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Tivadar Danka

@TivadarDanka

Jul 5 • 6 tweets • 2 min read • Read on X

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If I had to learn Math for Machine Learning from scratch, this is the roadmap I would follow:

1. Linear Algebra

These are non-negotiables:

• Vectors
• Matrices
• Equations
• Factorizations
• Matrices and graphs
• Linear transformations
• Eigenvalues and eigenvectors

Now you've learned how to represent and transform data.

2. Calculus

Don't skip any of these:

• Series
• Functions
• Sequences
• Integration
• Optimization
• Differentiation
• Limits and continuity

Now you understand the math behind algorithms like gradient descent and get a better feeling of what optimization is.

3. Multivariable Calculus

Here's how you start:

• Multivariable functions
• Derivatives and gradients
• Optimization in multiple variables

In real life, neural networks involve functions with thousands of parameters, and you need to know how they change together.

4. Probability Theory

Learn this:

• Distributions
• Expected values
• Random variables

Now you know how to model uncertainty, learn from data, and make predictions.

If you are looking for a single resource containing all these topics, I have good news for you.

I have packed 20 years of math studies into 700 pages full of intuitive and application-oriented lessons, the ultimate learning resource for you.

Get it now: amazon.com/Mathematics-Ma…

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Tivadar Danka

@TivadarDanka

Jul 30

One of the coolest ideas in mathematics is the estimation of a shape's area by throwing random points at it.

Don't believe this works? Check out the animation below, where I show the method on the unit circle. (Whose area equals to π.)

Here is what's behind the magic:

Let's make this method precise!

The first step is to enclose our shape S in a square.

You can imagine this as a rectangular dartboard.

Now, we select random points from the board and count how many hit the target.

Again, you can imagine this as closing your eyes, doing a 360° spin, then launching a dart.

(Suppose that you always hit the board. Yes, I know. But in math, reality doesn't limit imagination.)

Read 14 tweets

Tivadar Danka

@TivadarDanka

Jul 29

What is common between the Fourier series and the Cartesian coordinate system?

More than you think: they are (almost) the same.

Let me explain why!

Let's start with the basics: the inner product.

In the Euclidean plane, it can be calculated using the "magnitude x magnitude x cosine" formula, also known as the geometric definition.

Now, let's project x to y!

With basic trigonometry, we can see that the inner product is related to the length of the projection.

Read 15 tweets

Tivadar Danka

@TivadarDanka

Jul 27

One of my favorite formulas is the closed-form of the geometric series.

I am amazed by its ubiquity: whether we are solving basic problems or pushing the boundaries of science, the geometric series often makes an appearance.

Here is how to derive it from first principles:

Let’s start with the basics: like any other series, the geometric series is the limit of its partial sums.

Our task is to find that limit.

There is an issue: the number of terms depends on N.

Thus, we can’t take the limit term by term.

Read 13 tweets

Tivadar Danka

@TivadarDanka

Jul 26

Matrices + the Gram-Schmidt process = magic.

This magic is called the QR decomposition, and it's behind the famous eigenvalue-finding QR algorithm.

Here is how it works:

In essence, the QR decomposition factors an arbitrary matrix into the product of an orthogonal and an upper triangular matrix.

(We’ll illustrate everything with the 3 x 3 case, but everything works as is in general as well.)

First, some notations:

Every matrix can be thought of as a sequence of column vectors.

Trust me, this simple observation is the foundation of many-many Eureka moments in mathematics.

Read 13 tweets

Tivadar Danka

@TivadarDanka

Jul 25

Summing numbers is more exciting than you think.

For instance, summing the same alternating sequence of 1s and (-1)s can either be zero or one, depending on how we group the terms. What's wrong?

I'll explain. Enter the beautiful world of infinite series.

Let’s go back to square one: the sum of infinitely many terms is called an infinite series. (Or series in short.)

Infinite series form the foundations of mathematics.

Do infinite series make sense? Sure.

Take a look at the geometric series: summing the positive powers of 1/2 adds up to one.

Here is a visual proof to convince you.

Read 24 tweets

Tivadar Danka

@TivadarDanka

Jul 24

I have spent at least 50% of my life studying, practicing, and teaching mathematics.

The most common misconceptions I encounter:

• Mathematics is useless
• You must be good with numbers
• You must be talented to do math

These are all wrong. Here's what math is really about:

Let's start with a story.

There’s a reason why the best ideas come during showers or walks. They allow the mind to wander freely, unchained from the restraints of focus.

One particular example is graph theory, born from the regular daily walks of the legendary Leonhard Euler.

Here is the map of Königsberg (now known as Kaliningrad, Russia), where these famous walks took place.

This part of the city is interrupted by several rivers and bridges.

(I cheated a little and drew the bridges that were there in Euler's time, but not now).

Read 17 tweets

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