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1. Matrix Compression via Randomized Low Rank and Low Precision Factorization

   Saha, Rajarshi (October 2023, arXivorg)

   Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner. However, modern matrices can involve billions of elements, making their storage and processing quite demanding in terms of computational resources and memory usage. Although prohibitively large, such matrices are often approximately low rank. We propose an algorithm that exploits this structure to obtain a low rank decomposition of any matrix A as A≈LR, where L and R are the low rank factors. The total number of elements in L and R can be significantly less than that in A. Furthermore, the entries of L and R are quantized to low precision formats −− compressing A by giving us a low rank and low precision factorization. Our algorithm first computes an approximate basis of the range space of A by randomly sketching its columns, followed by a quantization of the vectors constituting this basis. It then computes approximate projections of the columns of A onto this quantized basis. We derive upper bounds on the approximation error of our algorithm, and analyze the impact of target rank and quantization bit-budget. The tradeoff between compression ratio and approximation accuracy allows for flexibility in choosing these parameters based on specific application requirements. We empirically demonstrate the efficacy of our algorithm in image compression, nearest neighbor classification of image and text embeddings, and compressing the layers of LlaMa-7b. Our results illustrate that we can achieve compression ratios as aggressive as one bit per matrix coordinate, all while surpassing or maintaining the performance of traditional compression techniques. 

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2. Blended Coarse Gradient Descent for Full Quantization of Deep Neural Networks

   https://doi.org/DOI:10.1007/s40687-018-0177-6  

   Yin, P; Zhang, S; Lyu, L; Osher, S; Qi, Y-Y; Xin, J (January 2019, Research in the mathematical sciences)

   Quantized deep neural networks (QDNNs) are attractive due to their much lower memory storage and faster inference speed than their regular full-precision counterparts. To maintain the same performance level especially at low bit-widths, QDNNs must be retrained. Their training involves piece-wise constant activation functions and discrete weights; hence, mathematical challenges arise. We introduce the notion of coarse gradient and propose the blended coarse gradient descent (BCGD) algorithm, for training fully quantized neural networks. Coarse gradient is generally not a gradient of any function but an artificial ascent direction. The weight update of BCGD goes by coarse gradient correction of a weighted average of the full-precision weights and their quantization (the so-called blending), which yields sufficient descent in the objective value and thus accelerates the training. Our experiments demonstrate that this simple blending technique is very effective for quantization at extremely low bit-width such as binarization. In full quantization of ResNet-18 for ImageNet classification task, BCGD gives 64.36% top-1 accuracy with binary weights across all layers and 4-bit adaptive activation. If the weights in the first and last layers are kept in full precision, this number increases to 65.46%. As theoretical justification, we show convergence analysis of coarse gradient descent for a two-linear-layer neural network model with Gaussian input data and prove that the expected coarse gradient correlates positively with the underlying true gradient. 

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3. Quantization and Training of Low Bit-Width Convolutional Neural Networks for Object Detection

   https://doi.org/10.4208/jcm.1803-m2017-0301.  

   Yin, P; Zhang, S.; Qi, Y-Y; Xin, J (January 2019, Journal of Computational Mathematics)

   We present LBW-Net, an efficient optimization based method for quantization and training of the low bit-width convolutional neural networks (CNNs). Specifically, we quantize the weights to zero or powers of 2 by minimizing the Euclidean distance between full-precision weights and quantized weights during back-propagation (weight learning). We characterize the combinatorial nature of the low bit-width quantization problem. For 2-bit (ternary) CNNs, the quantization of N weights can be done by an exact formula in O(N log N) complexity. When the bit-width is 3 and above, we further propose a semi-analytical thresholding scheme with a single free parameter for quantization that is computationally inexpensive. The free parameter is further determined by network retraining and object detection tests. The LBW-Net has several desirable advantages over full-precision CNNs, including considerable memory savings, energy efficiency, and faster deployment. Our experiments on PASCAL VOC dataset show that compared with its 32-bit floating-point counterpart, the performance of the 6-bit LBW-Net is nearly lossless in the object detection tasks, and can even do better in real world visual scenes, while empirically enjoying more than 4× faster deployment. 

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4. Post-training Quantization with Multiple Points: Mixed Precision without Mixed Precision

   Liu, Xingchao; Ye, Mao; Zhou, Dengyong; Liu, Qiang (January 2021, AAAI Conference on Artificial Intelligence (AAAI))

   We consider the post-training quantization problem, which discretizes the weights of pre-trained deep neural networks without re-training the model. We propose multipoint quantization, a quantization method that approximates a full-precision weight vector using a linear combination of multiple vectors of low-bit numbers; this is in contrast to typical quantization methods that approximate each weight using a single low precision number. Computationally, we construct the multipoint quantization with an efficient greedy selection procedure, and adaptively decides the number of low precision points on each quantized weight vector based on the error of its output. This allows us to achieve higher precision levels for important weights that greatly influence the outputs, yielding an 'effect of mixed precision' but without physical mixed precision implementations (which requires specialized hardware accelerators). Empirically, our method can be implemented by common operands, bringing almost no memory and computation overhead. We show that our method outperforms a range of state-of-the-art methods on ImageNet classification and it can be generalized to more challenging tasks like PASCAL VOC object detection. 

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5. Accelerating AWP-ODC for Large-scale Earthquake Simulations Using MVAPICH2

   Palla, A; Wang, S; Zhang, T; Cui, Y (August 2025, Annual MVAPICH User Group (MUG) Conference)

   Accurate simulation of earthquake scenarios is essential for advancing seismic hazard analysis and risk mitigation strategies. At the San Diego Supercomputer Center (SDSC), our research focuses on optimizing the performance and reliability of large-scale earthquake simulations using the AWP-ODC software. By implementing GPU-aware MPI calls, we enable direct data processing within GPU memory, eliminating the need for explicit data transfers between CPU and GPU. This GPU-aware MPI achieves nearly ideal parallel efficiency at full scale across both Nvidia and AMD GPUs, leveraging the MVAPICH-PLUS support on Frontier at Oak Ridge National Laboratory and Vista at the Texas Advanced Computing Center. We utilized the MVAPICH-Plus 4.0 compiler to enable ZFP compression, which significantly enhances inter-node communication efficiency – a critical improvement given the communication bottleneck inherent in large-scale simulations. Our GPU-aware AWP-ODC versions include linear forward, topography and nonlinear Iwan-type solvers with discontinuous mesh support. On the Frontier system with MVAPICH 4.0, Hip-aware MPI calls on MI250X GPUs deliver nearly ideal weak-scaling speedup up to 8,192 nodes for both linear and topography versions. On TACC’s Vista system, CUDA-aware MPI calls on GH200 GPUs substantially outperform their non-GPU-aware counterparts across all three solver versions. This poster will present a detailed evaluation of GPU-aware AWP-ODC using MVAPICH, including the impact of ZFP message compression compared to the native versions. Our results highlight the importance of Mvapich support for GPU-ware MPI and on-the-fly compression techniques for accelerating and scaling earthquake simulations. 

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