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SeRpEnt: Selective Resampling for Expressive State Space Models

Stefano Rando, Luca Romani, Matteo Migliarini, Luca Franco, Denis Gudovskiy, Fabio Galasso

TL;DR

The paper addresses the limitations of attention-based Transformers in sequence modeling by leveraging State Space Models (SSMs) and, in particular, analyzes the selectivity mechanism of Mamba as an information-processing tool. It introduces SeRpEnt, a selective resampling architecture that compresses sequences based on information content before feeding them to an SSM, with Gaussian basis expansion for time-difference features and a tunable compression rate. Theoretical insights show that learned selectivity acts as a linear approximation of information, justifying resampling, while empirical results demonstrate gains on the Long Range Arena and WikiText-103-v1 language modeling tasks when SeRpEnt is combined with SSM backbones like Mamba or S4. Overall, SeRpEnt offers a practical, orthogonal enhancement to existing SSM variants, enabling more efficient handling of long-range dependencies without relying on attention, and holds promise for broader adoption in sequence modeling workflows.

Abstract

State Space Models (SSMs) have recently enjoyed a rise to prominence in the field of deep learning for sequence modeling, especially as an alternative to Transformers. Their success stems from avoiding two well-known drawbacks of attention-based models: quadratic complexity with respect to the sequence length and inability to model long-range dependencies. The SSM variant Mamba has demonstrated performance comparable to Transformers without any form of attention, thanks to the use of a selective mechanism for the state parameters. Selectivity, however, is only evaluated empirically and the reasons of its effectiveness remain unclear. In this work, we show how selectivity is related to the sequence processing. Our analysis shows that selective time intervals in Mamba act as linear approximators of information. Then, we propose our SeRpEnt architecture, a SSM that further exploits selectivity to compress sequences in an information-aware fashion. It employs a resampling mechanism that aggregates elements based on their information content. Our empirical results in the Long Range Arena benchmark and other language modeling tasks show benefits of the SeRpEnt's resampling mechanism.

SeRpEnt: Selective Resampling for Expressive State Space Models

TL;DR

The paper addresses the limitations of attention-based Transformers in sequence modeling by leveraging State Space Models (SSMs) and, in particular, analyzes the selectivity mechanism of Mamba as an information-processing tool. It introduces SeRpEnt, a selective resampling architecture that compresses sequences based on information content before feeding them to an SSM, with Gaussian basis expansion for time-difference features and a tunable compression rate. Theoretical insights show that learned selectivity acts as a linear approximation of information, justifying resampling, while empirical results demonstrate gains on the Long Range Arena and WikiText-103-v1 language modeling tasks when SeRpEnt is combined with SSM backbones like Mamba or S4. Overall, SeRpEnt offers a practical, orthogonal enhancement to existing SSM variants, enabling more efficient handling of long-range dependencies without relying on attention, and holds promise for broader adoption in sequence modeling workflows.

Abstract

State Space Models (SSMs) have recently enjoyed a rise to prominence in the field of deep learning for sequence modeling, especially as an alternative to Transformers. Their success stems from avoiding two well-known drawbacks of attention-based models: quadratic complexity with respect to the sequence length and inability to model long-range dependencies. The SSM variant Mamba has demonstrated performance comparable to Transformers without any form of attention, thanks to the use of a selective mechanism for the state parameters. Selectivity, however, is only evaluated empirically and the reasons of its effectiveness remain unclear. In this work, we show how selectivity is related to the sequence processing. Our analysis shows that selective time intervals in Mamba act as linear approximators of information. Then, we propose our SeRpEnt architecture, a SSM that further exploits selectivity to compress sequences in an information-aware fashion. It employs a resampling mechanism that aggregates elements based on their information content. Our empirical results in the Long Range Arena benchmark and other language modeling tasks show benefits of the SeRpEnt's resampling mechanism.
Paper Structure (30 sections, 1 theorem, 27 equations, 4 figures, 3 tables)

This paper contains 30 sections, 1 theorem, 27 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Attention Mechanism in Transformers
  4. State Space Models
  5. Background
  6. State Space Models
  7. LTI SSMs as recurrent and convolutional neural networks.
  8. Initialization of parameters.
  9. Selective SSMs.
  10. Learned Selectivity as a Linear Approximation of Information
  11. Proposed Method
  12. Sequence Compression
  13. Selective Resampling for Compression
  14. Gaussian basis expansion of time differences.
  15. Resampling and compression rate.
  16. ...and 15 more sections

Key Result

Proposition 1

For the discrete SSM defined by equations (eq:discrete_ssm_ode-eq:discrete_ssm_output) and (eq:zoh_disc), when the matrix ${\bm{A}}$ is diagonal, the distance $\| {\bm{h}}_L - {\bm{h}}_L^{i} \|$ is asymptotically linear with respect to $\Delta_i$ as $\Delta_i \to 0$ i.e.

Figures (4)

  • Figure 1: (Top Left) A linear time-invariant SSM with uniform sampling assumes a sequence $\{ x_l \}$ is sampled from an underlying continuous function which incurs a loss of precision due to the constant time interval $\Delta$. (Top Right) The Mamba computes time-variable intervals $\Delta_l$ that are dependent on the sequence elements: the sequence is assumed to be non-uniformly sampled from the underlying function. (Bottom) Our SeRpEnt assumes time-dependent intervals $\Delta_l$ and, additionally, resamples the sequence through interpolation by aggregating together elements whose information is related.
  • Figure 2: (Compression) During the compression procedure through selective resampling a time interval $\Delta_l$ is computed for each element $x_l$ of the sequence. Then, the intervals are used to obtain the times at which elements sampled from the underlying function. Based on the $\Delta_l$ value and $\Delta$, elements are resampled and processed with a linear transformation together with their $K$ closest neighbors. Finally, the compressed sequence is the input to a SSM.
  • Figure 3: (Architecture) Our SeRpEnt block works by compressing an input sequence $\{ x_l \}_{l \leq L}$ at different compression rates. It separately processes each of the compressed inputs using a SSM layer, decompresses the outputs, and concatenates them back into a single sequence with a skip connection.
  • Figure 4: Training losses during training for different tasks

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • proof