A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees

Kyng, Rasmus; Song, Zhao

doi:10.1109/focs.2018.00043

Cited by 16 publications

(33 citation statements)

References 48 publications

(53 reference statements)

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“…We obtain as corollary that O(log |V |) random spanning trees gives a spectral sparsifier of a (weighted, undirected) graph G = (V, E) whp improving on the O(log 2 |V |) bound given in [KS18]. Our result thus matches the lower bound from [KS18] (see their Theorem 1.8). In Appendix B, we sketch a proof of the corollary and provide some relevant preliminaries.…”

Section: Introductionsupporting

confidence: 70%

“…When specialized to distributions with the SCP, Theorem 1.2 answers positively the question posed by Kyng and Song [KS18] on whether the log(k) factor in the exponent that appears in their matrix Chernoff bound for Strongly Rayleigh distributions can be removed. We obtain as corollary that O(log |V |) random spanning trees gives a spectral sparsifier of a (weighted, undirected) graph G = (V, E) whp improving on the O(log 2 |V |) bound given in [KS18]. Our result thus matches the lower bound from [KS18] (see their Theorem 1.8).…”

Section: Introductionmentioning

confidence: 95%

“…For example, in spectral graph theory, the bound [Tro12] can be used to prove that in a graph with |E| edges and |V | vertices, we can obtain a spectral sparsifier of the graph Laplacian using Θ(|V | log |V |) edges. Similarly, [KS18] shows as a corollary that a spectral sparsifier of the graph Laplacian can be constructed using Θ(log 2 |V |) random spanning trees of the graph.…”

Section: Introductionmentioning

confidence: 96%

“…Again, a lower tail bound also holds after replacing λ max (•) replacing λ min (•). These bounds of [Tro12] and [KS18] that we call Chernoff-like bounds work particularly well when λ max (E [f (ξ)]) is small, which happens when the "mass" of the distribution is well-spread out across all matrix "directions". An example of this is matrix-valued random variables with isotropic mean or covariance matrix.…”

Section: Introductionmentioning

confidence: 97%

“…This means the bound can only prove that O(|V | log 2 |V |) random spanning trees give a spectral sparsifier of a graph. If, instead, we take some care to modify their bound to treat a union of independent SCP distributions more carefully, we can reduce this to O( |V | log |V |) spanning trees to build a spectral sparsifier -but this is still exponentially worse than [KS18].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Scalar and Matrix Chernoff Bounds from $\ell_{\infty}$-Independence

Kaufman¹,

Kyng²,

Soldà³

2021

Preprint

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We show new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over {0, 1} n . Building on developments in high-dimensional expanders (Kaufman and Mass ITCS'17, Dinur and Kaufman FOCS'17, Kaufman and Oppenheim Combinatorica'20) and matroid theory (Adiprasito et al. Ann. Math.'18), a breakthrough result of Anari, Liu, Oveis and Vinzant (STOC '19) showed that the up-down random walk on matroid bases has polynomial mixing time -making it possible to efficiently sample from the (weighted) uniform distribution over matroid bases. Since then, there has been a flurry of related work proving polynomial mixing times for random walks used to sample from a wide range of discrete probability distributions. Many works have observed that as a corollary of their mixing time analysis, one can obtain scalar concentration for 1-Lipschitz functions of samples from the stable distribution via standard arguments that convert bounds on the modified log-Sobolev (MLS) or Poincaré constant of a random walk into concentration results for the associated stable distribution of the random walk, see for example Hermon and Salez (arXiv'19). Several recent works have considered a matrix analog of the Poincaré inequality for a random walk with an associated stable distribution. Using this matrix Poincaré inequality, these works have derived a concentration result for matrix-valued spectral norm-Lipschitz functions of samples from the distribution, see for example Auon et al. (Adv. Math.'20). Unfortunately, these bounds are weak in many important regimes.A recently developed strategy for analyzing up-down walks is based around a novel notion of spectral independence, which quantifies the dependence between variables in a distribution over {0, 1} n using the largest eigenvalue of an associated pairwise influence matrix I, see Anari et al. (FOCS'20). Many works on spectral independence have in fact bounded a stronger quantity I ∞→∞ ≥ λ max (I), which we call ℓ ∞ -independence. We show that any distribution over {0, 1} n which has bounded ℓ ∞ -independence satisfies a matrix Chernoff bound that in key regimes is much stronger than spectral norm-Lipschitz function concentration bounds derived from matrix Poincaré inequalities. Our bounds match the matrix Chernoff bound for independent random variables due to Tropp, which is the strongest known in many cases and is essentially tight for several key settings. For spectral graph sparsification, our matrix concentration results are exponentially stronger than those obtained from matrix Poincaré inequalities. Our matrix Chernoff bound is a broad generalization and strengthening of the matrix Chernoff bound of Kyng and Song (FOCS'18). Using our bound, we can conclude as a corollary that a union of O(log |V |) random spanning trees gives a spectral graph sparsifier of a graph with |V | vertices with high probability, matching results for independent edge sampling, and matching lower bounds from Kyng and Song. This improves on the O(log 2 |V |) spanning tr...

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Section: Introductionsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Scalar and Matrix Chernoff Bounds from $\ell_{\infty}$-Independence

Kaufman¹,

Kyng²,

Soldà³

2021

Preprint

Self Cite

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On sampling determinantal and Pfaffian point processes on a quantum computer

Bardenet,

Fanuel,

Feller

2024

J. Phys. A: Math. Theor.

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DPPs were introduced by Macchi as a model in quantum optics the 1970s. Since then, they have been widely used as models and subsamplingtools in statistics and computer science. Most applications requiresampling from a DPP, and given their quantum origin, it is natural towonder whether sampling a DPP on a quantum computer is easier thanon a classical one. We focus here on DPPs over a finite state space, whichare distributions over the subsets of {1, . . . ,N} parametrized by an N×NHermitian kernel matrix. Vanilla sampling consists in two steps, of respectivecosts O(N3) and O(Nr2) operations on a classical computer, wherer is the rank of the kernel matrix. A large first part of the current paperconsists in explaining why the state-of-the-art in quantum simulationof fermionic systems already yields quantum DPP sampling algorithms.We then modify existing quantum circuits, and discuss their insertion ina full DPP sampling pipeline that starts from practical kernel specifications.The bottom line is that, with P (classical) parallel processors, wecan divide the preprocessing cost by P and build a quantum circuit withO(Nr) gates that sample a given DPP, with depth varying from O(N) toO(r logN) depending on qubit-communication constraints on the targetmachine. We also connect existing work on the simulation of superconductorsto Pfaffian point processes, which generalize DPPs and would bea natural addition to the machine learner’s toolbox. In particular, we describe ``projective'' Pfaffian point processes, the cardinality of which has constantparity, almost surely. Finally, the circuitsare empirically validated on a classical simulator and on 5-qubit machines.

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