If you cant do it right, do it random!

I was trying to understand how random projections can be used for dimensionality reduction instead of PCA. It was not obvious how random projections could even attempt to match projections computed through Eigen decomposition of the covariance matrix (which sounds computationally demanding!). After a discussion with my friend, I realised that random projections were essentially similar to finding the Eigen vector through the Power method. I was really amazed as I thought of Power method only as a numerical method.

Now for some philosophy :-)

I was trying to build some intuition / wondering where else this method could be applied. Probably, I am stretching my analogy, but I couldnt help notice a strong connection to life. We all want to find something that we are good at, a job/lifestyle that is well oriented ( Now, thats more like it :-) ) with our skills and interests. How do we find it ?
You start doing something random, see if that suits you, and based on how you feel, you update what you are doing. If you persevere (iterate) enough, your lifestyle will converge to what you truly want!

If you cant do it right, do it random! And hope that the dots will connect! :-)

(Yet another) History of life as we know it.

Benford's law

Counter-intuitive, highly interesting law ! :)

http://en.wikipedia.org/wiki/Benford%27s_law

Love ? probably :-)

Yet another conversation

1: Did you see Sonam in Masakalli song ? Tharumaru da :)
2: Already fell in love da.. Ava ippo un anni
1: Ada pavi :) .. Neraiya one side love-a irukku.. Namba side dhan onnum vara mattengudhu
2: You mean the priors are high, but the likelihoods are pretty low ? :)
1: Dei, I am a freq freak. Epdi ellam bayesian analysis panna try pannadha
2: Frequentist ellam seri.. Adhukkaga oru interval-a vidama un confidence interval-a pala actress-ku extend panriye .. Overa therila ?:)
1: Hee hee.. By the way, did you read about (..theorem..stat..math ..mumble..)

:-)

Top 10 Reasons to Become a Statistician

Top 10 Reasons to Become a Statistician ( of course, in random order )

7. Deviation is considered normal
4. We feel complete and sufficient
1. We are 'mean' lovers
9. Statisticians do it discretely and continuously
5. We are right 95% of the time
8. We can legally comment on posterior distributions
10.We may not be normal, but we are transformable
2. We never have to say we are certain
6. We are honestly significantly different
3. Statisticians may be dull but we have our moments

:-)

A math love song ! :-)

http://www.youtube.com/watch?v=UTby_e4-Rhg

Listen to the song first! Then enjoy the lyrics :)

The path of love is never smooth
But mine's continuous for you
You're the upper bound in the chains of my heart
You're my Axiom of Choice, you know it's true

But lately our relation's not so well-defined
And I just can't function without you
I'll prove my proposition and I'm sure you'll find
We're a finite simple group of order two

I'm losing my identity
I'm getting tensor every day
And without loss of generality
I will assume that you feel the same way

Since every time I see you, you just quotient out
The faithful image that I map into
But when we're one-to-one you'll see what I'm about
'Cause we're a finite simple group of order two

Our equivalence was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now everything is so complexified

When we first met, we simply connected
My heart was open but too dense
Our system was already directed
To have a finite limit, in some sense

I'm living in the kernel of a rank-one map
From my domain, its image looks so blue,
'Cause all I see are zeroes, it's a cruel trap
But we're a finite simple group of order two

I'm not the smoothest operator in my class,
But we're a mirror pair, me and you,
So let's apply forgetful functors to the past
And be a finite simple group, a finite simple group,
Let's be a finite simple group of order two
(Oughter: "Why not three?")

I've proved my proposition now, as you can see,
So let's both be associative and free
And by corollary, this shows you and I to be
Purely inseparable. Q. E. D.

The top ten things that math probability says about the real world

A great read:

http://www.stat.berkeley.edu/~aldous/Top_Ten/talk.html

Characteristic function and Fourier transform

We were trying to prove that Linear combination of I.I.D Gaussian Rvs produce Gaussian Rvs( not necessarily independent though).

The MGF of Gaussian distribution is as follows:




The usual trick is to calculate the MGF of the sum of the IID Gaussian RVs and after a bit of re-arranging, it can be shown that this MGF has the form of that of MGF of a Gaussian RV.Since the MGF of the sum has the form of that of a Gaussian RV, he concluded that the sum is indeed a RV. I was not quite convinced as to how the inverse would be unique. I asked my prof if he could give some intuition as to how the pdf-MGF pair was unique. He said that one can look at the characteristic function as the Fourier transform of the pdf. Oh boy ! I had never looked at this way. I feel this interpretation is cool. You can read more abt this at

http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/SHUTLER3/node2.html

Undergraduation in ECE is not entirely useless after all ! :-)

Copyright & information theory

Came across this at http://yann.lecun.com/ex/fun/index.html

"All copyrights and trademarks are property of their respective owners".
Bleeding-edge research in a new branch of mathematics (the application of non-standard elliptical analysis to information theory), has recently proved that the above sentence carries exactly 0 bit of information.
Researchers in yet another field of applied mathematics (applications of non-standard elliptical analysis to decision making) have conjectured that only the fear of lawsuits will drive people to write sentences with provably zero information content !


I was not able to track the original paper, anyway I think this is cool !

36 Methods of Mathematical Proof

36 Methods of Mathematical Proof
http://osl.cs.uiuc.edu/~ksen/proof.html

Some are particularly nice ;-)

Proof by obviousness
"The proof is so clear that it need not be mentioned."

Proof by general agreement
"All in favor?. . . "

Proof by intimidation
"Trivial."

Proof by mumbo-jumbo

" (B Ì P ) , $ (C Î W )

Proof by profanity
(example omitted)

Proof by illegibility
(scribble, scribble) QED

Proof by clever variable choice
"Let A be the number such that this proof works. . "

Proof by intuition
"I just have this gut feeling. . ."