Multi-Matrix Factorization Attention (MFA)

• Multi-Matrix Factorization Attention (MFA) is an attention mechanism that factorizes the query-key interaction using multiple low-rank matrices to reduce computational costs and memory usage.
• It extends traditional multi-head attention by incorporating MFA-Key-Reuse, which reuses key representations for value projections to significantly cut down on KV cache requirements.
• Empirical evaluations demonstrate that MFA and its extensions maintain competitive performance while achieving remarkable efficiency, making them ideal for large-scale and resource-constrained deployments.

Multi-Matrix Factorization Attention (MFA) is an attention mechanism that extends the conventional multi-head attention paradigm by introducing multi-matrix low-rank factorization in the query-key (QK) interaction phase of transformer models (Hu et al., 26 Dec 2024). This design enhances both the representational capacity and the computational efficiency of attention layers, particularly under tight key-value (KV) cache constraints, and enables further optimization via architectural extensions such as MFA-Key-Reuse (MFA-KR).

1. Conceptual Foundation and Motivation

Attention mechanisms form the backbone of modern transformer architectures by enabling models to selectively focus on different parts of the input sequence. In classical Multi-Head Attention (MHA), each attention head computes full-rank projections for queries (Q), keys (K), and values (V), resulting in an attention matrix per head. However, as the number and dimensionality of heads scale, especially for large context windows or resource-limited environments, the memory requirements for the key-value (KV) cache and the computational costs become prohibitive.

MFA addresses this by rethinking the QK product. Rather than computing an attention matrix via a single large linear transformation per head, MFA leverages multiple low-rank factorized projections that collectively approximate the full QK interaction. This approach reduces parameter redundancy and memory usage, while preserving modeling capacity, particularly when operating under stringent KV cache budgets.

2. Architectural Design and Mathematical Formulation

At the heart of MFA is the replacement of standard full-rank QK projections with multi-matrix low-rank factorizations. The standard QK interaction for head i in MHA is:

$$A_i = \text{softmax}\left(\frac{Q W_i^{Q} (K W_i^{K})^\top}{\sqrt{d}}\right)$$

MFA replaces each full-rank $W_i^{QK}$ with a composition of multiple matrices $U$ and $V$, yielding a low-rank factorization:

$Q W^{QK} K^\top \approx Q U V^\top K^\top$

or with explicit low-rank projections,

$$A = \text{softmax}\left(\frac{(Q W^U) (V W^V)^\top}{\sqrt{r}}\right)$$

where $r$ is the rank of the factorization, and $W^U$, $W^V$ are learnable projection matrices. Each attention head is thus informed by a combination of low-rank matrix products that efficiently encode the query-key affinity. The composition of multiple factorized terms enables the model to capture richer interactions as the number of factors increases.

The MFA-KR (Key-Reuse) extension further optimizes resource usage by repurposing the key cache as the value projection. Instead of computing and storing separate key and value projections, MFA-KR introduces shared, re-parameterized representations such that:

• Key representations are computed once and reused in value computation.
• Value projections leverage the already cached key data, reducing redundancy and cumulative KV cache footprint.

This is achieved through reparameterization in the projection layer, enabling significant memory savings during both training and inference.

3. Computational and Memory Efficiency

MFA’s use of low-rank factorization brings a substantial reduction in both the parameter count and the operational footprint within the attention module. The explicit benefit is most visible in the size of the KV cache required for long-context processing:

• Standard MHA: Each head stores full-dimensional keys and values; cache requirements scale linearly with sequence length, number of heads, and head dimension.
• MFA: By factorizing QK into lower-dimensional projected factors, the size and update cost of the KV cache is proportionally reduced, with the modeling effects of multiple heads retained.
• MFA-KR: By reusing key representations for the value projection, MFA-KR reduces the cache size even further (by up to 93.7% compared to MHA, and up to 56% compared to MLA), enabling efficient inference for long sequences and large batch scenarios.

In addition, these reductions enable the scaling of head dimension and count without the multiplicative memory increase seen in classic MHA settings.

4. Empirical Performance and Comparative Analysis

Extensive empirical evaluations benchmark MFA and MFA-KR against Multi-Linear Attention (MLA) and standard MHA on large-scale tasks (Hu et al., 26 Dec 2024). Main findings include:

• Model capacity: MFA and MFA-KR match or surpass MHA and MLA in benchmark performance despite significantly reduced memory usage.
• KV cache constraints: MFA maintains strong or superior accuracy under strict cache budgets, where standard approaches degrade notably.
• Throughput and latency: MFA and MFA-KR demonstrate favorable speedups due to the reduced number of operations and parameter-sharing mechanisms.

These results confirm that factorizing the QK computation does not impair, and may indeed enhance, the expressive power of the attention mechanism under practical resource constraints.

5. Applicability Across Domains

MFA’s reduced computational and memory footprint, coupled with strong model expressivity, makes it well-suited to diverse deployment environments:

• LLMs and NLP: Enabling longer input sequences and more attention heads per resource unit, which is critical for scaling up model context or batch size.
• Computer vision transformers and multimodal architectures: MFA allows high-resolution and multi-stream features to be processed without excessive growth in KV cache, directly addressing challenges in real-time or edge computing scenarios.
• Resource-constrained deployment: The architecture’s efficiency translates to improved inference speed and lower DRAM usage, facilitating deployment on edge devices, mobile hardware, or large-scale serving infrastructure.

6. Extensions and Future Directions

Several avenues for advancing MFA are highlighted:

• Dynamic rank selection: Investigating adaptive, data-dependent schemes for selecting the factorization rank $r$ could yield models with further enhanced efficiency-adaptivity trade-offs.
• Further integration with retrieval or sparse attention: Combining MFA’s low-memory cost with sparse or block-sparse attention architectures might unlock even more efficient long-context modeling.
• Nonlinear or hierarchical factorizations: Exploring nonlinear matrix factorization or scale-specific multi-matrix strategies may further increase the adaptability of MFA in heterogeneous input regimes.

A plausible implication is that MFA’s combination of low-rank efficiency and competitive accuracy could drive a new generation of transformer architectures optimized specifically for long-context and memory-bound applications.

7. Summary Table: MFA and Related Attention Variants

Attention Mechanism	QK Interaction Structure	KV Cache Efficiency	Performance Under Constraint
MHA	Full-rank per head	Baseline	High, degrades sharply when memory constrained
MLA	Multi-linear, structured	Intermediate	Degrades under stricter budgets
MFA	Multi-matrix low-rank factorization	High (up to 56% savings)	Maintains strong performance
MFA-KR	Key-reuse with reparam. value proj.	Very high (up to 93.7%)	Slight trade-off, still strong

MFA and MFA-KR distinguish themselves by optimal trade-offs between accuracy and resource usage, enabling state-of-the-art attention-based modeling across tightly constrained and large-scale environments.