densevector

A dense vector is an ordered collection of numbers that represents a point in a real coordinate space, typically R^n. In a dense vector, every component is stored explicitly, even if some components are zero. This contrasts with sparse representations, which store only nonzero elements and their positions to save memory when many entries are zero.

In practice, dense vectors are implemented as one-dimensional arrays of numbers, such as floating-point values, and are stored in contiguous memory to support efficient access and vectorized operations. Common operations include indexing, element-wise addition and subtraction, scalar multiplication, dot products, and computation of norms. Many numerical libraries provide optimized routines that take advantage of CPU features likeSIMD to accelerate these operations on dense vectors.

Dense vectors are widely used when the dimensionality is moderate and the majority of elements are nonzero, or when simplicity and speed are prioritized. They are easier to manipulate than sparse representations and often incur less overhead in memory management. However, they can be wasteful in memory when vectors are high-dimensional and contain many zeros, which is where sparse vectors are preferable.

In software and libraries, dense vectors appear under various names. For example, in Python’s NumPy, a dense vector is a one-dimensional array. In Apache Spark’s MLlib, a DenseVector stores elements as a primitive array of doubles and is used in many machine learning algorithms alongside a SparseVector option. Dense vectors are fundamental to linear algebra, machine learning workflows, and numerical simulations.