arXiv subjects

Yibo Gao

Publications and source records attributed to Yibo Gao.

At least 19 recordsLinked to original sources

arXiv2026

Dual Lattice Functions of Polytopes

We define the dual lattice function of a rational polytope $P$ via the discrete Laplace transform of the exponential of its support function. This definition is a discrete analogue of the dual volume function of a polytope that the authors studied in previous work. We show that the dual lattice function is valuative, and by multiplying with the torus form, it becomes the canonical form of the exponential polytope $\mathrm{exp}(P)$ as a positive geometry. This result suggests the study of the class of toric polytopes, which are certain semialgebraic subsets of projective toric varieties. Our work is a first step towards discretization of positive geometries in the simplest case of polytopes.

math.CO↗

arXiv2026

Realizable Standard Young Tableaux

Given two vectors $u$ and $v$, their outer sum is given by the matrix $A$ with entries $A_{ij} = u_{i} + v_{j}$. If the entries of $u$ and $v$ are increasing and sufficiently generic, the total ordering of the entries of the matrix is a standard Young tableau of rectangular shape. We call rectangular standard Young tableaux arising in this way realizable. The set of realizable tableaux was defined by Mallows and Vanderbei for studying a deconvolution algorithm, but we show they have appeared in many other contexts including sorting algorithms, quantum computing, random sorting networks, reflection arrangements, fiber polytopes, and Goodman and Pollack's theory of allowable sequences. In our work, we prove tight bounds on the asymptotic number of realizable rectangular tableaux. We also derive tight asymptotics for the number of realizable allowable sequences, which are in bijection with realizable staircase-shaped standard Young tableaux with the notion of realizability coming from the theory of sorting networks.

math.CO↗

arXiv2026

Billey-Postnikov posets, rationally smooth Schubert varieties, and Poincaré duality

Billey-Postnikov (BP) decompositions govern when Schubert varieties $X(w)$ decompose as bundles of smaller Schubert varieties. We further develop the theory of BP decompositions and show that, in finite type, they can be recognized by pattern conditions and are indexed by the order ideals of a poset $\mathsf{bp}(w)$ that we introduce; we conjecture that this holds in any Coxeter group. We then apply BP decompositions to show that, when $X(w)$ is rationally smooth and $W$ simply laced, the Schubert structure constants $c_{uv}^w$ satisfy a triangularity property, yielding a canonical involution on the Schubert cells of $X(w)$ respecting Poincaré duality. We also classify the rationally smooth Bruhat intervals in finite type (other than $E$) which admit generalized Lehmer codes, answering questions and conjectures of Billey-Fan-Losonczy, Bolognini-Sentinelli, and Bishop-Milićević-Thomas. Finally, we show that rationally smooth Schubert varieties in infinite type need not have Grassmannian BP decompositions, disproving conjectures of Richmond-Slofstra and Oh-Richmond.

math.CO↗

arXiv2026

Coxeter Condorcet domains

Condorcet domains are subsets of permutations that ensure pairwise majority voting yields acyclic outcomes, and they form an active area of research at the intersection of social choice theory and combinatorics. In this paper, we extend the theory of Condorcet domains to the broader setting of arbitrary finite Coxeter groups. The core contribution of our approach is the introduction of Condorcet root posets, defined on the chosen root systems. Notably, we establish a natural bijection between closed Condorcet domains and Condorcet root posets, which facilitates the study of Condorcet domains. Using this correspondence, we extend the median graph representation of closed Condorcet domains to arbitrary finite Coxeter groups, demonstrating that these domains can be characterized by the skeletons of their associated Condorcet root posets. These results are novel even in type $A$. Furthermore, these posets give a unified language that efficiently captures a wide range of desirable properties of Condorcet domains, such as being maximal, connected, peak-pit, and of tiling type. Using this framework, we strengthen and generalize several classical results: we establish that a maximal Condorcet domain is connected if and only if it is peak-pit; we prove that the tiling-type property is equivalent to the combination of being maximal and connected, and having maximal width; and we show that strictly positive voting profiles on connected Condorcet domains yield outcomes with only simple ties.

math.CO↗

arXiv2026

Generalized Evidential Deep Learning: From a Bayesian Perspective

Evidential Deep Learning (EDL) has emerged as an efficient, sampling-free strategy for uncertainty estimation. A series of EDL variants have been proposed to address specific limitations of the original framework, achieving notable success. However, the underlying theoretical structure of EDL and the relationships among these variants have received limited systematic investigation. In this work, we establish a principled theoretical foundation for EDL by interpreting it within a generalized Bayesian framework that includes prior specification, posterior update, and training objective. We further characterize evidential uncertainty from a Bayesian distributional uncertainty viewpoint, established via asymptotic analysis. Building on this perspective, we further propose Generalized Evidential Deep Learning (GEDL), a unified and extensible framework that explicitly disentangles the roles of individual components and systematically relates GEDL to existing variants. Extensive experiments demonstrate that GEDL yields comparable results on classification, uncertainty estimation and OOD detections, with theoretical grounding.

cs.LG↗

arXiv2026

Beyond Forgetting in Continual Medical Image Segmentation: A Comprehensive Benchmark Study

Continual learning (CL) is essential for deploying medical image segmentation models in clinical environments where imaging domains, anatomical targets, and diagnostic tasks evolve over time. However, continual segmentation still faces three main challenges. First, the scenarios for this task remain insufficiently standardized for real-world clinical settings. Second, existing research has been primarily focused on mitigating forgetting, overlooking the other essential properties such as plasticity. Third, a benchmark work with comprehensive evaluation on existing methods is stll desirable. To address these gaps, we present such benchmark study of continual medical image segmentation. We first define three clinically motivated scenarios, namely Domain-CL, Class-CL, and Organ-CL, to respectively capture the cross-center domain shift, the incremental anatomical structure segmentation, and the cross-organ segmentation. We then introduce an evaluation framework that measures not only general performance and forgetting, but also plasticity, forward generalizability, parameter efficiency, and replay burden. The results, from extensive experiments with representative CL methods, showed that it was still challenging to develop a model that could satisfy all the requirements simultaneously. Nevertheless, these studies also suggested that the replay-based methods achieve the best overall balance between stability and plasticity, the parameter-isolation methods should be effective at reducing forgetting, though at the cost of increased model size, and the forward generalizability remain a significantly understudied aspect of this research field. Finally, we discuss related learning paradigms and outline future directions for continual medical image segmentation.

cs.CV↗

arXiv2026

MacNeille completions of parabolic quotients

Alternating sign matrices (ASMs) arise as the Dedekind-MacNeille completion of the Bruhat order on the symmetric group. They enjoy fruitful combinatorial and geometric properties, with a particularly rich history on enumerations and bijections. In this paper, we explicitly describe the Dedekind-MacNeille completion of the Bruhat order on any parabolic quotients of the symmetric group. It is naturally a subposet of the alternating sign matrices, with different lattice operations. Moreover, we demonstrate the relations between the meet and join operations in this lattice with taking unions and intersections of the corresponding ASM varieties, respectively. Finally, we conclude with a more detailed discussion of special cases.

math.CO↗

arXiv2026

Tilted Richardson Varieties

The study of the flag variety $\mathrm{Fl}_n$ and its subvarieties, including Schubert and Richardson varieties, plays a fundamental role in algebraic geometry and algebraic combinatorics. In this paper, we introduce and develop the theory of tilted Richardson varieties $\mathrm{T}_{u,v}$, a new family of subvarieties of the flag variety that provides a geometric framework for the quantum Bruhat graphs. These varieties are defined for all pairs of permutations $u$ and $v$, extending the classical Richardson varieties in the case where $u\leq v$ in the Bruhat order. We establish their fundamental geometric properties, proving irreducibility and providing explicit dimension formulas. Moreover, we show that they have a well-defined stratification indexed by tilted Bruhat intervals, a generalization of classical Bruhat intervals previously introduced by Brenti, Fomin, and Postnikov. Additionally, we introduce a tilted generalization of the classical Deodhar decomposition of Richardson varieties, which leads to a combinatorial formula for tilted Kazhdan--Lusztig R-polynomials, a notion that arises naturally in our framework. We further develop a theory of total positivity for tilted Richardson varieties. In particular, we define and study the totally nonnegative parts of tilted Richardson varieties, proving they form a CW complex. This generalizes earlier results on the totally nonnegative flag variety and answers Björner's questions regarding geometric realizations of tilted Bruhat intervals. Finally, we establish explicit connections between tilted Richardson varieties and quantum Schubert calculus. Specifically, we prove that $\mathrm{T}_{u,v}$ coincides with minimal-degree two-point curve neighborhoods. As a result, we compute their cohomology classes and derive new relationships among Gromov--Witten invariants of the flag variety.

math.CO↗

arXiv2026

From Latent Signals to Reflection Behavior: Tracing Meta-Cognitive Activation Trajectory in R1-Style LLMs

R1-style LLMs have attracted growing attention for their capacity for self-reflection, yet the internal mechanisms underlying such behavior remain unclear. To bridge this gap, we anchor on the onset of reflection behavior and trace its layer-wise activation trajectory. Using the logit lens to read out token-level semantics, we uncover a structured progression: (i) Latent-control layers, where an approximate linear direction encodes the semantics of thinking budget; (ii) Semantic-pivot layers, where discourse-level cues, including turning-point and summarization cues, surface and dominate the probability mass; and (iii) Behavior-overt layers, where the likelihood of reflection-behavior tokens begins to rise until they become highly likely to be sampled. Moreover, our targeted interventions uncover a causal chain across these stages: prompt-level semantics modulate the projection of activations along latent-control directions, thereby inducing competition between turning-point and summarization cues in semantic-pivot layers, which in turn regulates the sampling likelihood of reflection-behavior tokens in behavior-overt layers. Collectively, our findings suggest a human-like meta-cognitive process-progressing from latent monitoring, to discourse-level regulation, and to finally overt self-reflection. Our analysis code can be found at https://github.com/DYR1/S3-CoT.

cs.CL↗

arXiv2026

S3-CoT: Self-Sampled Succinct Reasoning Enables Efficient Chain-of-Thought LLMs

Large language models (LLMs) equipped with chain-of-thought (CoT) achieve strong performance and offer a window into LLM behavior. However, recent evidence suggests that improvements in CoT capabilities often come with redundant reasoning processes, motivating a key question: Can LLMs acquire a fast-thinking mode analogous to human System 1 reasoning? To explore this, our study presents a self-sampling framework based on activation steering for efficient CoT learning. Our method can induce style-aligned and variable-length reasoning traces from target LLMs themselves without any teacher guidance, thereby alleviating a central bottleneck of SFT-based methods-the scarcity of high-quality supervision data. Using filtered data by gold answers, we perform SFT for efficient CoT learning with (i) a human-like dual-cognitive system, and (ii) a progressive compression curriculum. Furthermore, we explore a self-evolution regime in which SFT is driven solely by prediction-consistent data of variable-length variants, eliminating the need for gold answers. Extensive experiments on math benchmarks, together with cross-domain generalization tests in medicine, show that our method yields stable improvements for both general and R1-style LLMs. Our data and model checkpoints can be found at https://github.com/DYR1/S3-CoT.

cs.CL↗

arXiv2025

Thermalizing channel states for rapid qubit heating

Although known for negatively impacting the operation of superconducting qubits, thermal baths are shown to exert qubit control in a positive way, provided they are properly engineered. We demonstrate an experimental method to engineer the transduction of microwave driving into heat flow through a leaky resonator. Given the precise conversion, a qubit receiving the heat flow obtains a quasi-thermal equilibrium with arbitrary target temperature in hundreds of nanoseconds. We show that the dynamics of the quantum transducing process is described by thermalizing channel states, generated from the double dressings of the resonator by the semi-classical driving and the qubit-resonator coupling. Their spectrum, coupling, and driving strength determine the channel rate of energy flow, along with the relaxation rates of photon leakage into the bath. The analytical prediction is shown to match well with the experimental measurements on an Xmon qubit circuit.

quant-ph↗

arXiv2025

Boolean Schubert Structure Coefficients

The Schubert problem asks for combinatorial models to compute structure constants of the cohomology ring with respect to Schubert classes and has been an important open problem in algebraic geometry and combinatorics that guided fruitful research for decades. In this paper, we provide an explicit formula for the (equivariant) Schubert structure constants $c_{uv}^w$ across all Lie types when the elements $u,v,w$ are boolean. In particular, in type $A$, all Schubert structure constants on boolean elements are either $0$ or $1$.

math.CO↗

arXiv2025

Evaluating Foundation Models with Pathological Concept Learning for Kidney Cancer

To evaluate the translational capabilities of foundation models, we develop a pathological concept learning approach focused on kidney cancer. By leveraging TNM staging guidelines and pathology reports, we build comprehensive pathological concepts for kidney cancer. Then, we extract deep features from whole slide images using foundation models, construct pathological graphs to capture spatial correlations, and trained graph neural networks to identify these concepts. Finally, we demonstrate the effectiveness of this approach in kidney cancer survival analysis, highlighting its explainability and fairness in identifying low- and high-risk patients. The source code has been released by https://github.com/shangqigao/RadioPath.

cs.AI↗

arXiv2025

Introduction to the Cohomology of the Flag Variety

One hundred years ago, Hilbert gave a list of important open problems in mathematics. His 15th problem asked for the development of a rigorous calculus explaining Schubert's enumerative results for intersecting varieties defined by rank conditions on vector spaces. Today by way of many contributions in algebraic topology, geometry, and combinatorics, we consider this solved. Yet, deep questions remain about the subtleties of actually carrying out the process. In this chapter, we hope to summarize the rigorous development of what has become known as Schubert calculus, with an eye toward computation. We discuss Grassmannians and flag varieties and their cohomology rings, following Monk's constructive algebraic approach. We derive formulas for Schur and Schubert polynomials, which represent cohomology classes of Schubert varieties. We hint at the vast literature in this area and point to the other references in the Handbook for more information. Finally, we identify open problems that remain a challenge even with modern tools at our fingertips in hopes of inspiring further contributions in this fascinating field. This is intended as the first chapter of a book entitled "Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety", a compendium of topics in the area. The book is being edited by Erik Insko, Martha Precup, and Ed Richmond. In addition to this introductory chapter, others will cover more advanced topics such as Kazhdan-Lusztig varieties, generalized smooth Schubert varieties, Richardson varieties and positroid varieties, spherical and torus orbit closures, spanning line configurations, different types of Hessenberg varieties, and generalizations to Kac-Moody flag varieties, each written by experts in those areas. We hope you enjoy this chapter enough to seek out the others, and that you send us any comments or corrections you find as you read this article!

math.CO↗

arXiv2025

Graham positivity of triple Schubert calculus

We prove Samuel's conjecture on certain Graham positivity of the expansion coefficient of two double Schubert polynomials in three sets of variables by establishing a refined version of Graham's positivity theorem. As a corollary, we prove Kirillov's conjecture on the positivity of skew divided difference operators applied to Schubert polynomials.

math.CO↗

arXiv2025

Learning Concept-Driven Logical Rules for Interpretable and Generalizable Medical Image Classification

The pursuit of decision safety in clinical applications highlights the potential of concept-based methods in medical imaging. While these models offer active interpretability, they often suffer from concept leakages, where unintended information within soft concept representations undermines both interpretability and generalizability. Moreover, most concept-based models focus solely on local explanations (instance-level), neglecting the global decision logic (dataset-level). To address these limitations, we propose Concept Rule Learner (CRL), a novel framework to learn Boolean logical rules from binarized visual concepts. CRL employs logical layers to capture concept correlations and extract clinically meaningful rules, thereby providing both local and global interpretability. Experiments on two medical image classification tasks show that CRL achieves competitive performance with existing methods while significantly improving generalizability to out-of-distribution data. The code of our work is available at https://github.com/obiyoag/crl.

cs.CV↗

arXiv2025

Interlacing triangles, Schubert puzzles, and graph colorings

We show that interlacing triangular arrays, introduced by Aggarwal-Borodin-Wheeler to study certain probability measures, can be used to compute structure constants for multiplying Schubert classes in the $K$-theory of Grassmannians, in the cohomology of their cotangent bundles, and in the cohomology of partial flag varieties. Our results are achieved by establishing a splitting lemma, allowing for interlacing triangular arrays of high rank to be decomposed into arrays of lower rank, and by constructing a bijection between interlacing triangular arrays of rank 3 with certain proper vertex colorings of the triangular grid graph that factors through generalizations of Knutson-Tao puzzles. Along the way, we prove one enumerative conjecture of Aggarwal-Borodin-Wheeler and disprove another.

math.CO↗

arXiv2025

Empowering Medical Multi-Agents with Clinical Consultation Flow for Dynamic Diagnosis

Traditional AI-based healthcare systems often rely on single-modal data, limiting diagnostic accuracy due to incomplete information. However, recent advancements in foundation models show promising potential for enhancing diagnosis combining multi-modal information. While these models excel in static tasks, they struggle with dynamic diagnosis, failing to manage multi-turn interactions and often making premature diagnostic decisions due to insufficient persistence in information collection.To address this, we propose a multi-agent framework inspired by consultation flow and reinforcement learning (RL) to simulate the entire consultation process, integrating multiple clinical information for effective diagnosis. Our approach incorporates a hierarchical action set, structured from clinic consultation flow and medical textbook, to effectively guide the decision-making process. This strategy improves agent interactions, enabling them to adapt and optimize actions based on the dynamic state. We evaluated our framework on a public dynamic diagnosis benchmark. The proposed framework evidentially improves the baseline methods and achieves state-of-the-art performance compared to existing foundation model-based methods.

cs.AI↗