In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, \mathbf{x},\ \mathbf{y} \in \mathbb{R}^n, we say that \mathbf{x} weakly majorizes (or dominates) \mathbf{y} from below, commonly denoted \mathbf{x} \succ_w \mathbf{y}, when \sum_{i=1}^k x_i^{\downarrow} \geq \sum_{i=1}^k y_i^{\downarrow} for all k=1,\,\dots,\,n, where x_i^{\downarrow} denotes the ith largest entry of \mathbf{x}. If \mathbf{x}, \mathbf{y} further satisfy \sum_{i=1}^n x_i = \sum_{i=1}^n y_i, we say that \mathbf{x} majorizes (or dominates) \mathbf{y} , commonly denoted \mathbf{x} \succ \
In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, \mathbf{x},\ \mathbf{y} \in \mathbb{R}^n, we say that \mathbf{x} weakly majorizes (or dominates) \mathbf{y} from below, commonly denoted \mathbf{x} \succ_w \mathbf{y}, when \sum_{i=1}^k x_i^{\downarrow} \geq \sum_{i=1}^k y_i^{\downarrow} for all k=1,\,\dots,\,n, where x_i^{\downarrow} denotes the ith largest entry of \mathbf{x}. If \mathbf{x}, \mathbf{y} further satisfy \sum_{i=1}^n x_i = \sum_{i=1}^n y_i, we say that \mathbf{x} majorizes (or dominates) \mathbf{y} , commonly denoted \mathbf{x} \succ \mathbf{y}.
Both weak majorization and majorization are partial orders for vectors whose entries are non-decreasing, but only a preorder for general vectors, since majorization is agnostic to the ordering of the entries in vectors, e.g., the statement (1,2)\prec (0,3) is simply equivalent to (2,1)\prec (3,0).
Discovered by embedding cosine similarity (sentence-transformers MiniLM, 384-dim).