Journal · Jun 03, 2026
Build ML Model with Pen and Paper
Build ML Model with Pen and Paper
Build ML Model with Pen and Paper
A gentle introduction to ML and the underlying math — from matrix multiplication to the essence of deep learning.
A gentle introduction to ML and the underlying math — from matrix multiplication to the essence of deep learning.
Overview
Overview
Overview
A deep learning model takes data as input and outputs a prediction: identifying the mushroom in a picture, picking the next word in a sentence, or estimating a house’s price. It all comes down to three critical pieces:
A deep learning model takes data as input and outputs a prediction: identifying the mushroom in a picture, picking the next word in a sentence, or estimating a house’s price. It all comes down to three critical pieces:
Input – your data (say, an image), converted into a matrix
Weights – another matrix that gets multiplied with the input to form a prediction
Loss function – how far that prediction lands from the truth
Input – your data (say, an image), converted into a matrix
Weights – another matrix that gets multiplied with the input to form a prediction
Loss function – how far that prediction lands from the truth
It sounds complex, but it’s mostly matrix multiplication with a little calculus mixed in.
It sounds complex, but it’s mostly matrix multiplication with a little calculus mixed in.
Linear Algebra & Calculus Review
Linear Algebra & Calculus Review
Linear Algebra & Calculus Review
The part of linear algebra relevant to deep learning is matrix operations.
The part of linear algebra relevant to deep learning is matrix operations.
NOTE: GPUs became a critical part of deep learning because they pack thousands of small cores that run arithmetic in parallel – exactly what matrix multiplication needs.
NOTE: GPUs became a critical part of deep learning because they pack thousands of small cores that run arithmetic in parallel – exactly what matrix multiplication needs.
Even though modern tools such as PyTorch and TensorFlow are relatively forgiving for engineers without a strong math background, it’s still worth reviewing the basics to build intuition.
Even though modern tools such as PyTorch and TensorFlow are relatively forgiving for engineers without a strong math background, it’s still worth reviewing the basics to build intuition.
Matrix Multiplication
Matrix Multiplication
Matrix Multiplication
The building block is the dot product of two vectors. The result is the sum of element-wise products between a horizontal row vector and a vertical column vector.
The building block is the dot product of two vectors. The result is the sum of element-wise products between a horizontal row vector and a vertical column vector.
[1 2 3] · [4 5 6] = 1·4 + 2·5 + 3·6 = 32
[1 2 3] · [4 5 6] = 1·4 + 2·5 + 3·6 = 32
Note what happened to the shape: two vectors of length 3 collapse into one scalar, a single number. The vectors must be the same length, since every element needs a partner.
Note what happened to the shape: two vectors of length 3 collapse into one scalar, a single number. The vectors must be the same length, since every element needs a partner.
Matrix multiplication is just this dot product done many times. Each entry in the result is the dot product of a row from the left and a column from the right.
Matrix multiplication is just this dot product done many times. Each entry in the result is the dot product of a row from the left and a column from the right.