Normalizations and filters

In LatNet Builder, it is possible to apply transformations and filters to the merit values computed by the software.

Transformations and filters are only available for the lattice point set type.

The transformations have the following form:

\[ \mathcal E_q(P_n) = \frac{D_q(P_n)}{D^\ast_q(n)} \]

where \(D^\ast_q(n)\) is a normalization factor. This is why transformations implemented by LatNet Builder are also called normalizations. The normalization factor can be a bound on (or an estimate of) the best possible value of \(D_q(P_n)\), or a bound on the average of \(D_q(P_n)\) among all the possible point sets.

It is worth mentioning that the bounds are often a family of bounds parametrized by a scalar \(\lambda \in \mathcal A\) where \(\mathcal A\) is an admissible interval of the form \([\frac{1}{\alpha}, 1)\) which depends on the normalization. In LatNet Builder, the best (smallest) bound is selected with the Brent minimizer:

\[ D^\ast_q(n) = D^\ast_q(n, \lambda^\ast) = \min_{\lambda \in \mathcal A} D^\ast_q(n, \lambda). \]

This normalizations can be useful to define comparable measures of uniformity for point sets of different sizes and to combine them to measure the global uniformity of multilevel point sets.

Additionally, in the case of random searches, normalizations can be combined with a low-pass filter to eliminate the point sets whose quality is deemed too poor. Low-pass filters reject point sets whose merit value exceeds a threshold. Note that when the random variant of a search is used with a low-pass filter, the candidate samples that are rejected by the filter pipeline are not considered as valid samples, meaning that the number of random samples corresponds to the number of accepted candidates.

Additional details and references can be found in Section 3.4 of [18].

Normalizations and filters for multilevel point sets

For multilevel point sets, normalizations using a bound on the figure of merit are applied individually to each embedding level. Each embedding level \(m\) has a weight \(w_m = c_m^{1 / \lambda^\ast_m}\) , called the per-level weight. By default, all the \(c_m\) equal one over the number of embedding levels. Optionally, one can select a range of levels, in this case, \(c_m\) equal zero for the unselected levels and one over the number of selected levels for the selected levels.

Then, LatNet Builder selects \(\lambda^\ast_m\) as follows:

\[ \lambda^\ast_m = arg\,min_{\lambda \in \mathcal A} c_m^{-1/\lambda }D^\ast_q(n_m). \]

The normalized merit value of each embedding level is then computed as follows:

\[ \mathcal E_q(P_{n_m}) = w_m \frac{D_q(P_{n_m})}{D^\ast_q(n)}. \]

Optionally, multilevel low-pass filters are applied to reject the point sets for which at least one multilevel merit value is deemed too high. Then, the merit values are aggregated, either by taking their maximum or their sum:

\[ \bar{\mathcal E_q}(P_{n_1}, \dots, P_{n_k})^q = \max_{m = 1, \dots, k} \mathcal E_q(P_{n_m})^q, \]

or:

\[ \bar{\mathcal E_q}(P_{n_1}, \dots, P_{n_k})^q = \sum_{m = 1, \dots, k} \mathcal E_q(P_{n_m})^q. \]

Additional details and references can be found in Section 3.5 of [18].

List of normalizations

Currently, the following normalizations are available:

• for the \(P_\alpha\) discrepancy of ordinary lattices:
  • SL10: the general bound derived in Theorem 3 of [31] for projection-dependent weights,
  • DPW08: the bound derived in Theorem 10 of [7] for product weights.
• for the \(P_\alpha\) discrepancy of polynomial lattice rules:
  • the bound derived in Theorem 3 of [8] for projection-dependent weights.
• for the \( B_{\alpha, d, (1)}\) interlaced discrepancy from [10] :
  • the general bound derived in Theorem 4 of [10] for projection-dependent weights.
• for the \( B_{d, (2)}\) interlaced discrepancy from [10] :
  • the general bound derived in Theorem 5 of [10] for projection-dependent weights.