
	Constructor.
unsigned	alpha () const
Real	normType () const
Real	minExp () const
Real	maxExp () const
Real	value (Real lambda, const SizeParam< LR, L > &sizeParam, Dimension dimension, Real norm=1.0) const
	Returns the value of the bound.
Real	operator() (const SizeParam< LR, L > &sizeParam, Dimension dimension, Real norm=1.0) const
	Returns the smallest value of the bound for dimension dimension.
Real	minimum (const SizeParam< LR, L > &sizeParam, Dimension dimension, Real norm) const
	Returns the minimum value of the bound function.

Additional Inherited Members
Static Public Attributes inherited from LatBuilder::Norm::NormAlphaBase< PAlphaTilde >
static const unsigned	MINIMIZER_MAX_ITER
	Maximum number of iterations to be used with the minimizer.
static const int	MINIMIZER_PREC_BITS
	Relative precision on the minimum value to be used with the minimizer.

Detailed Description

Bound on the weighted \(\mathcal \tilde{P}_\alpha\) discrepancy.

This is the general bound derived in Theorem 3 of [rDIC15a] for projection-dependent weights. The theorem states that, for \(\mathcal D^2(\boldsymbol a_s, n) = \mathcal tilde{P}_\alpha(\boldsymbol a_s, n)\), there exists a generating vector \(\boldsymbol a_s\) such that

\[ \mathcal D^2(\boldsymbol a_s, n) \leq N_{n,s}(\lambda) \]

for any \(\lambda \in (1/\alpha,1]\), where

\[ N_{n,s}(\lambda) = \left[ \frac{2}{n} \sum_{\emptyset \neq \mathfrak u \subseteq \{1,\dots,s\}} \gamma_{\mathfrak u}^\lambda \, \left( \mu(\alpha\lambda) \right)^{|\mathfrak u|} \right]^{1/\lambda}, \]

in which \(\mu(x) = \frac{2^x}{2^x - 2} \). The normalization that is used is:

\[ \min_\lambda N_{n,s}(\lambda) \]

For order-dependent weights, the bound can be rewritten as:

\[ N_{n,s}(\lambda) = \left[ \frac{2}{n} \sum_{\ell=1}^s \Gamma_\ell^\lambda \, \frac{s!}{\ell! (s-\ell)!} \, \left( 2 \beta(\alpha\lambda) \right)^\ell \right]^{1/\lambda} \]

For product weights, it can be written as:

\[ N_{n,s}(\lambda) = \left\{ \frac{2}{n} \left[ \prod_{j=1}^s \left( 1 + \gamma_j^\lambda \, \mu(\alpha\lambda) \right) - 1 \right] \right\}^{1/\lambda}, \]

For product and order-dependent (POD) weights, the bound can be written as:

\[ N_{n,s}(\lambda) = \left[ \frac{2}{n} \sum_{\ell=1}^s \Gamma_\ell^\lambda \, e_\ell^d \left( \mu(\alpha\lambda) \gamma_1^\lambda, \dots, \mu(\alpha\lambda) \gamma_d^\lambda \right)^\ell \right]^{1/\lambda} \]

where \(e_\ell^d\) designates the elementary symmetric polynomial of degree \(l\) with \(d\) variables.

Constructor & Destructor Documentation

PAlphaTilde()

LatBuilder::Norm::PAlphaTilde::PAlphaTilde	(	unsigned int	alpha,
		const LatticeTester::Weights &	weights,
		Real	normType = 2 )

Constructor.

Parameters

alpha	Smoothness level \(\alpha\) of the class of functions.
weights	Projection-dependent weights \( \gamma_{\mathfrak u} \).
normType	Type of cross-projection norm used by the figure of merit.

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The documentation for this class was generated from the following file:

• include/latbuilder/Norm/PAlphaTilde.h