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Least squares problem

Solving a least squares problem of the form

\[\inf_{\alpha \in \mathbb{R}^d} \sum_{m=1}^M \left(\sum_{i=1}^d \alpha_i g_i\circ\varphi(x_m) - y_m\right)^2\]

is done by creating an instance of VSLeastSquares

VectorSpaceLeastSquares.VSLeastSquares — Method
VSLeastSquares(basis::Tb, transform::Tt=VoidTransformation(), Td::Type=Float64) where {Tb<:AbstractBasis, Tt<:AbstractTransformation}

Create a VSLeastSquares object from basis and transform and capable of handling Td typed data.

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Base.length — Method
length(vslsq::VSLeastSquares{Tb, Tt, Td}) where {Tb<:AbstractBasis, Tt<:AbstractTransformation, Td<:Real}

Return the number of functions of the basis used to solve the least squares problem

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VectorSpaceLeastSquares.nVariates — Method
nVariates(vslsq::VSLeastSquares{Tb, Tt, Td}) where {Tb<:AbstractBasis, Tt<:AbstractTransformation, Td<:Real}

Return the number of variates in the least squares problem

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Base.size — Method
size(vslsq::VSLeastSquares{Tb, Tt, Td}) where {Tb<:AbstractBasis, Tt<:AbstractTransformation, Td<:Real}

Return the tuple (nVariates, length)

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VectorSpaceLeastSquares.getCoefficients — Method
getCoefficients(vslsq::VSLeastSquares{Tb, Tt, Td}) where {Tb<:AbstractBasis, Tt<:AbstractTransformation, Td<:Real}

Return the coefficients solution to the least squares problem.

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VectorSpaceLeastSquares.getBasis — Method
getBasis(vslsq::VSLeastSquares{Tb, Tt, Td}) where {Tb<:AbstractBasis, Tt<:AbstractTransformation, Td<:Real}

Return the basis used to solve the least squares problem.

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An instance of VSLeastSquares is typically manipulated using the following methods to solve the least squares problem and compute a prediction

VectorSpaceLeastSquares.fit — Method
fit(vslsq::VSLeastSquares{Tb, Tt, Td}, x::AbstractVector{<:AbstractVector{Td}}, y::AbstractVector{Td}) where {Tb<:AbstractBasis, Tt<:AbstractTransformation, Td<:Real}

Solve the least squares problem.

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VectorSpaceLeastSquares.predict — Method
predict(vslsq::VSLeastSquares{Tb, Tt, Td}, x::AbstractVector{Td}) where {Tb<:AbstractBasis, Tt<:AbstractTransformation, Td<:Real}

Compute the value predicted by the least squares problem.

The method fit must have been called before.

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If the underlying basis of type Tb is differentiable the derivative of the prediction can be computed using

VectorSpaceLeastSquares.derivative — Method
derivative(vslsq::VSLeastSquares{Tb, Tt, Td}, x::AbstractVector{Td}, index::Integer) where {Tb<:AbstractBasis, Tt<:AbstractTransformation, Td<:Real}

Compute the partial derivative of the prediction w.r.t to the index variable.

The method fit must have been called before.

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VectorSpaceLeastSquares.gradient — Method
gradient(vslsq::VSLeastSquares{Tb, Tt, Td}, x::AbstractVector{Td}) where {Tb<:AbstractBasis, Tt<:AbstractTransformation, Td<:Real}

Compute the gradient of the prediction at x.

The method fit must have been called before.

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