Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid

Anari, Nima; Liu, Kuikui; Gharan, Shayan Oveis; Vinzant, Cynthia

doi:10.1145/3313276.3316385

Cited by 103 publications

(139 citation statements)

References 30 publications

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“…where π min = min x∈Ω π(x). is will improve the previous bound t mix (P BX,π , ε) r log 1 π min + log 1 ε due to Anari et al (2018a). Since π min is most commonly exponentially small in the input size (e.g.…”

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confidence: 85%

“…An important example of homogeneous strongly log-concave distributions is the uniform distribution over the bases of a matroid (Anari et al, 2018a;Brändén and Huh, 2019). is discovery leads to the breakthrough result that the exchange walk over the bases of a matroid is rapidly mixing (Anari et al, 2018a), which implies the existence of a fully polynomial-time randomised approximation scheme (FPRAS) for the number of bases of any matroid (given by an independence oracle). e bases-exchange walk, denoted by P BX , is defined as follows.…”

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confidence: 99%

“…In fact, we show more than Theorem 1. Following Anari et al (2018a) and Kaufman and Oppenheim (2018), we stratify independent sets of the matroid M by their sizes, and define two random walks for each level, depending on whether they add or delete an element first. For instance, the bases-exchange walk P BX,π is the "delete-add" or "down-up" walk for the top level.…”

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confidence: 99%

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Modified log-Sobolev Inequalities for Strongly Log-Concave Distributions

Cryan

Guo

Mousa

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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A. We show that the modified log-Sobolev constant for a natural Markov chain which converges to an r-homogeneous strongly log-concave distribution is at least 1/r. Applications include an asymptotically optimal mixing time bound for the bases-exchange walk for matroids, and a concentration bound for Lipschitz functions over these distributions.π is the uniform distribution), the improvement is usually a polynomial factor. Our bound is asymptotically optimal without further assumptions, as the upper bound is achieved when π is the uniform distribution over the bases of some matroids (Jerrum, 2003). 2 Our main improvement is a modified log-Sobolev inequality for π and P BX,π . To introduce this inequality, we define the Dirichlet form of a reversible Markov chain P, over state space Ω, aswhere f, g are two functions over Ω, and I denotes the identity matrix. Moreover, let the (normalised) relative entropy of f :where we follow the convention that 0 log 0 = 0. If we normalise E π f = 1, then Ent π (f) is the relative entropy (or Kullback-Leibler divergence) between π(·)f(·) and π(·).e modified log-Sobolev constant (Bobkov and Tetali, 2006) is defined asOur main theorem is the following, which is a special case of Theorem 7.

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confidence: 85%

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confidence: 99%

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confidence: 99%

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Modified log-Sobolev Inequalities for Strongly Log-Concave Distributions

Cryan

Guo

Mousa

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…the largest independent set in this matroid has size r). In an exciting recent work [ALGV18] it was proven that this complex is a 0-one-sided HDX. Oppenheim [Opp18b] proved that if we truncate this complex by keeping only faces of dimensions 0 i d then it becomes a 1/(r − d − 2)-two-sided HDX.…”

Section: Agreement On High Dimensional Expandersmentioning

confidence: 94%

Agreement Testing Theorems on Layered Set Systems

Dikstein

Dinur

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We introduce a framework of layered subsets, and give a sufficient condition for when a set system supports an agreement test. Agreement testing is a certain type of property testing that generalizes PCP tests such as the plane vs. plane test. Previous work has shown that high dimensional expansion is useful for agreement tests. We extend these results to more general families of subsets, beyond simplicial complexes. These include -Agreement tests for set systems whose sets are faces of high dimensional expanders. Our new tests apply to all dimensions of complexes both in case of two-sided expansion and in the case of one-sided partite expansion. This improves and extends an earlier work of Dinur and Kaufman (FOCS 2017) and applies to matroids, and potentially many additional complexes.-Agreement tests for set systems whose sets are neighborhoods of vertices in a high dimensional expander. This family resembles the expander neighborhood family used in the gap-amplification proof of the PCP theorem. This set system is quite natural yet does not sit in a simplicial complex, and demonstrates some versatility in our proof technique.-Agreement tests on families of subspaces (also known as the Grassmann poset). This extends the classical low degree agreement tests beyond the setting of low degree polynomials.Our analysis relies on a new random walk on simplicial complexes which we call the "complement random walk" and which may be of independent interest. This random walk generalizes the non-lazy random walk on a graph to higher dimensions, and has significantly better expansion than previously-studied random walks on simplicial complexes.

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“…In case of high-dimensional expanders, in addition to canonical walks described here, a "non-lazy" version of these walks (moving from s to t only if s = t) was also considered by Kaufman and Oppenheim [KO18b], Anari et al [ALGV18] and Dikstein et al [DDFH18]. The swap walks studied in this paper were also considered independently in a very recent work of Dikstein and Dinur [DD19] (under the name "complement walks").…”

Section: Introductionmentioning

confidence: 99%

Approximating Constraint Satisfaction Problems on High-Dimensional Expanders

Alev

Jeronimo

Tulsiani

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We consider the problem of approximately solving constraint satisfaction problems with arity k > 2 (k-CSPs) on instances satisfying certain expansion properties, when viewed as hypergraphs. Random instances of k-CSPs, which are also highly expanding, are wellknown to be hard to approximate using known algorithmic techniques (and are widely believed to be hard to approximate in polynomial time). However, we show that this is not necessarily the case for instances where the hypergraph is a high-dimensional expander.We consider the spectral definition of high-dimensional expansion used by Dinur and Kaufman [FOCS 2017] to construct certain primitives related to PCPs. They measure the expansion in terms of a parameter γ which is the analogue of the second singular value for expanding graphs. Extending the results by Barak, Raghavendra and Steurer [FOCS 2011] for 2-CSPs, we show that if an instance of MAX k-CSP over alphabet [q] is a highdimensional expander with parameter γ, then it is possible to approximate the maximum fraction of satisfiable constraints up to an additive error ε using q O(k) Based on our analysis, we also suggest a notion of threshold-rank for hypergraphs, which can be used to extend the results for approximating 2-CSPs on low threshold-rank graphs. We show that if an instance of MAX k-CSP has threshold rank r for a threshold τ = (ε/k) O(1) · (1/q) O(k) , then it is possible to approximately solve the instance up to additive error ε, using r · q O(k) · (k/ε) O(1) levels of the sum-of-squares hierarchy. As in the case of graphs, high-dimensional expanders (with sufficiently small γ) have threshold rank 1 according to our definition. spectral expander 1 . Here, the graph G(X a ) is defined to have the vertex set {i | a ∪ {i} ∈ X} and edge-set {i, j | a ∪ {i, j} ∈ X}. If the (normalized) second singular value of each of the neighborhood graphs is bounded by γ, X is said to be a γ-high-dimensional expander (γ-HDX).Note that (the downward closure of) a random sparse (d + 1)-uniform hypergraph, say with n vertices and c · n edges, is very unlikely to be a d-dimensional expander. With high probability, no two hyperedges share more than one vertex and thus for any i ∈ [n], the neighborhood graph G i is simply a disjoint union of cliques of size d, which is very far from an expander. While random hypergraphs do not yield high-dimensional expanders, such objects are indeed known to exists via (surprising) algebraic constructions [LSV05b, LSV05a, KO18a, CTZ18] and are known to have several interesting properties and applications [KKL16, DHK + 18, KM17, KO18b, DDFH18, DK17, PRT16].Expander graphs can simply be thought of as the one-dimensional case of the above definition. The results of Barak, Raghavendra and Steurer [BRS11] for 2-CSPs yield that if the constraint graph of a 2-CSP instance (with size n and alphabet size q) is a sufficiently good (one dimensional) spectral expander, then one can efficiently find solutions satisfying OPT − ε fraction of constraints, where OPT denotes the maximum fraction of ...

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Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid

Cited by 103 publications

References 30 publications

Modified log-Sobolev Inequalities for Strongly Log-Concave Distributions

Modified log-Sobolev Inequalities for Strongly Log-Concave Distributions

Agreement Testing Theorems on Layered Set Systems

Approximating Constraint Satisfaction Problems on High-Dimensional Expanders

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