# $$f(\cdot)$$ 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, $$\arg \max f(\cdot)$$ means the same as $$\mathop{\arg\max}\limits_x f(x)$$

# $$\nabla$$Gradient

The $$\nabla$$ symbol when seen in front of a vector-valued function like $$\nabla f(x)$$ 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. $$\sigma(x)$$, 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.