kl.py

"""
Approximating the cross-entropy between two Power Sphericals.

Uses a second-order Taylor expansion to approximate E[log(1+z)].
"""

# author: vlad n <[email protected]>
# license: mit
# documentation: https://hackmd.io/@vladn/SJ93wMevK

import numpy as np
import torch

from power_spherical import PowerSpherical, HypersphericalUniform


def ps_variance_dot(ps, y):
    """Computes the variance of dot(x, y) for x~ps and norm(y) = 1."""

    alpha = ps.base_dist.marginal_t.base_dist.concentration1
    beta = ps.base_dist.marginal_t.base_dist.concentration0
    ratio = (alpha + beta) / (2 * beta)
    t_var = ps.base_dist.marginal_t.variance
    dp = ps.loc @ y  # check dimension
    yy = 1  # yy = y @ y, but we know this to be 1.
    return t_var * ((1 - ratio) * dp ** 2 + ratio * yy) #  + mean_sq - mean_sq


def check_ps_variance_dot(
        dim=10,
        k=20,
        n_samples=1000):

    dim = torch.tensor(dim)
    k = torch.tensor(k)

    unif = HypersphericalUniform(dim=dim)
    mu_p = unif.rsample()
    mu_q = unif.rsample()

    p = PowerSpherical(loc=mu_p, scale=k)

    xp = p.rsample((n_samples,))

    z = xp @ mu_q

    # true mean
    z_mean = p.mean @ mu_q

    print("mean z true:", z_mean.item())
    print("mean z num: ", torch.mean(z).item())

    print("V[z] tru: ", ps_variance_dot(p, mu_q).item())
    print("V[z] num: ", torch.var(z).item())


def check_taylor(
        dim=10,
        k=3,
        n_samples=10000):

    dim = torch.tensor(dim)
    k = torch.tensor(k)

    unif = HypersphericalUniform(dim=dim)
    mu_p = unif.rsample()
    mu_q = unif.rsample()

    p = PowerSpherical(loc=mu_p, scale=k)

    xp = p.rsample((n_samples,))

    # approximate E[ log(1+z) ] where z = dot(x,y), x~ps
    z = xp @ mu_q
    z_mean = p.mean @ mu_q

    taylor_first = torch.log1p(z_mean)
    taylor_second = ps_variance_dot(p, mu_q) / (2 * (1 + z_mean) ** 2)
    taylor = taylor_first - taylor_second

    print("MC:  ", torch.mean(torch.log1p(z)).item())
    print("Tay: ", taylor.item())


def main():
    check_ps_variance_dot()
    check_taylor()


if __name__ == '__main__':
    main()