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Title: Department of Mathematics
Title: Department of Mathematics


Research Question: The main research question of this paper is to analyze the complexity of the simplex algorithm, a method used to solve linear programming problems. The authors want to know if this algorithm has polynomial smoothed complexity, which is a hybrid of worst-case and average-case analysis.
Research Question: How can the complexity of curve fitting algorithms be reduced while maintaining accuracy?


Methodology: The authors introduce the concept of smoothed analysis of algorithms, which measures the maximum performance of an algorithm under small random perturbations of the input. They use geometric definitions, vector and matrix norms, probability theory, and Gaussian random vectors to analyze the simplex algorithm. They also use changes of variables to simplify the problem.
Methodology: The study focuses on a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. The authors propose precise conditions under which this algorithm can be significantly simplified.


Results: The authors present a two-phase method for the simplex algorithm, which involves an initial pivot phase and a subsequent optimization phase. They prove that the shadow size, which is a measure of the difficulty of the problem, is bounded, leading to polynomial smoothed complexity for the simplex algorithm.
Results: The research reveals that this reduction in complexity is possible when fitting circles but not ellipses or hyperbolas. The authors introduce the concept of a gradient weight function, which is crucial for maintaining accuracy. They also provide insights into how to evaluate the distance from a point to the curve, making the algorithm more efficient.


Implications: The results of this paper have significant implications for the field of computer science and operations research. The polynomial smoothed complexity of the simplex algorithm suggests that this method can be used to solve large-scale linear programming problems efficiently. The authors also discuss practicality of their analysis, further analysis of the simplex algorithm, and open questions related to smoothed analysis.
Implications: The findings have significant implications for the field of curve fitting. The proposed method allows for a substantial reduction in computational complexity without compromising accuracy. This can lead to faster and more efficient algorithms, particularly useful for large datasets. Moreover, the research provides a clear understanding of the conditions under which such reductions are possible, which can guide future research in this area.


Conclusion: In conclusion, the authors have introduced a new method for analyzing the complexity of algorithms, called smoothed analysis, and applied it to the simplex algorithm. They have shown that the simplex algorithm has polynomial smoothed complexity, which has important implications for the field of computer science and operations research.
Link to Article: https://arxiv.org/abs/0308023v1
 
Link to Article: https://arxiv.org/abs/0111050v3
Authors:  
Authors:  
arXiv ID: 0111050v3
arXiv ID: 0308023v1


[[Category:Computer Science]]
[[Category:Computer Science]]
[[Category:Algorithm]]
[[Category:Research]]
[[Category:Simplex]]
[[Category:Can]]
[[Category:Smoothed]]
[[Category:Fitting]]
[[Category:Complexity]]
[[Category:Which]]
[[Category:Analysis]]
[[Category:This]]

Latest revision as of 14:07, 24 December 2023

Title: Department of Mathematics

Research Question: How can the complexity of curve fitting algorithms be reduced while maintaining accuracy?

Methodology: The study focuses on a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. The authors propose precise conditions under which this algorithm can be significantly simplified.

Results: The research reveals that this reduction in complexity is possible when fitting circles but not ellipses or hyperbolas. The authors introduce the concept of a gradient weight function, which is crucial for maintaining accuracy. They also provide insights into how to evaluate the distance from a point to the curve, making the algorithm more efficient.

Implications: The findings have significant implications for the field of curve fitting. The proposed method allows for a substantial reduction in computational complexity without compromising accuracy. This can lead to faster and more efficient algorithms, particularly useful for large datasets. Moreover, the research provides a clear understanding of the conditions under which such reductions are possible, which can guide future research in this area.

Link to Article: https://arxiv.org/abs/0308023v1 Authors: arXiv ID: 0308023v1

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