Tutorial on large deviations for the binomial distribution

Arratia, Richard; Gordon, Louis

doi:10.1007/bf02458840

Cited by 121 publications

(105 citation statements)

References 5 publications

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“…Using the same result as used for Eq. (68), we can bound the error to a constant value if k = nα for α > ηx [41]. This shows a relationship between distinguishability and loss similarly to, but not exactly the same, as the one found in [29]: the more distinguishable photons are, the more we can classically simulate photon loss, and vice versa.…”

Section: Incorporating Losssupporting

confidence: 69%

Classically simulating near-term partially-distinguishable and lossy boson sampling

Moylett

García−Patrón

Renema

et al. 2019

Quantum Sci. Technol.

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Boson Sampling is the problem of sampling from the same distribution as indistinguishable single photons at the output of a linear optical interferometer. It is an example of a non-universal quantum computation which is believed to be feasible in the near term and cannot be simulated on a classical machine. Like all purported demonstrations of "quantum supremacy", this motivates optimizing classical simulation schemes for a realistic model of the problem, in this case Boson Sampling when the implementations experience lost or distinguishable photons. Although current simulation schemes for sufficiently imperfect boson sampling are classically efficient, in principle the polynomial runtime can be infeasibly large. In this work, we develop a scheme for classical simulation of Boson Sampling under uniform distinguishability and loss, based on the idea of sampling from distributions where at most k photons are indistinguishable. We show that asymptotically this scheme can provide a polynomial improvement in the runtime compared to classically simulating idealised Boson Sampling. More significantly, we show that in the regime considered experimentally relevant, our approach gives an substantial improvement in runtime over other classical simulation approaches.

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Section: Incorporating Losssupporting

confidence: 69%

Classically simulating near-term partially-distinguishable and lossy boson sampling

Moylett

García−Patrón

Renema

et al. 2019

Quantum Sci. Technol.

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“…The cumulative conditional probability of the hypothesis that the noise is responsible for formation of a peak N θ 1 relatively to a valley N θ 2 can be expressed as (Arratia & Gordon, 1989):

P_{v} = \sum_{N = N_{θ_{1}}}^{N_{c}} \frac{P (N; N_{c}, p)}{P (N_{θ_{2}}; N_{c}, p)} ≲ \frac{1}{1 - r_{1}} \sqrt{\frac{N_{θ_{2}} false(N_{c} - N_{θ_{2}} false)}{N_{θ_{1}} false(N_{c} - N_{θ_{1}} false)}} e^{- N_{c} (H_{1} - H_{2})},

where…”

Section: Resultsmentioning

confidence: 99%

The Lamellar Structure of the Brain Fiber Pathways

Galinsky

Frank

2016

Neural Computation

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We present a quantitative statistical analysis of pairwise crossings for all fibers obtained from whole brain tractography that confirms with high confidence that the “brain grid theory” (Wedeen et al., Science, 30 March 2012, p. 1628) is not supported by the evidence. The overall fiber tracts structure appears to be more consistent with small angle tree-like branching of tracts rather than with near orthogonal grid-like crossing of fiber sheets. The analysis uses our new method for high resolution whole brain tractography that is capable of resolving fibers crossing of less than 10° and correctly following a continuous angular distribution of fibers even when the individual fiber directions are not resolved. This analysis also allows us to demonstrate that the whole brain fiber pathway system is very well approximated by a lamellar vector field, providing a concise and quantitative mathematical characterization of the structural connectivity of the human brain.

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“…To answer this question, we use bounds on large deviations for the binomial distribution (67) as follows. Let and S m is the number of quartets satisfied by T r .…”

Section: Resultsmentioning

confidence: 99%

Phylo SI: a new genome-wide approach for prokaryotic phylogeny

Shifman

Ninyo

Gophna

et al. 2013

Nucleic Acids Research

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The evolutionary history of all life forms is usually represented as a vertical tree-like process. In prokaryotes, however, the vertical signal is partly obscured by the massive influence of horizontal gene transfer (HGT). The HGT creates widespread discordance between evolutionary histories of different genes as genomes become mosaics of gene histories. Thus, the Tree of Life (TOL) has been questioned as an appropriate representation of the evolution of prokaryotes. Nevertheless a common hypothesis is that prokaryotic evolution is primarily tree-like, and a routine effort is made to place new isolates in their appropriate location in the TOL. Moreover, it appears desirable to exploit non–tree-like evolutionary processes for the task of microbial classification. In this work, we present a novel technique that builds on the straightforward observation that gene order conservation (‘synteny’) decreases in time as a result of gene mobility. This is particularly true in prokaryotes, mainly due to HGT. Using a ‘synteny index’ (SI) that measures the average synteny between a pair of genomes, we developed the phylogenetic reconstruction tool ‘Phylo SI’. Phylo SI offers several attractive properties such as easy bootstrapping, high sensitivity in cases where phylogenetic signal is weak and computational efficiency. Phylo SI was tested both on simulated data and on two bacterial data sets and compared with two well-established phylogenetic methods. Phylo SI is particularly efficient on short evolutionary distances where synteny footprints remain detectable, whereas the nucleotide substitution signal is too weak for reliable sequence-based phylogenetic reconstruction. The method is publicly available at http://research.haifa.ac.il/ssagi/software/PhyloSI.zip.

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Tutorial on large deviations for the binomial distribution

Cited by 121 publications

References 5 publications

Classically simulating near-term partially-distinguishable and lossy boson sampling

Classically simulating near-term partially-distinguishable and lossy boson sampling

The Lamellar Structure of the Brain Fiber Pathways

Phylo SI: a new genome-wide approach for prokaryotic phylogeny

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