# Understanding Input Selectivity in Mamba: Impact on Approximation Power, Memorization, and Associative Recall Capacity

arXiv: https://arxiv.org/abs/2506.11891 · HTML: https://mlsanity.com/paper/2506.11891/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Core SSM architectures for language, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 20 of its 33 references that are in the map.
- First posted on arXiv 13 Jun 2025.

## Placed in

- State-space sequence models (S4, Mamba) / [Core SSM architectures for language](https://mlsanity.com/ssm/core-ssm-architectures-for-language/)

## Builds on (most cited first, 20 of 20)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [Simple linear attention language models balance the recall-throughput tradeoff](https://mlsanity.com/paper/2402.18668/) (2024)
- [Zoology: Measuring and Improving Recall in Efficient Language Models](https://mlsanity.com/paper/2312.04927/) (2023)
- [The Illusion of State in State-Space Models](https://mlsanity.com/paper/2404.08819/) (2024)
- [Birth of a Transformer: A Memory Viewpoint](https://mlsanity.com/paper/2306.00802/) (2023)
- [Overcoming a Theoretical Limitation of Self-Attention](https://mlsanity.com/paper/2202.12172/) (2022)
- [Unlocking State-Tracking in Linear RNNs Through Negative Eigenvalues](https://mlsanity.com/paper/2411.12537/) (2024)
- [Transformers, parallel computation, and logarithmic depth](https://mlsanity.com/paper/2402.09268/) (2024)
- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [The Expressive Capacity of State Space Models: A Formal Language Perspective](https://mlsanity.com/paper/2405.17394/) (2024)
- [Theoretical limitations of multi-layer Transformer](https://mlsanity.com/paper/2412.02975/) (2024)
- [Universality of Linear Recurrences Followed by Non-linear Projections: Finite-Width Guarantees and Benefits of Complex Eigenvalues](https://mlsanity.com/paper/2307.11888/) (2023)
- [One-layer transformers fail to solve the induction heads task](https://mlsanity.com/paper/2408.14332/) (2024)
- [Pointer Value Retrieval: A new benchmark for understanding the limits of neural network generalization](https://mlsanity.com/paper/2107.12580/) (2021)
- [Inverse Approximation Theory for Nonlinear Recurrent Neural Networks](https://mlsanity.com/paper/2305.19190/) (2023)
- [On the Power of Convolution Augmented Transformer](https://mlsanity.com/paper/2407.05591/) (2024)

## Built on by (most cited first, 7 of 7)

- [ParaRNN: Unlocking Parallel Training of Nonlinear RNNs for Large Language Models](https://mlsanity.com/paper/2510.21450/) (2025)
- [A Theoretical Analysis of Mamba's Training Dynamics: Filtering Relevant Features for Generalization in State Space Models](https://mlsanity.com/paper/2602.12499/) (2026)
- [SHiPPO: Recurrent Memory with Transported Polynomial Projections](https://mlsanity.com/paper/2607.03055/) (2026)
- [On the Recall Scaling Laws in Mamba: A Theoretical and Mechanistic Study via Hashing](https://mlsanity.com/paper/2609.07681/) (2026)
- [Sessa: Selective State Space Attention](https://mlsanity.com/paper/2604.18580/) (2026)
- [On the Importance of Gating: Memorization vs. In-Context Learning in State Space Models](https://mlsanity.com/paper/2609.16540/) (2026)
- [A Comparative Analysis of Attention versus State-Space Models for In-Context Learning](https://mlsanity.com/paper/2609.32341/) (2026)

## Abstract (arXiv)

State-Space Models (SSMs), and particularly Mamba, have recently emerged as a promising alternative to Transformers. Mamba introduces input selectivity to its SSM layer (S6) and incorporates convolution and gating into its block definition. While these modifications do improve Mamba's performance over its SSM predecessors, it remains largely unclear how Mamba leverages the additional functionalities provided by input selectivity, and how these interact with the other operations in the Mamba architecture. In this work, we demystify the role of input selectivity in Mamba, investigating its impact on function approximation power, long-term memorization, and associative recall capabilities. In particular: (i) we prove that the S6 layer of Mamba can represent projections onto Haar wavelets, providing an edge over its Diagonal SSM (S4D) predecessor in approximating discontinuous functions commonly arising in practice; (ii) we show how the S6 layer can dynamically counteract memory decay; (iii) we provide analytical solutions to the MQAR associative recall task using the Mamba architecture with different mixers -- Mamba, Mamba-2, and S4D. We demonstrate the tightness of our theoretical constructions with empirical results on concrete tasks. Our findings offer a mechanistic understanding of Mamba and reveal opportunities for improvement.