Dear StackExchange users, while I was reading a research article on optimization approximation algorithm I came across the expression $O(2^{\log^{1-\epsilon}n})$. What is the meaning of the expression $\log^{1-\epsilon}n$?
I took logarithm as a natural logarithm and I write as $\log^{1-\epsilon}n=(\log n)^{1-\epsilon}$. Am I correct? Your comment is appreciated.
$\log^{1-\epsilon} n$ means $(\log(n))^a$ where $a$ is close to but strictly less than $1$. What exactly this $a$ is is presumably hard to determine and/or not important.
Returning to the original situation in which this appears as an exponent, whether the logarithm is $\ln$ or $\log_2$ is not apparent from the amount of context you provided, and this matters from the point of view of asymptotics (the constant factor becomes more than just a constant factor when it is inserted into the exponent).