Conference proceeding
Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification
2022 IEEE 63RD ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS), pp.378-388
Annual IEEE Symposium on Foundations of Computer Science
01/01/2022
DOI: 10.1109/FOCS54457.2022.00043
Abstract
We give an efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal (namely, near-quadratic, d <= 1/lambda(2+o(1))) tradeoff between (any desired) spectral expansion lambda and degree d. Furthermore, the algorithm is local: every vertex can compute its new neighbors as a subset of its original neighborhood of radius O(log(1/lambda)). The optimal quadratic trade-off is known as the Ramanujan bound, so our construction gives almost Ramanujan expanders from arbitrary expanders. The locality of the transformation preserves structural properties of the original graph, and thus has many consequences. Applied to Cayley graphs, our transformation shows that any expanding finite group has almost Ramanujan expanding generators. Similarly, one can obtain almost optimal explicit constructions of quantum expanders, dimension expanders, monotone expanders, etc., from existing (suboptimal) constructions of such objects. Another consequence is a "derandomized" random walk on the original (suboptimal) expander with almost optimal convergence rate. Our transformation also applies when the degree is not bounded or the expansion is not constant. We obtain our results by a generalization of Ta-Shma's technique in his breakthrough paper [STOC 2017], used to obtain explicit almost optimal binary codes. Specifically, our spectral amplification extends Ta-Shma's analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way. Curiously, while Ta-Shma's explicit bias amplification derandomizes a well-known probabilistic argument (underlying the Gilbert-Varshamov bound), there seems to be no known probabilistic (or other existential) way of achieving our explicit ("high-dimensional") spectral amplification.
Details
- Title: Subtitle
- Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification
- Creators
- Fernando Granha Jeronimo - Institute for Advanced StudyTushant Mittal - University of ChicagoSourya Roy - University of California, RiversideAvi Wigderson - Institute for Advanced Study
- Resource Type
- Conference proceeding
- Publication Details
- 2022 IEEE 63RD ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS), pp.378-388
- Publisher
- IEEE
- Series
- Annual IEEE Symposium on Foundations of Computer Science
- DOI
- 10.1109/FOCS54457.2022.00043
- ISSN
- 0272-5428
- Number of pages
- 11
- Grant note
- CCF-1900460 / NSF; National Science Foundation (NSF)
- Language
- English
- Date published
- 01/01/2022
- Academic Unit
- Computer Science
- Record Identifier
- 9984446527602771
Metrics
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