ApproximateVanishingIdeals.L2LossMethod

Creates and returns objective function, gradient and solution (through inversion) w.r.t. L2 loss.

Arguments

• 'data::Matrix{Float64}': evaluations of O_terms
• 'labels::Vector{Float64}': current border term evaluated
• 'lambda::Union{Float64, Int64}': regularization parameter
• 'data_squared::Matrix{Float64}': data' * data
• 'data_labels::Vector{Float64}': data' * labels
• 'labels_squared::Float64': labels' * labels
• 'datasquaredinverse::Union{Matrix{Float64}, Nothing}': inverse of data_squared for IHB, optional (default is nothing)

Returns

• 'solution::Vector{Float64}': solution to (unconstrained) minimization problem
• 'evaluate_function<:Function': objective function
ApproximateVanishingIdeals.abmMethod

Runs ABM algorithm to find coefficient vector and computes loss.

Arguments

• 'oracle_type::String': string denoting which oracle to construct
• 'data::Matrix{Float64}': data (O_evaluations)
• 'labels::Vector{Float64}': labels (term_evaluated)
• 'lambda::Union{Float64, Int64}': regularization parameter (if applicable)
• 'data_squared::Matrix{Float64}': squared data
• 'data_labels::Vector{Float64}': data' * labels
• 'labels_squared::Float64': labels' * labels
• 'datasquaredinverse::Union{Matrix{Float64}, Nothing}': inverse of data_squared (default is nothing)

Returns

• 'coefficient_vector::Vector{Float64}': coefficient vector minimizing ABM optimization problem
• 'loss::Float64': loss w.r.t. 'coefficient_vector'
ApproximateVanishingIdeals.conditional_gradientsMethod

Returns coefficient_vector and loss found through CG-based algorithm fit to 'data'.

Arguments

• 'oracle_type::String': type of CG-based algorithm (choice from 'CG', 'BCG', 'BPCG')
• 'data::Matrix{Float64}': evaluations of O_terms
• 'labels::Vector{Float64}': current border term evaluated
• 'lambda::Union{Float64, Int64}': regularization parameter
• 'data_squared::Matrix{Float64}': data' * data
• 'data_labels::Vector{Float64}': data' * labels
• 'labels_squared::Float64': labels' * labels
• 'datasquaredinverse::Union{Matrix{Float64}, Nothing}': inverse of data_squared for IHB, optional (default is nothing)
• 'psi::Float64': vanishing parameter, optional (default is 0.1)
• 'epsilon::Float64': solver accuracy, optional (default is 0.001)
• 'tau::Float64': bound on coefficient_vector norm, optional (default is 1000.)
• 'inversehessianboost::String': whether or not to use IHB (choice from 'false', 'weak', 'full'), optional (default is 'false')

Returns

• 'coefficientvector::Vector{Float64}': coefficientvector minimizing L2Loss over L1Ball
• 'loss::Float64': loss w.r.t. 'coefficient_vector'
ApproximateVanishingIdeals.construct_borderFunction

constructs the border of 'terms'

Arguments

• 'terms::Matrix{Int64}': Matrix with monomial terms as columns
• 'terms_evaluated::Matrix{Float64}': Matrix with evaluations of 'terms' over X
• 'X_train::Matrix{Float64}': data
• 'degree1terms::Matrix{Int64}': Matrix with degree 1 monomials as columns
• 'degree1termsevaluated::Matrix{Float64}': evaluations of 'degree1_terms' over X
• 'purging_terms::Matrix{Int64}': purge terms in 'terms' divisible by any of these

Returns

• 'bordertermsraw::Matrix{Int64}': non-purged border constructed from 'terms'
• 'borderevaluationsraw::Matrix{Float64}': non-purged evaluations of border terms over X
• 'nonpurgingindices::Vector{Int64}': array of non-purging indices
• 'raw_permutation::Vector{Int64}': array with deg-lex ordering permutation
ApproximateVanishingIdeals.evaluate_transformation_oaviMethod

Evaluates the transformation corresponding to the polynomials in Gcoefficientvectors.

Arguments

• 'sets::SetsOandG': instance of SetsOandG, containing transformation

Returns

• 'totalnumberofzeros::Int64': Sum of all zero entries in coefficientvectors in Gcoefficientvectors.
• 'totalnumberofentries::Int64': Total number of entries in coefficientvectors in Gcoefficientvectors.
• 'avgsparsity::Float64': The average sparsity of coefficientvectors in Gcoefficientvectors.
• 'numberofpolynomials::Int64': Number of polynomials in G.
• 'numberofterms::Int64': Number of terms in O.
• 'degree::Float64': Average degree of polynomials in G.
ApproximateVanishingIdeals.evaluate_transformation_vcaMethod

Evaluates the transformation corresponding to the polynomials in V w.r.t. the functions in F and C

Arguments

• 'sets::SetsVCA': instance of SetsVCA.

Returns

• 'totalnumberofzeros::Int64': Sum of all zero entries in coefficientvectors in Vcoefficientvectors.
• 'totalnumberofentries::Int64': Total number of entries in coefficientvectors in Vcoefficientvectors.
• 'avgsparsity::Float64': The average sparsity of coefficientvectors in Vcoefficientvectors.
• 'numberofpolynomials::Int64': Number of polynomials in Vcoefficientvectors.
• 'numberofterms::Int64': Number of non-vanishing terms.
• 'degree::Float64': Average degree of polynomials in V.
ApproximateVanishingIdeals.find_range_null_vcaMethod

Performs FindRangeNull (using SVD) for VCA. Reference: https://proceedings.mlr.press/v28/livni13.html

Arguments

• 'F::Matrix{Float64}': evaluation of F polynomials
• 'C::Matrix{Float64}': evaluation of C polynomials
• 'psi::Float64': vanishing parameter

Returns

• 'Vcoefficientvectors::Matrix{Float64}': Coefficient vectors of polynomials we append to V.
• 'Vevaluationvectors::Matrix{Float64}': Evaluation vectors of polynomials we append to V.
• 'Fcoefficientvectors::Matrix{Float64}': Coefficient vectors of polynomials we append to F.
• 'Fevaluationvectors::Matrix{Float64}': Evaluation vectors of polynomials we append to F.
ApproximateVanishingIdeals.fit_oaviMethod

Creates OAVI feature transformation fitted to X_train

Arguments

• 'X_train::Matrix{Float64}': training data
• 'max_degree::Int64': max degree of polynomials computed (default 10)
• 'psi::Float64': vanishing extent (default 0.1)
• 'epsilon::Float64': accuracy for convex optimizer (default 1.0e-7)
• 'tau::Union{Float64, Int64}': upper bound on norm of coefficient vector
• 'lambda::Float64': regularization parameter
• 'oracle::Union{String, <:Function}': string denoting which predefined constructor to use OR constructor function. (external constructor function MUST have 'data' and 'labels' as varargs)
• 'max_iters::Int64': max number of iterations in oracle
• 'inversehessianboost::String': whether or not to use IHB. Choose from "false", "weak" or "full".
• 'oracle_kwargs::Vector': Array containing keyword arguments for external constructor functions

Returns

• 'Xtraintransformed::Matrix{Float64}': transformed X_train
• 'sets::SetsOandG': instance of 'SetsOandG' keeping track of important sets
ApproximateVanishingIdeals.fit_vcaMethod

This function creates and applies a VCA transformation fitted to X.

Arguments

• 'X::Matrix{Float64}': data, stored row-wise
• 'psi::Float64': vanishing parameter
• 'max_degree::Int64': maximum degree to consider

Returns

• 'Xtraintransformed::Matrix{Float64}': X transformed according to transformation found by VCA
• 'sets_vca::SetsVCA': instance of SetsVCA containing relevant sets for VCA
ApproximateVanishingIdeals.print_polynomialsMethod

Prints the polynomials obtained through OAVI as a LaTeX string.

'digits' can be used to determine how many decimal places you want to round to. Terms with rounded coefficient values 0.0 are omitted and coefficients with value 1.0 are omitted.

'ret' can be used to return vector with poly strings, instead of printing the polynomials

ApproximateVanishingIdeals.purgeMethod

purges each term in 'terms' that is divisible by at least one term 'purging_terms'

Arguments

• 'terms::Matrix{Int64}': Matrix with monomial terms as columns
• 'terms_evaluated::Matrix{Float64}': evaluations of 'terms' over data
• 'purging_terms::Matrix{Int64}': Matrix with purging terms as columns

Returns

• 'terms[:, inidces]::Matrix{Int64}': purged version of terms
• 'terms_evaluated[:, indices]::Matrix{Float64}': purged evaluations
• 'indices::Vector{Int64}': array with non-purging indices