Can Transformers Do Enumerative Geometry?

• URL: http://arxiv.org/abs/2408.14915v1
• Date: Tue, 27 Aug 2024 09:44:01 GMT
• Title: Can Transformers Do Enumerative Geometry?
• Authors: Baran Hashemi, Roderic G. Corominas, Alessandro Giacchetto,
• Abstract summary: We introduce a new paradigm in computational enumerative geometry in analyzing the $psi$-class intersection numbers on the moduli space of curves. We develop a Transformer-based model for computing $psi$-class intersection numbers based on the underlying quantum Airy structure. We go beyond merely computing intersection numbers and explore the enumerative "world-model" of the Transformers.
• Score: 44.99833362998488
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: How can Transformers model and learn enumerative geometry? What is a robust procedure for using Transformers in abductive knowledge discovery within a mathematician-machine collaboration? In this work, we introduce a new paradigm in computational enumerative geometry in analyzing the $\psi$-class intersection numbers on the moduli space of curves. By formulating the enumerative problem as a continuous optimization task, we develop a Transformer-based model for computing $\psi$-class intersection numbers based on the underlying quantum Airy structure. For a finite range of genera, our model is capable of regressing intersection numbers that span an extremely wide range of values, from $10^{-45}$ to $10^{45}$. To provide a proper inductive bias for capturing the recursive behavior of intersection numbers, we propose a new activation function, Dynamic Range Activator (DRA). Moreover, given the severe heteroscedasticity of $\psi$-class intersections and the required precision, we quantify the uncertainty of the predictions using Conformal Prediction with a dynamic sliding window that is aware of the number of marked points. Next, we go beyond merely computing intersection numbers and explore the enumerative "world-model" of the Transformers. Through a series of causal inference and correlational interpretability analyses, we demonstrate that Transformers are actually modeling Virasoro constraints in a purely data-driven manner. Additionally, we provide evidence for the comprehension of several values appearing in the large genus asymptotic of $\psi$-class intersection numbers through abductive hypothesis testing.

Related papers

• The calculus of variations of the Transformer on the hyperspherical tangent bundle [0.0]
  We offer a theoretical mathematical background to Transformers through Lagrangian optimization across the token space.<n>The Transformer, as a flow map, exists in the tangent fiber for each token along the high-dimensional unit sphere.<n>We derive the Euler-Lagrange equation for the Transformer.
  arXiv  Detail & Related papers  (2025-07-21T09:43:33Z)
• Approximation Bounds for Transformer Networks with Application to Regression [9.549045683389085]
  We explore the approximation capabilities of Transformer networks for H"older and Sobolev functions. We establish novel upper bounds for standard Transformer networks approxing sequence-to-sequence mappings. We show that if the self-attention layer in a Transformer can perform column averaging, the network can approximate sequence-to-sequence H"older functions.
  arXiv  Detail & Related papers  (2025-04-16T15:25:58Z)
• (How) Can Transformers Predict Pseudo-Random Numbers? [7.201095605457193]
  We study the ability of Transformers to learn pseudo-random number sequences from linear congruential generators (LCGs) Our analysis reveals that Transformers can perform in-context prediction of LCG sequences with unseen moduli ($m$) and parameters ($a,c$)
  arXiv  Detail & Related papers  (2025-02-14T18:59:40Z)
• On the Role of Depth and Looping for In-Context Learning with Task Diversity [69.4145579827826]
  We study in-context learning for linear regression with diverse tasks. We show that multilayer Transformers are not robust to even distributional shifts as small as $O(e-L)$ in Wasserstein distance.
  arXiv  Detail & Related papers  (2024-10-29T03:27:56Z)
• Can Looped Transformers Learn to Implement Multi-step Gradient Descent for In-context Learning? [69.4145579827826]
  We show a fast flow on the regression loss despite the gradient non-ity algorithms for our convergence landscape. This is the first theoretical analysis for multi-layer Transformer in this setting.
  arXiv  Detail & Related papers  (2024-10-10T18:29:05Z)
• Bayesian Circular Regression with von Mises Quasi-Processes [57.88921637944379]
  In this work we explore a family of expressive and interpretable distributions over circle-valued random functions.<n>For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Gibbs sampling.<n>We present experiments applying this model to the prediction of wind directions and the percentage of the running gait cycle as a function of joint angles.
  arXiv  Detail & Related papers  (2024-06-19T01:57:21Z)
• Towards Understanding Inductive Bias in Transformers: A View From Infinity [9.00214539845063]
  We argue transformers tend to be biased towards more permutation symmetric functions in sequence space. We show that the representation theory of the symmetric group can be used to give quantitative analytical predictions. We argue WikiText dataset, does indeed possess a degree of permutation symmetry.
  arXiv  Detail & Related papers  (2024-02-07T19:00:01Z)
• Transolver: A Fast Transformer Solver for PDEs on General Geometries [66.82060415622871]
  We present Transolver, which learns intrinsic physical states hidden behind discretized geometries. By calculating attention to physics-aware tokens encoded from slices, Transovler can effectively capture intricate physical correlations. Transolver achieves consistent state-of-the-art with 22% relative gain across six standard benchmarks and also excels in large-scale industrial simulations.
  arXiv  Detail & Related papers  (2024-02-04T06:37:38Z)
• Efficient Nonparametric Tensor Decomposition for Binary and Count Data [27.02813234958821]
  We propose ENTED, an underlineEfficient underlineNon underlineTEnsor underlineDecomposition for binary and count tensors.
  arXiv  Detail & Related papers  (2024-01-15T14:27:03Z)
• Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learning [77.1421343649344]
  We propose a generalization of Transformers towards operating entirely on the product of constant curvature spaces. We also provide a kernelized approach to non-Euclidean attention, which enables our model to run in time and memory cost linear to the number of nodes and edges.
  arXiv  Detail & Related papers  (2023-09-08T02:44:37Z)
• Scalable Transformer for PDE Surrogate Modeling [9.438207505148947]
  Transformer has emerged as a promising tool for surrogate modeling of partial differential equations (PDEs) We propose Factorized Transformer (FactFormer), which is based on an axial factorized kernel integral. We showcase that the proposed model is able to simulate 2D Kolmogorov flow on a $256times 256$ grid and 3D smoke buoyancy on a $64times64times64$ grid with good accuracy and efficiency.
  arXiv  Detail & Related papers  (2023-05-27T19:23:00Z)
• Transformers Learn Shortcuts to Automata [52.015990420075944]
  We find that a low-depth Transformer can represent the computations of any finite-state automaton. We show that a Transformer with $O(log T)$ layers can exactly replicate the computation of an automaton on an input sequence of length $T$. We further investigate the brittleness of these solutions and propose potential mitigations.
  arXiv  Detail & Related papers  (2022-10-19T17:45:48Z)
• The Parallelism Tradeoff: Limitations of Log-Precision Transformers [29.716269397142973]
  We prove that transformers whose arithmetic precision is logarithmic in the number of input tokens can be simulated by constant-depth logspace-uniform threshold circuits. This provides insight on the power of transformers using known results in complexity theory.
  arXiv  Detail & Related papers  (2022-07-02T03:49:34Z)
• Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformers [50.85524803885483]
  This work proposes a formal definition of statistically meaningful (SM) approximation which requires the approximating network to exhibit good statistical learnability. We study SM approximation for two function classes: circuits and Turing machines.
  arXiv  Detail & Related papers  (2021-07-28T04:28:55Z)
• $O(n)$ Connections are Expressive Enough: Universal Approximability of Sparse Transformers [71.31712741938837]
  We show that sparse Transformers with only $O(n)$ connections per attention layer can approximate the same function class as the dense model with $n2$ connections. We also present experiments comparing different patterns/levels of sparsity on standard NLP tasks.
  arXiv  Detail & Related papers  (2020-06-08T18:30:12Z)
• Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomography [0.4110409960377149]
  We propose a novel convolutional neural network (CNN) designed for learning pseudodifferential operators ($Psi$DOs) in the context of linear inverse problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN. In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling and convolution, can be exactly determined by combining the convolutional nature of the limited angle X-ray transform and basic properties defining a wavelet system.
  arXiv  Detail & Related papers  (2020-06-02T14:03:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.