Lower Triangular Topology

lower_triangular([rng], [T], dims...; radius=1.0, sparsity=0.9, return_sparse=false)

Create and return a sparse reservoir matrix with a lower triangular topology (Cossu et al., Oct 2024). This function populates the main diagonal and the immediately adjacent lower sub-diagonals with random uniform weights in the range (-1, 1) until the target sparsity is reached. It guarantees structural symmetry by only adding complete diagonals.

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. Must be square.

Keyword arguments

• radius: The desired spectral radius of the reservoir. Defaults to 1.0.
• sparsity: The exact target fraction of zero elements in the matrix. To hit this target precisely while preventing spatial bias, any remaining weights needed for the final partial sub-diagonal are randomly distributed across its indices. Defaults to 0.9.
• return_sparse: Flag for returning a SparseMatrixCSC instead of a dense matrix. Setting to true requires SparseArrays to be loaded. Defaults to false.

Examples

Default call:

julia> W = lower_triangular(5, 5)
5×5 Matrix{Float32}:
 -0.214159   0.0         0.0        0.0        0.0
  0.710328   0.655184    0.0        0.0        0.0
 -0.912441   0.311545   -0.412411   0.0        0.0
  0.0        0.114112    0.551211   0.812451   0.0
  0.0        0.0        -0.741211   0.914141   0.115412

Returning a SparseMatrixCSC:

julia> using SparseArrays

julia> W_sparse = lower_triangular(6, 6; sparsity=0.8, return_sparse=true)
6×6 SparseMatrixCSC{Float32, Int64} with 7 stored entries:
  0.4512   ⋅        ⋅        ⋅        ⋅        ⋅
  0.1245  -0.8123   ⋅        ⋅        ⋅        ⋅
 -0.7612   0.9123  -0.1124   ⋅        ⋅        ⋅
  ⋅        ⋅        0.4412  -0.5123   ⋅        ⋅
  ⋅        ⋅        ⋅        ⋅        0.8812   ⋅
  ⋅        ⋅        ⋅        ⋅        ⋅        0.3314

Scaling to a custom spectral radius:

julia> W_scaled = lower_triangular(Float16, 4, 4; radius=2.5)
4×4 Matrix{Float16}:
  2.145   0.0     0.0     0.0
 -1.123   1.854   0.0     0.0
  0.954  -2.113  -1.542   0.0
  0.0     1.412   2.014  -1.953