• Corpus ID: 244728308

Hypercontractivity on High Dimensional Expanders: Approximate Efron-Stein Decompositions for $\epsilon$-Product Spaces

@article{Gur2021HypercontractivityOH,
  title={Hypercontractivity on High Dimensional Expanders: Approximate Efron-Stein Decompositions for \$\epsilon\$-Product Spaces},
  author={Tom Gur and Noam Lifshitz and Siqi Liu},
  journal={Electron. Colloquium Comput. Complex.},
  year={2021},
  volume={TR21}
}
Abstract. We prove hypercontractive inequalities on high dimensional expanders. As in the settings of the p-biased hypercube, the symmetric group, and the Grassmann scheme, our inequalities are effective for global functions, which are functions that are not significantly affected by a restriction of a small set of coordinates. As applications, we obtain Fourier concentration, small-set expansion, and Kruskal– Katona theorems for high dimensional expanders. Our techniques rely on a new… 

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

This work quantifies the advantage of different poset architectures in both a spectral and combinatorial sense, highlighting how regularity controls the spectral decay and edge-expansion of corresponding random walks.

References

SHOWING 1-10 OF 43 REFERENCES

Hypercontractivity on high dimensional expanders

This work develops a new theory of hypercontractivity on high dimensional expanders (HDX), an important class of expanding complexes that has recently seen similarly impressive applications in both coding theory and approximate sampling and leads to a new understanding of the structure of Boolean functions on HDX.

Hypercontractivity on the symmetric group

This work considers the symmetric group, S_n, one of the most basic non-product domains, and establishes hypercontractive inequalities on it, that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube.

High-Dimensional Expanders from Expanders

We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary

Mixing in High-Dimensional Expanders

This paper removes the assumption of a complete skeleton, showing that simultaneous concentration of the Laplace spectra in all dimensions implies pseudo-randomness in any complex.

Improved Product-Based High-Dimensional Expanders

  • Louis Golowich
  • Mathematics, Computer Science
    Electron. Colloquium Comput. Complex.
  • 2021
This paper introduces an improved combinatorial high-dimensional expander construction, by modifying a previous construction of Liu, Mohanty, and Yang (ITCS 2020), which is based on a high- dimensional variant of a tensor product.

Construction of new local spectral high dimensional expanders

This work constructs new families of bounded degree high dimensional expanders obeying the local spectral expansion property, a property that implies, geometric overlapping, fast mixing of high dimensional random walks, agreement testing and agreement expansion.

Hypercontractivity for global functions and sharp thresholds

The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the KKL

Boolean function analysis on high-dimensional expanders

The results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model.

Explicit SoS lower bounds from high-dimensional expanders

An explicit family of 3XOR instances which is hard for O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy is constructed, based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, and the analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov.

Testing Odd Direct Sums Using High Dimensional Expanders

This work shows that the property of k-direct-sum is testable for odd values of k, and is the first to combine the topological notion of high dimensional expansion (called co-systolic expansion) with the combinatorial/spectral notion ofHighdimensional expansion ( called colorful expansion) to obtain the result.