What characterizes a sparse matrix?

A sparse matrix is characterized by having most of its elements as zero. This feature is significant in various fields, particularly in data mining and machine learning, where data sets can have a large number of variables (or dimensions) but only a few of them are non-zero for any given example or instance.

The concept of sparsity is essential because it influences the efficiency of algorithms used to process the matrix. Sparse matrices allow for specialized storage schemes that save memory and can lead to faster computations, as operations can ignore the zero values. This characteristic is especially prevalent in scenarios such as recommendation systems, natural language processing (where term-document matrices can be sparse), and large-scale machine learning applications.

The other options do not accurately encapsulate the defining features of a sparse matrix. For instance, a sparse matrix is not characterized by containing numerous non-zero elements or being restricted to only representing numerical data, as it can represent categorical data as well. Additionally, it is not specifically used primarily for image processing, even though sparse representations can sometimes be advantageous in that field. These distinctions clarify why having most elements as zero is the defining feature of a sparse matrix.