Rank collapse, a phenomenon where embedding vectors in sequence models rapidly converge to a uniform token or equilibrium state, has recently gained at- tention in the deep learning literature. This phenomenon leads to reduced expres- sivity and potential training instabilities due to vanishing gradients. Empirical ev- idence suggests that architectural components like skip connections, LayerNorm, and MultiLayer Perceptrons (MLPs) play critical roles in mitigating rank collapse. While this issue is well-documented for transformers, alternative sequence mod- els, such as State Space Models (SSMs), which have recently gained prominence, have not been thoroughly examined for similar vulnerabilities. This paper extends the theory of rank collapse from transformers to SSMs using a unifying frame- work that captures both architectures. We introduce a modification in the skip connection component, termed lambda-skip connections, that provides guaran- tees for rank collapse prevention. We present, via analytical results, a sufficient condition to achieve the guarantee for all of the aforementioned architectures. We also study the necessity of this condition via ablation studies and analytical exam- ples. To our knowledge, this is the first study that provides a general guarantee to prevent rank collapse, and that investigates rank collapse in the context of SSMs, offering valuable understanding for both theoreticians and practitioners. Finally, we validate our findings with experiments demonstrating the crucial role of archi- tectural components in preventing rank collapse.

Recent work has revealed that state space models (SSMs), while efficient for long-sequence processing, are fundamentally limited in their ability to represent formal languages—particularly due to time-invariant and real-valued recurrence structures. In this work, we draw inspiration from adaptive and structured dynamics observed in biological neural systems and introduce the Adaptive Unitary State Space Model (AUSSM): a novel class of SSMs that leverages skew-symmetric, input-dependent recurrence to achieve unitary evolution and high expressive power. Using algebraic automata theory, we prove that AUSSM can perform modulo counting and simulate solvable group automata at precision logarithmically bounded in the input length, enabling SSMs to model a broad class of regular languages out of reach for other SSM architectures. To overcome the practical inefficiencies of adaptive recurrence, we develop a separable convolution formulation and a CUDA implementation that enables scalable parallel training. Empirically, we show that AUSSM and its hybrid variant—interleaved with Mamba—outperform prior SSMs on formal algorithmic tasks such as parity and modular arithmetic, and achieve competent performance on real-world long time-series classification benchmarks. Our results demonstrate that adaptive unitary recurrence provides a powerful and efficient inductive bias for both symbolic and continuous sequence modeling. The code is available at https://github.com/arjunkaruvally/AUSSM

The impressive capabilities of Large Language Models (LLMs) have led to various efforts in enabling robots to be controlled through natural language instructions, opening exciting possibilities for human-robot interaction. The goal is for the motor-control task to be performed accurately, efficiently and safely while also enjoying the flexibility imparted by LLMs to specify and adjust the task through natural language. In this work, we demonstrate how a careful layering of an LLM in combination with a Model Predictive Control (MPC) formulation allows for accurate and flexible robotic control via natural language while taking into consideration safety constraints. In particular, we rely on the LLM to effectively frame constraints and objective functions as mathematical expressions, which are later used in the motor-control module via MPC. The transparency of the optimization formulation allows for interpretability of the task and enables adjustments through human feedback. We demonstrate the validity of our method through extensive experiments on long-horizon reasoning, contact-rich, and multi-object interaction tasks. Our evaluations show that NARRATE outperforms current existing methods on these benchmarks and effectively transfers to the real world on two different embodiments.Videos, Code and Prompts at narrate-mpc.github.io

Softmax attention is the principle backbone of foundation models for various artificial intelligence applications, yet its quadratic complexity in sequence length can limit its inference throughput in long-context settings. To address this challenge, alternative architectures such as linear attention, State Space Models (SSMs), and Recurrent Neural Networks (RNNs) have been considered as more efficient alternatives. While connections between these approaches exist, such models are commonly developed in isolation and there is a lack of theoretical understanding of the shared principles underpinning these architectures and their subtle differences, greatly influencing performance and scalability. In this paper, we introduce the Dynamical Systems Framework (DSF), which allows a principled investigation of all these architectures in a common representation. Our framework facilitates rigorous comparisons, providing new insights on the distinctive characteristics of each model class. For instance, we compare linear attention and selective SSMs, detailing their differences and conditions under which both are equivalent. We also provide principled comparisons between softmax attention and other model classes, discussing the theoretical conditions under which softmax attention can be approximated. Additionally, we substantiate these new insights with empirical validations and mathematical arguments. This shows the DSF's potential to guide the systematic development of future more efficient and scalable foundation models.