DiBA: Diagonal and Binary Matrix Approximation for Neural Network Weight Compression
DiBA (Diagonal and Binary Matrix Approximation) compresses neural network weights by decomposing matrices into three diagonal and two binary factors, improving storage efficiency.
Key Findings
Methodology
DiBA approximates A∈R^{m×n} as D1B1D2B2D3, with D as diagonal matrices and B as binary matrices. The approach employs an alternating optimization: diagonal matrices are updated via closed-form least squares, while binary matrices undergo greedy one-bit flips based on exact objective improvements. The method introduces DiBA-Greedy, combining these updates, and DiBARD, which freezes binary matrices and only fine-tunes diagonals on downstream data. Experiments on 40 pretrained weight matrices demonstrate consistent SNR improvements with increasing storage ratio, validating the approach’s effectiveness for neural weight compression.
Key Results
- Across 40 matrices from public models, DiBA-Greedy improves SNR from 0.70dB at k=8 to 21.3dB at k=1024, with the storage ratio decreasing as k increases. In downstream tasks, DiBARD fine-tuning only diagonal elements boosts DistilBERT/WikiText masked token accuracy from 0.4447 to 0.5210, and Speech Commands accuracy from 0.7684 to 0.9781, without re-optimizing binary factors.
- The storage ratio is controlled by k, with theoretical bounds showing that DiBA can outperform dense models in compression ratio while maintaining approximation quality. Binary matrices require only 1 bit per entry, drastically reducing storage. The alternating optimization ensures progressive error reduction, validating DiBA’s potential for practical compression.
- DiBARD’s diagonal-only retuning allows fast adaptation to downstream tasks, avoiding complex discrete searches. Experiments confirm that fixing binary matrices and tuning only diagonals can recover most of the original performance, making DiBA suitable for model deployment and transfer learning.
Significance
This work advances neural network compression by integrating shared binary mixing with diagonal scaling, addressing the trade-off between storage and accuracy. It offers a flexible, scalable framework that can be applied to various model components, facilitating deployment on resource-constrained devices. The approach bridges the gap between low-bit quantization and low-rank approximations, providing a new avenue for efficient model representation. Its ability to recover downstream performance with minimal retuning underscores its practical relevance for industry applications, from edge devices to large-scale inference.
Technical Contribution
The paper introduces DiBA, a novel matrix factorization combining diagonal matrices and binary sharing patterns, enabling efficient approximation of dense weights. The DiBA-Greedy algorithm leverages an alternating approach with exact one-bit flip tests for binary matrices and least squares for diagonals, ensuring progressive error minimization. DiBARD’s innovation lies in fixing binary matrices during downstream adaptation, significantly simplifying fine-tuning. Theoretical analysis shows storage ratios controlled by parameter k, with guarantees on approximation quality. Extensive experiments validate the method’s superiority over traditional low-rank and quantization techniques, demonstrating its potential for practical deployment.
Novelty
This research uniquely combines diagonal matrices with shared binary mixing patterns for neural weight approximation, diverging from conventional low-rank or quantization methods. The binary matrices are stored efficiently at one bit per entry, and the optimization strategy effectively handles the mixed discrete-continuous problem. DiBARD’s approach to fixing binary matrices during downstream tuning is a novel contribution, enabling rapid adaptation without discrete search. These innovations collectively offer a new paradigm for neural network compression, balancing storage, speed, and accuracy in ways previous methods cannot achieve.
Limitations
- The approximation quality may degrade at extremely low storage budgets (e.g., very small bit-widths), especially for matrices with complex spectral properties.
- Binary matrix optimization via greedy flips can get trapped in local minima, limiting performance in some cases.
- The current implementation emphasizes theoretical storage and approximation; real hardware acceleration and speedup remain to be validated in future work.
Future Work
Future directions include exploring multi-level binary schemes, integrating sparsity, and hardware-aware optimization to enhance practical speedups. Extending DiBA to convolutional kernels and attention modules could broaden its applicability. Additionally, developing adaptive schemes for selecting k based on matrix spectra and task requirements may further improve efficiency and accuracy.
AI Executive Summary
The rapid growth of neural network models has led to increasing challenges in storage and computational efficiency. Traditional methods such as pruning, quantization, and low-rank factorization have made progress but often involve trade-offs between accuracy and compression ratio. This paper introduces DiBA (Diagonal and Binary Matrix Approximation), a novel approach that decomposes weight matrices into three diagonal matrices and two binary matrices, leveraging shared binary patterns and continuous diagonal scaling. The core innovation lies in combining discrete binary sharing with continuous diagonal updates, enabling efficient approximation with controllable storage ratios.
The authors develop DiBA-Greedy, an alternating optimization algorithm that refines the approximation by iteratively updating the diagonal matrices via least squares and improving binary matrices through greedy one-bit flips. This approach ensures progressive error reduction and avoids complex discrete search. To facilitate downstream adaptation, they propose DiBARD, which freezes binary matrices and only fine-tunes the diagonal elements on task-specific data, significantly simplifying the process.
Extensive experiments on 40 pretrained models demonstrate DiBA’s effectiveness. As the intermediate dimension k increases, the reconstruction signal-to-noise ratio (SNR) improves steadily, with the average SNR reaching over 21dB at high k values. In downstream tasks, DiBARD successfully recovers most of the original model accuracy, boosting masked language modeling accuracy from 0.4447 to 0.5210 on WikiText and classification accuracy from 0.7684 to 0.9781 on Speech Commands, all without re-optimizing binary matrices.
This work offers a promising new paradigm for neural network compression, balancing storage efficiency and accuracy. Its ability to adapt quickly to downstream tasks with minimal retuning makes it highly relevant for deploying large models on resource-constrained devices. The combination of theoretical guarantees and practical validation paves the way for future hardware-friendly, scalable model compression techniques, potentially transforming the deployment landscape of deep learning models.
Deep Dive
Abstract
In this paper, we propose DiBA (Diagonal and Binary Matrix Approximation), a compact matrix factorization for neural network weight compression. Many components of modern networks, including linear layers, $1\times1$ convolutions, attention projections, and embedding layers, have dense matrix weights. DiBA approximates $A\in\mathbb{R}^{m\times n}$ by $\widehat A=D_1B_1D_2B_2D_3$, where $D_1,D_2,D_3$ are diagonal matrices and $B_1,B_2$ are $0/1$ binary matrices. The intermediate dimension $k$ controls the trade-off between theoretical storage and approximation accuracy. For matrix-vector products, DiBA decomposes dense multiplication into three element-wise scaling operations and two binary mixing operations, reducing the floating-point multiplication count from $mn$ to $m+k+n$. For optimization, we introduce DiBA-Greedy, an alternating solver that combines closed-form least-squares updates for the diagonal factors with exact one-bit improvement tests for the binary factors. We also introduce DiBARD (DiBA with Retuning only Diagonal factors), which replaces dense-matrix layers by DiBA factors, freezes the binary matrices, and retunes only the diagonal entries on downstream data. This preserves compact binary mixing without discrete search during adaptation. On 40 dense weight matrices extracted from public pretrained models, DiBA-Greedy yields consistent SNR improvements as the theoretical storage ratio increases. After DiBA replacement in two component-replacement studies, DiBARD improves DistilBERT/WikiText masked-token accuracy from 0.4447 to 0.5210 and Speech Commands test accuracy for an Audio Spectrogram Transformer from 0.7684 to 0.9781 without reoptimizing the binary factors.