Beyond Position: the emergence of wavelet-like properties in Transformers

• URL: http://arxiv.org/abs/2410.18067v3
• Date: Tue, 21 Jan 2025 17:50:47 GMT
• Title: Beyond Position: the emergence of wavelet-like properties in Transformers
• Authors: Valeria Ruscio, Fabrizio Silvestri,
• Abstract summary: This paper studies how transformer models develop robust wavelet-like properties that effectively compensate for the theoretical limitations of Rotary Position Embeddings (RoPE) We show that attention heads naturally evolve to implement multi-resolution processing analogous to wavelet transforms.
• Score: 7.3645788720974465
• License:
• Abstract: This paper studies how transformer models develop robust wavelet-like properties that effectively compensate for the theoretical limitations of Rotary Position Embeddings (RoPE), providing insights into how these networks process sequential information across different scales. Through theoretical analysis and empirical validation across models ranging from 1B to 12B parameters, we show that attention heads naturally evolve to implement multi-resolution processing analogous to wavelet transforms. Our analysis establishes that attention heads consistently organize into complementary frequency bands with systematic power distribution patterns, and these wavelet-like characteristics become more pronounced in larger models. We provide mathematical analysis showing how these properties align with optimal solutions to the fundamental uncertainty principle between positional precision and frequency resolution. Our findings suggest that the effectiveness of modern transformer architectures stems significantly from their development of optimal multi-resolution decompositions that naturally address the theoretical constraints of position encoding.

Related papers

• Tight Stability, Convergence, and Robustness Bounds for Predictive Coding Networks [60.3634789164648]
  Energy-based learning algorithms, such as predictive coding (PC), have garnered significant attention in the machine learning community. We rigorously analyze the stability, robustness, and convergence of PC through the lens of dynamical systems theory.
  arXiv  Detail & Related papers  (2024-10-07T02:57:26Z)
• A Unified Framework for Interpretable Transformers Using PDEs and Information Theory [3.4039202831583903]
  This paper presents a novel unified theoretical framework for understanding Transformer architectures by integrating Partial Differential Equations (PDEs), Neural Information Flow Theory, and Information Bottleneck Theory. We model Transformer information dynamics as a continuous PDE process, encompassing diffusion, self-attention, and nonlinear residual components. Our comprehensive experiments across image and text modalities demonstrate that the PDE model effectively captures key aspects of Transformer behavior, achieving high similarity (cosine similarity > 0.98) with Transformer attention distributions across all layers.
  arXiv  Detail & Related papers  (2024-08-18T16:16:57Z)
• Dynamical Mean-Field Theory of Self-Attention Neural Networks [0.0]
  Transformer-based models have demonstrated exceptional performance across diverse domains. Little is known about how they operate or what are their expected dynamics. We use methods for the study of asymmetric Hopfield networks in nonequilibrium regimes.
  arXiv  Detail & Related papers  (2024-06-11T13:29:34Z)
• Function Approximation for Reinforcement Learning Controller for Energy from Spread Waves [69.9104427437916]
  Multi-generator Wave Energy Converters (WEC) must handle multiple simultaneous waves coming from different directions called spread waves. These complex devices need controllers with multiple objectives of energy capture efficiency, reduction of structural stress to limit maintenance, and proactive protection against high waves. In this paper, we explore different function approximations for the policy and critic networks in modeling the sequential nature of the system dynamics.
  arXiv  Detail & Related papers  (2024-04-17T02:04:10Z)
• Understanding the Expressive Power and Mechanisms of Transformer for Sequence Modeling [10.246977481606427]
  We study the mechanisms through which different components of Transformer, such as the dot-product self-attention, affect its expressive power. Our study reveals the roles of critical parameters in the Transformer, such as the number of layers and the number of attention heads.
  arXiv  Detail & Related papers  (2024-02-01T11:43:13Z)
• On the Convergence of Encoder-only Shallow Transformers [62.639819460956176]
  We build the global convergence theory of encoder-only shallow Transformers under a realistic setting. Our results can pave the way for a better understanding of modern Transformers, particularly on training dynamics.
  arXiv  Detail & Related papers  (2023-11-02T20:03:05Z)
• Variational waveguide QED simulators [58.720142291102135]
  Waveguide QED simulators are made by quantum emitters interacting with one-dimensional photonic band-gap materials. Here, we demonstrate how these interactions can be a resource to develop more efficient variational quantum algorithms.
  arXiv  Detail & Related papers  (2023-02-03T18:55:08Z)
• Convexifying Transformers: Improving optimization and understanding of transformer networks [56.69983975369641]
  We study the training problem of attention/transformer networks and introduce a novel convex analytic approach. We first introduce a convex alternative to the self-attention mechanism and reformulate the regularized training problem of transformer networks. As a byproduct of our convex analysis, we reveal an implicit regularization mechanism, which promotes sparsity across tokens.
  arXiv  Detail & Related papers  (2022-11-20T18:17:47Z)
• Transformer Meets Boundary Value Inverse Problems [4.165221477234755]
  Transformer-based deep direct sampling method is proposed for solving a class of boundary value inverse problem. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and reconstructed images.
  arXiv  Detail & Related papers  (2022-09-29T17:45:25Z)
• Multiplicative noise and heavy tails in stochastic optimization [62.993432503309485]
  empirical optimization is central to modern machine learning, but its role in its success is still unclear. We show that it commonly arises in parameters of discrete multiplicative noise due to variance. A detailed analysis is conducted in which we describe on key factors, including recent step size, and data, all exhibit similar results on state-of-the-art neural network models.
  arXiv  Detail & Related papers  (2020-06-11T09:58:01Z)
• Masked Language Modeling for Proteins via Linearly Scalable Long-Context Transformers [42.93754828584075]
  We present a new Transformer architecture, Performer, based on Fast Attention Via Orthogonal Random features (FAVOR) Our mechanism scales linearly rather than quadratically in the number of tokens in the sequence, is characterized by sub-quadratic space complexity and does not incorporate any sparsity pattern priors. It provides strong theoretical guarantees: unbiased estimation of the attention matrix and uniform convergence.
  arXiv  Detail & Related papers  (2020-06-05T17:09:16Z)

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.