Equation of Geometric Difference

[Squigspear]

Hi,

I would like to clarify the following code that was used to implement geometric difference from the paper: "The power of data in quantum machine learning".

def calculate_geometric_difference(k_1, k_2, normalization_lambda=0.001):
"""
Calculate the geometric difference g(K_1 || K_2), which is equation F9 in
"The power of data in quantum machine learning" (https://arxiv.org/abs/2011.01938)
and characterize the separation between classical and quantum kernels.
Args:
k_1: Quantum kernel Gram matrix
k_2: Classical kernel Gram matrix
normalization_lambda: normalization factor, must be close to zero
Returns:
geometric difference between the two kernel functions (float).
"""

In the paper, geometric difference between classical and quantum (g_cq ) is defined with k₁ = classical kernel and k₂ = quantum kernel. However, in your code implementation, k₁ is quantum and k₂ is classical. As the equation is asymmetric to the order of the matrices k₁ and k₂, could there be an error in the code implementation?

Thanks for taking the time to read this!

[Fradm98]

Hi @Squigspear, thanks for your insight, we fixed this issue in the branch package, we are going soon to merge it to the main one.

[qifengpan]

Hi,

I would like ask a follow up question for the function calculate_geometric_difference
In your implementation, you calculate the root of matrix in following form:

        # √K1
        k_1_sqrt = np.real(sqrtm(k_1))
        # √K2
        k_2_sqrt = np.real(sqrtm(k_2))

Since the result of sqrtm(k_1) can be a matrix containing complex number, direct cast it to a real number matrix would lose the imformation of imagenary part, and lead a wrong result. For e.g:

import tensorflow as tf
import numpy as np
from scipy.linalg import sqrtm
x = tf.convert_to_tensor([[1,2],[3,3]])
x_sqrt_scipy = np.real(sqrtm(x))
print(x_sqrt_scipy @ x_sqrt_scipy )

According to the definition, the result of x_sqrt_scipy @ x_sqrt_scipy should be x itself, but the output of this block is not identical to x. The reason behind is that matrix [1,2;3,3] is not semi-definite matrix. Therefore, the result of sqrtm contains complex number.
I'm not sure if I made some mistake for this, or misunderstanding for geometric difference. Or maybe you have other intention/implementation for that. Purely from math perspective, I would code this block as

        # √K1
        k_1_sqrt = sqrtm(k_1)
        # √K2
        k_2_sqrt = sqrtm(k_2)

Thanks for your time of reading/proofing, I'm happy to have further discussion for this.