arXiv subjects

Shanwen Hu

Publications and source records attributed to Shanwen Hu.

6 recordsLinked to original sources

arXiv2024

LegiLM: A Fine-Tuned Legal Language Model for Data Compliance

Ensuring compliance with international data protection standards for privacy and data security is a crucial but complex task, often requiring substantial legal expertise. This paper introduces LegiLM, a novel legal language model specifically tailored for consulting on data or information compliance. LegiLM leverages a pre-trained GDPR Fines dataset and has been fine-tuned to automatically assess whether particular actions or events breach data security and privacy regulations. By incorporating a specialized dataset that includes global data protection laws, meticulously annotated policy documents, and relevant privacy policies, LegiLM is optimized for addressing data compliance challenges. The model integrates advanced legal reasoning methods and information retrieval enhancements to enhance accuracy and reliability in practical legal consulting scenarios. Our evaluation using a custom benchmark dataset demonstrates that LegiLM excels in detecting data regulation breaches, offering sound legal justifications, and recommending necessary compliance modifications, setting a new benchmark for AI-driven legal compliance solutions. Our resources are publicly available at https://github.com/DAOLegalAI/LegiLM

cs.CL↗

arXiv2018

Convex hulls of unitary orbits of normal elements in $C^*$-algebras with tracial rank zero

Let $A$ be a unital separable simple $C^*$-algebra with tracial rank zero and let $x, \, y\in A$ be two normal elements. We show that $x$ is in the closure of the convex full of the unitary obit of $y$ if and only if there exists a sequence of unital completely positive linear maps $ϕ_n$ from $A$ to $A$ such that the sequence $ϕ_n(y)$ convergent to $x$ in norm and also approximately preserves the trace values. A purely measure theoretical description for normal elements in the closure of convex hull of unitary orbit of $y$ is also given. In the case that $A$ has a unique tracial state some classical results about the closure of the convex hull of the unitary orbits in von Neumann algebras are proved to be hold in $C^*$-algebras setting.

math.OA↗

arXiv2015

Distance between unitary orbits of normal elements in simple C*-algebras of real rank zero

Let $x, y$ be two normal elements in a unital simple C*-algebra $A.$ We introduce a function $D_c(x, y)$ and show that in a unital simple AF-algebra there is a constant $1>C>0$ such that $$ C\cdot D_c(x, y)\le {\rm dist}({\cal U}(x),{\cal U}(y))\le D_c(x,y), $$ where ${\cal U}(x)$ and ${\cal U}(y)$ are the closures of the unitary orbits of $x$ and of $y,$ respectively. We also generalize this to unital simple C*-algebras with real rank zero, stable rank one and weakly unperforated $K_0$-group. More complicated estimates are given in the presence of non-trivial $K_1$-information.

math.OA↗

arXiv2012

$C^*$-algebras generated by three projections

In this short note, we prove that for a $C^*$-algebra $å$ generated by $n$ elements, $M_{k}(\tildeå)$ is generated by $k$ mutually unitarily equivalent and almost mutually orthogonal projections for any $k\ge \de(n)=\min\big\{k\in\mathbb N\,|\,(k-1)(k-2)\ge 2n\big\}$. Then combining this result with recent works of Nagisa, Thiel and Winter on the generators of $C^*$--algebras, we show that for a $C^*$-algebra $å$ generated by finite number of elements, there is $d\ge 3$ such that $M_d(\tilde A)$ is generated by three mutually unitarily equivalent and almost mutually orthogonal projections. Furthermore, for certain separable purely infinite simple unital $C^*$--algebras and $AF$--algebras, we give some conditions that make them be generated by three mutually unitarily equivalent and almost mutually orthogonal projections.

math.OA↗

arXiv2012

Least Squares Problems in Orthornormalization

For any $n$-tuple $(α_1,...,α_n)$ of linearly independent vectors in Hilbert space $H$, we construct a unique orthonormal basis $(ε_1,...,ε_n)$ of $span\{α_1,...,α_n\}$ satisfying: $$\sum_{i=1}^n\|ε_i-α_i\|^2\le\sum_{i=1}^n\|β_i-α_i\|^2$$ for all orthonormal basis $(β_1,...,β_n)$ of $span\{α_1,...,α_n\}$. We study the stability of the orthornormalization and give some applications and examples.

math.FA↗

arXiv2012

Completeness of $n$--tuple of projections in $C^*$--algebras

Let $(P_1,...,P_n)$ be an $n$--tuple of projections in a unital $C^*$--algebra $å$. We say $\pn$ is complete in $å$ if $å$ is the linear direct sum of the closed subspaces $P_1å,...,P_nå$. In this paper, we give some necessary and sufficient conditions for the completeness of $\pn$ and discuss the perturbation problem and topology of the set of all complete $n$--tuple of projections in $å$. Some interesting and significant results are obtained in this paper.

math.OA↗