def test_cosine_analytical(d, allclose):
pytest.importorskip("scipy") # beta, betainc, betaincinv
dt = 0.0001
x = np.arange(-1 + dt, 1, dt)
def p(x, d):
# unnormalized CosineSimilarity distribution, derived by Eric H.
return (1 - x * x) ** ((d - 3) / 2.0)
dist = CosineSimilarity(d)
pdf_exp = dist.pdf(x)
pdf_act = p(x, d)
cdf_exp = dist.cdf(x)
cdf_act = np.cumsum(pdf_act) / np.sum(pdf_act)
# Check that we get the expected pdf after normalization
assert allclose(pdf_exp / np.sum(pdf_exp), pdf_act / np.sum(pdf_act), atol=0.01)
# Check that this accumulates to the expected cdf
assert allclose(cdf_exp, cdf_act, atol=0.01)
# Check that the inverse cdf gives back x
assert allclose(dist.ppf(cdf_exp), x, atol=0.01)
def test_cosine_intercept(d, p, rng, allclose):
"""Tests CosineSimilarity inverse cdf for finding intercepts."""
pytest.importorskip("scipy") # betaincinv
num_samples = 500
exp_dist = UniformHypersphere(surface=True)
act_dist = CosineSimilarity(d)
dots = exp_dist.sample(num_samples, d, rng=rng)[:, 0]
# Find the desired intercept so that dots >= c with probability p
c = act_dist.ppf(1 - p)
assert allclose(np.sum(dots >= c) / float(num_samples), p, atol=0.05)