Reading math notation: sets, functions, indices, sums and products

Before this lesson: school algebra (solve $2x + 3 = 7$). Lesson 01 is optional. Time: about 55 minutes, plus the notebook. This lesson has many symbols: do not try to learn the tables at once. Add the recall cards to your review; they bring the symbols back over the next weeks.

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Why this matters #

Here is the mean squared error of a linear model, as it appears in many books and papers:

To read it you need sums ($\sum$), indices ($x^{(i)}$, $y^{(i)}$), vectors and the transpose ($w^\top x$), functions of parameters ($L(w, b)$) and the idea of a mean ($\frac{1}{m}\sum$). All later tracks write their ideas this way: the gradient in Calculus, the likelihood in Probability and Statistics, attention in Linear Algebra lesson 15. Notation is the entry ticket. The good news: the set of symbols is small, and every symbol is a short sentence in plain words.

1. Predict first #

2. The picture #

How to read a formula #

Take the prediction of a linear model, $\hat y = w^\top x + b$. Read it in three passes.

1. Say it aloud: "y hat equals w transpose x plus b".
2. Find the roles. The input $x$ changes from example to example. The parameters $w$ and $b$ are fixed while we make a prediction (training chooses them). The output $\hat y$ is what we compute. The hat says "an estimate".
3. Check the shapes. If $x$ has $n$ entries, $w$ must have $n$ entries too, and $w^\top x$ is one number. So $\hat y$ is one number.

When a formula looks hard, do these three passes, then compute it for a tiny case (two or three numbers). The figures below do this for the most common symbols.

Sums: $\Sigma$ is a loop #

The capital Greek letter $\Sigma$ (sigma) means "add up". In $\sum_{i=1}^{n} a_i$:

• $i$ is the index (the loop variable). It starts at the lower bound 1 (below $\Sigma$) and goes up by 1 to the upper bound $n$ (above $\Sigma$);
• $a_i$ is the term: for each value of $i$, compute it and add it to the total.

For example,

In Python it is a for loop with a running total.

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Indices: which number in the table? #

Data in ML is a table. Each row is one example (a sample), for example one house. Each column is one feature, for example the area. With $m$ samples and $n$ features the table is an $m \times n$ matrix $X$.

• $X_{ij}$ is the entry in row $i$ and column $j$: feature $j$ of sample $i$. Rows come first.
• $x^{(i)}$ is sample $i$: the whole row $i$, a list (vector) of $n$ numbers. The superscript in parentheses is a label, not a power.
• $x^{(i)}_j$ is entry $j$ of sample $i$, the same number as $X_{ij}$.

Math counts from 1. Python counts from 0. So $X_{ij}$ is X[i-1, j-1] in NumPy (lesson 03 explains the syntax).

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max and argmax: a value and a place #

For a function $f$ on an interval, $\max_x f(x)$ is the largest value of $f$: a height. The $\arg\max_x f(x)$ is the set of inputs where this largest value is reached: places on the horizontal axis. "Arg" is short for argument, another word for input.

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3. Definitions and main results #

Numbers and sets #

A set is a collection of distinct objects, its elements. Order and repeats do not matter: $\{1, 2, 2, 3\} = \{3, 1, 2\}$, a set with 3 elements.

Symbol	Read it as	Example
$\mathbb{N}$	the natural numbers	$\{1, 2, 3, \dots\}$ (some books include 0)
$\mathbb{Z}$	the integers	$\{\dots, -2, -1, 0, 1, 2, \dots\}$
$\mathbb{R}$	the real numbers	$-1.5$, $0$, $\sqrt 2$, $\pi$
$x \in A$	$x$ is an element of $A$	$2 \in \{1, 2, 3\}$
$x \notin A$	$x$ is not an element of $A$	$5 \notin \{1, 2, 3\}$
$A \subseteq B$	$A$ is a subset of $B$: every element of $A$ is in $B$	$\{1, 2\} \subseteq \{1, 2, 3\}$
$A \cup B$	union: in $A$ or in $B$ (or both)	$\{1, 2\} \cup \{2, 5\} = \{1, 2, 5\}$
$A \cap B$	intersection: in $A$ and in $B$	$\{1, 2\} \cap \{2, 5\} = \{2\}$
$A \setminus B$	difference: in $A$ but not in $B$	$\{1, 2\} \setminus \{2, 5\} = \{1\}$
$\emptyset$	the empty set	$\{1\} \cap \{2\} = \emptyset$
$\lvert A \rvert$	the size (number of elements)	$\lvert \{1, 2, 2, 3\} \rvert = 3$
$A \times B$	Cartesian product: all pairs $(a, b)$	$\{1, 2\} \times \{0\} = \{(1, 0), (2, 0)\}$

Set-builder notation describes a set by a rule: $\{x \in \mathbb{R} : x > 0\}$ is read "the set of all real $x$ such that $x > 0$". The colon (some books use a bar $\mid$) means "such that".

Intervals are sets of real numbers between two ends. A square bracket includes the end, a round bracket excludes it: $[a, b] = \{x \in \mathbb{R} : a \le x \le b\}$, $(a, b] = \{x \in \mathbb{R} : a < x \le b\}$, and $(0, \infty) = \{x \in \mathbb{R} : x > 0\}$ ($\infty$ is never included).

$\mathbb{R}^n$ is the set of all lists of $n$ real numbers, $\mathbb{R}^n = \mathbb{R} \times \dots \times \mathbb{R}$. So $(3, -1) \in \mathbb{R}^2$ is a point of the plane, and a house with 3 features is a point of $\mathbb{R}^3$. The complement of $A$ is $A^c = U \setminus A$, where $U$ is the set of everything we talk about (the universe).

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The rule in the caption is the inclusion–exclusion rule for two finite sets. Why it is true: $\lvert A \rvert + \lvert B \rvert$ counts each element of $A \cap B$ twice (once in $A$, once in $B$) and every other element of $A \cup B$ once. Subtracting $\lvert A \cap B \rvert$ removes the second count. $\blacksquare$

Functions #

A function $f: X \to Y$ assigns to each input $x \in X$ exactly one output $f(x) \in Y$. $X$ is the domain (allowed inputs), $Y$ is the codomain (the type of the outputs). The range (or image) is the set of outputs that really occur: $\{f(x) : x \in X\}$.

• $f: \mathbb{R} \to \mathbb{R}$, $x \mapsto x^2$ is read "$f$ maps $x$ to $x^2$". Its codomain is $\mathbb{R}$, but its range is $[0, \infty)$, because a square is never negative.
• $f$ versus $f(x)$. $f$ is the function (the whole rule); $f(x)$ is one number, the output at $x$.
• A rule that sends one input to two outputs is not a function. "$x \mapsto$ a number whose square is $x$" sends 4 to both 2 and $-2$; the function $\sqrt{x}$ fixes this by choosing the non-negative one.
• Several inputs. $f: \mathbb{R}^2 \to \mathbb{R}$, $f(x_1, x_2) = x_1 + 2x_2$ takes two numbers (one point of the plane) and gives one number.
• Parameters after a semicolon. $f(x; \theta)$ is a function of the input $x$ whose shape is set by the parameters $\theta$. A linear model is $f(x; w, b) = w^\top x + b$. Training changes $\theta$; prediction changes $x$. A model is a function.

Composition. $(f \circ g)(x) = f(g(x))$: first apply $g$, then apply $f$ to the result. Read $f \circ g$ from right to left: the function next to $x$ acts first. In general $f \circ g \ne g \circ f$. A neural network is a composition of layers: $f(x) = f_3(f_2(f_1(x)))$, written $f_3 \circ f_2 \circ f_1$.

Inverse. If every $y$ in the range comes from exactly one $x$, then $f$ has an inverse function $f^{-1}$ with $f^{-1}(f(x)) = x$ for every $x$ in the domain. For $f(x) = 2x + 1$: solve $y = 2x + 1$ for $x$ to get $f^{-1}(y) = (y - 1)/2$. The function $x \mapsto x^2$ on $\mathbb{R}$ has no inverse, since 2 and $-2$ both give 4; on the domain $[0, \infty)$ its inverse is $\sqrt{y}$. Note: $f^{-1}$ is not $1/f$.

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Indices, vectors and matrices #

• A vector $x \in \mathbb{R}^n$ is a list of $n$ numbers $x_1, \dots, x_n$ (subscripts count the entries). In this site, as in most ML books, a vector is a column; to save space we write $x = (x_1, \dots, x_n)$ in a line.
• The transpose $x^\top$ turns the column into a row. For two vectors of the same length, $w^\top x = w_1 x_1 + \dots + w_n x_n = \sum_{j=1}^{n} w_j x_j$, one number (the dot product; books also write $w \cdot x$ or $\langle w, x \rangle$).
• A matrix $A \in \mathbb{R}^{m \times n}$ has $m$ rows and $n$ columns. Rows first: $A_{ij}$ is in row $i$, column $j$. In NumPy, A.shape == (m, n) and $A_{ij}$ is A[i-1, j-1].
• The data matrix $X \in \mathbb{R}^{m \times n}$ has the samples as rows: $X_{ij} = x^{(i)}_j$. The labels (targets) form a vector $y = (y^{(1)}, \dots, y^{(m)})$.
• $x^{(i)}$ is not a power. $x^{(2)}$ is sample 2. A square is written without parentheses, $x^2$, or, for the square of an entry, $\bigl(x^{(i)}_j\bigr)^2$.

Linear Algebra lesson 01 shows what matrices do; here we only need to read them.

Sums #

Definition. For numbers $a_m, a_{m+1}, \dots, a_n$ (with integers $m \le n$),

If $n < m$ there are no terms, and the empty sum is 0. The index is a dummy variable: the name does not matter, $\sum_{i=1}^{n} a_i = \sum_{k=1}^{n} a_k$, and $i$ has no meaning outside the sum. A sum over a set is written $\sum_{i \in S} a_i$: one term for each element of $S$.

Rules. For numbers $a_i$, $b_i$ and a constant $c$ (a number that does not depend on $i$):

Rule	Formula	Small check
constant term	$\sum_{i=1}^{n} c = n c$	$\sum_{i=1}^{4} 5 = 20$
linearity	$\sum_{i=1}^{n} (c\,a_i + b_i) = c \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i$	$\sum_{i=1}^{3} (2i + 1) = 2 \cdot 6 + 3 = 15$
splitting	$\sum_{i=1}^{n} a_i = \sum_{i=1}^{k} a_i + \sum_{i=k+1}^{n} a_i$ for $1 \le k < n$	$\sum_{i=1}^{4} i = 3 + 7$
shifting the index	$\sum_{i=1}^{n} a_i = \sum_{k=0}^{n-1} a_{k+1}$ (put $k = i - 1$)	the Python loop for k in range(n)
swapping a double sum	$\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} = \sum_{j=1}^{n} \sum_{i=1}^{m} a_{ij}$	add a table by rows or by columns
product of sums	$\sum_{i=1}^{m} \sum_{j=1}^{n} a_i b_j = \bigl(\sum_{i=1}^{m} a_i\bigr) \bigl(\sum_{j=1}^{n} b_j\bigr)$	$a = (1, 2)$, $b = (1, 2, 3)$: $18 = 3 \cdot 6$

Proof of linearity. Write the sum out and regroup (finite sums can be reordered):

The constant-term rule is the case $a_i = 1$, $b_i = 0$: $\sum c = c \cdot n$. $\blacksquare$

Proof of the product of sums. The factor $a_i$ does not depend on $j$, so linearity (pull out a constant) gives $\sum_{j} a_i b_j = a_i \sum_{j} b_j$. Then $B = \sum_j b_j$ does not depend on $i$, so $\sum_i a_i B = B \sum_i a_i$. Together: $\sum_i \sum_j a_i b_j = \sum_i a_i B = \bigl(\sum_i a_i\bigr) B$. $\blacksquare$

A warning. The product of sums has all pairs $(i, j)$. The sum $\sum_i a_i b_i$ has only the pairs with $i = j$. They are different: for $a = (1, 2)$, $b = (3, 4)$ we get $\sum_i a_i b_i = 3 + 8 = 11$, but $\bigl(\sum a_i\bigr)\bigl(\sum b_i\bigr) = 3 \cdot 7 = 21$.

Two closed forms. $\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}$ and $\sum_{i=1}^{n} i^2 = \dfrac{n(n+1)(2n+1)}{6}$.

Why $1 + 2 + \dots + n = n(n+1)/2$

1. Call the sum $T$. Write it forwards and backwards: $T = 1 + 2 + \dots + n$ and $T = n + (n-1) + \dots + 1$.
2. Add the two lines term by term. Each pair adds to $n + 1$: $1 + n$, $2 + (n - 1)$, and so on.
3. There are $n$ pairs, so $2T = n(n+1)$, and $T = n(n+1)/2$. $\blacksquare$

For $n = 100$: $T = 5050$. Lesson 09 proves the same formula a second way, by induction; the same method proves the formula for $\sum i^2$. The notebook checks both formulas for many $n$.

Sums in ML. The dot product $w^\top x = \sum_{j=1}^{n} w_j x_j$ (one sum over features). The mean $\bar x = \frac{1}{m} \sum_{i=1}^{m} x^{(i)}$ (one sum over samples). The mean squared error of "Why this matters" has a sum over samples, and inside it a dot product, which is a sum over features.

Products #

$\prod_{i=1}^{n} a_i = a_1 \cdot a_2 \cdots a_n$ ($\Pi$ is the capital Greek letter pi). The empty product is 1. The factorial is $n! = \prod_{i=1}^{n} i$, so $5! = 120$ and $0! = 1$.

In ML: the likelihood. If the samples are independent, the probability of the whole data set is the product of the probabilities of the samples: $L(\theta) = \prod_{i=1}^{m} p(x^{(i)}; \theta)$. For a coin with $P(\text{heads}) = 0.7$ and the data heads, heads, tails: $L = 0.7 \cdot 0.7 \cdot 0.3 = 0.147$. Products of many small numbers become tiny, so ML works with the logarithm, which turns the product into a sum: $\log \prod_i a_i = \sum_i \log a_i$ for positive $a_i$. Here $\log 0.147 = 2 \log 0.7 + \log 0.3 \approx -1.917$ (lesson 04 explains the logarithm).

max, min, argmax, argmin #

For a function $f$ on a set $D$ (often an interval):

• $\max_{x \in D} f(x)$ is the largest value of $f$ on $D$, if there is one: a number $M$ with $f(x) \le M$ for every $x \in D$ and $f(x_0) = M$ for some $x_0 \in D$. $\min$ is defined the same way.
• $\arg\max_{x \in D} f(x) = \{x \in D : f(x) = \max_{z \in D} f(z)\}$ is the set of places where the maximum is reached. It can have several elements (a tie). When it has one element, books write $x^\star = \arg\max_x f(x)$ for that element.

A maximum may not exist. On the open interval $(0, 1)$, $f(x) = x$ gets as close to 1 as you like but never reaches it, so there is no maximum. On a closed interval $[a, b]$, a continuous function always has a maximum and a minimum (the extreme value theorem); they can be at the ends.

Two patterns appear everywhere in ML:

• Training: $\theta^\star = \arg\min_\theta L(\theta)$, the parameters with the smallest loss. We want the place $\theta^\star$ (the model), not the value $L(\theta^\star)$.
• Prediction: a classifier computes a probability $p_k$ for each class $k$ and predicts $\hat y = \arg\max_k p_k$, the class (a place), not the probability (a value).

NumPy's np.argmax returns only the first index of a tie; the math argmax is the whole set.

Logic shorthand and decorations #

Symbol	Read it as	Example
$\forall$	for all	$\forall x \in \mathbb{R}: x^2 \ge 0$
$\exists$	there exists	$\exists x \in \mathbb{R}: x^2 = 2$
$\Rightarrow$	implies	$x > 2 \Rightarrow x^2 > 4$
$\Leftrightarrow$, "iff"	if and only if	$x^2 = 0 \Leftrightarrow x = 0$
$:=$	is defined as	$\bar x := \frac{1}{m}\sum_i x^{(i)}$
$\approx$	is approximately	$\pi \approx 3.14$
$\propto$	is proportional to (equal up to a constant factor)	$p(\theta \mid D) \propto p(D \mid \theta)\, p(\theta)$
s.t.	such that; in optimization, "subject to"	$\min_x f(x)$ s.t. $x \ge 0$
$\hat y$	y hat: an estimate or a prediction	$\hat y = w^\top x + b$
$\bar x$	x bar: a mean	$\bar x = 5$
$x^\star$	x star: the best (optimal) value	$\theta^\star = \arg\min_\theta L(\theta)$
$\tilde x$	x tilde: a changed or noisy version of $x$	$\tilde x = x + \varepsilon$

Note that $\Rightarrow$ goes one way only: $x > 2 \Rightarrow x^2 > 4$ is true, but $x^2 > 4$ does not imply $x > 2$ (take $x = -3$). The bars $\lvert \cdot \rvert$ have several meanings: $\lvert -3 \rvert = 3$ (absolute value), $\lvert S \rvert$ (size of a set), and in some books $\lvert A \rvert$ (determinant of a matrix). The length (norm) of a vector has double bars: $\lVert x \rVert$.

Greek letters in ML #

Letter	Name	Usual meaning in ML
$\alpha$	alpha	a step size or learning rate; a significance level in statistics
$\beta$	beta	coefficients of a regression; momentum factors $\beta_1, \beta_2$ in Adam
$\gamma$	gamma	the discount factor in reinforcement learning; a scale parameter
$\delta$, $\Delta$	delta	a small number; $\Delta$ = a change or difference, $\Delta x = x_{\text{new}} - x_{\text{old}}$
$\varepsilon$	epsilon	a small positive number; a noise term
$\eta$	eta	the learning rate
$\theta$	theta	the parameters of a model
$\lambda$	lambda	the strength of a regularization term; an eigenvalue
$\mu$	mu	a mean
$\nu$	nu	degrees of freedom; a second mean or measure
$\pi$	pi	the number $3.14159\dots$; a policy in reinforcement learning
$\rho$	rho	a correlation coefficient
$\sigma$, $\Sigma$	sigma	$\sigma$: standard deviation, or the sigmoid function; $\Sigma$: a sum, or a covariance matrix
$\tau$	tau	a temperature; a time constant
$\phi$, $\varphi$	phi	a feature map; the density of the standard normal distribution
$\psi$	psi	a second feature map or network
$\omega$, $\Omega$	omega	a frequency; $\Omega$: a set of all outcomes

The same letter can mean different things. $\sigma(z)$ with an argument is usually the sigmoid function; $\sigma$ alone, next to $\mu$, is a standard deviation. $\Sigma$ with bounds below and above is a sum; $\Sigma$ alone, next to $\mu$ in a Gaussian, is a covariance matrix. $\lambda$ is a regularization strength in a loss and an eigenvalue in linear algebra. The context decides; good papers define each symbol when it first appears.

Two more symbols: expectation and big-O

Expectation. $P(A)$ is the probability of an event $A$. For a random number $X$ with finitely many values, the expectation (mean) is $\mathbb{E}[X] = \sum_x x \, P(X = x)$, a weighted sum. For a fair die, $\mathbb{E}[X] = (1 + 2 + \dots + 6)/6 = 3.5$. Lesson 08 introduces these ideas.

Big-O. $f(n) = O(g(n))$ means: for large $n$, $f(n)$ is at most a constant times $g(n)$ (there are $C > 0$ and $n_0$ with $\lvert f(n) \rvert \le C\, g(n)$ for all $n \ge n_0$). It describes how cost grows. Adding $n$ numbers costs $O(n)$ operations; multiplying two $n \times n$ matrices with the usual loops costs $n^3$ multiplications, $O(n^3)$. $3n^2 + 10n = O(n^2)$, because $3n^2 + 10n \le 4n^2$ for $n \ge 10$.

Answers to section 1. (1) $3 + 5 + 7 = 15$. (2) No: $\sum_i a_i b_i = 1 \cdot 3 + 2 \cdot 4 = 11$, but $(1 + 2)(3 + 4) = 21$. The product of sums has all four pairs $a_i b_j$; the first sum has only the pairs with $i = j$. (3) No. In ML papers $x^{(2)}$ is sample number 2, a label in parentheses; the square is $x^2$.

4. Worked example #

(a) Read and compute a loss. Three houses, two features each, and their prices:

Compute $L(w, b) = \frac{1}{m} \sum_{i=1}^{m} \bigl(w^\top x^{(i)} + b - y^{(i)}\bigr)^2$.

1. Shapes. $X \in \mathbb{R}^{3 \times 2}$, so $m = 3$ samples and $n = 2$ features. $w$ has 2 entries, so $w^\top x^{(i)}$ is defined.
2. Samples. $x^{(1)} = (1, 0)$, $x^{(2)} = (2, 1)$, $x^{(3)} = (0, 3)$. For example $x^{(2)}_1 = X_{21} = 2$.
3. Predictions $\hat y^{(i)} = w^\top x^{(i)} + b = w_1 x^{(i)}_1 + w_2 x^{(i)}_2 + b$:
   • $\hat y^{(1)} = 1 \cdot 1 + 1 \cdot 0 + 1 = 2$;
   • $\hat y^{(2)} = 1 \cdot 2 + 1 \cdot 1 + 1 = 4$;
   • $\hat y^{(3)} = 1 \cdot 0 + 1 \cdot 3 + 1 = 4$.
4. Errors $\hat y^{(i)} - y^{(i)}$: $2 - 2 = 0$, $4 - 4 = 0$, $4 - 5 = -1$.
5. Squares: $0$, $0$, $1$. Sum: $\sum_{i=1}^{3} (\dots)^2 = 1$.
6. Mean: $L(w, b) = \frac{1}{3} \cdot 1 = \frac13 \approx 0.333$.

(b) A double sum. $a = (1, 2)$, $b = (1, 2, 3)$. Expand $\sum_{i=1}^{2} \sum_{j=1}^{3} a_i b_j$ row by row:

• $i = 1$: $a_1 b_1 + a_1 b_2 + a_1 b_3 = 1 + 2 + 3 = 6$;
• $i = 2$: $a_2 b_1 + a_2 b_2 + a_2 b_3 = 2 + 4 + 6 = 12$;
• total: $6 + 12 = 18$, and indeed $(1 + 2)(1 + 2 + 3) = 3 \cdot 6 = 18$.

(c) max and argmax on a grid. Let $L(\theta) = (\theta - 2)^2 + 1$ on the grid $\theta \in \{0, 1, 2, 3, 4\}$. The values are $5, 2, 1, 2, 5$. So $\min_\theta L(\theta) = 1$ and $\arg\min_\theta L(\theta) = \{2\}$; $\max_\theta L(\theta) = 5$ and $\arg\max_\theta L(\theta) = \{0, 4\}$ (a tie). Now take $L(\theta) = (\theta - 1.5)^2$ on the same grid: the values are $2.25, 0.25, 0.25, 2.25, 6.25$, so the minimum is $0.25$ and the argmin is $\{1, 2\}$.

5. In code #

Python counts from 0, so every math index is shifted by one. sum(...) and np.sum play the role of $\Sigma$. (Lesson 03 teaches NumPy properly.)

import numpy as np

X = np.array([[1, 0], [2, 1], [0, 3]])   # 3 samples (rows), 2 features (columns)
y = np.array([2, 4, 5])
w = np.array([1, 1]); b = 1
m, n = X.shape
print(m, n)                    # 3 2
print(X[1, 0])                 # x^(2)_1 = 2   (row 2 -> index 1, column 1 -> index 0)

# the loss, written as the two sums of the formula
total = 0
for i in range(m):                                      # sum over samples
    y_hat = sum(w[j] * X[i, j] for j in range(n)) + b   # w^T x^(i) + b: sum over features
    total += (y_hat - y[i]) ** 2
print(total / m)               # 0.3333333333333333

# a double sum equals the product of two sums
a, bb = np.array([1, 2]), np.array([1, 2, 3])
print(sum(a[i] * bb[j] for i in range(2) for j in range(3)), np.sum(a) * np.sum(bb))   # 18 18

# argmin on a grid: np.argmin returns only the FIRST index of a tie
grid = np.array([0, 1, 2, 3, 4])
L = (grid - 1.5) ** 2
print(L.min(), grid[np.argmin(L)], grid[L == L.min()])   # 0.25 1 [1 2]

6. Where it shows up #

• Every loss function is a mean over samples: $\frac{1}{m} \sum_{i=1}^{m} \ell\bigl(f(x^{(i)}; \theta), y^{(i)}\bigr)$, where $\ell$ is the loss of one sample. Mean squared error for regression, cross-entropy for classification (Information Theory).
• Training is $\theta^\star = \arg\min_\theta L(\theta)$, found with gradient descent (Optimization); prediction with a classifier is $\hat y = \arg\max_k p_k$.
• Maximum likelihood maximizes $\prod_i p(x^{(i)}; \theta)$, in practice the sum $\sum_i \log p(x^{(i)}; \theta)$ (Probability and Statistics).
• Neural networks are compositions $f_L \circ \dots \circ f_1$ of layers, each a function with its own parameters (Deep Learning).
• Attention in transformers is written $\operatorname{softmax}\!\bigl(QK^\top / \sqrt{d_k}\bigr) V$: matrices, a transpose and a function applied row by row. Each entry of $QK^\top$ is a dot product, a sum over the $d_k$ dimensions: $(QK^\top)_{st} = \sum_{k=1}^{d_k} Q_{sk} K_{tk}$.
• ML for science: a simulation grid or a time series is indexed the same way: $u_{ij}$ at grid point $(i, j)$, $x_t$ at time step $t$; numerical schemes are sums over neighbours.

7. Common mistakes #

• Reading $x^{(i)}$ as a power. It is sample $i$. The parentheses are the signal.
• $\sum_i a_i b_i = \bigl(\sum_i a_i\bigr)\bigl(\sum_i b_i\bigr)$. False. The product of sums has all pairs $a_i b_j$. Check with $a = (1, 2)$, $b = (3, 4)$: 11 versus 21.
• Adding a constant once. $\sum_{i=1}^{n} (a_i + c) = \sum_i a_i + nc$, not $\sum_i a_i + c$.
• Confusing max and argmax. The max is a value (a height); the argmax is a set of places. The trained model is the argmin, not the minimum loss.
• Reading $f \circ g$ from left to right. $(f \circ g)(x) = f(g(x))$: $g$ acts first.
• Off-by-one indices. Math $X_{ij}$ is Python X[i-1, j-1]; $\sum_{i=1}^{n}$ is range(n) with the index shifted, or range(1, n + 1).
• Rows and columns swapped. $A \in \mathbb{R}^{m \times n}$ has $m$ rows. $A_{ij}$: row first.
• Writing a set with order or repeats. $\{1, 2, 2\} = \{2, 1\}$ has 2 elements. Use a list or a vector $(1, 2, 2)$ when order and repeats matter.
• $f^{-1}(x) = 1/f(x)$. The inverse function undoes $f$; it is not the reciprocal.

8. Check yourself #

1. Read aloud and compute: $\lvert \{x \in \mathbb{Z} : -2 \le x < 3\} \rvert$.
   Answer

   "The number of integers $x$ with $-2 \le x < 3$." The set is $\{-2, -1, 0, 1, 2\}$ (3 is excluded), so the size is 5.
2. Compute $\sum_{i=1}^{4} (3i - 1)$ in two ways: term by term, and with the rules.
   Answer

   Term by term: $2 + 5 + 8 + 11 = 26$. With the rules: $3 \sum_{i=1}^{4} i - \sum_{i=1}^{4} 1 = 3 \cdot 10 - 4 = 26$.
3. Let $f(x) = 2x$ and $g(x) = x + 5$. Give formulas for $f \circ g$ and $g \circ f$.
   Answer

   $(f \circ g)(x) = f(x + 5) = 2x + 10$. $(g \circ f)(x) = g(2x) = 2x + 5$. They differ.
4. What is the range of $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^2 + 1$? What is $\arg\min_x f(x)$?
   Answer

   The range is $[1, \infty)$, because $x^2 \ge 0$ and every value $y \ge 1$ is reached at $x = \sqrt{y - 1}$. The minimum value is 1 and the argmin is $\{0\}$.
5. $X \in \mathbb{R}^{100 \times 3}$ is a data matrix. How many samples and features? Write $x^{(7)}_2$ as a NumPy expression.
   Answer

   100 samples (rows) and 3 features (columns). $x^{(7)}_2 = X_{72}$ is `X[6, 1]`.
6. Compute $\prod_{i=1}^{4} \frac{i}{i+1}$.
   Answer

   $\frac12 \cdot \frac23 \cdot \frac34 \cdot \frac45 = \frac{1 \cdot 2 \cdot 3 \cdot 4}{2 \cdot 3 \cdot 4 \cdot 5} = \frac15$. Every factor except the first numerator and the last denominator cancels.

Recall cardsAnswer in your head first, then reveal. Add the cards to your review deck to see them again just before you would forget.

1. How do you read $\{x \in \mathbb{R} : x > 0\}$, and which interval is it?

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2. State the linearity rule of sums.

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3. What is $\sum_{i=1}^{n} c$ for a constant $c$?

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4. What does $\sum_{i=1}^{m}\sum_{j=1}^{n} a_i b_j$ equal, and why?

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5. Compute $\prod_{i=1}^{4} i$.

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6. What is the difference between $\max_x f(x)$ and $\arg\max_x f(x)$?

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7. In ML notation, what are $x^{(i)}$ and $x^{(i)}_j$?

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8. $A \in \mathbb{R}^{m \times n}$. What is its NumPy shape, and which Python index gives $A_{ij}$?

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9. In $(f \circ g)(x)$, which function acts first?

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10. Give two meanings of $\sigma$ in ML texts.

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9. Interview angle #

• "Write the mean squared error of linear regression." — $L(w, b) = \frac{1}{m}\sum_{i=1}^{m}\bigl(w^\top x^{(i)} + b - y^{(i)}\bigr)^2$; in matrix form $\frac{1}{m}\lVert Xw + b\mathbf{1} - y \rVert^2$. Say what $m$, $x^{(i)}$ and $w^\top x$ are.
• "What is the difference between max and argmax? What does np.argmax do with ties?" — The max is a value, the argmax a set of places; np.argmax returns the first index of a tie.
• "Why do we maximize the log-likelihood instead of the likelihood?" — The log turns the product over samples into a sum, which avoids numbers that underflow to 0 and is easier to differentiate; the log is increasing, so the argmax does not change (lesson 04).
• "What is the shape of $XW$ for $X \in \mathbb{R}^{m \times n}$ and $W \in \mathbb{R}^{n \times k}$?" — $m \times k$. Shape questions are the most common notation questions in coding interviews.

10. Practice #

The notebook for this lesson runs in Colab or in your browser. It uses only plain Python, np.array and np.sum, and every line has a comment. Five short exercises, each with a check cell and a folded solution:

1. Write $\sum_{i=1}^{n} i^2$ as a loop and with sum(...), and check the closed form for many $n$.
2. Check that $\sum_i \sum_j a_i b_j = (\sum a)(\sum b)$ and that $\sum_i a_i b_i$ is different.
3. Read entries, samples and features of a data table with 1-based math indices.
4. Compute union, intersection, difference and complement with Python sets, and check inclusion–exclusion.
5. Find the maximum and the whole argmax of a function on a grid, including a tie.

11. Watch and read #

Sigma notation for sums | Sequences, series and induction | Precalculus | Khan AcademyKhan Academy · 4 min

How to read the index, the bounds and the term of a Σ-sum, with examples.

Intersection and union of sets | Probability and Statistics | Khan AcademyKhan Academy · 8 min

Union and intersection with Venn diagrams, as in the set figure of this lesson.

Introduction to function composition | Functions and their graphs | Algebra II | Khan AcademyKhan Academy · 6 min

Composition as one function feeding into another; watch the order of evaluation.

MaDL - The Argmin and Argmax OperatorsTübingen Machine Learning · 5 min

The argmin and argmax operators as used in machine learning.

• Goodfellow, Bengio and Courville, Deep Learning, Notation: the notation page of a standard ML textbook; read it after this lesson and check that you know each line.
• Deisenroth, Faisal and Ong, Mathematics for Machine Learning (free PDF): the table of symbols at the start and the beginning of chapter 2.
• Lehman, Leighton and Meyer, Mathematics for Computer Science (free PDF), chapter 4 (sets, sequences, functions) and chapter 14 on sums.
• Andrew Ng and Tengyu Ma, CS229 lecture notes, part I: the $x^{(i)}$, $y^{(i)}$ notation for training data, used in many ML courses.