Machinelearning

Diffusion is a physical process that models random motion, first analyzed by Brown when studying pollen grains in water. In this section, we will first analyze a simplified 1-dimensional version, and then delve into diffusion models for images, the ones closest to (Ho et al. 2020). The Diffusion Process This note follows original Einstein’s presentation, here we have a simplified version. Let’s suppose we have a particle at $t = 0$ at some position $i$. We have a probability of jumping to the left of $p$ to right of $q$, the rest is staying at the same position. ...

The DP (Dirichlet Processes) is part of family of models called non-parametric models. Non parametric models concern learning models with potentially infinite number of parameters. One of the classical application is unsupervised techniques like clustering. Intuitively, clustering concerns in finding compact subsets of data, i.e. finding groups of points in the space that are particularly close by some measure. The Dirichlet Process See Beta and Dirichlet Distributions for the definition and intuition of these two distributions. One quite important thing that Dirichlet allows to do is the ability of assigning an ever growing number of clusters to data. This models are thus quite flexible to change and growth. ...

The idea of ensemble methods goes back to Sir Francis Galton. In 787, he noted that although not every single person got the right value, the average estimate of a crowd of people predicted quite well. The main idea of ensemble methods is to combine relatively weak classifiers into a highly accurate predictor. The motivation for boosting was a procedure that combines the outputs of many “weak” classifiers to produce a powerful “committee.” ...

A simple motivation Fisher’s Linear Discriminant is a simple idea used to linearly classify our data. The image above, taken from (Bishop 2006), is the summary of the idea. We clearly see that if we first project using the direction of maximum variance (See Principal Component Analysis) then the data is not linearly separable, but if we take other notions into consideration, then the idea becomes much more cleaner. ...

Gaussian processes can be viewed through a Bayesian lens of the function space: rather than sampling over individual data points, we are now sampling over entire functions. They extend the idea of bayesian linear regression by introducing an infinite number of feature functions for the input XXX. In geostatistics, Gaussian processes are referred to as kriging regressions, and many other models, such as Kalman Filters or radial basis function networks, can be understood as special cases of Gaussian processes. In this framework, certain functions are more likely than others, and we aim to model this probability distribution. ...

Introduction to the course Machine learning offers a new way of thinking about reality: rather than attempting to directly capture a fragment of reality, as many traditional sciences have done, we elevate to the meta-level and strive to create an automated method for capturing it. This first lesson will be more philosophical in nature. We are witnessing a paradigm shift in the sense described by Thomas Kuhn in his theory of scientific revolutions. But what drives such a shift, and how does it unfold? ...

As we will briefly see, Kernels will have an important role in many machine learning applications. In this note we will get to know what are Kernels and why are they useful. Intuitively they measure the similarity between two input points. So if they are close the kernel should be big, else it should be small. We briefly state the requirements of a Kernel, then we will argue with a simple example why they are useful. ...

We will present some methods related to regression methods for data analysis. Some of the work here is from (Hastie et al. 2009). This note does not treat the bayesian case, you should see Bayesian Linear Regression for that. Problem setting $$ Y = \beta_{0} + \sum_{j = 1}^{d} X_{j}\beta_{j} $$We usually don’t know the distribution of $P(X)$ or $P(Y \mid X)$ so we need to assume something about these distributions. ...

In this note we will first talk about briefly some of the main differences of the three main approaches regarding statistics: the bayesian, the frequentist and the statistical learning methods and then present the concept of the estimator, compare how the approaches differ from method to method, we will explain maximum likelihood estimator and the Rao-Cramer Bound. Short introduction to the statistical methods Bayesian 🟩 $$ p(\theta \mid X) = \frac{1}{z}p(X \mid \theta) p(\theta) $$The quantity $P(X \mid \theta)$ could be very complicated if our model is complicated. ...

PAC Learning is one of the most famous theories in learning theory. Learning theory concerns in answering questions like: What is learnable? Somewhat akin to La macchina di Turing for computability theory. How well can you learn something? PAC is a framework that allows to formally answer these questions. Now there is also a bayesian version of PAC in which there is a lot of research. Some definitions Empirical Risk Minimizer and Errors $$ \arg \min_{\hat{c} \in \mathcal{H}} \hat{R}_{n}(\hat{c}) $$ Where the inside is the empirical error. ...