Convex combinations

In [2]: 
import numpy as np
import matplotlib.pyplot as pt
In [3]: 
n_vectors = 3
vecs = np.random.randn(2, n_vectors)

Let's make an array of random coefficients, in coeffs.

In [4]: 
n_coeffs = n_vectors
In [5]: 
# Random numbers between 0 and 1. Don't confuse with randn.
coeffs = np.random.rand(n_coeffs, 10000)

sum_of_first_two = np.sum(coeffs[:2], axis=0)
coeffs[2] = 1-sum_of_first_two # (*)

# First two coefficients guaranteed to be positive
# Third coefficcient could be negative
# All coefficients guaranteed to add up to one because of (*)
good_coeffs = coeffs[2] >= 0
print(good_coeffs)

coeffs = coeffs[:, good_coeffs]

[ True  True False ...,  True  True  True]

Now assemble the convex combinations:

In [6]: 
n_combinations = coeffs.shape[-1]
print(n_combinations)

combinations = np.zeros((2, n_combinations))

5050

Shape summary:

  • combinations (output): 2 (X and Y) $\times$ n_combinations
  • coeffs (input): 3 (# of coefficients in conv. comb) $\times$ n_combinations
  • vecs (input): 2 (X and Y) $\times$ 3 (# of vectors in conv. comb.)
In [7]: 
for i in range(n_combinations):
    for j in range(n_vectors):
        combinations[:, i] += coeffs[j, i]*vecs[:, j]

(edit this cell for solution)

In [8]: 
pt.plot(combinations[0], combinations[1], "o")
Out[8]:
[<matplotlib.lines.Line2D at 0x7f90ab743dd8>]

/usr/lib/python3/dist-packages/matplotlib/backends/backend_agg.py:517: DeprecationWarning: npy_PyFile_Dup is deprecated, use npy_PyFile_Dup2
  filename_or_obj, self.figure.dpi)

Now do the same thing with a numpy one-liner:

In [9]: 
combinations = np.einsum("ji,kj->ki", coeffs, vecs)

(edit this cell for solution)

In [10]: 
pt.plot(combinations[0], combinations[1], "o")
Out[10]:
[<matplotlib.lines.Line2D at 0x7f90ab692d30>]