arXiv subjects

Meghana Bhat

Publications and source records attributed to Meghana Bhat.

7 recordsLinked to original sources

arXiv2025

Waring decompositions of the product of two quadrics: the small rank cases

In this paper we study forms of the type $(x_1^2+ \cdots +x_m^2)(y_1^2+ \cdots+y_n^2)$ using projections. For $m=1, m=2$, and for any $n$ we describe: the forbidden locus, the structure and the Hilbert function of all minimal apolar sets. In particular, we show that every minimal apolar ideal has the same Hilbert function. Further, we compute the cactus rank, a bound on the border rank, and the dimension of the Variety of Sums of Powers. For $m,n \geq 3,$ we provide new lower and upper bounds for the Waring rank.

math.AC↗

arXiv2025

Symmetric decomposition of the Hilbert function of an ideal

Let $(R, \mathcal{M})$ be a local ring over a field $k$ with $k = R/\mathcal M$ and $J$ an ideal in $R$ such that $A =R/J$ is an Artinian Gorenstein (AG) $k$-algebra. In 1989, A. Iarrobino introduced the symmetric decomposition of the Hilbert function of $A$. This became a very powerful tool for classifying the Hilbert functions of AG $k$-algebras. In this article, we introduce the symmetric decomposition of the Hilbert function of any ideal $I$ in $A.$ Our hope is that this result will be useful in classifying the possible Hilbert function of an ideal in an AG $k$-algebra. We illustrate this by giving a complete list of $2$-admissible sequences of length at most $3$ and with $h_0=2$ that are realizable by an ideal in an AG $k$-algebra.

math.AC↗

arXiv2024

On Gorensteinness of associated graded rings of filtrations

Let $(A, \mathfrak{m})$ be a Gorenstein local ring, and $\mathcal{F} =\{F_n \}_{n\in \mathbb{Z}}$ a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of $\mathcal{F}$ in terms of the Hilbert coefficients of $\mathcal{F}$ in some cases. As a consequence we recover and extend a result proved by Okuma, Watanabe and Yoshida. Further, we present ring-theoretic properties of the normal tangent cone of the maximal ideal of $A=S/(f)$ where $S=K[\![x_0,x_1,\ldots, x_m]\!]$ is a formal power series ring over an algebraically closed field $K$, and $f=x_0^a-g(x_1,\ldots,x_m)$, where $g$ is a polynomial with $g \in (x_1,\ldots,x_m)^b \setminus (x_1,\ldots,x_m)^{b+1}$, and $a, \, b, \, m$ are integers. We show that the normal tangent cone $\overline{G}(\mathfrak{m})$ is Cohen-Macaulay if $A$ is normal and $a \le b$. Moreover, we give a criterion of the Gorensteinness of $\overline{G}(\mathfrak{m})$.

math.AC↗

arXiv2023

XGen-7B Technical Report

Large Language Models (LLMs) have become ubiquitous across various domains, transforming the way we interact with information and conduct research. However, most high-performing LLMs remain confined behind proprietary walls, hindering scientific progress. Most open-source LLMs, on the other hand, are limited in their ability to support longer sequence lengths, which is a key requirement for many tasks that require inference over an input context. To address this, we have trained XGen, a series of 7B parameter models on up to 8K sequence length for up to 1.5T tokens. We have also finetuned the XGen models on public-domain instructional data, creating their instruction-tuned counterparts (XGen-Inst). We open-source our models for both research advancements and commercial applications. Our evaluation on standard benchmarks shows that XGen models achieve comparable or better results when compared with state-of-the-art open-source LLMs. Our targeted evaluation on long sequence modeling tasks shows the benefits of our 8K-sequence models over 2K-sequence open-source LLMs.

cs.CL↗

arXiv2023

Large Language Models as Tax Attorneys: A Case Study in Legal Capabilities Emergence

Better understanding of Large Language Models' (LLMs) legal analysis abilities can contribute to improving the efficiency of legal services, governing artificial intelligence, and leveraging LLMs to identify inconsistencies in law. This paper explores LLM capabilities in applying tax law. We choose this area of law because it has a structure that allows us to set up automated validation pipelines across thousands of examples, requires logical reasoning and maths skills, and enables us to test LLM capabilities in a manner relevant to real-world economic lives of citizens and companies. Our experiments demonstrate emerging legal understanding capabilities, with improved performance in each subsequent OpenAI model release. We experiment with retrieving and utilising the relevant legal authority to assess the impact of providing additional legal context to LLMs. Few-shot prompting, presenting examples of question-answer pairs, is also found to significantly enhance the performance of the most advanced model, GPT-4. The findings indicate that LLMs, particularly when combined with prompting enhancements and the correct legal texts, can perform at high levels of accuracy but not yet at expert tax lawyer levels. As LLMs continue to advance, their ability to reason about law autonomously could have significant implications for the legal profession and AI governance.

cs.CL↗

arXiv2023

Few-shot Unified Question Answering: Tuning Models or Prompts?

Question-answering (QA) tasks often investigate specific question types, knowledge domains, or reasoning skills, leading to specialized models catering to specific categories of QA tasks. While recent research has explored the idea of unified QA models, such models are usually explored for high-resource scenarios and require re-training to extend their capabilities. To overcome these drawbacks, the paper explores the potential of two paradigms of tuning, model, and prompts, for unified QA under a low-resource setting. The paper provides an exhaustive analysis of their applicability using 16 QA datasets, revealing that prompt tuning can perform as well as model tuning in a few-shot setting with a good initialization. The study also shows that parameter-sharing results in superior few-shot performance, simple knowledge transfer techniques for prompt initialization can be effective, and prompt tuning achieves a significant performance boost from pre-training in a low-resource regime. The research offers insights into the advantages and limitations of prompt tuning for unified QA in a few-shot setting, contributing to the development of effective and efficient systems in low-resource scenarios.

cs.CL↗

arXiv2022

AugTriever: Unsupervised Dense Retrieval and Domain Adaptation by Scalable Data Augmentation

Dense retrievers have made significant strides in text retrieval and open-domain question answering. However, most of these achievements have relied heavily on extensive human-annotated supervision. In this study, we aim to develop unsupervised methods for improving dense retrieval models. We propose two approaches that enable annotation-free and scalable training by creating pseudo querydocument pairs: query extraction and transferred query generation. The query extraction method involves selecting salient spans from the original document to generate pseudo queries. On the other hand, the transferred query generation method utilizes generation models trained for other NLP tasks, such as summarization, to produce pseudo queries. Through extensive experimentation, we demonstrate that models trained using these augmentation methods can achieve comparable, if not better, performance than multiple strong dense baselines. Moreover, combining these strategies leads to further improvements, resulting in superior performance of unsupervised dense retrieval, unsupervised domain adaptation and supervised finetuning, benchmarked on both BEIR and ODQA datasets. Code and datasets are publicly available at https://github.com/salesforce/AugTriever.

cs.CL↗