DeltaProduct: Improving State-Tracking in Linear RNNs via Householder Products

Julien Siems, Timur Carstensen, Arber Zela, Frank Hutter, Massimiliano Pontil, Riccardo Grazzi

Keywords: linear rnn, linear attention, state-space models, state-tracking, negative eigenvalues, householder, formal languages, regular languages, stability, finite-state automata

TL;DR: We extend DeltaNet by using products of householders as state-transition matrices allowing us to trade-off expressivity and computational complexity.

Abstract:

Linear Recurrent Neural Networks (linear RNNs) have emerged as competitive alternatives to Transformers for sequence modeling, offering efficient training and linear-time inference. However, existing architectures face a fundamental trade-off between expressivity and efficiency, dictated by the structure of their state-transition matrices. Diagonal matrices, used in models such as Mamba, GLA, or mLSTM, yield fast runtime but have limited expressivity. To address this, recent architectures such as DeltaNet and RWKV-7 adopted a diagonal plus rank-1 structure, which allows simultaneous token and channel mixing,
improving associative recall and, as recently shown, state-tracking when
allowing negative eigenvalues in the state-transition matrices.
Building on the interpretation of DeltaNet's recurrence as performing one step of online gradient descent per token on an associative recall loss, we introduce DeltaProduct, which instead takes multiple ($n_h$) steps per token. This naturally leads to diagonal plus rank-$n_h$ state-transition matrices, formed as products of $n_h$ generalized Householder transformations, providing a tunable mechanism to balance expressivity and efficiency.
We provide a detailed theoretical characterization of the state-tracking capability of DeltaProduct in finite precision and how it improves by increasing $n_h$.
Our extensive experiments demonstrate that DeltaProduct outperforms DeltaNet in both state-tracking and language modeling, while also showing significantly improved length extrapolation capabilities.

Primary Area: Deep learning (e.g., architectures, generative models, optimization for deep networks, foundation models, LLMs)

Submission Number: 22323