Lean Copilot uses LLMs to assist theorem proving in Lean, improving efficiency and automation.
problem Challenges in using existing neural theorem provers to prove novel theorems autonomously.
method Introduces Lean Copilot, a framework for integrating LLMs into Lean's theorem proving process.
result Lean Copilot automates 74.2% of proof steps on average, significantly improving over existing methods.
Report on formalizing differential geometry in Lean.
problem Formalizing differential geometry in a proof assistant.
method Lean's type theory approach to formalization.
result Surprising differences between formal and informal proofs.
Formalizes synthetic differential geometry in Lean.
problem Formalizing synthetic differential geometry in a proof assistant.
method Formalization of synthetic differential geometry with Lean and mathlib.
result Proves a Taylor theorem for functions of several variables.
Lean 4 formalizes Stokes' theorem for smooth singular cubes.
problem Formalizing Stokes' theorem for singular cubes in arbitrary dimensions.
method Using true differential-form pullback via Frechet derivative, bridging to mathlib4's extDeriv.
result d^2=0 for singular cubical chains, chain-level Stokes extended.
Lean 4 library formalizes mathematical finance, verifying over 200 theorems.
problem Formal verification of complex financial mathematics.
method Lean 4 proof assistant, Mathlib, and BrownianMotion package.
result Formal verification yields certified unification of known results.
Scores political leanings in Web3 betting markets.
problem Understanding political motivations in decentralized prediction markets.
method Constructing PBLS from Polymarket data, analyzing 15k addresses, 4k events, 8k markets.
result Validated PBLS through internal and external comparisons, revealing political and profit motives.
LeanML reduces machine learning project waste by estimating best performance without training models.
problem Avoidable wastes in machine learning projects.
method Lean design pattern based on mutual information and performance metrics.
result Estimating best performance without training models is faster and cheaper.
A study ranks critical Lean Six Sigma tools for implementation in Portuguese companies.
problem Identifying the most important tools for successful Lean Six Sigma implementation in Portugal.
method An online survey with Portuguese consultants evaluated 37 tools based on frequency of use, difficulty, importance, and impact. A ranking was developed using a procedure to assess consultants' know-how.
result Honshin Kanri, VOC, VSM were identified as the most important tools for Lean Six Sigma implementation.
This paper formalizes Q-learning and linear TD convergence using Lean 4.
problem Formalizing convergence properties of Q-learning and linear TD learning. method Formal verification using Lean 4 theorem prover and Mathlib library.
result Formalized almost sure convergence of Q-learning and linear TD learning. Formalizes integral curves on Banach manifolds in Lean.
problem Existence and uniqueness of integral curves on Banach manifolds.
method Formalized differential equations on Banach spaces, then generalized to Banach manifolds.
result Established theorems for integral curves on Banach manifolds.
Lean 4 library formalizes mathematical finance, verifying over 200 theorems.
problem Formal verification of complex financial mathematics.
method Lean 4 proof assistant, Mathlib, BrownianMotion package, formal verification of over 200 theorems.
result Formal verification yields certified unification of known financial results.
Transformed geometry into algebra to prove Pick's theorem efficiently.
problem Translating geometric Pick's theorem into formal algebraic proof.
method Formalized geometric Pick's theorem into algebraic proof using Lean.
result Efficient formal proof of Pick's theorem.
Formalizes vNM utility theorem using Lean 4, proving existence and uniqueness.
problem Formalizing and proving the von Neumann-Morgenstern utility theorem.
method Implement classical axioms in Lean 4, formalizing preference relations over lotteries.
result Machine-verified proofs of existence and uniqueness of utility representations.
Formalizes the Fundamental Theorem of Asset Pricing in Lean 4.
problem Formalizing the Fundamental Theorem of Asset Pricing in a proof assistant.
method Formalization in Lean 4 over Mathlib, covering three market settings.
result Constructs the equivalent martingale measure explicitly and proves its properties.
LeanDojo removes barriers to theorem proving with open-source tools and data.
problem Difficulty in reproducing and building on existing theorem proving methods.
method Introduces LeanDojo, an open-source Lean playground with toolkits, data, models, and benchmarks.
result ReProver, an LLM-based prover augmented with retrieval, outperforms non-retrieval baselines and GPT-4.
Paper formalizes Simon's satisficing through FFSD, proving its equivalence to expected utility theory.
problem Formalizing Herbert Simon's bounded rationality concept in economic decision-making.
method Developed FFSD framework using Lean 4 theorem prover, proving equivalence to expected utility theory.
result Equivalence theorem linking FFSD to expected utility maximization for approximate indicator functions.
Understanding the representational power of Restricted Boltzmann Machines (RBMs) with multiple layers is an ill-understood problem and is an area of active research. Motivated from the approach of \emph{Inherent Structure formalism} (Stillinger & Weber, 1982), extensively used in analysing Spin Glasses, we propose a no…
Tests validity of DML estimators without assumptions.
problem Validating DML estimators without making assumptions.
method Develops tests to falsify assumptions for DML estimators.
result Falsifies assumptions for DML estimators with non-trivial power.
In this article we show that if a knot diagram admits a non-trivial coloring modulo 13 then there is an equivalent diagram which can be colored with 5 colors. Leaning on known results, this implies that the minimum number of colors modulo 13 is 5.
We propose a novel learning method for multilayered neural networks which uses feedforward supervisory signal and associates classification of a new input with that of pre-trained input. The proposed method effectively uses rich input information in the earlier layer for robust leaning and revising internal representat…
New method for valid inference from ML-predicted data.
problem Invalid scientific conclusions from ML-predicted outcomes.
method Assumption-Lean and Data-Adaptive Post-Prediction Inference (PSPA)
result Valid and powerful inference based on ML-predicted data.
A machine-checked Itô calculus for Brownian motion on [0,T]
problem Developing an L2 Itô calculus for Brownian motion method Formalized in Lean 4 on top of Mathlib and the BrownianMotion package
result First machine-checked proof of Itô's formula and construction of Itô integral as martingale-valued process
Paper formalizes multi-dimensional FSD using geometric methods.
problem Complex measure theory and calculus barriers to formalization in proof assistants.
method Geometric framework for first-order stochastic dominance in N dimensions.
result Geometric approach bypasses complex integration theory for direct comparison of survival probabilities.
We introduce trading fees into AMM models and analyze their impact on swap rates and profits.
problem The impact of trading fees on AMM models and users' trading strategies.
method We extend a foundational AMM model by introducing a trading fee parameter and analyze the model using economic and mathematical rigor.
result Trading fees affect the additivity of swap rates and can lead to greater profits from larger trades.
Convolutional Neural Networks (CNNs) have become indispensable for solving machine learning tasks in speech recognition, computer vision, and other areas that involve high-dimensional data. A CNN filters the input feature using a network containing spatial convolution operators with compactly supported stencils. In pra…
This paper formalizes the h-principle and sphere eversion in differential topology.
problem Formalizing the h-principle and sphere eversion in differential topology.
method Lean formalization of the local h-principle for first-order partial differential relations, using convex integration.
result Reproves Smale's sphere eversion theorem and formalizes advanced mathematics.
We present a formal proof in Lean of probably approximately correct (PAC) learnability of the concept class of decision stumps. This classic result in machine learning theory derives a bound on error probabilities for a simple type of classifier. Though such a proof appears simple on paper, analytic and measure-theoret…
Proposes new methods for inference in GLMs without assuming model correctness.
problem Inference for GLMs assumes model correctness, leading to uncertainty and bias.
method Develops nonparametric estimands and uses influence curves with flexible procedures.
result Inference for GLM parameters is improved without model correctness assumptions.
Benchmarking recursive collapse claims with a new framework under false-positive control.
problem Evaluating recursive systems for failure patterns and warning claims.
method Developed Loopzero framework for testing recursive failures, specified claim boundaries in Lean, evaluated under FP constraint, and compared with standard detectors.
result No standard detectors or Loopzero's pre-registered quantile detector achieved the required operating point under the false-positive contract.
Optimal scheduling of hydrogen production in dynamic pricing power market can maximize the profit of hydrogen producer; however, it highly depends on the accurate forecast of hydrogen consumption. In this paper, we propose a deep leaning based forecasting approach for predicting hydrogen consumption of fuel cell vehicl…
No feature ranking can be faithful, stable, and complete when features are collinear.
problem The impossibility of creating a feature ranking that is simultaneously faithful, stable, and complete when features are collinear.
method Proving the impossibility, quantifying it for four model classes, resolving it via ensemble averaging (DASH), and machine-verifying it with Lean 4 theorems.
result No method lies outside the dichotomy of faithful-complete methods (unstable, with rankings that flip up to 50% of the time) and ensemble methods (stable, reporting ties for symmetric features).
Dictionary leaning (DL) and dimensionality reduction (DR) are powerful tools to analyze high-dimensional noisy signals. This paper presents a proposal of a novel Riemannian joint dimensionality reduction and dictionary learning (R-JDRDL) on symmetric positive definite (SPD) manifolds for classification tasks. The joint…
TADA detects anomalies in time series using topological data analysis.
problem Detecting global changes in dependency structure between channels in multivariate time series.
method Topological Data Analysis for detecting anomalies in multivariate time series.
result The approach is more suitable for detecting global changes of correlation structures than existing methods.
Examines learning efficiency in neural networks and related models.
problem Analyzing efficiency in deep learning models with singular learning coefficients.
method Examined learning coefficients in neural networks and three-layer neural networks with ReLU units.
result Extended results to include Softmax function, providing a broader understanding of learning efficiency.
We investigate the learning rate of multiple kernel leaning (MKL) with elastic-net regularization, which consists of an ℓ1-regularizer for inducing the sparsity and an ℓ2-regularizer for controlling the smoothness. We focus on a sparse setting where the total number of kernels is large but the number of non…
Convolutional Neural Networks (CNNs) filter the input data using spatial convolution operators with compact stencils. Commonly, the convolution operators couple features from all channels, which leads to immense computational cost in the training of and prediction with CNNs. To improve the efficiency of CNNs, we introd…
We provide a lean, non-technical exposition on the pricing of path-dependent and European-style derivatives in the Cox-Ross-Rubinstein (CRR) pricing model. The main tool used in the paper for cleaning up the reasoning is applying static hedging arguments. This can be accomplished by taking various routes through some a…
Researchers solved a number-theoretic hypothesis to determine the spin parity of k-differentials.
problem Determining the spin parity of k-differentials on Riemann surfaces of genus zero and one.
method Proved a number-theoretic hypothesis (Conjecture A.10) by reformulating it in terms of Jacobi symbols and reducing it to a combinatorial identity.
result The spin parity of k-differentials on Riemann surfaces of genus zero and one was completely determined.
In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type (2, n) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on …
Developed a machine-checked Itô calculus for Brownian motion.
problem Formal verification of Itô calculus for Brownian motion.
method Machine-checked formalization in Lean over Mathlib.
result First machine-checked constructions of the Itô integral and Itô's formula.
Build-to-order (BTO) supply chains have become common-place in industries such as electronics, automotive and fashion. They enable building products based on individual requirements with a short lead time and minimum inventory and production costs. Due to their nature, they differ significantly from traditional supply …
Anomaly detection is the process of finding data points that deviate from a baseline. In a real-life setting, anomalies are usually unknown or extremely rare. Moreover, the detection must be accomplished in a timely manner or the risk of corrupting the system might grow exponentially. In this work, we propose a two lev…
Paper presents a new insurance model equation for diverse structures.
problem Handling diverse insurance models with a single equation.
method Developed a canonical model construction and stochastic backward equations.
result Comparison theorems for different models follow from the new equation.
A new method for safer statistical inference after predictions.
problem Statistical inference with pseudo-outcomes from machine learning predictions.
method Prediction De-Correlated Inference (PDC) framework.
result PDC consistently outperforms supervised methods and can adapt to any model.
The paper develops methods to price options under rough volatility models using BSPDEs.
problem Pricing options in models with non-Markovian dynamics.
method Backward stochastic partial differential equations (BSPDEs) and deep learning for numerical approximations.
result Existence and uniqueness of weak solutions for general nonlinear BSPDEs.
Deep generative models are tremendously successful in learning low-dimensional latent representations that well-describe the data. These representations, however, tend to much distort relationships between points, i.e. pairwise distances tend to not reflect semantic similarities well. This renders unsupervised tasks, s…
Paper presents a privacy-preserving algorithm for estimating peer effects using the Ising model.
problem Privacy concerns in estimating peer effects using network data.
method Developed a (ε,δ)-differentially private algorithm using Ising model. result Established regret bounds and validated performance on synthetic and real-world networks.
Four-dimensional scanning transmission electron microscopy (4D-STEM) of local atomic diffraction patterns is emerging as a powerful technique for probing intricate details of atomic structure and atomic electric fields. However, efficient processing and interpretation of large volumes of data remain challenging, especi…