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Large Language Models (LLMs) excel in general tasks, but adapting them to specialized domains relies on high-quality supervised fine-tuning (SFT) data. Although existing methods can identify subsets of high-quality data and reduce training cost to some extent, their selection process still suffers from over-reliance on LLMs' internal knowledge, weak interpretability, and limited generalization. To address these limitations, we propose THTB (The Harder The Better), a cognitive science-inspired framework for instruction data selection and annotation guidance. THTB prioritizes higher-level cognitive instructions by combining quality filtering with intrinsic and extrinsic hardness scoring, offering interpretable and quantifiable criteria for efficient SFT, both in data selection and annotation guidance. Experiments show that THTB enables models trained on only 5% of the data to outperform full-dataset training, while achieving superior generalization compared with LLM-only selection. In addition, THTB provides effective annotation guidance in vertical domains, enabling a model trained on just 2% of the data to surpass models trained on much larger datasets, demonstrating strong potenti

We prove that for any simply connected isotropic reductive group G over a Dedekind domain D, any Zariski-locally trivial principal G-bundle over D is trivial. The corresponding result for quasi-split groups was proved in 1967 by G. Harder.

In this article, we study the notion of semi-stability and the Harder-Narasimhan filtration from a game-theoretic point of view. This allows us to provide a unified proof for the existence and uniqueness of the Harder-Narasimhan filtration in various settings and offer a conceptual interpretation for semi-stability conditions. As an application, we establish the existence and uniqueness of a coprimary filtration for modules of finite type over a commutative Noetherian ring and interpret it as a Harder-Narasimhan filtration.

For any flat family of pure-dimensional coherent sheaves on a family of projective schemes, the Harder-Narasimhan type (in the sense of Gieseker semistability) of its restriction to each fiber is known to vary semicontinuously on the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder-Narasimhan filtration with a given Harder-Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder-Narasimhan type. The above schematic stratification induces a stacky stratification on the algebraic stack of pure-dimensional coherent sheaves. We deduce that coherent sheaves of a fixed Harder-Narasimhan type form an algebraic stack in the sense of Artin.

While generating better negative samples for contrastive learning has been widely studied in the areas of CV and NLP, very few work has focused on graph-structured data. Recently, Mixup has been introduced to synthesize hard negative samples in graph contrastive learning (GCL). However, due to the unsupervised learning nature of GCL, without the help of soft labels, directly mixing representations of samples could inadvertently lead to the information loss of the original hard negative and further adversely affect the quality of the newly generated harder negative. To address the problem, in this paper, we propose a novel method DropMix to synthesize harder negative samples, which consists of two main steps. Specifically, we first select some hard negative samples by measuring their hardness from both local and global views in the graph simultaneously. After that, we mix hard negatives only on partial representation dimensions to generate harder ones and decrease the information loss caused by Mixup. We conduct extensive experiments to verify the effectiveness of DropMix on six benchmark datasets. Our results show that our method can lead to better GCL performance. Our data and cod

Knowledge graph embedding~(KGE) aims to represent entities and relations into low-dimensional vectors for many real-world applications. The representations of entities and relations are learned via contrasting the positive and negative triplets. Thus, high-quality negative samples are extremely important in KGE. However, the present KGE models either rely on simple negative sampling methods, which makes it difficult to obtain informative negative triplets; or employ complex adversarial methods, which requires more training data and strategies. In addition, these methods can only construct negative triplets using the existing entities, which limits the potential to explore harder negative triplets. To address these issues, we adopt mixing operation in generating harder negative samples for knowledge graphs and introduce an inexpensive but effective method called MixKG. Technically, MixKG first proposes two kinds of criteria to filter hard negative triplets among the sampled negatives: based on scoring function and based on correct entity similarity. Then, MixKG synthesizes harder negative samples via the convex combinations of the paired selected hard negatives. Experiments on two p

The notion of Harder-Narasimhan filtration was firstly introduced by Harder and Narasimhan in the setting of vector bundles on a non-singular projective curve. Curiously, analogous constructions have been discovered in other branches of mathematics which motivate categorical constructions of Harder-Narasimhan filtration. In this article, we introduce a categorical construction of Harder-Narasimhan filtration via slope function method which does not need the additive condition in degree function. We also give a method to prove the existence and uniqueness theorem of the Harder-Narasimhan filtration intrinsically from a set E and a admissible collection of subsets of E, which does not need sub-quotient structure.

We propose a generalization of Quillen's exact category -- arithmetic exact category and we discuss conditions on such categories under which one can establish the notion of Harder-Narasimhan filtrations and Harder-Narsimhan polygons. Furthermore, we show the functoriality of Harder-Narasimhan filtrations (indexed by $\mathbb R$), which can not be stated in the classical setting of Harder and Narasimhan's formalism.

We establish in this article convergence results of normalized Harder-Narasimhan polygons both in geometric and in arithmetic frameworks by introducing the Harder-Narasimhan filtration indexed by $\mathbb R$ and the associated Borel probability measure.

Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a Kac-Moody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on the cohomology of quotients in 'linear algebra type' situations.

The paper gives an example of a tree language G that is recognised by an unambiguous parity automaton and is analytic-complete as a set in Cantor space. This already shows that the unambiguous languages are topologically more complex than the deterministic ones, that are all coanalytic. Using set G as a building block we construct an unambiguous language that is topologically harder than any countable boolean combination of analytic and coanalytic sets. In particular the language is harder than any set in difference hierarchy of analytic sets considered by O.Finkel and P.Simonnet in the context of nondeterministic automata.

Triplet loss is a widely adopted loss function in ReID task which pulls the hardest positive pairs close and pushes the hardest negative pairs far away. However, the selected samples are not the hardest globally, but the hardest only in a mini-batch, which will affect the performance. In this report, a hard batch mining method is proposed to mine the hardest samples globally to make triplet harder. More specifically, the most similar classes are selected into a same mini-batch so that the similar classes could be pushed further away. Besides, an adversarial scene removal module composed of a scene classifier and an adversarial loss is used to learn scene invariant feature representations. Experiments are conducted on dataset MSMT17 to prove the effectiveness, and our method surpasses all of the previous methods and sets state-of-the-art result.