## Optimal quantization of the standard multivariate normal distribution

 one_dim_1_1000.zip one_dim_1001_5999.zip mult_dimensional_grids.zip

The compressed folder one_dim_1_1000.zip contains optimal quantization grids of the standard univariate normal distribution of size to , and one_dim_1001_5999.zip from the grids of size to .

The compressed folder mult_dimensional_grids.zip contains optimized quantization grids of the standard multivariate normal distribution of size between and and dimension between and . (It does not contain the one-dimensional grids which are available separately.)

The files are in text format. In every case, the filename is N_d_nopti where N is the quantizer size and d is the dimension.

For a given size , the text files are organized as follows. It presents in the form of a matrix with rows and columns.

• On row : element of the grid and its companion parameters.
• On last row :
• In particular we can verify that
Methodology:
• The multi-dimensional grids were obtained by an incremental "splitting" method based on an optimization by a mixed Lloyd-CLVQ algorithm. The splitting method consists in appending to an optimized grid of elements random points to get the starting point for the optimization procedure for a quantizer of size .

Note that the CLVQ procedure is only used for small values of .

• The one-dimensional grids where obtained by deterministic methods. This is to directly minimize the quadratic distortion seen as a function of values. Several methods are available as a tridiagonal Newton-Raphson method or a semi-closed Lloyd's algorithm. See article [1] for more details on these algorithms.
In the one dimensional setting, 32 significant figures are available.

### References

1. Gilles Pagès, and Jacques Printems, Monte Carlo Methods and Applications, vol. 9, pp. 135–166, 2003.