Chang Meng

Clarissa Garvey, Chang Meng, James G. Nagy

In this paper we propose an approach to approximate a truncated singular value decomposition of a large structured matrix. By first decomposing the matrix into a sum of Kronecker products, our approach can be used to approximate a large number of singular values and vectors more efficiently than other well known schemes, such as randomized matrix algorithms or iterative algorithms based on Golub-Kahan bidiagonalization. We provide theoretical results and numerical experiments to demonstrate the accuracy of our approximation and show how the approximation can be used to solve large scale ill-posed inverse problems, either as an approximate filtering method, or as a preconditioner to accelerate iterative algorithms.

Chang Meng, Wayne Burleson, Giovanni De Micheli

Approximate multipliers (AppMults) are widely used in deep learning accelerators to reduce their area, delay, and power consumption. However, AppMults introduce arithmetic errors into deep learning models, necessitating a retraining process to recover accuracy. A key step in retraining is computing the gradient of the AppMult, i.e., the partial derivative of the approximate product with respect to each input operand. Existing approaches typically estimate this gradient using that of the accurate multiplier (AccMult), which can lead to suboptimal retraining results. To address this, we propose two methods to obtain more precise gradients of AppMults. The first, called LUT-2D, characterizes the AppMult gradient with 2-dimensional lookup tables (LUTs), providing fine-grained estimation and achieving the highest retraining accuracy. The second, called LUT-1D, is a compact and more efficient variant that stores gradient values in 1-dimensional LUTs, achieving comparable retraining accuracy with shorter runtime. Experimental results show that on CIFAR-10 with convolutional neural networks, our LUT-2D and LUT-1D methods improve retraining accuracy by 3.83% and 3.72% on average, respectively. On ImageNet with vision transformer models, our LUT-1D method improves retraining accuracy by 23.69% on average, compared to a state-of-the-art retraining framework.