• 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…

• 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…