arXiv subjects

Max Hopkins

Publications and source records attributed to Max Hopkins.

At least 19 recordsLinked to original sources

arXiv2026

Toward a KKL Theorem for any HDX

The KKL Theorem, a seminal result in boolean function analysis, characterizes the structure of low-influence (non-expanding) functions on the hypercube. While recent years have seen breakthrough results across a variety of areas relying on analogs of the KKL Theorem beyond the cube (e.g., on product spaces, Grassmann graphs), further progress has been inhibited by our poor understanding of the phenomenon across more general domains. Motivated in this context, Bafna, Hopkins, Kaufman, and Lovett (STOC 2022) and Gur, Lifshitz, and Liu (STOC 2022) proved a generalized KKL-type Theorem for spectral high dimensional expanders (HDX). Their results, however, remain highly restricted due to strong quantitative expansion requirements on the underlying complex. In this work, we introduce a simple local-to-global method for analyzing low influence functions on simplicial complexes. Using this method we prove a local-to-global KKL-type Theorem: any simplicial complex whose links satisfy a KKL-Theorem also satisfies such a result globally. Building on Gotlib and Kaufman (RANDOM 2023), we also prove a weaker dimension-dependent KKL-type Theorem for simplicial complexes with any non-trivial (two-sided) expansion. As concrete applications of our framework, we give the first characterization of non-expanding functions on `combinatorial' HDX such as dense clique complexes and a corresponding Kruskal-Katona Theorem, as well as a small-set expansion theorem for the Ramanujan Complexes of Lubotzky, Samuels, and Vishne (EJC '05).

math.CO↗

arXiv2026

A Simple Sub-Polynomial Degree Coboundary Expander

High dimensional expanders simultaneously satisfying spectral and combinatorial (coboundary) expansion have recently played a major role in breakthroughs in PCP and coding theory, but the only known construction of such complexes is extremely involved, requiring deep algebraic number theory. In this work, we give an extremely simple combinatorial construction of a sub-polynomial degree complex based on projections of the flags complex (subspace chains) that is (i) a local spectral expander, (ii) a coboundary expander, and (iii) a swap coboundary expander. As a corollary, we also give the first near-linear size combinatorial hypergraphs with good agreement tests in the '1%' regime, and a simple PCP construction with near-linear size.

math.CO↗

arXiv2026

Non-Signaling Locality Lower Bounds for Dominating Set

Minimum dominating set is a basic local covering problem and a core task in distributed computing. Despite extensive study, in the classic LOCAL model there exist significant gaps between known algorithms and lower bounds. Chang and Li prove an $\Omega(\log n)$-locality lower bound for a constant factor approximation, while Kuhn--Moscibroda--Wattenhofer gave an algorithm beating this bound beyond $\log \Delta$-approximation, along with a weaker lower bound for this degree-dependent setting scaling roughly with $\min\{\log \Delta/\log\log \Delta,\sqrt{\log n/\log\log n}\}$. Unfortunately, this latter bound is weak for small $\Delta$, and never recovers the Chang--Li bound, leaving central questions: does $O(\log \Delta)$-approximation require $\Omega(\log n)$ locality, and do such bounds extend beyond LOCAL? In this work, we take a major step toward answering these questions in the non-signaling model, which strictly subsumes the LOCAL, quantum-LOCAL, and bounded-dependence settings. We prove every $O(\log\Delta)$-approximate non-signaling distribution for dominating set requires locality $\Omega(\log n/(\log\Delta \cdot \mathrm{poly}\log\log\Delta))$. Further, we show for some $\beta \in (0,1)$, every $O(\log^\beta \Delta)$-approximate non-signaling distribution requires locality $\Omega(\log n/\log\Delta)$, which combined with the KMW bound yields a degree-independent $\Omega(\sqrt{\log n/\log\log n})$ quantum-LOCAL lower bound for $O(\log^\beta\Delta)$-approximation algorithms. The proof is based on two new low-soundness sensitivity lower bounds for label cover, one via Impagliazzo--Kabanets--Wigderson-style parallel repetition with degree reduction and one from a sensitivity-preserving reworking of the Dinur--Harsha framework, together with the reductions from label cover to set cover to dominating set and the sensitivity-to-locality transfer theorem of Fleming and Yoshida.

cs.DS↗

arXiv2026

High Rate Efficient Local List Decoding from HDX

We construct the first (locally computable, approximately) locally list decodable codes with rate, efficiency, and error tolerance approaching the information theoretic limit, a core regime of interest for the complexity theoretic task of hardness amplification. Our algorithms run in polylogarithmic time and sub-logarithmic depth, which together with classic constructions in the unique decoding (low-noise) regime leads to the resolution of several long-standing problems in coding and complexity theory: 1. Near-optimally input-preserving hardness amplification (and corresponding fast PRGs) 2. Constant rate codes with $\log(N)$-depth list decoding (RNC$^1$) 3. Complexity-preserving distance amplification Our codes are built on the powerful theory of (local-spectral) high dimensional expanders (HDX). At a technical level, we make two key contributions. First, we introduce a new framework for ($\mathrm{polylog(N)}$-round) belief propagation on HDX that leverages a mix of local correction and global expansion to control error build-up while maintaining high rate. Second, we introduce the notion of strongly explicit local routing on HDX, local algorithms that given any two target vertices, output a random path between them in only polylogarithmic time (and, preferably, sub-logarithmic depth). Constructing such schemes on certain coset HDX allows us to instantiate our otherwise combinatorial framework in polylogarithmic time and low depth, completing the result.

cs.CC↗

arXiv2025

Approximate Replicability in Learning

Replicability, introduced by (Impagliazzo et al. STOC '22), is the notion that algorithms should remain stable under a resampling of their inputs (given access to shared randomness). While a strong and interesting notion of stability, the cost of replicability can be prohibitive: there is no replicable algorithm, for instance, for tasks as simple as threshold learning (Bun et al. STOC '23). Given such strong impossibility results we ask: under what approximate notions of replicability is learning possible? In this work, we propose three natural relaxations of replicability in the context of PAC learning: (1) Pointwise: the learner must be consistent on any fixed input, but not across all inputs simultaneously, (2) Approximate: the learner must output hypotheses that classify most of the distribution consistently, (3) Semi: the algorithm is fully replicable, but may additionally use shared unlabeled samples. In all three cases, for constant replicability we obtain close to sample-optimal agnostic PAC learners: 1) and 2) are achievable using $O(d/\alpha^2 + 1/\alpha^{4})$ samples, while 3) requires $\Theta(d^2/\alpha^2)$ labeled samples.

cs.LG↗

arXiv2025

From Generative to Episodic: Sample-Efficient Replicable Reinforcement Learning

The epidemic failure of replicability across empirical science and machine learning has recently motivated the formal study of replicable learning algorithms [Impagliazzo et al. (2022)]. In batch settings where data comes from a fixed i.i.d. source (e.g., hypothesis testing, supervised learning), the design of data-efficient replicable algorithms is now more or less understood. In contrast, there remain significant gaps in our knowledge for control settings like reinforcement learning where an agent must interact directly with a shifting environment. Karbasi et. al show that with access to a generative model of an environment with $S$ states and $A$ actions (the RL 'batch setting'), replicably learning a near-optimal policy costs only $\tilde{O}(S^2A^2)$ samples. On the other hand, the best upper bound without a generative model jumps to $\tilde{O}(S^7 A^7)$ [Eaton et al. (2024)] due to the substantial difficulty of environment exploration. This gap raises a key question in the broader theory of replicability: Is replicable exploration inherently more expensive than batch learning? Is sample-efficient replicable RL even possible? In this work, we (nearly) resolve this problem (for low-horizon tabular MDPs): exploration is not a significant barrier to replicable learning! Our main result is a replicable RL algorithm on $\tilde{O}(S^2A)$ samples, bridging the gap between the generative and episodic settings. We complement this with a matching $\tilde{\Omega}(S^2A)$ lower bound in the generative setting (under the common parallel sampling assumption) and an unconditional lower bound in the episodic setting of $\tilde{\Omega}(S^2)$ showcasing the near-optimality of our algorithm with respect to the state space $S$.

cs.LG↗

arXiv2025

The Role of Randomness in Stability

Stability is a central property in learning and statistics promising the output of an algorithm $A$ does not change substantially when applied to similar datasets $S$ and $S'$. It is an elementary fact that any sufficiently stable algorithm (e.g.\ one returning the same result with high probability, satisfying privacy guarantees, etc.) must be randomized. This raises a natural question: can we quantify how much randomness is needed for algorithmic stability? We study the randomness complexity of two influential notions of stability in learning: replicability, which promises $A$ usually outputs the same result when run over samples from the same distribution (and shared random coins), and differential privacy, which promises the output distribution of $A$ remains similar under neighboring datasets. The randomness complexity of these notions was studied recently in (Dixon et al. ICML 2024) and (Cannone et al. ITCS 2024) for basic $d$-dimensional tasks (e.g. estimating the bias of $d$ coins), but little is known about the measures more generally or in complex settings like classification. Toward this end, we prove a `weak-to-strong' boosting theorem for stability: the randomness complexity of a task $M$ (either under replicability or DP) is tightly controlled by the best replication probability of any deterministic algorithm solving the task, a weak measure called `global stability' that is universally capped at $\frac{1}{2}$ (Chase et al. FOCS 2023). Using this, we characterize the randomness complexity of PAC Learning: a class has bounded randomness complexity iff it has finite Littlestone dimension, and moreover scales at worst logarithmically in the excess error of the learner. This resolves a question of (Chase et al. STOC 2024) who asked for such a characterization in the equivalent language of (error-dependent) `list-replicability'.

cs.LG↗

arXiv2024

Hypercontractivity on HDX II: Symmetrization and q-Norms

Bourgain's symmetrization theorem is a powerful technique reducing boolean analysis on product spaces to the cube. It states that for any product $\Omega_i^{\otimes d}$, function $f: \Omega_i^{\otimes d} \to \mathbb{R}$, and $q > 1$: $$||T_{\frac{1}{2}}f(x)||_q \leq ||\tilde{f}(r,x)||_{q} \leq ||T_{c_q}f(x)||_q$$ where $T_{\rho}f = \sum\limits \rho^Sf^{=S}$ is the noise operator and $\widetilde{f}(r,x) = \sum\limits r_Sf^{=S}(x)$ `symmetrizes' $f$ by convolving its Fourier components $\{f^{=S}\}_{S \subseteq [d]}$ with a random boolean string $r \in \{\pm 1\}^d$. In this work, we extend the symmetrization theorem to high dimensional expanders (HDX). Building on (O'Donnell and Zhao 2021), we show this implies nearly-sharp $(2{\to}q)$-hypercontractivity for partite HDX. This resolves the main open question of (Gur, Lifshitz, and Liu STOC 2022) and gives the first fully hypercontractive subsets $X \subset [n]^d$ of support $n\cdot\exp(\text{poly}(d))$, an exponential improvement over Bafna, Hopkins, Kaufman, and Lovett's $n\cdot\exp(\exp(d))$ bound (BHKL STOC 2022). Adapting (Bourgain JAMS 1999), we also give the first booster theorem for HDX, resolving a main open question of BHKL. Our proof is based on two elementary new ideas in the theory of high dimensional expansion. First we introduce `$q$-norm HDX', generalizing standard spectral notions to higher moments, and observe every spectral HDX is a $q$-norm HDX. Second, we introduce a simple method of coordinate-wise analysis on HDX which breaks high dimensional random walks into coordinate-wise components and allows each component to be analyzed as a $\textit{$1$-dimensional}$ operator locally within $X$. This allows for application of standard tricks such as the replacement method, greatly simplifying prior analytic techniques.

cs.CC↗

arXiv2024

Replicability in High Dimensional Statistics

The replicability crisis is a major issue across nearly all areas of empirical science, calling for the formal study of replicability in statistics. Motivated in this context, [Impagliazzo, Lei, Pitassi, and Sorrell STOC 2022] introduced the notion of replicable learning algorithms, and gave basic procedures for $1$-dimensional tasks including statistical queries. In this work, we study the computational and statistical cost of replicability for several fundamental high dimensional statistical tasks, including multi-hypothesis testing and mean estimation. Our main contribution establishes a computational and statistical equivalence between optimal replicable algorithms and high dimensional isoperimetric tilings. As a consequence, we obtain matching sample complexity upper and lower bounds for replicable mean estimation of distributions with bounded covariance, resolving an open problem of [Bun, Gaboardi, Hopkins, Impagliazzo, Lei, Pitassi, Sivakumar, and Sorrell, STOC2023] and for the $N$-Coin Problem, resolving a problem of [Karbasi, Velegkas, Yang, and Zhou, NeurIPS2023] up to log factors. While our equivalence is computational, allowing us to shave log factors in sample complexity from the best known efficient algorithms, efficient isoperimetric tilings are not known. To circumvent this, we introduce several relaxed paradigms that do allow for sample and computationally efficient algorithms, including allowing pre-processing, adaptivity, and approximate replicability. In these cases we give efficient algorithms matching or beating the best known sample complexity for mean estimation and the coin problem, including a generic procedure that reduces the standard quadratic overhead of replicability to linear in expectation.

stat.ML↗

arXiv2024

Chernoff Bounds and Reverse Hypercontractivity on HDX

We prove optimal concentration of measure for lifted functions on high dimensional expanders (HDX). Let $X$ be a $k$-dimensional HDX. We show for any $i\leq k$ and $f:X(i)\to [0,1]$: \[\Pr_{s\in X(k)}\left[\left|\underset{{t\subseteq s}}{\mathbb{E}}[f(t)]-\mu\right|\geq\varepsilon\right]\leq exp\left(-\varepsilon^2\frac{k}{i}\right).\] Using this fact, we prove that high dimensional expanders are reverse hypercontractive, a powerful functional inequality from discrete analysis implying that for any sets $A,B \subset X(k)$, the probability a $\rho$-correlated pair passes between them is at least \[\Pr_{s,s' \sim T_\rho}[s \in A, s' \in B] \geq \Pr[A]^{O(1)} \Pr[B]^{O(1)}.\] Our results hold under weak spectral assumptions on $X$. Namely we prove exponential concentration of measure for any complex below the `Trickling-Down Threshold' (beyond which concentration may be arbitrarily poor), and optimal concentration for $\sqrt{k}$-skeletons of such complexes. We also show optimal bounds for the top dimension of stronger HDX among other settings. We leverage our inequalities to prove several new agreement testing theorems on high dimensional expanders, including a new 99%-regime test for subsets, and a variant of the `Z-test' achieving inverse exponential soundness under the stronger assumption of $\ell_\infty$-expansion. The latter gives rise to the first optimal testers beyond the complete complex and products, a stepping stone toward the use of HDX in strong soundness PCPs. We also give applications within expansion, analysis, combinatorics, and coding theory, including a proof that two-sided HDX have optimal geometric overlap (giving the first explicit bounded-degree construction), near-optimal double samplers, new super-exponential degree lower bounds for certain HDX, distance-amplified list-decodable and locally testable codes, a Frankl-R\"odl Theorem and more.

cs.CC↗

arXiv2023

Stability is Stable: Connections between Replicability, Privacy, and Adaptive Generalization

The notion of replicable algorithms was introduced in Impagliazzo et al. [STOC '22] to describe randomized algorithms that are stable under the resampling of their inputs. More precisely, a replicable algorithm gives the same output with high probability when its randomness is fixed and it is run on a new i.i.d. sample drawn from the same distribution. Using replicable algorithms for data analysis can facilitate the verification of published results by ensuring that the results of an analysis will be the same with high probability, even when that analysis is performed on a new data set. In this work, we establish new connections and separations between replicability and standard notions of algorithmic stability. In particular, we give sample-efficient algorithmic reductions between perfect generalization, approximate differential privacy, and replicability for a broad class of statistical problems. Conversely, we show any such equivalence must break down computationally: there exist statistical problems that are easy under differential privacy, but that cannot be solved replicably without breaking public-key cryptography. Furthermore, these results are tight: our reductions are statistically optimal, and we show that any computational separation between DP and replicability must imply the existence of one-way functions. Our statistical reductions give a new algorithmic framework for translating between notions of stability, which we instantiate to answer several open questions in replicability and privacy. This includes giving sample-efficient replicable algorithms for various PAC learning, distribution estimation, and distribution testing problems, algorithmic amplification of $\delta$ in approximate DP, conversions from item-level to user-level privacy, and the existence of private agnostic-to-realizable learning reductions under structured distributions.

cs.LG↗

arXiv2023

Do PAC-Learners Learn the Marginal Distribution?

The Fundamental Theorem of PAC Learning asserts that learnability of a concept class $H$ is equivalent to the $\textit{uniform convergence}$ of empirical error in $H$ to its mean, or equivalently, to the problem of $\textit{density estimation}$, learnability of the underlying marginal distribution with respect to events in $H$. This seminal equivalence relies strongly on PAC learning's `distribution-free' assumption, that the adversary may choose any marginal distribution over data. Unfortunately, the distribution-free model is known to be overly adversarial in practice, failing to predict the success of modern machine learning algorithms, but without the Fundamental Theorem our theoretical understanding of learning under distributional constraints remains highly limited. In this work, we revisit the connection between PAC learning, uniform convergence, and density estimation beyond the distribution-free setting when the adversary is restricted to choosing a marginal distribution from a known family $\mathscr{P}$. We prove that while the traditional Fundamental Theorem indeed fails, a finer-grained connection between the three fundamental notions continues to hold: 1. PAC-Learning is strictly sandwiched between two refined models of density estimation, differing only in whether the learner $\textit{knows}$ the set of well-estimated events in $H$. 2. Under reasonable assumptions on $H$ and $\mathscr{P}$, density estimation is equivalent to $\textit{uniform estimation}$, a relaxation of uniform convergence allowing non-empirical estimators. Together, our results give a clearer picture of how the Fundamental Theorem extends beyond the distribution-free setting and shed new light on the classically challenging problem of learning under arbitrary distributional assumptions.

cs.LG↗

arXiv2022

Robust Empirical Risk Minimization with Tolerance

Developing simple, sample-efficient learning algorithms for robust classification is a pressing issue in today's tech-dominated world, and current theoretical techniques requiring exponential sample complexity and complicated improper learning rules fall far from answering the need. In this work we study the fundamental paradigm of (robust) $\textit{empirical risk minimization}$ (RERM), a simple process in which the learner outputs any hypothesis minimizing its training error. RERM famously fails to robustly learn VC classes (Montasser et al., 2019a), a bound we show extends even to `nice' settings such as (bounded) halfspaces. As such, we study a recent relaxation of the robust model called $\textit{tolerant}$ robust learning (Ashtiani et al., 2022) where the output classifier is compared to the best achievable error over slightly larger perturbation sets. We show that under geometric niceness conditions, a natural tolerant variant of RERM is indeed sufficient for $\gamma$-tolerant robust learning VC classes over $\mathbb{R}^d$, and requires only $\tilde{O}\left( \frac{VC(H)d\log \frac{D}{\gamma\delta}}{\epsilon^2}\right)$ samples for robustness regions of (maximum) diameter $D$.

cs.LG↗

arXiv2022

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

We study the relationship between the underlying structure of posets and the spectral and combinatorial properties of their higher-order random walks. While fast mixing of random walks on hypergraphs has led to myriad breakthroughs throughout theoretical computer science in the last five years, many other important applications (e.g. locally testable codes, 2-2 games) rely on the more general non-simplicial structures. 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 architectures, highlighting how structural regularity controls the spectral decay and edge-expansion of corresponding random walks. In particular, we show the spectra of walks on expanding posets (Dikstein, Dinur, Filmus, Harsha RANDOM 2018) concentrate in strips around a small number of approximate eigenvalues controlled by the poset's regularity. This gives a simple condition to identify architectures (e.g. the Grassmann) that exhibit fast (exponential) decay of eigenvalues, versus architectures like hypergraphs with slow (linear) decay -- 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 variance-based characterization of edge-expansion on eposets generalizing (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 used in the proof of the 2-2 Games Conjecture which relies on $\ell_\infty$ rather than $\ell_2$-structure.

math.CO↗

arXiv2022

Explicit Lower Bounds Against $\Omega(n)$-Rounds of Sum-of-Squares

We construct an explicit family of 3-XOR instances hard for $\Omega(n)$-levels of the Sum-of-Squares (SoS) semi-definite programming hierarchy. Not only is this the first explicit construction to beat brute force search (beyond low-order improvements (Tulsiani 2021, Pratt 2021)), combined with standard gap amplification techniques it also matches the (optimal) hardness of random instances up to imperfect completeness (Grigoriev TCS 2001, Schoenebeck FOCS 2008). Our result is based on a new form of small-set high dimensional expansion (SS-HDX) inspired by recent breakthroughs in locally testable and quantum LDPC codes. Adapting the recent framework of Dinur, Filmus, Harsha, and Tulsiani (ITCS 2021) for SoS lower bounds from the Ramanujan complex to this setting, we show any (bounded-degree) SS-HDX can be transformed into a highly unsatisfiable 3-XOR instance that cannot be refuted by $\Omega(n)$-levels of SoS. We then show Leverrier and Z\'emor's (Arxiv 2022) recent qLDPC construction gives the desired explicit family of bounded-degree SS-HDX. Incidentally, this gives the strongest known form of bi-directional high dimensional expansion to date.

cs.CC↗

arXiv2022

Sampling Equilibria: Fast No-Regret Learning in Structured Games

Learning and equilibrium computation in games are fundamental problems across computer science and economics, with applications ranging from politics to machine learning. Much of the work in this area revolves around a simple algorithm termed \emph{randomized weighted majority} (RWM), also known as "Hedge" or "Multiplicative Weights Update," which is well known to achieve statistically optimal rates in adversarial settings (Littlestone and Warmuth '94, Freund and Schapire '99). Unfortunately, RWM comes with an inherent computational barrier: it requires maintaining and sampling from a distribution over all possible actions. In typical settings of interest the action space is exponentially large, seemingly rendering RWM useless in practice. In this work, we refute this notion for a broad variety of \emph{structured} games, showing it is possible to efficiently (approximately) sample the action space in RWM in \emph{polylogarithmic} time. This gives the first efficient no-regret algorithms for problems such as the \emph{(discrete) Colonel Blotto game}, \emph{matroid congestion}, \emph{matroid security}, and basic \emph{dueling games}. As an immediate corollary, we give a polylogarithmic time meta-algorithm to compute approximate Nash Equilibria for these games that is exponentially faster than prior methods in several important settings. Further, our algorithm is the first to efficiently compute equilibria for more involved variants of these games with general sums, more than two players, and, for Colonel Blotto, multiple resource types.

cs.GT↗

arXiv2022

Active Learning Polynomial Threshold Functions

We initiate the study of active learning polynomial threshold functions (PTFs). While traditional lower bounds imply that even univariate quadratics cannot be non-trivially actively learned, we show that allowing the learner basic access to the derivatives of the underlying classifier circumvents this issue and leads to a computationally efficient algorithm for active learning degree-$d$ univariate PTFs in $\tilde{O}(d^3\log(1/\varepsilon\delta))$ queries. We also provide near-optimal algorithms and analyses for active learning PTFs in several average case settings. Finally, we prove that access to derivatives is insufficient for active learning multivariate PTFs, even those of just two variables.

cs.LG↗

arXiv2021

Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments

Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the $p$-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of Khot's 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). In this work, we develop 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. Our results lead to a new understanding of the structure of Boolean functions on HDX, including a tight analog of the KKL Theorem and a new characterization of non-expanding sets. Unlike previous settings satisfying hypercontractivity, HDX can be asymmetric, sparse, and very far from products, which makes the application of traditional proof techniques challenging. We handle these barriers with the introduction of two new tools of independent interest: a new explicit combinatorial Fourier basis for HDX that behaves well under restriction, and a new local-to-global method for analyzing higher moments. Interestingly, unlike analogous second moment methods that apply equally across all types of expanding complexes, our tools rely inherently on simplicial structure. This suggests a new distinction among high dimensional expanders based upon their behavior beyond the second moment.

cs.DM↗