Computing Kurdyka-Lojasiewicz (KL) exponent of sum of two KL functions.

[Vulture1991]

Two KL functions $f_1:\mathbb{R}^n\rightarrow \mathbb{R}$ and $f_2:\mathbb{R}^n\rightarrow \mathbb{R}$ are given which have KL exponent $\alpha_1$ and $\alpha_2$. What is the KL exponent of $f_1+f_2$?

 

[jvicens]

The KL exponent of $f_1+f_2$ is the maximum of $\alpha_1$ and $\alpha_2$. This is because the KL exponent measures the rate of convergence for the KL divergence between two probability distributions, and when two functions are added, their KL divergence is also added. Therefore, the KL exponent of $f_1+f_2$ should be the maximum of the KL exponents of $f_1$ and $f_2$. This means that the convergence rate for the KL divergence between the distributions of $f_1+f_2$ and the true distribution will be determined by the function with the faster convergence rate, which is represented by the larger KL exponent.