Daily Papers

kooplearn is a machine-learning library that implements linear, kernel, and deep-learning estimators of dynamical operators and their spectral decompositions. kooplearn can model both discrete-time evolution operators (Koopman/Transfer) and continuous-time infinitesimal generators. By learning these operators, users can analyze dynamical systems via spectral methods, derive data-driven reduced-order models, and forecast future states and observables. kooplearn's interface is compliant with the scikit-learn API, facilitating its integration into existing machine learning and data science workflows. Additionally, kooplearn includes curated benchmark datasets to support experimentation, reproducibility, and the fair comparison of learning algorithms. The software is available at https://github.com/Machine-Learning-Dynamical-Systems/kooplearn.

We introduce Infinigen, a procedural generator of photorealistic 3D scenes of the natural world. Infinigen is entirely procedural: every asset, from shape to texture, is generated from scratch via randomized mathematical rules, using no external source and allowing infinite variation and composition. Infinigen offers broad coverage of objects and scenes in the natural world including plants, animals, terrains, and natural phenomena such as fire, cloud, rain, and snow. Infinigen can be used to generate unlimited, diverse training data for a wide range of computer vision tasks including object detection, semantic segmentation, optical flow, and 3D reconstruction. We expect Infinigen to be a useful resource for computer vision research and beyond. Please visit https://infinigen.org for videos, code and pre-generated data.

We present Infinity, a Bitwise Visual AutoRegressive Modeling capable of generating high-resolution, photorealistic images following language instruction. Infinity redefines visual autoregressive model under a bitwise token prediction framework with an infinite-vocabulary tokenizer & classifier and bitwise self-correction mechanism, remarkably improving the generation capacity and details. By theoretically scaling the tokenizer vocabulary size to infinity and concurrently scaling the transformer size, our method significantly unleashes powerful scaling capabilities compared to vanilla VAR. Infinity sets a new record for autoregressive text-to-image models, outperforming top-tier diffusion models like SD3-Medium and SDXL. Notably, Infinity surpasses SD3-Medium by improving the GenEval benchmark score from 0.62 to 0.73 and the ImageReward benchmark score from 0.87 to 0.96, achieving a win rate of 66%. Without extra optimization, Infinity generates a high-quality 1024x1024 image in 0.8 seconds, making it 2.6x faster than SD3-Medium and establishing it as the fastest text-to-image model. Models and codes will be released to promote further exploration of Infinity for visual generation and unified tokenizer modeling.

We propose a new approach for generative modeling based on training a neural network to be idempotent. An idempotent operator is one that can be applied sequentially without changing the result beyond the initial application, namely f(f(z))=f(z). The proposed model f is trained to map a source distribution (e.g, Gaussian noise) to a target distribution (e.g. realistic images) using the following objectives: (1) Instances from the target distribution should map to themselves, namely f(x)=x. We define the target manifold as the set of all instances that f maps to themselves. (2) Instances that form the source distribution should map onto the defined target manifold. This is achieved by optimizing the idempotence term, f(f(z))=f(z) which encourages the range of f(z) to be on the target manifold. Under ideal assumptions such a process provably converges to the target distribution. This strategy results in a model capable of generating an output in one step, maintaining a consistent latent space, while also allowing sequential applications for refinement. Additionally, we find that by processing inputs from both target and source distributions, the model adeptly projects corrupted or modified data back to the target manifold. This work is a first step towards a ``global projector'' that enables projecting any input into a target data distribution.

We introduce InfinityStar, a unified spacetime autoregressive framework for high-resolution image and dynamic video synthesis. Building on the recent success of autoregressive modeling in both vision and language, our purely discrete approach jointly captures spatial and temporal dependencies within a single architecture. This unified design naturally supports a variety of generation tasks such as text-to-image, text-to-video, image-to-video, and long interactive video synthesis via straightforward temporal autoregression. Extensive experiments demonstrate that InfinityStar scores 83.74 on VBench, outperforming all autoregressive models by large margins, even surpassing some diffusion competitors like HunyuanVideo. Without extra optimizations, our model generates a 5s, 720p video approximately 10x faster than leading diffusion-based methods. To our knowledge, InfinityStar is the first discrete autoregressive video generator capable of producing industrial level 720p videos. We release all code and models to foster further research in efficient, high-quality video generation.

In this paper, we present NUWA-Infinity, a generative model for infinite visual synthesis, which is defined as the task of generating arbitrarily-sized high-resolution images or long-duration videos. An autoregressive over autoregressive generation mechanism is proposed to deal with this variable-size generation task, where a global patch-level autoregressive model considers the dependencies between patches, and a local token-level autoregressive model considers dependencies between visual tokens within each patch. A Nearby Context Pool (NCP) is introduced to cache-related patches already generated as the context for the current patch being generated, which can significantly save computation costs without sacrificing patch-level dependency modeling. An Arbitrary Direction Controller (ADC) is used to decide suitable generation orders for different visual synthesis tasks and learn order-aware positional embeddings. Compared to DALL-E, Imagen and Parti, NUWA-Infinity can generate high-resolution images with arbitrary sizes and support long-duration video generation additionally. Compared to NUWA, which also covers images and videos, NUWA-Infinity has superior visual synthesis capabilities in terms of resolution and variable-size generation. The GitHub link is https://github.com/microsoft/NUWA. The homepage link is https://nuwa-infinity.microsoft.com.

Recent advancements in Chain-of-Thoughts (CoT) and Program-of-Thoughts (PoT) methods have greatly enhanced language models' mathematical reasoning capabilities, facilitating their integration into instruction tuning datasets with LLMs. However, existing methods for large-scale dataset creation require substantial seed data and high computational costs for data synthesis, posing significant challenges for scalability. We introduce InfinityMATH, a scalable instruction tuning dataset for programmatic mathematical reasoning. The construction pipeline emphasizes decoupling numbers from mathematical problems to synthesize number-independent programs, enabling efficient and flexible scaling while minimizing dependency on specific numerical values. Fine-tuning experiments with open-source language and code models, such as Llama2 and CodeLlama, demonstrate the practical benefits of InfinityMATH. These fine-tuned models, showed significant relative improvements on both in-domain and out-of-domain benchmarks, ranging from 184.7% to 514.3% on average. Additionally, these models exhibited high robustness on the GSM8K+ and MATH+ benchmarks, which are enhanced version of test sets with simply the number variations. InfinityMATH ensures that models are more versatile and effective across a broader range of mathematical problems. The data is available at https://huggingface.co/datasets/flagopen/InfinityMATH.

Modularization is a cornerstone of computer science, abstracting complex functions into atomic building blocks. In this paper, we introduce a new level of modularization by abstracting generative models into atomic generative modules. Analogous to fractals in mathematics, our method constructs a new type of generative model by recursively invoking atomic generative modules, resulting in self-similar fractal architectures that we call fractal generative models. As a running example, we instantiate our fractal framework using autoregressive models as the atomic generative modules and examine it on the challenging task of pixel-by-pixel image generation, demonstrating strong performance in both likelihood estimation and generation quality. We hope this work could open a new paradigm in generative modeling and provide a fertile ground for future research. Code is available at https://github.com/LTH14/fractalgen.

Scientists often infer abstract procedures from specific instances of problems and use the abstractions to generate new, related instances. For example, programs encoding the formal rules and properties of a system have been useful in fields ranging from RL (procedural environments) to physics (simulation engines). These programs can be seen as functions which execute to different outputs based on their parameterizations (e.g., gridworld configuration or initial physical conditions). We introduce the term EFA (Executable Functional Abstraction) to denote such programs for math problems. EFA-like constructs have been shown to be useful for math reasoning as problem generators for stress-testing models. However, prior work has been limited to abstractions for grade-school math (whose simple rules are easy to encode in programs), while generating EFAs for advanced math has thus far required human engineering. We explore the automatic construction of EFAs for advanced math problems. We operationalize the task of automatically constructing EFAs as a program synthesis task, and develop EFAGen, which conditions an LLM on a seed math problem and its step-by-step solution to generate candidate EFA programs that are faithful to the generalized problem and solution class underlying the seed problem. Furthermore, we formalize properties any valid EFA must possess in terms of executable unit tests, and show how the tests can be used as verifiable rewards to train LLMs to become better writers of EFAs. We demonstrate that EFAs constructed by EFAGen behave rationally by remaining faithful to seed problems, produce learnable problem variations, and that EFAGen can infer EFAs across multiple diverse sources of competition-level math problems. Finally, we show downstream uses of model-written EFAs e.g. finding problem variations that are harder or easier for a learner to solve, as well as data generation.

This paper introduces a novel training methodology that enables a small Transformer model to generalize the addition of two-digit numbers to numbers with unseen lengths of digits. The proposed approach employs an autoregressive generation technique, processing from right to left, which mimics a common manual method for adding large numbers. To the best of my knowledge, this methodology has not been previously explored in the literature. All results are reproducible, and the corresponding R code is available at: https://github.com/AGPatriota/ALGA-R/.

In this article, we propose a new counter-based implementation of John von Neumann's middle-square random number generator (RNG). Several rounds of squaring are applied to a counter to produce a random output. We discovered that four rounds are sufficient to provide satisfactory data. Two versions of the RNG are presented, a 4-round version with 32-bit output and a 5-round version with 64-bit output. Both pass stringent tests of randomness and may be the fastest counter-based generators.

Formal proofs are challenging to write even for experienced experts. Recent progress in Neural Theorem Proving (NTP) shows promise in expediting this process. However, the formal corpora available on the Internet are limited compared to the general text, posing a significant data scarcity challenge for NTP. To address this issue, this work proposes Alchemy, a general framework for data synthesis that constructs formal theorems through symbolic mutation. Specifically, for each candidate theorem in Mathlib, we identify all invocable theorems that can be used to rewrite or apply to it. Subsequently, we mutate the candidate theorem by replacing the corresponding term in the statement with its equivalent form or antecedent. As a result, our method increases the number of theorems in Mathlib by an order of magnitude, from 110k to 6M. Furthermore, we perform continual pretraining and supervised finetuning on this augmented corpus for large language models. Experimental results demonstrate the effectiveness of our approach, achieving a 5% absolute performance improvement on Leandojo benchmark. Additionally, our synthetic data achieve a 2.5% absolute performance gain on the out-of-distribution miniF2F benchmark. To provide further insights, we conduct a comprehensive analysis of synthetic data composition and the training paradigm, offering valuable guidance for developing a strong theorem prover.

We introduce infty-Diff, a generative diffusion model which directly operates on infinite resolution data. By randomly sampling subsets of coordinates during training and learning to denoise the content at those coordinates, a continuous function is learned that allows sampling at arbitrary resolutions. In contrast to other recent infinite resolution generative models, our approach operates directly on the raw data, not requiring latent vector compression for context, using hypernetworks, nor relying on discrete components. As such, our approach achieves significantly higher sample quality, as evidenced by lower FID scores, as well as being able to effectively scale to higher resolutions than the training data while retaining detail.

Toward infinite-scale 3D city synthesis, we propose a novel framework, InfiniCity, which constructs and renders an unconstrainedly large and 3D-grounded environment from random noises. InfiniCity decomposes the seemingly impractical task into three feasible modules, taking advantage of both 2D and 3D data. First, an infinite-pixel image synthesis module generates arbitrary-scale 2D maps from the bird's-eye view. Next, an octree-based voxel completion module lifts the generated 2D map to 3D octrees. Finally, a voxel-based neural rendering module texturizes the voxels and renders 2D images. InfiniCity can thus synthesize arbitrary-scale and traversable 3D city environments, and allow flexible and interactive editing from users. We quantitatively and qualitatively demonstrate the efficacy of the proposed framework. Project page: https://hubert0527.github.io/infinicity/

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. We demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We showcase an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

Can one inject new concepts into an already trained generative model, while respecting its existing structure and knowledge? We propose a new task - domain expansion - to address this. Given a pretrained generator and novel (but related) domains, we expand the generator to jointly model all domains, old and new, harmoniously. First, we note the generator contains a meaningful, pretrained latent space. Is it possible to minimally perturb this hard-earned representation, while maximally representing the new domains? Interestingly, we find that the latent space offers unused, "dormant" directions, which do not affect the output. This provides an opportunity: By "repurposing" these directions, we can represent new domains without perturbing the original representation. In fact, we find that pretrained generators have the capacity to add several - even hundreds - of new domains! Using our expansion method, one "expanded" model can supersede numerous domain-specific models, without expanding the model size. Additionally, a single expanded generator natively supports smooth transitions between domains, as well as composition of domains. Code and project page available at https://yotamnitzan.github.io/domain-expansion/.

We present Generative Logic (GL), a deterministic architecture that begins from user-supplied axiomatic definitions -- written in a minimalist Mathematical Programming Language (MPL) -- and systematically explores their deductive neighborhood. Definitions are compiled into a distributed grid of simple Logic Blocks (LBs) that exchange messages; any time several expressions unify under an inference rule, a new fact is emitted with full provenance to its sources, yielding replayable, auditable proof graphs. A prototype software implementation instantiates the workflow on first-order Peano arithmetic. Starting only from the Peano axioms, GL enumerates candidate implications, applies normalization and type filters, and automatically reconstructs machine-checkable proofs of foundational arithmetic laws including associativity and commutativity of addition, associativity and commutativity of multiplication, and distributivity. Generated proofs export to navigable HTML so that every inference step can be inspected independently. We outline a hardware-software co-design path toward massively parallel realizations and describe prospective integration with probabilistic models (e.g., Large Language Models (LLMs)) for autoformalization and conjecture seeding. The Python and MPL code to reproduce the Peano experiments, along with the full HTML proof graphs, are available in the project's GitHub repository at https://github.com/Generative-Logic/GL/tree/35a111ea9ba53afe051703d6050be0c3923e9724 and are permanently archived at https://doi.org/10.5281/zenodo.16408441. We invite community feedback and collaboration.

In the last three years, the largest dense deep learning models have grown over 1000x to reach hundreds of billions of parameters, while the GPU memory has only grown by 5x (16 GB to 80 GB). Therefore, the growth in model scale has been supported primarily though system innovations that allow large models to fit in the aggregate GPU memory of multiple GPUs. However, we are getting close to the GPU memory wall. It requires 800 NVIDIA V100 GPUs just to fit a trillion parameter model for training, and such clusters are simply out of reach for most data scientists. In addition, training models at that scale requires complex combinations of parallelism techniques that puts a big burden on the data scientists to refactor their model. In this paper we present ZeRO-Infinity, a novel heterogeneous system technology that leverages GPU, CPU, and NVMe memory to allow for unprecedented model scale on limited resources without requiring model code refactoring. At the same time it achieves excellent training throughput and scalability, unencumbered by the limited CPU or NVMe bandwidth. ZeRO-Infinity can fit models with tens and even hundreds of trillions of parameters for training on current generation GPU clusters. It can be used to fine-tune trillion parameter models on a single NVIDIA DGX-2 node, making large models more accessible. In terms of training throughput and scalability, it sustains over 25 petaflops on 512 NVIDIA V100 GPUs(40% of peak), while also demonstrating super linear scalability. An open source implementation of ZeRO-Infinity is available through DeepSpeed, a deep learning optimization library that makes distributed training easy, efficient, and effective.

The rapid advancement of large language models (LLMs) such as GPT-3, PaLM, and Llama has significantly transformed natural language processing, showcasing remarkable capabilities in understanding and generating language. However, these models often struggle with tasks requiring complex reasoning, particularly in mathematical problem-solving, due in part to the scarcity of large-scale, high-quality, domain-specific datasets necessary for training sophisticated reasoning abilities. To address this limitation, we introduce Template-based Data Generation (TDG), a novel approach that leverages LLMs (GPT-4) to automatically generate parameterized meta-templates, which are then used to synthesize a vast array of high-quality problems and solutions. Leveraging TDG, we create TemplateMath Part I: TemplateGSM, a dataset comprising over 7 million synthetically generated grade school math problems--each accompanied by code-based and natural language solutions--with the potential to generate an effectively unlimited number more. This dataset alleviates the scarcity of large-scale mathematical datasets and serves as a valuable resource for pre-training, fine-tuning, and evaluating LLMs in mathematical reasoning. Our method not only enables the generation of virtually infinite data but also elevates data augmentation to a new level by using GPT-4 for meta-template generation, ensuring diverse and high-quality problem structures. The TemplateMath Part I: TemplateGSM dataset is publicly available at https://huggingface.co/datasets/math-ai/TemplateGSM. The code is available at https://github.com/iiis-ai/TemplateMath.

Automated theorem proving (ATP) has become an appealing domain for exploring the reasoning ability of the recent successful generative language models. However, current ATP benchmarks mainly focus on symbolic inference, but rarely involve the understanding of complex number combination reasoning. In this work, we propose TRIGO, an ATP benchmark that not only requires a model to reduce a trigonometric expression with step-by-step proofs but also evaluates a generative LM's reasoning ability on formulas and its capability to manipulate, group, and factor number terms. We gather trigonometric expressions and their reduced forms from the web, annotate the simplification process manually, and translate it into the Lean formal language system. We then automatically generate additional examples from the annotated samples to expand the dataset. Furthermore, we develop an automatic generator based on Lean-Gym to create dataset splits of varying difficulties and distributions in order to thoroughly analyze the model's generalization ability. Our extensive experiments show our proposed TRIGO poses a new challenge for advanced generative LM's including GPT-4 which is pre-trained on a considerable amount of open-source formal theorem-proving language data, and provide a new tool to study the generative LM's ability on both formal and mathematical reasoning.

We study the problem of extrapolative controlled generation, i.e., generating sequences with attribute values beyond the range seen in training. This task is of significant importance in automated design, especially drug discovery, where the goal is to design novel proteins that are better (e.g., more stable) than existing sequences. Thus, by definition, the target sequences and their attribute values are out of the training distribution, posing challenges to existing methods that aim to directly generate the target sequence. Instead, in this work, we propose Iterative Controlled Extrapolation (ICE) which iteratively makes local edits to a sequence to enable extrapolation. We train the model on synthetically generated sequence pairs that demonstrate small improvement in the attribute value. Results on one natural language task (sentiment analysis) and two protein engineering tasks (ACE2 stability and AAV fitness) show that ICE considerably outperforms state-of-the-art approaches despite its simplicity. Our code and models are available at: https://github.com/vishakhpk/iter-extrapolation.

In recent years, there have been remarkable advancements in the performance of Transformer-based Large Language Models (LLMs) across various domains. As these LLMs are deployed for increasingly complex tasks, they often face the needs to conduct longer reasoning processes or understanding larger contexts. In these situations, the length generalization failure of LLMs on long sequences become more prominent. Most pre-training schemes truncate training sequences to a fixed length (such as 2048 for LLaMa). LLMs often struggle to generate fluent texts, let alone carry out downstream tasks, after longer contexts, even with relative positional encoding which is designed to cope with this problem. Common solutions such as finetuning on longer corpora often involves daunting hardware and time costs and requires careful training process design. To more efficiently leverage the generation capacity of existing LLMs, we theoretically and empirically investigate the main out-of-distribution (OOD) factors contributing to this problem. Inspired by this diagnosis, we propose a simple yet effective solution for on-the-fly length generalization, LM-Infinite, which involves only a Lambda-shaped attention mask and a distance limit while requiring no parameter updates or learning. We find it applicable to a variety of LLMs using relative-position encoding methods. LM-Infinite is computational efficient with O(n) time and space, and demonstrates consistent fluency and generation quality to as long as 32k tokens on ArXiv and OpenWebText2 datasets, with 2.72x decoding speedup. On downstream task such as passkey retrieval, it continues to work on inputs much longer than training lengths where vanilla models fail immediately.

AlphaEvolve is a generic evolutionary coding agent that combines the generative capabilities of LLMs with automated evaluation in an iterative evolutionary framework that proposes, tests, and refines algorithmic solutions to challenging scientific and practical problems. In this paper we showcase AlphaEvolve as a tool for autonomously discovering novel mathematical constructions and advancing our understanding of long-standing open problems. To demonstrate its breadth, we considered a list of 67 problems spanning mathematical analysis, combinatorics, geometry, and number theory. The system rediscovered the best known solutions in most of the cases and discovered improved solutions in several. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. These results demonstrate that large language model-guided evolutionary search can autonomously discover mathematical constructions that complement human intuition, at times matching or even improving the best known results, highlighting the potential for significant new ways of interaction between mathematicians and AI systems. We present AlphaEvolve as a powerful new tool for mathematical discovery, capable of exploring vast search spaces to solve complex optimization problems at scale, often with significantly reduced requirements on preparation and computation time.

Advancements in DNA sequencing technologies have significantly improved our ability to decode genomic sequences. However, the prediction and interpretation of these sequences remain challenging due to the intricate nature of genetic material. Large language models (LLMs) have introduced new opportunities for biological sequence analysis. Recent developments in genomic language models have underscored the potential of LLMs in deciphering DNA sequences. Nonetheless, existing models often face limitations in robustness and application scope, primarily due to constraints in model structure and training data scale. To address these limitations, we present GENERator, a generative genomic foundation model featuring a context length of 98k base pairs (bp) and 1.2B parameters. Trained on an expansive dataset comprising 386B bp of eukaryotic DNA, the GENERator demonstrates state-of-the-art performance across both established and newly proposed benchmarks. The model adheres to the central dogma of molecular biology, accurately generating protein-coding sequences that translate into proteins structurally analogous to known families. It also shows significant promise in sequence optimization, particularly through the prompt-responsive generation of promoter sequences with specific activity profiles. These capabilities position the GENERator as a pivotal tool for genomic research and biotechnological advancement, enhancing our ability to interpret and predict complex biological systems and enabling precise genomic interventions.

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for pi, ln(2), Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

Compression is at the heart of intelligence. A theoretically optimal way to compress any sequence of data is to find the shortest program that outputs that sequence and then halts. However, such 'Kolmogorov compression' is uncomputable, and code generating LLMs struggle to approximate this theoretical ideal, as it requires reasoning, planning and search capabilities beyond those of current models. In this work, we introduce the KoLMogorov-Test (KT), a compression-as-intelligence test for code generating LLMs. In KT a model is presented with a sequence of data at inference time, and asked to generate the shortest program that produces the sequence. We identify several benefits of KT for both evaluation and training: an essentially infinite number of problem instances of varying difficulty is readily available, strong baselines already exist, the evaluation metric (compression) cannot be gamed, and pretraining data contamination is highly unlikely. To evaluate current models, we use audio, text, and DNA data, as well as sequences produced by random synthetic programs. Current flagship models perform poorly - both GPT4-o and Llama-3.1-405B struggle on our natural and synthetic sequences. On our synthetic distribution, we are able to train code generation models with lower compression rates than previous approaches. Moreover, we show that gains on synthetic data generalize poorly to real data, suggesting that new innovations are necessary for additional gains on KT.

We introduce Functional Diffusion Processes (FDPs), which generalize score-based diffusion models to infinite-dimensional function spaces. FDPs require a new mathematical framework to describe the forward and backward dynamics, and several extensions to derive practical training objectives. These include infinite-dimensional versions of Girsanov theorem, in order to be able to compute an ELBO, and of the sampling theorem, in order to guarantee that functional evaluations in a countable set of points are equivalent to infinite-dimensional functions. We use FDPs to build a new breed of generative models in function spaces, which do not require specialized network architectures, and that can work with any kind of continuous data. Our results on real data show that FDPs achieve high-quality image generation, using a simple MLP architecture with orders of magnitude fewer parameters than existing diffusion models.

Despite their strong performances on many generative tasks, diffusion models require a large number of sampling steps in order to generate realistic samples. This has motivated the community to develop effective methods to distill pre-trained diffusion models into more efficient models, but these methods still typically require few-step inference or perform substantially worse than the underlying model. In this paper, we present Score Implicit Matching (SIM) a new approach to distilling pre-trained diffusion models into single-step generator models, while maintaining almost the same sample generation ability as the original model as well as being data-free with no need of training samples for distillation. The method rests upon the fact that, although the traditional score-based loss is intractable to minimize for generator models, under certain conditions we can efficiently compute the gradients for a wide class of score-based divergences between a diffusion model and a generator. SIM shows strong empirical performances for one-step generators: on the CIFAR10 dataset, it achieves an FID of 2.06 for unconditional generation and 1.96 for class-conditional generation. Moreover, by applying SIM to a leading transformer-based diffusion model, we distill a single-step generator for text-to-image (T2I) generation that attains an aesthetic score of 6.42 with no performance decline over the original multi-step counterpart, clearly outperforming the other one-step generators including SDXL-TURBO of 5.33, SDXL-LIGHTNING of 5.34 and HYPER-SDXL of 5.85. We will release this industry-ready one-step transformer-based T2I generator along with this paper.

We introduce Goedel-Prover, an open-source large language model (LLM) that achieves the state-of-the-art (SOTA) performance in automated formal proof generation for mathematical problems. The key challenge in this field is the scarcity of formalized math statements and proofs, which we tackle in the following ways. We train statement formalizers to translate the natural language math problems from Numina into formal language (Lean 4), creating a dataset of 1.64 million formal statements. LLMs are used to check that the formal statements accurately preserve the content of the original natural language problems. We then iteratively build a large dataset of formal proofs by training a series of provers. Each prover succeeds in proving many statements that the previous ones could not, and these new proofs are added to the training set for the next prover. The final prover outperforms all existing open-source models in whole-proof generation. On the miniF2F benchmark, it achieves a 57.6% success rate (Pass@32), exceeding the previous best open-source model by 7.6%. On PutnamBench, Goedel-Prover successfully solves 7 problems (Pass@512), ranking first on the leaderboard. Furthermore, it generates 29.7K formal proofs for Lean Workbook problems, nearly doubling the 15.7K produced by earlier works.

Straight line equation y=mx with slope m, when singularly perturbed as ay^3+y=mx with a positive parameter a, results in S-shaped curves or S-curves on a real plane. As arightarrow 0, we get back y=mx which is a cumulative distribution function of a continuous uniform distribution that describes the occurrence of every event in an interval to be equally probable. As arightarrowinfty, the derivative of y has finite support only at y=0 resembling a degenerate distribution. Based on these arguments, in this work, we propose that these S-curves can represent maximum entropy uniform distribution to a zero entropy single value. We also argue that these S-curves are superposable as they are only parametrically nonlinear but fundamentally linear. So far, the superposed forms have been used to capture the patterns of natural systems such as nonlinear dynamics of biological growth and kinetics of enzyme reactions. Here, we attempt to use the S-curve and its superposed form as statistical models. We fit the models on a classical dataset containing flower measurements of iris plants and analyze their usefulness in pattern recognition. Based on these models, we claim that any non-uniform pattern can be represented as a singular perturbation to uniform distribution. However, our parametric estimation procedure have some limitations such as sensitivity to initial conditions depending on the data at hand.

Large language models have made significant progress in mathematical reasoning, which serves as an important testbed for AI and could impact scientific research if further advanced. By scaling reasoning with reinforcement learning that rewards correct final answers, LLMs have improved from poor performance to saturating quantitative reasoning competitions like AIME and HMMT in one year. However, this approach faces fundamental limitations. Pursuing higher final answer accuracy doesn't address a key issue: correct answers don't guarantee correct reasoning. Moreover, many mathematical tasks like theorem proving require rigorous step-by-step derivation rather than numerical answers, making final answer rewards inapplicable. To push the limits of deep reasoning, we believe it is necessary to verify the comprehensiveness and rigor of mathematical reasoning. Self-verification is particularly important for scaling test-time compute, especially for open problems without known solutions. Towards self-verifiable mathematical reasoning, we investigate how to train an accurate and faithful LLM-based verifier for theorem proving. We then train a proof generator using the verifier as the reward model, and incentivize the generator to identify and resolve as many issues as possible in their own proofs before finalizing them. To maintain the generation-verification gap as the generator becomes stronger, we propose to scale verification compute to automatically label new hard-to-verify proofs, creating training data to further improve the verifier. Our resulting model, DeepSeekMath-V2, demonstrates strong theorem-proving capabilities, achieving gold-level scores on IMO 2025 and CMO 2024 and a near-perfect 118/120 on Putnam 2024 with scaled test-time compute.

We propose Stable Video Infinity (SVI) that is able to generate infinite-length videos with high temporal consistency, plausible scene transitions, and controllable streaming storylines. While existing long-video methods attempt to mitigate accumulated errors via handcrafted anti-drifting (e.g., modified noise scheduler, frame anchoring), they remain limited to single-prompt extrapolation, producing homogeneous scenes with repetitive motions. We identify that the fundamental challenge extends beyond error accumulation to a critical discrepancy between the training assumption (seeing clean data) and the test-time autoregressive reality (conditioning on self-generated, error-prone outputs). To bridge this hypothesis gap, SVI incorporates Error-Recycling Fine-Tuning, a new type of efficient training that recycles the Diffusion Transformer (DiT)'s self-generated errors into supervisory prompts, thereby encouraging DiT to actively identify and correct its own errors. This is achieved by injecting, collecting, and banking errors through closed-loop recycling, autoregressively learning from error-injected feedback. Specifically, we (i) inject historical errors made by DiT to intervene on clean inputs, simulating error-accumulated trajectories in flow matching; (ii) efficiently approximate predictions with one-step bidirectional integration and calculate errors with residuals; (iii) dynamically bank errors into replay memory across discretized timesteps, which are resampled for new input. SVI is able to scale videos from seconds to infinite durations with no additional inference cost, while remaining compatible with diverse conditions (e.g., audio, skeleton, and text streams). We evaluate SVI on three benchmarks, including consistent, creative, and conditional settings, thoroughly verifying its versatility and state-of-the-art role.

Mathematical formulas are a fundamental and widely used component in various scientific fields, serving as a universal language for expressing complex concepts and relationships. While state-of-the-art transformer models excel in processing and understanding natural language, they encounter challenges with mathematical notation, which involves a complex structure and diverse representations. This study focuses on the development of specialized training datasets to enhance the encoding of mathematical content. We introduce Math Mutator (MAMUT), a framework capable of generating equivalent and falsified versions of a given mathematical formula in LaTeX notation, effectively capturing the mathematical variety in notation of the same concept. Based on MAMUT, we have generated four large mathematical datasets containing diverse notation, which can be used to train language models with enhanced mathematical embeddings.

Autoregressive (AR) models have achieved state-of-the-art performance in text and image generation but suffer from slow generation due to the token-by-token process. We ask an ambitious question: can a pre-trained AR model be adapted to generate outputs in just one or two steps? If successful, this would significantly advance the development and deployment of AR models. We notice that existing works that try to speed up AR generation by generating multiple tokens at once fundamentally cannot capture the output distribution due to the conditional dependencies between tokens, limiting their effectiveness for few-step generation. To address this, we propose Distilled Decoding (DD), which uses flow matching to create a deterministic mapping from Gaussian distribution to the output distribution of the pre-trained AR model. We then train a network to distill this mapping, enabling few-step generation. DD doesn't need the training data of the original AR model, making it more practical.We evaluate DD on state-of-the-art image AR models and present promising results on ImageNet-256. For VAR, which requires 10-step generation, DD enables one-step generation (6.3times speed-up), with an acceptable increase in FID from 4.19 to 9.96. For LlamaGen, DD reduces generation from 256 steps to 1, achieving an 217.8times speed-up with a comparable FID increase from 4.11 to 11.35. In both cases, baseline methods completely fail with FID>100. DD also excels on text-to-image generation, reducing the generation from 256 steps to 2 for LlamaGen with minimal FID increase from 25.70 to 28.95. As the first work to demonstrate the possibility of one-step generation for image AR models, DD challenges the prevailing notion that AR models are inherently slow, and opens up new opportunities for efficient AR generation. The project website is at https://imagination-research.github.io/distilled-decoding.

Recent image generative models typically capture the image distribution in a pre-constructed latent space, relying on a frozen image tokenizer. However, there exists a significant discrepancy between the reconstruction and generation distribution, where current tokenizers only prioritize the reconstruction task that happens before generative training without considering the generation errors during sampling. In this paper, we comprehensively analyze the reason for this discrepancy in a discrete latent space, and, from which, we propose a novel tokenizer training scheme including both main-training and post-training, focusing on improving latent space construction and decoding respectively. During the main training, a latent perturbation strategy is proposed to simulate sampling noises, \ie, the unexpected tokens generated in generative inference. Specifically, we propose a plug-and-play tokenizer training scheme, which significantly enhances the robustness of tokenizer, thus boosting the generation quality and convergence speed, and a novel tokenizer evaluation metric, \ie, pFID, which successfully correlates the tokenizer performance to generation quality. During post-training, we further optimize the tokenizer decoder regarding a well-trained generative model to mitigate the distribution difference between generated and reconstructed tokens. With a sim400M generator, a discrete tokenizer trained with our proposed main training achieves a notable 1.60 gFID and further obtains 1.36 gFID with the additional post-training. Further experiments are conducted to broadly validate the effectiveness of our post-training strategy on off-the-shelf discrete and continuous tokenizers, coupled with autoregressive and diffusion-based generators.

Periodic signals play an important role in daily lives. Although conventional sequential models have shown remarkable success in various fields, they still come short in modeling periodicity; they either collapse, diverge or ignore details. In this paper, we introduce a novel framework inspired by Fourier series to generate periodic signals. We first decompose the given signals into multiple sines and cosines and then conditionally generate periodic signals with the output components. We have shown our model efficacy on three tasks: reconstruction, imputation and conditional generation. Our model outperforms baselines in all tasks and shows more stable and refined results.

Large language models (LLMs) have exhibited great potential in mathematical reasoning. However, there remains a performance gap in this area between existing open-source models and closed-source models such as GPT-4. In this paper, we introduce MathGenie, a novel method for generating diverse and reliable math problems from a small-scale problem-solution dataset (denoted as seed data). We augment the ground-truth solutions of our seed data and train a back-translation model to translate the augmented solutions back into new questions. Subsequently, we generate code-integrated solutions for the new questions. To ensure the correctness of the code-integrated solutions, we employ rationale-based strategy for solution verification. Various pretrained models, ranging from 7B to 70B, are trained on the newly curated data to test the effectiveness of the proposed augmentation technique, resulting in a family of models known as MathGenieLM. These models consistently outperform previous open-source models across five representative mathematical reasoning datasets, achieving state-of-the-art performance. In particular, MathGenieLM-InternLM2 achieves an accuracy of 87.7% on GSM8K and 55.7% on MATH, securing the best overall score among open-source language models.

Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics, for example aiding in proofs of irrationality of constants. However, the discovery of such formulas has historically remained scarce, often perceived as an act of mathematical genius by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to automate the discovery of formulas for mathematical constants, such as the Ramanujan Machine project, relied on exhaustive search. Despite several successful discoveries, exhaustive search remains limited by the space of options that can be covered and by the need for vast amounts of computational resources. Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences. We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA) algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns in integer sequences that represent mathematical constants. The ESMA algorithm found various known formulas for e, e^2, tan(1), and ratios of values of Bessel functions. The algorithm further discovered a large number of new conjectures for these constants, some providing simpler representations and some providing faster numerical convergence than the corresponding simple continued fractions. Along with the algorithm, we present mathematical tools for manipulating continued fractions. These connections enable us to characterize what space of constants can be found by ESMA and quantify its algorithmic advantage in certain scenarios. Altogether, this work continues in the development of augmenting mathematical intuition by computer algorithms, to help reveal mathematical structures and accelerate mathematical research.

Large language models have transformed natural language processing, yet supervised fine-tuning (SFT) remains computationally intensive. This paper formally proves that capabilities acquired through SFT can be approximated by a base transformer model using inference-time techniques, specifically in-context learning (ICL), without altering model parameters, under idealized assumptions including unbounded computational resources and access to the fine-tuning dataset. We extend these results to practical scenarios with finite context lengths and partial dataset access. For text generation tasks with fixed output length l, datasets of size Oleft( m V{varepsilon^2} log m{delta} right) or, with bounded context, Oleft( l log V{varepsilon^2} log 1{delta} right) suffice to approximate fine-tuned behavior across m contexts within error varepsilon, where V is the vocabulary size and delta is the failure probability. For linear classification, datasets of size Oleft( d{varepsilon} right) or, with fixed context, Oleft( 1{varepsilon^2} log 1{delta} right) are sufficient, where d is the input dimension. Grounded in the Turing completeness of transformers, these results provide a theoretical foundation for resource-efficient deployment of large language models, with practical techniques like retrieval-augmented generation bridging theory to real-world applications.

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were developed over the centuries by mathematicians, who emphasized the attainability of arbitrary precision. Computers, however, operate on few limited precision types, such as the popular float32. In this study, we show that when aiming for limited precision, existing approximation methods can be outperformed by programs automatically discovered from scratch by a simple evolutionary algorithm. In particular, over real numbers, our method can approximate the exponential function reaching orders of magnitude more precision for a given number of operations when compared to previous approaches. More practically, over float32 numbers and constrained to less than 1 ULP of error, the same method attains a speedup over baselines by generating code that triggers better XLA/LLVM compilation paths. In other words, in both cases, evolution searched a vast space of possible programs, without knowledge of mathematics, to discover previously unknown optimized approximations to high precision, for the first time. We also give evidence that these results extend beyond the exponential. The ubiquity of transcendental functions suggests that our method has the potential to reduce the cost of scientific computing applications.

Generating realistic and controllable 3D human avatars is a long-standing challenge, particularly when covering broad attribute ranges such as ethnicity, age, clothing styles, and detailed body shapes. Capturing and annotating large-scale human datasets for training generative models is prohibitively expensive and limited in scale and diversity. The central question we address in this paper is: Can existing foundation models be distilled to generate theoretically unbounded, richly annotated 3D human data? We introduce InfiniHuman, a framework that synergistically distills these models to produce richly annotated human data at minimal cost and with theoretically unlimited scalability. We propose InfiniHumanData, a fully automatic pipeline that leverages vision-language and image generation models to create a large-scale multi-modal dataset. User study shows our automatically generated identities are undistinguishable from scan renderings. InfiniHumanData contains 111K identities spanning unprecedented diversity. Each identity is annotated with multi-granularity text descriptions, multi-view RGB images, detailed clothing images, and SMPL body-shape parameters. Building on this dataset, we propose InfiniHumanGen, a diffusion-based generative pipeline conditioned on text, body shape, and clothing assets. InfiniHumanGen enables fast, realistic, and precisely controllable avatar generation. Extensive experiments demonstrate significant improvements over state-of-the-art methods in visual quality, generation speed, and controllability. Our approach enables high-quality avatar generation with fine-grained control at effectively unbounded scale through a practical and affordable solution. We will publicly release the automatic data generation pipeline, the comprehensive InfiniHumanData dataset, and the InfiniHumanGen models at https://yuxuan-xue.com/infini-human.

Can we build continuous generative models which generalize across scales, can be evaluated at any coordinate, admit calculation of exact derivatives, and are conceptually simple? Existing MLP-based architectures generate worse samples than the grid-based generators with favorable convolutional inductive biases. Models that focus on generating images at different scales do better, but employ complex architectures not designed for continuous evaluation of images and derivatives. We take a signal-processing perspective and treat continuous image generation as interpolation from samples. Indeed, correctly sampled discrete images contain all information about the low spatial frequencies. The question is then how to extrapolate the spectrum in a data-driven way while meeting the above design criteria. Our answer is FunkNN -- a new convolutional network which learns how to reconstruct continuous images at arbitrary coordinates and can be applied to any image dataset. Combined with a discrete generative model it becomes a functional generator which can act as a prior in continuous ill-posed inverse problems. We show that FunkNN generates high-quality continuous images and exhibits strong out-of-distribution performance thanks to its patch-based design. We further showcase its performance in several stylized inverse problems with exact spatial derivatives.

Recent advances in large language models have brought immense value to the world, with their superior capabilities stemming from the massive number of parameters they utilize. However, even the GPUs with the highest memory capacities, currently peaking at 80GB, are far from sufficient to accommodate these vast parameters and their associated optimizer states when conducting stochastic gradient descent-based optimization. One approach to hosting such huge models is to aggregate device memory from many GPUs. However, this approach introduces prohibitive costs for most academic researchers, who always have a limited budget for many high-end GPU servers. In this paper, we focus on huge model fine-tuning on a single, even low-end, GPU in a commodity server, which is accessible to most AI researchers. In such a scenario, the state-of-the-art work ZeRO-Infinity suffers from two severe issues when running in a commodity server: 1) low GPU utilization due to inefficient swapping, and 2) limited trainable model size due to CPU memory capacity. The underlying reason is that ZeRO-Infinity is optimized for running on high-end GPU servers. To this end, we present Fuyou, a low-cost training framework that enables efficient 100B huge model fine-tuning on a low-end server with a low-end GPU and limited CPU memory capacity. The key idea is to add the SSD-CPU communication as an optimization dimension and thus carefully co-optimize computation and data swapping from a systematic approach to maximize GPU utilization. The experimental results show that 1) Fuyou is able to fine-tune 175B GPT-3 on a consumer GPU RTX 4090 with high GPU utilization, while ZeRO-Infinity fails to fine-tune; and 2) when training a small GPT-3 13B model, Fuyou achieves 156 TFLOPS on an RTX 4090 GPU while ZeRO-Infinity only achieves 45 TFLOPS.

Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs are strictly more expressive efficient than many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework that unifies vastly different existing models, but also show great potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks. We also highlight PGCs' connection to the theory of strongly Rayleigh distributions.

The advancement in generative AI could be boosted with more accessible mathematics. Beyond human-AI chat, large language models (LLMs) are emerging in programming, algorithm discovery, and theorem proving, yet their genomics application is limited. This project introduces Math Agents and mathematical embedding as fresh entries to the "Moore's Law of Mathematics", using a GPT-based workflow to convert equations from literature into LaTeX and Python formats. While many digital equation representations exist, there's a lack of automated large-scale evaluation tools. LLMs are pivotal as linguistic user interfaces, providing natural language access for human-AI chat and formal languages for large-scale AI-assisted computational infrastructure. Given the infinite formal possibility spaces, Math Agents, which interact with math, could potentially shift us from "big data" to "big math". Math, unlike the more flexible natural language, has properties subject to proof, enabling its use beyond traditional applications like high-validation math-certified icons for AI alignment aims. This project aims to use Math Agents and mathematical embeddings to address the ageing issue in information systems biology by applying multiscalar physics mathematics to disease models and genomic data. Generative AI with episodic memory could help analyse causal relations in longitudinal health records, using SIR Precision Health models. Genomic data is suggested for addressing the unsolved Alzheimer's disease problem.

This article establishes a complete approximate axiomatization for the real-closed field R expanded with all differentially-defined functions, including special functions such as sin(x), cos(x), e^x, dots. Every true sentence is provable up to some numerical approximation, and the truth of such approximations converge under mild conditions. Such an axiomatization is a fragment of the axiomatization for differential dynamic logic, and is therefore a finite extension of the axiomatization of real-closed fields. Furthermore, the numerical approximations approximate formulas containing special function symbols by FOL_{R} formulas, improving upon earlier decidability results only concerning closed sentences.

The Omega numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-L{\"o}f random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-L{\"o}f random left-c.e. real alpha, a universal prefix-free machine U whose halting probability is alpha. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real alpha, one cannot uniformly produce a left-c.e. real beta such that alpha -- beta is neither left-c.e. nor right-c.e.

Nowadays, formal theorem provers have made monumental progress on high-school and competition-level mathematics, but few of them generalize to more advanced mathematics. In this paper, we present REAL-Prover, a new open-source stepwise theorem prover for Lean 4 to push this boundary. This prover, based on our fine-tuned large language model (REAL-Prover-v1) and integrated with a retrieval system (Leansearch-PS), notably boosts performance on solving college-level mathematics problems. To train REAL-Prover-v1, we developed HERALD-AF, a data extraction pipeline that converts natural language math problems into formal statements, and a new open-source Lean 4 interactive environment (Jixia-interactive) to facilitate synthesis data collection. In our experiments, our prover using only supervised fine-tune achieves competitive results with a 23.7% success rate (Pass@64) on the ProofNet dataset-comparable to state-of-the-art (SOTA) models. To further evaluate our approach, we introduce FATE-M, a new benchmark focused on algebraic problems, where our prover achieves a SOTA success rate of 56.7% (Pass@64).

Recent image generation schemes typically capture image distribution in a pre-constructed latent space relying on a frozen image tokenizer. Though the performance of tokenizer plays an essential role to the successful generation, its current evaluation metrics (e.g. rFID) fail to precisely assess the tokenizer and correlate its performance to the generation quality (e.g. gFID). In this paper, we comprehensively analyze the reason for the discrepancy of reconstruction and generation qualities in a discrete latent space, and, from which, we propose a novel plug-and-play tokenizer training scheme to facilitate latent space construction. Specifically, a latent perturbation approach is proposed to simulate sampling noises, i.e., the unexpected tokens sampled, from the generative process. With the latent perturbation, we further propose (1) a novel tokenizer evaluation metric, i.e., pFID, which successfully correlates the tokenizer performance to generation quality and (2) a plug-and-play tokenizer training scheme, which significantly enhances the robustness of tokenizer thus boosting the generation quality and convergence speed. Extensive benchmarking are conducted with 11 advanced discrete image tokenizers with 2 autoregressive generation models to validate our approach. The tokenizer trained with our proposed latent perturbation achieve a notable 1.60 gFID with classifier-free guidance (CFG) and 3.45 gFID without CFG with a sim400M generator. Code: https://github.com/lxa9867/ImageFolder.

Even when massively overparameterized, deep neural networks show a remarkable ability to generalize. Research on this phenomenon has focused on generalization within distribution, via smooth interpolation. Yet in some settings neural networks also learn to extrapolate to data far beyond the bounds of the original training set, sometimes even allowing for infinite generalization, implying that an algorithm capable of solving the task has been learned. Here we undertake a case study of the learning dynamics of recurrent neural networks (RNNs) trained on the streaming parity task in order to develop an effective theory of algorithm development. The streaming parity task is a simple but nonlinear task defined on sequences up to arbitrary length. We show that, with sufficient finite training experience, RNNs exhibit a phase transition to perfect infinite generalization. Using an effective theory for the representational dynamics, we find an implicit representational merger effect which can be interpreted as the construction of a finite automaton that reproduces the task. Overall, our results disclose one mechanism by which neural networks can generalize infinitely from finite training experience.

Recent advances in large language models show strong promise for formal reasoning. However, most LLM-based theorem provers have long been constrained by the need for expert-written formal statements as inputs, limiting their applicability to real-world problems expressed in natural language. We tackle this gap with Mathesis, the first end-to-end theorem proving pipeline processing informal problem statements. It contributes Mathesis-Autoformalizer, the first autoformalizer using reinforcement learning to enhance the formalization ability of natural language problems, aided by our novel LeanScorer framework for nuanced formalization quality assessment. It also proposes a Mathesis-Prover, which generates formal proofs from the formalized statements. To evaluate the real-world applicability of end-to-end formal theorem proving, we introduce Gaokao-Formal, a benchmark of 488 complex problems from China's national college entrance exam. Our approach is carefully designed, with a thorough study of each component. Experiments demonstrate Mathesis's effectiveness, with the autoformalizer outperforming the best baseline by 22% in pass-rate on Gaokao-Formal. The full system surpasses other model combinations, achieving 64% accuracy on MiniF2F with pass@32 and a state-of-the-art 18% on Gaokao-Formal.

We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for N dimensional data by embedding paths in N{+}D dimensional space while still controlling the progression with a simple scalar norm of the D additional variables. The new models reduce to PFGM when D{=}1 and to diffusion models when D{to}infty. The flexibility of choosing D allows us to trade off robustness against rigidity as increasing D results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of D, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models (D{to} infty) to any finite D values. Our experiments show that models with finite D can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ 64{times}64 datasets, with FID scores of 1.91/2.43 when D{=}2048/128. In class-conditional setting, D{=}2048 yields current state-of-the-art FID of 1.74 on CIFAR-10. In addition, we demonstrate that models with smaller D exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp

Text-to-motion generation holds potential for film, gaming, and robotics, yet current methods often prioritize short motion generation, making it challenging to produce long motion sequences effectively: (1) Current methods struggle to handle long motion sequences as a single input due to prohibitively high computational cost; (2) Breaking down the generation of long motion sequences into shorter segments can result in inconsistent transitions and requires interpolation or inpainting, which lacks entire sequence modeling. To solve these challenges, we propose InfiniMotion, a method that generates continuous motion sequences of arbitrary length within an autoregressive framework. We highlight its groundbreaking capability by generating a continuous 1-hour human motion with around 80,000 frames. Specifically, we introduce the Motion Memory Transformer with Bidirectional Mamba Memory, enhancing the transformer's memory to process long motion sequences effectively without overwhelming computational resources. Notably our method achieves over 30% improvement in FID and 6 times longer demonstration compared to previous state-of-the-art methods, showcasing significant advancements in long motion generation. See project webpage: https://steve-zeyu-zhang.github.io/InfiniMotion/

The math abilities of large language models can represent their abstract reasoning ability. In this paper, we introduce and open-source our math reasoning LLMs InternLM-Math which is continue pre-trained from InternLM2. We unify chain-of-thought reasoning, reward modeling, formal reasoning, data augmentation, and code interpreter in a unified seq2seq format and supervise our model to be a versatile math reasoner, verifier, prover, and augmenter. These abilities can be used to develop the next math LLMs or self-iteration. InternLM-Math obtains open-sourced state-of-the-art performance under the setting of in-context learning, supervised fine-tuning, and code-assisted reasoning in various informal and formal benchmarks including GSM8K, MATH, Hungary math exam, MathBench-ZH, and MiniF2F. Our pre-trained model achieves 30.3 on the MiniF2F test set without fine-tuning. We further explore how to use LEAN to solve math problems and study its performance under the setting of multi-task learning which shows the possibility of using LEAN as a unified platform for solving and proving in math. Our models, codes, and data are released at https://github.com/InternLM/InternLM-Math.

With infinitely many high-quality data points, infinite computational power, an infinitely large foundation model with a perfect training algorithm and guaranteed zero generalization error on the pretext task, can the model be used for everything? This question cannot be answered by the existing theory of representation, optimization or generalization, because the issues they mainly investigate are assumed to be nonexistent here. In this paper, we show that category theory provides powerful machinery to answer this question. We have proved three results. The first one limits the power of prompt-based learning, saying that the model can solve a downstream task with prompts if and only if the task is representable. The second one says fine tuning does not have this limit, as a foundation model with the minimum required power (up to symmetry) can theoretically solve downstream tasks for the category defined by pretext task, with fine tuning and enough resources. Our final result can be seen as a new type of generalization theorem, showing that the foundation model can generate unseen objects from the target category (e.g., images) using the structural information from the source category (e.g., texts). Along the way, we provide a categorical framework for supervised and self-supervised learning, which might be of independent interest.

A fundamental dilemma in generative modeling persists: iterative diffusion models achieve outstanding fidelity, but at a significant computational cost, while efficient few-step alternatives are constrained by a hard quality ceiling. This conflict between generation steps and output quality arises from restrictive training objectives that focus exclusively on either infinitesimal dynamics (PF-ODEs) or direct endpoint prediction. We address this challenge by introducing an exact, continuous-time dynamics equation that analytically defines state transitions across any finite time interval. This leads to a novel generative paradigm, Transition Models (TiM), which adapt to arbitrary-step transitions, seamlessly traversing the generative trajectory from single leaps to fine-grained refinement with more steps. Despite having only 865M parameters, TiM achieves state-of-the-art performance, surpassing leading models such as SD3.5 (8B parameters) and FLUX.1 (12B parameters) across all evaluated step counts. Importantly, unlike previous few-step generators, TiM demonstrates monotonic quality improvement as the sampling budget increases. Additionally, when employing our native-resolution strategy, TiM delivers exceptional fidelity at resolutions up to 4096x4096.

Arbitrary resolution image generation provides a consistent visual experience across devices, having extensive applications for producers and consumers. Current diffusion models increase computational demand quadratically with resolution, causing 4K image generation delays over 100 seconds. To solve this, we explore the second generation upon the latent diffusion models, where the fixed latent generated by diffusion models is regarded as the content representation and we propose to decode arbitrary resolution images with a compact generated latent using a one-step generator. Thus, we present the InfGen, replacing the VAE decoder with the new generator, for generating images at any resolution from a fixed-size latent without retraining the diffusion models, which simplifies the process, reducing computational complexity and can be applied to any model using the same latent space. Experiments show InfGen is capable of improving many models into the arbitrary high-resolution era while cutting 4K image generation time to under 10 seconds.

A very recent trend in generative modeling is building 3D-aware generators from 2D image collections. To induce the 3D bias, such models typically rely on volumetric rendering, which is expensive to employ at high resolutions. During the past months, there appeared more than 10 works that address this scaling issue by training a separate 2D decoder to upsample a low-resolution image (or a feature tensor) produced from a pure 3D generator. But this solution comes at a cost: not only does it break multi-view consistency (i.e. shape and texture change when the camera moves), but it also learns the geometry in a low fidelity. In this work, we show that it is possible to obtain a high-resolution 3D generator with SotA image quality by following a completely different route of simply training the model patch-wise. We revisit and improve this optimization scheme in two ways. First, we design a location- and scale-aware discriminator to work on patches of different proportions and spatial positions. Second, we modify the patch sampling strategy based on an annealed beta distribution to stabilize training and accelerate the convergence. The resulted model, named EpiGRAF, is an efficient, high-resolution, pure 3D generator, and we test it on four datasets (two introduced in this work) at 256^2 and 512^2 resolutions. It obtains state-of-the-art image quality, high-fidelity geometry and trains {approx} 2.5 times faster than the upsampler-based counterparts. Project website: https://universome.github.io/epigraf.

Large Language Models (LLMs) demonstrate strong performance in real-world applications, yet existing open-source instruction datasets often concentrate on narrow domains, such as mathematics or coding, limiting generalization and widening the gap with proprietary models. To bridge this gap, we introduce Infinity-Instruct, a high-quality instruction dataset designed to enhance both foundational and chat capabilities of LLMs through a two-phase pipeline. In Phase 1, we curate 7.4M high-quality foundational instructions (InfInstruct-F-7.4M) from over 100M samples using hybrid data selection techniques. In Phase 2, we synthesize 1.5M high-quality chat instructions (InfInstruct-G-1.5M) through a two-stage process involving instruction selection, evolution, and diagnostic filtering. We empirically evaluate Infinity-Instruct by fine-tuning several open-source models, including Mistral, LLaMA, Qwen, and Yi, and observe substantial performance gains across both foundational and instruction following benchmarks, consistently surpassing official instruction-tuned counterparts. Notably, InfInstruct-LLaMA3.1-70B outperforms GPT-4-0314 by 8.6\% on instruction following tasks while achieving comparable foundational performance. These results underscore the synergy between foundational and chat training and offer new insights into holistic LLM development. Our datasethttps://huggingface.co/datasets/BAAI/Infinity-Instruct and codeshttps://gitee.com/li-touch/infinity-instruct have been publicly released.

The grand vision of enabling persistent, large-scale 3D visual geometry understanding is shackled by the irreconcilable demands of scalability and long-term stability. While offline models like VGGT achieve inspiring geometry capability, their batch-based nature renders them irrelevant for live systems. Streaming architectures, though the intended solution for live operation, have proven inadequate. Existing methods either fail to support truly infinite-horizon inputs or suffer from catastrophic drift over long sequences. We shatter this long-standing dilemma with InfiniteVGGT, a causal visual geometry transformer that operationalizes the concept of a rolling memory through a bounded yet adaptive and perpetually expressive KV cache. Capitalizing on this, we devise a training-free, attention-agnostic pruning strategy that intelligently discards obsolete information, effectively ``rolling'' the memory forward with each new frame. Fully compatible with FlashAttention, InfiniteVGGT finally alleviates the compromise, enabling infinite-horizon streaming while outperforming existing streaming methods in long-term stability. The ultimate test for such a system is its performance over a truly infinite horizon, a capability that has been impossible to rigorously validate due to the lack of extremely long-term, continuous benchmarks. To address this critical gap, we introduce the Long3D benchmark, which, for the first time, enables a rigorous evaluation of continuous 3D geometry estimation on sequences about 10,000 frames. This provides the definitive evaluation platform for future research in long-term 3D geometry understanding. Code is available at: https://github.com/AutoLab-SAI-SJTU/InfiniteVGGT

Convolution is a broadly useful operation with applications including signal processing, machine learning, probability, optics, polynomial multiplication, and efficient parsing. Usually, however, this operation is understood and implemented in more specialized forms, hiding commonalities and limiting usefulness. This paper formulates convolution in the common algebraic framework of semirings and semimodules and populates that framework with various representation types. One of those types is the grand abstract template and itself generalizes to the free semimodule monad. Other representations serve varied uses and performance trade-offs, with implementations calculated from simple and regular specifications. Of particular interest is Brzozowski's method for regular expression matching. Uncovering the method's essence frees it from syntactic manipulations, while generalizing from boolean to weighted membership (such as multisets and probability distributions) and from sets to n-ary relations. The classic trie data structure then provides an elegant and efficient alternative to syntax. Pleasantly, polynomial arithmetic requires no additional implementation effort, works correctly with a variety of representations, and handles multivariate polynomials and power series with ease. Image convolution also falls out as a special case.

Large language models have demonstrated impressive capabilities across various natural language processing tasks, especially in solving mathematical problems. However, large language models are not good at math theorem proving using formal languages like Lean. A significant challenge in this area is the scarcity of training data available in these formal languages. To address this issue, we propose a novel pipeline that iteratively generates and filters synthetic data to translate natural language mathematical problems into Lean 4 statements, and vice versa. Our results indicate that the synthetic data pipeline can provide useful training data and improve the performance of LLMs in translating and understanding complex mathematical problems and proofs. Our final dataset contains about 57K formal-informal question pairs along with searched proof from the math contest forum and 21 new IMO questions. We open-source our code at https://github.com/InternLM/InternLM-Math and our data at https://huggingface.co/datasets/InternLM/Lean-Workbook.

We propose an online training procedure for a transformer-based automated theorem prover. Our approach leverages a new search algorithm, HyperTree Proof Search (HTPS), inspired by the recent success of AlphaZero. Our model learns from previous proof searches through online training, allowing it to generalize to domains far from the training distribution. We report detailed ablations of our pipeline's main components by studying performance on three environments of increasing complexity. In particular, we show that with HTPS alone, a model trained on annotated proofs manages to prove 65.4% of a held-out set of Metamath theorems, significantly outperforming the previous state of the art of 56.5% by GPT-f. Online training on these unproved theorems increases accuracy to 82.6%. With a similar computational budget, we improve the state of the art on the Lean-based miniF2F-curriculum dataset from 31% to 42% proving accuracy.

Idempotence is the stability of image codec to re-compression. At the first glance, it is unrelated to perceptual image compression. However, we find that theoretically: 1) Conditional generative model-based perceptual codec satisfies idempotence; 2) Unconditional generative model with idempotence constraint is equivalent to conditional generative codec. Based on this newfound equivalence, we propose a new paradigm of perceptual image codec by inverting unconditional generative model with idempotence constraints. Our codec is theoretically equivalent to conditional generative codec, and it does not require training new models. Instead, it only requires a pre-trained mean-square-error codec and unconditional generative model. Empirically, we show that our proposed approach outperforms state-of-the-art methods such as HiFiC and ILLM, in terms of Fr\'echet Inception Distance (FID). The source code is provided in https://github.com/tongdaxu/Idempotence-and-Perceptual-Image-Compression.

For decades, procedural worlds have been built on procedural noise functions such as Perlin noise, which are fast and infinite, yet fundamentally limited in realism and large-scale coherence. We introduce Terrain Diffusion, an AI-era successor to Perlin noise that bridges the fidelity of diffusion models with the properties that made procedural noise indispensable: seamless infinite extent, seed-consistency, and constant-time random access. At its core is InfiniteDiffusion, a novel algorithm for infinite generation, enabling seamless, real-time synthesis of boundless landscapes. A hierarchical stack of diffusion models couples planetary context with local detail, while a compact Laplacian encoding stabilizes outputs across Earth-scale dynamic ranges. An open-source infinite-tensor framework supports constant-memory manipulation of unbounded tensors, and few-step consistency distillation enables efficient generation. Together, these components establish diffusion models as a practical foundation for procedural world generation, capable of synthesizing entire planets coherently, controllably, and without limits.

Transformers have been shown to emulate logical deduction over natural language theories (logical rules expressed in natural language), reliably assigning true/false labels to candidate implications. However, their ability to generate implications of a theory has not yet been demonstrated, and methods for reconstructing proofs of answers are imperfect. In this work we show that a generative model, called ProofWriter, can reliably generate both implications of a theory and the natural language proof(s) that support them. In particular, iterating a 1-step implication generator results in proofs that are highly reliable, and represent actual model decisions (rather than post-hoc rationalizations). On the RuleTaker dataset, the accuracy of ProofWriter's proofs exceed previous methods by +9% absolute, and in a way that generalizes to proof depths unseen in training and on out-of-domain problems. We also show that generative techniques can perform a type of abduction with high precision: Given a theory and an unprovable conclusion, identify a missing fact that allows the conclusion to be proved, along with a proof. These results significantly improve the viability of neural methods for systematically reasoning over natural language.

We address the problem of determining the Hausdorff dimension of sets consisting of complex irrationals whose complex continued fraction digits satisfy prescribed restrictions and growth conditions. For the Hurwitz continued fraction, we confirm Hirst's conjecture, as a complex analogue of the result of Wang and Wu [Bull. Lond. Math. Soc. {\bf 40} (2008), no. 1, 18--22] for the regular continued fraction. We also prove a complex analogue of the second-named author's result on the Hausdorff dimension of sets with restricted slowly growing digits [Proc. Amer. Math. Soc. {\bf 151} (2023), no. 9, 3645--3653]. To these ends, we exploit an infinite conformal iterated function system associated with the Hurwitz continued fraction.

Recent breakthroughs in large language models (LLMs) on reasoning tasks rely heavily on massive, high-quality datasets-typically human-annotated and thus difficult to scale. While data synthesis or distillation offers a promising alternative, existing methods struggle with inconsistent data quality and an inability to dynamically adapt to the evolving capabilities of the model, leading to suboptimal training signals. To address these limitations, we introduce Socratic-Zero, a fully autonomous framework that generates high-quality training data from minimal seed examples through the co-evolution of three agents: the Teacher, the Solver, and the Generator. The Solver continuously refines its reasoning by learning from preference feedback on both successful and failed trajectories; the Teacher adaptively crafts increasingly challenging questions based on the Solver's weaknesses; and the Generator distills the Teacher's question-design strategy to enable scalable, high-fidelity curriculum generation. This closed-loop system produces a self-improving curriculum-requiring no pre-existing tasks or labels. Remarkably, starting from only 100 seed questions, our Socratic-Solver-8B achieves an average gain of +20.2 percentage points over prior data synthesis methods across seven mathematical reasoning benchmarks (AMC23, AIME24-25, Olympiad, MATH-500, Minerva, and GSM8K), with consistent gains on both Qwen3 and GLM4 series models. Even more surprisingly, synthetic data from Socratic-Generator-32B enables student LLMs to achieve superior performance compared to other state-of-the-art (SOTA) commercial LLMs on these benchmarks, including Qwen3-235B-A22B, DeepSeek-V3.1-671B, GPT-5, Gemini-2.5-Pro, Grok-4, and Claude-4.1-Opus.

This paper shows that two commonly used evaluation metrics for generative models, the Fr\'echet Inception Distance (FID) and the Inception Score (IS), are biased -- the expected value of the score computed for a finite sample set is not the true value of the score. Worse, the paper shows that the bias term depends on the particular model being evaluated, so model A may get a better score than model B simply because model A's bias term is smaller. This effect cannot be fixed by evaluating at a fixed number of samples. This means all comparisons using FID or IS as currently computed are unreliable. We then show how to extrapolate the score to obtain an effectively bias-free estimate of scores computed with an infinite number of samples, which we term textrm{FID}_infty and textrm{IS}_infty. In turn, this effectively bias-free estimate requires good estimates of scores with a finite number of samples. We show that using Quasi-Monte Carlo integration notably improves estimates of FID and IS for finite sample sets. Our extrapolated scores are simple, drop-in replacements for the finite sample scores. Additionally, we show that using low discrepancy sequence in GAN training offers small improvements in the resulting generator.

Recent theoretical work has identified surprisingly simple reasoning problems, such as checking if two nodes in a graph are connected or simulating finite-state machines, that are provably unsolvable by standard transformers that answer immediately after reading their input. However, in practice, transformers' reasoning can be improved by allowing them to use a "chain of thought" or "scratchpad", i.e., generate and condition on a sequence of intermediate tokens before answering. Motivated by this, we ask: Does such intermediate generation fundamentally extend the computational power of a decoder-only transformer? We show that the answer is yes, but the amount of increase depends crucially on the amount of intermediate generation. For instance, we find that transformer decoders with a logarithmic number of decoding steps (w.r.t. the input length) push the limits of standard transformers only slightly, while a linear number of decoding steps, assuming a slight generalization to standard pre-norm, adds a clear new ability (under standard complexity conjectures): recognizing all regular languages. Our results also imply that linear steps keep transformer decoders within context-sensitive languages, and polynomial steps with generalized pre-norm make them recognize exactly the class of polynomial-time solvable problems -- the first exact characterization of a type of transformers in terms of standard complexity classes. Together, our results provide a nuanced framework for understanding how the length of a transformer's chain of thought or scratchpad impacts its reasoning power.

The recent rise of large language models (LLMs) has resulted in increased efforts towards running LLMs at reduced precision. Running LLMs at lower precision supports resource constraints and furthers their democratization, enabling users to run billion-parameter LLMs on their personal devices. To supplement this ongoing effort, we propose INT-FP-QSim: an open-source simulator that enables flexible evaluation of LLMs and vision transformers at various numerical precisions and formats. INT-FP-QSim leverages existing open-source repositories such as TensorRT, QPytorch and AIMET for a combined simulator that supports various floating point and integer formats. With the help of our simulator, we survey the impact of different numerical formats on the performance of LLMs and vision transformers at 4-bit weights and 4-bit or 8-bit activations. We also compare recently proposed methods like Adaptive Block Floating Point, SmoothQuant, GPTQ and RPTQ on the model performances. We hope INT-FP-QSim will enable researchers to flexibly simulate models at various precisions to support further research in quantization of LLMs and vision transformers.

Existing private synthetic data generation algorithms are agnostic to downstream tasks. However, end users may have specific requirements that the synthetic data must satisfy. Failure to meet these requirements could significantly reduce the utility of the data for downstream use. We introduce a post-processing technique that improves the utility of the synthetic data with respect to measures selected by the end user, while preserving strong privacy guarantees and dataset quality. Our technique involves resampling from the synthetic data to filter out samples that do not meet the selected utility measures, using an efficient stochastic first-order algorithm to find optimal resampling weights. Through comprehensive numerical experiments, we demonstrate that our approach consistently improves the utility of synthetic data across multiple benchmark datasets and state-of-the-art synthetic data generation algorithms.

The infinitely wide neural network has been proven a useful and manageable mathematical model that enables the understanding of many phenomena appearing in deep learning. One example is the convergence of random deep networks to Gaussian processes that allows a rigorous analysis of the way the choice of activation function and network weights impacts the training dynamics. In this paper, we extend the seminal proof of Matthews et al. (2018) to a larger class of initial weight distributions (which we call PSEUDO-IID), including the established cases of IID and orthogonal weights, as well as the emerging low-rank and structured sparse settings celebrated for their computational speed-up benefits. We show that fully-connected and convolutional networks initialized with PSEUDO-IID distributions are all effectively equivalent up to their variance. Using our results, one can identify the Edge-of-Chaos for a broader class of neural networks and tune them at criticality in order to enhance their training. Moreover, they enable the posterior distribution of Bayesian Neural Networks to be tractable across these various initialization schemes.

Conditional neural text generation models generate high-quality outputs, but often concentrate around a mode when what we really want is a diverse set of options. We present a search algorithm to construct lattices encoding a massive number of generation options. First, we restructure decoding as a best-first search, which explores the space differently than beam search and improves efficiency by avoiding pruning paths. Second, we revisit the idea of hypothesis recombination: we can identify pairs of similar generation candidates during search and merge them as an approximation. On both summarization and machine translation, we show that our algorithm encodes thousands of diverse options that remain grammatical and high-quality into one lattice. This algorithm provides a foundation for building downstream generation applications on top of massive-scale diverse outputs.

P. Kabamba developed generation theory as a tool for studying self-reproducing systems. We provide an alternative definition of a generation system and give a complete solution to the problem of finding optimal seeds for a finite self-replicating system. We also exhibit examples illustrating a connection between self-replication and fixed-point theory.

We propose a novel theoretical framework of analysis for Generative Adversarial Networks (GANs). We reveal a fundamental flaw of previous analyses which, by incorrectly modeling GANs' training scheme, are subject to ill-defined discriminator gradients. We overcome this issue which impedes a principled study of GAN training, solving it within our framework by taking into account the discriminator's architecture. To this end, we leverage the theory of infinite-width neural networks for the discriminator via its Neural Tangent Kernel. We characterize the trained discriminator for a wide range of losses and establish general differentiability properties of the network. From this, we derive new insights about the convergence of the generated distribution, advancing our understanding of GANs' training dynamics. We empirically corroborate these results via an analysis toolkit based on our framework, unveiling intuitions that are consistent with GAN practice.

This paper is the second in the series Commutative Scaling of Width and Depth (WD) about commutativity of infinite width and depth limits in deep neural networks. Our aim is to understand the behaviour of neural functions (functions that depend on a neural network model) as width and depth go to infinity (in some sense), and eventually identify settings under which commutativity holds, i.e. the neural function tends to the same limit no matter how width and depth limits are taken. In this paper, we formally introduce and define the commutativity framework, and discuss its implications on neural network design and scaling. We study commutativity for the neural covariance kernel which reflects how network layers separate data. Our findings extend previous results established in [55] by showing that taking the width and depth to infinity in a deep neural network with skip connections, when branches are suitably scaled to avoid exploding behaviour, result in the same covariance structure no matter how that limit is taken. This has a number of theoretical and practical implications that we discuss in the paper. The proof techniques in this paper are novel and rely on tools that are more accessible to readers who are not familiar with stochastic calculus (used in the proofs of WD(I))).

Software engineers mainly write code by editing existing programs. In contrast, large language models (LLMs) autoregressively synthesize programs in a single pass. One explanation for this is the scarcity of open-sourced edit data. While high-quality instruction data for code synthesis is already scarce, high-quality edit data is even scarcer. To fill this gap, we develop a synthetic data generation algorithm called LintSeq. This algorithm refactors existing code into a sequence of code edits by using a linter to procedurally sample across the error-free insertions that can be used to sequentially write programs. It outputs edit sequences as text strings consisting of consecutive program diffs. To test LintSeq, we use it to refactor a dataset of instruction + program pairs into instruction + program-diff-sequence tuples. Then, we instruction finetune a series of smaller LLMs ranging from 2.6B to 14B parameters on both the re-factored and original versions of this dataset, comparing zero-shot performance on code synthesis benchmarks. We show that during repeated sampling, edit sequence finetuned models produce more diverse programs than baselines. This results in better inference-time scaling for benchmark coverage as a function of samples, i.e. the fraction of problems "pass@k" solved by any attempt given "k" tries. For example, on HumanEval pass@50, small LLMs finetuned on synthetic edit sequences are competitive with GPT-4 and outperform models finetuned on the baseline dataset by +20% (+/-3%) in absolute score. Finally, we also pretrain our own tiny LMs for code understanding. We show that finetuning tiny models on synthetic code edits results in state-of-the-art code synthesis for the on-device model class. Our 150M parameter edit sequence LM matches or outperforms code models with twice as many parameters, both with and without repeated sampling, including Codex and AlphaCode.

In order to avoid well-know paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type_0 : Type_1 : cdots . Such type systems are called cumulative if for any type A we have that A : Type_{i} implies A : Type_{i+1}. The predicative calculus of inductive constructions (pCIC) which forms the basis of the Coq proof assistant, is one such system. In this paper we present and establish the soundness of the predicative calculus of cumulative inductive constructions (pCuIC) which extends the cumulativity relation to inductive types.

Retrieval-Augmented Generation (RAG) prevails in Large Language Models. It mainly consists of retrieval and generation. The retrieval modules (a.k.a. retrievers) aim to find useful information used to facilitate generation modules (a.k.a. generators). As such, generators' performance largely depends on the effectiveness and efficiency of retrievers. However, the retrieval paradigm that we design and use remains flat, which treats the retrieval procedures as a one-off deal with constant granularity. Despite effectiveness, we argue that they suffer from two limitations: (1) flat retrieval exerts a significant burden on one retriever; (2) constant granularity limits the ceiling of retrieval performance. In this work, we propose a progressive retrieval paradigm with coarse-to-fine granularity for RAG, termed FunnelRAG, so as to balance effectiveness and efficiency. Specifically, FunnelRAG establishes a progressive retrieval pipeline by collaborating coarse-to-fine granularity, large-to-small quantity, and low-to-high capacity, which can relieve the burden on one retriever and also promote the ceiling of retrieval performance. Extensive experiments manifest that FunnelRAG achieves comparable retrieval performance while the time overhead is reduced by nearly 40 percent.

PySR is an open-source library for practical symbolic regression, a type of machine learning which aims to discover human-interpretable symbolic models. PySR was developed to democratize and popularize symbolic regression for the sciences, and is built on a high-performance distributed back-end, a flexible search algorithm, and interfaces with several deep learning packages. PySR's internal search algorithm is a multi-population evolutionary algorithm, which consists of a unique evolve-simplify-optimize loop, designed for optimization of unknown scalar constants in newly-discovered empirical expressions. PySR's backend is the extremely optimized Julia library SymbolicRegression.jl, which can be used directly from Julia. It is capable of fusing user-defined operators into SIMD kernels at runtime, performing automatic differentiation, and distributing populations of expressions to thousands of cores across a cluster. In describing this software, we also introduce a new benchmark, "EmpiricalBench," to quantify the applicability of symbolic regression algorithms in science. This benchmark measures recovery of historical empirical equations from original and synthetic datasets.

Autoregressive models have achieved promising results in natural language processing. However, for image generation tasks, they encounter substantial challenges in effectively capturing long-range dependencies, managing computational costs, and most crucially, defining meaningful autoregressive sequences that reflect natural image hierarchies. To address these issues, we present Next-Frequency Image Generation (NFIG), a novel framework that decomposes the image generation process into multiple frequency-guided stages. Our approach first generates low-frequency components to establish global structure with fewer tokens, then progressively adds higher-frequency details, following the natural spectral hierarchy of images. This principled autoregressive sequence not only improves the quality of generated images by better capturing true causal relationships between image components, but also significantly reduces computational overhead during inference. Extensive experiments demonstrate that NFIG achieves state-of-the-art performance with fewer steps, offering a more efficient solution for image generation, with 1.25times speedup compared to VAR-d20 while achieving better performance (FID: 2.81) on the ImageNet-256 benchmark. We hope that our insight of incorporating frequency-domain knowledge to guide autoregressive sequence design will shed light on future research. We will make our code publicly available upon acceptance of the paper.

As synthetic data becomes higher quality and proliferates on the internet, machine learning models are increasingly trained on a mix of human- and machine-generated data. Despite the successful stories of using synthetic data for representation learning, using synthetic data for generative model training creates "self-consuming loops" which may lead to training instability or even collapse, unless certain conditions are met. Our paper aims to stabilize self-consuming generative model training. Our theoretical results demonstrate that by introducing an idealized correction function, which maps a data point to be more likely under the true data distribution, self-consuming loops can be made exponentially more stable. We then propose self-correction functions, which rely on expert knowledge (e.g. the laws of physics programmed in a simulator), and aim to approximate the idealized corrector automatically and at scale. We empirically validate the effectiveness of self-correcting self-consuming loops on the challenging human motion synthesis task, and observe that it successfully avoids model collapse, even when the ratio of synthetic data to real data is as high as 100%.

Formula-driven supervised learning (FDSL) is a pre-training method that relies on synthetic images generated from mathematical formulae such as fractals. Prior work on FDSL has shown that pre-training vision transformers on such synthetic datasets can yield competitive accuracy on a wide range of downstream tasks. These synthetic images are categorized according to the parameters in the mathematical formula that generate them. In the present work, we hypothesize that the process for generating different instances for the same category in FDSL, can be viewed as a form of data augmentation. We validate this hypothesis by replacing the instances with data augmentation, which means we only need a single image per category. Our experiments shows that this one-instance fractal database (OFDB) performs better than the original dataset where instances were explicitly generated. We further scale up OFDB to 21,000 categories and show that it matches, or even surpasses, the model pre-trained on ImageNet-21k in ImageNet-1k fine-tuning. The number of images in OFDB is 21k, whereas ImageNet-21k has 14M. This opens new possibilities for pre-training vision transformers with much smaller datasets.

SDForger is a flexible and efficient framework for generating high-quality multivariate time series using LLMs. Leveraging a compact data representation, SDForger provides synthetic time series generation from a few samples and low-computation fine-tuning of any autoregressive LLM. Specifically, the framework transforms univariate and multivariate signals into tabular embeddings, which are then encoded into text and used to fine-tune the LLM. At inference, new textual embeddings are sampled and decoded into synthetic time series that retain the original data's statistical properties and temporal dynamics. Across a diverse range of datasets, SDForger outperforms existing generative models in many scenarios, both in similarity-based evaluations and downstream forecasting tasks. By enabling textual conditioning in the generation process, SDForger paves the way for multimodal modeling and the streamlined integration of time series with textual information. SDForger source code will be open-sourced soon.

Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies to compute forces and higher-order derivatives. Furthermore, such densities are often defined on non-trivial topologies. A recent example are Boltzmann Generators for generating 3D-structures of peptides and small proteins. These generative models leverage the space of internal coordinates (dihedrals, angles, and bonds), which is a product of hypertori and compact intervals. In this work, we introduce a class of smooth mixture transformations working on both compact intervals and hypertori. Mixture transformations employ root-finding methods to invert them in practice, which has so far prevented bi-directional flow training. To this end, we show that parameter gradients and forces of such inverses can be computed from forward evaluations via the inverse function theorem. We demonstrate two advantages of such smooth flows: they allow training by force matching to simulation data and can be used as potentials in molecular dynamics simulations.

Recent large language models (LLMs) have witnessed significant advancement in various tasks, including mathematical reasoning and theorem proving. As these two tasks require strict and formal multi-step inference, they are appealing domains for exploring the reasoning ability of LLMs but still face important challenges. Previous studies such as Chain-of-Thought (CoT) have revealed the effectiveness of intermediate steps guidance. However, such step-wise annotation requires heavy labor, leading to insufficient training steps for current benchmarks. To fill this gap, this work introduces MUSTARD, a data generation framework that masters uniform synthesis of theorem and proof data of high quality and diversity. MUSTARD synthesizes data in three stages: (1) It samples a few mathematical concept seeds as the problem category. (2) Then, it prompts a generative language model with the sampled concepts to obtain both the problems and their step-wise formal solutions. (3) Lastly, the framework utilizes a proof assistant (e.g., Lean Prover) to filter the valid proofs. With the proposed MUSTARD, we present a theorem-and-proof benchmark MUSTARDSAUCE with 5,866 valid data points. Each data point contains an informal statement, an informal proof, and a translated formal proof that passes the prover validation. We perform extensive analysis and demonstrate that MUSTARD generates validated high-quality step-by-step data. We further apply the MUSTARDSAUCE for fine-tuning smaller language models. The fine-tuned Llama 2-7B achieves a 15.41% average relative performance gain in automated theorem proving, and 8.18% in math word problems. Codes and data are available at https://github.com/Eleanor-H/MUSTARD.

This paper presents a quantitative program verification infrastructure for discrete probabilistic programs. Our infrastructure can be viewed as the probabilistic analogue of Boogie: its central components are an intermediate verification language (IVL) together with a real-valued logic. Our IVL provides a programming-language-style for expressing verification conditions whose validity implies the correctness of a program under investigation. As our focus is on verifying quantitative properties such as bounds on expected outcomes, expected run-times, or termination probabilities, off-the-shelf IVLs based on Boolean first-order logic do not suffice. Instead, a paradigm shift from the standard Boolean to a real-valued domain is required. Our IVL features quantitative generalizations of standard verification constructs such as assume- and assert-statements. Verification conditions are generated by a weakest-precondition-style semantics, based on our real-valued logic. We show that our verification infrastructure supports natural encodings of numerous verification techniques from the literature. With our SMT-based implementation, we automatically verify a variety of benchmarks. To the best of our knowledge, this establishes the first deductive verification infrastructure for expectation-based reasoning about probabilistic programs.

We introduce the concept of a generative infinite game, a video game that transcends the traditional boundaries of finite, hard-coded systems by using generative models. Inspired by James P. Carse's distinction between finite and infinite games, we leverage recent advances in generative AI to create Unbounded: a game of character life simulation that is fully encapsulated in generative models. Specifically, Unbounded draws inspiration from sandbox life simulations and allows you to interact with your autonomous virtual character in a virtual world by feeding, playing with and guiding it - with open-ended mechanics generated by an LLM, some of which can be emergent. In order to develop Unbounded, we propose technical innovations in both the LLM and visual generation domains. Specifically, we present: (1) a specialized, distilled large language model (LLM) that dynamically generates game mechanics, narratives, and character interactions in real-time, and (2) a new dynamic regional image prompt Adapter (IP-Adapter) for vision models that ensures consistent yet flexible visual generation of a character across multiple environments. We evaluate our system through both qualitative and quantitative analysis, showing significant improvements in character life simulation, user instruction following, narrative coherence, and visual consistency for both characters and the environments compared to traditional related approaches.

We present LRM-Zero, a Large Reconstruction Model (LRM) trained entirely on synthesized 3D data, achieving high-quality sparse-view 3D reconstruction. The core of LRM-Zero is our procedural 3D dataset, Zeroverse, which is automatically synthesized from simple primitive shapes with random texturing and augmentations (e.g., height fields, boolean differences, and wireframes). Unlike previous 3D datasets (e.g., Objaverse) which are often captured or crafted by humans to approximate real 3D data, Zeroverse completely ignores realistic global semantics but is rich in complex geometric and texture details that are locally similar to or even more intricate than real objects. We demonstrate that our LRM-Zero, trained with our fully synthesized Zeroverse, can achieve high visual quality in the reconstruction of real-world objects, competitive with models trained on Objaverse. We also analyze several critical design choices of Zeroverse that contribute to LRM-Zero's capability and training stability. Our work demonstrates that 3D reconstruction, one of the core tasks in 3D vision, can potentially be addressed without the semantics of real-world objects. The Zeroverse's procedural synthesis code and interactive visualization are available at: https://desaixie.github.io/lrm-zero/.

For the evolutionary Stokes problem with dynamic boundary condition we show maximal regularity of weak solutions in time. Due to the characteriation of R-sectorial operators on Hilbert spaces, the proof reduces to finding the correct functional analytic setting and proving that an operator is sectorial, i.e. generates an analytic semigroup.

Formal methods is pivotal for verifying the reliability of critical systems through rigorous mathematical proofs. However, its adoption is hindered by labor-intensive manual proofs and the expertise required to use theorem provers. Recent advancements in large language models (LLMs) offer new opportunities for automated theorem proving. Two promising approaches are generating tactics step by step and generating a whole proof directly with an LLM. However, existing work makes no attempt to combine the two approaches. In this work, we introduce HybridProver, a dual-model proof synthesis framework that combines tactic-based generation and whole-proof synthesis to harness the benefits of both approaches. HybridProver generates whole proof candidates for evaluation directly, then extracts proof sketches from those candidates. It then uses a tactic-based generation model that integrates automated tools to complete the sketches via stepwise refinement. We implement HybridProver for the Isabelle theorem prover and fine-tune LLMs on our optimized Isabelle datasets. Evaluation on the miniF2F dataset illustrates HybridProver's effectiveness. We achieve a 59.4% success rate on miniF2F, where the previous SOTA is 56.1%. Our ablation studies show that this SOTA result is attributable to combining whole-proof and tactic-based generation. Additionally, we show how the dataset quality, training parameters, and sampling diversity affect the final result during automated theorem proving with LLMs. All of our code, datasets, and LLMs are open source.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.

The reasoning capabilities of large language models (LLMs) have been a longstanding focus of research. Recent works have further enhanced these capabilities using reinforcement learning (RL), with many new methods claiming significant improvements with minimal or no external supervision. Surprisingly, some studies even suggest that random or incorrect reward signals can enhance reasoning performance. However, these breakthroughs are mostly reported on the Qwen2.5 model family and evaluated on well-known benchmarks such as MATH-500, AMC, and AIME, while failing to achieve similar gains on other models like Llama, which warrants further investigation. Our analysis shows that although Qwen2.5 achieves strong mathematical reasoning performance, its pretraining on large-scale web corpora makes it vulnerable to data contamination in popular benchmarks. As a result, results derived from these benchmarks may be unreliable. To address this, we introduce a generator that produces fully synthetic arithmetic problems of arbitrary length and difficulty, yielding a clean dataset we call RandomCalculation. Using these leakage-free datasets, we show that only accurate reward signals consistently improve performance, while noisy or incorrect signals do not. We advocate for evaluating RL methods on uncontaminated benchmarks and across diverse model families to ensure trustworthy conclusions.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

Leveraging mathematical Large Language Models (LLMs) for proof generation is a fundamental topic in LLMs research. We argue that the ability of current LLMs to prove statements largely depends on whether they have encountered the relevant proof process during training. This reliance limits their deeper understanding of mathematical theorems and related concepts. Inspired by the pedagogical method of "proof by counterexamples" commonly used in human mathematics education, our work aims to enhance LLMs' ability to conduct mathematical reasoning and proof through counterexamples. Specifically, we manually create a high-quality, university-level mathematical benchmark, CounterMATH, which requires LLMs to prove mathematical statements by providing counterexamples, thereby assessing their grasp of mathematical concepts. Additionally, we develop a data engineering framework to automatically obtain training data for further model improvement. Extensive experiments and detailed analyses demonstrate that CounterMATH is challenging, indicating that LLMs, such as OpenAI o1, have insufficient counterexample-driven proof capabilities. Moreover, our exploration into model training reveals that strengthening LLMs' counterexample-driven conceptual reasoning abilities is crucial for improving their overall mathematical capabilities. We believe that our work offers new perspectives on the community of mathematical LLMs.

In this paper, we study the infinite-depth limit of finite-width residual neural networks with random Gaussian weights. With proper scaling, we show that by fixing the width and taking the depth to infinity, the pre-activations converge in distribution to a zero-drift diffusion process. Unlike the infinite-width limit where the pre-activation converge weakly to a Gaussian random variable, we show that the infinite-depth limit yields different distributions depending on the choice of the activation function. We document two cases where these distributions have closed-form (different) expressions. We further show an intriguing change of regime phenomenon of the post-activation norms when the width increases from 3 to 4. Lastly, we study the sequential limit infinite-depth-then-infinite-width and compare it with the more commonly studied infinite-width-then-infinite-depth limit.

This paper presents Representation-Conditioned image Generation (RCG), a simple yet effective image generation framework which sets a new benchmark in class-unconditional image generation. RCG does not condition on any human annotations. Instead, it conditions on a self-supervised representation distribution which is mapped from the image distribution using a pre-trained encoder. During generation, RCG samples from such representation distribution using a representation diffusion model (RDM), and employs a pixel generator to craft image pixels conditioned on the sampled representation. Such a design provides substantial guidance during the generative process, resulting in high-quality image generation. Tested on ImageNet 256times256, RCG achieves a Frechet Inception Distance (FID) of 3.31 and an Inception Score (IS) of 253.4. These results not only significantly improve the state-of-the-art of class-unconditional image generation but also rival the current leading methods in class-conditional image generation, bridging the long-standing performance gap between these two tasks. Code is available at https://github.com/LTH14/rcg.

Fuzz testing, or "fuzzing," refers to a widely deployed class of techniques for testing programs by generating a set of inputs for the express purpose of finding bugs and identifying security flaws. Grey-box fuzzing, the most popular fuzzing strategy, combines light program instrumentation with a data driven process to generate new program inputs. In this work, we present a machine learning approach that builds on AFL, the preeminent grey-box fuzzer, by adaptively learning a probability distribution over its mutation operators on a program-specific basis. These operators, which are selected uniformly at random in AFL and mutational fuzzers in general, dictate how new inputs are generated, a core part of the fuzzer's efficacy. Our main contributions are two-fold: First, we show that a sampling distribution over mutation operators estimated from training programs can significantly improve performance of AFL. Second, we introduce a Thompson Sampling, bandit-based optimization approach that fine-tunes the mutator distribution adaptively, during the course of fuzzing an individual program. A set of experiments across complex programs demonstrates that tuning the mutational operator distribution generates sets of inputs that yield significantly higher code coverage and finds more crashes faster and more reliably than both baseline versions of AFL as well as other AFL-based learning approaches.

In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of zeta(3). Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

We introduce Score identity Distillation (SiD), an innovative data-free method that distills the generative capabilities of pretrained diffusion models into a single-step generator. SiD not only facilitates an exponentially fast reduction in Fr\'echet inception distance (FID) during distillation but also approaches or even exceeds the FID performance of the original teacher diffusion models. By reformulating forward diffusion processes as semi-implicit distributions, we leverage three score-related identities to create an innovative loss mechanism. This mechanism achieves rapid FID reduction by training the generator using its own synthesized images, eliminating the need for real data or reverse-diffusion-based generation, all accomplished within significantly shortened generation time. Upon evaluation across four benchmark datasets, the SiD algorithm demonstrates high iteration efficiency during distillation and surpasses competing distillation approaches, whether they are one-step or few-step, data-free, or dependent on training data, in terms of generation quality. This achievement not only redefines the benchmarks for efficiency and effectiveness in diffusion distillation but also in the broader field of diffusion-based generation. The PyTorch implementation is available at https://github.com/mingyuanzhou/SiD

We present Open-MAGVIT2, a family of auto-regressive image generation models ranging from 300M to 1.5B. The Open-MAGVIT2 project produces an open-source replication of Google's MAGVIT-v2 tokenizer, a tokenizer with a super-large codebook (i.e., 2^{18} codes), and achieves the state-of-the-art reconstruction performance (1.17 rFID) on ImageNet 256 times 256. Furthermore, we explore its application in plain auto-regressive models and validate scalability properties. To assist auto-regressive models in predicting with a super-large vocabulary, we factorize it into two sub-vocabulary of different sizes by asymmetric token factorization, and further introduce "next sub-token prediction" to enhance sub-token interaction for better generation quality. We release all models and codes to foster innovation and creativity in the field of auto-regressive visual generation.

Recent work has shown the immense potential of synthetically generated datasets for training large language models (LLMs), especially for acquiring targeted skills. Current large-scale math instruction tuning datasets such as MetaMathQA (Yu et al., 2024) and MAmmoTH (Yue et al., 2024) are constructed using outputs from closed-source LLMs with commercially restrictive licenses. A key reason limiting the use of open-source LLMs in these data generation pipelines has been the wide gap between the mathematical skills of the best closed-source LLMs, such as GPT-4, and the best open-source LLMs. Building on the recent progress in open-source LLMs, our proposed prompting novelty, and some brute-force scaling, we construct OpenMathInstruct-1, a math instruction tuning dataset with 1.8M problem-solution pairs. The dataset is constructed by synthesizing code-interpreter solutions for GSM8K and MATH, two popular math reasoning benchmarks, using the recently released and permissively licensed Mixtral model. Our best model, OpenMath-CodeLlama-70B, trained on a subset of OpenMathInstruct-1, achieves a score of 84.6% on GSM8K and 50.7% on MATH, which is competitive with the best gpt-distilled models. We release our code, models, and the OpenMathInstruct-1 dataset under a commercially permissive license.

Neural networks have a reputation for being better at solving statistical or approximate problems than at performing calculations or working with symbolic data. In this paper, we show that they can be surprisingly good at more elaborated tasks in mathematics, such as symbolic integration and solving differential equations. We propose a syntax for representing mathematical problems, and methods for generating large datasets that can be used to train sequence-to-sequence models. We achieve results that outperform commercial Computer Algebra Systems such as Matlab or Mathematica.

We show that feedforward neural networks with ReLU activation generalize on low complexity data, suitably defined. Given i.i.d. data generated from a simple programming language, the minimum description length (MDL) feedforward neural network which interpolates the data generalizes with high probability. We define this simple programming language, along with a notion of description length of such networks. We provide several examples on basic computational tasks, such as checking primality of a natural number, and more. For primality testing, our theorem shows the following. Suppose that we draw an i.i.d. sample of Theta(N^{delta}ln N) numbers uniformly at random from 1 to N, where deltain (0,1). For each number x_i, let y_i = 1 if x_i is a prime and 0 if it is not. Then with high probability, the MDL network fitted to this data accurately answers whether a newly drawn number between 1 and N is a prime or not, with test error leq O(N^{-delta}). Note that the network is not designed to detect primes; minimum description learning discovers a network which does so.

Recently, 3D Gaussian splatting (3D-GS) has achieved great success in reconstructing and rendering real-world scenes. To transfer the high rendering quality to generation tasks, a series of research works attempt to generate 3D-Gaussian assets from text. However, the generated assets have not achieved the same quality as those in reconstruction tasks. We observe that Gaussians tend to grow without control as the generation process may cause indeterminacy. Aiming at highly enhancing the generation quality, we propose a novel framework named GaussianDreamerPro. The main idea is to bind Gaussians to reasonable geometry, which evolves over the whole generation process. Along different stages of our framework, both the geometry and appearance can be enriched progressively. The final output asset is constructed with 3D Gaussians bound to mesh, which shows significantly enhanced details and quality compared with previous methods. Notably, the generated asset can also be seamlessly integrated into downstream manipulation pipelines, e.g. animation, composition, and simulation etc., greatly promoting its potential in wide applications. Demos are available at https://taoranyi.com/gaussiandreamerpro/.

Code has been shown to be effective in enhancing the mathematical reasoning abilities of large language models due to its precision and accuracy. Previous works involving continued mathematical pretraining often include code that utilizes math-related packages, which are primarily designed for fields such as engineering, machine learning, signal processing, or module testing, rather than being directly focused on mathematical reasoning. In this paper, we introduce a novel method for generating mathematical code accompanied with corresponding reasoning steps for continued pretraining. Our approach begins with the construction of a high-quality mathematical continued pretraining dataset by incorporating math-related web data, code using mathematical packages, math textbooks, and synthetic data. Next, we construct reasoning steps by extracting LaTeX expressions, the conditions needed for the expressions, and the results of the expressions from the previously collected dataset. Based on this extracted information, we generate corresponding code to accurately capture the mathematical reasoning process. Appending the generated code to each reasoning step results in data consisting of paired natural language reasoning steps and their corresponding code. Combining this data with the original dataset results in a 19.2B-token high-performing mathematical pretraining corpus, which we name MathCode-Pile. Training several popular base models with this corpus significantly improves their mathematical abilities, leading to the creation of the MathCoder2 family of models. All of our data processing and training code is open-sourced, ensuring full transparency and easy reproducibility of the entire data collection and training pipeline. The code is released at https://github.com/mathllm/MathCoder2 .

We explore the application of transformer-based language models to automated theorem proving. This work is motivated by the possibility that a major limitation of automated theorem provers compared to humans -- the generation of original mathematical terms -- might be addressable via generation from language models. We present an automated prover and proof assistant, GPT-f, for the Metamath formalization language, and analyze its performance. GPT-f found new short proofs that were accepted into the main Metamath library, which is to our knowledge, the first time a deep-learning based system has contributed proofs that were adopted by a formal mathematics community.

Autoformalization addresses the scarcity of data for Automated Theorem Proving (ATP) by translating mathematical problems from natural language into formal statements. Efforts in recent work shift from directly prompting large language models to training an end-to-end formalizer model from scratch, achieving remarkable advancements. However, existing formalizer still struggles to consistently generate valid statements that meet syntactic validity and semantic consistency. To address this issue, we propose the Autoformalizer with Tool Feedback (ATF), a novel approach that incorporates syntactic and consistency information as tools into the formalization process. By integrating Lean 4 compilers for syntax corrections and employing a multi-LLMs-as-judge approach for consistency validation, the model is able to adaptively refine generated statements according to the tool feedback, enhancing both syntactic validity and semantic consistency. The training of ATF involves a cold-start phase on synthetic tool-calling data, an expert iteration phase to improve formalization capabilities, and Direct Preference Optimization to alleviate ineffective revisions. Experimental results show that ATF markedly outperforms a range of baseline formalizer models, with its superior performance further validated by human evaluations. Subsequent analysis reveals that ATF demonstrates excellent inference scaling properties. Moreover, we open-source Numina-ATF, a dataset containing 750K synthetic formal statements to facilitate advancements in autoformalization and ATP research.

Recent advancements in Large Language Models (LLMs) have significantly enhanced their ability to process long contexts, yet a notable gap remains in generating long, aligned outputs. This limitation stems from a training gap where pre-training lacks effective instructions for long-text generation, and post-training data primarily consists of short query-response pairs. Current approaches, such as instruction backtranslation and behavior imitation, face challenges including data quality, copyright issues, and constraints on proprietary model usage. In this paper, we introduce an innovative iterative training framework called Self-Lengthen that leverages only the intrinsic knowledge and skills of LLMs without the need for auxiliary data or proprietary models. The framework consists of two roles: the Generator and the Extender. The Generator produces the initial response, which is then split and expanded by the Extender. This process results in a new, longer response, which is used to train both the Generator and the Extender iteratively. Through this process, the models are progressively trained to handle increasingly longer responses. Experiments on benchmarks and human evaluations show that Self-Lengthen outperforms existing methods in long-text generation, when applied to top open-source LLMs such as Qwen2 and LLaMA3. Our code is publicly available at https://github.com/QwenLM/Self-Lengthen.

We extend Textual Inversion to learn pseudo-words that represent a concept at different resolutions. This allows us to generate images that use the concept with different levels of detail and also to manipulate different resolutions using language. Once learned, the user can generate images at different levels of agreement to the original concept; "A photo of S^*(0)" produces the exact object while the prompt "A photo of S^*(0.8)" only matches the rough outlines and colors. Our framework allows us to generate images that use different resolutions of an image (e.g. details, textures, styles) as separate pseudo-words that can be composed in various ways. We open-soure our code in the following URL: https://github.com/giannisdaras/multires_textual_inversion

Theorem proving in natural mathematical language - the mixture of symbolic and natural language used by humans - plays a central role in mathematical advances and education, and tests aspects of reasoning that are core to intelligence. Yet it has remained underexplored with modern generative models. We study large-scale language models on two new generation tasks: suggesting the next step in a mathematical proof, and full proof generation. We develop NaturalProver, a language model that generates proofs by conditioning on background references (e.g. theorems and definitions that are either retrieved or human-provided), and optionally enforces their presence with constrained decoding. On theorems from the NaturalProofs benchmark, NaturalProver improves the quality of next-step suggestions and generated proofs over fine-tuned GPT-3, according to human evaluations from university-level mathematics students. NaturalProver is capable of proving some theorems that require short (2-6 step) proofs, and providing next-step suggestions that are rated as correct and useful over 40% of the time, which is to our knowledge the first demonstration of these capabilities using neural language models.

Generating long-term texts such as novels using artificial intelligence has always been a challenge. A common approach is to use large language models (LLMs) to construct a hierarchical framework that first plans and then writes. Despite the fact that the generated novels reach a sufficient length, they exhibit poor logical coherence and appeal in their plots and deficiencies in character and event depiction, ultimately compromising the overall narrative quality. In this paper, we propose a method named Extracting Excelsior and Expanding. Ex3 initially extracts structure information from raw novel data. By combining this structure information with the novel data, an instruction-following dataset is meticulously crafted. This dataset is then utilized to fine-tune the LLM, aiming for excelsior generation performance. In the final stage, a tree-like expansion method is deployed to facilitate the generation of arbitrarily long novels. Evaluation against previous methods showcases Ex3's ability to produce higher-quality long-form novels.

In order to properly train a machine learning model, data must be properly collected. To guarantee a proper data collection, verifying that the collected data set holds certain properties is a possible solution. For example, guaranteeing that the data set contains samples across the whole input space, or that the data set is balanced w.r.t. different classes. We present a formal approach for verifying a set of arbitrarily stated properties over a data set. The proposed approach relies on the transformation of the data set into a first order logic formula, which can be later verified w.r.t. the different properties also stated in the same logic. A prototype tool, which uses the z3 solver, has been developed; the prototype can take as an input a set of properties stated in a formal language and formally verify a given data set w.r.t. to the given set of properties. Preliminary experimental results show the feasibility and performance of the proposed approach, and furthermore the flexibility for expressing properties of interest.

The massive and large-scale design of foundational semiconductor integrated circuits (ICs) is crucial to sustaining the advancement of many emerging and future technologies, such as generative AI, 5G/6G, and quantum computing. Excitingly, recent studies have shown the great capabilities of foundational models in expediting the design of digital ICs. Yet, applying generative AI techniques to accelerate the design of analog ICs remains a significant challenge due to critical domain-specific issues, such as the lack of a comprehensive dataset and effective representation methods for analog circuits. This paper proposes, AnalogGenie, a textbf{Gen}erattextbf{i}ve textbf{e}ngine for automatic design/discovery of textbf{Analog} circuit topologies--the most challenging and creative task in the conventional manual design flow of analog ICs. AnalogGenie addresses two key gaps in the field: building a foundational comprehensive dataset of analog circuit topology and developing a scalable sequence-based graph representation universal to analog circuits. Experimental results show the remarkable generation performance of AnalogGenie in broadening the variety of analog ICs, increasing the number of devices within a single design, and discovering unseen circuit topologies far beyond any prior arts. Our work paves the way to transform the longstanding time-consuming manual design flow of analog ICs to an automatic and massive manner powered by generative AI. Our source code is available at https://github.com/xz-group/AnalogGenie.

With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research-level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

Symbolic execution is a powerful technique for software testing, but suffers from limitations when encountering external functions, such as native methods or third-party libraries. Existing solutions often require additional context, expensive SMT solvers, or manual intervention to approximate these functions through symbolic stubs. In this work, we propose a novel approach to automatically generate symbolic stubs for external functions during symbolic execution that leverages Genetic Programming. When the symbolic executor encounters an external function, AutoStub generates training data by executing the function on randomly generated inputs and collecting the outputs. Genetic Programming then derives expressions that approximate the behavior of the function, serving as symbolic stubs. These automatically generated stubs allow the symbolic executor to continue the analysis without manual intervention, enabling the exploration of program paths that were previously intractable. We demonstrate that AutoStub can automatically approximate external functions with over 90% accuracy for 55% of the functions evaluated, and can infer language-specific behaviors that reveal edge cases crucial for software testing.

Despite their spectacular progress, language models still struggle on complex reasoning tasks, such as advanced mathematics. We consider a long-standing open problem in mathematics: discovering a Lyapunov function that ensures the global stability of a dynamical system. This problem has no known general solution, and algorithmic solvers only exist for some small polynomial systems. We propose a new method for generating synthetic training samples from random solutions, and show that sequence-to-sequence transformers trained on such datasets perform better than algorithmic solvers and humans on polynomial systems, and can discover new Lyapunov functions for non-polynomial systems.

Automated code generation with large language models has gained significant traction, but there remains no guarantee on the correctness of generated code. We aim to use formal verification to provide mathematical guarantees that the generated code is correct. However, generating formally verified code with LLMs is hindered by the scarcity of training data and the complexity of formal proofs. To tackle this challenge, we introduce AlphaVerus, a self-improving framework that bootstraps formally verified code generation by iteratively translating programs from a higher-resource language and leveraging feedback from a verifier. AlphaVerus operates in three phases: exploration of candidate translations, Treefinement -- a novel tree search algorithm for program refinement using verifier feedback, and filtering misaligned specifications and programs to prevent reward hacking. Through this iterative process, AlphaVerus enables a LLaMA-3.1-70B model to generate verified code without human intervention or model finetuning. AlphaVerus shows an ability to generate formally verified solutions for HumanEval and MBPP, laying the groundwork for truly trustworthy code-generation agents.

Conducting contamination-free evaluation of mathematical capabilities can be difficult for two reasons: models may memorize a test set once it is made public, and current mathematical benchmarks are prone to overfitting due to having limited diversity of symbols and rules, coupled with closed-ended answers. This paper proposes a method to leverage these shortcomings as useful features to a construct dynamic, counterfactual benchmark, which can be used to both reveal overfitting and measure true reasoning. We demonstrate this via MatheMagic, which generates math test instances with the interpretations of numbers and operators altered, yet has automatically verifiable answers. Test instances are randomly seeded and constructed at test time to evaluate a model's induction or deduction capability, offering stability, extensibility, comparability, and robustness to overfitting. Our experiments find that models solve deduction more easily than induction, but they revert to standard math. Further analysis reveals that math-adapted models fail to exhibit a general "skill" of reasoning, and fine-tuning on induction tasks generalizes poorly.