This paper introduces SICSM, a novel structural inference framework that integrates Selective State Space Models (selective SSMs) with Generative Flow Networks (GFNs) to handle the challenges posed by dynamical systems with irregularly sampled trajectories and partial observations. By utilizing the robust temporal modeling capabilities of selective SSMs, our approach learns input-dependent transition functions that adapt to non-uniform time intervals, thereby enhancing the accuracy of structural inference. By aggregating dynamics across diverse temporal dependencies and channeling them into the GFN, the SICSM adeptly approximates the posterior distribution of the system's structure. This process not only enables precise inference of complex interactions within partially observed systems but also ensures the seamless integration of prior knowledge, enhancing the model's accuracy and robustness.Extensive evaluations on sixteen diverse datasets demonstrate that SICSM outperforms existing methods, particularly in scenarios characterized by irregular sampling and incomplete observations, which highlight its potential as a reliable tool for scientific discovery and system diagnostics in disciplines that demand precise modeling of complex interactions.

This paper introduces SICSM, a novel structural inference framework that integrates Selective State Space Models (selective SSMs) with Generative Flow Networks (GFNs) to handle the challenges posed by dynamical systems with irregularly sampled trajectories and partial observations. By utilizing the robust temporal modeling capabilities of selective SSMs, our approach learns input-dependent transition functions that adapt to non-uniform time intervals, thereby enhancing the accuracy of structural inference. By aggregating dynamics across diverse temporal dependencies and channeling them into the GFN, the SICSM adeptly approximates the posterior distribution of the system's structure. This process not only enables precise inference of complex interactions within partially observed systems but also ensures the seamless integration of prior knowledge, enhancing the model's accuracy and robustness.Extensive evaluations on sixteen diverse datasets demonstrate that SICSM outperforms existing methods, particularly in scenarios characterized by irregular sampling and incomplete observations, which highlight its potential as a reliable tool for scientific discovery and system diagnostics in disciplines that demand precise modeling of complex interactions.

Recent advancements in language modeling tasks have been driven by architectures such as Transformers and, more recently, by Selective State Space Models (SSMs). In this paper, we introduce an alternative selection mechanism inspired by control theory methodologies. Specifically, we propose a novel residual generator for selection, drawing an analogy to fault detection strategies in Linear Time-Invariant (LTI) systems. Unlike Mamba, which utilizes Linear Time-Varying (LTV) systems, our approach combines multiple LTI systems, preserving their beneficial properties during training while achieving comparable selectivity. To evaluate the effectiveness of the proposed architecture, we test its performance on synthetic tasks. While these tasks are not inherently critical, they serve as benchmarks to test the selectivity properties of different cores architecture. This work highlights the potential of integrating theoretical insights with experimental advancements, offering a complementary perspective to deep learning innovations at the intersection of control theory and machine learning.

Deep Selective State-Space Models (SSMs), characterized by input-dependent, time-varying parameters, offer significant expressive power but pose challenges for stability analysis, especially with discontinuous gating signals. In this paper, we investigate the stability and regularity properties of continuous-time selective SSMs through the lens of passivity and Input-to-State Stability (ISS). We establish that intrinsic energy dissipation guarantees exponential forgetting of past states. Crucially, we prove that the unforced system dynamics possess an underlying minimal quadratic energy function whose defining matrix exhibits robust $\text{AUC}_{\text{loc}}$ regularity, accommodating discontinuous gating. Furthermore, assuming a universal quadratic storage function ensures passivity across all inputs, we derive parametric LMI conditions and kernel constraints that limit gating mechanisms, formalizing "irreversible forgetting" of recurrent models. Finally, we provide sufficient conditions for global ISS, linking uniform local dissipativity to overall system robustness. Our findings offer a rigorous framework for understanding and designing stable and reliable deep selective SSMs.

State Space Models (SSMs) have emerged as a promising alternative to the popular transformer-based models and have been increasingly gaining attention. Compared to transformers, SSMs excel at tasks with sequential data or longer contexts, demonstrating comparable performances with significant efficiency gains. In this survey, we provide a coherent and systematic overview for SSMs, including their theoretical motivations, mathematical formulations, comparison with existing model classes, and various applications. We divide the SSM series into three main sections, providing a detailed introduction to the original SSM, the structured SSM represented by S4, and the selective SSM typified by Mamba. We put an emphasis on technicality, and highlight the various key techniques introduced to address the effectiveness and efficiency of SSMs. We hope this manuscript serves as an introduction for researchers to explore the theoretical foundations of SSMs.

State-space models (SSMs) are a new class of foundation models that have emerged as a compelling alternative to Transformers and their attention mechanisms for sequence processing tasks. This paper provides a detailed theoretical analysis of selective SSMs, the core components of the Mamba and Mamba-2 architectures. We leverage the connection between selective SSMs and the self-attention mechanism to highlight the fundamental similarities between these models. Building on this connection, we establish a length independent covering number-based generalization bound for selective SSMs, providing a deeper understanding of their theoretical performance guarantees. We analyze the effects of state matrix stability and input-dependent discretization, shedding light on the critical role played by these factors in the generalization capabilities of selective SSMs. Finally, we empirically demonstrate the sequence length independence of the derived bounds on two tasks.

Personalized federated learning (PFL) studies effective model personalization to address the data heterogeneity issue among clients in traditional federated learning (FL). Existing PFL approaches mainly generate personalized models by relying solely on the clients' latest updated models while ignoring their previous updates, which may result in suboptimal personalized model learning. To bridge this gap, we propose a novel framework termed pFedSeq, designed for personalizing adapters to fine-tune a foundation model in FL. In pFedSeq, the server maintains and trains a sequential learner, which processes a sequence of past adapter updates from clients and generates calibrations for personalized adapters. To effectively capture the cross-client and cross-step relations hidden in previous updates and generate high-performing personalized adapters, pFedSeq adopts the powerful selective state space model (SSM) as the architecture of sequential learner. Through extensive experiments on four public benchmark datasets, we demonstrate the superiority of pFedSeq over state-of-the-art PFL methods.

Selective state-space models (SSMs) are an emerging alternative to the Transformer, offering the unique advantage of parallel training and sequential inference. Although these models have shown promising performance on a variety of tasks, their formal expressiveness and length generalization properties remain underexplored. In this work, we provide insight into the workings of selective SSMs by analyzing their expressiveness and length generalization performance on regular language tasks, i.e., finite-state automaton (FSA) emulation. We address certain limitations of modern SSM-based architectures by introducing the Selective Dense State-Space Model (SD-SSM), the first selective SSM that exhibits perfect length generalization on a set of various regular language tasks using a single layer. It utilizes a dictionary of dense transition matrices, a softmax selection mechanism that creates a convex combination of dictionary matrices at each time step, and a readout consisting of layer normalization followed by a linear map. We then proceed to evaluate variants of diagonal selective SSMs by considering their empirical performance on commutative and non-commutative automata. We explain the experimental results with theoretical considerations. Our code is available at https://github.com/IBM/selective-dense-state-space-model.

In this paper, we analyze the computational limitations of Mamba and State-space Models (SSMs) by using the circuit complexity framework. Despite Mamba's stateful design and recent attention as a strong candidate to outperform Transformers, we have demonstrated that both Mamba and SSMs with $\mathrm{poly}(n)$-precision and constant-depth layers reside within the $\mathsf{DLOGTIME}$-uniform $\mathsf{TC}^0$ complexity class. This result indicates Mamba has the same computational capabilities as Transformer theoretically, and it cannot solve problems like arithmetic formula problems, boolean formula value problems, and permutation composition problems if $\mathsf{TC}^0 \neq \mathsf{NC}^1$. Therefore, it challenges the assumption Mamba is more computationally expressive than Transformers. Our contributions include rigorous proofs showing that Selective SSM and Mamba architectures can be simulated by $\mathsf{DLOGTIME}$-uniform $\mathsf{TC}^0$ circuits, and they cannot solve problems outside $\mathsf{TC}^0$.

Rank collapse, a phenomenon where embedding vectors in sequence models rapidly converge to a uniform token or equilibrium state, has recently gained attention in the deep learning literature. This phenomenon leads to reduced expressivity and potential training instabilities due to vanishing gradients. Empirical evidence 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 models, 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 framework that captures both architectures. We study how a parametrized version of the classic skip connection component, which we call \emph{lambda-skip connections}, provides guarantees for rank collapse prevention. Through analytical results, we present a sufficient condition to guarantee prevention of rank collapse across all the aforementioned architectures. We also study the necessity of this condition via ablation studies and analytical examples. 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 architectural components such as skip connections and gating mechanisms in preventing rank collapse.