API

hankel_correlation_cleaner(
    noisy_correlations::AbstractVector{T};
    # KEYWORD ARGUMENTS
    maxiter::Int,
    tol::T,
    positive_curvature::Bool = true,
    fixed_endpoints::Bool = true,
    symmetric::Bool = false,
    verbose::Bool = false
) where {T<:AbstractFloat}

Denoise imaginary-time correlation data using the Hankel projection method. Returns the tuple (clean_correlations iter, err) where clean_correlation::Vector{T} is a vector of the cleaned correlations, iter::Int is the number of iterations and err::T is the final error.

Arguments

• noisy_correlations::AbstractVector{T}: Vector of imaginary time correlation data to be cleaned.

Keyword Arguments

• maxiter::T: Maximum number of iteration used in Dykstra's algorithm.
• tol::T: Tolerance threshold used in Dykstra's algorithm.
• positive_curvature::Bool = false: Whether to project onto a matrix with average anti-diagonals that have strictly positive curvature.
• fixed_endpoints::Bool = true: Whether to fix the correlation values at $\tau = 0$ and $\tau = \beta$.
• symmetric::Bool = false: Whether the imaginary-time correlation data is symmetric about $\tau = \beta/2$.
• verbose::Bool = false: Whether to print statements tracking the convergence of the denoising process.

hankel_correlation_cleaner!(
    correlations::AbstractVector{T};
    # KEYWORD ARGUMENTS
    maxiter::Int,
    tol::T,
    positive_curvature::Bool = true,
    fixed_endpoints::Bool = true,
    symmetric::Bool = false,
    verbose::Bool = false
) where {T<:AbstractFloat}

Denoise imaginary-time correlation data using the Hankel projection method. Also return the tuple (iter, err) where iter::Int is the number of iterations and err::T is the final error.

Arguments

• correlations::AbstractVector{T}: Vector of imaginary time correlation data to be cleaned.

Keyword Arguments

• maxiter::T: Maximum number of iteration used in Dykstra's algorithm.
• tol::T: Tolerance threshold used in Dykstra's algorithm.
• positive_curvature::Bool = false: Whether to project onto a matrix with average anti-diagonals that have strictly positive curvature.
• fixed_endpoints::Bool = true: Whether to fix the correlation values at $\tau = 0$ and $\tau = \beta$.
• symmetric::Bool = false: Whether the imaginary-time correlation data is symmetric about $\tau = \beta/2$.
• verbose::Bool = false: Whether to print statements tracking the convergence of the denoising process.

hankel_correlation_cleaner!(
    clean_correlations::AbstractVector{T},
    noisy_correlations::AbstractVector{T};
    # KEYWORD ARGUMENTS
    maxiter::Int,
    tol::T,
    positive_curvature::Bool = true,
    fixed_endpoints::Bool = true,
    symmetric::Bool = false,
    verbose::Bool = false
) where {T<:AbstractFloat}

Denoise imaginary-time correlation data using the Hankel projection method. Also return the tuple (iter, err) where iter::Int is the number of iterations and err::T is the final error.

Arguments

• clean_correlations::AbstractVector{T}: Vector to contain cleaned/denoised imaginary-time correlation data.
• noisy_correlations::AbstractVector{T}: Noisy imaginary-time correlation data.

Keyword Arguments

• maxiter::T: Maximum number of iteration used in Dykstra's algorithm.
• tol::T: Tolerance threshold used in Dykstra's algorithm.
• positive_curvature::Bool = false: Whether to project onto a matrix with average anti-diagonals that have strictly positive curvature.
• fixed_endpoints::Bool = true: Whether to fix the correlation values at $\tau = 0$ and $\tau = \beta$.
• symmetric::Bool = false: Whether the imaginary-time correlation data is symmetric about $\tau = \beta/2$.
• verbose::Bool = false: Whether to print statements tracking the convergence of the denoising process.

jackknife_hankel_correlation_cleaner(;
    # KEYWORD ARGUMENTS
    correlation_bins::AbstractMatrix{T},
    sign_bins::AbstractVector{T} = ones(eltype(correlation_bins), size(correlation_bins, 2)),
    maxiter::Int = 1000,
    tol::T = 1e-4,
    positive_curvature::Bool = false,
    fixed_endpoints::Bool = true,
    symmetric::Bool = false,
    covariance::Bool = false
) where {T<:AbstractFloat}

Denoise binned imaginary-time correlation data, propagating the error using the jackknife algorithm. Returns the mean and error of the denoised imaginary-time correlation data.

Arguments

• clean_correlations_mean::AbstractVector{T}: Vector to contain denoised imaginary-time correlations.
• clean_correlations_error::AbstractVector{T}: Vector to contain error associated with denoised imaginary-time correlations.
• correlation_bins::AbstractMatrix{T}: Binned imaginary-time correlations where the columns correspond to the bins and the rows to the imaginary time slices.

Keyword Arguments

• sign_bins::AbstractVector{T} = ones(eltype(correlation_bins), size(correlation_bins, 2)): For data generated via a quantum Monte Carlo simulation with a sign problem, the average sign associated with each bin.
• maxiter::T: Maximum number of iteration used in Dykstra's algorithm.
• tol::T: Tolerance threshold used in Dykstra's algorithm.
• positive_curvature::Bool = false: Whether to project onto a matrix with average anti-diagonals that have strictly positive curvature.
• fixed_endpoints::Bool = true: Whether to fix the correlation values at $\tau = 0$ and $\tau = \beta$.
• symmetric::Bool = false: Whether the imaginary-time correlation data is symmetric about $\tau = \beta/2$.
• covariance::Bool = false: If true, this method also returns the covariance matrix.

jackknife_hankel_correlation_cleaner!(
    clean_correlations_mean::AbstractVector{T},
    clean_correlations_error::AbstractVector{T},
    correlation_bins::AbstractMatrix{T};
    # KEYWORD ARGUMENTS
    sign_bins::AbstractVector{T} = ones(eltype(correlation_bins), size(correlation_bins, 2)),
    maxiter::Int = 1000,
    tol::T = 1e-4,
    positive_curvature::Bool = false,
    fixed_endpoints::Bool = true,
    symmetric::Bool = false,
    covariance::Bool = false
) where {T<:AbstractFloat}

Denoise binned imaginary-time correlation data, propagating the error using the jackknife algorithm.

Arguments

• clean_correlations_mean::AbstractVector{T}: Vector to contain denoised imaginary time correlations.
• clean_correlations_error::AbstractVector{T}: Vector to contain error associated with denoised imaginary time correlations.
• correlation_bins::AbstractMatrix{T}: Binned imaginary time correlations where the columns correspond to the bins and the rows to the imaginary time slices.

Keyword Arguments

• sign_bins::AbstractVector{T} = ones(eltype(correlation_bins), size(correlation_bins, 2)): For data generated via a quantum Monte Carlo simulation with a sign problem, the average sign associated with each bin.
• maxiter::T: Maximum number of iteration used in Dykstra's algorithm.
• tol::T: Tolerance threshold used in Dykstra's algorithm.
• positive_curvature::Bool = false: Whether to project onto a matrix with average anti-diagonals that have strictly positive curvature.
• fixed_endpoints::Bool = true: Whether to fix the correlation values at $\tau = 0$ and $\tau = \beta$.
• symmetric::Bool = false: Whether the imaginary-time correlation data is symmetric about $\tau = \beta/2$.
• covariance::Bool = false: If true, this method also returns the covariance matrix.

bootstrap_hankel_correlation_cleaner(;
    # KEYWORD ARGUMENTS
    correlation_bins::AbstractMatrix{T},
    sign_bins::AbstractVector{T} = ones(eltype(correlation_bins), size(correlation_bins, 2)),
    N_bootstrap::Int = 100,
    maxiter::Int = 1000,
    tol::T = 1e-4,
    positive_curvature::Bool = false,
    fixed_endpoints::Bool = true,
    symmetric::Bool = false,
    covariance::Bool = false
) where {T<:AbstractFloat}

Denoise binned imaginary-time correlation data, propagating the error via bootstrapping.

Arguments

• correlation_bins::AbstractMatrix{T}: Binned imaginary time correlations where the columns correspond to the bins and the rows to the imaginary time slices.

Keyword Arguments

• sign_bins::AbstractVector{T} = ones(eltype(correlation_bins), size(correlation_bins, 2)): For data generated via a quantum Monte Carlo simulation with a sign problem, the average sign associated with each bin.
• N_bootstrap::Int = 100: Number of bootstrap samples used to calculate the error.
• maxiter::T: Maximum number of iteration used in Dykstra's algorithm.
• tol::T: Tolerance threshold used in Dykstra's algorithm.
• positive_curvature::Bool = false: Whether to project onto a matrix with average anti-diagonals that have strictly positive curvature.
• fixed_endpoints::Bool = true: Whether to fix the correlation values at $\tau = 0$ and $\tau = \beta$.
• symmetric::Bool = false: Whether the imaginary-time correlation data is symmetric about $\tau = \beta/2$.
• covariance::Bool = false: If true, this method also returns the covariance matrix.

bootstrap_hankel_correlation_cleaner!(;
    # KEYWORD ARGUMENTS
    clean_correlations_mean::AbstractVector{T},
    clean_correlations_error::AbstractVector{T},
    correlation_bins::AbstractMatrix{T},
    sign_bins::AbstractVector{T} = ones(eltype(correlation_bins), size(correlation_bins, 2)),
    N_bootstrap::Int = 100,
    maxiter::Int = 1000,
    tol::T = 1e-4,
    positive_curvature::Bool = false,
    fixed_endpoints::Bool = true,
    symmetric::Bool = false,
    covariance::Bool = false
) where {T<:AbstractFloat}

Denoise binned imaginary-time correlation data, propagating the error via bootstrapping.

Arguments

• clean_correlations_mean::AbstractVector{T}: Vector to contain denoised imaginary time correlations.
• clean_correlations_error::AbstractVector{T}: Vector to contain error associated with denoised imaginary time correlations.
• correlation_bins::AbstractMatrix{T}: Binned imaginary time correlations where the columns correspond to the bins and the rows to the imaginary time slices.

Keyword Arguments

• sign_bins::AbstractVector{T} = ones(eltype(correlation_bins), size(correlation_bins, 2)): For data generated via a quantum Monte Carlo simulation with a sign problem, the average sign associated with each bin.
• N_bootstrap::Int = 100: Number of bootstrap samples used to calculate the error.
• maxiter::T: Maximum number of iteration used in Dykstra's algorithm.
• tol::T: Tolerance threshold used in Dykstra's algorithm.
• positive_curvature::Bool = false: Whether to project onto a matrix with average anti-diagonals that have strictly positive curvature.
• fixed_endpoints::Bool = true: Whether to fix the correlation values at $\tau = 0$ and $\tau = \beta$.
• symmetric::Bool = false: Whether the imaginary-time correlation data is symmetric about $\tau = \beta/2$.
• covariance::Bool = false: If true, this method also returns the covariance matrix.

dykstra!(;
    # KEYWORD ARGUMENTS
    H::AbstractArray{T},
    projections!::Vector{Function},
    tol::T,
    maxiter::Int,
    H0::AbstractArray{T} = zero(H),
    P0::AbstractArray{T} = zero(H),
    Ps::AbstractArray{T} = zeros(eltype(H), size(H)..., length(projections!)),
    verbose::Bool = false,
    projection_names::Vectors{String} = String[]
) where {T<:AbstractFloat}

Perform Dykstra's algorithm on an input array H given a vector projections! for projection functions that modify H in place. This method is based on the implementation found in the Dykstra python package.

project_psd!(
    H::AbstractMatrix{T},
    A::AbstractMatrix{T} = similar(H)
) where {T<:AbstractFloat}

Modifying H in-place, project it onto the nears positive semidefinite matrix by setting all the negative eigenvalues to zero.

project_hankel!(
    H::AbstractMattrix{T},
    endpoints::Tuple{T,T} = (NaN, NaN)
) where {T<:AbstractFloat}

Modifying the matrix H in-place, project it onto the closest Hankel matrix as defined by the Frobenius norm, by replacing each anti-diagonal by its average value. If finite endpoints are passed, then the values of H[1,1] and H[end,end] are fixed to those values respectively.

project_positive_curvature!(
    H::AbstractMatrix{T},
    endpoints::Tuple{T,T} = (NaN, NaN)
) where {T<:AbstractFloat}

Project the matrix H onto the nearest matrix, as measured by the Frobenius norm, that has anti-diagonal average values that have strictly positive curvature i.e. if you take the second finite difference of anti-diagonal average values the result values are strictly positive. If the endpoints are real numbers, then the H[1,1] and H[end:end] set set equal to the passed values.

init_hankel_matrix(
    correlations::AbstractVector{T}
) where {T<:AbstractFloat}

Allocate, initialize and return the Hankel matrix based on the vector of correlations.

" inithankelmatrix!( H::AbstractMatrix{T}, correaltions::AbstractVector{T} ) where {T<:AbstractFloat}

Initialize the Hankel matrix H in-place based on the vector of correlations.