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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

One of the most important properties of high dimensional expanders is that high dimensional random walks converge rapidly. This property has proven to be extremely useful in a variety of fields in the theory of computer science from agreement testing to sampling, coding theory and more. In this paper we present a state of the art result in a line of works analyzing the convergence of high dimensional random walks [Tali Kaufman and David Mass, 2017; Irit Dinur and Tali Kaufman, 2017; Tali Kaufman and Izhar Oppenheim, 2018; Vedat Levi Alev and Lap Chi Lau, 2020], by presenting a structured version of the result of [Vedat Levi Alev and Lap Chi Lau, 2020]. While previous works examined the expansion in the viewpoint of the worst possible eigenvalue, in this work we relate the expansion of a function to the entire spectrum of the random walk operator using the structure of the function; We call such a theorem a Fine Grained High Order Random Walk Theorem. In sufficiently structured cases the fine grained result that we present here can be much better than the worst case while in the worst case our result is equivalent to [Vedat Levi Alev and Lap Chi Lau, 2020].
In order to prove the Fine Grained High Order Random Walk Theorem we introduce a way to bootstrap the expansion of random walks on the vertices of a complex into a fine grained understanding of higher order random walks, provided that the expansion is good enough.
In addition, our single bootstrapping theorem can simultaneously yield our Fine Grained High Order Random Walk Theorem as well as the well known Trickling down Theorem. Prior to this work, High order Random walks theorems and Tricking down Theorem have been obtained from different proof methods.

Roy Gotlib and Tali Kaufman. Fine Grained Analysis of High Dimensional Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 49:1-49:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gotlib_et_al:LIPIcs.APPROX/RANDOM.2023.49, author = {Gotlib, Roy and Kaufman, Tali}, title = {{Fine Grained Analysis of High Dimensional Random Walks}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {49:1--49:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.49}, URN = {urn:nbn:de:0030-drops-188740}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.49}, annote = {Keywords: High Dimensional Expanders, High Dimensional Random Walks, Local Spectral Expansion, Local to Global, Trickling Down} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

One of the key components in PCP constructions are agreement tests. In agreement test the tester is given access to subsets of fixed size of some set, each equipped with an assignment. The tester is then tasked with testing whether these local assignments agree with some global assignment over the entire set. One natural generalization of this concept is the case where, instead of a single assignment to each local view, the tester is given access to l different assignments for every subset. The tester is then tasked with testing whether there exist l global functions that agree with all of the assignments of all of the local views. In this work we present sufficient condition for a set system to exhibit this generalized definition of list agreement expansion. This is, to our knowledge, the first work to consider this natural generalization of agreement testing.
Despite initially appearing very similar to agreement expansion in definition, proving that a set system exhibits list agreement expansion seem to require a different set of techniques. This is due to the fact that the natural extension of agreement testing (i.e. that there exists a pairing of the lists such that each pair agrees with each other) does not suffice when testing for list agreement as list agreement crucially relies on a global structure. It follows that if a local assignments satisfy list agreement they must not only agree locally but also exhibit some additional structure. In order to test for the existence of this additional structure we use the connection between covering spaces of a high dimensional complex and its coboundaries. Specifically, we use this connection as a form of "decoupling".
Moreover, we show that any set system that exhibits list agreement expansion also supports direct sum testing. This is the first scheme for direct sum testing that works regardless of the parity of the sizes of the local sets. Prior to our work the schemes for direct sum testing were based on the parity of the sizes of the local tests.

Roy Gotlib and Tali Kaufman. List Agreement Expansion from Coboundary Expansion. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 61:1-61:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gotlib_et_al:LIPIcs.ITCS.2023.61, author = {Gotlib, Roy and Kaufman, Tali}, title = {{List Agreement Expansion from Coboundary Expansion}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {61:1--61:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.61}, URN = {urn:nbn:de:0030-drops-175647}, doi = {10.4230/LIPIcs.ITCS.2023.61}, annote = {Keywords: High dimensional Expanders, Property Testing, Agreement Testing} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Local to global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland [Garland, 1973] to our days. In this work we develop a local to global machinery for general posets. We show that the high dimensional expansion notions and many recent expansion results have a generalization to posets. Examples are fast convergence of high dimensional random walks generalizing [Kaufman et al., 2020], [Alev and Lau, 2020], an equivalence with a global random walk definition, generalizing [Dikstein et al., 2018] and a trickling down theorem, generalizing [Oppenheim, 2018].
In particular, we show that some posets, such as the Grassmannian poset, exhibit qualitatively stronger trickling down effect than simplicial complexes.
Using these methods, and the novel idea of posetification to Ramanujan complexes [Lubotzky et al., 2005a], [Lubotzky et al., 2005b], we construct a constant degree expanding Grassmannian poset, and analyze its expansion. This it the first construction of such object, whose existence was conjectured in [Dikstein et al., 2018].

Tali Kaufman and Ran J. Tessler. Garland’s Technique for Posets and High Dimensional Grassmannian Expanders. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 78:1-78:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kaufman_et_al:LIPIcs.ITCS.2023.78, author = {Kaufman, Tali and Tessler, Ran J.}, title = {{Garland’s Technique for Posets and High Dimensional Grassmannian Expanders}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {78:1--78:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.78}, URN = {urn:nbn:de:0030-drops-175819}, doi = {10.4230/LIPIcs.ITCS.2023.78}, annote = {Keywords: High dimensional Expanders, Posets, Grassmannian, Garland Method} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Recent works have shown that expansion of pseudorandom sets is of great importance. However, all current works on pseudorandom sets are limited only to product (or approximate product) spaces, where Fourier Analysis methods could be applied. In this work we ask the natural question whether pseudorandom sets are relevant in domains where Fourier Analysis methods cannot be applied, e.g., one-sided local spectral expanders.
We take the first step in the path of answering this question. We put forward a new definition for pseudorandom sets, which we call "double balanced sets". We demonstrate the strength of our new definition by showing that small double balanced sets in one-sided local spectral expanders have very strong expansion properties, such as unique-neighbor-like expansion. We further show that cohomologies in cosystolic expanders are double balanced, and use the newly derived strong expansion properties of double balanced sets in order to obtain an exponential improvement over the current state of the art lower bound on their minimal distance.

Tali Kaufman and David Mass. Double Balanced Sets in High Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 3:1-3:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kaufman_et_al:LIPIcs.APPROX/RANDOM.2022.3, author = {Kaufman, Tali and Mass, David}, title = {{Double Balanced Sets in High Dimensional Expanders}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {3:1--3:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.3}, URN = {urn:nbn:de:0030-drops-171257}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.3}, annote = {Keywords: High dimensional expanders, Double balanced sets, Pseudorandom functions} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

In this work, we define a notion of local testability of codes that is strictly stronger than the basic one (studied e.g., by recent works on high rate LTCs), and we term it amplified local testability. Amplified local testability is a notion close to the result of optimal testing for Reed-Muller codes achieved by Bhattacharyya et al.
We present a scheme to get amplified locally testable codes from high dimensional expanders. We show that single orbit Affine invariant codes, and in particular Reed-Muller codes, can be described via our scheme, and hence are amplified locally testable. This gives the strongest currently known testability result of single orbit affine invariant codes, strengthening the celebrated result of Kaufman and Sudan.

Tali Kaufman and Izhar Oppenheim. High Dimensional Expansion Implies Amplified Local Testability. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 5:1-5:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kaufman_et_al:LIPIcs.APPROX/RANDOM.2022.5, author = {Kaufman, Tali and Oppenheim, Izhar}, title = {{High Dimensional Expansion Implies Amplified Local Testability}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {5:1--5:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.5}, URN = {urn:nbn:de:0030-drops-171276}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.5}, annote = {Keywords: Locally testable codes, High dimensional expanders, Amplified testing} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Fast mixing of random walks on hypergraphs (simplicial complexes) has recently led to myriad breakthroughs throughout theoretical computer science. Many important applications, however, (e.g. to LTCs, 2-2 games) rely on a more general class of underlying structures called posets, and crucially take advantage of non-simplicial structure. These works make it clear that the global expansion properties of posets depend strongly on their underlying architecture (e.g. simplicial, cubical, linear algebraic), but the overall phenomenon remains poorly understood. In this work, we quantify 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.
We show that the spectra of walks on expanding posets (Dikstein, Dinur, Filmus, Harsha APPROX-RANDOM 2018) concentrate in strips around a small number of approximate eigenvalues controlled by the regularity of the underlying poset. This gives a simple condition to identify poset architectures (e.g. the Grassmann) that exhibit strong (even exponential) decay of eigenvalues, versus architectures like hypergraphs whose eigenvalues decay linearly - a crucial distinction in applications to hardness of approximation and agreement testing such as the recent proof of the 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). We show these results lead to a tight characterization of edge-expansion on expanding posets in the 𝓁₂-regime (generalizing recent work of Bafna, Hopkins, Kaufman, and Lovett (SODA 2022)), and pay special attention to the case of the Grassmann where we show our results are tight for a natural set of sparsifications of the Grassmann graphs. We note for clarity that our results do not recover the characterization of expansion used in the proof of the 2-2 Games Conjecture which relies on 𝓁_∞ rather than 𝓁₂-structure.

Jason Gaitonde, Max Hopkins, Tali Kaufman, Shachar Lovett, and Ruizhe Zhang. Eigenstripping, Spectral Decay, and Edge-Expansion on Posets. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 16:1-16:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gaitonde_et_al:LIPIcs.APPROX/RANDOM.2022.16, author = {Gaitonde, Jason and Hopkins, Max and Kaufman, Tali and Lovett, Shachar and Zhang, Ruizhe}, title = {{Eigenstripping, Spectral Decay, and Edge-Expansion on Posets}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {16:1--16:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.16}, URN = {urn:nbn:de:0030-drops-171381}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.16}, annote = {Keywords: High-dimensional expanders, posets, eposets} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

In recent years, high dimensional expanders have been found to have a variety of applications in theoretical computer science, such as efficient CSPs approximations, improved sampling and list-decoding algorithms, and more. Within that, an important high dimensional expansion notion is cosystolic expansion, which has found applications in the construction of efficiently decodable quantum codes and in proving lower bounds for CSPs.
Cosystolic expansion is considered with systems of equations over a group where the variables and equations correspond to faces of the complex. Previous works that studied cosystolic expansion were tailored to the specific group 𝔽₂. In particular, Kaufman, Kazhdan and Lubotzky (FOCS 2014), and Evra and Kaufman (STOC 2016) in their breakthrough works, who solved a famous open question of Gromov, have studied a notion which we term "parity" expansion for small sets. They showed that small sets of k-faces have proportionally many (k+1)-faces that contain an odd number of k-faces from the set. Parity expansion for small sets could then be used to imply cosystolic expansion only over 𝔽₂.
In this work we introduce a stronger unique-neighbor-like expansion for small sets. We show that small sets of k-faces have proportionally many (k+1)-faces that contain exactly one k-face from the set. This notion is fundamentally stronger than parity expansion and cannot be implied by previous works.
We then show, utilizing the new unique-neighbor-like expansion notion introduced in this work, that cosystolic expansion can be made group-independent, i.e., unique-neighbor-like expansion for small sets implies cosystolic expansion over any group.

Tali Kaufman and David Mass. Unique-Neighbor-Like Expansion and Group-Independent Cosystolic Expansion. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 56:1-56:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kaufman_et_al:LIPIcs.ISAAC.2021.56, author = {Kaufman, Tali and Mass, David}, title = {{Unique-Neighbor-Like Expansion and Group-Independent Cosystolic Expansion}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {56:1--56:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.56}, URN = {urn:nbn:de:0030-drops-154898}, doi = {10.4230/LIPIcs.ISAAC.2021.56}, annote = {Keywords: High dimensional expanders, Unique-neighbor-like expansion, Cosystolic expansion} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to spectral expansion. In higher dimensions this is not the case: a simplicial complex can be spectrally expanding but not have high dimensional edge-expansion. The phenomenon of high dimensional edge expansion in higher dimensions is much more involved than spectral expansion, and is far from being understood. In particular, prior to this work, the only known bounded degree cosystolic expanders were derived from the theory of buildings that is far from being elementary.
In this work we study high dimensional complexes which are strongly symmetric. Namely, there is a group that acts transitively on top dimensional cells of the simplicial complex [e.g., for graphs it corresponds to a group that acts transitively on the edges]. Using the strong symmetry, we develop a new machinery to prove coboundary and cosystolic expansion.
It was an open question whether the recent elementary construction of bounded degree spectral high dimensional expanders based on coset complexes give rise to bounded degree cosystolic expanders. In this work we answer this question affirmatively. We show that these complexes give rise to bounded degree cosystolic expanders in dimension two, and that their links are (two-dimensional) coboundary expanders. We do so by exploiting the strong symmetry properties of the links of these complexes using a new machinery developed in this work.
Previous works have shown a way to bound the co-boundary expansion using strong symmetry in the special situation of "building like" complexes. Our new machinery shows how to get coboundary expansion for general strongly symmetric coset complexes, which are not necessarily "building like", via studying the (Dehn function of the) presentation of the symmetry group of these complexes.

Tali Kaufman and Izhar Oppenheim. Coboundary and Cosystolic Expansion from Strong Symmetry. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 84:1-84:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kaufman_et_al:LIPIcs.ICALP.2021.84, author = {Kaufman, Tali and Oppenheim, Izhar}, title = {{Coboundary and Cosystolic Expansion from Strong Symmetry}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {84:1--84:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.84}, URN = {urn:nbn:de:0030-drops-141539}, doi = {10.4230/LIPIcs.ICALP.2021.84}, annote = {Keywords: High dimensional expanders, Cosystolic expansion, Coboundary expansion} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We generalize the expander Chernoff bound to high-dimensional expanders. The expander Chernoff bound is an essential property of expanders, first proved by Gillman [Gillman, 1993]. Given a graph G and a function f on the vertices, it states that the probability of f’s mean sampled via a random walk on G to deviate from its actual mean, has a bound that depends on the spectral gap of the walk and decreases exponentially as the walk’s length increases.
We are interested in obtaining an analog Chernoff bound for high order walks on high-dimensional expanders. A naive generalization of the expander Chernoff bound from expander graphs to high-dimensional expanders gives a very poor bound due to obstructions that occur in high-dimensional expanders and are not present in (one-dimensional) expander graphs. Because of these obstructions, the spectral gap of high-order random walks is inherently small.
A natural question that arises is how to get a meaningful Chernoff bound for high-dimensional expanders. In this paper, we manage to get a strong Chernoff bound for high-dimensional expanders by looking beyond the spectral gap.
First, we prove an expander Chernoff bound that depends on a notion that we call the "shrinkage of a function" instead of the spectral gap. In one-dimensional expanders, the shrinkage of any function with zero-mean is bounded by λ(M). Therefore, the spectral gap is just the one-dimensional manifestation of the shrinkage.
Next, we show that in good high-dimensional expanders, the shrinkage of functions that "do not come from below" is good. A function does not come from below if from any local point of view (called "link") its mean is zero.
Finally, we prove a high-dimensional Chernoff bound that captures the expansion of the complex. When the function on the faces has a small variance and does not "come from below", our bound is better than the naive high-dimensional expander Chernoff bound.

Tali Kaufman and Ella Sharakanski. Chernoff Bound for High-Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 25:1-25:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kaufman_et_al:LIPIcs.APPROX/RANDOM.2020.25, author = {Kaufman, Tali and Sharakanski, Ella}, title = {{Chernoff Bound for High-Dimensional Expanders}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {25:1--25:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.25}, URN = {urn:nbn:de:0030-drops-126287}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.25}, annote = {Keywords: High Dimensional Expanders, Random Walks, Tail Bounds} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Agreement expansion is concerned with set systems for which local assignments to the sets with almost perfect pairwise consistency (i.e., most overlapping pairs of sets agree on their intersections) implies the existence of a global assignment to the ground set (from which the sets are defined) that agrees with most of the local assignments.
It is currently known that if a set system forms a two-sided or a partite high dimensional expander then agreement expansion is implied. However, it was not known whether agreement expansion can be implied for one-sided high dimensional expanders.
In this work we show that agreement expansion can be deduced for one-sided high dimensional expanders assuming that all the vertices' links (i.e., the neighborhoods of the vertices) are agreement expanders. Thus, for one-sided high dimensional expander, an agreement expansion of the large complicated complex can be deduced from agreement expansion of its small simple links.
Using our result, we settle the open question whether the well studied Ramanujan complexes are agreement expanders. These complexes are neither partite nor two-sided high dimensional expanders. However, they are one-sided high dimensional expanders for which their links are partite and hence are agreement expanders. Thus, our result implies that Ramanujan complexes are agreement expanders, answering affirmatively the aforementioned open question.
The local-to-global agreement expansion that we prove is based on the variance method that we develop. We show that for a high dimensional expander, if we define a function on its top faces and consider its local averages over the links then the variance of these local averages is much smaller than the global variance of the original function. This decreasing in the variance enables us to construct one global agreement function that ties together all local agreement functions.

Tali Kaufman and David Mass. Local-To-Global Agreement Expansion via the Variance Method. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 74:1-74:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kaufman_et_al:LIPIcs.ITCS.2020.74, author = {Kaufman, Tali and Mass, David}, title = {{Local-To-Global Agreement Expansion via the Variance Method}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {74:1--74:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.74}, URN = {urn:nbn:de:0030-drops-117597}, doi = {10.4230/LIPIcs.ITCS.2020.74}, annote = {Keywords: Agreement testing, High dimensional expanders, Local-to-global, Variance method} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

In this work, using methods from high dimensional expansion, we show that the property of k-direct-sum is testable for odd values of k . Previous work of [Kaufman and Lubotzky, 2014] could inherently deal only with the case that k is even, using a reduction to linearity testing. Interestingly, our work is the first to combine the topological notion of high dimensional expansion (called co-systolic expansion) with the combinatorial/spectral notion of high dimensional expansion (called colorful expansion) to obtain the result.
The classical k-direct-sum problem applies to the complete complex; Namely it considers a function defined over all k-subsets of some n sized universe. Our result here applies to any collection of k-subsets of an n-universe, assuming this collection of subsets forms a high dimensional expander.

Roy Gotlib and Tali Kaufman. Testing Odd Direct Sums Using High Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 50:1-50:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gotlib_et_al:LIPIcs.APPROX-RANDOM.2019.50, author = {Gotlib, Roy and Kaufman, Tali}, title = {{Testing Odd Direct Sums Using High Dimensional Expanders}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {50:1--50:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.50}, URN = {urn:nbn:de:0030-drops-112651}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.50}, annote = {Keywords: High Dimensional Expanders, Property Testing, Direct Sum} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

A local tester for an error-correcting code is a probabilistic procedure that queries a small subset of coordinates, accepts codewords with probability one, and rejects non-codewords with probability proportional to their distance from the code. The local tester is robust if for non-codewords it satisfies the stronger property that the average distance of local views from accepting views is proportional to the distance from the code. Robust testing is an important component in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests.
In this work we show that for certain codes, any (natural) local tester can be converted to a roubst tester with roughly the same number of queries. Our result holds for the class of affine-invariant lifted codes which is a broad class of codes that includes Reed-Muller codes, as well as recent constructions of high-rate locally testable codes (Guo, Kopparty, and Sudan, ITCS 2013). Instantiating this with known local testing results for lifted codes gives a more direct proof that improves some of the parameters of the main result of Guo, Haramaty, and Sudan (FOCS 2015), showing robustness of lifted codes.
To obtain the above transformation we relate the notions of local testing and robust testing to the notion of agreement testing that attempts to find out whether valid partial assignments can be stitched together to a global codeword. We first show that agreement testing implies robust testing, and then show that local testing implies agreement testing. Our proof is combinatorial, and is based on expansion / sampling properties of the collection of local views of local testers. Thus, it immediately applies to local testers of lifted codes that query random affine subspaces in F_q^m, and moreover seems amenable to extension to other families of locally testable codes with expanding families of local views.

Irit Dinur, Prahladh Harsha, Tali Kaufman, and Noga Ron-Zewi. From Local to Robust Testing via Agreement Testing. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 29:1-29:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dinur_et_al:LIPIcs.ITCS.2019.29, author = {Dinur, Irit and Harsha, Prahladh and Kaufman, Tali and Ron-Zewi, Noga}, title = {{From Local to Robust Testing via Agreement Testing}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {29:1--29:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.29}, URN = {urn:nbn:de:0030-drops-101221}, doi = {10.4230/LIPIcs.ITCS.2019.29}, annote = {Keywords: Local testing, Robust testing, Agreement testing, Affine-invariant codes, Lifted codes} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

We study high order random walks on high dimensional expanders on simplicial complexes (i.e., hypergraphs). These walks walk from a k-face (i.e., a k-hyperedge) to a k-face if they are both contained in a k+1-face (i.e, a k+1 hyperedge). This naturally generalizes the random walks on graphs that walk from a vertex (0-face) to a vertex if they are both contained in an edge (1-face).
Recent works have studied the spectrum of high order walks operators and deduced fast mixing. However, the spectral gap of high order walks operators is inherently small, due to natural obstructions (called coboundaries) that do not happen for walks on expander graphs.
In this work we go beyond spectral gap, and relate the expansion of a function on k-faces (called k-cochain, for k=0, this is a function on vertices) to its structure.
We show a Decomposition Theorem: For every k-cochain defined on high dimensional expander, there exists a decomposition of the cochain into i-cochains such that the square norm of the k-cochain is a sum of the square norms of the i-chains and such that the more weight the k-cochain has on higher levels of the decomposition the better is its expansion, or equivalently, the better is its shrinkage by the high order random walk operator.
The following corollaries are implied by the Decomposition Theorem:
- We characterize highly expanding k-cochains as those whose mass is concentrated on the highest levels of the decomposition that we construct. For example, a function on edges (i.e. a 1-cochain) which is locally thin (i.e. it contains few edges through every vertex) is highly expanding, while a function on edges that contains all edges through a single vertex is not highly expanding.
- We get optimal mixing for high order random walks on Ramanujan complexes. Ramanujan complexes are recently discovered bounded degree high dimensional expanders. The optimality in their mixing that we prove here, enable us to get from them more efficient Two-Layer-Samplers than those presented by the previous work of Dinur and Kaufman.

Tali Kaufman and Izhar Oppenheim. High Order Random Walks: Beyond Spectral Gap. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 47:1-47:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kaufman_et_al:LIPIcs.APPROX-RANDOM.2018.47, author = {Kaufman, Tali and Oppenheim, Izhar}, title = {{High Order Random Walks: Beyond Spectral Gap}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {47:1--47:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.47}, URN = {urn:nbn:de:0030-drops-94516}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.47}, annote = {Keywords: High Dimensional Expanders, Simplicial Complexes, Random Walk} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Random walks on bounded degree expander graphs have numerous applications, both in theoretical and practical computational problems. A key property of these walks is that they converge rapidly to their stationary distribution.
In this work we define high order random walks: These are generalizations of random walks on graphs to high dimensional simplicial complexes, which are the high dimensional analogues of graphs. A simplicial complex of dimension d has vertices, edges, triangles, pyramids, up to d-dimensional cells. For any 0 \leq i < d, a high order random walk on dimension i moves between neighboring i-faces (e.g., edges) of the complex, where two i-faces are considered neighbors if they share a common (i+1)-face (e.g., a triangle). The case of i=0 recovers the well studied random walk on graphs.
We provide a local-to-global criterion on a complex which implies rapid convergence of all high order random walks on it. Specifically, we prove that if the 1-dimensional skeletons of all the links of a complex are spectral expanders, then for all 0 \le i < d the high order random walk on dimension i converges rapidly to its stationary distribution.
We derive our result through a new notion of high dimensional combinatorial expansion of complexes which we term colorful expansion. This notion is a natural generalization of combinatorial expansion of graphs and is strongly related to the convergence rate of the high order random walks.
We further show an explicit family of bounded degree complexes which satisfy this criterion. Specifically, we show that Ramanujan complexes meet this criterion, and thus form an explicit family of bounded degree high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution.

Tali Kaufman and David Mass. High Dimensional Random Walks and Colorful Expansion. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 4:1-4:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kaufman_et_al:LIPIcs.ITCS.2017.4, author = {Kaufman, Tali and Mass, David}, title = {{High Dimensional Random Walks and Colorful Expansion}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {4:1--4:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.4}, URN = {urn:nbn:de:0030-drops-81838}, doi = {10.4230/LIPIcs.ITCS.2017.4}, annote = {Keywords: High dimensional expanders, expander graphs, random walks} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map X -> R^d there exists a point p in R^d whose preimage intersects a positive fraction mu > 0 of the d-cells of X. More generally, the conclusion holds if R^d is replaced by any d-dimensional piecewise-linear (PL) manifold M, with a constant \mu that depends only on d and on the expansion properties of X, but not on M.

Dominic Dotterrer, Tali Kaufman, and Uli Wagner. On Expansion and Topological Overlap. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 35:1-35:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dotterrer_et_al:LIPIcs.SoCG.2016.35, author = {Dotterrer, Dominic and Kaufman, Tali and Wagner, Uli}, title = {{On Expansion and Topological Overlap}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {35:1--35:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.35}, URN = {urn:nbn:de:0030-drops-59270}, doi = {10.4230/LIPIcs.SoCG.2016.35}, annote = {Keywords: Combinatorial Topology, Selection Lemmas, Higher-Dimensional Expanders} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8341, Sublinear Algorithms (2008)

For Boolean functions that are $epsilon$-far from the set of linear functions,
we study the lower bound on the rejection probability (denoted by $extsc{rej}(epsilon)$) of the linearity test suggested by Blum, Luby and Rubinfeld.
This problem is arguably the most fundamental and extensively studied problem in property testing of Boolean functions.
The previously best bounds for $extsc{rej}(epsilon)$ were obtained by Bellare,
Coppersmith, H{{a}}stad, Kiwi and Sudan. They used Fourier analysis
to show that $ extsc{rej}(epsilon) geq e$ for every $0 leq epsilon leq
frac{1}{2}$. They also conjectured that this bound might not be tight for
$epsilon$'s which are close to $1/2$. In this paper we show that this indeed is
the case. Specifically, we improve the lower bound of $ extsc{rej}(epsilon) geq
epsilon$ by an additive constant that depends only on $epsilon$:
$extsc{rej}(epsilon) geq epsilon + min {1376epsilon^{3}(1-2epsilon)^{12},
frac{1}{4}epsilon(1-2epsilon)^{4}}$, for every $0 leq epsilon leq frac{1}{2}$.
Our analysis is based on a relationship between $extsc{rej}(epsilon)$ and the
weight distribution of a coset of the Hadamard code. We use both Fourier
analysis and coding theory tools to estimate this weight distribution.

Tali Kaufman, Simon Litsyn, and Ning Xie. Breaking the $\epsilon$-Soundness Bound of the Linearity Test over GF(2). In Sublinear Algorithms. Dagstuhl Seminar Proceedings, Volume 8341, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{kaufman_et_al:DagSemProc.08341.3, author = {Kaufman, Tali and Litsyn, Simon and Xie, Ning}, title = {{Breaking the \$\backslashepsilon\$-Soundness Bound of the Linearity Test over GF(2)}}, booktitle = {Sublinear Algorithms}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8341}, editor = {Artur Czumaj and S. Muthu Muthukrishnan and Ronitt Rubinfeld and Christian Sohler}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08341.3}, URN = {urn:nbn:de:0030-drops-16971}, doi = {10.4230/DagSemProc.08341.3}, annote = {Keywords: Linearity test, Fourier analysis, coding theory} }

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