### Summary

A Sparse Vector is a vector where most entries are zero. This form is popular in machine learning because it requires less storage and computational resources, especially for dealing with large data with many features yet only a few non-zero values.

### ELI5

Imagine you have a long list where you’ve put lots of zeros and a few numbers here and there, perhaps representing some characteristics of things we’re interested in. This sort of list is what we call a Sparse Vector.

### In-depth explanation

In a mathematical context, a vector is a one-dimensional array of numbers. A sparse vector is a kind of vector where majority of the elements are zero. This sort of data structure is very common in real-world systems and computations, particularly in machine learning and data science, due to how beneficial they are when dealing with enormous data sets an high dimensions.

Sparse vectors are instrumental in handling data with high dimensionality. In a machine learning context, if you were creating a model based on a large number of features, but each individual instance only has a handful of non-zero feature values, it would make a great deal of sense to use a sparse representation. For instance, in natural language processing (NLP), a common approach to represent a document as a set of features is by using a structure known as a sparse matrix. Each row corresponds to a document, and each column corresponds to a word, and the value in a cell represents the frequency of that word in that document. Since a single document will contain only a small subset of the total vocabulary, these vectors will largely be filled with zeros.

Another feather in the cap for sparse vectors is computational efficiency. It is computationally cheaper to multiply and add zeros than non-zero elements. Furthermore, storage is another concern; a sparse vector needs substantially less memory storage. For this reason, there are specialized formats to store and deal with sparse data such as Compressed Sparse Column(CSC) and Compressed Sparse Row (CSR).

Still, while sparse vectors have their strengths, they come with their own challenges. For instance, computation with sparse vectors often leads to denser vectors (populated with more non-zeros). This phenomenon, named “fill-in”, can significantly dampen the computational and storage advantages of sparsity. Careful handling is needed to optimize computations and ensure that sparse vectors remain advantageous.

Sparse Matrix, High Dimensionality, Feature Vector, Natural Language Processing (NLP), Compressed Sparse Column (CSC), Compressed Sparse Row (CSR), One-hot Encoding