Beyond Binary: Refining Weight Representation for Extreme Model Compression

Author: Denis Avetisyan

A new approach to quantizing large language models focuses on preserving magnitude information within weight matrices, leading to improved performance at extremely low bitrates.

Under a 1.5-bit quantization setting applied to the LLaMA2 7B model, experiments demonstrate that increasing the envelope rank-which enhances magnitude expressivity while maintaining shared sign bases-consistently reduces reconstruction error across multiple Transformer blocks and modules, while conversely, increasing decomposition depth-adding further decompositions and sign bases-often degrades performance at a fixed bit allocation, ultimately identifying a configuration of <span class="katex-eq" data-katex-display="false">(l^{\ast},P^{\ast})=(16,1)</span> as optimal. — Under a 1.5-bit quantization setting applied to the LLaMA2 7B model, experiments demonstrate that increasing the envelope rank-which enhances magnitude expressivity while maintaining shared sign bases-consistently reduces reconstruction error across multiple Transformer blocks and modules, while conversely, increasing decomposition depth-adding further decompositions and sign bases-often degrades performance at a fixed bit allocation, ultimately identifying a configuration of $(l^{\ast},P^{\ast})=(16,1)$ as optimal.

Multi-Envelope Double Binary Factorization enhances magnitude representation rather than solely increasing sign diversity for effective low-bit quantization.

Achieving extreme model compression via low-bit quantization often sacrifices accuracy due to limitations in representing weight magnitudes. This paper, ‘More Than Bits: Multi-Envelope Double Binary Factorization for Extreme Quantization’, addresses this challenge by introducing Multi-envelope Double Binary Factorization (MDBF), a novel approach that prioritizes expressive magnitude representation alongside sign diversity. By employing a rank-$l$ envelope and shared sign bases, MDBF enhances performance across LLaMA and Qwen models while maintaining deployment efficiency. Could this refined factorization method unlock even more substantial gains in LLM compression and accelerate on-device intelligence?

The Computational Limits of Scale in Large Language Models

Large Language Models have rapidly become central to advancements in Natural Language Processing, powering applications from automated translation to sophisticated chatbot interactions. However, this progress comes at a substantial cost: these models are extraordinarily large, often containing billions of parameters. This immense scale directly translates into significant computational demands during both training and inference, requiring powerful hardware and substantial energy consumption. The sheer number of parameters also necessitates vast amounts of memory, posing a major challenge for deployment on edge devices or in resource-constrained environments. Consequently, while LLMs demonstrate impressive capabilities, their practical application is frequently limited by these escalating computational and memory requirements, driving ongoing research into model compression, quantization, and efficient architectures.

The escalating complexity of large language models presents a substantial barrier to widespread accessibility and continued advancement. Each additional parameter within these models-often numbering in the billions-demands greater computational power and memory, effectively excluding deployment on common devices like smartphones or embedded systems. This limitation isn’t merely about hardware; it also constrains the pursuit of deeper reasoning capabilities. Scaling to models with even more parameters, theoretically capable of more nuanced understanding, quickly becomes impractical due to the exponential growth in resource requirements. Consequently, research is increasingly focused on model compression, efficient architectures, and algorithmic innovations to circumvent this bottleneck and unlock the full potential of artificial intelligence without being perpetually bound by computational limits.

Deconstructing Scale: The Foundation of Double Binary Factorization

Double Binary Factorization (DBF) is a model compression technique that reduces the size of weight matrices by representing them with the product of two binary matrices. This process leverages low-rank approximation; a weight matrix $W \in \mathbb{R}^{m \times n}$ is decomposed into two binary matrices, $B_1 \in \{-1, 1\}^{m \times k}$ and $B_2 \in \{-1, 1\}^{k \times n}$ , where $k$ is a significantly smaller rank than either $m$ or $n$ . The original weight matrix is then approximated as $W \approx B_1 B_2$ . By representing weights with binary values, the number of parameters required to store the model is substantially reduced – from $m \times n$ to $m \times k + k \times n$ – leading to decreased memory usage and potentially faster inference speeds, particularly on hardware optimized for binary operations.

Low-rank approximation techniques reduce the dimensionality of weight matrices by representing them with a product of smaller matrices, effectively decreasing the total number of trainable parameters. For a weight matrix $W \in \mathbb{R}^{m \times n}$ , standard matrix factorization aims to approximate it as $W \approx UV^T$ , where $U \in \mathbb{R}^{m \times k}$ and $V \in \mathbb{R}^{n \times k}$ , with $k << min(m, n)$ . This reduction in parameters directly translates to a smaller model size, requiring less memory for storage and decreasing computational demands during inference, as fewer multiplications and additions are needed to perform calculations. The extent of parameter reduction is proportional to the rank $k$ chosen relative to the original matrix dimensions.

The Single Envelope Constraint in standard Double Binary Factorization (DBF) restricts the binary factor matrices to share a common scaling factor, thereby limiting the expressiveness of the low-rank approximation. This constraint forces a uniform scaling across all singular values, preventing the model from accurately capturing weight distributions with varying magnitudes. Consequently, complex weight matrices requiring diverse scaling factors for effective representation are poorly approximated, leading to increased approximation error and potential degradation in model performance. The limitation arises from the imposition of a single envelope – a single set of bounds – on both binary factor matrices, hindering the ability to represent weights with significantly different importance levels.

Analysis of LLaMA2 13B’s Transformer layers using entropy-based effective rank reveals that the demodulated envelopes |U| and |V| consistently maintain a rank greater than one, suggesting multiple magnitude modes and supporting the removal of a single-envelope constraint. — Analysis of LLaMA2 13B’s Transformer layers using entropy-based effective rank reveals that the demodulated envelopes $|U|$ and $|V|$ consistently maintain a rank greater than one, suggesting multiple magnitude modes and supporting the removal of a single-envelope constraint.

Expanding Representational Power: The Architecture of Multi-Envelope Double Binary Factorization

Multi-Envelope Double Binary Factorization (MDBF) builds upon Double Binary Factorization (DBF) by representing weight matrices with multiple demodulated envelope modes. Traditional DBF utilizes a single envelope to capture magnitude information, whereas MDBF decomposes the magnitude into several distinct envelope representations. This allows for a more granular and accurate approximation of the original weight matrix, particularly in scenarios where the magnitude distribution is complex or multi-modal. Each envelope is a binary matrix, and the combination of these multiple binary envelopes, along with the binary sign matrix inherent in DBF, forms the complete factored representation. This increased representational capacity enables MDBF to achieve higher approximation accuracy with a comparable number of parameters, or equivalent accuracy with fewer parameters, than standard DBF.

MDBF utilizes Entropy-Based Effective Rank to dynamically determine the number of envelopes required for weight matrix factorization, achieving a balance between model compression and accuracy preservation. This metric quantifies the information content retained during factorization; analysis demonstrates a positive correlation between envelope rank and entropy rank. Specifically, increasing the number of envelopes consistently results in a higher entropy rank, indicating that multiple magnitude modes are being effectively captured within the factorization. This adaptive approach avoids the need for pre-defined envelope counts, allowing the method to tailor the factorization complexity to the inherent structure of the weight matrix and maintain a higher degree of representational fidelity.

The Alternating Direction Method of Multipliers (ADMM) is employed to optimize the Multi-Envelope Double Binary Factorization (MDBF) process by breaking down the original factorization problem into smaller, more manageable sub-problems. This decomposition facilitates parallel computation and allows for efficient handling of large-scale weight matrices. ADMM iteratively solves these sub-problems while coordinating updates through penalty terms and dual variables, ultimately converging towards a refined factorization that minimizes reconstruction error and improves model performance. The method’s ability to effectively manage the constraints inherent in the factorization process contributes to a more stable and accurate solution compared to direct optimization techniques.

Refining the Foundation: Initialization and Truncated Singular Value Decomposition

Effective initialization is a critical component of the Multi-Dimensional Bilinear Factorization (MDBF) process, directly influencing both convergence speed and the quality of the resulting factorization. Closed-Form Initialization addresses this by providing a deterministic and computationally efficient method to establish an initial approximation of the weight matrix. Unlike random initialization, this approach leverages the inherent structure within the data to generate a starting point closer to the optimal solution, thereby reducing the number of iterations required for convergence and mitigating the risk of converging to a local minimum. This initial approximation is subsequently refined through techniques like Truncated Singular Value Decomposition (TSVD), but its robustness significantly improves the overall stability and performance of the MDBF algorithm.

The initialization process for MDBF utilizes Multi-Envelope Singular Value Decomposition (MSVID) to generate a preliminary low-rank approximation of the weight matrix. MSVID extends standard Singular Value Decomposition (SVD) by applying it to multiple “envelopes” or subsets of the data, effectively creating a series of SVD factorizations. These individual factorizations are then combined to produce an initial approximation that captures significant variance within the weight matrix. This approach improves the robustness of the factorization, particularly when dealing with high-dimensional or noisy data, by providing a more informed starting point for subsequent refinement steps such as Truncated SVD.

Truncated Singular Value Decomposition (TSVD) operates by identifying and retaining only the most significant singular values and corresponding singular vectors from the Singular Value Decomposition of the weight matrix. This process reduces dimensionality and noise, creating a low-rank approximation that minimizes the Frobenius norm of the error between the original matrix and its reconstruction. The number of retained singular values determines the rank of the approximation; selecting an appropriate rank balances model complexity and accuracy in representing the original weight matrix. By discarding smaller singular values, TSVD effectively filters out potentially spurious or irrelevant information, resulting in a more robust and generalized factorization for use in subsequent MDBF computations.

Towards Ubiquitous Intelligence: Democratizing Access with Efficient Large Language Models

Model Decomposition with Block Factorization (MDBF) represents a substantial stride toward democratizing access to large language models. By strategically decomposing these models, MDBF dramatically lowers both computational demands and memory requirements – critical limitations that previously restricted deployment to high-end hardware. This efficiency unlocks the potential for running sophisticated language processing applications on resource-constrained devices, such as smartphones, tablets, and embedded systems. Consequently, individuals and communities lacking access to powerful computing infrastructure can now benefit from the capabilities of LLMs, fostering broader participation in the development and application of artificial intelligence. The technique doesn’t simply shrink the model; it restructures it for optimal performance within limited environments, paving the way for more inclusive and sustainable AI solutions.

Root Mean Square Layer Normalization, or RMSNorm, represents a critical refinement in the architecture of large language models, directly addressing challenges related to training stability and computational demands. By normalizing the layer inputs based on the root mean square of their values, RMSNorm effectively mitigates the vanishing or exploding gradient problems that often plague deep neural networks. This stabilization allows for faster convergence during training and enables the use of larger learning rates, ultimately accelerating the development process. The resulting models exhibit improved robustness and generalization capabilities, translating to a more consistent and reliable user experience – responses are generated with greater accuracy and coherence, and the model is less prone to unpredictable behavior or failures.

The pursuit of efficient large language models is fundamentally about broadening access and fostering sustainability within the field of artificial intelligence. Recent advancements detailed in this work demonstrate a pathway toward this goal, showing that techniques like MDBF, when optimized by increasing the envelope rank, consistently deliver improved performance – evidenced by reductions in reconstruction error and gains in both perplexity and zero-shot accuracy, as detailed in Appendix C. This isn’t merely a technical refinement; it represents a crucial step toward democratizing powerful language processing capabilities, enabling deployment on resource-constrained devices and ultimately making these technologies available to a wider audience and reducing the environmental impact of computationally intensive AI.

The pursuit of extreme quantization, as demonstrated by Multi-Envelope Double Binary Factorization, necessitates a careful consideration of representation. It’s not merely about reducing the number of bits, but about preserving essential information within that constraint. As Ken Thompson observed, “Simplicity scales, cleverness does not.” This sentiment directly aligns with the paper’s focus on magnitude representation; by prioritizing a clear and scalable method for encoding weight significance, MDBF avoids the pitfalls of overly complex approaches that might offer marginal gains in sign diversity at the expense of overall performance and generalizability. The method’s elegance lies in its deliberate simplicity, acknowledging that a robust system thrives on fundamental clarity rather than intricate embellishments.

What Lies Ahead?

The pursuit of extreme quantization, as exemplified by Multi-Envelope Double Binary Factorization, reveals a fundamental tension. The field initially fixated on maximizing the diversity of sign, a seemingly logical approach. This work subtly redirects attention, suggesting that a refined representation of magnitude may be the more fruitful path – a humbling reminder that obvious solutions are often insufficient. The improvements demonstrated are encouraging, but the underlying question remains: how much information can truly be discarded without fundamentally altering the expressive capacity of these models? Future investigation must address this directly, moving beyond empirical gains to a more theoretically grounded understanding of information preservation in highly quantized networks.

Current methods, including MDBF, often treat weight matrices as isolated entities. Yet, a Large Language Model is not a collection of independent components; it is a complex, interconnected system. The interplay between layers, the impact of quantization on activation functions, and the propagation of errors – these systemic effects remain largely unexplored. A holistic view, perhaps incorporating insights from dynamical systems theory, may reveal emergent properties and limitations not apparent in isolated weight analysis.

The elegance of a truly compressed model lies not merely in its reduced size, but in its ability to retain performance with minimal resources. The improvements offered by MDBF are steps in this direction, but the ultimate goal – a genuinely efficient and robust architecture – remains elusive. Good architecture is invisible until it breaks, and only then is the true cost of decisions visible.

Original article: https://arxiv.org/pdf/2512.24545.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/