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a2bdc816e98acacfa9c628b80ac2c6ac9fe3f2e7 | subsection | 46 | 92 | Rates and Domains of Convergence | Lastly,the final row shows the large q\gg 1 leading order approximationsof \alpha _{c} and Q^{*}.]In order to asses the applicability of SCE for anharmonicities with
q\ne 4, we list the values of the critical \alpha _{c} and the
optimal \alpha ^{*} for the first few integer perturbation powers
q>2 in REF . \alpha _{c} ... | {
"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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43e01ad36375afcdffab3c80ba16a6b0f74a7310 | subsection | 47 | 92 | Rates and Domains of Convergence | However, it should be noted that these bounds are not particularly
tight as compared with the numerically fitted values (as demonstrated
in for q=4; additional numerical
results are listed in the table). Furthermore, they also do not reflect
the additional convergent factor of \left(1-1/G\right)^{N}, giving
a stretched... | {
"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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f4db80e2a2cf2d3cbbaab5a03b6a0abb84349b03 | subsection | 48 | 92 | SCE in the Complex Plane: Oscillatory
Integrals and Stokes Phenomenon | Following the success of the SCE in treating the anharmonic oscillator,
we wish to elucidate other properties of this technique. We would
like to explore how the SCE carries over to complex functions, and
in particular, oscillatory integrals. Using the results of the previous
section for q=3, we will treat the Airy fun... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1142/p345",
"end": 463,
"openalex_id": "https://openalex.org/W1552566358",
"raw": "O. Vallée and M. Soares, Airy Functions and Applications to Physics (Imperial College Press, London, 2010).",
"source_ref_id": "1f105dd802fca08d5714... | 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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... |
850912496192738e005bb0e22f4680aec90c8698 | subsection | 49 | 92 | The SCE of | The Airy function can be put into the integral representation \mathrm {Ai}\left(z\right) & =\frac{e^{-\frac{2}{3}z^{3/2}}}{\pi }\int _{0}^{\infty }e^{-z^{1/2}t^{2}}\cos \left(\frac{t^{3}}{3}\right)dt\sim \frac{e^{-\frac{2}{3}z^{3/2}}}{\pi z^{\frac{1}{4}}}\int _{-\infty e^{\frac{1}{4}\arg z}}^{\infty e^{\frac{1}{4}\arg ... | {
"cite_spans": [
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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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90ce36059837790fcd38000905fcf307b0772b77 | subsection | 50 | 92 | The SCE of | As we saw in the case of the anharmonic oscillator,
it was this independence that allowed us to demonstrate uniform and
exponential convergence. A lesson that is learned from this is that
one must take care not to introduce artificial symmetries into the
problem when applying the SCE: Originally, the relation between t... | {
"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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7dd791ce7e199145d7d0ab3f6f69aaade337dd34 | subsection | 51 | 92 | The SCE of | The integrand e^{-\left(t^{\prime }\right)^{2}} is entire and decays to
zero at infinity for \left|\arg t^{\prime }\right|<\frac{\pi }{4}, and since
we assumed \left|\arg G_{\Delta }\right|<\pi , we may deform the
integration path to again run over the positive real axis. This leads
to\tilde{\mathrm {Ai}}_{\Delta }^{\l... | {
"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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] | 2,018 | en | Physics | [
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e8a42cced32d07a52c9e3e50b278066d5d316e80 | subsection | 52 | 92 | The SCE of | In other words, we treat the SCE for the Airy function as the sum
of two separate SCEs, whose combined numeric value gives \mathrm {Ai}\left(z\right).
The condition for each G_{\Delta } is now0 & =M\left[1-\left(3i\Delta G_{\Delta }^{\frac{3}{4}}\left(1-\frac{\sqrt{z}}{\sqrt{G_{\Delta }}}\right)\right)^{-1}\frac{C_{3}\... | {
"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.... |
ca089f19f3fe9bdb91ea8632ad28dceb1170a365 | subsection | 53 | 92 | The SCE of | This finally yields\tilde{\mathrm {Ai}}_{\Delta }^{\left(N\right)}\left(z\right) & =\frac{1}{2}\left(\frac{z}{G_{\Delta }}\right)^{\frac{1}{4}}\sum _{n=0}^{N}\frac{1}{n!}\left(1-\frac{\sqrt{z}}{\sqrt{G_{\Delta }}}\right)^{n}\sum _{l=0}^{n}\binom{n}{l}\left(-\frac{C_{3}\left(M\right)}{M}\right)^{-l}\Gamma \left(\frac{2n... | {
"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... |
701f58ccb9c8f13bfd05a1ae2c1c99f825bb5682 | subsection | 54 | 92 | The SCE of | For general z,
one finds that G_{-}^{\frac{1}{4}}\left(z\right)=\left(G_{+}^{\frac{1}{4}}\left(z^{*}\right)\right)^{*},
which quickly leads to the conclusion that the expansion satisfies
\tilde{\mathrm {Ai}}^{\left(N\right)}\left(z^{*}\right)=\left[\tilde{\mathrm {Ai}}^{\left(N\right)}\left(z\right)\right]^{*}
for all ... | {
"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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... |
6715ec1987fa8031bef61c1dc4775a014602f6b9 | subsection | 55 | 92 | Analytic Properties and Solutions for | Recall that each of the Airy SCEs \tilde{\mathrm {Ai}}_{\pm }\left(z\right)
is in essence a complex extension of the partition function of an
anharmonic oscillator with an x^{3} perturbation, as analyzed in
. In particular, this implies that
the coefficients of the SCE expansion, when viewed as power series
in \left(1-... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1450,
"openalex_id": "https://openalex.org/W1965926499",
"raw": "G. G. Stokes, Trans. Camb. Phil. Soc. 10, 105 (1864).",
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"start": 1077
},
{
"ar... | 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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dd78ae7507f021135691ed53924b91720057a398 | subsection | 56 | 92 | Analytic Properties and Solutions for | Thus, the lines of phase \arg z=0,\pm \frac{\pi }{3},\pm \frac{2\pi }{3}
and \pi , which are called Stokes linesWe will not draw the distinction between Stokes and anti-Stokes lines.
define six different wedges in the complex plane with three distinct
behaviors of \mathrm {Ai}\left(z\right) — exponential decay, growth,... | {
"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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... |
6969d36bcf8b7b59cf067d0e0ee4bed3d9a5e207 | subsection | 57 | 92 | Analytic Properties and Solutions for | Note that as M\rightarrow \infty
this root tends to the principal root of \@root 3 \of {C_{3}\left(M\right)/3iM}.
However, this property is suddenly violated once we cross the Stokes
line at \arg z=\frac{2\pi }{3}. To see this, we substitute y=rz^{\frac{1}{4}}
into () to obtain3\Delta ir(r^{2}-1)z^{\frac{3}{4}}=C_{3}\... | {
"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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32cc20c10d95a4632a66ee16cd8c23400c8cd5e8 | subsection | 58 | 92 | Analytic Properties and Solutions for | If instead
\delta \varphi >0, then our privileged root scatters from r=1
up to r\propto e^{+\frac{i\pi }{3}} and G_{+}^{\frac{1}{4}} tends
to the line e^{+\frac{i\pi }{3}+\frac{1}{4}\frac{2\pi i}{3}}=e^{+\frac{i\pi }{2}}.
The same logic applies for the root G_{-}, with all the signs of
the arguments above negated.We no... | {
"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.... |
e0ea8b58571a76762810e284e6df78d238dada49 | subsection | 59 | 92 | Analytic Properties and Solutions for | REF and REF ,
respectively.Recalling that C_{3}\left(M\right)/3M\sim \sqrt{M}, it asymptotically
behaves as\left(1-\frac{\sqrt{z}}{\sqrt{G_{\Delta }}}\right)^{N}\sim \left(1-\frac{\sqrt{z}}{M^{\frac{1}{3}}3^{-\frac{2}{3}}e^{-\frac{\Delta }{3}\pi i}}\right)^{N}\sim e^{-\left(\frac{9}{\alpha }\right)^{\frac{1}{3}}e^{\fra... | {
"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... |
0fb3dfcaabaa3e6947a40635cd6e9b171e2d9e9f | subsection | 60 | 92 | Analytic Properties and Solutions for | However, the actual exponential convergence rate is roughly 10^{-0.5N},
so the scaling of the cusp is closer to 10^{-0.20N}, or 10^{-0.27\left|z\right|^{\frac{3}{2}}}
if we substitute N=M/\alpha , with M given approximately by
(), and \alpha =\alpha ^{*}\approx 1. This
implies that the exponential convergence is always... | {
"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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d720745cbd0b72a0b86e08cad9eb6f0dad66b3e8 | subsection | 61 | 92 | Analytic Properties and Solutions for | Note theeye formed for N<30; for N>30, the error transitions smoothlyacross both Stokes lines, showing that the SCE smooths out the Stokeslines at large order. Lastly, the lower and upper dashed lines representthe uniform exponential convergence component in the \left|\arg z\right|<\frac{\pi }{3}and \frac{\pi }{3}<\lef... | {
"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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... |
e2a5c12580b454d60e97bcf40f2dbcd16f2e9b50 | subsection | 62 | 92 | Analytic Properties and Solutions for | It is only once
N is reduced, that the roots split discontinuously, with \sqrt{G}\rightarrow \sqrt{z}
or \sqrt{G}\rightarrow 0, depending on whether \arg z is larger
or smaller than \frac{2\pi }{3}. This occurs abruptly, at the value
of N at which the “collision event” depicted in REF
takes place, and predicted by () ... | {
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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",
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0.008729354478418827,
-0.023669980466365814,
0.007149829994887114,
0.04511183872818947,
-0.010186790488660336,
0.03152945265173912,
-0.02803465910255909,
0.... |
482f2a9f6d9d3efbf147c630d97d38c91530fd84 | subsection | 63 | 92 | Analytic Properties and Solutions for | However for any z, it may be bounded
by \left|1-\sqrt{z}/\sqrt{G}\right|^{N}<2^{N}\approx 10^{0.30N}
[see REF ], which
is weaker than the exponentially convergent component (both its bound
and its rate in practice), and convergence is formally still uniform.(iii) For \frac{2\pi }{3}<\left|\arg z\right|\le \pi , SCE pro... | {
"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 | [
-0.026354489848017693,
0.038455888628959656,
-0.04318657144904137,
-0.016328491270542145,
-0.016053806990385056,
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0.00107203412335366,
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0.022600464522838593,
-0.008515232242643833,
0.04410218819975853,
0.011391793377697468,
... |
88365840aaa46ee8b953ddfe48df452cda173702 | subsection | 64 | 92 | Further Numerical Results | Recalling that z=0 corresponds to an infinite anharmonicity [cf.
the rhs of ()], and so
represents an extreme test case, we first examine the convergence
of the SCE to \mathrm {Ai}\left(0\right). We take z=0 as having
phase zero, that is, lying on the positive real line, and so it is
contained in the first Stokes wedge... | {
"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 | [
-0.03130427002906799,
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0.00023610256903339177,
0.002059491351246834,
0... |
3e2efa5f8ad8c6cc25c3d5e75ba843b4d764bd84 | subsection | 65 | 92 | Further Numerical Results | Similarly to ,
the SCE exhibits performance much better then superasymptotics, and
comparable with hyperasymptotics, though in principle the hyperasymptotic
expansion is of order 2N_{0}.
[Figure: Comparison of the Airy SCE versus the Padé and Chebyshev \tau approximations (a) The relative accuracy of the three methods ... | {
"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... |
f5846ddd3f97fd4bbf6bd90a40e3339b57a763a3 | subsection | 66 | 92 | Extension to Multiple Degrees of Freedom | In the case of an oscillator in more than one spatial dimension, or
alternatively that of several coupled oscillators, we may examine
a potential of the generic formV\left(\left\lbrace x_{i}\right\rbrace \right)=\frac{1}{2}\sum \gamma _{ij}x_{i}x_{j}+\sum g_{ijkl}x_{i}x_{j}x_{k}x_{l}\,,for which we assume that \gamma _... | {
"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.... |
66f1d6b45cabe72d982369ea996bf177b07e99b7 | subsection | 67 | 92 | Extension to Multiple Degrees of Freedom | \gamma \left(\Omega \right) and g\left(\Omega \right) are the quadratic
and quartic coefficients of the potential along the ray defined by
\Omega , which by assumption are positive and non-negative, respectively.
By satisfying first-order self-consistency for \left\langle r^{2M}\right\rangle ,
we find that\mathcal {Z}_... | {
"cite_spans": [
{
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"doi": "",
"end": 1694,
"openalex_id": "",
"raw": "D. S. Rosa, R. L. S. Farias, and R. O. Ramos, Physica A 464, 11 (2016).",
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"start": 1551
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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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... |
6ff06f56fa5e20fea1afb36847bb710dd5ff7732 | subsection | 68 | 92 | Conclusions and Outlook | In this paper we have investigated the analytical properties of the
SCE by applying it to the toy model of the classical anharmonic oscillator
in thermal equilibrium. We utilized the benefit of an explicit closed-form
expansion to show that for this model the SCE is exponentially and
uniformly convergent for any positi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2894,
"openalex_id": "",
"raw": "I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993).",
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"start": 2507
}
]
} | 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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0.01... |
0d4b90a3b249c3962bcc5988f14850c21b795004 | subsection | 69 | 92 | Conclusions and Outlook | Its strength is exemplified by
the remarkable result that in the SCE, optimal convergence is achieved
repeatedly in the linear scaling M\left(N\right)\sim N, for any
possible anharmonicity.These results provide fertile grounds for additional inquiry: We have
seen that for a given system, the SCE is not unique — neither... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2261,
"openalex_id": "",
"raw": "C. M. Bender, A. Duncan, and H. F. Jones, Phys. Rev. D 49, 4219 (1994).",
"source_ref_id": "b203fbf35013eb199eb167eef3477df300270d64",
"start": 2066
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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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0.0... |
5eaf9be27e500216d26602d8661fa2f013cac479 | subsection | 70 | 92 | Conclusions and Outlook | Most recently,
Serone, Spada, and Villadoro introduced Exact Perturbation Theory
(EPT) , an approach
in which the problem is framed as a particular realization in the
parameter space of a more general model with a Borel-resummable PT,
and the coupling-dependent interpolation within this space provides
the non-perturbat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 339,
"openalex_id": "",
"raw": "M. Serone, G. Spada, and G. Villadoro, J. High Energy Phys. 05, 56 (2017a).",
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"start": 0
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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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0.016766920685768127,
... |
d3e08287b2e8d0d053f5aa1b21d8768c3aeb3152 | subsection | 71 | 92 | Direct Estimation of the Quartic
SCE Coefficients | The specific case of the quartic anharmonicity admits a direct error
estimation by inspection of the SCE series coefficients. Given by
the sum over l in (), these coefficients may
be expressed asS_{n,K} & =\sum _{l=0}^{n}\binom{n}{l}\frac{\Gamma \left(n+l+\frac{1}{2}\right)}{n!\left(-K\right)^{l}}=\frac{\left(-1\right)... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 500,
"openalex_id": "https://openalex.org/W3089116201",
"raw": "DLMF, “NIST Digital Library of Mathematical Functions,” http://dlmf.nist.gov/.",
"source_ref_id": "1bad14f3312569fc1768be8ccfc8645c155416c3",
"start": 126... | 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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0.02921321429312229,
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0.023626986891031265,
-0.04279720410704613,
0.00... |
8d393af7fd5759ed1d91640b97261bb2cbb658e8 | subsection | 72 | 92 | Direct Estimation of the Quartic
SCE Coefficients | This reproduces the bounding form we used to prove case 3 of Proposition REF
at the end of .Writing down the sum represented by _{1}F_{1} explicitly, we haveS_{n,K} & =\frac{\left(-1\right)^{n}\pi e^{-K}}{K^{n}n!\Gamma \left(\frac{1}{2}-n\right)}\sum _{s=0}^{\infty }\frac{K^{s}\Gamma \left(-n+s+\frac{1}{2}\right)}{s!\... | {
"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... |
b664f28de71a78ddf5d5949147648132fbec7153 | subsection | 73 | 92 | Direct Estimation of the Quartic
SCE Coefficients | They occur for integer values of s which are closest
to the two solutions of\frac{K\left(n-s-\frac{1}{2}\right)}{\left(s+1\right)\left(2n-s-\frac{1}{2}\right)}=1\,,which are given bys_{max}^{\pm } & =\frac{1}{4}\left[\left(4n+1\right)+\left(2K-1\right)-3\pm \sqrt{\left(4n+1\right)^{2}+\left(2K-1\right)^{2}-1}\right]\ap... | {
"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.... |
c76d058fd88a6fbdd4c73cab181933746777f5c0 | subsection | 74 | 92 | Direct Estimation of the Quartic
SCE Coefficients | Recalling that K=M+2 and substituting M=\alpha N
into the approximate roots in (), we find the
following limits:\ln Q_{-} & \equiv \lim _{N\rightarrow \infty }\frac{1}{N}\ln P_{+}=-\frac{1}{2}\left[\alpha +\sqrt{\alpha ^{2}+4}+4\mathrm {tanh}^{-1}\left(\frac{\alpha }{2}-\frac{1}{2}\sqrt{4+\alpha ^{2}}\right)\right]\,,\... | {
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"start": 1166
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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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a7e277f60e30943d9968a84e4f3ede6bd7fa1996 | subsection | 75 | 92 | Direct Estimation of the Quartic
SCE Coefficients | Thus, the expansion must diverge for this scaling of M
with 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",
"math.MP"
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... |
d67e74a1fc6cddf8c6645f96614dd98f21bffdd8 | subsection | 76 | 92 | Summary of Competing Asymptotic and
Numerical Methods | In we compared the SCE with other
asymptotic methods for the case of g\ll 1, while in the strongly
coupled regime g\gg 1 we compared it against numerical approximation
schemes. In this Appendix, we will briefly describe each. | {
"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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ec0fd710147409deb8e748bbc10e0fb6dc9bd0e7 | subsection | 77 | 92 | Superasymptotics | The superasymptotic expansion of \mathcal {Z} is defined by terminating its
usual asymptotic series at its least term . This
truncation usually depends on the value of g. We can find the general
superasymptotic form of \mathcal {Z} by standard PT,\mathcal {Z}_{SA}\left(g\right) & =\sum _{n=0}^{N_{0}}\int _{-\infty }^{\... | {
"cite_spans": [
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"doi": "10.1201/... | 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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797275bd1f38f8d7098d368e29371f8f88513eee | subsection | 78 | 92 | Hyperasymptotics of | Past the optimal truncation of superasymptotics, one may find an asymptotic
expansion for the remainder. Truncation of this series at its least
term will yield a new, smaller remainder. This process can be iterated
systematically to improve upon superasymptotics. This technique it
called hyperasymptotics, and was devel... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 347,
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"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"
] | [
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b7437d758f82344a704e249665294653d1aedc47 | subsection | 79 | 92 | Hyperasymptotics of | Taking the limit of u\rightarrow +\infty with k=1,
we see that all paths end at complex infinity,z_{1}\left(u\right)\rightarrow \left\lbrace -\infty ,-i\infty \right\rbrace ,\ z_{2}\left(u\right)\rightarrow \left\lbrace -\infty ,\infty \right\rbrace ,\ z_{3}\left(u\right)\rightarrow \left\lbrace \infty ,i\infty \right\... | {
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df8aa6ad9aae844c02135950a6a2f43720a3fc27 | subsection | 80 | 92 | Hyperasymptotics of | We thus define\forall z\in C_{2}:\qquad \left[k\left(f\left(z\right)-f_{2}\right)\right]^{\frac{1}{2}}\equiv k^{\frac{1}{2}}z\sqrt{\frac{1}{2}+gz^{2}}\,,where k^{\frac{1}{2}} is taken on the smooth manifold of the square
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Exponentially-Convergent Self-Consistent Expansion | [
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90f545211874d254c40ced2d6ef4085ee04dcaa7 | subsection | 81 | 92 | Hyperasymptotics of | However, 2 is adjacent to both. It is easily verified that for
k=-1 (or in general, k with argument \pm \pi ), the path z_{2}\left(u\right)
in () coincides partially with z_{1,3}\left(u^{\prime }\right)
for some real u^{\prime }<u (i.e. z_{2}\left(u\right) “arrives late”
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0d64835944b755ab62569c2e1be4438bf330f6d7 | subsection | 82 | 92 | Hyperasymptotics of | Using our definition (REF ), one observes
that the following holds:\forall z\in C_{1}\left(+\pi \right):\qquad \arg \left\lbrace z\sqrt{\frac{1}{2}+gz^{2}}\right\rbrace =-\frac{\pi }{2}\ ,\\
\forall z\in C_{3}\left(-\pi \right):\qquad \arg \left\lbrace z\sqrt{\frac{1}{2}+gz^{2}}\right\rbrace =+\frac{\pi }{2}\ .Thus, si... | {
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a0efd98881eced2b96c88047c8ba51f2b0d19d1a | subsection | 83 | 92 | Hyperasymptotics of | The coefficients at z_{1} are thenT_{r}^{\left(1\right)} & =\frac{\left(r-\frac{1}{2}\right)!}{2\pi i}\oint _{z_{1}}\frac{\left[\left(f\left(z\right)-f_{1}\right)\right]^{\frac{1}{2}}dz}{\left[f\left(z\right)-f\left(z_{1}\right)\right]^{r+1}}\\
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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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6a861cffa1eade59172ba4c8178ebba74ff580f1 | subsection | 84 | 92 | Hyperasymptotics of | The saddles z_{1} and z_{3} are not adjacent to each other:
The contour C_{3}\left(\theta _{k}\right) never leaves the top half
of the complex plane, except for \arg \,k which is a multiple of
2\pi , when it coincides with half of the real line. The same applies
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e686f187ee73b535cb56a09e8ee5814726c87753 | subsection | 85 | 92 | Hyperasymptotics of | We haveN\left(2\right)=\frac{1}{16g}=10,\quad N\left(21\right)=N\left(23\right)=5,\quad N\left(212\right)=N\left(232\right)=2,\\
N\left(2121\right)=N\left(2123\right)=N\left(2321\right)=N\left(2323\right)=1\,.where N\left(nm\ldots \right) is the truncation of the scattering
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ff22deb2dc8473ef24e42c3912b0337e227a1b73 | subsection | 86 | 92 | Hyperasymptotics of | We
thus obtain\mathcal {Z}_{HA} & =\sum _{r=0}^{N\left(2\right)}T_{r}^{\left(2\right)}+2\sum _{r=0}^{N\left(23\right)}K^{\left(23\right)}T_{r}^{\left(3\right)}+2\sum _{r=0}^{N\left(232\right)}K^{\left(232\right)}T_{r}^{\left(2\right)}+4\sum _{r=0}^{N\left(2323\right)}K^{\left(2323\right)}T_{r}^{\left(3\right)}\,.The re... | {
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9e2454ac66c0f0de7979e7aa8f5160970d2ba3e8 | subsection | 87 | 92 | Hyperasymptotics of | Setting g=\frac{1}{160} and S=3 (and dividing
by the exact \mathcal {Z}\left(g\right)), we obtain an expected relative
error of 4.0\cdot 10^{-12}, in close agreement with our numerical
result.It is worth noting that for g much smaller than \frac{1}{160},
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d59f599cd686521113c5b25cf76a9df4f22b08ed | subsection | 88 | 92 | Chebyshev Polynomial Approximation by Lanczos's | With the definition (), we have\frac{d\mathcal {Z}}{dg}=-\int _{-\infty }^{\infty }x^{4}e^{-\left[\frac{1}{2}x^{2}+gx^{4}\right]}dx\,.However, performing the rescaling y=g^{\frac{1}{4}}x, we have\mathcal {Z}\left(g\right)g^{\frac{1}{4}}=\int _{-\infty }^{\infty }e^{-\left[\frac{1}{2}g^{-\frac{1}{2}}y^{2}+y^{4}\right]}d... | {
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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",
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6627eea341b45e0f9c664b6aef153f9b37ccaaf0 | subsection | 89 | 92 | Chebyshev Polynomial Approximation by Lanczos's | Note that g=0 is a singular point of (),
so we do not necessarily require a second boundary condition.Next, we outline the procedure of the \tau method, due to Lanczos
, : To obtain the N-th order
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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 | [
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b6319681719df14d73d407fc7e4b6938d64913c5 | subsection | 90 | 92 | Chebyshev Polynomial Approximation by Lanczos's | This \tau approximation can
be computed systematically with ease by any computer algebra software.In the case of the Airy function, we note that \mathrm {Ai}\left(z\right)
satisfies the second-order differential equation \frac{d^{2}\mathrm {Ai}\left(z\right)}{dz^{2}}=z\cdot \mathrm {Ai}\left(z\right)\,.Defining \tilde{... | {
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64eb25c6a6097aadc669456d55df2c41f03f060a | subsection | 91 | 92 | Padé Approximants | With the usual perturbative expansion of \mathcal {Z} in Subsection REF ,
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5d2c42caad1c82d981f70c4565bae818a707b3e7 | abstract | 0 | 47 | Abstract | We consider the problem of link prediction, based on partial observation of a
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e9c417385da5f74fce12b1d19cf82c55940e51a0 | subsection | 1 | 47 | Introduction | In the field of network analysis, the task of link prediction consists in predicting the presence or absence of edges in a large graph, based on the observations of some of its edges, and on side information. Network analysis has become a growing inspiration for statistical problems. Indeed, one of the main characteris... | {
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b17b86e6b16c19d724a7234f0b2259bbde6ddd82 | subsection | 2 | 47 | Introduction | The key assumption in this model is that the network is a consequence of the information, but not necessarily based on similarity: it is possible to model more complex interactions, e.g. where opposites attract.The focus on a high-dimensional setting is another aspect of this work that is also motivated by modern appli... | {
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63ec92c490b4b54bd72be0f0f5b8896dd8bd1e49 | subsection | 3 | 47 | Introduction | Furthermore, we show in Section that the minimax rate cannot be attained by a (randomised) polynomial-time algorithm, and we identify a corresponding computational lower bound.
The proof of this bound is based on a reduction scheme from the so-called dense subgraph detection problem.
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60e8e35c902b48c0cb10f8d801c2ee969b7c6813 | subsection | 4 | 47 | Generative model | For a set of vertices V = [n] and explanatory variables X_i \in \mathbf {R}^d associated to each i \in V, a random graph G=(V,E) is generated by the following model. For all i,j \in V, variables X_i,X_j \in \mathbf {R}^d and an unknown matrix \Theta _\star \in \mathbf {S}_d, an edge connects the two vertices i and j in... | {
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60924cc3df18cd20fb67ca24553f962d12d3bc2e | subsection | 5 | 47 | Generative model | Indeed, writing \textbf {vec}(A) \in \mathbf {R}^{d^2} for the vectorized form of a matrix A\in \mathbf {S}^d, we have thatX_i^\top \Theta _\star X_j = \mathbf {Tr}(X_j X_i^\top \Theta _\star ) = \langle \textbf {vec}(X_j X_i^\top ), \textbf {vec}(\Theta _\star ) \rangle \, .The vector of observation Y \in \mathbf {R}^... | {
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b69782d0e3e4114808550f3805c76c40bd97895b | subsection | 6 | 47 | Comparaison with other models | This model can be compared to other settings in the statistical and learning literature.Generalised linear model.
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468dbd39e3aba01f7b5f79f6d56f9a3b711431c9 | subsection | 7 | 47 | Parameter space | The unknown predictor matrix \Theta _\star describes the relationship between the observed features X_i and the probabilities of connection \pi _{ij}(\Theta _\star ) = \sigma (X_i^\top \Theta _\star X_j) following Definition REF . We focus on the high-dimensional setting where d^2 \gg N: the number of features for each... | {
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2671350bdbdff0231ead65efed0a6ccd42794b86 | subsection | 8 | 47 | Parameter space | The effect of these projections on the affinity is weighted by the \lambda _\ell . By allowing for negative eigenvalues, we allow our model to go beyond a geometric description, where close or similar Xs are more likely to be connected. This can be used to model interactions where opposites attract.The assumption of si... | {
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9983fb612906361b91e6a6115caa5f82130a0e3d | subsection | 9 | 47 | Explanatory variables | As mentioned above, this problem is different from tasks such as metric learning, where the objective is to estimate the X_i with no side information. Here they are seen as covariates, allowing us to infer from the observation on the graph the predictor variable \Theta _\star . For this task to be even possible in a hi... | {
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1d4a8263ada8634f349db107f6fdbac7922a9168 | subsection | 10 | 47 | Explanatory variables | These different measures of restricted isometry are related, as shown in the following lemmaLemma 5
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d9cc1a241d8bee126fc89be8a97daf2eb0d6f7a1 | subsection | 11 | 47 | Explanatory variables | Similarly to , Assumption REF can be shown to be redundant
for minimax optimal prediction,
because the log-likelihood function in the matrix logistic regression model satisfies the so-called self-concordant property.
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efe49b8c864389cb63526e0cc3bd6407392a883c | subsection | 12 | 47 | Random designs | For random designs, we require the block isometry property to hold with high probability.
Then the results in this article carry over directly
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a01a1ec321cc7115e0e60ab9102fb15b10d5d6a8 | subsection | 13 | 47 | Matrix Logistic Regression | The log-likelihood for this problem is\ell _Y(\Theta ) = -\sum _{(i,j) \in \Omega } \xi (s_{(i,j)} X_i^\top \Theta X_j)\, ,where s_{(i,j)} = 2 Y_{(i,j)} -1 is a sign variable that depends on the observations Y and \xi : x \mapsto \log (1+e^x) is a softmax function, convex on \mathbf {R}. As a consequence, the negative ... | {
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27a4b364531f4ac0e0f9cf996b61f865cf963ef8 | subsection | 14 | 47 | Penalized logistic loss | In a classical setting where d is fixed and N grows, the maximiser of \ell _Y - the maximum likelihood estimator - is an accurate estimator of \Theta _\star , provided that it is possible to identify \Theta from \mathbf {P}_{\Theta } (i.e. if the X_i are well conditioned). We are here in a high-dimensional setting wher... | {
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b0ab7b3ebbd90a670bf6956a1800a1010dfa10a7 | subsection | 15 | 47 | Penalized logistic loss | It can formally be shown using standard conditioning arguments .Corollary 12 Assume the design matrix \mathbb {X} satisfies the block isometry property from Definition REF and \max _{(i,j) \in \Omega } |X_i^\top \Theta _\star X_j| < M for some M > 0 and all \Theta _\star in a given class, and the penalty term p(\Theta ... | {
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f8e9554a20a413f4c561dbe5e0535370387edb34 | subsection | 16 | 47 | Penalized logistic loss | In particular,
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a22a8348c27dc2264dc203bd22a7bbaeb21182cd | subsection | 17 | 47 | Convex relaxation | In practice, computation of the estimator (REF ) is often infeasible.
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Sophisticated step-wise model selection procedures allow to reduce the number of analysed models.
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4613cf77e961917fbd02ef3a0cc4131bc1658030 | subsection | 18 | 47 | Prediction | In applications, as new users join the network, we are interested in predicting
the probabilities of the links between them and the existing users.
It is natural to measure the
prediction error of an estimator \hat{\Theta } by {\rm I}\hspace{-1.79993pt}{\rm E}\big [ \sum _{(i,j)\in \Omega } (\pi _{ij}(\hat{\Theta }) - ... | {
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e92c5048bcee1566ef7e9b6f23f08273d19a94f4 | subsection | 19 | 47 | Information-theoretic lower bounds | The following result demonstrates
that the minimax lower bound on the rate of estimation matches the upper bound in Theorem REF
implying that the rate of estimation is minimax optimal.Theorem 17
Let the design matrix \mathbb {X} satisfy the block isometry property. Then for estimating \Theta _\star \in \mathcal {P}_{... | {
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8742f1d089a45ded28e9261ec16d4763b0886e93 | subsection | 20 | 47 | Computational lower bounds | In this section, we investigate whether the lower bound in Theorem REF can be achieved with an estimator
computable in polynomial time. The fastest rate of estimation attained by a (randomised) polynomial-time algorithm
in the worst-case scenario is usually referred to as a computational lower bound.
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71c8e2af440f8e34d77666ef269c29426be228ee | subsection | 21 | 47 | The dense subgraph detection problem | Although our work is related to the study of graphs, we recall for absolute clarity the following
notions from graph theory.
A graph G = (V,E ) is a non-empty set V of vertices, together with a set
E of distinct unordered pairs \lbrace i,j \rbrace with i,j \in V, i \ne j. Each element
\lbrace i,j \rbrace of E is an edg... | {
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"arxiv_id": "",
"doi": "10.1007/978-1-4684-2001-2_9",
"end": 870,
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"raw": "Karp, R. (1972). Reducibility among combinatorial problems. Springer.",
"source_ref_id": "f7ac7dcf0077ac5ad31e27a7f242d716bb72464b",
... | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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6c3d626562324690bcb86ffd0dcb9605efd456e2 | subsection | 22 | 47 | The dense subgraph detection problem | Its variations have been used extensively as computational hardness assumptions in statistical problems, see , , , .The Planted Clique problem can be reduced to the so-called dense subgraph detection problem of testing the null hypothesis in (REF ) against the alternative H_1: A \sim G(n,1/2,k,q), where q \in (1/2,1]. ... | {
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"raw": "Berthet, Q. and Rigollet, P. (2013b). Optimal detection of sparse principal components in high dimension. The Annals of Statistics 41 1780–1815.",
... | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
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] | 2,018 | en | Mathematics | [
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cb614d8be56f1ca19c85376a944dae8dafbbc8bf | subsection | 23 | 47 | Reduction to the dense subgraph detection problem and a computational lower bound | Consider the vectors of explanatory variables X_i = N^{1/4} e_i, i =1,..., n and assume without loss of generality that
the observed set of edges \Omega in the matrix logistic regression model consists of the interactions of the
n nodes X_i, i.e. it holds N = |\Omega | ={n 2} .
It follows from the matrix logistic regre... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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... | |
aef8cbab88b1a343068858fbf097d893480b34f1 | subsection | 24 | 47 | Reduction to the dense subgraph detection problem and a computational lower bound | Let
c > 0 be a positive constant and f(k,d,N) be a real-valued function
satisfying f(k,d,N) \le ck^2/N for k = k_{n} < n^{\beta }, 0 < \beta < 1/2 and a sequence d = d_{n},
for all n> m_0 \in {\rm I}\hspace{-1.79993pt}{\rm N}.
If Conjecture REF holds, for some the design \mathbb {X} that fulfils the block isometry prop... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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eaea17fbb88cb9a2762cd0f3e6887d223e786c7c | subsection | 25 | 47 | Reduction to the dense subgraph detection problem and a computational lower bound | Assume that there exists a hypothetical estimator \hat{\Theta } computable in polynomial time
that attains the rate f(k,d,N) for the prediction
error, i.e. such that it holds that\limsup _{n \rightarrow \infty } \frac{1}{f(k,d,N)} \sup _{\Theta _\star \in \mathcal {F}_k}
\frac{1}{N}{\rm I}\hspace{-1.79993pt}{\rm E}\big... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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... | |
eeb57c4df6ffe7f1fa977843c47c3c7ba1c93df2 | subsection | 26 | 47 | Reduction to the dense subgraph detection problem and a computational lower bound | For the type II error, we obtain\sup _{\Theta \in \mathcal {G}_k^{\alpha _N}} \mathbf {P}_{\Theta } (\psi = 0) & =
\sup _{\Theta \in \mathcal {G}_k^{\alpha _N}} \mathbf {P}_{\Theta } \big (\Vert \hat{\Theta }\Vert _{F,\Omega } < \tau _{d,k}(u)\big ) \\
& \le \sup _{\Theta \in \mathcal {G}_k^{\alpha _N}} \mathbf {P}_{\T... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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... | |
ca032493822e7fa1962a58a3357476e707ded86f | subsection | 27 | 47 | Concluding remarks | Our results shed further light on the emerging topic of statistical and computational trade-offs in high-dimensional estimation.
The matrix logistic regression model is very natural to study the connection between statistical accuracy and computational efficiency
as the model is based on the study of a generative model... | {
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"doi": "10.1214/15-aoas842",
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"raw": "Bogdan, M., van den Berg, E., Sabatti, C., Su, W. and Candès, E. (2015). Slope—adaptive variable selection via convex optimization. The Annals of A... | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
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8d453cd4d8f4a2f36c4f1adde42d69e8953775f5 | subsection | 28 | 47 | Some geometric properties of the likelihood | Let us recall the stochastic component of the likelihood function\zeta (\Theta ) = \ell _Y(\Theta ) - \ell (\Theta ) =
\sum _{(i,j) \in \Omega } \big (Y_{(i,j)} - \pi _{ij}(\Theta _\star ) \big )X_i^\top \Theta X_j \,,which is a linear function in \Theta .
The deviation of the gradient \nabla \zeta of the stochastic co... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
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"stat.TH"
] | 2,018 | en | Mathematics | [
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0.047155145555734634,
-0.029056115075945854... | |
80c97962427f7379b6e2c334c394e5f774978978 | subsection | 29 | 47 | Some geometric properties of the likelihood | In this notation, we have \zeta (\Theta ) = \savebox {
}{\m@th {\langle }}\mathopen {\copy
\hspace{1.111pt}\hspace{0.0pt}\usebox {
}}\zeta , \Theta \savebox {
}{\m@th {\rangle }}\mathclose {\copy
\hspace{1.111pt}\hspace{0.0pt}\usebox {
}}_F,
with\nabla \zeta = \sum _{(i,j) \in \Omega } \varepsilon _{i,j} X_j X_i^\top... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.022798502817749977,
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0.002977615687996149,
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... | |
67ed0f90664cf77368678f3db6c7dcbdfe4c5f3e | subsection | 30 | 47 | Some geometric properties of the likelihood | Furthermore, using that \sup _{t\in \mathbf {R}}\sigma ^\prime (t) \le 1/4 , we obtain for all
\Theta \in \mathcal {P}(M)\ell (\Theta _\star ) - \ell (\Theta )\le \frac{1}{8} \Vert \mathbb {X}^\top ( \Theta _\star - \Theta ) \mathbb {X}\Vert _{F,\Omega }^2 \,.We shall also be using the bounds\max _{(i,j) \in \Omega } \... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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b7df72d49cbb5ad156ef871f17a5337661cb601f | subsection | 31 | 47 | Entropy bounds for some classes of matrices | Recall that an \varepsilon -net of a bounded subset \mathbf {K} of some metric space with a metric \rho is a collection \lbrace K_1,...,K_{N_\varepsilon } \rbrace \in \mathbf {K}
such that for each K \in \mathbf {K}, there exists i \in \lbrace 1,...,N_\varepsilon \rbrace such that \rho (K,K_i) \le \varepsilon . The \va... | {
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"raw": "Candes, E. and Plan, Y. (2011). Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements. IEE... | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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784fa8634095de313e816232a33b2a28e5da71fd | subsection | 32 | 47 | Proof of Theorem | It suffices to show the following uniform deviation inequality\sup _{\Theta _\star \in \mathcal {P}_{k,r}(M)}
\mathbf {P}_{\Theta _\star }\big ( \ell (\Theta _\star ) -\ell (\hat{\Theta }) + p(\hat{\Theta }) > 2 p(\Theta _\star ) + R_t^2\big ) \le e^{- c R_t}\,,for any R_t > 0 and some numeric constant c >0 . Indeed, t... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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a4620aa059109f512b6e51ec19be6074bbcc8c08 | subsection | 33 | 47 | Proof of Theorem | In view of \ell _Y(\hat{\Theta }) - p(\hat{\Theta }) \ge \ell _Y(\Theta _\star ) - p(\Theta _\star ), we have on the complement:\savebox {
}{\m@th {\langle }}\mathopen {\copy
\hspace{1.111pt}\hspace{0.0pt}\usebox {
}}\mathcal {E}_\Omega ,\mathbb {X}^\top (\hat{\Theta }- \Theta _\star )\mathbb {X}\savebox {
}{\m@th {\r... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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04046d2262ef555b3dac0add8a44138e635232d8 | subsection | 34 | 47 | Proof of Theorem | It follows,&\mathbf {P}_{\Theta _\star }\Big (\sup _{\tau (\Theta ; \Theta _\star ) \ge R_t} \frac{\savebox {
}{\m@th {\langle }}\mathopen {\copy
\hspace{1.111pt}\hspace{0.0pt}\usebox {
}}\mathcal {E}_\Omega ,\mathbb {X}^\top (\Theta - \Theta _\star )\mathbb {X}\savebox {
}{\m@th {\rangle }}\mathclose {\copy
\hspace{... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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1cc4dbb5f8213e267891390f4e2035f3e73aa78d | subsection | 35 | 47 | Proof of Theorem | Then following the lines of the proof of Lemma REF and using the singular value decomposition, we deriveH(\varepsilon , \lbrace \mathbb {X}^\top \Theta ^\prime \mathbb {X}:\Theta ^\prime \in \mathcal {G}_1, \Vert \mathbb {X}^\top \Theta ^\prime \mathbb {X}\Vert _{F,\Omega } \le B \rbrace , \Vert \cdot \Vert _{F,\Omega ... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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cabe88d61e9b2e6a5da708d78f9a6d9e71e1bd0d | subsection | 36 | 47 | Proof of Theorem | By Dudley's entropy integral bound, see and for a more recent reference, we then have{\rm I}\hspace{-1.79993pt}{\rm E}\big [&\sup _{{\Theta \in G_{2^s R_t}(\Theta _\star ) \\k(\Theta )=K, \, \operatorname{\mathbf {rank}}(\Theta ) = R}} \savebox {
}{\m@th {\langle }}\mathopen {\copy
\hspace{1.111pt}\hspace{0.0pt}\usebo... | {
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"raw": "Dudley, R. (1967). The sizes of compact subsets of hilbert space and continuity of gaussian processes. Journal of Functional Analysis 1 29... | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.02402760088443756,
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8e488bd3666e57f6757e2cef15f048c5158583f4 | subsection | 37 | 47 | Proof of Theorem | Furthermore, by Bousquet's version of Talagrand's inequality, see Theorem REF , in view of the bounds (REF ) and (), we have
for all u > 0\mathbf {P}_{\Theta _\star }\Big (&\sup _{{\Theta \in G_{2^s R_t}(\Theta _\star ) \\k(\Theta )=K, \, \operatorname{\mathbf {rank}}(\Theta ) = R}} \savebox {
}{\m@th {\langle }}\matho... | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0.003... | |
f6e1d8bb34e0ba20b33a0a09443e6734073561e2 | subsection | 38 | 47 | Proof of Theorem | Plugging this back into (REF ) and using (REF ), we obtain\mathbf {P}_{\Theta _\star }\Big (\sup _{{\Theta \in G_{2^s R_t}(\Theta _\star ) \\k(\Theta )=K, \, \operatorname{\mathbf {rank}}(\Theta ) = R}} \savebox {
}{\m@th {\langle }}\mathopen {\copy
\hspace{1.111pt}\hspace{0.0pt}\usebox {
}}\mathcal {E}_\Omega ,\mathb... | {
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"raw": "Bousquet, O. (2002). A bennett concentration inequality and its application to suprema of empirical processes. Comptes Rendus Mathematiq... | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0.0050537800416350365,
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4c5c55f14dd142048e8db69735c8507a8106239d | subsection | 39 | 47 | Proof of Theorem | Let \mathcal {F} be a countable set of measurable
real-valued functions on B such that f(\varepsilon _i)\le b < \infty a.s. and
{\rm I}\hspace{-1.79993pt}{\rm E}f(\varepsilon _i) = 0 for all i = 1,...,n, f \in \mathcal {F} . | {
"cite_spans": []
} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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1fc690d6db4bb794b78896c16a7866fe452f807e | subsection | 40 | 47 | Proof of Theorem | LetS := \sup _{f \in \mathcal {F}}\sum _{i=1}^{n}f(\varepsilon _i)\,, \quad \quad v:= \sup _{f \in \mathcal {F}}\sum _{i=1}^{n} {\rm I}\hspace{-1.79993pt}{\rm E}[f^2(\varepsilon _i)] .Then for all u > 0, it holds that\mathbf {P}\Big (S - {\rm I}\hspace{-1.79993pt}{\rm E}[S] \ge \sqrt{2(v + 2b{\rm I}\hspace{-1.79993pt}{... | {
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5e47898508ea742a63ebbd2894682fbe42a17972 | subsection | 41 | 47 | Proof of Theorem | Plugging this bound back into (REF ) and using the block isometry property yields the desired assertion.The proof is split into two parts. First, we show a lower bound of the order kr and then a lower bound of the order k \log (de/k).
A simple inequality
(a + b)/2 \le \max \lbrace a,b\rbrace for all a, b > 0 then comp... | {
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8451a9bc4f194b99211d2e3ab675d4d36fa9c78d | subsection | 42 | 47 | Proof of Theorem | Thus \mathcal {B}^\circ
is a 2\delta -separated set in the Frobenius metric with \delta ^2 = \frac{kr}{64} \big ( \frac{\alpha _N}{2}\big )^2\lfloor \frac{k}{r} \rfloor .
The Kullback-Leibler divergence between the measures \mathbf {P}_{\Theta _{u}} and \mathbf {P}_{\Theta _{v}},
\Theta _{u}, \Theta _{v} \in \mathcal ... | {
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} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
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91a6c851015db5ed5ca3c67dac40d5f1060a0953 | subsection | 43 | 47 | Proof of Theorem | Using simple calculations, we then have
(2k -1)\alpha _N^2 \le \Vert \Theta _u - \Theta _v \Vert _F^2 \le 2 k^2\alpha _N^2
for all \Theta _u, \Theta _v \in \mathcal {G}_k^{\alpha _N}, u\ne v. Furthermore, according to Lemma REF to follow, there exists a subset
\mathcal {G}^{\alpha _N, 0}_{k} \subset \mathcal {G}_k^{\al... | {
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} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
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159171ec3053c1c65c58ddb17199d87d98843384 | subsection | 44 | 47 | Proof of Theorem | As in the first part of the proof, taking \gamma > 0 small enough, we obtain\frac{1+ \Delta _{\Omega ,2k}(\mathbb {X})}{4} k^2\alpha _N^2N +\log 2
&= k \gamma \log (2) \log \big (\frac{de}{k}\big ) +\log 2
= \log (2^{k \gamma \log (de/k) + 1})\\
&< \log (2^{\rho k \log (de/k)} + 1) \,.The desired lower bound then follo... | {
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6502149a6ae6b12f5692afc9bef483bef274a173 | subsection | 45 | 47 | Proof of Theorem | Then there exists a subset \mathcal {G}^0 = \lbrace E^{(0)},..., E^{(J)}\rbrace \subset \mathcal {G} of cardinality\log J := \log (\operatorname{\mathsf {card}}(\mathcal {G}^0 ))\ge \rho k \log (\frac{de}{k})\,,where \rho = \frac{\alpha }{- \log (\alpha \beta )}(- \log \beta +\beta -1 )
such that\rho _H(E^{(k)}, E^{(l)... | {
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} | 1803.07054 | Optimal link prediction with matrix logistic regression | [
"Nicolai Baldin",
"Quentin Berthet"
] | [
"math.ST",
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430cf471ee160cfd1fffd373649928a2c2d87834 | subsection | 46 | 47 | Proof of Theorem | For instance, in order to get the distance between matrices 2k^2, i.e. c = 2
we need to shift all the k columns (and consequently rows) and so the number of distinct columns of a matrix is m = k, and in order to get the minimal possible distance
4k - 2, i.e. c = (4k - 2)/k^2 we need to shift only one column and a corre... | {
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"doi": "10.1007/978-3-540-48503-2",
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"raw": "Massart, P. (2007). Concentration inequalities and model selection, vol. 6. Springer.",
"source_ref_id": "a37e3e039449b03ed507a8ba3c7c5... | 1803.07054 | Optimal link prediction with matrix logistic regression | [
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87727d5f3716d801c3a685abe211f6d9ca73ac94 | abstract | 0 | 41 | Abstract | We study the question of whether coherent neutrino scattering can occur on
macroscopic scales, leading to a significant increase of the detection cross
section. We concentrate on radiative neutrino scattering on atomic electrons
(or on free electrons in a conductor). Such processes can be coherent provided
that the net... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
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942a4b289a566d0c4a7113fd93bab68b86aa48dc | subsection | 1 | 41 | Introduction | Recently, the COHERENT collaboration has reported the first observation of
coherent elastic neutrino–nucleus scattering , , a process
predicted over forty years ago , .
This observation completed the standard-model picture of neutrino interactions
with nucleons and nuclei and opened up a new window to probe physics bey... | {
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"doi": "10.1126/science.aao0990",
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neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
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] | 2,018 | en | Physics | [
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612a56c86645a7afadc0196f355b93cb68557a70 | subsection | 2 | 41 | Macroscopic coherence? | How about scattering with coherence on macroscopic scales? Clearly,
this would require measuring even much smaller recoil energies and so
does not look practical. It is interesting, however, to inquire what could be
the increase of the detection cross sections if such measurements were
possible, leaving for the moment ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
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] | 2,018 | en | Physics | [
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f9b7d5f68b4a469b9c8b02cbb90d36a34cf67403 | subsection | 3 | 41 | Macroscopic coherence? | For q_0\sim (1~{\rm cm})^{-1}\simeq 2\times 10^{-5} eV and the
total mass of particles in the coherent volume m_t\sim 1\,g, one finds
E_{rec}\sim \vec{q_0}^2/2m_t\sim 10^{-43}~{\rm eV}, the quantity which
is not going to be ever measured. To give just one reason for that, in order
to measure recoil energy of this magni... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1689,
"openalex_id": "",
"raw": "J. Weber, “Gravitons, neutrinos, and anti-neutrinos,” Found. Phys. 14 (1984) 1185.",
"source_ref_id": "c7af87d093b27031c136a3221ce2bbadd7a633c9",
"start": 1620
},
{
"arxiv... | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
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] | 2,018 | en | Physics | [
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... |
8030f965cc8a1e9d8fa07a936997220207d5fc00 | subsection | 4 | 41 | Weber's approach and structure factors | Weber suggested to
detect neutrinos through their coherent scattering on crystals in torsion
balance experiments. This approach combines two interesting ideas. First,
as the force coincides with momentum transfer per unit time, the force
neutrinos impinge on a crystal is directly related to the momentum transfer
to the... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-1-4684-8911-8_1",
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"raw": "P. G. Debrunner and H. Frauenfelder, Introduction to the Mössbauer effect, in “An introduction to Mössbauer spectroscopy”, Ed. L. May, Plenum... | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
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43203d8df8f1780a1170400efb6101f5057eb4ef | subsection | 5 | 41 | Weber's approach and structure factors | For elastic neutrino scattering
the structure factor is given byF(\vec{k}-\vec{k}^{\prime })
=\sum _{i=1}^N e^{i(\vec{k}-\vec{k}^{\prime })\vec{r}_i}\,,where \vec{k} and \vec{k}^{\prime } are the momenta of the incident and scattered
neutrinos,
\vec{r}_i is the coordinate of the ith
scatterer and N is the total number ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
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"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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3996ee64dad96ec382293c575815928a7f0f6b65 | subsection | 6 | 41 | Weber's approach and structure factors | This leads to the well known Bragg condition for diffraction on
crystals,2d\sin \vartheta = n\lambda \,,where d is the interplanar distance in the crystal, \vartheta is the angle
between the neutrino momentum and the atomic plane (the scattering angle being
\theta =2\vartheta ), \lambda =2\pi /|\vec{k}| and n is an int... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 763,
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"raw": "L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media. Second Edition revised and enlarged by E. M. Lifshitz and L. P. Pitaevsky, Pergamon Press, 1984,... | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
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] | 2,018 | en | Physics | [
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