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. 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< IAAlpha > | |
| 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< 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 [10] (with the minimization on \(\lambda\) step proposed in Algorithm 2 in [7]).
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.
The documentation for this class was generated from the following file:
- include/latbuilder/Norm/IAAlpha.h
