Lecture

Convex Sets and Functions

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In course

MGT-448: Statistical inference and machine learning

This course aims to provide graduate students a thorough grounding in the methods, theory, mathematics and algorithms needed to do research and applications in machine learning. The course covers topics from machine learning, classical statistics, and data ...

Description

This lecture covers the concepts of convex sets and functions, defining convex sets as those where a linear combination of two points remains in the set, and convex functions as those satisfying a specific inequality. It also discusses minimizers of convex functions, optimality conditions, and the characterization of convexity using first and second-order conditions. Examples of affine and quadratic functions are provided, along with the Cauchy-Schwarz inequality and the Descent Lemma for continuously differentiable functions with Lipschitz gradient.

Instructor

Negar Kiyavash

Teaching & PhD PhD Students Mohammad Shahverdikondori, Badri Mohammad Reza, Arefe Boushehrian, Sepehr Elahi, Mohammadsadegh Khorasani, Mohammadsaeed Masiha, Amir Mohammad Aboueimehrizi Past EPFL PhD Students Sina Akbari, Ehsan Mokhtarian, Yaroslav Kivva, Fateme Jamshidi Courses Causal inference MGT-416 Students will learn the core concepts and techniques of network analysis with emphasis on causal inference. Theory andapplication will be balanced, with students working directly with networ

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Convex set

In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve.

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.

Convex function

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain.

Linear span

In mathematics, the linear span (also called the linear hull or just span) of a set S of vectors (from a vector space), denoted span(S), is defined as the set of all linear combinations of the vectors in S. For example, two linearly independent vectors span a plane. The linear span can be characterized either as the intersection of all linear subspaces that contain S, or as the smallest subspace containing S. The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules.

Convex hull

In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact.

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