permutation_init

permutation_init([rng], [T], dims...;
    weight=0.1, permutation_matrix=nothing, return_sparse=false,
    radius=nothing, kwargs...)

Creates a permutation reservoir as described in (Boedecker et al., 2009), by first initializing a scaled identity (self-loops) and then applying a column permutation.

This construction yields:

\[ \widehat{W} = \lambda P\]

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 reservoir matrix.

Keyword arguments

• weight: Weight used for the initial self-loop initialization (and the magnitude of the nonzeros after permutation). Default is 0.1.
• permutation_matrix: Optional permutation matrix to apply. If nothing, a random permutation is generated (using rng) and applied.
• return_sparse: flag for returning a sparse matrix. true requires SparseArrays to be loaded. Default is false.
• radius: The desired spectral radius of the reservoir. If nothing is passed, no scaling takes place. Defaults to nothing.
• sampling_type: Sampling that decides the distribution of weight negative numbers. If set to :no_sample the sign is unchanged. If set to :bernoulli_sample! then each forward_weight can be positive with a probability set by positive_prob. If set to :irrational_sample! the weight is negative if the decimal number of the irrational number chosen is odd. If set to :regular_sample!, each weight will be assigned a negative sign after the chosen strides. strides can be a single number or an array. Default is :no_sample.
• positive_prob: probability of the weight being positive when sampling_type is set to :bernoulli_sample!. Default is 0.5.
• irrational: Irrational number whose decimals decide the sign of weight. Default is pi.
• start: Which place after the decimal point the counting starts for the irrational sign counting. Default is 1.
• strides: number of strides for assigning negative value to a weight. It can be an integer or an array. Default is 2.

Examples

Default kwargs:

julia> reservoir_matrix = permutation_init(5, 5)
5×5 Matrix{Float32}:
 0.0  0.0  0.0  0.0  0.1
 0.0  0.1  0.0  0.0  0.0
 0.1  0.0  0.0  0.0  0.0
 0.0  0.0  0.1  0.0  0.0
 0.0  0.0  0.0  0.1  0.0

Changing the weights magnitudes to a different unique value:

julia> reservoir_matrix = permutation_init(5, 5; weight=0.99)
5×5 Matrix{Float32}:
 0.0   0.0   0.0   0.0   0.99
 0.0   0.99  0.0   0.0   0.0
 0.99  0.0   0.0   0.0   0.0
 0.0   0.0   0.99  0.0   0.0
 0.0   0.0   0.0   0.99  0.0

Changing the weights signs with different sampling techniques:

julia> reservoir_matrix = permutation_init(5, 5; sampling_type=:bernoulli_sample!)
5×5 Matrix{Float32}:
 0.0  0.1   0.0  0.0   0.0
 0.0  0.0  -0.1  0.0   0.0
 0.1  0.0   0.0  0.0   0.0
 0.0  0.0   0.0  0.0  -0.1
 0.0  0.0   0.0  0.1   0.0

Changing the weights to random numbers. Note that the length of the given array must be at least as long as the subdiagonal one wants to fill:

julia> reservoir_matrix = permutation_init(5, 5; weight=rand(Float32, 5))
5×5 Matrix{Float32}:
 0.0       0.0       0.0       0.0       0.0263106
 0.0       0.923927  0.0       0.0       0.0
 0.255075  0.0       0.0       0.0       0.0
 0.0       0.0       0.585589  0.0       0.0
 0.0       0.0       0.0       0.353418  0.0

Returning a sparse matrix:

julia> reservoir_matrix = permutation_init(5, 5; return_sparse=true)
5×5 SparseMatrixCSC{Float32, Int64} with 5 stored entries:
  ⋅    ⋅    ⋅    ⋅   0.1
  ⋅   0.1   ⋅    ⋅    ⋅
 0.1   ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅   0.1   ⋅    ⋅
  ⋅    ⋅    ⋅   0.1   ⋅