Lecture

Vector Differentiation and Convex Functions

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In course

EE-566: Adaptation and learning

In this course, students learn to design and master algorithms and core concepts related to inference and learning from data and the foundations of adaptation and learning theories with applications.

Description

This lecture covers the concepts of vector differentiation and convex functions. It explains the importance of gradient vectors and Hessian matrices in iterative algorithms for inference and learning. The lecture defines first and second-order differentiation for functions of vector arguments and introduces related differentiation operations.

Instructor

Ali H. Sayed

Ali H. Sayed is Dean of Engineering at EPFL, Switzerland, where he also leads the Adaptive Systems Laboratory. He has also served as Distinguished Professor and Chairman of Electrical Engineering at UCLA. He is recognized as a Highly Cited Researcher and is a member of the US National Academy of Engineering. He is also a member of the World Academy of Sciences and served as President of the IEEE Signal Processing Society during 2018 and 2019. Dr. Sayed is an author/co-author of over 570 scholarly publications and six books. His research involves several areas including adaptation and learning theories, data and network sciences, statistical inference, and multiagent systems. His work has been recognized with several major awards including the 2022 IEEE Fourier Award, the 2020 Norbert Wiener Society Award and the 2015 Education Award from the IEEE Signal Processing Society, the 2014 Papoulis Award from the European Association for Signal Processing, the 2013 Meritorious Service Award and the 2012 Technical Achievement Award from the IEEE Signal Processing Society, the 2005 Terman Award from the American Society for Engineering Education, the 2005 Distinguished Lecturer from the IEEE Signal Processing Society, the 2003 Kuwait Prize, and the 1996 IEEE Donald G. Fink Prize. His publications have been awarded several Best Paper Awards from the IEEE (2002, 2005, 2012, 2014) and EURASIP (2015). He is a Fellow of IEEE, EURASIP, and the American Association for the Advancement of Science (AAAS); the publisher of the journal Science.

Official source

Related concepts (32)

Notation for differentiation

In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.

Vector calculus

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations.

Gradient

In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) whose value at a point is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative.

Automatic differentiation

In mathematics and computer algebra, automatic differentiation (auto-differentiation, autodiff, or AD), also called algorithmic differentiation, computational differentiation, is a set of techniques to evaluate the partial derivative of a function specified by a computer program. Automatic differentiation exploits the fact that every computer calculation, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.

Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space . A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

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