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

$$\begin{align} \theta_{\text{MAP …

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$\newcommand{\triple}{(\Omega, \mathcal{F}, \mathbf{P})}$$\newcommand{\P}{\mathbf{P}}$

Recap

If you remember the note on random variables, then this picks up exactly where that left off.

Definition 1.1: A Measurable space $(X,\Sigma)$ consists of a set $X$ and a $\sigma$-algebra $\Sigma$ defined on $X …

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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 …

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

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$\newcommand{\triple}{(\Omega, \mathcal{F}, \mathbf{P})}$$\newcommand{\P}{\mathbf{P}}$ This note on random variables follows as a result of confusing notation in several math textbooks. I'll explain random variables (in measure theoretic terms) as verbosely as I can, and then prove some results. This article assumes that the …

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I'll begin this article by brushing up a few definitions:

A function is a mapping between two sets: the domain D and the codomain C.

It's very important to note here that the function is the mapping itself, and not an element in the codomain or the domain. The function …

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$\newcommand{\pdv}[2]{\frac{\partial{#1}}{\partial{#2}}}$ $\newcommand{\ah}{\pmb{a} + \pmb{h}}$ $\newcommand{\a}{\pmb{a}}$ $\newcommand{\h}{\pmb{h}}$

The Jacobian Matrix

Consider a function that maps reals to reals, $f:\Bbb{R} \to \Bbb{R}$. The linear approximation of this function is given by $$f(a …

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Just after 10th and as I was beginning my JEE Preparation, I felt the need for an extensible note-taking apparatus. Online notes would be too much trouble and would keep me hooked to the computer. Notebooks were also difficult, as I wanted my notes to be extensible; adding pages in …

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This beauty came in the 2nd September shift 2 paper:

Let $A = \{ X = (x, y, z)^T : PX = 0 \text{ and } x^2+y^2+z^2=1 \}$, where $$P = \left[ \begin{array}{l l l}1&2&1 \\ -2&3&-4 \\ 1&9&-1 \end{array}\right]$$ then the …

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This is a small set of posts that come into the category of "notes": self-explanations of concepts that I have recently picked up and found interesting.

Lagrange Interpolation

Lagrange Interpolation is a concept that allows us to find a polynomial of least degree passing through a given set of points …

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My old man always used to say that there are two ways to solve a problem: there's the horse way, and there's the donkey way. The donkey way involves tedious calculations, whereas the horse way 'cuts' through the problem. Take this problem as an example:

For each positive integer $n …

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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 …

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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 …

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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 …

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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 …

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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 …

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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 …

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

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This is a regression analysis attempt for the COVID-19 spread data. The graphs represent total cases per day. The Orange graph is USA, the smaller graph on the left is China and the graph on the right is India. A Standard gaussian curve of the form $Ae^{-b(x-c)^2 …

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Define $ f(x) = ax^2 + bx + c , a,b,c \in \mathbb{R}$ and $ f^n(x) = f(f^{n-1}(x)), n>1 $. Prove that the real roots of $ f^n(x) $ are symmetric about the vertical line passing through vertex i.e. $ x = \frac{-b}{2a} $

This seems like …

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