I came across the proof of Perron Frobenius theorem here and it used Banach fixed point theorem. So I spent some time to understand the fixed point theorem from the wiki. Theorem Banach fixed point theorem considers a complete metric space (with metric ) and for any contraction mapping , there will exist a…
I thought I studied this before but can’t find a post from my blog. Byzantine generals problem was named by Leslie Lamport, the creator of LaTex. They consider a distributed consensus problem with the allegory that every general will need to agree on a broadcast message so that they can unify their attacks. It seems…
My Gigabyte hanged again, probably because of overheating. Chrome became super slow after rebooting. It was eventually fixed after reinstalling. That is, sudo apt purge google-chrome-stable rm -rf ~/.config/google-chrome/ and then reinstall by downloading Chrome from another browser. Update: the problem does not seem to be fixed somehow… But turned off hardware acceleration seems…
IMHO, the Levy process is just a generalization of the Poisson process. For example, consider $latex X_t$ as the number of arrivals by time $latex t$ as in the Poisson process. We expect the followings will be satisfied $latex X_t =0$ for $latex t = 0$ Random variables $latex X_{t_2} – X_{t_1}$ and $latex X_{t_4}-X_{t_3}$…
I think I knew how to show it before. But suddenly just realize I can’t. Say let’s define $latex F(w) = \frac{1}{2 \pi} \int f(t) e^{-i wt} dt$ and the inverse transform $latex \hat{f}(t)= \int F(w) e^{i w t}dw = f(t)$. So we expect $latex f(t)=\frac{1}{2 \pi} \int \int f(\tau) e^{-i w\tau} d\tau e^{iwt}dw$ $latex…
By Shannon-Nyquist theorem, a bandlimited signal can be completely reconstructed with $latex 2W$ samples per second. Think of the opposite way, this means that each second we have $latex 2W$ independent components that we can vary for each second. The total noise power per second is $latex 2 W (N_0/2) = N_0 W$ and as…
I guess I seem to be confused with the partition function from evidence. When they appear together, I don’t seem to be confused. But when they appeared in different contexts, I just got confused like falling into some illusion tricks. Given data and parameter , the evidence is simply . And , where the denominator…
It sounds like a misnomer to me. I probably will just call it a “vector” representation. It doesn’t have the “distributed” meaning of scattering information into different places. For example, to recognize a cat with “distributed” representation, we may distribute features into like “does it has a tail?”, “does it have four legs?”, and “does…
LeCun has a new course on deep learning this spring. I found two things he mentioned that worth jotting down. First, natural data lives in low-dimensional manifold. Probably I should have came across that before but it didn’t register earlier. Come to think of it. This is a very important fact. Second, as it is…
Asian disease problem illustrated that framing can alter one’s decision based on if we are emphasizing gain or loss. Prospect theory is just a fancy name to conjecture what happens when the utility function is indeed what economists believe.