“There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man’s life to know them the little new that each man gets from life is very costly and the only heritage he has to leave.” - Ernest Hemingway (More…)
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Virtual Network Embedding
Virtual Network Embedding # Virtual Network Embedding (VNE) is the problem of mapping virtual network components (nodes and links) onto a physical network infrastructure, optimizing resource utilization and quality of service. Recent Literature # Deep Reinforcement Learning for Virtual Network Embedding ACM SIGCOMM, 2020. paper, code Zhu Ren, Liang Hong, Wei Zhang GNN-based Reinforcement Learning for Virtual Network Embedding IEEE ICDCS, 2021. paper Yikang Wang, Zhu Ren, Mingwei Xu, Wei Zhang Neural Network Assisted Heuristics for Virtual Network Embedding IEEE INFOCOM, 2021. paper
A lemma of J. L. Lions
This post explores J. L. Lions’ lemma about Banach spaces with compact injection, including applications to functional analysis. Lemma statement: Let $X$, $Y$, and $Z$ be three Banach spaces with norms $|| \cdot ||_X$, $|| \cdot ||_Y$, and $|| \cdot ||_Z$. Assume that $X \subset Y$ with compact injection and that $Y \subset Z$ with continuous injection. Prove that $$ \forall \varepsilon > 0, \exists C_\varepsilon > 0 \text{ satisfying } || u ||_Y \leq \varepsilon || u ||_X + C _{\varepsilon}|| u ||_Z,\quad \forall u \in X $$
Complex Hahn-Banach Theorem
Let $X$ be a complex vector space, $X_0$ one of its subspaces, $p: X \to \mathbb{R}_+$ such that $$ p(\lambda x) = |\lambda| p(x), \quad \forall \lambda \in \mathbb{C}, x \in X \text{ and } p(x + y) \leq p(x) + p(y), \quad \forall x, y \in X, $$ satisfying $|f(x)| \leq p(x)$, $\forall x \in X_0$, where $f: X_0 \to \mathbb{C}$ is linear. Under these conditions, there exists a linear functional $F: X \to \mathbb{C}$ such that $F|_{X_0} = f$ and
Real Hahn-Banach Theorem
Suppose $X$ is a vector space over $\mathbb{R}$, $p: X \to \mathbb{R}$ has the following properties: $p(X) = \lambda p(x)$, $\forall x \in X$, $\lambda \in \mathbb{R}_+$ and $p(x + y) \leq p(x) + p(y)$, $\forall x, y \in X$. Let $X_0$ be a subspace of $X$ and $u: X_0 \to \mathbb{R}$ a linear functional such that $u(x) \leq p(x)$, $\forall x \in X_0$. Then we can find $f: X \to \mathbb{R}$ a linear functional such that $f|_{X_0} = u$ and $f(x) \leq u(x)$, $\forall x \in X$.
Riesz Representation Theorem
1. Riesz Representation Theorem # Let $H$ be a Hilbert space over $\mathbb{R}$ or $\mathbb{C}$, and $T$ be a bounded linear functional on $H$ (a bounded operator from $H$ to the field $\mathbb{R}$ or $\mathbb{C}$, where $H$ is defined over that field). The following is known as the Riesz Representation Theorem: Theorem 1: If $T$ is a bounded linear functional on the Hilbert space $H$, then there exists $g \in H$ such that for every $f \in H$, we have: $$ T(f) = \langle f, g \rangle. $$