Tivadar Danka Profile picture
Apr 13, 2021 9 tweets 3 min read Read on X
Convolution is not the easiest operation to understand: it involves functions, sums, and two moving parts.

However, there is an illuminating explanation — with probability theory!

There is a whole new aspect of convolution that you (probably) haven't seen before.

🧵 👇🏽
In machine learning, convolutions are most often applied for images, but to make our job easier, we shall take a step back and go to one dimension.

There, convolution is defined as below.
Now, let's forget about these formulas for a while, and talk about a simple probability distribution: we toss two 6-sided dices and study the resulting values.

To formalize the problem, let 𝑋 and 𝑌 be two random variables, describing the outcome of the first and second toss.
Just for fun, let's also assume that the dices are not fair, and they don't follow a uniform distribution.

The distributions might be completely different.

We only have a single condition: 𝑋 and 𝑌 are independent.
What is the distribution of the sum 𝑋 + 𝑌?

Let's see a simple example. What is the probability that the sum is 4?

That can happen in three ways:

𝑋 = 1 and 𝑌 = 3,
𝑋 = 2 and 𝑌 = 2,
𝑋 = 3 and 𝑌 = 1.
Since 𝑋 and 𝑌 are independent, the joint probability can be calculated by multiplying the individual probabilities together.

Moreover, the three possible ways are disjoint, so the probabilities can be summed up.
In the general case, the formula is the following.

(Don't worry about the index going from minus infinity to infinity. Except for a few terms, all others are zero, so they are eliminated.)

Is it getting familiar?
This is convolution.

We can immediately see this by denoting the individual distributions with 𝑓 and 𝑔.

The same explanation works when the random variables are continuous, or even multidimensional.

Only thing that is required is independence.
Even though images are not probability distributions, this viewpoint helps us make sense of the moving parts and the everyone-with-everyone sum.

If you would like to see an even simpler visualization, just think about this:

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More from @TivadarDanka

Jul 3
Behold one of the mightiest tools in mathematics: the camel principle.

I am dead serious. Deep down, this tiny rule is the cog in many methods. Ones that you use every day.

Here is what it is, how it works, and why it is essential. Image
First, the story.

The old Arab passes away, leaving half of his fortune to his eldest son, third to his middle son, and ninth to his smallest.

Upon opening the stable, they realize that the old man had 17 camels. Image
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So, they turn to the wise neighbor for advice. Image
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Jul 3
The single biggest argument about statistics: is probability frequentist or Bayesian?

It's neither, and I'll explain why.

Buckle up. Deep-dive explanation incoming. Image
First, let's look at what is probability.

Probability quantitatively measures the likelihood of events, like rolling six with a dice. It's a number between zero and one.

This is independent of interpretation; it’s a rule set in stone. Image
In the language of probability theory, the events are formalized by sets within an event space.

The event space is also a set, usually denoted by Ω.) Image
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Jul 2
Matrix multiplication is not easy to understand.

Even looking at the definition used to make me sweat, let alone trying to comprehend the pattern. Yet, there is a stunningly simple explanation behind it.

Let's pull back the curtain! Image
First, the raw definition.

This is how the product of A and B is given. Not the easiest (or most pleasant) to look at.

We are going to unwrap this. Image
Here is a quick visualization before the technical details.

The element in the i-th row and j-th column of AB is the dot product of A's i-th row and B's j-th column. Image
Read 16 tweets
Jul 1
The single most undervalued fact of linear algebra: matrices are graphs, and graphs are matrices.

Encoding matrices as graphs is a cheat code, making complex behavior simple to study.

Let me show you how! Image
If you looked at the example above, you probably figured out the rule.

Each row is a node, and each element represents a directed and weighted edge. Edges of zero elements are omitted.

The element in the 𝑖-th row and 𝑗-th column corresponds to an edge going from 𝑖 to 𝑗.
To unwrap the definition a bit, let's check the first row, which corresponds to the edges outgoing from the first node. Image
Read 18 tweets
Jun 30
In calculus, going from a single variable to millions of variables is hard.

Understanding the three main types of functions helps make sense of multivariable calculus.

Surprisingly, they share a deep connection. Let's see why! Image
In general, a function assigns elements of one set to another.

This is too abstract for most engineering applications. Let's zoom in a little! Image
As our measurements are often real numbers, we prefer functions that operate on real vectors or scalars.

There are three categories:

1. vector-scalar,
2. vector-vector,
3. and scalar-vector. Image
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Jun 30
Neural networks are stunningly powerful.

This is old news: deep learning is state-of-the-art in many fields, like computer vision and natural language processing. (But not everywhere.)

Why are neural networks so effective? I'll explain. Image
First, let's formulate the classical supervised learning task!

Suppose that we have a dataset D, where xₖ is a data point and yₖ is the ground truth. Image
The task is simply to find a function g(x) for which

• g(xₖ) is approximately yₖ,
• and g(x) is computationally feasible.

To achieve this, we fix a parametrized family of functions. For instance, linear regression uses this function family: Image
Read 19 tweets

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