V2EX outer product

Outer Product

定义 Definition

outer product外积(也常称“张量积”的一种具体形式)。指两个向量相乘后得到一个矩阵:若 \(a\) 是 \(m\) 维列向量、\(b\) 是 \(n\) 维列向量,则外积 \(ab^{\mathsf T}\) 是一个 \(m\times n\) 的矩阵,其第 \((i,j)\) 个元素为 \(a_i b_j\)。在机器学习中常用于构造秩为 1(rank-1)的矩阵或做特征交互。

发音 Pronunciation

/atr prdkt/

例句 Examples

The outer product of two vectors is a matrix.
两个向量的外积是一个矩阵。

In the model, we approximate the weight matrix by an outer product \(uv^{\mathsf T}\) to obtain a rank1 representation.
在该模型中,我们用外积 \(uv^{\mathsf T}\) 来近似权重矩阵,从而得到一个秩为 1 的表示。

词源 Etymology

“outer”意为“外部的/外面的”,这里强调结果不再是一个标量(相对 inner product“内积”),而是“向外扩展”为一个矩阵或更高阶张量;“product”来自拉丁语 productus(“产生、引出”),在数学中泛指“乘积/运算结果”。

相关词 Related Words

文学与著作 Literary Works

  • Gilbert Strang, Introduction to Linear Algebra(线性代数教材中用于介绍 \(ab^{\mathsf T}\) 与秩 1 矩阵)
  • Gene H. Golub & Charles F. Van Loan, Matrix Computations(矩阵计算与分解中涉及外积更新)
  • Roger A. Horn & Charles R. Johnson, Matrix Analysis(矩阵理论语境下讨论外积与秩、投影等)
  • Ian Goodfellow, Yoshua Bengio & Aaron Courville, Deep Learning(神经网络与表示学习中常出现外积/张量积相关用法)
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