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bafca359044939e94287ac03f4235c88b2ae306a | subsection | 23 | 26 | Comparison of the estimated averages of subsidies amounts under the two policies | Given that the rules for the determination of benefit amount are well known and that we had informations about employees' wages, we were able to compare the amount of subsidies given under the two laws. To do it, we used informations on wages to estimate approximativelyWhere it is approximatively because the rate of wa... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
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a435e70dd467eab1612e90194ed00ddd8ce283f9 | subsection | 24 | 26 | Comparison of the distribution of treated and controls among non-CICO-detected categories | In the following graph are represented the distributions, among different non-CICO-detected categories, of individuals with 23 and individuals with 24 months of unemployment. In order to build these distributions we used RTFL data. The last is the result of a quarterly survey conducted by ISTAT (Italian National Instit... | {
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} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
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683534aafa560018b45144e8d834e5b3bff39ce5 | subsection | 25 | 26 | Bandwidth Changes | Note that the different bandwidths we tested for, are all smaller than the chosen bandwidth. Indeed, given the method we used in bandwidth selection, in a bandwidth bigger than the chosen ones there wouldn't be balance between treated and controls. The estimation of treatment effect is not very sensitive to bandwidth s... | {
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7b56df4224eec0dfad0f5675dadeec0f33b1308a | abstract | 0 | 6 | Abstract | Three-dimensional, compressible, magnetohydrodynamic turbulence of an
isothermal, self-gravitating fluid is analyzed using two-point statistics in
the asymptotic limit of large Reynolds numbers (both kinetic and magnetic).
Following an alternative formulation proposed by S. Banerjee and S. Galtier
(Phys. Rev. E,93, 033... | {
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} | 10.1103/PhysRevE.97.023107 | 1802.03491 | Energy transfer in compressible magnetohydrodynamic turbulence for
isothermal self-gravitating fluids | [
"Supratik Banerjee",
"Alexei G. Kritsuk"
] | [
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3e5bf1373fa9e9c240cf652594c1cafed1969f40 | subsection | 1 | 6 | Introduction | Turbulence is a non-linear phenomenon omnipresent in nature. For a fully developed turbulence in the limit of infinitely large Reynolds numbers, the fluid flow contains fluctuations, populating a wide range of length and time scales. In the inertial range sufficiently decoupled from the large-scale forcing and small-sc... | {
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} | 10.1103/PhysRevE.97.023107 | 1802.03491 | Energy transfer in compressible magnetohydrodynamic turbulence for
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"Supratik Banerjee",
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a7621c9432f70d078af92815b8df8cbd767240d0 | subsection | 2 | 6 | Introduction | In contrast to the previous flux-source form, the current formulation casts the invariant flux rate \varepsilon in terms of mixed second order structure functions, associating the fluctuations of the density (\rho ), the fluid velocity ({u}), the vorticity ({\omega }), the magnetic field ({B}), the current (\nabla \tim... | {
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} | 10.1103/PhysRevE.97.023107 | 1802.03491 | Energy transfer in compressible magnetohydrodynamic turbulence for
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f5666fb22e01b00e81d307703911b406a524de57 | subsection | 3 | 6 | Basic Equations | In this paper, we are interested in a self-gravitating isothermal magnetohydrodynamic fluid. The basic equations are given as\partial _t \rho +
\leavevmode \nabla \cdotj= 0,t j+
\nabla \cdot \nabla \cdot (ju) = -
\nabla p\nabla p + jbb+ g + dH + f,t b= ( ub) + dM,g = - 4 G (- 0 ),
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} | 10.1103/PhysRevE.97.023107 | 1802.03491 | Energy transfer in compressible magnetohydrodynamic turbulence for
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0a29fca090edf54b7cb12f669a69ffc2fd101e7e | subsection | 4 | 6 | Basic Equations | We have (assuming periodic boundary conditions or zero velocity on the boundary surface)\int _V {\partial _t} \left(\frac{\rho {u}^2}{2}\right) dx &= \int _V \left[ {{j}}\cdot {g}- {u}\cdot \nabla p + {u}\cdot \left( {{j_b}}\times {b}\right) \right] dx, \\
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} | 10.1103/PhysRevE.97.023107 | 1802.03491 | Energy transfer in compressible magnetohydrodynamic turbulence for
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5bcc8a088a57af9c3c341d8deb6aef032d582c18 | subsection | 5 | 6 | Derivation of the exact relation | Unlike the exact relations for compressible isothermal MHD turbulence which have been derived , , in this paper we propose an alternative form which is an extension of in case of compressible fluids. Under this formalism, the final extact relation is simpler and hence for a physical system, the calculation of the energ... | {
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} | 10.1103/PhysRevE.97.023107 | 1802.03491 | Energy transfer in compressible magnetohydrodynamic turbulence for
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af7b1879853993359736b36bd996230520eeb3cd | abstract | 0 | 45 | Abstract | Every student in statistics or data science learns early on that when the
sample size largely exceeds the number of variables, fitting a logistic model
produces estimates that are approximately unbiased. Every student also learns
that there are formulas to predict the variability of these estimates which are
used for t... | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
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c8a2ccd7996feafb811f81149db544fd5284474f | subsection | 1 | 45 | Logistic regression: classical theory and practice | Logistic regression
, ,
is by and large the most frequently used model to estimate the
probability of a binary response from the value of multiple
features/predictor variables. It is used all the time in the social
sciences, the finance industry, the medical sciences, and so on. As an
example, a typical application of... | {
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b3981ed50350d5634204b294034c47e233b9769b | subsection | 2 | 45 | Logistic regression: classical theory and practice | The coefficient estimates\hat{\beta }_j are obtained by maximum likelihood, and for eachvariable, R provides an estimate of the standard deviation of\hat{\beta _j} as well as a p-value for testing whether\beta _j = 0 or not.]Another well-known result in logistic regression is Wilks'
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6c9e66ed99d7f3c0231561b0da0de00a6f292402 | subsection | 3 | 45 | Failures in moderately large dimensions | In modern-day data analysis, new technologies now produce extremely
large datasets, often with huge numbers of features on each of a
comparatively small number of experimental units. Yet, software
packages and practitioners continue to perform calculations as if
classical theory applies and, therefore, the main issue i... | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
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0a4904a54875f2e3c3264f477d78d25af17ce477 | subsection | 4 | 45 | Unbiasedness? | Figure REF plots the true and fitted coefficients
in the setting where one quarter of the regression coefficients have a
magnitude equal to 10, and the rest are zero. Half of the nonzero
coefficients are positive and the other half are negative. A striking
feature is that the black curve does not pass through the cente... | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
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d74aa765a5aefbc030f4fdd6cb8c71b3a54ee321 | subsection | 5 | 45 | Unbiasedness? | This
is illustrated in Figure REF (b). Observe that when
f({X}_*) < 1/2, lots of predictions tend to be close to zero, even
when f({X}_*) is nowhere near zero. A similar behavior is obtained
by symmetry; when f({X}_*) > 1/2, we see a shrinkage toward the
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
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434243ab7049eabaf0b76d0f8419c81c025c0cb6 | subsection | 6 | 45 | Accuracy of classical standard errors? | Consider the same matrix {X} as before and regression coefficients
now sampled as follows: half of the \beta _j's are i.i.d. draws from
{\mathcal {N}}(7,1), and the other half vanish. Figure
REF (a) shows standard errors computed via Monte Carlo
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ccfe10bea07be0545861970bc23e504940ef6a0c | subsection | 7 | 45 | Distribution of the LRT? | By now, the reader should be
suspicious that the chi-square approximation for the distribution of
the likelihood-ratio test holds in higher dimensions. Indeed, it does
not and this actually is not a new observation. In
, the authors established that for a class of
logistic regression models, the LRT converges weakly to... | {
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b5d0ed845fa4a908af43f3f4a76ce4b990717d07 | subsection | 8 | 45 | Summary. | We have hopefully made the case that classical
results, which software packages continue to rely upon, are downright
erroneous in higher dimensions.Estimates seem systematically biased in the sense that effect magnitudes are overestimated.
Estimates are far more variable than classically predicted.
Inference measures... | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
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896da0353778d69ac43f50aaba2bd1355ea14bb2 | subsection | 9 | 45 | Our contribution | Our contribution is to develop a brand new theory, which applies to
high-dimensional logistic regression models with independent
variables, and is capable of accurately describing all the phenomena
we have discussed. Taking them one by one, the theory from this paper
predicts:the bias of the MLE;
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0.013364670798182487,
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aff74e4ade9a18cc1046965f9806f083ffe256f4 | subsection | 10 | 45 | Prior work | Asymptotic properties of M-estimators in the context of linear
regression have been extensively studied in diverging dimensions
starting from , followed by
and . These
papers investigated the consistency and asymptotic normality
properties of M-estimators in a regime where p = o(n^{\alpha }), for
some \alpha < 1. Late... | {
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{
"arxiv_id": "",
"doi": "10.1214/aos/1176342503",
"end": 162,
"openalex_id": "https://openalex.org/W2065742895",
"raw": "Peter J Huber. Robust regression: asymptotics, conjectures and Monte Carlo. The Annals of Statistics, pages 799–821, 1973.",
"source_ref_i... | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
"Emmanuel J. Candes"
] | [
"math.ST",
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] | 2,018 | en | Mathematics | [
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6c9cf2317ec04752c7f68a2c86077401a6abbb89 | subsection | 11 | 45 | Prior work | The rule of thumb is usually 10 events per variable
(EPV) or more as mentioned in
, , while a later study
suggested that it could be even
less. As we clearly see in this paper, such a rule is not at all valid
when the number of features is large.
contested the previously established
10 EPV rule.To the best of our kno... | {
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"raw": "Peter Peduzzi, John Concato, Elizabeth Kemper, Theodore R Holford, and Alvan R Feinstein. A simulation study of the number of events per ... | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
"Emmanuel J. Candes"
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d05176f7da7cacb4c07f63c19281cae9dc7f9ae6 | subsection | 12 | 45 | Setting. | We describe the asymptotic properties of the MLE
and the LRT in a high-dimensional regime, where n and p both go to
infinity in such a way that p/n \rightarrow \kappa . We work with
independent observations \lbrace {X}_i, y_i\rbrace from a logistic model such
that \mathbb {P}(y_i = 1 \, | \, {X}_i) = \rho ^{\prime }({X... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
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cc0ec8dc6d9bce82fe8882796d92d43e072d269d | subsection | 13 | 45 | When does the MLE exist? | The MLE \hat{{\beta }} is the minimizer of the negative log-likelihood
\ell defined via (observe that the sigmoid is the first derivative
of \rho )\ell ({b})=\sum _{i=1}^n \lbrace \rho ({X}_i^{\prime }{b}) - y_i \, ({X}_i^{\prime }{b}) \rbrace , \qquad \rho (t) = \log (1+e^t).A first important remark is that in high di... | {
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regression | [
"Pragya Sur",
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510b6b19035e3d04b428bbecd25c67d843df873e | subsection | 14 | 45 | A system of nonlinear equations | As we shall soon see, the asymptotic behavior of both the MLE and the
LRT is characterized by a system of equations in three variables
(\alpha ,\sigma ,\lambda ):\left\lbrace \begin{aligned}\sigma ^2 & = \frac{1}{\kappa ^2} \operatorname{\mathbb {E}}\left[2\rho ^{\prime }(Q_1)\left(\lambda \rho ^{\prime }(\mathsf {prox... | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
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f735e5bc223653da51dd0f4edc5fe33f59d20e58 | subsection | 15 | 45 | A system of nonlinear equations | It
turns out that the system admits a unique solution if and only if
(\kappa ,\gamma ) is in the region where the MLE asymptotically
exists!It is instructive to note that in the case where the signal strength
vanishes, \gamma =0, the system of equations (REF ) reduces
to the following two-dimensional system:\left\lbrac... | {
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737049059c80cda18446873ffe286895a005edd7 | subsection | 16 | 45 | The average behavior of the MLE | Our first main result characterizes the `average' behavior of the MLE.Theorem 2
Assume the dimensionality and signal strength parameters \kappa
and \gamma are such that \gamma < g_{\text{MLE}}(\kappa ) (the
region where the MLE exists asymptotically and shown in Figure
REF ). Assume the logistic model described above... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
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c835e8defb9c7798eeeb08cee034544bad3ed74a | subsection | 17 | 45 | The average behavior of the MLE | This can be seen by
taking \psi (t,u) = t^2, which leads to
\frac{1}{p} \sum _{j=1}^p (\hat{\beta }_j - \alpha _{\star }\beta _{j})^2 \,\,
{\stackrel{\text{a.s.}}{\longrightarrow }} \, \,
\sigma _{\star }^2.
As before, this can also be seen from the empirical results from the
previous section. When \kappa = 0.2 and
\... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
"Emmanuel J. Candes"
] | [
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a1c5bd301d8f15c4f2db20b91423fdf3462722ef | subsection | 18 | 45 | The average behavior of the MLE | Note that there
are several ways of determining \alpha _{\star } empirically. For instance,
the limit (REF ) directly suggests taking the ratio
\sum _j \hat{\beta }_j/\sum _j \beta _j. An alternative is to consider
taking the ratio when restricting the summation to nonzero indices.
Empirically, we find there is not muc... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
"Emmanuel J. Candes"
] | [
"math.ST",
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1d8a197a311d0f11c2db362645e434023b4529dd | subsection | 19 | 45 | The distribution of the null MLE coordinates | Whereas Theorem REF describes the average or bulk behavior
of the MLE across all of its entries, our next result provides the
explicit distribution of \hat{\beta }_j whenever \beta _j = 0,
i.e. whenever the j-th variable is independent from the response
y.Theorem 3
Let j be any variable such that \beta _j = 0. Then in... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
"Emmanuel J. Candes"
] | [
"math.ST",
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59326984f27e62cd45ab5dbcad9f6bf04db793ff | subsection | 20 | 45 | The distribution of the LRT | We finally turn our attention to the distribution of the likelihood
ratio statistic for testing \beta _j = 0.Theorem 4
Consider the LLR
\Lambda _j = \min _{{b}\, : \, b_j = 0} \ell ({b}) - \min _{{b}}
\ell ({b}) for testing \beta _j = 0. In the setting of Theorem
REF , twice the LLR is asymptotically distributed as a
... | {
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regression | [
"Pragya Sur",
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2af5fb1dcabae5536a84a547f9cbada6662e003a | subsection | 21 | 45 | The distribution of the LRT | We plot the empirical cdf of the p-values, zooming in
the tail, in Figure REF (d). We find that the
rescaled chi-squared approximation works extremely well even in the
tails of the distribution.
[Table: P-value probabilities estimated over 500,000replicates with standard errors in parentheses. Here, \kappa =0.1 and the... | {
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regression | [
"Pragya Sur",
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70bc9b7bc774476e55eeb5774c85fce0216d5f6a | subsection | 22 | 45 | Other scalings | Throughout this section, we worked under the assumption that
\lim _{n \rightarrow \infty } \text{Var}({X}_i^{\prime }{\beta }) = \gamma ^2,
which does not depend on n, and we explained that this is the only
scaling that makes sense to avoid a trivial problem. We set the
variables to have variance 1/n but this is of cou... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
"Emmanuel J. Candes"
] | [
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be7ca00e8f2dac044721fed50dd39ccbf8c13379 | subsection | 23 | 45 | Non-Gaussian covariates | Our model assumes that the features are Gaussian, yet, we expect that
the same results hold under other distributions with the proviso that
they have sufficiently light tails. In this section, we empirically
study the applicability of our results for certain non-Gaussian
features.
[Figure: Simulation for a non-Gaussian... | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
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7fbcfdd9fd91e9afa53dc9b6150c9d2e7e728cd4 | subsection | 24 | 45 | Adjusting Inference by Estimating the Signal Strength | All of our asymptotic results, namely, the average behavior of the
MLE, the asymptotic distribution of a null coordinate, and the LLR,
depend on the unknown signal strength \gamma . In this section, we
describe a simple procedure for estimating this single parameter from
an idea proposed by Boaz Nadler and Rina Barber ... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
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] | [
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93dcb07d4065b0d965336c82de79bc4b7d85b771 | subsection | 25 | 45 | ProbeFrontier: estimating | We estimate the signal strength by actually using the predictions from
our theory, namely, the fact that we have asymptotically characterized
in Section 2 the region where the MLE exists. We know from Theorem
REF that for each \gamma , there is a maximum
dimensionality g_{\text{MLE}}^{-1}(\gamma ) at which the MLE ceas... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
"Emmanuel J. Candes"
] | [
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cb733ef7495851ffe68c0545088abbfc53ec6306 | subsection | 26 | 45 | Empirical performance of adjusted inference | We demonstrate the accuracy of ProbeFrontier via some empirical
results. We begin by generating 4000 i.i.d. observations
(y_i,{X}_i) using the same setup as in Figure REF
(\kappa =0.1 and half of the regression coefficients are null). We
work with a sequence \lbrace \kappa _j\rbrace of points spaced apart by
10^{-3} a... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
"Emmanuel J. Candes"
] | [
"math.ST",
"stat.ME",
"stat.TH"
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31d41cea9104d0981a2246c7577d95d6d093bc80 | subsection | 27 | 45 | Empirical performance of adjusted inference | We report an average over 6000 replicateswith the std. dev. between parentheses.]Finally, we focus on the estimation accuracy for a particular
(\kappa ,\gamma ) pair across several replicates. In the setting of
Figure REF , we generate 6000 samples and obtain
estimates of bias (\hat{\alpha }), std. dev. (\hat{\sigma })... | {
"cite_spans": []
} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
"Emmanuel J. Candes"
] | [
"math.ST",
"stat.ME",
"stat.TH"
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9d91a0c0d8bfa95413e37825b83e018d288bb38c | subsection | 28 | 45 | De-biasing the MLE and its predictions | We have seen that maximum likelihood produces biased
coefficient estimates and predictions. The question is how
precisely can our proposed theory and methods correct this. Recall
the example from Figure REF , where the theoretical
prediction for the bias is \alpha _{\star }= 1.499. For this dataset,
ProbeFrontier yield... | {
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c016e9021dad3acce48b046c8e24a238485b43d3 | subsection | 29 | 45 | Main Mathematical Ideas | As we mentioned earlier, we do not provide detailed proofs in
this paper. The reader will find them in and the first author's Ph. D. thesis.
However, in this section we give some
of the key mathematical ideas and explain some of the main steps in
the arguments, relying as much as possible on published results from
. | {
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655b23333cef48372ee51b88fdec62d8296aae37 | subsection | 30 | 45 | The bulk distribution of the MLE | To analyze the MLE, we introduce an approximate message passing (AMP)
algorithm that tracks the MLE in the limit of large n and p. Our
purpose is a little different from the work in
which, in the context of generalized
linear models, proposed AMP algorithms for Bayesian posterior
inference, and whose properties have l... | {
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a02ad46e9f28fdfd579bd9ef1f41f56324437e1a | subsection | 31 | 45 | The bulk distribution of the MLE | Then starting from an initial pair
\alpha _0, \sigma _0, for t = 0, 1, \ldots , we inductively define
\lambda _t as the solution to\operatorname{\mathbb {E}}\left[\frac{2\rho ^{\prime }(Q_1^t)}{1+\lambda \rho ^{\prime \prime }(\mathsf {prox}_{\lambda \rho }(Q_2^t))} \right] = 1- \kappaand the extra parameters \alpha _{... | {
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3b386914bd304242b92e4203e23b72d462e90b97 | subsection | 32 | 45 | The bulk distribution of the MLE | From now on, we use
\alpha _0 = \alpha _{\star }, \sigma _0 = \sigma _{\star } so that the sequence
\lbrace \alpha _t, \sigma _t, \lambda _t\rbrace is stationary; i. e. for all t \ge 0,\alpha _t = \alpha _{\star }, \quad \sigma _t = \sigma _{\star }, \quad \lambda _t = \lambda _{\star }.With this stationary sequence of... | {
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cd3e521ef6359abe51ccb493578d4f57912ff310 | subsection | 33 | 45 | The bulk distribution of the MLE | From here on, an analysis similar to that in
establishes that in the limit of large
iteration counts, the AMP iterates converge to the MLE, that is,\lim _{t \rightarrow \infty } \lim _{n \rightarrow \infty }\frac{1}{p} \sum _{j=1}^{p}\psi (\hat{\beta }_{j}^{t } - \alpha _{\star }\beta _{j}, \beta _j) = \lim _{n \right... | {
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25c15e70564e2f37bb12c5c791a28ec7b7c242cf | subsection | 34 | 45 | The distribution of a null coordinate | We sketch the proof of Theorem REF in the case
where the empirical limiting distribution \Pi has a point mass at
zero. The analysis in the general case, where the number of
vanishing coefficients is arbitrary, and in particular, o(n), is
very different and may be found in Appendix .Now consider Theorem REF with
\psi (t... | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
"Pragya Sur",
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318d99676d9d632bae1c9514d74235284a970a18 | subsection | 35 | 45 | The distribution of a null coordinate | From
(REF ) and the weak law of large numbers, we have
\Vert \hat{{\beta }}_{[k]} \Vert /\Vert {Z}\Vert \stackrel{\mathbb {P}}{\rightarrow }\sigma _{\star }, leading to
\hat{\beta }_j \stackrel{\mathrm {d}}{\rightarrow }{\mathcal {N}}(0, \sigma _{\star }^2). | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
regression | [
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113ec7616b927f76713c342c4e0dbbe19c70b050 | subsection | 36 | 45 | Broader Implications and Future Directions | This paper shows that in high-dimensions, classical ML theory is
unacceptable. Among other things, classical theory predicts that the
MLE is approximately unbiased when in reality it seriously
overestimates effect magnitudes. Since the purpose of logistic
modeling is to estimate the risk of a specific disease given a
p... | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
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d4ac2d041fd525f34c46dedbcdb3f3d0344d21cb | subsection | 37 | 45 | Fisher information | We work with the model from Section and introduce the
Fisher information matrix defined asI({\beta }) = \operatorname{\mathbb {E}}\left[ \sum _i \rho ^{\prime \prime }({X}_i^{\prime }{\beta }) {X}_i {X}_i^{\prime }\right] = n \operatorname{\mathbb {E}}\left[ \rho ^{\prime \prime }({X}_i^{\prime }{\beta }) {X}_i {X}_i^{... | {
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
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9259f34d69a3c46036a7b35802e514c793e06f8e | subsection | 38 | 45 | Properties of fixed points of the AMP algorithm | In this section, we elaborate on the connection between the fixed points of (REF ) and the MLE \hat{{\beta }}. From (REF ), we immediately see that if (\hat{{\beta }}_{\star },{S}_{\star }) is a fixed point, the pair satisfies{X}^{\prime }\lbrace {y}- \rho ^{\prime }(\mathsf {prox}_{\lambda _{\star }\rho }(\lambda _{\s... | {
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"Pragya Sur",
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b8ae8bbb2c82bbe79851040e606fa717c324d7d5 | subsection | 39 | 45 | Refined analysis of the distribution of a null coordinate | The AMP analysis is useful to analyze the bulk behavior of the MLE;
i.e. the expected behavior when averaging over all coordinates. It
also helps in characterizing the distribution of a null coordinate
when the limiting empirical cdf does not have a point mass at zero, as
we have seen in Section REF . However, the stud... | {
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4812d04ea4f786c6a2e5a38028fb2ff3160d7d17 | subsection | 40 | 45 | Refined analysis of the distribution of a null coordinate | On the other hand, we can subtract the
two score equations \nabla \ell (\hat{{\beta }})=0 and
\nabla {\ell _{[-j]}} (\hat{{\beta }}_{[-j]})=0 (\ell _{[-j]} is the
negative log-likelihood for the reduced model), which upon separating
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} | 10.1073/pnas.1810420116 | 1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic
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"Pragya Sur",
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a04dbaf3fadfb122cb42226525ea048301ddc23d | subsection | 41 | 45 | Refined analysis of the distribution of a null coordinate | It was established in that the denominator above is equal to \kappa /\lambda _{[-j]} + o_P(1), where, we have see in Section REF that\lambda _{[-j]} := \frac{1}{n} \mathrm {Tr}[\nabla ^2(\ell _{[-j]}(\hat{{\beta }}_{[-j]}))^{-1}].Note that since \beta _j = 0, {y} and {X}_{\bullet -j}, \hat{{\beta }}_{[-j]}
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4102805becd2ce480867275df5ea61e602edf2a7 | subsection | 42 | 45 | Refined analysis of the distribution of a null coordinate | Denote by
\nabla {\ell }_{[-i],[-j]}(\tilde{{\beta }}_{[-i],[-j]})=0 the reduced score
equation and subtract it from the score equation for \hat{{\beta }} to
obtain{X}_{i,-j}\left(y_i - \rho ^{\prime }({X}_{i,-j}^{\prime } \hat{{\beta }}_{[-j]} ) \right) + \sum _{k \ne i} {X}_{k,-j}\left( \rho ^{\prime }({X}_{k,-j}^{\p... | {
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8f329321c397818a2e6bf882a21b11fb0f798ed9 | subsection | 43 | 45 | Refined analysis of the distribution of a null coordinate | Hence, the fitted values can be approximated
as{X}_{i,-j}^{\prime }\hat{{\beta }}_{[-j]} \approx {X}_{i,-j}^{\prime } \hat{{\beta }}_{[-i],[-j]} + \lambda _{[-j]} \left(y_i - \rho ^{\prime }({X}_{i,-j}^{\prime }\hat{{\beta }}_{[-j]}) \right).Recalling the definition of the proximal mapping operator, \mathsf {prox}_{\la... | {
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f54855125b4b6e08778e420da40bb5e4bda1fc7c | subsection | 44 | 45 | Refined analysis of the distribution of a null coordinate | Instead, we find the joint distribution of
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495b3e97ceba37a90794f46502bf4d1a63a0732e | abstract | 0 | 92 | Abstract | For many nonlinear physical systems, approximate solutions are pursued by
conventional perturbation theory in powers of the non-linear terms.
Unfortunately, this often produces divergent asymptotic series, collectively
dismissed by Abel as "an invention of the devil." An alternative method, the
self-consistent expansio... | {
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} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
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8d0bdfc9bc66076309355578791d716c69a8cd5a | abstract | 1 | 92 | Abstract | These results allow us to treat the Airy function, and to see the
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"Benjamin Remez",
"Moshe Goldstein"
] | [
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6eab5c9b8de3287e0af3641dcc2b7f131c29beaa | subsection | 2 | 92 | Introduction | A recurring theme in physics is the necessity of approximations. Exactly
solvable systems are rare in the landscape of modern science, and
the majority of physical problems cannot be assailed directly without
resorting to any estimates or simplifications. First-principle calculations
must often be replaced by phenomeno... | {
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... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
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"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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1ef7d78a05d7e27bd2c18c7b11cdb9c005923224 | subsection | 3 | 92 | Introduction | This method was subsequently applied to more complicated
variants of the KPZ equation , , , , ,
and to other problems such as fracture, turbulence, and the XY model
, , , .The SCE’s central idea is to smartly choose the zeroth-order
system around which one expands, so that the zeroth-order system is
as close as possibl... | {
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... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
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"hep-th",
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] | 2,018 | en | Physics | [
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db60a38eb401ccad841e53ca1f8e1c1c70c78933 | subsection | 4 | 92 | Introduction | Moreover, our simplified approach will allow us to then extend our
treatment to the double well case, higher anharmonicities, many coupled
oscillators, and, most importantly, the complex coupling case vis-à-vis
the Stokes phenomenon.Our main result is a proof that by imposing self-consistency (i.e.,
determining the spl... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
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d23502b607c5892c2a13e19047d6e048487aad46 | subsection | 5 | 92 | Divergence of PT | A canonical example of the limitation of PT is found in the anharmonic
oscillator. The simplest anharmonicity that keeps global stability is a quartic
perturbation, which we writeV(x)=\frac{1}{2}\gamma x^{2}+g_{0}x^{4}\,,with g_{0}>0. This potential refines models of ideal binding interactions
and also lends itself to ... | {
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"source_ref_id": "d88a70f14cea1e8d188fa... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
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850cebbf692bc8649e5750a7d739fa45c890c194 | subsection | 6 | 92 | Divergence of PT | Second, the application of the resummation
procedure presents its own challenges, such as knowledge of the late
PT terms, which may not be available. Therefore, we will use the anharmonic
oscillator as a test-bed and show that a convergent series may instead
be obtained by applying the SCE. | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
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3fce5083130f12a953aa82df19e3c45f4840e73d | subsection | 7 | 92 | SCE Around a Modified Oscillator | Ref. offers a treatment of the anharmonic oscillator
by expanding its Fokker-Planck equation of motion. Here we pursue
another approach: In the language of the SCE, instead of expanding
the system around the harmonic term, we expand around a modified harmonic
potential, whose strength is consistently varied to obtain a... | {
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... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
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692fdf559d02e725a5c606073c29b864525e6583 | subsection | 8 | 92 | SCE Around a Modified Oscillator | Binomial expansion then gives:\mathcal {Z}^{\left(N\right)} & =\int _{-\infty }^{\infty }e^{-\frac{1}{2}G\left(N\right)x^{2}}\sum _{n=0}^{N}\frac{\left(-1\right)^{n}}{n!}\sum _{l=0}^{n}\binom{n}{l}\left[\frac{1}{2}(1-G\left(N)\right)x^{2}\right]^{n-l}\left[gx^{4}\right]^{l}dx\\
& =\sum _{n=0}^{N}\frac{\left(-1\right)^{... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
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"hep-th",
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8c4123d281dc856fac447b3750b4f7cf37d6a00f | subsection | 9 | 92 | The Self-Consistent Criteria for | Lastly, we require a choice of the function G\left(N\right). In
the ODM it is determined solely based on mathematical convergence
properties, while for the OPT and the LDE this is usually done by
one of two common criteria: the principle of minimal sensitivity (PMS),
or the principle of fastest apparent convergence (FA... | {
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{
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"end":... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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61ea928bf4ec78564d1cff3e1beb199a03f9fcc9 | subsection | 10 | 92 | The Self-Consistent Criteria for | To first order in the SCE perturbation, the partition function
is\mathcal {Z}^{\left(1\right)} & =\sqrt{\frac{2}{G}}\Gamma \left(\frac{1}{2}\right)\left(1+\frac{1}{2}\left(\left(1-\frac{1}{G}\right)-\frac{4g}{G}\frac{3}{2}\right)\right)\,,while the corresponding moment x^{2M} would be\left[\mathcal {Z}\cdot \left\langl... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
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114cea3f49e10d4f1574ef2130418dd5fc48867e | subsection | 11 | 92 | The Self-Consistent Criteria for | A useful inequality for G is then:\max \left(1,\,\frac{1}{2}+2\sqrt{gK}\right)<G\left(K\right)<1+2\sqrt{gK}\,.A particular convenience of this choice is that\frac{(1-G)G}{4g}=\frac{1}{16g}\left(1-\sqrt{1+16Kg}\right)\left(1+\sqrt{1+16Kg}\right)=-K\,,so now the expansion takes the form\mathcal {Z}^{\left(N\right)}=\sqrt... | {
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"start": 548... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
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f01e87292b9fff954b7e8324f44ae49ec08cfb51 | subsection | 12 | 92 | Convergence Properties of the SCE | Let us state and prove our main result for the convergence properties
of the SCE; its relation to the convergence properties and their proofs
for the related schemes mentioned above will be discussed at the end
of this Section. We will show that the following proposition holds:Proposition 1 Let the self-consistently co... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
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"hep-th",
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f7355b8b314531eef42ff67d95df91e7b225b812 | subsection | 13 | 92 | Convergence to | For our proof we will denote explicitly the limiting operations involved
in the definitions of the summations of infinite series, in order
to emphasize that we never stumble into the same pitfall as regular
perturbation theory. Our proof begins by examining:\lim _{N\rightarrow \infty }\mathcal {Z}^{\left(N\right)} & =\... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
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f3c83441a9912ffa9d18189a512d8f871862436c | subsection | 14 | 92 | Convergence to | Thus, the error associated with the expansion is due to the
remainder of the Taylor series, which is truncated before integration.We note that the integrand has three distinct regions where its behavior
is qualitatively different:\mathcal {D}_{1}=\left[0,1\right],\qquad \mathcal {D}_{2}=\left[1,\infty \right)=\mathcal ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
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ec3a6242c187e67ad32c77eb3d567f9d0f7891a0 | subsection | 15 | 92 | The Domain | In this domain, the remainder can be bounded explicitly. The error
is negative, and is the sum of the truncated terms in the exponent's
Taylor series:-R_{1}^{\left(N\right)} & =\sqrt{\frac{2K\left(N\right)}{G\left(K\left(N\right)\right)}}\int _{0}^{1}e^{-Kv}\left[\lim _{L\rightarrow \infty }\sum _{n=N+1}^{L}\frac{K^{n}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
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f0559fcf7f5bab0ed1348923fd36f6662ca88a66 | subsection | 16 | 92 | The Domain | We then find-R_{1}^{\left(N\right)} & =\sqrt{\frac{2K\left(N\right)}{G\left(K\left(N\right)\right)}}\sum _{n=N+1}^{\infty }\left(1-\frac{1}{G\left(K\left(N\right)\right)}\right)^{n}\frac{K^{n}\left(N\right)}{n!}\int _{0}^{1}e^{-Kv}v^{n}\left(1-v\right)^{n}\frac{dv}{\sqrt{v}}\\
& <\sqrt{\frac{2K\left(N\right)}{G\left(K\... | {
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Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
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6e4f428bb90cf00f6f78af4e460c926aeff66eb7 | subsection | 17 | 92 | The Domain | With x=n and s=\frac{1}{2} it reads \frac{\Gamma \left(n+\frac{1}{2}\right)}{\Gamma \left(n+1\right)}<n^{-\frac{1}{2}},
leaving us with-R_{1}^{\left(N\right)} & <\sqrt{\frac{2}{G\left(K\left(N\right)\right)}}\sum _{n=N+1}^{\infty }\left(1-\frac{1}{G\left(K\left(N\right)\right)}\right)^{n}\left[\frac{K\left(N\right)}{K\... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
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-... |
dbcb2e2714755d47dd1b6dbfcbd8aba6f719e117 | subsection | 18 | 92 | The Domain | We thus have\left(-1\right)^{N}R_{2}^{\left(N\right)} & <\sqrt{\frac{2K\left(N\right)}{G\left(K\left(N\right)\right)}}\int _{1}^{\infty }e^{-Kv}\frac{K^{N^{\prime }}\left(N\right)}{N^{\prime }!}\left(1-\frac{1}{G\left(K\left(N\right)\right)}\right)^{N^{\prime }}v^{N^{\prime }-\frac{1}{2}}\left(v-1\right)^{N^{\prime }}d... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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ca0a257e098ce2f61833a8b3a312a7c6c289ae5e | subsection | 19 | 92 | The Domain | However, it turns out that this bound is a bit looser, and only shows
convergence for M/N>1.04 . We go the extra mile so we can show
that M=N, as used in Ref. , leads to convergence
as well., which occurs atFor K>2N, this maximum lies outside the domain of integration,
as 2N/K<1. However, this maximum still bounds the ... | {
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"raw": "M. Schwartz and E. Katzav, J. Stat. Mech: Theory Exp. 2008, P04023 (2008).",
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"start": 97
}
]
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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1cb1e4e28d100bf2f3fca18aab855b16c2fad0e1 | subsection | 20 | 92 | The Domain | We
so proceed to bound\ln \left(v\right)+\ln \left(v+1\right)\le \ln \left(v_{0}\right)+\ln \left(v_{0}+1\right)+\left(\frac{1}{v_{0}}+\frac{1}{v_{0}+1}\right)\left(v-v_{0}\right)\,.In total, we now get\int _{1}^{\infty }v^{N^{\prime }} & \left(v+1\right)^{N^{\prime }}e^{-Kv}dv\le \int _{1}^{\infty }v_{0}^{N^{\prime }}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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e73c1f33a0cbe1cb54f144ba56f317bfca9e943b | subsection | 21 | 92 | The Domain | If q<4 then r<1, and v^{r} is
a concave function which on the interval \left[0,1\right] is simply
bounded by v. This will reproduce the eventual \left(\frac{K}{K+N^{\prime }}\right)^{N}
result obtained for q=4. Let us then focus on q>4, so henceforth
r>1: As v^{r} is now a convex function, it is bounded by any
line dra... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0.0009052050299942493,
0... |
68623fc5876c9277a033f6c74237527437003a2d | subsection | 22 | 92 | The Domain | However, since e^{\left(r-1\right)v_{0}^{r}} can be
expanded in powers of v_{0}^{r}, the first non-vanishing derivative
of the lhs is the \left(r-1\right)-th, giving N^{\prime }\left(-r!\right)<0.
Thus, this equation is satisfied at least in the neighborhood of 0^{+}.
Indeed, for v_{0}\ll 1, we can expandK\left(r-1\rig... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0... |
5a6ae90c35f19714484f137ea42d5c21c83a96ce | subsection | 23 | 92 | The Domain | Inserting it into the error bound, we have-R_{1,q}^{\left(N\right)} & <\sqrt{\frac{2K}{GN^{\prime }\left(K+N^{\prime }rv_{0}^{r-1}\right)}}\left(1-\frac{1}{G}\right)^{N^{\prime }}\sum _{n=N^{\prime }}^{\infty }Q_{1}^{n}=\sqrt{\frac{2Kv_{0}}{G\left(N^{\prime }\right)^{2}}}\left(1-\frac{1}{G}\right)^{N^{\prime }}\frac{Q_... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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3933b535e612269e51b8a08ebd68a4d124575210 | subsection | 24 | 92 | The Domain | We may write\left|R_{2,q}^{\left(N\right)}\right| & =\sqrt{\frac{2K}{G}}\frac{K^{N^{\prime }}}{N^{\prime }!}\left(1-\frac{1}{G}\right)^{N^{\prime }}\int _{1}^{\infty }e^{-Kv}v^{N^{\prime }}\left(v^{r}-1\right)^{N^{\prime }}\frac{dv}{\sqrt{v}}\\
& <\sqrt{\frac{2K}{G}}\frac{K^{N^{\prime }}}{N^{\prime }!}\left(1-\frac{1}{... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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... |
8ccbd575aff625edccb412d673ddb98aac007268 | subsection | 25 | 92 | The Domain | Note that we now have to evaluate
the same expression for R_{2}^{\left(N\right)} as we did for q=4
in (), only multiplied by r^{N^{\prime }}. Since q<4,
we have r=\frac{q-2}{2}<1, and thus r^{N^{\prime }}<1. This means that
the requirements imposed on K\left(N\right), and consequently
on M\left(N\right), are more relax... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0.0... |
06f70aa5ec03e522d01f94f9c9ea8ab99bca05ea | subsection | 26 | 92 | Domain of Convergence | Summing the magnitudes of the remainders from both domains, we get
a total error bound ofR^{\left(N\right)} & =\left|\mathcal {Z}^{\left(N\right)}-\mathcal {Z}\right|=\left|R_{1}^{\left(N\right)}+R_{2}^{\left(N\right)}\right|<\left|R_{1}^{\left(N\right)}\right|+\left|R_{2}^{\left(N\right)}\right|\\
& <\sqrt{\frac{2}{G}... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0.0... |
b4f861e2bb9db4b8cf14e5e3c93ca2dd74de4692 | subsection | 27 | 92 | Domain of Convergence | This would require us to pick a minimal value of \alpha >\ln 2\approx 0.693.(iii) The last term scales as\frac{2^{N}N^{N}}{N!}\left(\frac{2N}{M}+1\right)^{N}e^{-2N-\frac{M}{2}-\frac{M^{2}}{2N+M}} & \sim \left(2\left(\frac{2}{\alpha }+1\right)e^{-1-\frac{\alpha }{2}-\frac{\alpha ^{2}}{\alpha +2}}\right)^{N}\,.The expone... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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-... |
85992660db5844583d29c1bef905c38c91599725 | subsection | 28 | 92 | Domain of Convergence | This case is discussed further in .In total, we expect the large-N error to scale asymptotically
asR^{\left(N\right)}=\mathcal {O}\left(10^{-A\left(\alpha \right)N-B\left(\alpha ,g\right)\sqrt{N}}\right)\,,where the bound on A\left(\alpha \right) isA\left(\alpha \right)={\left\lbrace \begin{array}{ll}
-\log _{10}\left(... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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... |
2649538645782baf39aaf7f9c357ab5cf694991e | subsection | 29 | 92 | Domain of Convergence | The error
will then be dominated by the first term, and have the functional
form\sqrt{\frac{2}{G}}\sqrt{\frac{M^{2}}{N^{3}}}\left(1-\frac{1}{G}\right)^{N}\left[\frac{M}{M+N}\right]^{N} & \sim N^{-\frac{p}{4}}N^{p-\frac{3}{2}}\left(1-\frac{1}{N^{p/2}}\right)^{N}\left(1-\frac{N}{N^{p}}\right)^{N}\sim N^{\frac{3}{4}\left(... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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... |
519bf8792c08c12f6a4f02eee056bdf4dfb2fc46 | subsection | 30 | 92 | Domain of Convergence | It is bounded by the first term in the sum, l=0 (which is positive),
so we have\mathcal {Z}^{\left(N\right)} & <\sqrt{\frac{2}{G\left(K\left(N\right)\right)}}\sum _{n=0}^{N}\left[1-\frac{1}{G\left(K\left(N\right)\right)}\right]^{n}\frac{\Gamma \left(n+\frac{1}{2}\right)}{n!}\\
& <\sqrt{\frac{2}{G\left(K\left(N\right)\r... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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846090ac89211e59f073a134c2cde736c1d59153 | subsection | 31 | 92 | Comparison with Results for Related Methods | It is instructive to compare the results we have shown here with those
obtained for the ODM/OPT/LDE schemes. Following arguments by Zinn-Justin
and Seznec , Buckley, Duncan, and Jones
showed that the sequence \mathcal {Z}^{\left(N\right)}
converges to \mathcal {Z}\left(g\right) if the modified harmonic coefficient
G sc... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 462,
"openalex_id": "",
"raw": "J. Zinn-Justin and R. Seznec, J. Math. Phys. 20, 1398 (1979).",
"source_ref_id": "2595fec7292f1ea07fbdfe7f791c394666cd058c",
"start": 109
},
{
"arxiv_id": "",
"doi": ... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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aefdfdc242036052ebd1d1bf02371e7a56c68891 | subsection | 32 | 92 | Numerical Results | In order to demonstrate the properties of SCE, the expansion in ()
was evaluated in Mathematica , and was compared
against a direct evaluation of (). Mathematica
was chosen by virtue of its ability to evaluate both to arbitrary
numerical precision . However, this precluded the
usage of floating-point values of g and \a... | {
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... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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376a7b25c0132b91b5e010c0b5a46eb36aa3cce5 | subsection | 33 | 92 | Numerical Results | The minimal error shown is roughly 1.5\times 10^{-92}.Note that in order to obtain this many accurate digits, the intermediatecalculations had to be performed with over 300 digit precision.]REF depicts the convergence properties of the
SCE as a function of \alpha . It shows the error of the expansion
for two orders, N=... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1808,
"openalex_id": "",
"raw": "I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993).",
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"start": 1469
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{
"arxiv_id": "... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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8bb31c7c95b458c6c1dcb9df799b0d1198948a14 | subsection | 34 | 92 | Numerical Results | The fitted values for the parameter A are 0.072, 0.200 and
0.283, for \alpha =1 , 2, and \frac{4}{3}, respectively.
These are in agreement with the bounding values 0.018, 0.176,
and 0.243, given by ().We continue with a comparison of the SCE with other asymptotic and
numerical approximation schemes. These include the m... | {
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{
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"end": 460,
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... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0.023965885862708092,
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0.01... |
67a94c836df64df06f945b02bda0f50324817e5c | subsection | 35 | 92 | Numerical Results | To make a “fair” comparison, the
SCE, Padé, and \tau methods are evaluated at N=N_{0}, though
in principle they converge as N\rightarrow \infty .A striking result of REF is the similarity
of the errors produced by SCE (at order N_{0}=\frac{1}{16g}) and
hyperasymptotics. This can be explained by the error estimates of
b... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1570,
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"raw": "M. V. Berry and C. J. Howls, Proc. R. Soc. A 430, 653 (1990).",
"source_ref_id": "dfee5969e4d320f8d915ee156a5c53b119ff41a5",
"start": 1346
},
{
"arxiv_id": "",
"doi"... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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-0... |
992395e2da0863f469dc27142f7ec55486b34969 | subsection | 36 | 92 | Numerical Results | If compared with hyperasymptotics
when carried through to its conclusion, halting at roughly 2N_{0}
terms with an error of order e^{-2.386N_{0}}, then at order 2N_{0}
the SCE would result in an error of order \left(1-\frac{1}{\frac{1}{2}+\frac{1}{2}\sqrt{1+2\times \alpha }}\right)^{2N_{0}}\left(\frac{\alpha }{\alpha +1... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0.0... |
8e1cca9317678f4c41c09e8928c9f55a1e9430d6 | subsection | 37 | 92 | Numerical Results | Both the Padé approximants and
the \tau approximations are rational functions of g, where the
numerator and denominator are of identical order in g; as such,
for a fixed order N, they will tend to a constant as g\rightarrow \infty .
\mathcal {Z}\left(g\right) decays to zero for g\rightarrow \infty (the
anharmonicity co... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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... |
bd9f07cbd2769728639ac849eabd5179adeb2066 | subsection | 38 | 92 | Non-Perturbative SCE: The Double-Well Potential | Now that the convergent nature of the SCE has been observed, we can
examine a more intricate case, that of the double-well potential,
corresponding to a negative quadratic part of the potential (\gamma <0
in ()). This has the effect of flipping
the sign of the quadratic term in (). It is an interesting test case, since... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1119/1.1972842",
"end": 909,
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"raw": "M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Vol. 55 (Dover, New York, 1964).",
"source_ref_id": "d88a70f14cea1e8d188fa5... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0.051127854734659195,
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-0.... |
ff489270a87249c80d7670628755b22614e2670e | subsection | 39 | 92 | Non-Perturbative SCE: The Double-Well Potential | This is of course unavoidable, since
at g=0 the potential is not bound from below and the partition
function cannot be defined.We attribute the initial divergence of the SCE at low orders to an
incompatibility between two conflicting goals. The first is that the
SCE zeroth order potential V_{0}\left(x\right)=\frac{1}{2... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
-0.0254458487033844,
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0.04280638322234154,
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0.008573480881750584,
-0.011372829787433147,
-0.... |
243db9d9f54163b940b86e57934cdb6c715eb8a8 | subsection | 40 | 92 | Non-Perturbative SCE: The Double-Well Potential | As \gamma \rightarrow 0^{+}, g\rightarrow \infty
and so does G. As \gamma crosses zero, G\rightarrow \infty
for \gamma \rightarrow 0^{-}, but as \left|\gamma \right| increases,
G descends from infinity along the double-well branch in ().Strikingly, we have shown that the SCE provides a means to write down
a perturbat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 915,
"openalex_id": "",
"raw": "I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993).",
"source_ref_id": "939c4a588d4b82d0d306dfa9bae7c7c6c042f3ca",
"start": 711
},
{
"arxiv_id": "",... | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
-0.04498652368783951,
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-0.03882147744297981,
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0.04095787927508354,
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ea4ab117d34fe5db450019dfb7b1fcaf1bfba31d | subsection | 41 | 92 | General Power-Law Perturbations | A SCE may be performed in the case of a general perturbation g\left|x\right|^{q}.
We assume that q>2, and that g may be complex but has a positive
real part. Expanding again around a modified \frac{1}{2}Gx^{2}
harmonic oscillator, the partition function is now\mathcal {Z}^{\left(N\right)}=\sqrt{\frac{2}{G}}\sum _{n=0}^... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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... |
3b73acc8e1485e1ddfba0c7762e1763bc574e513 | subsection | 42 | 92 | General Power-Law Perturbations | If G-1 is small but M\gg q (i.e.,
gM^{\frac{q}{2}-1} is small but M is large), then a simpler
limit applies,G\approx 1+2^{\frac{q}{2}}gM^{\frac{q}{2}-1}\,.Going back to the expansion for \mathcal {Z}, it now becomes\mathcal {Z}^{\left(N\right)}=\sqrt{\frac{2}{G}}\sum _{n=0}^{N}\left(1-\frac{1}{G}\right)^{n}\sum _{l=0}^... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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-0.007159051485359669,... |
cf97a06d501bcc59bd22b1abd191b2e76e0a3de7 | subsection | 43 | 92 | General Power-Law Perturbations | The error can once
again be calculated by\lim _{N\rightarrow \infty }\mathcal {Z}^{\left(N\right)} & =\lim _{N\rightarrow \infty }\int _{-\infty }^{\infty }e^{-\frac{1}{2}Gx^{2}}\sum _{n=0}^{N}\frac{1}{n!}\left(-\left[\frac{1}{2}\left(1-G\right)x^{2}+gx^{q}\right]\right)^{n}dx\\
& =\lim _{N\rightarrow \infty }\sqrt{\fr... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0.... |
d95227f31d23e2e1188a373cd95dc99fcfa9401b | subsection | 44 | 92 | Rates and Domains of Convergence | Collecting the contributions from all domains and cases, we find the
total bound on the error to beR_{q}^{\left(N\right)} & <\sqrt{\frac{2}{G}}\left(1-\frac{1}{G}\right)^{N^{\prime }}{\left\lbrace \begin{array}{ll}
\sqrt{\frac{Kv_{0}}{\left(N^{\prime }\right)^{2}}}\frac{Q_{1}^{N^{\prime }}}{1-Q_{1}}+\frac{\left(\left(r... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0.03338555246591568,
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... |
5e47ed164de88f68dee7fc998600a6d5da7a7abc | subsection | 45 | 92 | Rates and Domains of Convergence | For r>1, from \mathcal {D}_{1} we
have Q_{1}<1 given by (), while from
\mathcal {D}_{2} we haveQ_{2}=\alpha ^{-r}\left(\frac{r+1}{e}\right)^{r+1}e\Rightarrow \alpha _{c}=\frac{1}{e}\left(r+1\right)^{1+\frac{1}{r}}\,.For r\le 1, Q_{1}=\frac{\alpha }{\alpha +1} and \mathcal {D}_{2}
has two components, which areQ_{2A}=e^{... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.056017 | 1803.06631 | A Deal with the Devil: From Divergent Perturbation Theory to an
Exponentially-Convergent Self-Consistent Expansion | [
"Benjamin Remez",
"Moshe Goldstein"
] | [
"math-ph",
"cond-mat.stat-mech",
"hep-th",
"math.MP"
] | 2,018 | en | Physics | [
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0.... |
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