Einstein Finsler Metrics and Killing Vector Fields on Riemannian Manifolds

In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${\rm Ric} = 2 F^2$, ${\rm Ric}=0$ and ${\rm Ric}=- 2 F^2$, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not

On the flag curvature of Finsler metrics of scalar curvature

The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In this paper, we study Finsler metrics of scalar curvature (i.e., the flag curvature is a scalar function on the slit tangent bundle) and partially determine the flag curvature when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, we classify locally projectively flat Randers metrics with isotropic S-curvature.Comment: 23 page

"Do Anything Now": Characterizing and Evaluating In-The-Wild Jailbreak Prompts on Large Language Models

The misuse of large language models (LLMs) has garnered significant attention from the general public and LLM vendors. In response, efforts have been made to align LLMs with human values and intent use. However, a particular type of adversarial prompts, known as jailbreak prompt, has emerged and continuously evolved to bypass the safeguards and elicit harmful content from LLMs. In this paper, we conduct the first measurement study on jailbreak prompts in the wild, with 6,387 prompts collected from four platforms over six months. Leveraging natural language processing technologies and graph-based community detection methods, we discover unique characteristics of jailbreak prompts and their major attack strategies, such as prompt injection and privilege escalation. We also observe that jailbreak prompts increasingly shift from public platforms to private ones, posing new challenges for LLM vendors in proactive detection. To assess the potential harm caused by jailbreak prompts, we create a question set comprising 46,800 samples across 13 forbidden scenarios. Our experiments show that current LLMs and safeguards cannot adequately defend jailbreak prompts in all scenarios. Particularly, we identify two highly effective jailbreak prompts which achieve 0.99 attack success rates on ChatGPT (GPT-3.5) and GPT-4, and they have persisted online for over 100 days. Our work sheds light on the severe and evolving threat landscape of jailbreak prompts. We hope our study can facilitate the research community and LLM vendors in promoting safer and regulated LLMs

Confidence bands for a log-concave density

We present a new approach for inference about a log-concave distribution: Instead of using the method of maximum likelihood, we propose to incorporate the log-concavity constraint in an appropriate nonparametric confidence set for the cdf $F$. This approach has the advantage that it automatically provides a measure of statistical uncertainty and it thus overcomes a marked limitation of the maximum likelihood estimate. In particular, we show how to construct confidence bands for the density that have a finite sample guaranteed confidence level. The nonparametric confidence set for $F$ which we introduce here has attractive computational and statistical properties: It allows to bring modern tools from optimization to bear on this problem via difference of convex programming, and it results in optimal statistical inference. We show that the width of the resulting confidence bands converges at nearly the parametric $n^{-\frac{1}{2}}$ rate when the log density is $k$-affine.Comment: Added more experiments, other minor change

In ChatGPT We Trust? Measuring and Characterizing the Reliability of ChatGPT

The way users acquire information is undergoing a paradigm shift with the advent of ChatGPT. Unlike conventional search engines, ChatGPT retrieves knowledge from the model itself and generates answers for users. ChatGPT's impressive question-answering (QA) capability has attracted more than 100 million users within a short period of time but has also raised concerns regarding its reliability. In this paper, we perform the first large-scale measurement of ChatGPT's reliability in the generic QA scenario with a carefully curated set of 5,695 questions across ten datasets and eight domains. We find that ChatGPT's reliability varies across different domains, especially underperforming in law and science questions. We also demonstrate that system roles, originally designed by OpenAI to allow users to steer ChatGPT's behavior, can impact ChatGPT's reliability. We further show that ChatGPT is vulnerable to adversarial examples, and even a single character change can negatively affect its reliability in certain cases. We believe that our study provides valuable insights into ChatGPT's reliability and underscores the need for strengthening the reliability and security of large language models (LLMs)