Maths-as-Code for Machine Learning

Turn the maths you see in ML papers into programmer-friendly code patterns.

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Why “Maths-as-Code”?

ML papers speak in symbols. Engineers speak in loops, arrays, and functions. This guide acts as the Rosetta Stone between the two:

Keep this page open when reading a paper; copy the snippets into a notebook and try them on a tiny dataset.

Mental model: A tensor is just an n-dimensional array.

Math	Meaning	Python / NumPy	Common Use
`x`	Scalar (0-D)	`x = 3.14`	Learning rate, bias
`𝐱 = [x₁, …, xₙ]`	Vector (1-D)	`x = np.array([1,2,3])`	Features, weights
`𝐗 = [[xᵢⱼ]]`	Matrix (2-D)	`X = np.array([[1,2],[3,4]])`	Weights, batches
`𝓧`	Tensor (n-D)	`np.array([...])`	Images, embeddings
`𝐗ᵀ`	Transpose	`X.T`	Switch row/col orientation

import numpy as np
X = np.array([[1, 2], [3, 4]])
XT = X.T

…and thought “that looks like pseudocode,” this guide is for you.

Math	Meaning	Python / NumPy	ML Example
`x`	Scalar — single number	`x = 3.14`	Learning rate, bias
`𝐱 = [x₁, …, xₙ]`	Vector — ordered list	`x = np.array([1,2,3])`	Features, weights
`𝐗 = [[xᵢⱼ]]`	Matrix — 2D grid	`X = np.array([[1,2],[3,4]])`	Weight matrices
`𝓧`	Tensor — n-D array	`np.array([...])` / `tf.constant([...])`	Images, embeddings
`𝐗ᵀ`	Transpose — swap rows/cols	`X.T`	Switching dimensions

💡 Intuition:
Scalars → Vectors → Matrices → Tensors
Every ML data structure is one of these.

import numpy as np
X = np.array([[1, 2], [3, 4]])
print("Matrix:", X)
print("Transpose:", X.T)

Mental model: Σ means loop and add. E[X] means average (expected value).

Math	Meaning	Code Equivalent	ML Example
`Σᵢ₌₁ⁿ xᵢ`	Sum elements	`np.sum(x)`	Total loss
`E[X] = (1/n) Σ xᵢ`	Expectation / mean	`np.mean(x)`	Batch average
`∏ᵢ₌₁ⁿ xᵢ`	Product	`np.prod(x)`	Likelihood
`Var(X)`	Variance	`np.var(x)`	Feature scaling

import numpy as np
x = np.array([1, 2, 3, 4])
np.sum(x), np.mean(x), np.var(x)

Mental model: Matrices transform vectors. Neural networks are chains of transforms + non-linearities.

Math	Meaning	Code Equivalent	ML Use
`𝐱·𝐲 = Σ xᵢyᵢ`	Dot product	`np.dot(x, y)`	Similarity, regression
`𝐗𝐘`	Matrix multiply	`X @ Y` or `np.matmul(X,Y)`	Forward pass
`		𝐱		₂ = √Σ xᵢ²`	L2 norm	`np.linalg.norm(x)`	Normalization
`𝐈`	Identity matrix	`np.eye(n)`	Initialization
`𝐗⁻¹`	Inverse	`np.linalg.inv(X)`	Solving linear systems

import numpy as np
X = np.array([[1, 2], [3, 4]])
W = np.array([[0.5], [0.2]])
y = X @ W
np.linalg.norm(y)