Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Gradient and Hessian of Pullback

In course

MATH-512: Optimization on manifolds

We develop, analyze and implement numerical algorithms to solve optimization problems of the form min f(x) where x is a point on a smooth manifold. To this end, we first study differential and Riemannian geometry (with a focus dictated by computational con ...

Description

This lecture explains the relationship between the Riemannian gradients and Hessians of a function f and the Euclidean gradient and Hessian of the pullback at a given point x. By examining the Taylor expansion of the pullback, the instructor demonstrates how these quantities are related, particularly at critical points. The lecture concludes by highlighting the practical implications of this relationship for further analysis.

Instructor

Nicolas Boumal

Official source

Related concepts (18)

Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.

Series expansion

In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division). The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.

Asymptotic expansion

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

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.

Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation.

Related lectures (30)

Holomorphic Functions: Taylor Series Expansion

Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.

Limit Calculations Applications MOOC: Analysis I MOOC: Analysis I (part 6) : Study of functions, limited developments

Demonstrates efficient limit calculations using Taylor expansions and basic algebraic operations.

Composition of Taylor Expansions

Covers the computation of Taylor expansions composition up to order 4.

Differentiating vector fields: Why do it?

Explores the importance of differentiating vector fields and the correct methodology to achieve it, emphasizing the significance of going beyond the first order.

Laplace's Method: Exercises