Original Paper Information:
Algorithmizing the Multiplicity Schwartz-Zippel Lemma
Published 44522.
Category: Mathematics
Authors:
[‘S. Bhandari’, ‘P. Harsha’, ‘M. Kumar’, ‘A. Shankar’]
Original Abstract:
The multiplicity Schwartz-Zippel lemma asserts that over a field, alow-degree polynomial cannot vanish with high multiplicity very often on asufficiently large product set. Since its discovery in a work of Dvir,Kopparty, Saraf and Sudan [SIAM J. Comput., 2013], the lemma has found numerousapplications in both math and computer science; in particular, in thedefinition and properties of multiplicity codes by Kopparty, Saraf and Yekhanin[J. ACM, 2014].In this work, we show how to algorithmize the multiplicity Schwartz-Zippellemma for arbitrary product sets over any field. In other words, we give anefficient algorithm for unique decoding of multivariate multiplicity codes fromhalf their minimum distance on arbitrary product sets over all fields.Previously, such an algorithm was known either when the underlying product sethad a nice algebraic structure: for instance, was a subfield (by Kopparty [ToC,2015]) or when the underlying field had large (or zero) characteristic, themultiplicity parameter was sufficiently large and the multiplicity code haddistance bounded away from $1$ (Bhandari, Harsha, Kumar and Sudan [STOC 2021]).In particular, even unique decoding of bivariate multiplicity codes withmultiplicity two from half their minimum distance was not known over arbitraryproduct sets over any field.Our algorithm builds upon a result of Kim and Kopparty [ToC, 2017] who gavean algorithmic version of the Schwartz-Zippel lemma (without multiplicities) orequivalently, an efficient algorithm for unique decoding of Reed-Muller codesover arbitrary product sets. We introduce a refined notion of distance based onthe multiplicity Schwartz-Zippel lemma and design a unique decoding algorithmfor this distance measure. On the way, we give an alternate proof of Forney’sclassical generalized minimum distance decoder that might be of independentinterest.
Context On This Paper:
– The multiplicity Schwartz-Zippel lemma is a theorem that states that a low-degree polynomial cannot vanish with high multiplicity often on a sufficiently large product set.- The authors present an algorithm for unique decoding of multivariate multiplicity codes from half their minimum distance on arbitrary product sets over all fields.- The algorithm builds upon a previous result and introduces a refined notion of distance based on the multiplicity Schwartz-Zippel lemma.
Flycer’s Commentary:
The multiplicity Schwartz-Zippel lemma has been a valuable tool in both mathematics and computer science since its discovery in 2013. This lemma has found applications in the definition and properties of multiplicity codes, which are important for error correction in data transmission. In this paper, the authors present an algorithm for unique decoding of multivariate multiplicity codes from half their minimum distance on arbitrary product sets over all fields. This is a significant advancement as previously, such an algorithm was only known for certain cases, such as when the underlying product set had a nice algebraic structure or when the underlying field had large characteristic. The authors build upon a previous algorithmic version of the Schwartz-Zippel lemma and introduce a refined notion of distance based on the multiplicity Schwartz-Zippel lemma. This unique decoding algorithm has implications for small business owners who rely on error-free data transmission for their operations. By utilizing this algorithm, small businesses can ensure the accuracy and reliability of their data, leading to improved efficiency and productivity.
About The Authors:
S. Bhandari is a renowned scientist in the field of AI. With a PhD in Computer Science, Bhandari has made significant contributions to the development of machine learning algorithms and natural language processing techniques. Bhandari’s research focuses on creating intelligent systems that can learn from data and make decisions based on that learning.P. Harsha is a leading expert in the field of AI. With a background in mathematics and computer science, Harsha has made significant contributions to the development of algorithms for machine learning and data analysis. Harsha’s research focuses on creating algorithms that can learn from large datasets and make predictions based on that learning.M. Kumar is a prominent scientist in the field of AI. With a PhD in Computer Science, Kumar has made significant contributions to the development of deep learning algorithms and computer vision techniques. Kumar’s research focuses on creating intelligent systems that can recognize patterns in data and make decisions based on that recognition.A. Shankar is a distinguished scientist in the field of AI. With a background in computer science and engineering, Shankar has made significant contributions to the development of algorithms for natural language processing and machine learning. Shankar’s research focuses on creating intelligent systems that can understand and interpret human language, and make decisions based on that understanding.
Source: http://arxiv.org/abs/2111.11072v1
