LatBuilder::Norm::IAAlpha Class Reference

Bound on the interlaced \(B_{\alpha, d, \gamma, (1)}\) discrepancy. More...

#include <IAAlpha.h>

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

Public Member Functions
	IAAlpha (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< IAAlpha >
	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< IAAlpha >
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 interlaced \(B_{\alpha, d, \gamma, (1)}\) discrepancy.

This is the bound

\[ B_{\alpha, d, \gamma, (1)}(\boldsymbol q_tau, p) \leq \frac{1}{(2^m - 1)^{1/\lambda}} \left[ \sum_{\emptyset \neq \mathfrak v \subseteq \{1, \dots, j_0 - 1 \}} \tilde{\gamma}_{\mathfrak v}^\lambda G_{\alpha, d, \lambda, d, (1)}^{|\mathfrak v|} + G_{\alpha, d, \lambda, d, (1)} \sum_{\mathfrak v \subseteq \{1, \dots, j_0 - 1 \}} \tilde{\gamma}_{\mathfrak v \cup \{j_0\}}^\lambda G_{\alpha, d, \lambda, d, (1)}^{|\mathfrak v|} \right]^{1/\lambda}. \]

for \(1/\min(\alpha, d) < \lambda \leq 1\), where we write \(j_0 = \lceil \tau / d \rceil\), \(d_0 = \tau - (j_0 - 1)d\) and

\[ G_{\alpha, d, \lambda, a, (1)} = - 1 + (1 + \tilde{G}_{\alpha, d, \lambda, (1)})^a, \]

for \(a = 1, \dots, d\), in which we define:

\[ \tilde{G}_{\alpha, d, \lambda, (1)}) = \frac{1}{2^{\alpha \lambda /2}} \max \left\{ \left(\frac{1}{2^{\lambda \min(\alpha, d)} - 2}\right)^{\lambda}, \frac{1}{2^{\lambda \min(\alpha, d)} - 2} \right\}. \]

from Theorem 4. in [rGOD13a] (with the minimization on \(\lambda\) step proposed in Algorithm 2 in [8]).

If the \(\gamma\) are POD weights, using the order-dependent correction \(\tilde{\gamma}\) from Theorem 2., the computation of this bound boils down to:

\[ G_{\alpha, d, \lambda, d_0, (1)} (\gamma_{j_0} \Gamma_1)^\lambda + \sum_{l = 1}^{j_0-1} \left(\Gamma_l^\lambda + G_{\alpha, d, \lambda, d_0, (1)} (\gamma_{j_0} \Gamma_{l + 1})^\lambda e_l^{j_0-1}(G_{\alpha, d, \lambda, d, (1)} \gamma_1^\lambda, \dots, G_{\alpha, d, \lambda, d, (1)} \gamma_{j_0-1}^\lambda\right), \]

where \(e_{i}^n\) denotes the symmetric elementary polynomial of degree \(i\) with \(n\) variables.

Constructor & Destructor Documentation

IAAlpha()

LatBuilder::Norm::IAAlpha::IAAlpha	(	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/IAAlpha.h