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Alex Chandler

Publications and source records attributed to Alex Chandler.

8 recordsLinked to original sources

Detecting Errors through Ensembling Prompts (DEEP): An End-to-End LLM Framework for Detecting Factual Errors

Accurate text summarization is one of the most common and important tasks performed by Large Language Models, where the costs of human review for an entire document may be high, but the costs of errors in summarization may be even greater. We propose Detecting Errors through Ensembling Prompts (DEEP) - an end-to-end large language model framework for detecting factual errors in text summarization. Our framework uses a diverse set of LLM prompts to identify factual inconsistencies, treating their outputs as binary features, which are then fed into ensembling models. We then calibrate the ensembled models to produce empirically accurate probabilities that a text is factually consistent or free of hallucination. We demonstrate that prior models for detecting factual errors in summaries perform significantly worse without optimizing the thresholds on subsets of the evaluated dataset. Our framework achieves state-of-the-art (SOTA) balanced accuracy on the AggreFact-XSUM FTSOTA, TofuEval Summary-Level, and HaluEval Summarization benchmarks in detecting factual errors within transformer-generated text summaries. It does so without any fine-tuning of the language model or reliance on thresholding techniques not available in practical settings.

cs.CL

Structures in HOMFLY-PT homology

We study the structure of triply graded Khovanov-Rozansky homology using both the data recently computed by Nakagane and Sano for knots up to 11 crossings, and the $\mathfrak{sl}(2)$ action defined by the second author, Hogancamp and Mellit. In particular, we compute the HOMFLY-PT $S$-invariant for all knots in the dataset, and compare it to the $\mathfrak{sl}(N)$ concordance invariants.

math.GT

Type $A$ DAHA and Doubly Periodic Tableaux

Analogously to the construction of Suzuki and Vazirani, we construct representations of the $GL_m$-type Double Affine Hecke Algebra at roots of unity. These representations are graded and the weight spaces for the $X$-variables are parametrized by the combinatorial objects we call doubly periodic tableaux. We show that our representations exhaust all graded $X$-semisimple representations, and the direct sum of all our representations is faithful. Analogously to the construction of Jordan and Vazirani of rectangular DAHA representations, we show that our representations can be interpreted in terms of ribbon fusion categories associated to $U_q(\mathfrak{gl}_N)$ at roots of unity. Combining the ribbon structure with faithfulness we deduce a conjecture of Morton and Samuelson about realization of DAHA as a skein algebra of the torus with base string modulo certain local relations.

math.RT

Thin Posets, CW Posets, and Categorification

Motivated by generalizing Khovanov's categorification of the Jones polynomial, we study functors $F$ from thin posets $P$ to abelian categories $\mathcal{A}$. Such functors $F$ produce cohomology theories $H^*(P,\mathcal{A},F)$. We find that CW posets, that is, face posets of regular CW complexes, satisfy conditions making them particularly suitable for the construction of such cohomology theories. We consider a category of tuples $(P,\mathcal{A},F,c)$, where $c$ is a certain $\{1,-1\}$-coloring of the cover relations in $P$, and show the cohomology arising from a tuple $(P,\mathcal{A},F,c)$ is functorial, and independent of the coloring $c$ up to natural isomorphism. Such a construction provides a framework for the categorification of a variety of familiar topological/combinatorial invariants: anything expressible as a rank-alternating sum over a thin poset.

math.CO

A Broken Circuit Model for Chromatic Homology Theories

Using the tools of algebraic Morse theory, and the thin poset approach to constructing homology theories, we give a categorification of Whitney's broken circuit theorem for the chromatic polynomial, and for Stanley's chromatic symmetric function.

math.CO

On the Strength of Chromatic Symmetric Homology for graphs

In this paper, we investigate the strength of chromatic symmetric homology as a graph invariant. Chromatic symmetric homology is a lift of the chromatic symmetric function for graphs to a homological setting, and its Frobenius characteristic is a q,t generalization of the chromatic symmetric function. We exhibit three pairs of graphs where each pair has the same chromatic symmetric function but distinct homology. We also show that integral chromatic symmetric homology contains torsion, and based on computations, conjecture that Z_2-torsion in bigrading (1,0) detects nonplanarity in the graph.

math.CO

Torsion in thin regions of Khovanov homology

In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported in two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb{Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of 3-braids, strictly containing all 3-strand torus links, thus giving a partial answer to Sazdanovic and Przytycki's conjecture that 3-braids have only $\mathbb{Z}_2$ torsion in Khovanov homology. We also give explicit computations of integral Khovanov homology for all links in this family.

math.GT

A Categorification of the Vandermonde Determinant

In the spirit of Bar Natan's construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers $\vec{x}=(x_1,...,x_n)$, we construct a commutative diagram in the shape of the Bruhat order on $S_n$ whose nodes are colored smoothings of the $2$-strand torus link $T_{2,n}$, and whose arrows are colored cobordisms. An application of a TQFT to this diagram yields a chain complex whose Euler characteristic is the Vandermonde determinant evaluated at $\vec{x}$. A generalization to arbitrary link diagrams is given, producing categorifications of certain generalized Vandermonde determinants. We also address functoriality of this construction.

math.CO