Vector transformation technique

Question

I have vectors (embeddings) in a particular space and I wish to calculate the distance between them. So far, chordal distance has been the most helpful. However, I'd like to map these vectors so that my similarity metric (chordal distance) in the original space is equivalent to dot product search in this new mapped space. What are the approaches to do this?

Answer 1

There are several methods to map high-dimensional vectors to a new space while preserving the similarity metric:

Isometric Mapping: Isometric Mapping (also known as Isomap) uses a combination of multidimensional scaling and graph theory to create a low-dimensional representation of the original data while preserving the geometric relationships between the points.

Multidimensional Scaling: Multidimensional Scaling (MDS) is a method to convert a dissimilarity matrix (representing the distance between points) into a set of coordinates in a lower-dimensional space while preserving the relationships as best as possible.

t-SNE: t-SNE (t-distributed Stochastic Neighbor Embedding) is a dimensionality reduction technique that is specifically designed to handle dense and highly non-linear data. It maps the data to a low-dimensional space while preserving the relationships between the points.

PCA: Principal Component Analysis (PCA) is a linear dimensionality reduction technique that projects the data onto a lower-dimensional space by maximizing the variance of the projected data.

LLE: Locally Linear Embedding (LLE) is a non-linear dimensionality reduction technique that maps the data to a lower-dimensional space while preserving the relationships between nearby points.

All of these methods have their own advantages and limitations, so you would have to choose the one that best suits your specific requirements.