Theoretical Foundations and Applications of Deep Selective State-Space Models

Concetti Chiave

The authors explore the theoretical underpinnings of deep selective state-space models, highlighting their expressive power and efficiency in capturing high-order statistics. They demonstrate how input-controlled transitions enhance model capabilities beyond linear filtering.

Sintesi

The content delves into the theoretical foundations of deep selective state-space models, emphasizing their ability to capture non-linear interactions between tokens at different timescales. The authors discuss the advantages of using diagonal recurrences for computational efficiency but highlight the limitations in capturing higher order statistics compared to non-diagonal counterparts. Additionally, they propose a chaining approach to regain expressivity without sacrificing computational advantages.
Key points include:

Introduction to sequence-to-sequence blocks in deep learning.
Comparison between attention-based transformers and state-space models (SSMs).
Theoretical grounding using tools from Rough Path Theory.
Analysis of linear controlled differential equations (CDEs) and their expressivity.
Discussion on stability, efficiency, and path-to-path learning capabilities.
Empirical validation through anti-symmetric signature prediction tasks.
Exploration of chaining diagonal CDEs for enhanced expressivity.
Consideration of path-to-path learning models and their universal approximation properties.
Overall, the content provides a comprehensive overview of the theoretical foundations and practical applications of deep selective state-space models.

Statistiche

SSMs achieve state-of-the-art results on long-range-reasoning benchmarks (Tay et al., 2020).
Computational complexity scales linearly in sequence length for SSMs compared to quadratic scaling for attention mechanisms.
Mamba architecture achieves efficient input selectivity with reduced computational cost.
Linear CDEs can be expanded as linear combinations of terms in the signature transform.

Citazioni

"Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states, then the resulting architecture can surpass attention-powered foundation models trained on text." - Content
"Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants." - Content

Approfondimenti chiave tratti da

Theoretical Foundations of Deep Selective State-Space Models

by Nicola Muca ... alle arxiv.org 03-01-2024

https://arxiv.org/pdf/2402.19047.pdf

Domande più approfondite

How do input-controlled transitions enhance model capabilities beyond linear filtering?

Input-controlled transitions in models like Mamba go beyond simple linear filtering by allowing the system to filter out irrelevant information and remember relevant information indefinitely. These transitions enable the model to capture high-order statistics of the input data, providing a more nuanced understanding of the relationships between different features over time. By incorporating selectivity mechanisms based on inputs, these models can perform content-based reasoning efficiently, making them powerful tools for tasks requiring complex interactions between tokens at distinct timescales.

What are the implications of using diagonal recurrences for computational efficiency but limitations in capturing higher order statistics?

Using diagonal recurrences in models like S4 or Mamba offers significant computational advantages due to their ability to scale linearly with sequence length compared to dense matrices that have a complexity of O(N^2L). However, there are limitations when it comes to capturing higher-order statistics. Diagonal recurrences restrict the model's capacity to extract non-linear interactions and dependencies among different features since they can only capture symmetric parts of signatures. This limitation hinders their expressivity in learning complex patterns that require understanding beyond linear relationships within sequences.

How does chaining diagonal CDEs help regain expressivity without sacrificing computational advantages?

Chaining diagonal CDEs is a strategy that helps regain expressivity while maintaining computational efficiency. By sequentially applying multiple layers of diagonal recurrent structures, each layer can capture additional depth in feature interactions and dependencies within sequences. This approach allows for recovering higher-order statistics from input data step by step, enabling the model to learn more intricate patterns and relationships present in the data. Chaining these structures enhances expressive power without compromising on computational advantages such as parallelization and reduced training complexity associated with diagonality.