直交射影は基本的な概念です 線形代数 and geometry, used to project vectors onto a specific subspace. In simpler terms, it refers to the process of dropping a perpendicular (orthogonal) line from a point (or vector) to a line (or plane) in a vector space, resulting in the closest point in that line or plane.
数学的には、ベクトルに対して v in a vector space and a subspace defined by an 直交正規基底, the orthogonal projection is calculated using the formula:
projW(v) = Σ (v · wi) wi, where wi are the orthonormal basis vectors of the subspace W.
この射影は最小化します ユークリッド距離 between the original vector v and its projection onto the subspace, ensuring that the resulting vector is as close as possible to v ただし、依然として部分空間内にあります。
直交射影はさまざまな分野で広く使用されています、例えば コンピュータグラフィックス, where they help in rendering scenes by projecting 3D points onto 2D planes for display on screens. Additionally, they play a significant role in data science, particularly in dimensionality reduction techniques such as 主成分分析 (PCA), where data is projected onto lower-dimensional subspaces to reveal patterns and structures.