Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Convex combination
Summary
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the count of the weights as in a standard weighted average.
More formally, given a finite number of points in a real vector space, a convex combination of these points is a point of the form
where the real numbers satisfy and
As a particular example, every convex combination of two points lies on the line segment between the points.
A set is convex if it contains all convex combinations of its points.
The convex hull of a given set of points is identical to the set of all their convex combinations.
There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval is convex but generates the real-number line under linear combinations. Another example is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).
A random variable is said to have an -component finite mixture distribution if its probability density function is a convex combination of so-called component densities.
A conical combination is a linear combination with nonnegative coefficients. When a point is to be used as the reference origin for defining displacement vectors, then is a convex combination of points if and only if the zero displacement is a non-trivial conical combination of their respective displacement vectors relative to .
Weighted means are functionally the same as convex combinations, but they use a different notation. The coefficients (weights) in a weighted mean are not required to sum to 1; instead the weighted linear combination is explicitly divided by the count of the weights.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related courses (6)
MGT-418: Convex optimization
This course introduces the theory and application of modern convex optimization from an engineering perspective.
MATH-261: Discrete optimization
This course is an introduction to linear and discrete optimization.
Warning: This is a mathematics course! While much of the course will be algorithmic in nature, you will still need to be able to p
EE-556: Mathematics of data: from theory to computation
This course provides an overview of key advances in continuous optimization and statistical analysis for machine learning. We review recent learning formulations and models as well as their guarantees
Related publications (46)
Related concepts (11)
Conical combination
Given a finite number of vectors in a real vector space, a conical combination, conical sum, or weighted sum of these vectors is a vector of the form where are non-negative real numbers. The name derives from the fact that a conical sum of vectors defines a cone (possibly in a lower-dimensional subspace). The set of all conical combinations for a given set S is called the conical hull of S and denoted cone(S) or coni(S). That is, By taking k = 0, it follows the zero vector (origin) belongs to all conical hulls (since the summation becomes an empty sum).
Affine combination
In mathematics, an affine combination of x1, ..., xn is a linear combination such that Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients are elements of K. The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the are elements of K (or for a Euclidean space), and the affine combination is also a point. See for the definition in this case.
Convex cone
In linear algebra, a cone—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under positive scalar multiplication; that is, C is a cone if implies for every positive scalar s. When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a positive scalar.