$f(\cdot )$ Center-dot notation in functions

Some papers will use a center dot (or interpunct) to represent that any value can be inserted as the value of a function.

For example, $\arg maxf(\cdot )$ means the same as $\underset{x}{\arg max}f\left(x\right)$

See Mathematics StackExchange for more information.

$\nabla$ Gradient

The $\nabla$ symbol when seen in front of a vector-valued function like $\nabla f\left(x\right)$ refers to the function’s gradient.

$\odot$ Hadamard Product

This circle-dot symbol can mean a few different things, depending on context. Typically in machine learning literature, it refers to the Hadamard product (component-wise multiplication for matrices).

$\sigma$ Sigma

The $\sigma$ symbol is often used to represent the standard deviation of a probability distribution.

When used as a function that takes in a value, e.g. $\sigma \left(x\right)$, then it refers to the logistic sigmoid function.

Wikipedia has a more complete list of the symbol’s uses

$\oplus$ XOR, exclusive-or, exclusive disjunction

The XOR operator is a logical operator that takes two terms. It returns true when both inputs differ.

For example, $a\oplus b$ is only true when $a$ is true and $b$ is false, or when $b$ is false and when $a$ is true.

Put another way, $a\oplus b$ will be false when both $a$ and $b$ are false, or when both $a$ and $b$ are true. See Exclusive or on Wikipedia for more information.

$[x{]}_{+}$ , bracket operator, max operator

The bracket operator $[x{]}_{+}$ where an expression $x$ is surrounded by a plus sign and a subscript is equivalent to $max(0,X)$. See 8 for more information.