Sparse vector

See also: Machine learning terms

Introduction

In the field of machine learning, a sparse vector is a vector representation of data that contains a significant number of zero-valued elements. Sparse vectors are widely used in various applications, such as natural language processing, information retrieval, and recommender systems, to name a few. This article will discuss the concept of sparse vectors, their properties, and applications in machine learning.

Sparse Vectors and their Properties

Definition

A sparse vector is a mathematical construct that represents a high-dimensional vector with a large number of zero-valued elements. In other words, only a few elements in the vector are non-zero, while the majority are zeros. Mathematically, a sparse vector can be denoted as v = (v₁, v₂, ..., vₙ), where n is the dimension of the vector, and most of the elements vᵢ are zero.

Sparsity

The concept of sparsity refers to the proportion of zero elements in a vector or matrix. The degree of sparsity can be measured using various metrics, such as the sparsity ratio, which is defined as the number of zero elements divided by the total number of elements in the vector or matrix. A higher sparsity ratio indicates that the data is more sparse, while a lower ratio signifies denser data.

Applications of Sparse Vectors in Machine Learning

Sparse vectors play a crucial role in many machine learning applications. Some of the most common applications are:

Natural Language Processing

In natural language processing (NLP), sparse vectors are commonly used to represent text documents in the form of bag-of-words or term frequency-inverse document frequency (TF-IDF) representations. These representations are sparse because the vocabulary of a language is vast, and each document contains only a small subset of the possible words.

Information Retrieval

In information retrieval systems, sparse vectors are used to represent documents and queries. The documents and queries are represented as high-dimensional vectors, where each dimension corresponds to a unique term or concept. Due to the vast number of possible terms, most of the elements in these vectors are zero, making them sparse.

Recommender Systems

In recommender systems, sparse vectors are used to represent user preferences and item features. Users typically rate or interact with only a small fraction of the available items, resulting in a sparse representation of their preferences. Similarly, items usually have only a small number of features or attributes that are relevant, leading to sparse feature vectors.

Explain Like I'm 5 (ELI5)

Imagine you have a long row of buckets, and each bucket can hold a number. Most of the buckets are empty (have a zero in them), but a few buckets have some numbers in them. This row of buckets is like a sparse vector.

In machine learning, we use sparse vectors to represent things like words in a document or users' preferences. Since there are so many possible words or preferences, and each document or user only has a few of them, we end up with lots of empty buckets (zeros) and just a few buckets with numbers (non-zero elements).