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Increase the input dimension of a function

Description

We want to build a function f : \mathbb{R}^d \mapsto \mathbb{R} from p functions f_i: \mathbb{R} \mapsto \mathbb{R}.

We can do that:

  • Case 1: using the tensor product of the functions f_i,

  • Case 2: by addition of the functions f_i.

We need to implement both basic steps:

  • Step 1: creation of the projection function: \pi_i : (x_1, \dots, x_d) \mapsto x_i,

  • Step 2: creation of the composed function: g_i = f_i \circ \pi_i : (x_1, \dots, x_d) \mapsto f_i(x_i).

Step 1: Creation of the projection function

The projection function is defined by:

\pi_i : (x_1, \dots, x_d) \mapsto x_i

We can do that using:

Method 1: We use the SymbolicFunction class.

import openturns as ot


def buidProjSymbolic(p, i):
    # R^p --> R
    # (x1, ..., xp) --> xi
    inputVar = ot.Description.BuildDefault(p, "x")
    return ot.SymbolicFunction(inputVar, [inputVar[i]])


d = 2
all_projections = [buidProjSymbolic(d, i) for i in range(d)]
print(
    "Input dimension = ",
    all_projections[0].getInputDimension(),
    "Output dimension = ",
    all_projections[0].getOutputDimension(),
)

Input dimension =  2 Output dimension =  1

Method 2: We use the LinearFunction class.

The function \pi_i(\vect{x}) = \mat{A}\left(\vect{x}-\vect{c}\right) + \vect{b}.

def buildProjLinear(d, i):
    # R^d --> R
    # (x1, ..., xd) --> xi
    matA = ot.Matrix(1, d)
    matA[0, i] = 1.0
    cVect = [0.0] * d
    bVect = [0.0]
    return ot.LinearFunction(cVect, bVect, matA)


all_projections = [buildProjLinear(d, i) for i in range(d)]

Step 2: Creation of the composed function

The composed function is defined by: g_i = f_i \circ \pi_i defined by:

g_i: (x_1, \dots, x_d) \mapsto f_i(x_i)

We use the ComposedFunction class.

f1 = ot.SymbolicFunction(["x1"], ["x1^2"])
f2 = ot.SymbolicFunction(["x2"], ["3*x2"])
fi_list = [f1, f2]
all_g = [ot.ComposedFunction(f, proj) for (f, proj) in zip(fi_list, all_projections)]
print(all_g[0].getInputDimension(), all_g[0].getOutputDimension())

2 1

Case 1: Tensor product

We want to build the function f : \mathbb{R}^d \mapsto \mathbb{R} defined by:

f: (x_1, \dots, x_d) \mapsto \prod_{i=1}^d f_i(x_i)

As the operator * can only be applied to functions sharing the same input space, we need to use the projection function \pi_i and the functions g_i all defined on \mathbb{R}^d.

def tensorProduct(factors):
    prod = factors[0]
    for i in range(1, len(factors)):
        prod = prod * factors[i]
    return prod


f = tensorProduct(all_g)
print("input dimension =", f.getInputDimension())
print("output dimension =", f.getOutputDimension())
print("f([1.0, 2.0]) = ", f([1.0, 2.0]))

input dimension = 2
output dimension = 1
f([1.0, 2.0]) =  [6]

Case 2: Sum

We want to build the function f : \mathbb{R}^d \mapsto \mathbb{R} defined by:

f: (x_1, \dots, x_d) \mapsto \sum_{i=1}^d f_i(x_i)

We use the LinearCombinationFunction class.

coef = [1.0, 1.0]
f = ot.LinearCombinationFunction(all_g, [1.0] * len(all_g))
print("input dimension =", f.getInputDimension())
print("output dimension =", f.getOutputDimension())

input dimension = 2
output dimension = 1