Higher-order function

In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following: takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself a procedure), returns a function as its result. All other functions are first-order functions. In mathematics higher-order functions are also termed operators or functionals. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation). In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions that take one function as argument are values with types of the form . map function, found in many functional programming languages, is one example of a higher-order function. It takes as arguments a function f and a collection of elements, and as the result, returns a new collection with f applied to each element from the collection. Sorting functions, which take a comparison function as a parameter, allowing the programmer to separate the sorting algorithm from the comparisons of the items being sorted. The C standard function qsort is an example of this. filter fold apply Function composition Integration Callback Tree traversal Montague grammar, a semantic theory of natural language, uses higher-order functions The examples are not intended to compare and contrast programming languages, but to serve as examples of higher-order function syntax In the following examples, the higher-order function takes a function, and applies the function to some value twice. If has to be applied several times for the same it preferably should return a function rather than a value. This is in line with the "don't repeat yourself" principle.

Efficient Continual Finite-Sum Minimization

Volkan Cevher, Efstratios Panteleimon Skoulakis, Leello Tadesse Dadi

Given a sequence of functions

f_1,\ldots,f_n

with

f_i:\mathcal{D}\mapsto \mathbb{R}

, finite-sum minimization seeks a point

{x}^\star \in \mathcal{D}

minimizing

\sum_{j=1}^nf_j(x)/n

. In this work, we propose a key twist into the finite-sum minimizat ...

2024

Efficient Continual Finite-Sum Minimization

DFT plus U-type functional derived to explicitly address the flat plane condition

On algebraic array theories

DFT plus U-type functional derived to explicitly address the flat plane condition

Efficient Continual Finite-Sum Minimization

On algebraic array theories