arXiv subjects

Boyan Duan

Publications and source records attributed to Boyan Duan.

8 recordsLinked to original sources

arXiv2025

Gold-Medal-Level Olympiad Geometry Solving with Efficient Heuristic Auxiliary Constructions

Automated theorem proving in Euclidean geometry, particularly for International Mathematical Olympiad (IMO) level problems, remains a major challenge and an important research focus in Artificial Intelligence. In this paper, we present a highly efficient method for geometry theorem proving that runs entirely on CPUs without relying on neural network-based inference. Our initial study shows that a simple random strategy for adding auxiliary points can achieve silver-medal level human performance on IMO. Building on this, we propose HAGeo, a Heuristic-based method for adding Auxiliary constructions in Geometric deduction that solves 28 of 30 problems on the IMO-30 benchmark, achieving gold-medal level performance and surpassing AlphaGeometry, a competitive neural network-based approach, by a notable margin. To evaluate our method and existing approaches more comprehensively, we further construct HAGeo-409, a benchmark consisting of 409 geometry problems with human-assessed difficulty levels. Compared with the widely used IMO-30, our benchmark poses greater challenges and provides a more precise evaluation, setting a higher bar for geometry theorem proving.

cs.AI↗

arXiv2024

VC-dimensions Between Partially Ordered Sets and Totally Ordered Sets

We say that two partial orders on $[n]$ are compatible if there exists a partial order that refines both of them. This compatibility relation induces a natural set system structure between the collection $\mathcal{F}$ of all partial orders and the collection $\mathcal{G}$ of all total orders on $[n]$, where each order is associated with the set of orders compatible with it. In this note, we determine the VC-dimension of $\mathcal{F}$ with respect to $\mathcal{G}$, proving that $\operatorname{VC}_{\mathcal{G}}(\mathcal{F}) = \lfloor\frac{n^2}{4}\rfloor$ for $n \ge 4$. We also establish bounds on the dual VC-dimension, showing that $2(n-3) \le \operatorname{VC}_{\mathcal{F}}(\mathcal{G}) \le n \log_2 n$ for all $n \ge 1$.

math.CO↗

arXiv2024

Interactive identification of individuals with positive treatment effect while controlling false discoveries

Out of the participants in a randomized experiment with anticipated heterogeneous treatment effects, is it possible to identify which subjects have a positive treatment effect? While subgroup analysis has received attention, claims about individual participants are much more challenging. We frame the problem in terms of multiple hypothesis testing: each individual has a null hypothesis (stating that the potential outcomes are equal, for example) and we aim to identify those for whom the null is false (the treatment potential outcome stochastically dominates the control one, for example). We develop a novel algorithm that identifies such a subset, with nonasymptotic control of the false discovery rate (FDR). Our algorithm allows for interaction -- a human data scientist (or a computer program) may adaptively guide the algorithm in a data-dependent manner to gain power. We show how to extend the methods to observational settings and achieve a type of doubly-robust FDR control. We also propose several extensions: (a) relaxing the null to nonpositive effects, (b) moving from unpaired to paired samples, and (c) subgroup identification. We demonstrate via numerical experiments and theoretical analysis that the proposed method has valid FDR control in finite samples and reasonably high identification power.

stat.ME↗

arXiv2023

Data fission: splitting a single data point

Suppose we observe a random vector $X$ from some distribution $P$ in a known family with unknown parameters. We ask the following question: when is it possible to split $X$ into two parts $f(X)$ and $g(X)$ such that neither part is sufficient to reconstruct $X$ by itself, but both together can recover $X$ fully, and the joint distribution of $(f(X),g(X))$ is tractable? As one example, if $X=(X_1,\dots,X_n)$ and $P$ is a product distribution, then for any $m<n$, we can split the sample to define $f(X)=(X_1,\dots,X_m)$ and $g(X)=(X_{m+1},\dots,X_n)$. Rasines and Young (2022) offers an alternative approach that uses additive Gaussian noise -- this enables post-selection inference in finite samples for Gaussian distributed data and asymptotically when errors are non-Gaussian. In this paper, we offer a more general methodology for achieving such a split in finite samples by borrowing ideas from Bayesian inference to yield a (frequentist) solution that can be viewed as a continuous analog of data splitting. We call our method data fission, as an alternative to data splitting, data carving and p-value masking. We exemplify the method on a few prototypical applications, such as post-selection inference for trend filtering and other regression problems.

stat.ME↗

arXiv2022

Interactive rank testing by betting

In order to test if a treatment is perceptibly different from a placebo in a randomized experiment with covariates, classical nonparametric tests based on ranks of observations/residuals have been employed (eg: by Rosenbaum), with finite-sample valid inference enabled via permutations. This paper proposes a different principle on which to base inference: if -- with access to all covariates and outcomes, but without access to any treatment assignments -- one can form a ranking of the subjects that is sufficiently nonrandom (eg: mostly treated followed by mostly control), then we can confidently conclude that there must be a treatment effect. Based on a more nuanced, quantifiable, version of this principle, we design an interactive test called i-bet: the analyst forms a single permutation of the subjects one element at a time, and at each step the analyst bets toy money on whether that subject was actually treated or not, and learns the truth immediately after. The wealth process forms a real-valued measure of evidence against the global causal null, and we may reject the null at level $α$ if the wealth ever crosses $1/α$. Apart from providing a fresh "game-theoretic" principle on which to base the causal conclusion, the i-bet has other statistical and computational benefits, for example (A) allowing a human to adaptively design the test statistic based on increasing amounts of data being revealed (along with any working causal models and prior knowledge), and (B) not requiring permutation resampling, instead noting that under the null, the wealth forms a nonnegative martingale, and the type-1 error control of the aforementioned decision rule follows from a tight inequality by Ville. Further, if the null is not rejected, new subjects can later be added and the test can be simply continued, without any corrections (unlike with permutation p-values).

stat.ME↗

arXiv2021

Familywise Error Rate Control by Interactive Unmasking

We propose a method for multiple hypothesis testing with familywise error rate (FWER) control, called the i-FWER test. Most testing methods are predefined algorithms that do not allow modifications after observing the data. However, in practice, analysts tend to choose a promising algorithm after observing the data; unfortunately, this violates the validity of the conclusion. The i-FWER test allows much flexibility: a human (or a computer program acting on the human's behalf) may adaptively guide the algorithm in a data-dependent manner. We prove that our test controls FWER if the analysts adhere to a particular protocol of "masking" and "unmasking". We demonstrate via numerical experiments the power of our test under structured non-nulls, and then explore new forms of masking.

stat.ME↗

arXiv2021

Interactive Martingale Tests for the Global Null

Global null testing is a classical problem going back about a century to Fisher's and Stouffer's combination tests. In this work, we present simple martingale analogs of these classical tests, which are applicable in two distinct settings: (a) the online setting in which there is a possibly infinite sequence of $p$-values, and (b) the batch setting, where one uses prior knowledge to preorder the hypotheses. Through theory and simulations, we demonstrate that our martingale variants have higher power than their classical counterparts even when the preordering is only weakly informative. Finally, using a recent idea of "masking" $p$-values, we develop a novel interactive test for the global null that can take advantage of covariates and repeated user guidance to create a data-adaptive ordering that achieves higher detection power against structured alternatives.

stat.ME↗

arXiv2020

Cross-Spectral Face Hallucination via Disentangling Independent Factors

The cross-sensor gap is one of the challenges that have aroused much research interests in Heterogeneous Face Recognition (HFR). Although recent methods have attempted to fill the gap with deep generative networks, most of them suffer from the inevitable misalignment between different face modalities. Instead of imaging sensors, the misalignment primarily results from facial geometric variations that are independent of the spectrum. Rather than building a monolithic but complex structure, this paper proposes a Pose Aligned Cross-spectral Hallucination (PACH) approach to disentangle the independent factors and deal with them in individual stages. In the first stage, an Unsupervised Face Alignment (UFA) module is designed to align the facial shapes of the near-infrared (NIR) images with those of the visible (VIS) images in a generative way, where UV maps are effectively utilized as the shape guidance. Thus the task of the second stage becomes spectrum translation with aligned paired data. We develop a Texture Prior Synthesis (TPS) module to achieve complexion control and consequently generate more realistic VIS images than existing methods. Experiments on three challenging NIR-VIS datasets verify the effectiveness of our approach in producing visually appealing images and achieving state-of-the-art performance in HFR.

cs.CV↗