Beginner Explanation
Imagine you have a huge box of colorful marbles, and you want to find out how similar they are to each other. But it’s hard to compare them directly because there are so many colors and sizes. Fixed random projection is like taking a special camera that takes a picture of the marbles from a certain angle, simplifying the view. Instead of looking at every detail of each marble, you just see a simpler version that still shows you how similar they are. This way, you can quickly find out which marbles are alike without getting lost in all the details.Technical Explanation
Fixed Random Projection (FRP) is a dimensionality reduction technique that projects high-dimensional data into a lower-dimensional space using a fixed random matrix. The key idea is to use a matrix with entries drawn from a random distribution (e.g., Gaussian) that is fixed across different runs. This allows for efficient distance computations while preserving the relative distances between points. The projection can be defined mathematically as follows: If X is our original data matrix, the projection Y is given by Y = X * R, where R is a random matrix with dimensions (d x k) for d original dimensions and k projected dimensions. This method is particularly useful in applications like nearest neighbor search in high-dimensional spaces, where traditional distance computations become computationally expensive. Here’s a simple Python example using NumPy: “`python import numpy as np # Original data points (5 points in 10 dimensions) X = np.random.rand(5, 10) # Random projection matrix (10 dimensions to 3 dimensions) R = np.random.randn(10, 3) # Projected data points Y = X @ R “`Academic Context
Fixed Random Projection is grounded in the theory of dimensionality reduction and the Johnson-Lindenstrauss lemma, which states that a set of points in high-dimensional space can be embedded into a lower-dimensional space while approximately preserving distances. The fixed aspect of the projection matrix ensures that the same transformation is applied consistently across different datasets or iterations, which can be advantageous for reproducibility and interpretability. Key papers include ‘Random Projection and Approximation of the Johnson-Lindenstrauss Lemma’ by Achlioptas (2003) and ‘Dimensionality Reduction: A Comparative Review’ by van der Maaten et al. (2009), which discuss the theoretical underpinnings and practical implications of random projections in machine learning.Code Examples
Example 1:
import numpy as np
# Original data points (5 points in 10 dimensions)
X = np.random.rand(5, 10)
# Random projection matrix (10 dimensions to 3 dimensions)
R = np.random.randn(10, 3)
# Projected data points
Y = X @ RExample 2:
import numpy as np
# Original data points (5 points in 10 dimensions)
X = np.random.rand(5, 10)
View Source: https://arxiv.org/abs/2511.16575v1