## Batch Normalization

Posted on Thu 16 February 2023 in Mathematics • Tagged with Mathematics, Machine Learning, Deep Learning

$$\require{physics} \newcommand{B}{\mathcal{B}}$$

Batch Normalization was proposed by Ioffe and Szegedy in 2015, and it spawned several normalization techniques that are used in SOTA models today (layer norm, weight norm, etc). Batch normalization normalizes the output of each layer based on the mean and variance of the …

## L2 regularization intuition

Posted on Sun 22 January 2023 in Mathematics • Tagged with Mathematics, Machine Learning, Deep Learning

A nice intuition for L2 regularization comes from having a prior on the distribution of parameters: the prior assumes that the parameters are close to zero. Let's assume that the prior is $\mathcal{N}(0, \Sigma)$. The MAP estimate of the parameters would then be

## The JEE never fails to amaze

Posted on Mon 19 September 2022 in Mathematics • Tagged with Mathematics, JEE

I promised I wouldn't look at the JEE 2022 paper, but couldn't help sneaking a peek. As always, the professors currently making life difficult occasionally get a chance to make the life of all of India's students difficult, and they never cease to amaze.

Welp, this will be a short …

## A Note on Conditional Probability

Posted on Sat 25 December 2021 in Mathematics • Tagged with Mathematics

$\newcommand{\cE}{\mathbf{E}(#1\ |\ #2)}$$\newcommand{\cP}{\mathbf{P}(#1\ |\ #2)}$$\renewcommand{\P}{\mathbf{P}(#1)}$$\newcommand{\E}{\mathbf{E}(#1)}$$\newcommand{\F}{\mathcal{F}}$$\newcommand{\G}{\mathcal{G}}$$\newcommand{\ind}{\mathbf{1}_{#1}}$ To motivate this note, I’ll pose the following …

## A Note on Random Variables

Posted on Fri 03 December 2021 in Mathematics • Tagged with Mathematics

## ISC 2020 Analysis: An application of Data Science and Statistics

Posted on Sat 11 July 2020 in Mathematics • Tagged with Mathematics, ISC, Data Science

The ISC Exam results were released on 10th June, 3 PM IST. In this article, I'll be analyzing the results of the 117 students in the science stream of my school and showing how they performed. I'll also weave a story with the data and point out things that could …

## Solving the African Integral (from YG file cover)

Posted on Thu 09 July 2020 in Mathematics • Tagged with Mathematics

A long time ago, I gave my friend (Let's call him C) an integral to solve. He came back to me a few days later and the conversation went something like this:

C: Debu, I couldn't solve that African integral you gave me.
Me: Ok, but why are you calling …

## A Compilation of hard limits

Posted on Fri 05 June 2020 in Mathematics • Tagged with Mathematics

This list consists of the limits that I found most challenging.

1. $$\lim_{n \to \infty} \left( \frac{n!}{n^n} \right) ^\frac 1n$$
2. $$\lim_{x \to 0} \left( \frac{1}{\ln(x + \sqrt{x^2+1})} - \frac 1{\ln(x+1)} \right)$$
3. \lim_{n \to \infty} \frac{n + n^2 …

## Limit involving higher order infinitesimals

Posted on Fri 05 June 2020 in Mathematics • Tagged with Mathematics

Simple limit problems consist of the form $\lim_{x \to 0}\frac{O_1(x)O_2(x)..}{O_a(x)O_b(x)..}$, such as $\lim_{x \to 0} \frac{\sin 3x \tan 2x \tan^{-1} 5x}{x^2 \ln(1+x)}$. Here, the infinitesimals are well defined and cancel out easily. Some …

## COVID-19 USA Analysis: effects of the lack of lockdown

Posted on Fri 29 May 2020 in Mathematics • Tagged with Mathematics, COVID 19

The USA currently stands at 1.76 million COVID-19 cases. That's more than the next 5 nations combined. A large number of these cases are due to government inaction against the virus. The lack of a concerted lockdown across the country is also to blame. Here's a graph showing the …

## COVID 19 Regression analysis Update

Posted on Tue 19 May 2020 in Mathematics • Tagged with Mathematics, COVID 19

The Previous COVID regression analysis was fairly accurate. However, the opening of lockdown offset the statistics a bit and now there are more number of projected cases. Here is a recomputation of the statistics, which projects an average of 172,000 cases by June 1 and 520,000 cases overall …

## No square ends in 3

Posted on Fri 15 May 2020 in Mathematics • Tagged with Mathematics

This is an interesting number theory fact that seems strange when taken at face value. Here's a small proof of it: