• The Tensor Product: from vector spaces to categories
    Mathematics

    The Tensor Product: from vector spaces to categories

      OUTLINE Introduction General Definition The tensor product is not the same as the Cartesian product A jump to categories Applications TensorFlow Artificial Intelligence Quantum Optics   SUMMARY:¬†We show why the tensor product is not the same as the Cartesian product, and we extend that result to categories. We then mention the use of the tensor product in applications, such as artificial intelligence and quantum optics. Introduction For any vectors $\left\langle x_{1},x_{2}\right\rangle $ of $\mathbb{R}^{2}$ and $\left\langle y_{1},y_{2},y_{3}\right\rangle $ of $\mathbb{R}^{3}$, a product of these two vectors, which is denoted as $\left\langle x_{1},x_{2}\right\rangle \otimes\left\langle y_{1},y_{2},y_{3}\right\rangle $, is defined as the matrix \[ \left[\begin{array}{ccc} x_{1}y_{1} & x_{1}y_{2} & x_{1}y_{_{3}}\\ x_{2}y_{1} &…

  • math convex functions
    Mathematics

    A Generalization of Convex Sets and Convex Functions

      OUTLINE Introduction The setting: a vector space over a field with a partial order A Convex set is the same as an order-convex set A convex function defined in terms of a convex set A convex function in terms of order-convexity Convex sets and convex functions in machine learning Convex Optimization Software Libraries   SUMMARY: We define a convex set in a general framework of a vector space over a field with a partial order, and we show how the general notion is related to the usual notion of a convex set. Then we define a convex function in terms of that general notion of a convex set, and…