Linear-Time Sequence Modeling: The Post-Transformer Era and the Rise of State Space Architectures

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1. Introduction: The Quadratic Wall and the Imperative for Linearity

The trajectory of artificial intelligence over the past decade has been defined, almost exclusively, by the ascendancy of the Transformer architecture. Since its introduction in 2017, the Transformer has displaced Recurrent Neural Networks (RNNs) and Convolutional Neural Networks (CNNs) to become the de facto standard for natural language processing, computer vision, and multimodal learning. Its success is predicated on the self-attention mechanism, which allows for the modeling of global dependencies across an entire sequence, granting the model a comprehensive “receptive field” that previous architectures struggled to achieve. This capability has powered the revolution in Large Language Models (LLMs), enabling emergent behaviors in reasoning, coding, and creative generation.

However, this dominance conceals a fundamental algorithmic inefficiency that has become the primary bottleneck in the next phase of AI scaling: the quadratic complexity of the attention mechanism. The core operation of self-attention—computing the pairwise interactions between every token in a sequence—requires determining similarity scores for the query ($Q$) and key ($K$) vectors. This results in an attention matrix of size $N \times N$, where $N$ is the sequence length. Consequently, both the computational operations (FLOPs) and the memory required to store the Key-Value (KV) cache scale as $O(N^2)$.1

For the relatively short sequences that characterized early NLP tasks (e.g., 512 or 2,048 tokens), this quadratic cost was negligible compared to the benefits of parallelization. Yet, as the field moves toward “long-context” applications—processing entire genomic sequences, analyzing legal or financial repositories with millions of tokens, or maintaining persistent memory in conversational agents—the quadratic cost becomes prohibitive. Processing a sequence of 100,000 tokens requires not 100 times more compute than a sequence of 1,000 tokens, but 10,000 times more. This scaling law effectively imposes a “Quadratic Wall,” limiting the feasibility of Transformers for truly long-range sequence modeling on commercially available hardware.3

This report provides an exhaustive analysis of the architectural paradigm shift currently underway to dismantle this wall. We investigate the resurgence of State Space Models (SSMs) and their derivatives—architectures that promise the “holy grail” of deep learning: the modeling capability of Transformers combined with the linear $O(N)$ scaling of RNNs. We dissect the theoretical foundations of models like Mamba, RWKV, Hyena, and RetNet, analyzing how they navigate the trade-offs between memory efficiency, training parallelization, and recall capability. Furthermore, we scrutinize the emergence of Hybrid Architectures like Jamba, which seek to bridge the gap between the two paradigms, and evaluate the theoretical limitations of SSMs regarding the “Recall Problem.” Through a synthesis of recent technical reports, benchmark data, and theoretical proofs, this document delineates the post-Transformer landscape.

 

2. The Computational Dynamics of Sequence Modeling

To understand the significance of State Space Models, one must first deeply understand the inefficiencies they aim to correct. The limitation of the Transformer is not merely theoretical; it manifests as concrete bottlenecks in memory bandwidth and latency during both training and inference.

 

2.1 The Attention Mechanism and the KV Cache Bottleneck

In a standard Transformer, the attention mechanism computes outputs based on the entire history of inputs. During training, this is parallelizable, as all tokens are available simultaneously. This “training parallelism” was the primary advantage of Transformers over RNNs, which required sequential processing that left GPUs underutilized. However, during inference (text generation), the Transformer becomes autoregressive. To generate the next token $x_{t+1}$, the model must attend to all previous tokens $x_0, \dots, x_t$.

To avoid recomputing the representations for previous tokens at every step, these representations are stored in the Key-Value (KV) Cache. As the sequence length $N$ grows, this cache grows linearly in size but must be accessed in its entirety for every new token generated. This creates a memory bandwidth bottleneck. The GPU spends more time moving the massive KV cache from High Bandwidth Memory (HBM) to the compute units than it does performing the actual matrix multiplications.2

For extremely long sequences (e.g., 1 million tokens), the KV cache can exceed the VRAM capacity of even the most advanced data center GPUs (like the NVIDIA H100), necessitating complex distributed inference setups purely to hold the model’s “short-term memory.” This phenomenon effectively renders standard Transformers unusable for tasks requiring massive context windows without significant approximations (like sparse attention) that often degrade performance.5

 

2.2 The Promise of Linear Recurrence

 

The alternative paradigm is Linear Recurrence. Recurrent Neural Networks (RNNs) like LSTMs process sequences sequentially, maintaining a hidden state $h_t$ that acts as a compressed summary of the past. To generate the next token, the RNN only needs the current input $x_t$ and the previous state $h_{t-1}$. The inference cost is $O(1)$—constant time—regardless of whether the sequence length is 10 or 10 million. The memory footprint is also constant, determined by the fixed size of the hidden state vector.6

Historically, RNNs failed to scale because:

  1. Sequential Training: They could not be trained in parallel, making them excruciatingly slow to train on large datasets.
  2. The Vanishing Gradient: Compressing a long sequence into a fixed state vector caused information from early tokens to decay or “vanish,” preventing the model from learning long-range dependencies.

Modern State Space Models (SSMs) are designed to solve these two specific historical failures while retaining the inference efficiency of RNNs. They achieve this through a combination of control theory (to handle memory) and novel algorithms (to enable parallel training).8

 

3. Theoretical Foundations: From Continuous Systems to Discrete Selection

 

The theoretical lineage of modern SSMs like Mamba does not trace back to the LSTM or the GRU, but rather to classical control theory and signal processing. Understanding Mamba requires understanding the continuous-time dynamics that underpin it.

 

3.1 The Continuous-Time State Space Model

 

A State Space Model describes a physical system where an input signal $x(t)$ is mapped to an output signal $y(t)$ through a latent state $h(t)$. In continuous time, this is defined by the linear ordinary differential equation (ODE):

 

$$h'(t) = A h(t) + B x(t)$$

 

$$y(t) = C h(t) + D x(t)$$

Here, $h(t)$ is the state vector (the “memory” of the system).

  • $A$ is the State Matrix (evolution parameter), governing how the state evolves over time.
  • $B$ is the Input Matrix, governing how the input influences the state.
  • $C$ is the Output Matrix, governing how the state translates to the output.
  • $D$ is the Feedthrough Matrix, representing a direct connection from input to output (often a residual connection).

In classical engineering, $A, B, C, D$ are fixed matrices defining a static system (like a spring-mass damper). In Deep Learning, specifically in models like the Structured State Space Sequence (S4) model, these matrices are learned parameters of a neural network.8

 

3.2 Discretization: The Bridge to Digital Sequences

 

Since language text, DNA sequences, and audio samples are discrete data points rather than continuous signals, the continuous ODE must be discretized. This transformation is critical; it provides the mathematical link between the continuous theory (which offers elegant properties for handling long-range dependencies) and the discrete implementation required for digital computers.

Discretization introduces a Time Scale parameter $\Delta$ (delta). Using the Zero-Order Hold (ZOH) method, the continuous parameters ($A, B$) are transformed into discrete parameters ($\bar{A}, \bar{B}$) 2:

 

$$\bar{A} = \exp(\Delta A)$$

 

$$\bar{B} = (\Delta A)^{-1} (\exp(\Delta A) – I) \cdot \Delta B$$

The resulting discrete system follows the recurrence relation:

 

$$h_t = \bar{A} h_{t-1} + \bar{B} x_t$$

 

$$y_t = C h_t$$

This equation highlights the dual nature of SSMs:

  1. Recurrent View: $h_t$ depends on $h_{t-1}$. This allows for $O(1)$ inference, identical to an RNN.
  2. Convolutional View: If the matrices $\bar{A}$ and $\bar{B}$ are constant across time (Linear Time Invariant, or LTI), the recurrence can be unrolled into a single global convolution. The state $h$ essentially “convolves” the input sequence $x$ with a filter derived from $A$ and $B$. This allows for $O(N \log N)$ training using Fast Fourier Transforms (FFT), solving the “Sequential Training” bottleneck of traditional RNNs.6

 

3.3 The HiPPO Matrix: Solving Long-Term Memory

 

The “Vanishing Gradient” problem in standard RNNs arose because repeated multiplication by a matrix (during recurrence) causes gradients to explode or vanish if the eigenvalues are not carefully controlled. The S4 model addressed this using the HiPPO (High-Order Polynomial Projection Operators) matrix.

The HiPPO matrix initializes the state matrix $A$ in a specific mathematical form that guarantees the state $h(t)$ acts as an optimal compression of the history of inputs $x(t)$ using orthogonal polynomials (like Legendre polynomials). This provides a mathematical guarantee that the system preserves information over extremely long timescales, allowing S4 models to handle dependencies spanning tens of thousands of steps—something LSTMs could never achieve reliably.9

 

4. Mamba: The Selective State Space Model

 

While S4 solved the memory and training issues, it had a critical flaw for language modeling: it was Linear Time Invariant (LTI). In S4, the matrices $A, B, C$ are constant. The model processes every token with the exact same dynamics. This is insufficient for language, which requires Content-Based Reasoning. A model needs to react differently to the word “not” than to the word “apple.” It needs to selectively “remember” a name mentioned 5,000 tokens ago while “forgetting” the filler words in between.

Mamba, introduced by Gu and Dao (2023), represents a paradigm shift by introducing Selectivity into the SSM framework.3

 

4.1 The Selection Mechanism

 

In Mamba, the parameters $B, C,$ and $\Delta$ are no longer static learned weights. They are functions of the current input $x_t$.

 

$$B_t = \text{Linear}(x_t)$$

 

$$C_t = \text{Linear}(x_t)$$

 

$$\Delta_t = \text{Softplus}(\text{Parameter} + \text{Linear}(x_t))$$

This simple change has profound implications:

  • Dynamic Focus: The model can modulate $\Delta_t$ to control the “step size” of the memory update. A large $\Delta_t$ means the model focuses heavily on the current input $x_t$ and overwrites the old state (short-term focus). A small $\Delta_t$ means the model ignores the current input and preserves the existing state (long-term memory).13
  • Input-Dependent Gating: The interaction between $B_t$ (input projection) and $C_t$ (output projection) allows the model to selectively admit information into the state or filter it out based on the content of the token itself.

This mechanism aligns Mamba’s capabilities closer to the Gated Recurrent Unit (GRU) or LSTM, but with the rigorous long-memory foundations of the SSM. However, making the parameters time-varying destroys the Convolutional equivalence. Since the kernel changes at every timestep, FFTs can no longer be used for parallel training.

 

4.2 The Hardware-Aware Parallel Scan

 

To retain the training speed of Transformers without the convolution trick, Mamba employs a Hardware-Aware Parallel Scan algorithm. A “scan” (or prefix sum) operation is associative, meaning the order of operations can be grouped:

 

$$(a \cdot b) \cdot c = a \cdot (b \cdot c)$$

 

This property allows the sequential recurrence to be parallelized across the GPU threads using a tree-based reduction algorithm.14

Crucially, the Mamba implementation uses Kernel Fusion. The naive approach would be to compute the dynamic matrices $B_t, C_t, \Delta_t$ for all time steps and store them in GPU HBM (High Bandwidth Memory). This would be prohibitively slow due to memory I/O. Mamba’s kernel loads the inputs into the GPU’s ultra-fast SRAM (Static Random Access Memory), performs the discretization and parallel scan entirely within SRAM, and writes only the final result back to HBM. This avoids the memory bandwidth bottleneck that plagues Transformers, where the massive $N \times N$ attention matrix must be materialized.11

This architectural choice allows Mamba to achieve 5x higher inference throughput than Transformers and scale linearly with sequence length, effectively breaking the quadratic wall.13

 

4.3 The Unified Mamba Block

 

The Mamba architecture also simplifies the neural network block structure. A standard Transformer block consists of two distinct sub-blocks: Multi-Head Attention (MHA) and a Feed-Forward Network (MLP). Mamba consolidates this into a single Unified Mamba Block.3

The data flow in a Mamba block is as follows:

  1. Input Projection: The input sequence (dimension $D$) is expanded to a larger dimension (usually $2D$) via a linear projection.
  2. Convolution: A short 1D convolution (kernel size usually 3 or 4) is applied. This replaces the positional encodings of Transformers, providing the model with local context awareness before the state space processing.
  3. SSM Core: The selective state space operation is applied (the parallel scan).
  4. Gating: A multiplicative gate (SiLU activation) modulates the flow of information, akin to the Gated Linear Unit (GLU).
  5. Output Projection: The dimension is projected back down to $D$.

This homogeneous architecture simplifies model design and scaling, as there is no need to balance the ratio of Attention to MLP parameters.

 

5. Mamba-2: State Space Duality and Tensor Optimization

 

While Mamba-1 proved that SSMs could compete with Transformers, Mamba-2 (released in 2024) refined the theoretical understanding and computational efficiency further through the framework of Structured State Space Duality (SSD).16

 

5.1 The Theory of Duality

 

Mamba-2 posits that Structured State Space Models and Linear Attention are not merely competitors but are mathematically dual forms of the same underlying operation. The authors demonstrate that the recurrence of an SSM can be rewritten as a matrix multiplication involving a specific type of structured matrix: a semiseparable matrix.10

In Linear Attention, the attention score is computed without the Softmax normalization (or with a simplified kernel). Mamba-2 restricts the state matrix $A$ to be a diagonal structure (specifically, scalar times identity in the SSD formulation). This restriction allows the interaction matrix to be decomposed into block-diagonal components. A lower-triangular matrix is $N$-semiseparable if all submatrices contained in its lower triangular part are of at most rank $N$. Mamba-2 utilizes 1-semiseparable matrices, which implies a high degree of structure and redundancy that can be exploited for efficiency.10

This duality allows Mamba-2 to flexibly choose the most efficient computation mode based on the task:

  • Linear Mode (Recurrent): Efficient for autoregressive inference ($O(1)$ per step), functioning like an SSM.
  • Quadratic Mode (Dual): Efficient for training short sequences using massive matrix multiplication, functioning like Linear Attention.

 

5.2 Block Matrix Decomposition and Tensor Cores

 

A limitation of Mamba-1’s scan algorithm was its reliance on element-wise operations. Modern GPUs (like the NVIDIA H100) are specialized for Matrix-Matrix multiplications (GEMM) using Tensor Cores, which offer vastly higher throughput than standard arithmetic units. Mamba-1’s scan could not fully utilize these Tensor Cores.17

Mamba-2’s SSD formulation allows the computation to be cast as a series of block matrix multiplications (Block Matrix Decomposition). The algorithm breaks the sequence into chunks, computes local interactions within chunks using Attention-like matrix multiplies (utilizing Tensor Cores), and then propagates the state between chunks using the SSM recurrence.16

This shift allows Mamba-2 to achieve significantly higher training throughput than Mamba-1, despite similar theoretical complexity. The implementation is also drastically simplified; the “SSD Minimal” code fits in roughly 25 lines of PyTorch, eschewing the complex custom CUDA kernels required for Mamba-1.8

Additionally, Mamba-2 reintroduces a “head” dimension ($P$), similar to Multi-Head Attention. By processing independent state spaces in parallel heads (e.g., 64 or 128 heads), Mamba-2 aligns more closely with the memory layout optimizations of FlashAttention, further boosting speed.19

 

6. The Cambrian Explosion of Linear Architectures

 

Mamba is the most prominent, but not the only, architecture vying to replace the Transformer. The field is currently witnessing a “Cambrian Explosion” of linear-time architectures, each taking a slightly different mathematical path to the same goal.

 

6.1 RWKV: The RNN That Thinks It’s a Transformer

 

RWKV (Receptance Weighted Key Value) is a parallelizable RNN that has gained significant traction in the open-source community, currently in its 6th iteration (RWKV-6 “Finch”).9

Mechanism: RWKV reformulates the linear attention mechanism into a recurrence using four primary vectors: $R$ (Receptance), $W$ (Weight/Decay), $K$ (Key), and $V$ (Value).

  • WKV Operator: The core is the “WKV” mechanism, which accumulates information with a time-decay factor $w = \exp(-\text{decay})$. This ensures older information fades exponentially, maintaining stability.21
  • Token Shift: A unique feature of RWKV is “Token Shift,” where the input at step $t$ is linearly mixed with the input at $t-1$. This acts as a lightweight “time-mixing” mechanism that requires negligible compute but provides local context, similar to the convolution in Mamba.20
  • Architecture: RWKV blocks are divided into Time-Mixing (analogous to Attention, handling temporal dependencies) and Channel-Mixing (analogous to Feed-Forward Networks, handling feature transformation). Both operate in linear time.23

Evolution to RWKV-6: Early versions of RWKV had static decay rates, limiting their ability to recall specific information (similar to S4). RWKV-6 introduces Data-Dependent Decays (using Low-Rank Adaptation, or LoRA, to modulate the weights based on context), significantly boosting expressivity and aligning it closer to Mamba’s selective capability.20 Benchmarks in vision tasks (RWKV-SAM) indicate that RWKV can outperform Mamba in inference speed for high-resolution image segmentation due to its simpler CUDA kernel implementation.24

 

6.2 Hyena: Implicit Global Convolutions

 

Hyena takes a different approach, aiming to replace Attention entirely with Long Global Convolutions.25

Mechanism: Instead of a recurrent state space, Hyena learns a convolution filter that is as long as the input sequence. Learning a filter of size $N$ parameter-by-parameter is inefficient. Hyena solves this by Implicit Parametrization: the filter weights are generated by a small neural network (an MLP) based on positional embeddings. This allows an effectively infinite context window defined by a small number of parameters.26

  • Gating: Hyena interleaves these long convolutions with element-wise multiplication gates (data-controlled gating), mimicking the “projection” aspect of Attention without the cost of the $N \times N$ matrix.

Comparison to Mamba: While Hyena is sub-quadratic ($O(N \log N)$ due to FFTs), it is fundamentally a convolutional model. This means that during inference, it requires a cache of the previous inputs (size $N$) to compute the convolution, whereas Mamba compresses the history into a fixed state (size $D$). Thus, Hyena has a larger memory footprint during inference than Mamba or RWKV.27 However, for “prefill” or parallel processing of long prompts, Hyena operators are claimed to be 100x faster than FlashAttention at sequence lengths of 100k.26

 

6.3 RetNet: The “Successor” Architecture

 

RetNet (Retentive Network), proposed by Microsoft Research, introduces the “Retention” mechanism.28

Mechanism: Retention is designed to support three computation paradigms: parallel (for training), recurrent (for inference), and chunkwise recurrent (for long-sequence efficiency).

  • Multi-Scale Decay: Unlike Mamba’s input-dependent $\Delta$, RetNet uses a fixed, multi-scale decay structure. Different heads have different fixed decay rates (e.g., one head decays quickly, another slowly). This makes the model structurally simpler and easier to optimize but potentially less expressive than Mamba’s fully dynamic selection.28
  • Positioning: RetNet is often described as “Attention with a complex-valued exponential decay bias.” It retains the closest architectural similarity to Transformers, positioning it as a conservative but stable “successor”.30

 

6.4 Comparative Analysis of Architectures

 

The following table summarizes the key differences between these architectures:

Feature Transformer Mamba (SSM) RWKV-6 (RNN) Hyena (Conv) RetNet
Time Complexity $O(N^2)$ $O(N)$ $O(N)$ $O(N \log N)$ $O(N)$
Inference State Linear ($N$) Constant ($D$) Constant ($D$) Linear ($N$) Constant ($D$)
Core Mechanism Attention ($QK^T$) Selective Scan WKV + Token Shift Implicit Convolution Retention
Input-Dependent? Yes (Full) Yes (Selection) Yes (Decay LoRA) Yes (Gating) Partial (Fixed Decay)
Training Parallel Parallel (Scan) Parallel Parallel (FFT) Parallel
Key Strength Exact Retrieval Efficiency + Selectivity Production Ready Long Context Stability

 

7. The Hybrid Frontier: Jamba and the “Best of Both Worlds”

 

Despite the theoretical elegance of pure SSMs, empirical rigor has revealed limitations. Pure SSMs struggle with tasks requiring precise retrieval of information from the distant past (“associative recall”) and “in-context learning” (ICL) tasks where the model must learn a pattern from the prompt itself.27 This has led to the emergence of Hybrid Architectures, which interleave SSM layers with traditional Attention layers to balance efficiency with recall capability.

 

7.1 The Jamba Architecture

 

Jamba, developed by AI21 Labs, is the premier example of this hybrid approach.8 Jamba recognizes that while Mamba is excellent at compressing context, Attention is superior at retrieving specific details.

Architecture Specifics:

  • Layer Ratio: Jamba employs a generic 1:7 ratio (specifically in Jamba 1.5, there is one Transformer layer for every seven Mamba layers).33 This ratio was empirically determined to be the “sweet spot” where the computational cost remains dominated by the efficient Mamba layers, but the few Attention layers provide enough “global checkpoints” to maintain high-quality retrieval.
  • Mixture-of-Experts (MoE): To scale parameters without exploding the active compute cost, Jamba utilizes a MoE architecture. The Jamba 1.5 Large model has 398 billion total parameters but only 94 billion active parameters per token. It uses 16 experts and routes tokens to the top 2 experts.34 This allows the model to have massive capacity while fitting on standard hardware clusters.

Performance and Quantization:

  • Context Window: The hybrid design enables an effective context length of 256,000 tokens.33
  • ExpertsInt8: Jamba 1.5 introduces a novel quantization technique called ExpertsInt8. Since 85% of the weights are in the MoE layers (which are bandwidth-bound), quantizing these to Int8 while keeping the critical Mamba/Attention weights in higher precision allows the 94B active parameter model to fit on a single machine with 8x 80GB GPUs.34
  • Throughput: Jamba 1.5 demonstrates a 3x throughput advantage over Mixtral 8x7B on long contexts and achieves a 10x reduction in KV cache memory compared to a dense Transformer.33

 

7.2 Pure vs. Hybrid: The Falcon-Mamba Debate

 

While Jamba advocates for hybrids, the Falcon-Mamba 7B model (released by the Technology Innovation Institute, TII) argues that pure SSMs can still be competitive.37

The Argument: Falcon-Mamba is a pure Mamba architecture (specifically an architecture similar to Mamba-1/Mamba-2 hybrid concepts but without Attention layers). TII claims it outperforms Llama 3.1 8B on general leaderboards and is the first “pure” SSM to do so.38

The Benchmark Reality:

While Falcon-Mamba performs well on standard language modeling tasks (Arc, Hellaswag), a deeper look at the Open LLM Leaderboard reveals significant gaps. On tasks like IFEval (Instruction Following Evaluation) and GPQA (Graduate-Level Google-Proof Q&A), Falcon-Mamba lags significantly behind hybrid models and Transformers.39

  • IFEval Score: Falcon-Mamba 7B scores roughly 33.36, while Llama-3-8B scores significantly higher (often >60).
  • Implication: This reinforces the theory that while SSMs capture syntax, semantics, and “vibes” well, they struggle with complex, logic-heavy instruction following that benefits from the global comparison capabilities of Attention.

 

7.3 Distillation: “Mamba-in-the-Llama”

 

Another avenue for creating efficient models is Distillation. The “Mamba-in-the-Llama” research explores distilling a large, pre-trained Transformer (the teacher) into a hybrid Linear RNN (the student).41

Process:

  1. Seed Prompt Generation: Create a diverse set of prompts.
  2. Supervised Finetuning: Train the student model to mimic the teacher’s outputs.
  3. Distilled Alignment: Use the teacher to score the student’s responses, refining the student’s policy.

Findings: A hybrid Mamba model retaining just 50% of the Attention layers matches the teacher’s performance on chat benchmarks (MT-Bench). However, reducing the Attention ratio to 12.5% or 0% (pure Mamba) causes a degradation in performance, further supporting the hybrid hypothesis.42

 

8. The “Recall Problem”: Theoretical Limits and Solutions

 

To provide a nuanced understanding, one must address why SSMs are not yet a universal replacement for Transformers. The core issue lies in the State Capacity and the Recall Problem.

 

8.1 Associative Recall vs. Joint Recall

 

A standard synthetic benchmark for testing sequence models is Associative Recall (AR). The task is to retrieve a value associated with a key provided earlier in the sequence (e.g., “A:1, B:2,… A:?”).

  • Transformers: Solve this easily. The attention mechanism acts as a “Look-Up Table,” comparing the query “A” with all previous keys to find “A” and return “1”. This is an $O(1)$ operation in terms of “reasoning steps” (one layer) and is robust to sequence length.27
  • SSMs: Must compress the pair “A:1” into the fixed-size state $h_t$. As the sequence length grows and more pairs are added, the state $h_t$ becomes “saturated.” The noise generated by intervening tokens can wash out the signal of “A:1” unless the selection mechanism is perfectly tuned.27

However, recent research argues that AR is too simple. Real-world complexity is better modeled by Joint Recall, where the value of a key depends on the context (e.g., “In Context 1, A=1. In Context 2, A=2…. Context 1, A=?”).

  • Theoretical Proof: Zhan et al. (2025) mathematically prove that pure SSMs cannot solve the Multi-Query Joint Recall task in sub-quadratic time. They lack the expressiveness to dynamically route information based on multiple previous contexts simultaneously without a massive expansion of the state dimension, which would negate their efficiency capability.44

 

8.2 The “Copying” Gap and “Zoology”

 

Empirical studies, such as those in the “Zoology” paper, confirm that Gated Convolutions (H3, Hyena) and even Mamba struggle with Copying Tasks—replicating a random string of characters verbatim from the context—when the string length exceeds the training distribution.12

  • Induction Heads: Transformers develop “Induction Heads”—specialized attention heads that look for the previous occurrence of the current token and copy the token that followed it. This mechanism is trivially easy for Attention but difficult for a recurrent state to simulate perfectly over long distances.45

 

8.3 Solution: Context-Dependent Sparse Attention (CDSA)

 

To bridge this gap without reverting to full quadratic Attention, researchers have proposed Context-Dependent Sparse Attention (CDSA) and architectures like HAX (Hashing Attention with sparse Key Selection).44

Mechanism:

  • HAX: This architecture integrates an SSM (for general context) with a sparse attention mechanism based on Locality Sensitive Hashing (LSH).
  • Logic: The SSM handles the “gist” and syntax of the document. When the model encounters a query requiring precise retrieval, the LSH mechanism hashes the query and retrieves a small bucket of relevant keys from the history.
  • Efficiency: This allows for retrieval of specific “hard” memories in $O(N \log N)$ or near-linear time, solving the Joint Recall problem theoretically and empirically outperforming pure SSMs on benchmarks.44

Associative Treecall (ATR): Further theoretical work introduces “Associative Treecall,” a task requiring the model to infer hierarchical structure (like a parse tree) from the sequence. Experiments show that while Transformers and Mamba (due to its selection) can handle this, LTI SSMs (like H3) fail completely, highlighting the necessity of the input-dependent selection mechanism for understanding language structure.12

 

9. Benchmarking and Performance Characterization

 

A systematic characterization of these models on practical hardware reveals the trade-offs between theoretical complexity and real-world throughput.

 

9.1 Throughput and Latency on Hardware (H100 vs A100)

 

The choice of hardware significantly impacts the performance gain of SSMs.

  • A100 GPU: On an NVIDIA A100, Mamba achieves roughly 5x higher throughput than Transformers for long sequences. The primary bottleneck for Transformers is the memory bandwidth required to load the KV cache.48
  • H100 GPU: The NVIDIA H100 has vastly more memory bandwidth and specialized Transformer Engines. However, Mamba-2’s optimization for Tensor Cores allows it to scale even better on H100s. For a batch size of 64, an H100 can process tokens significantly faster than an A100, but the relative gap between Mamba and Transformers widens as sequence length increases because the H100’s HBM is still finite.
  • Latency: For short sequences (<2k tokens), Mamba and Transformers have comparable latency. In some unoptimized implementations, Mamba can be slightly slower due to the overhead of launching custom kernels (kernel launch latency). However, for sequences >10k tokens, Mamba’s latency remains flat while Transformer latency spikes.49

 

9.2 Memory Efficiency and the KV Cache

 

  • Transformer: For a Llama-3-70B model with a 128k context, the KV cache can consume over 100GB of VRAM. This necessitates multi-GPU setups (model parallelism) just to hold the cache, even if the weights fit on one GPU.
  • SSM: Mamba requires a fixed-size state (e.g., 16MB) regardless of sequence length. This allows running very long context jobs on consumer hardware (e.g., RTX 4090) or single server-grade GPUs. This is the “killer feature” for edge AI and decentralized inference.4

 

9.3 Domain-Specific Performance

 

  • Genomics: In genomic analysis, where sequences (DNA strands) can be millions of tokens long, Mamba outperforms Transformers significantly. The dependencies in DNA are often extremely long-range, and the fixed vocabulary size makes the “copying” issue less prevalent than in natural language.50
  • Audio Generation (SC09): Mamba has been shown to excel in audio generation benchmarks like SC09, where it matches the performance of complex Transformers but generates samples much faster. The continuous nature of audio signals aligns well with the control-theory roots of SSMs.51
  • Edge AI: For ultra-low-power edge devices (e.g., BrainChip’s neuromorphic hardware), SSMs are ideal because they do not require massive external memory access (DRAM) to fetch a KV cache. The state can be kept in on-chip SRAM, drastically reducing energy consumption per token.52

 

10. Conclusion

 

The era of “Attention Is All You Need” is evolving into a more nuanced landscape where “Attention is Expensive, and State Space is Efficient.”

State Space Models, led by Mamba and Mamba-2, have successfully dismantled the Quadratic Wall that threatened to stall the scaling of context windows. By reintroducing recurrence with a hardware-aware, input-dependent twist, they offer a path to processing millions of tokens on a single GPU. RWKV proves that this can be done with the stability of standard RNNs, while Hyena explores the limits of convolutional scaling.

However, the “Free Lunch” theorem applies. The theoretical inability of pure SSMs to perform exact Joint Recall and Copying is a hard limit for applications requiring perfect fidelity (like citing specific case law or debugging code). This reality has crowned Hybrid Architectures like Jamba as the current pragmatic state-of-the-art for enterprise applications. By mixing a heavy dose of Mamba for compression with a light sprinkling of Attention for retrieval, hybrids achieve the “best of both worlds.”

Looking forward, the convergence of these architectures—formalized in the SSD framework—suggests that future models will not be “Transformers” or “SSMs,” but composite systems. They will likely employ dynamic routing (MoE) to switch between linear scan modes for rote generation and quadratic attention modes for complex reasoning, all accelerated by hardware that is finally catching up to the math of recurrence. The post-Transformer era has not eliminated the Transformer; it has simply contextualized it as one powerful tool in a rapidly expanding linear arsenal.