Pseudo-Normalization via Integer Fast Inverse Square Root and Its Application to Fast Computation without Division
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors1. Summary
This paper presents a novel pseudo-normalization algorithm that achieves vector normalization through integer arithmetic, thereby avoiding the computationally expensive division operation. By using only multiply-add operations and bit shifts, the algorithm demonstrates improvements in computation speed and energy efficiency in embedded systems. The key innovation lies in extending a floating-point algorithm to integer arithmetic, which enhances computational efficiency.
2. Strengths:
+ The paper proposes a new pseudo-normalization algorithm that achieves vector normalization using integer arithmetic. This method has significant advantages in embedded systems as it avoids time-consuming division operations, instead using multiply-add operations and bit shifts to improve computation speed and energy efficiency.
+ The paper provides the description of the algorithm's design process, including the residual correction method for approximating square roots and reciprocals. It also discusses the implementation of the algorithm in Verilog, demonstrating its feasibility in hardware.
3. Weakness：
- The method is innovative for embedded systems, but similar optimization algorithms already exist in the broader field of computing. The paper should discuss the differences and advantages of this method compared to existing algorithms more thoroughly. It is also recommended to include comparative experiments with other normalization algorithms to highlight the advantages in performance and energy efficiency.
- The paper should discuss how the proposed algorithm can be effectively used in machine learning, such as improving the training efficiency of related models, preferably with experimental demonstrations. The introduction states that “the proposed algorithm can be used in various vector normalization applications”, but the paper only provides experimental results for computing the atan2 function using integer arithmetic. This limited result does not fully support the work's conclusion.
- While the Verilog implementation is described in detail, the discussion on hardware performance is insufficient. It is recommended to include actual hardware test data to validate the algorithm's practical application. The experiments are mainly based on simulations and lack results from real hardware environments. Supplementing performance tests on real embedded devices would verify the effectiveness in practical applications. Additionally, the paper should present complete experimental results to discuss the specific speed improvements.
- Some typos need to be corrected. For example, "for normalized" in line 137 should be "for normalization", "the following a linear expression" around equation 8 should be "the following linear expression", and "must be normalizeed" in line 203 should be "must be normalized".
Comments on the Quality of English LanguageSome typos need to be corrected. For example, "for normalized" in line 137 should be "for normalization", "the following a linear expression" around equation 8 should be "the following linear expression", and "must be normalizeed" in line 203 should be "must be normalized".
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe research introduces a method for vector normalization using integer arithmetic, there are several notable issues and areas as per the following:
1 1. More technical details and the underlying principles of the algorithm is needed to provide a clearer understanding.
2 2. Quantitative data comparing the proposed method with existing techniques (e.g., floating-point normalization or FISR) is needed to strengthen the claims.
3. Explaining the boundary conditions, applicability range, and potential edge cases which is beneficial.
4. The applications in signal processing, AI training, and graphics rendering are mentioned, specific case studies are needed to provide better insights into the method’s effectiveness.
5. A detailed comparison or analysis of how the proposed method stacks up against FISR in terms of accuracy, speed, computational complexity, and energy efficiency.
6 6. A detailed analysis of the accuracy of the arctangent computation and overall vector normalization should be provided. This would include comparisons with standard floating-point implementations to highlight any trade-offs.
7. The research lacks specific measurements or metrics illustrating the power savings achieved by this method compared to conventional approaches.
8. The overall organization of the document can be improved. ensuring each section logically follows from the previous one would enhance readability.
9 9. The claim that the approach can be applied to a wide range of fields (e.g., signal processing, AI, graphics) needs more concrete evidence or examples demonstrating this versatility. Discussing potential challenges or limitations when adapting the method to different applications would also be helpful.
Comments on the Quality of English LanguageSome sentences should be re-structured with proper grammar.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors proposed a method for vector normalization using integer arithmetic, ensuring efficient and robust operations for low-performance embedded systems. They demonstrated the effectiveness of this method through simulations, highlighting its potential applications in IMU sensor data processing, AI training, and computer graphics rendering.
1- The introduction effectively highlights the growing demand for advanced computational processing due to the rise of IoT and AI. The focus on reducing power consumption is relevant, but the introduction could benefit from a clearer connection between this demand and the proposed solution.
2- The authors reference their previous work on fast computation algorithms for embedded systems. While this establishes credibility, it would be helpful to provide a brief summary of those findings to better contextualize the current research.
3- The proposed fixed-point vector normalization method is explained, but the introduction could be clearer about the novelty and advantages of this method over existing solutions. Additionally, the term "pseudo-normalization" is introduced without sufficient explanation of its limitations.
4- There is some redundancy in the text, particularly in discussing the limitations of division and the benefits of multiply-add operations.
5- The simulations effectively demonstrate the behavior of the proposed algorithm, but more details on real-world testing would strengthen the findings.
6- The explanation of the recursive structure and its convergence to Newton's method is insightful. However, the practical implications of using this method versus others (e.g., Taylor expansion) could be further explored.
Comments on the Quality of English LanguageMinor editing of English language required
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsAuthors have addressed the issues, the manuscript may be considered for the publication.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors addressed the comments thoroughly, and I would like to see the paper published. Good work!
