Aaron Chan, Laurent Demonet
For a finite-dimensional gentle algebra, it is already known that the functorially finite torsion classes of its category of finite-dimensional modules can be classified using a combinatorial interpretation, called maximal non-crossing sets of strings, of the corresponding support $τ$-tilting module (or equivalently, two-term silting complexes). In the topological interpretation of gentle algebras via marked surfaces, such a set can be interpreted as a dissection (or partial triangulation), or equivalently, a lamination that does not contain a closed curve. We will refine this combinatorics, which gives us a classification of torsion classes in the category of finite length modules over a (possibly infinite-dimensional) gentle algebra. As a consequence, our result also unifies the functorially finite torsion class classification of finite-dimensional gentle algebras with certain classes of special biserial algebras - such as Brauer graph algebras.
Aaron Chan, Jiashu Xu, Boyuan Long, Soumya Sanyal, Tanishq Gupta, Xiang Ren
Augmenting pre-trained language models with knowledge graphs (KGs) has achieved success on various commonsense reasoning tasks. However, for a given task instance, the KG, or certain parts of the KG, may not be useful. Although KG-augmented models often use attention to focus on specific KG components, the KG is still always used, and the attention mechanism is never explicitly taught which KG components should be used. Meanwhile, saliency methods can measure how much a KG feature (e.g., graph, node, path) influences the model to make the correct prediction, thus explaining which KG features are useful. This paper explores how saliency explanations can be used to improve KG-augmented models' performance. First, we propose to create coarse (Is the KG useful?) and fine (Which nodes/paths in the KG are useful?) saliency explanations. Second, to motivate saliency-based supervision, we analyze oracle KG-augmented models which directly use saliency explanations as extra inputs for guiding their attention. Third, we propose SalKG, a framework for KG-augmented models to learn from coarse and/or fine saliency explanations. Given saliency explanations created from a task's training set, SalKG jointly trains the model to predict the explanations, then solve the task by attending to KG features highlighted by the predicted explanations. On three commonsense QA benchmarks (CSQA, OBQA, CODAH) and a range of KG-augmented models, we show that SalKG can yield considerable performance gains -- up to 2.76% absolute improvement on CSQA.
Aaron Chan, Rene Marczinzik
Gendo-symmetric algebras were introduced by Fang and Koenig as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost $ν$-stable derived equivalences, introduced by Hu and Xi, between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost $ν$-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.
Aaron Chan
We study the relation between simple-minded systems and two-term tilting complexes for self-injective Nakayama algebras. More precisely, we show that any simple-minded system of a self-injective Nakayama algebra is the image of the set of simple modules under a stable equivalence, which is given by the restriction of a standard derived equivalence induced by a two-term tilting complex. We achieve this by exploiting and connecting the mutation theories from the combinatorics of Brauer tree, configurations of stable translations quivers of type A, and triangulations of a punctured convex regular polygon.
Aaron Chan, Osamu Iyama, Rene Marczinzik
We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga-Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras - called the replicated algebras - given by certain subquotients of its repetitive algebra. We calculate the self-injective dimension and dominant dimension of all such replicated algebras and determine which of them are minimal Auslander-Gorenstein, i.e. when the two dimensions are finite and equal to each other. In particular, we show that there exist infinitely many minimal Auslander-Gorenstien algebras in such a series if, and only if, the Serre-formal algebra is twisted fractionally Calabi-Yau. We apply these results to a construction of algebras from Yamagata, called SGC extensions, given by iteratively taking the endomorphism ring of the smallest generator-cogenerator. We give a sufficient condition so that the SGC extensions and replicated algebras coincide. Consequently, in such a case, we obtain explicit formulae for the self-injective dimension and dominant dimension of the SGC extension algebras.
Aaron Chan
We show that taking the wreath product of a quasi-hereditary algebra with symmetric group inherits several homological properties of the original algebra, namely BGG duality, standard Koszulity, balancedness as well as a condition which makes the Ext-algebra of its standard modules a Koszul algebra.
Aaron Chan, Zhiyuan Zeng, Wyatt Lake, Brihi Joshi, Hanjie Chen, Xiang Ren
Language models (LMs) have yielded impressive results on many language reasoning tasks, but their unexpected errors raise doubts about their reasoning abilities. In light of this, there is growing interest in finetuning/prompting LMs with both task instances and their associated free-text rationales (FTRs), which explain the correct reasoning process for predicting the correct task output (i.e., how to be "right for the right reasons"). However, existing finetuning methods fail to improve LM performance, while prompting needs prohibitively large (i.e., >50B) LMs to work well. We propose KNIFE, which shows that reasoning knowledge can be effectively distilled from FTRs into a small (i.e., <1B) LM and improve the LM's performance. First, KNIFE finetunes a teacher LM (given task input and FTR) to predict the task output, transferring reasoning knowledge from the FTRs to the teacher's hidden states. Second, KNIFE finetunes a student LM (given task input only) such that its hidden states are aligned with the teacher's. Thus, the student is endowed with reasoning knowledge but can be used for inference without direct FTR input. On two question-answering datasets, KNIFE outperforms various finetuning and prompting baselines in fully-supervised and low-resource settings. Also, we observe that FTR quality is crucial to KNIFE's performance.
Aaron Chan, Maziar Sanjabi, Lambert Mathias, Liang Tan, Shaoliang Nie, Xiaochang Peng, Xiang Ren, Hamed Firooz
An extractive rationale explains a language model's (LM's) prediction on a given task instance by highlighting the text inputs that most influenced the prediction. Ideally, rationale extraction should be faithful (reflective of LM's actual behavior) and plausible (convincing to humans), without compromising the LM's (i.e., task model's) task performance. Although attribution algorithms and select-predict pipelines are commonly used in rationale extraction, they both rely on certain heuristics that hinder them from satisfying all three desiderata. In light of this, we propose UNIREX, a flexible learning framework that generalizes rationale extractor optimization as follows: (1) specify architecture for a learned rationale extractor; (2) select explainability objectives (i.e., faithfulness and plausibility criteria); and (3) jointly the train task model and rationale extractor on the task using the selected objectives. UNIREX enables replacing prior works' heuristic design choices with a generic learned rationale extractor in (1) and optimizing it for all three desiderata in (2)-(3). To facilitate comparison between methods with respect to multiple desiderata, we introduce the Normalized Relative Gain (NRG) metric. Across five text classification datasets, our best UNIREX configuration outperforms baselines by an average of 32.9% NRG. Plus, we find that UNIREX-trained rationale extractors can even generalize to unseen datasets and tasks.
Aaron Chan, Osamu Iyama, Rene Marczinzik
We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules. We show that every trivial extension algebra $T(A)$ of a $d$-representation-finite algebra A admits a mixed cluster tilting module and show that this can be seen as a generalisation of the well known result that $d$-representation-finite algebras are fractionally Calabi-Yau. We show that iterated SGC-extensions of a gendo-symmetric dominant Auslander-Gorenstein algebra admit mixed precluster tilting modules.
Mrigank Raman, Aaron Chan, Siddhant Agarwal, Peifeng Wang, Hansen Wang, Sungchul Kim, Ryan Rossi, Handong Zhao, Nedim Lipka, Xiang Ren
Knowledge graphs (KGs) have helped neural models improve performance on various knowledge-intensive tasks, like question answering and item recommendation. By using attention over the KG, such KG-augmented models can also "explain" which KG information was most relevant for making a given prediction. In this paper, we question whether these models are really behaving as we expect. We show that, through a reinforcement learning policy (or even simple heuristics), one can produce deceptively perturbed KGs, which maintain the downstream performance of the original KG while significantly deviating from the original KG's semantics and structure. Our findings raise doubts about KG-augmented models' ability to reason about KG information and give sensible explanations.
Aaron Chan, Shaoliang Nie, Liang Tan, Xiaochang Peng, Hamed Firooz, Maziar Sanjabi, Xiang Ren
Following how humans communicate, free-text rationales aim to use natural language to explain neural language model (LM) behavior. However, free-text rationales' unconstrained nature makes them prone to hallucination, so it is important to have metrics for free-text rationale quality. Existing free-text rationale metrics measure how consistent the rationale is with the LM's predicted label, but there is no protocol for assessing such metrics' reliability. Thus, we propose FRAME, a framework for evaluating rationale-label consistency (RLC) metrics for free-text rationales. FRAME is based on three axioms: (1) good metrics should yield highest scores for reference rationales, which maximize RLC by construction; (2) good metrics should be appropriately sensitive to semantic perturbation of rationales; and (3) good metrics should be robust to variation in the LM's task performance. Across three text classification datasets, we show that existing RLC metrics cannot satisfy all three FRAME axioms, since they are implemented via model pretraining which muddles the metric's signal. Then, we introduce a non-pretraining RLC metric that greatly outperforms baselines on (1) and (3), while performing competitively on (2). Finally, we discuss the limitations of using RLC to evaluate free-text rationales.
Takahide Adachi, Takuma Aihara, Aaron Chan
Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.
Aaron Chun Shing Chan
This paper provides an alternate proof to parts of the Goulden-Slofstra formula for enumerating two vertex maps by genus, which is an extension of the famous Harer-Zagier formula that computes the Euler characteristic of the moduli space of curves. This paper also shows a further simplification to the Goulden-Slofstra formula. Portions of this alternate proof will be used in a subsequent paper, where it forms a basis for a more general result that applies for a certain class of maps with an arbitrary number of vertices.
Aaron Chun Shing Chan
This paper provides the generating series for the embedding of tree-like graphs of arbitrary number of vertices, accourding to their genus. It applies and extends the techniques of Chan, where it was used to give an alternate proof of the Goulden and Slofstra formula. Furthermore, this greatly generalizes the famous Harer-Zagier formula, which computes the Euler characteristic of the moduli space of curves, and is equivalent to the computation of one vertex maps.
Véronique Bazier-Matte, Aaron Chan, Kayla Wright
We initiate the investigation of representation theory of non-orientable surfaces. As a first step towards finding an additive categorification of Dupont and Palesi's quasi-cluster algebras associated marked non-orientable surfaces, we study a certain modification on the objects of the cluster category associated to the orientable double covers in the unpunctured case. More precisely, we consider symmetric representation theory studied by Derksen-Weyman and Boos-Cerulli Irelli, and lift it to the cluster category. This gives a way to consider `indecomposable orbits of objects' under a contravariant duality functor. Hence, we can assign curves on a non-orientable surface $(\mathbb{S}, \mathbb{M})$ to indecomposable symmetric objects. Moreover, we define a new notion of symmetric extension, and show that the arcs and quasi-arcs on $(\mathbb{S}, \mathbb{M})$ correspond to the indecomposable symmetric objects without symmetric self-extension. Consequently, we show that quasi-triangulations of $(\mathbb{S}, \mathbb{M})$ correspond to a symmetric analogue of cluster tilting objects.
Aaron Chan, Osamu Iyama, Rene Marczinzik
Auslander and Reiten called a finite dimensional algebra $A$ over a field Cohen-Macaulay if there is an $A$-bimodule $W$ which gives an equivalence between the category of finitely generated $A$-modules of finite projective dimension and the category of finitely generated $A$-modules of finite injective dimension. For example, Iwanaga-Gorenstein algebras and algebras with finitistic dimension zero on both sides are Cohen-Macaulay, and tensor products of Cohen-Macaulay algebras are again Cohen-Macaulay. They seem to be all of the known examples of Cohen-Macaulay algebras. In this paper, we give the first non-trivial class of Cohen-Macaulay algebras by showing that all contracted preprojective algebras of Dynkin type are Cohen-Macaulay. As a consequence, for each simple singularity $R$ and a maximal Cohen-Macaulay $R$-module $M$, the stable endomorphism algebra $\underline{End}_R(M)$ is Cohen-Macaulay. We also give a negative answer to a question of Auslander-Reiten asking whether the category $CM A$ of Cohen-Macaulay $A$-modules coincides with the category of $d$-th syzygies, where $d\ge1$ is the injective dimension of $W$. In fact, if $A$ is a Cohen-Macaulay algebra that is additionally $d$-Gorenstein in the sense of Auslander, then $CM A$ always coincides with the category of $d$-th syzygies.
Guoxin Zheng, Yuan Zhu, Shirin Mozaffari, Ning Mao, Kuan-Wen Chen, Kaila Jenkins, Dechen Zhang, Aaron Chan, Hasitha W. Suriya Arachchige, Richa P. Madhogaria, Matthew Cothrine, William R. Meier, Yang Zhang, David Mandrus, Lu Li
Metals with kagome lattice provide bulk materials to host both the flat-band and Dirac electronic dispersions. A new family of kagome metals is recently discovered in AV6Sn6. The Dirac electronic structures of this material need more experimental evidence to confirm. In the manuscript, we investigate this problem by resolving the quantum oscillations in both electrical transport and magnetization in ScV6Sn6. The revealed orbits are consistent with the electronic band structure models. Furthermore, the Berry phase of a dominating orbit is revealed to be around $π$, providing direct evidence for the topological band structure, which is consistent with calculations. Our results demonstrate a rich physics and shed light on the correlated topological ground state of this kagome metal.
Dechen Zhang, Kuan-Wen Chen, Guoxin Zheng, Fanghang Yu, Mengzhu Shi, Yuan Zhu, Aaron Chan, Kaila Jenkins, Jianjun Ying, Ziji Xiang, Xianhui Chen, Lu Li
The thermal Hall effect recently provided intriguing probes to the ground state of exotic quantum matters. These observations of transverse thermal Hall signals lead to the debate on the fermionic versus bosonic origins of these phenomena. The recent report of quantum oscillations (QOs) in Kitaev spin liquid points to a possible resolution. The Landau level quantization would most likely capture only the fermionic thermal transport effect. However, the QOs in the thermal Hall effect are generally hard to detect. In this work, we report the observation of a large oscillatory thermal Hall effect of correlated Kagome metals. We detect a 180-degree phase change of the oscillation and demonstrate the phase flip as an essential feature for QOs in the thermal transport properties. More importantly, the QOs in the thermal Hall channel are more profound than those in the electrical Hall channel, which strongly violates the Wiedemann Franz (WF) law for QOs. This result presents the oscillatory thermal Hall effect as a powerful probe to the correlated quantum materials.
Aaron Chan, Volodymyr Mazorchuk
In this paper we introduce and investigate the notions of diagrams and discrete extensions in the study of finitary $2$-representations of finitary $2$-categories.
Takahide Adachi, Aaron Chan, Yuta Kimura, Mayu Tsukamoto
A quasi-hereditary algebra is an algebra equipped with a certain partial order $\unlhd$ on its simple modules. Such a partial order -- called a quasi-hereditary structure -- gives rise to a characteristic tilting module $T_{\unlhd}$ by a classical result due to Ringel. A fundamental question is to determine which tilting modules can be realised as characteristic tilting modules. We answer this question by using the notion of IS-tilting module, which is a pair $(T,\unlhd)$ of a tilting module $T$ and a partial order $\unlhd$ on its direct summands such that iterative idempotent truncation along $\unlhd$ always reveals a simple direct summand. Specifically, we show that a tilting module $T$ is characteristic if, and only if, there is some $\unlhd$ so that $(T,\unlhd)$ is IS-tilting; in which case, we have $T=T_{\unlhd}$. This result enables us to study quasi-hereditary structures using tilting theory. As an application of the above result, we show that, for an algebra $A$, all tilting modules are characteristic if, and only if, $A$ is a quadratic linear Nakayama algebra. Furthermore, for such an $A$, we provide a decomposition of the set of its tilting modules that can be used to derive a recursive formula for enumerating its quasi-hereditary structures. Finally, we describe the quasi-hereditary structures of $A$ via `nodal gluing' and binary tree sequences.