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.

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.

Singular Value Decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square matrix by extending the concept to asymmetric or rectangular matrices, which cannot be diagonalized directly using eigendecomposition. The SVD aims to find the following decomposition of a real-valued matrix $A$: $$A = U\Sigma V^T,$$ where $U$ and $V$ are orthogonal (orthonormal) matrices, and $\Sigma$ is a diagonal matrix.

Deep Dive into Regression: Recursive Least Squares Explained (Part 3) Introduction to Recursive Least Squares Ordinary least squares assumes that all data is available at once, but in practice, this isn’t always the case. Often, measurements are obtained sequentially, and we need to update our estimates as new data comes in. Simply augmenting the data matrix $\mathbf{X}$ each time a new measurement arrives can become computationally expensive, especially when dealing with a large number of measurements.

An Introductory Guide (Part 2) Understanding Ridge Regression In machine learning, one of the key challenges is finding the right balance between underfitting and overfitting a model. Overfitting occurs when a model is too complex and captures not only the underlying patterns in the training data but also the noise. This results in a model that performs well on the training data but poorly on new, unseen data. Underfitting, on the other hand, happens when a model is too simple to capture the underlying patterns in the data, leading to poor performance both on the training data and on new data.