Nonelementwise

Nonelementwise is a term used in mathematics and computer science to describe an operation on arrays, tensors, or matrices whose output at a given position depends on more than just the elements at the corresponding positions of the inputs. In contrast, an elementwise operation applies a function independently to each element pair by pair, producing outputs that align element-for-element with the inputs.

In practice, nonelementwise operations include many core linear algebra and tensor operations. Matrix multiplication (and its generalized forms, such as tensor contractions) combines information across rows and columns, so each output element depends on many input elements. Other examples are dot products, outer products, solving linear systems, computing determinants, and matrix inverses. Convolutions in many neural network contexts also fall into this category, as the result at a location depends on a neighborhood of input elements through a sliding window.

Some ordinary-looking transforms and normalizations are nonelementwise as well. For instance, softmax involves exponentiating each input element and then normalizing by a shared sum over all elements, coupling all outputs together rather than treating each position independently. Reductions, aggregations, and operations that mix indices across axes—such as summing along an axis, reshaping combined with contractions, or performing a matrix rank operation—are typically nonelementwise.

Understanding whether an operation is elementwise or nonelementwise is important for performance and differentiability, especially in numerical libraries and automatic differentiation frameworks. Many frameworks implement elementwise operations efficiently as straightforward per-element computations, while nonelementwise operations often require more complex indexing, memory access patterns, and potentially different gradient rules. See also: elementwise operations, broadcasting, tensor contractions.