Algorithmic Truncation of MiniMax Polynomial Coefficients

Tawfik, Sara; Fahmy, H. M.

doi:10.1109/iscas.2006.1693111

Cited by 5 publications

(4 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Coefficient fine-tuning means the quantization of the polynomial coefficients does not directly use the round operation but makes some adjustments. There are many types of adjustments, such as the MILP or ILP method used in [23] and [24]. However, as both ILP and MILP are NP-hard, these solutions will become impractical when the polynomial order is higher.…”

Section: A Methodology Overviewmentioning

confidence: 99%

“…Another shortcoming of methods proposed in [16]- [19] is that although they considered the quantization effect, they truncate the polynomial coefficients by direct rounding. Tawfik et al [23] used a mixed-integer linear programming (MILP) method and got a significant increase in accuracy over direct rounding. Caro et al [24] proposed another truncation method for linear and quadratic fitting functions by using an integer linear programming (ILP) method.…”

Section: Introductionmentioning

confidence: 99%

“…However, both the ILP and MILP problem are NP-hard, hence for higher precisions or higher-order polynomial approximations, these solutions will become impractical as the computing time is difficult to predict. Moreover, both [23] and [24] divided the target fitting interval equally, which greatly increases the hardware area.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

QPA: A Quantization-Aware Piecewise Polynomial Approximation Methodology for Hardware-Efficient Implementations

Geng¹,

Chen²,

Zhao³

et al. 2022

Preprint

View full text Add to dashboard Cite

<p>Piecewise polynomial approximation on non-linear functions plays an important role in neural network accelerators and digital signal processing. In this paper, we proposed QPA, a quantization-aware piecewise polynomial approximation methodology, to generate the optimized coefficients for hardware implementations targeting any polynomial order. QPA incorporated several key features to minimize the fitting error and the hardware cost, including using the Remez algorithm to compute the min-max fitting polynomial, combining the fitting and quantization operations to get an error-flattened characteristic, assigning specific coefficient bit width to each multiplier to reduce the hardware cost, and fine-tuning the truncated coefficients to further reduce the fitting error. We applied the proposed methodology to piecewise linear (PWL) and piecewise quadratic (PWQ) approximations. Experimental results showed that QPA consistently achieved the lowest fitting error compared with the state-of-the-art methods. We synthesized the proposed designs with 28nm TSMC CMOS technology. The synthesis results showed the proposed designs achieved up to 43.8% area reduction and 37.5% fitting error reduction compared to state-of-the-art PWL designs, up to 22.1% area reduction and 33.1% fitting error reduction compared to state-of-the-art PWQ designs respectively.</p>

show abstract

Section: A Methodology Overviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

QPA: A Quantization-Aware Piecewise Polynomial Approximation Methodology for Hardware-Efficient Implementations

Geng¹,

Chen²,

Zhao³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…If this difference is at its minimum, p * (x) is regarded as the optimum polynomial approximation scheme, which was adopted in this study. The commonly used polynomial approximation schemes are Taylor, minimax error [35], and Chebyshev [36]. The Taylor polynomial approximation algorithm approximately calculates the function value through its Taylor expansion.…”

Section: Polynomial Approximationmentioning

confidence: 99%

swAFL: A Library of High-Performance Activation Function for the Sunway Architecture

Xu¹,

Li²,

Hou³

et al. 2022

Electronics

View full text Add to dashboard Cite

The Sunway supercomputers have recently attracted considerable attention to execute neural networks. Meanwhile, activation functions help extend the applicability of neural networks to nonlinear models by introducing nonlinear factors. Despite the numerous activation function-supported AI frameworks, only PyTorch and TensorFlow were ported to the Sunway platforms. Although these libraries can meet the minimum functional requirements to deploy a neural network on the Sunway machines, there still exist some drawbacks including the limited number of usable functions and unsatisfactory performances remaining unresolved. Therefore, two activation function algorithms with different computing accuracies were developed in this study, and an efficient implementation scheme was designed using the single instruction/multiple data extension and multiply–add instructions of the platform. Finally, an efficient library-swAFL-composed of 48 function interfaces was designed and implemented on the Sunway platforms. Experimental results indicate that swAFL outperformed PyTorch and TensorFlow by 19.5 and 23 times, respectively, on average.

show abstract

Very low resource table-based FPGA evaluation of elementary functions

Neto

Véstias

2013

2013 International Conference on Reconfigurable Computing and FPGAs (ReConFig)

View full text Add to dashboard Cite

Algorithmic Truncation of MiniMax Polynomial Coefficients

Cited by 5 publications

References 16 publications

QPA: A Quantization-Aware Piecewise Polynomial Approximation Methodology for Hardware-Efficient Implementations

QPA: A Quantization-Aware Piecewise Polynomial Approximation Methodology for Hardware-Efficient Implementations

swAFL: A Library of High-Performance Activation Function for the Sunway Architecture

Very low resource table-based FPGA evaluation of elementary functions

Contact Info

Product

Resources

About