Mathematical Symbols

$\sum$

Summation

Add terms over an index, e.g. a loss $\sum_{i=1}^{n} \ell_i$ over the batch.

$\prod$

Product

Multiply terms over an index, e.g. likelihood $\prod_i p(x_i)$.

$\nabla$

Nabla / gradient

Vector of partial derivatives $\nabla_\theta L$ used in gradient descent.

$\partial$

Partial derivative

$\partial L / \partial w$: rate of change holding other variables fixed.

$\times$

Cross / times

Scalar multiplication, or a Cartesian/shape product like $m \times n$.

$\cdot$

Dot

Dot product $a \cdot b$ or plain multiplication between scalars.

$\odot$

Hadamard product

Element-wise multiply of equal-shaped tensors (gating, masks).

$\lVert \cdot \rVert$

Norm

Length/magnitude of a vector, e.g. $\lVert x \rVert_2$ for L2.

$\approx$

Approximately equal

Value is close but not exact, e.g. $\pi \approx 3.14$.

$\propto$

Proportional to

$p(\theta \mid x) \propto p(x \mid \theta)\,p(\theta)$: equal up to a constant factor.

$\ll,\ \gg$

Much less / greater than

Order-of-magnitude comparison, e.g. learning rate $\eta \ll 1$.

$\equiv$

Equivalent / defined as

Two expressions are identical or a definition.

$\triangleq$

Defined as

Left side is defined to equal the right side.

$\to$

Tends to / maps to

Limit $x \to 0$ or a function type $f: \mathbb{R}^n \to \mathbb{R}$.

$\in$

Element of

$x \in \mathbb{R}^d$: $x$ is a $d$-dimensional real vector.

$\notin$

Not an element of

$0 \notin \{1,2,3\}$.

$\subset$

Subset

$A \subset B$: every element of $A$ is in $B$ (train set inside data).

$\cup,\ \cap$

Union / intersection

Combine or overlap sets of samples or features.

$\forall$

For all

$\forall i,\ x_i \ge 0$: statement holds for every index.

$\exists$

There exists

At least one element satisfies the condition.

$\Rightarrow$

Implies

$a \Rightarrow b$: if $a$ holds then $b$ holds.

$\iff$

If and only if

$a \iff b$: each implies the other (equivalence).

$\emptyset$

Empty set

The set with no elements.

$\mathbb{R},\ \mathbb{N}$

Number sets

Reals and natural numbers; $\mathbb{R}^{m\times n}$ is a matrix space.