• Mathematics

    The Tensor Product: from vector spaces to categories

    Introduction General Definition The tensor product is not the same as the Cartesian product A jump to categories Applications TensorFlow Artificial Intelligence 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 \[ \left[\begin{array}{ccc} x_{1}y_{1} & x_{1}y_{2} & x_{1}y_{_{3}}\\ x_{2}y_{1} & x_{2}y_{2} & x_{2}y_{3} \end{array}\right]. \] The collection of all the products $\left\langle x_{1},x_{2}\right\rangle \otimes\left\langle y_{1},y_{2},y_{3}\right\rangle $, up to some equivalence, is called the tensor product of $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ and is denoted by $\mathbb{R}^{2}\otimes\mathbb{R}^{3}$. General Definition For any vector spaces $U,V,W$ over…

  • Mathematics

    A Generalization of Convex Sets and Convex Functions

    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 Introduction Convex sets are thought of as subsets of $\mathbb{R}^{n}$, with $n$ a nonnegative integer, and convex functions as real-valued functions on convex subsets of $\mathbb{R}^{n}$. For instance, see this paper, p. 11 and this one. We’ll show you that convex sets and convex functions can be seen in a more general framework. The setting: a vector space…

  • Mathematics

    The Rank of a Matrix with some Association with Machine Learning and Quantum Computing

    Introduction General Definition Some results involving the rank of a matrix Theorem 5.1 (Ivanyos et al., 2018) Lemma 5.2 (Ivanyos et al., 2018) Rank of a matrix in machine learning Low-rank matrix approximation  Apache Spark: a computer framework for matrix computation  Rank of a matrix in quantum information Kraus Operators   Introduction (back to outline) For a matrix \[A=\left[\begin{array}{ccc}0 & -1 & 1\\1 & 2 & -1\\1 & 1 & 3\end{array}\right],\] where each entry is a real number, the columns of $A$ form a linearly independent set of vectors of $\mathbb{R}^{3}$, so the subspace generated by the three columns is of dimension $3$, which is isomorphic to $\mathbb{R}^{3}$. The dimension…