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Mathematical Notation

\(f(\cdot)\) 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 \(\mathop{\arg\max}\limits_x f(x)\)

See Mathematics StackExchange for more information.

\(\nabla\) 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.