Compressed sensing is a signal processing technique that allows for the reconstruction of a signal from a small set of measurements. The key idea behind compressed sensing is that many real-world signals are inherently sparse, meaning that they can be efficiently represented in a different space with only a few components compared to their original space representation. In this paper we will explore the mathematical formulation behind compressed sensing, its logic and pathologies, and apply compressed sensing to real world signals.

In this note research and publications by Zdzisław Pawlak and his collaborators from 1970s and 1981 are recalled. Focus is placed on the sources of inspiration which one can identify on the basis of those publications. Finally, developments from 1981 related to rough sets and information systems are outlined.

This paper introduces a rigorous mathematical framework for neural network explainability, and more broadly for the explainability of equivariant operators called Group Equivariant Operators (GEOs) based on Group Equivariant Non-Expansive Operators (GENEOs) transformations. The central concept involves quantifying the distance between GEOs by measuring the non-commutativity of specific diagrams. Additionally, the paper proposes a definition of interpretability of GEOs according to a complexity measure that can be defined according to each user preferences. Moreover, we explore the formal properties of this framework and show how it can be applied in classical machine learning scenarios, like image classification with convolutional neural networks.

This note provides a critical review of the mathematical concepts underlying the generalized diffusion denoising implicit model (gDDIM) and the exponential integrator (EI) scheme. We present enhanced mathematical results, including an exact expression for the reverse trajectory in the probability flow ODE and an exact expression for the covariance matrix in the gDDIM scheme. Furthermore, we offer an improved understanding of the EI scheme's efficiency in terms of the change of variables. The noising process in DDIM is analyzed from the perspective of non-equilibrium statistical physics. Additionally, we propose a new scheme for DDIM, called the principal-axis DDIM (paDDIM).

Understand the fundamentals of linear algebra and calculus, critical mathematical subjects underlying all of machine learning and data science Manipulate tensors using all three of the most important Python tensor libraries: NumPy, TensorFlow, and PyTorch How to apply all of the essential vector and matrix operations for machine learning and data science Reduce the dimensionality of complex data to the most informative elements with eigenvectors, SVD, and PCA Solve for unknowns with both simple techniques (e.g., elimination) and advanced techniques (e.g., pseudoinversion) Appreciate how calculus works, from first principles, via interactive code demos in Python Intimately understand advanced differentiation rules like the chain rule Compute the partial derivatives of machine-learning cost functions by hand as well as with TensorFlow and PyTorch Grasp exactly what gradients are and appreciate why they are essential for enabling ML via gradient descent Use integral calculus to determine the area under any given curve Be able to more intimately grasp the details of cutting-edge machine learning papers Develop an understanding of what's going on beneath the hood of machine learning algorithms, including those used for deep learning Solve for unknowns with both simple techniques (e.g., elimination) and advanced techniques (e.g., pseudoinversion) Develop an understanding of what's going on beneath the hood of machine learning algorithms, including those used for deep learning All code demos will be in Python so experience with it or another object-oriented programming language would be helpful for following along with the hands-on examples. Familiarity with secondary school-level mathematics will make the class easier to follow along with. If you are comfortable dealing with quantitative information -- such as understanding charts and rearranging simple equations -- then you should be well-prepared to follow along with all of the mathematics. All code demos will be in Python so experience with it or another object-oriented programming language would be helpful for following along with the hands-on examples. Familiarity with secondary school-level mathematics will make the class easier to follow along with.

Sometimes, you see a diagram and it gives you an'aha ha' moment I saw it on Frederick kratzert's blog Using the input variables x and y, The forwardpass (left half of the figure) calculates output z as a function of x and y i.e. f(x,y) The right side of the figures shows the backwardpass. Receiving dL/dz (the derivative of the total loss with respect to the output z), we can calculate the individual gradients of x and y on the loss function by applying the chain rule, as shown in the figure. This post is a part of my forthcoming book on Mathematical foundations of Data Science. The goal of the neural network is to minimise the loss function for the whole network of neurons. Hence, the problem of solving equations represented by the neural network also becomes a problem of minimising the loss function for the entire network.

However, the development of a rigorous mathematical foundation is still at an early stage. In this survey article, which is based on an invited lecture at the International Congress of Mathematicians 2022, we will in particular focus on the current "workhorse" of artificial intelligence, namely deep neural networks. We will present the main theoretical directions along with several exemplary results and discuss key open problems.

Mathematics forms the core of data science and machine learning. Thus, to be the best data scientist you can be, you must have a working understanding of the most relevant math. Getting started in data science is easy thanks to high-level libraries like Scikit-learn and Keras. But understanding the math behind the algorithms in these libraries opens an infinite number of possibilities up to you. From identifying modeling issues to inventing new and more powerful solutions, understanding the math behind it all can dramatically increase the impact you can make over the course of your career.

To be a good data scientist, you need to know how to use data science and machine learning libraries and algorithms, such as Scikit-learn, TensorFlow, and PyTorch, to solve whatever problem you have at hand. To be an excellent data scientist, you need to know how those libraries and algorithms work under the hood. This is where our "Machine Learning & Data Science Foundations Masterclass" comes in. Led by deep learning guru Dr. Jon Krohn, this course provides a firm grasp of the underlying mathematics, such as linear algebra, tensors, and eigenvectors, that operate behind the most important Python libraries, machine learning algorithms, and data science models. While the above sections constitute a standalone, introductory course on linear algebra all on their own, we're not stopping there!

Mathematics forms the core of data science and machine learning. Thus, to be the best data scientist you can be, you must have a working understanding of the most relevant math. Getting started in data science is easy thanks to high-level libraries like Scikit-learn and Keras. But understanding the math behind the algorithms in these libraries opens an infinite number of possibilities up to you. From identifying modeling issues to inventing new and more powerful solutions, understanding the math behind it all can dramatically increasing the impact you can make over the course of your career.