chebyshev_mapping

chebyshev_mapping([rng], [T], dims...;
    amplitude=one(T), sine_divisor=one(T),
    chebyshev_parameter=one(T), return_sparse=false)

Generate a Chebyshev-mapped matrix (Xie et al., 2024). The first row is initialized using a sine function and subsequent rows are iteratively generated via the Chebyshev mapping. The first row is defined as:

\[ W[1, j] = \text{amplitude} \cdot \sin(j \cdot \pi / (\text{sine_divisor} \cdot \text{n_cols}))\]

for j = 1, 2, …, ncols (with ncols typically equal to K+1, where K is the number of input layer neurons). Subsequent rows are generated by applying the mapping:

\[ W[i+1, j] = \cos( \text{chebyshev_parameter} \cdot \acos(W[pi, j]))\]

Arguments

• rng: Random number generator. Default is Utils.default_rng() from WeightInitializers.
• T: Type of the elements in the reservoir matrix. Default is Float32.
• dims: Dimensions of the matrix. Should follow res_size x in_size. res_size is assumed to be K+1.

Keyword arguments

• amplitude: Scaling factor used to initialize the first row. This parameter adjusts the amplitude of the sine function. Default value is one.
• sine_divisor: Divisor applied in the sine function's phase. Default value is one.
• chebyshev_parameter: Control parameter for the Chebyshev mapping in subsequent rows. This parameter influences the distribution of the matrix elements. Default is one.
• return_sparse: If true, the function returns the matrix as a sparse matrix. Default is false.

Examples

julia> input_matrix = chebyshev_mapping(10, 3)
10×3 Matrix{Float32}:
 0.866025  0.866025   1.22465f-16
 0.866025  0.866025  -4.37114f-8
 0.866025  0.866025  -4.37114f-8
 0.866025  0.866025  -4.37114f-8
 0.866025  0.866025  -4.37114f-8
 0.866025  0.866025  -4.37114f-8
 0.866025  0.866025  -4.37114f-8
 0.866025  0.866025  -4.37114f-8
 0.866025  0.866025  -4.37114f-8
 0.866025  0.866025  -4.37114f-8