Bound on the weighted \(\mathcal P_\alpha\)PLR discrepancy. More...
#include <PAlphaPLR.h>
Inherits LatBuilder::Norm::NormAlphaBase< PAlphaPLR >.
Public Member Functions | |
| PAlphaPLR (unsigned int alpha, const LatticeTester::Weights &weights, Real normType=2) | |
| Constructor. More... | |
| 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< PAlphaPLR > | |
| NormAlphaBase (unsigned int alpha, Real normType) | |
| Constructor. More... | |
| 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. More... | |
| Real | operator() (const SizeParam< LR, L > &sizeParam, Dimension dimension, Real norm=1.0) const |
Returns the smallest value of the bound for dimension dimension. More... | |
| Real | minimum (const SizeParam< LR, L > &sizeParam, Dimension dimension, Real norm) const |
| Returns the minimum value of the bound function. More... | |
Additional Inherited Members | |
| Static Public Attributes inherited from LatBuilder::Norm::NormAlphaBase< PAlphaPLR > | |
| 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 \(\mathcal P_\alpha\)PLR discrepancy.
This is the general bound derived in Theorem 3 of [8] for projection-dependent weights. The theorem states that, for \(\mathcal D^2(\boldsymbol a_s, n) = \mathcal P_\alphaPLR(\boldsymbol a_s, n)\), 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
◆ PAlphaPLR()
| LatBuilder::Norm::PAlphaPLR::PAlphaPLR | ( | 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.
The documentation for this class was generated from the following file:
- include/latbuilder/Norm/PAlphaPLR.h
