Higher-order derivatives can capture information about a function that first-order derivatives on their own cannot capture.  First-order derivatives can capture important information, such as the rate of change, but on their own they cannot distinguish between local minima or maxima, where the rate of change is zero for both. Several optimization algorithms address this limitation by exploiting the use of higher-order derivatives, such as in Newton’s method where the second-order derivatives are used to reach the local minimum of an […]

In the literature, the term Jacobian is often interchangeably used to refer to both the Jacobian matrix or its determinant.  Both the matrix and the determinant have useful and important applications: in machine learning, the Jacobian matrix aggregates the partial derivatives that are necessary for backpropagation; the determinant is useful in the process of changing between variables. In this tutorial, you will review a gentle introduction to the Jacobian.  After completing this tutorial, you will know: The Jacobian matrix collects […]

Hessian matrices belong to a class of mathematical structures that involve second order derivatives. They are often used in machine learning and data science algorithms for optimizing a function of interest.  In this tutorial, you will discover Hessian matrices, their corresponding discriminants, and their significance. All concepts are illustrated via an example. After completing this tutorial, you will know: Hessian matrices Discriminants computed via Hessian matrices What information is contained in the discriminant Let’s get started.

The Laplace operator was first applied to the study of celestial mechanics, or the motion of objects in outer space, by Pierre-Simon de Laplace, and as such has been named after him.  The Laplace operator has since been used to describe many different phenomena, from electric potentials, to the diffusion equation for heat and fluid flow, and quantum mechanics. It has also been recasted to the discrete space, where it has been used in applications related to image processing and […]

Whether it is a supervised learning problem or an unsupervised problem, there will be some optimization algorithm working in the background. Almost any classification, regression or clustering problem can be cast as an optimization problem. In this tutorial, you will discover what is optimization and concepts related to it. After completing this tutorial, you will know: What is Mathematical programming or optimization Difference between a maximization and minimization problems Difference between local and global optimal solutions Difference between constrained and […]

The method of Lagrange multipliers is a simple and elegant method of finding the local minima or local maxima of a function subject to equality or inequality constraints. Lagrange multipliers are also called undetermined multipliers. In this tutorial we’ll talk about this method when given equality constraints.  In this tutorial, you will discover the method of Lagrange multipliers and how to find the local minimum or maximum of a function when equality constraints are present. After completing this tutorial, you […]

The chain rule allows us to find the derivative of composite functions. It is computed extensively by the backpropagation algorithm, in order to train feedforward neural networks. By applying the chain rule in an efficient manner while following a specific order of operations, the backpropagation algorithm calculates the error gradient of the loss function with respect to each weight of the network.  In this tutorial, you will discover the chain rule of calculus for univariate and multivariate functions. After completing […]

The chain rule is an important derivative rule that allows us to work with composite functions. It is essential in understanding the workings of the backpropagation algorithm, which applies the chain rule extensively in order to calculate the error gradient of the loss function with respect to each weight of a neural network. We will be building on our earlier introduction to the chain rule, by tackling more challenging functions.  In this tutorial, you will discover how to apply the […]

When it comes to machine learning tasks such as classification or regression, approximation techniques play a key role in learning from the data. Many machine learning methods approximate a function or a mapping between the inputs and outputs via a learning algorithm. In this tutorial, you will discover what is approximation and its importance in machine learning and pattern recognition. After completing this tutorial, you will know:  What is approximation Importance of approximation in machine learning Let’s get started.

A Gentle Introduction to Taylor Series Taylor series expansion is an awesome concept, not only the world of mathematics, but also in optimization theory, function approximation and machine learning. It is widely applied in numerical computations when estimates of a function’s values at different points are required. In this tutorial, you will discover Taylor series and how to approximate the values of a function around different points using its Taylor series expansion. After completing this tutorial, you will know: Taylor […]