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

TL;DR

DeltaProduct enhances linear RNNs by using multiple Householder transformations per token, improving state-tracking and language modeling performance while balancing efficiency and expressivity.

Contribution

It introduces DeltaProduct, a novel architecture that employs multiple Householder transformations to improve state-tracking in linear RNNs.

Findings

Outperforms DeltaNet in state-tracking and language modeling.

Shows improved length extrapolation capabilities.

Provides theoretical analysis of state-tracking in finite precision.

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 state-transition matrices to have negative eigenvalues. 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, showing 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.