arXiv subjects
Publications and source records attributed to Victor Wang.
We present a short, digested, human-verified version of the recent OpenAI-generated counterexample to the Erd\H{o}s unit distance conjecture, and a sequence of reflections on it. The argument relies crucially on ideas that may, at least in retrospect, be attributed to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.
Email importance labeling has long been a critical yet challenging problem for businesses and individuals. Traditional approaches; such as keyword matching, user-defined rules, and sender-based heuristics; demand extensive manual feature engineering and fail to scale effectively or generalize. Recent advances in large language models (LLMs) demonstrate strong potential and a natural fit for this task, offering deep contextual understanding and superior labeling quality. However, using LLM models like GPT-4.1 at enterprise email volumes incurs prohibitive computational costs and hinders real-world deployment. We explore the trade-off space of using alternative labeling schemes as opposed to GPT4.1 scale LLMs, with the goal of achieving near GPT level labeling quality with significantly lower cost. We develop Argo, an enterprise email labeling framework, where we construct a profiler to efficiently search the cost quality trade-off space of labeling and identify cost-efficient alternatives to labeling emails. Additionally, we design an on-demand provisioning scheme to intelligently scale Argo with real time load, to minimize cost increases during peak load inference. Over 3 open-source email datasets, Argo achieves 148-167X inference cost reduction with negligible quality degradation and 20-640000X lower profiling costs, making large-scale, context-aware email labeling practical for enterprises.
Calibrated confidence estimates are necessary for large language model (LLM) outputs to be trusted by human users. While LLMs can express their confidence in human-interpretable ways, verbalized LLM-generated confidence scores have empirically been found to be miscalibrated, reporting high confidence on instances with low accuracy and thereby harming trust and safety. We hypothesize that this overconfidence often stems from a given LLM's heightened suggestibility when faced with claims that it encodes little information about; we empirically validate this hypothesis, finding more suggestibility on lower-accuracy claims. Building on this finding, we introduce Distractor-Normalized Coherence (DINCO), which estimates and accounts for an LLM's suggestibility bias by having the model verbalize its confidence independently across several self-generated distractors (i.e. alternative claims), and normalizes by the total verbalized confidence. To further improve calibration, we leverage generator-validator disagreement, augmenting normalized validator confidence with a consistency-based estimate of generator confidence. Here, we frame the popular approach of self-consistency as leveraging coherence across sampled generations, and normalized verbalized confidence as leveraging coherence across validations on incompatible claims, allowing us to integrate these complementary dimensions of coherence into DINCO. Moreover, our analysis shows that DINCO provides less saturated -- and therefore more usable -- confidence estimates, and that further sampling alone cannot close the gap between DINCO and baselines, with DINCO at 10 inference calls outperforming self-consistency at 100.
We study an alternating sum involving factorials and Stirling numbers of the first kind. We give an exponential generating function for these numbers and show they are nonnegative and enumerate the number of increasing trees on $n$ vertices that are won by the second of two players when interpreted as a game tree. We also give a simple description of the quotient from the weak order to the Tamari lattice in terms of plane trees, and give bijections between plane trees, 213-avoiding permutations, and 312-avoiding permutations. Finally, for a rooted tree, we give equivalent characterizations of when it describes a game won by the first or second player in terms of the rank-generating function of the lattice of prunings and the Euler characteristic of an associated real variety.
We prove a simple formula expanding the unicellular LLT polynomials of a class of graphs we call two-headed melting lollipops into ribbon Schur functions. Our work extends the Schur expansion originally found for melting lollipop graphs by Huh, Nam, and Yoo.
Using language models to scalably approximate human preferences on text quality (LLM-as-a-judge) has become a standard practice applicable to many tasks. A judgment is often extracted from the judge's textual output alone, typically with greedy decoding. However, LLM judges naturally provide distributions over judgment tokens, inviting a breadth of inference methods for extracting fine-grained preferences. We find that taking the mean of the judgment distribution consistently outperforms taking the mode (i.e. greedy decoding) in all evaluation settings (i.e. pointwise, pairwise, and listwise). We further explore novel methods of deriving preferences from judgment distributions, and find that methods incorporating risk aversion often improve performance. Lastly, we analyze LLM-as-a-judge paired with chain-of-thought (CoT) prompting, showing that CoT can collapse the spread of the judgment distribution, often harming performance. Our findings show that leveraging distributional output improves LLM-as-a-judge, as opposed to using the text interface alone.
We develop the theory of weighted P-partitions, which generalises the theory of P-partitions from labelled posets to weighted labelled posets. We define the related generating functions in the natural way and compute their product, coproduct and other properties. As an application we introduce the basis of combinatorial power sums for the Hopf algebra of quasisymmetric functions and the reverse basis, both of which refine the power sum symmetric functions. These bases share many properties with the type 1 and type 2 quasisymmetric power sums introduced by Ballantine, Daugherty, Hicks, Mason and Niese, and moreover expand into the monomial basis of quasisymmetric functions with nonnegative integer coefficients. We prove formulas for products, coproducts and classical quasisymmetric involutions via the combinatorics of P-partitions, and give combinatorial interpretations for the coefficients when expanded into the monomial and fundamental bases.
We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the weighted Laplacian also satisfies a vertex analogue of deletion-contraction. We give applications of weighted Laplacian eigenvalues to sparse cuts, independent sets and graph colouring, and establish new cases of a conjecture of Stanley on distinguishing nonisomorphic trees.
We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of e-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function $Y_G$ in noncommuting variables in this quotient algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph G centred at a vertex v. We then apply our methods to $y_{G :v}$ and find new families of unit interval graphs that are (e)-positive, a stronger condition than classical e-positivity, thus confirming new cases of the (3+1)-free conjecture of Stanley and Stembridge. In our study of $y_{G : v}$, we also describe methods of constructing new e-positive graphs from given $(e)$-positive graphs and classify the (e)-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of $y_{G : v}$ to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.
We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specializes to (extended) $k$-deletion, and we give two methods to obtain numerous new bases from weighted graphs for the algebra of symmetric functions. Moreover, we classify when two weighted paths have equal extended chromatic symmetric functions by proving this is equivalent to the classification of equal ribbon Schur functions. This latter classification is dependent on the operation composition of compositions, which we generalize to composition of graphs. We then apply our generalization to obtain infinitely many families of weighted graphs whose members have equal extended chromatic symmetric functions.