DDS does not work for lp-degree less than 1.0

It appears DDS does not work for lp-degree less than 1.0. Here's a reproducible example that shows the evaluation of a canonical polynomial does not coincide with the evaluation of the corresponding Newton polynomial:

import numpy as np
import minterpy as mp

mi = mp.MultiIndexSet.from_degree(2, 3, 0.5)  # No interaction between the two variables

def fun(xx):
    """Polynomial test function where there is no interaction between variables."""
    return 10 + 1 * xx[:,0] + 2 * xx[:,0]**2 + 3 * xx[:,0]**3 + 1 * xx[:,1] + 2 * xx[:,1]**2 + 3 * xx[:,1]**3

xx_test = -1 + 2 * np.random.rand(1000, 2)

can_poly = mp.CanonicalPolynomial(mi, coeffs=np.array([10., 1., 2., 3., 1., 2., 3.]))

nwt_poly = mp.CanonicalToNewton(can_poly)()

# This assertion is successful
assert np.allclose(can_poly(xx_test), fun(xx_test))

# But this assertion will fail
assert np.allclose(nwt_poly(xx_test), can_poly(xx_test))

Or exemplified another way, the Newton polynomial evaluation on the unisolvent nodes does not coincide with the Lagrange coefficients:

# or if we create a Lagrange polynomial first
grd = mp.Grid(mi)
lag_poly = mp.LagrangePolynomial(mi, fun(grd.unisolvent_nodes))
nwt_poly = mp.LagrangeToNewton(lag_poly)()

# This assertion will also fail (the Newton polynomial evaluation on the unisolvent nodes does not coincide with the Lagrange polynomial coefficients, which is troubling)
assert np.allclose(nwt_poly(grd.unisolvent_nodes), lag_poly.coeffs)
# nwt_poly eval  : array([14.       ,  6.       ,  9.375    ,  7.625    , 22.       , 16.3046875, 20.1015625]),
# lag_poly coeffs: array([14.   ,  6.   ,  9.375,  7.625, 22.   , 15.625, 17.375]))

This error can also be captured by including any lp-degree of less than 1.0 in the test fixture lp_degree in the conftest.py file.

In the applied metamodeling literature, lp-degree of less than 1.0 is known as the hyperbolic truncation scheme. It is of interest in many engineering models as these models often exhibit sparsity-of-effects where high-degree interactions between input variables are not observed. For such models, we might get away with a more compact polynomial model.

As far as I understand things, there is no reason for DDS to not work in this case as the resulting multi-index is still downward close.

After a preliminary investigation, I think I know what's the problem. There seems to be a problem with the current implementation of tree construction. I'm already working on a solution for this and currently testing its performance. The merge request for a fix is coming in few days.