Related papers: Matrix Calculus (for Machine Learning and Beyond)
We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces…
All machine learning algorithms use a loss, cost, utility or reward function to encode the learning objective and oversee the learning process. This function that supervises learning is a frequently unrecognized hyperparameter that…
Contribution: We demonstrate that it is feasible to include field specific problems in introductory mathematics courses to motivate engineering students. This is done in a way that still allows large parts of the course to be common to all…
Recent times are witnessing rapid development in machine learning algorithm systems, especially in reinforcement learning, natural language processing, computer and robot vision, image processing, speech, and emotional processing and…
Artificial intelligence has made remarkable progress in handling complex tasks, thanks to advances in hardware acceleration and machine learning algorithms. However, to acquire more accurate outcomes and solve more complex issues,…
Matrix inequalities that extend certain scalar ones have been at the center of numerous researchers' attention. In this article, we explore the celebrated subadditive inequality for matrices via concave functions and present a reversed…
The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…
Distance metric learning is a branch of machine learning that aims to learn distances from the data, which enhances the performance of similarity-based algorithms. This tutorial provides a theoretical background and foundations on this…
In a variety of problems originating in supervised, unsupervised, and reinforcement learning, the loss function is defined by an expectation over a collection of random variables, which might be part of a probabilistic model or the external…
The solution of systems of non-autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in…
In this paper, we consider characterisations of the class of unitary matrix integrals $\big\langle (\det U)^q {\rm e}^{s^{1/2} \operatorname{Tr}(U + U^\dagger)} \big\rangle_{U(l)}$ in terms of a first-order matrix linear differential…
Model-Agnostic Meta-Learning (MAML) is a versatile meta-learning framework applicable to both supervised learning and reinforcement learning (RL). However, applying MAML to meta-reinforcement learning (meta-RL) presents notable challenges.…
Neural ordinary differential equations (Neural ODEs) propose the idea that a sequence of layers in a neural network is just a discretisation of an ODE, and thus can instead be directly modelled by a parameterised ODE. This idea has had…
The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It…
The calculus of finite differences is a solid foundation for the development of operations such as the derivative and the integral for infinite sequences. Here we showed a way to extend it for finite sequences. We could then define…
Gradients of neural networks can be computed efficiently for any architecture, but some applications require differential operators with higher time complexity. We describe a family of restricted neural network architectures that allow…
Motivated by applications to stochastic programming, we introduce and study the expected-integral functionals, which are mappings given in an integral form depending on two variables, the first a finite dimensional decision vector and the…
Low rank approximation of a matrix (hereafter LRA) is a highly important area of Numerical Linear and Multilinear Algebra and Data Mining and Analysis. One can operate with an LRA at sublinear cost -- by using much fewer memory cells and…
We discuss a version of the fundamental theorem of calculus in several variables and some applications, of potential interest as a teaching material in undergraduate courses.
The method of choice to study one-dimensional strongly interacting many body quantum systems is based on matrix product states and operators. Such method allows to explore the most relevant, and numerically manageable, portion of an…