Concepts inEfficient projections onto the l1-ball for learning in high dimensions
Curse of dimensionality
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing high-dimensional spaces (often with hundreds or thousands of dimensions) that do not occur in low-dimensional settings such as the physical space commonly modeled with just three dimensions. There are multiple phenomena referred to by this name in domains such as sampling, combinatorics, machine learning and data mining.
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Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in the Euclidean plane, for example, is the same thing as a disk, the area bounded by a circle.
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Interior point method
Interior point methods (also referred to as barrier methods) are a certain class of algorithms to solve linear and nonlinear convex optimization problems. The interior point method was invented by John von Neumann. Von Neumann suggested a new method of linear programming, using the homogeneous linear system of Gordan (1873) which was later popularized by Karmarkar's algorithm, developed by Narendra Karmarkar in 1984 for linear programming.
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Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase. A generalization of the gradient for functions on a Euclidean space that have values in another Euclidean space is the Jacobian. A further generalization for a function from one Banach space to another is the Fréchet derivative.
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Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an (ordered) -tuple is a sequence (or ordered list) of elements, where is a positive integer. There is also one 0-tuple, an empty sequence. An -tuple is defined inductively using the construction of an ordered pair. Tuples are usually written by listing the elements within parentheses "" and separated by commas; for example, denotes a 5-tuple.
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Array data structure
In computer science, an array data structure or simply array is a data structure consisting of a collection of elements, each identified by at least one array index or key. An array is stored so that the position of each element can be computed from its index tuple by a mathematical formula.
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Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term ¿Euclidean¿ distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria.
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Mathematical optimization
In mathematics, computer science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function.
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