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).

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. The columns of $U$ are called the left singular vectors of $A$, the columns of $V$ are called the right singular vectors, and the diagonal elements of $\Sigma$ are called the singular values.

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. This is where Recursive Least Squares (RLS) comes into play.

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.

An Introductory Guide (Part 1)

Even with the rapid advancements in deep learning, regression continues to be widely used across various fields (e.g., finance, data science, statistics, and so on), maintaining its importance as a fundamental algorithm. That’s why I’ve decided to share this post, which is the first article in a dedicated series on regression. This series is designed to provide a thorough review while offering a gentle and accessible introduction.