I recently bought a mini PC because I wanted a lightweight machine that I can easily carry anywhere. Arch Linux’s minimalistic, rolling-release approach aligns perfectly with my love for a Vim-based workflow and a highly customizable setup. While the process can seem intimidating at first, it’s an incredibly rewarding experience that offers complete control over your system.

Installing Arch Linux (UEFI or BIOS)

Arch Linux is well-known for giving users full control over their system. This guide walks you through a fresh Arch Linux installation. While it is detailed, always refer to the official Arch Wiki for up-to-date information.

Introduction to Asymmetric Kernels

Recall that the dual form of LS-SVM is given by \begin{align*} \begin{bmatrix} 0 & y^T \\ y & \Omega + \frac{1}{\gamma} I \end{bmatrix} \begin{bmatrix} b \\ \alpha \end{bmatrix} = \begin{bmatrix} 0 \\ e \end{bmatrix} \end{align*} An interesting point here is that using an asymmetric kernel in LS-SVM will not reduce to its symmetrization and asymmetric information can be learned. Then we can develop asymmetric kernels in the LS-SVM framework in a straightforward way.

Introduction to Least-Square SVM

Introduction

Least Squares Support Vector Machine (LS-SVM) is a modified version of the traditional Support Vector Machine (SVM) that simplifies the quadratic optimization problem by using a least squares cost function. LS-SVM transforms the quadratic programming problem in classical SVM into a set of linear equations, which are easier and faster to solve.

Optimization Problem (Primal Problem)

\begin{align*} &\min_{w, b, e} \frac{1}{2} \lVert w\rVert^2 + \frac{\gamma}{2} \sum_{i=1}^N e_i^2,\\ &\text{subject to } y_i (w^T \phi(x_i) + b) = 1 - e_i, \ \forall i \end{align*} where:

Support Vector Machine

Introduction

Support Vector Machines (SVMs) are among the most effective and versatile tools in machine learning, widely used for various tasks. SVMs work by finding the optimal boundary, or hyperplane, that separates different classes of data with the maximum margin, making them highly reliable for classification, especially with complex datasets.

What truly sets SVMs apart is their ability to handle both linear and non-linear data through the kernel trick, allowing them to adapt to a wide range of problems with impressive accuracy. In this blog post, we’ll delve into how SVMs work and gently explore the mathematical foundations behind their powerful performance.

On Gradient Descent

Gradient descent is an optimization algorithm used to minimize a function by iteratively moving towards the function’s minimum value. It is a fundamental concept in machine learning, particularly in training models such as neural networks. The gradient is a vector that represents the direction of the steepest increase of the function at a given point. For example, for a convex function $z = ax^2 + by^2$, the gradient is $[2ax, 2by]$, which points in the direction of the steepest ascent.

Latent Variable Modeling

Motivation of Latent Variable Modeling

Let’s say we want to classify some data. If we had access to a corresponding latent variable for each observation $ \mathbf{x}_i $, modeling would be more straightforward. To illustrate this, consider the challenge of finding the latent variable (i.e., the true class of $ \mathbf{x} $). It can be expressed like $ z^* = \argmax_{z} p(\mathbf{x} | z) $. It is hard to identify the true clusters without prior knowledge about them. For example, we can cluster like Fig. (b) or (c).