High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games

Mitali Bafna, Max Hopkins, Tali Kaufman, Shachar Lovett

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations


Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible amount of study and application since their introduction by Kaufman and Mass (ITCS 2016), yet their broader combinatorial and spectral properties remain poorly understood. We develop a combinatorial characterization of the spectral structure of HD-walks on two-sided local-spectral expanders (Dinur and Kaufman FOCS 2017), which offer a broad generalization of the well-studied Johnson and Grassmann graphs. Our characterization, which shows that the spectra of HD-walks lie tightly concentrated in a few combinatorially structured strips, leads to novel structural theorems such as a tight ℓ2-characterization of edge-expansion, as well as to a new understanding of local-to-global graph algorithms on HDX. Towards the latter, we introduce a novel spectral complexity measure called Stripped Threshold Rank, and show how it can replace the (much larger) threshold rank as a parameter controlling the performance of algorithms on structured objects. Combined with a sum-of-squares proof for the former ℓ2-characterization, we give a concrete application of this framework to algorithms for unique games on HD-walks, where in many cases we improve the state of the art (Barak, Raghavendra, and Steurer FOCS 2011, and Arora, Barak, and Steurer JACM 2015) from nearly-exponential to polynomial time (e.g. for sparsifications of Johnson graphs or of slices of the q-ary hypercube). Our characterization of expansion also holds an interesting connection to hardness of approximation, where an ℓ∞-variant for the Grassmann graphs was recently used to resolve the 2-2 Games Conjecture (Khot, Minzer, and Safra FOCS 2018). We give a reduction from a related ℓ∞-variant to our ℓ2-characterization, but it loses factors in the regime of interest for hardness where the gap between ℓ2 and ℓ∞ structure is large. Nevertheless, our results open the door for further work on the use of HDX in hardness of approximation and their general relation to unique games.

Original languageEnglish
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2022
PublisherAssociation for Computing Machinery
Number of pages60
ISBN (Electronic)9781611977073
StatePublished - 2022
Externally publishedYes
Event33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 - Alexander, United States
Duration: 9 Jan 202212 Jan 2022

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022
Country/TerritoryUnited States

Bibliographical note

Publisher Copyright:
Copyright © 2021 by SIAM Unauthorized reproduction of this article is prohibited.


∗Department of Computer Science, Harvard University, MA 02138. Email: [email protected]. Supported in part by the Simons Investigator Award to Madhu Sudan. †Department of Computer Science and Engineering, UCSD, CA 92092. Email: [email protected]. Supported by NSF Award DGE-1650112. ‡Department of Computer Science, Bar-Ilan University. Email: [email protected]. Supported by ERC and BSF. §Department of Computer Science and Engineering, UCSD, CA 92092. Email: [email protected]. Supported by NSF Award CCF-1953928.

FundersFunder number
National Science FoundationDGE-1650112
Bloom's Syndrome FoundationCCF-1953928
European Research Council


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