A sparse vector is an array or list where most elements have a value of zero. Unlike dense vectors, which store mostly non-zero values, sparse vectors optimize storage and computation by focusing only on the elements that contain meaningful data. This is particularly useful in scenarios where data dimensions are high but actual values are rare, such as in natural language processing (NLP) or recommendation systems. For example, a text document represented as a bag-of-words vector might have thousands of dimensions (one per word), but only a few dozen non-zero values corresponding to words present in the document. Sparse vectors avoid wasting memory on zeros by storing only the indices and values of non-zero elements.
Sparse vectors are typically represented using data structures that prioritize efficiency. Common formats include Coordinate List (COO), Compressed Sparse Row (CSR), and Dictionary of Keys (DOK). For instance, in Python, a sparse vector might be stored as a dictionary where keys are indices and values are the non-zero entries. Libraries like SciPy provide optimized sparse matrix classes that handle these representations internally. For example, a vector with non-zero values at positions 3 and 7 (values 5 and 9) could be stored as {3: 5, 7: 9} instead of [0, 0, 0, 5, 0, 0, 0, 9]. This reduces memory usage and speeds up operations like dot products, as only non-zero elements are processed.
The primary advantages of sparse vectors are reduced memory overhead and faster computation for certain operations. In machine learning, algorithms like TF-IDF or one-hot encoding often produce sparse data, and using sparse vectors avoids unnecessary storage costs. For example, training a classifier on text data with 10,000 features per sample would require 99% less memory if 99% of features are zeros. Similarly, mathematical operations like vector addition or matrix multiplication can skip zero entries, leading to performance gains. However, sparse vectors are not universally optimal—operations requiring full-vector scans (e.g., finding all values above a threshold) may become slower due to the need to track indices. Choosing between sparse and dense representations depends on the specific use case and the sparsity level of the data.