5657:
5230:
7889:
7011:
5241:
4904:
4824:
6441:
5652:{\displaystyle {\begin{aligned}g_{jk}(\theta )=4h_{jk}^{\mathrm {fisher} }&=4h\left({\frac {\partial }{\partial \theta _{j}}},{\frac {\partial }{\partial \theta _{k}}}\right)\\&=\sum _{i}p_{i}(\theta )\;{\frac {\partial \log p_{i}(\theta )}{\partial \theta _{j}}}\;{\frac {\partial \log p_{i}(\theta )}{\partial \theta _{k}}}\\&=\mathrm {E} \left\end{aligned}}}
7362:
6727:
1374:
5225:{\displaystyle {\begin{aligned}h&={\frac {1}{4}}\sum _{i}p_{i}(\theta )\;d(\log p_{i}(\theta ))\;d(\log p_{i}(\theta ))\\&={\frac {1}{4}}\sum _{jk}\sum _{i}p_{i}(\theta )\;{\frac {\partial \log p_{i}(\theta )}{\partial \theta _{j}}}{\frac {\partial \log p_{i}(\theta )}{\partial \theta _{k}}}d\theta _{j}d\theta _{k}\end{aligned}}}
7339:
3851:
3154:
4551:
6296:
924:
7884:{\displaystyle {\begin{aligned}h_{jk}={}&h\left({\frac {\partial }{\partial \theta _{j}}},{\frac {\partial }{\partial \theta _{k}}}\right)\\={}&{\frac {1}{4}}\mathrm {E} \left\\{}&+\mathrm {E} \left-\mathrm {E} \left\mathrm {E} \left\\&{}+{\frac {i}{2}}\mathrm {E} \left\end{aligned}}}
7006:{\displaystyle {\begin{aligned}ds^{2}={}&{\frac {1}{4}}\int _{X}(\delta \log p)^{2}\;p\,dx\\{}&+\int _{X}(\delta \alpha )^{2}\;p\,dx-\left(\int _{X}\delta \alpha \;p\,dx\right)^{2}\\&{}+{\frac {i}{2}}\int _{X}(\delta \log p\delta \alpha -\delta \alpha \delta \log p)\;p\,dx\end{aligned}}}
2397:
5790:
is not discrete, but continuous, the argument still holds. This can be seen in one of two different ways. One way is to carefully recast all of the above steps in an infinite-dimensional space, being careful to define limits appropriately, etc., in order to make sure that all manipulations are
4443:
This embedding induces a metric on the sphere, it is inherited directly from the
Euclidean metric on the ambient space. It takes exactly the same form as the above, taking care to ensure that the coordinates are constrained to lie on the surface of the sphere. This can be done, e.g. with the
1194:
7920:. Equivalently, the Fubini–Study metric can be understood as the metric on complex projective Hilbert space that is induced by the complex extension of the flat Euclidean metric. The difference between this, and the Bures metric, is that the Bures metric is written in terms of mixed states.
527:
3686:
3431:
5767:; whereas the non-coordinate form is the same as the Euclidean (flat-space) metric. That is, the Fisher information metric on a statistical manifold is simply (four times) the Euclidean metric restricted to the positive orthant of the sphere, after appropriate changes of variable.
4342:
2729:
2005:
1648:
8392:
7045:
in the above makes it clear that the first term is (one-fourth of) the Fisher information metric. The full form of the above can be made slightly clearer by changing notation to that of standard
Riemannian geometry, so that the metric becomes a symmetric
7122:
3705:
2950:
6602:
4819:{\displaystyle {\begin{aligned}h&=\sum _{i}dy_{i}\;dy_{i}=\sum _{i}d{\sqrt {p_{i}}}\;d{\sqrt {p_{i}}}\\&={\frac {1}{4}}\sum _{i}{\frac {dp_{i}\;dp_{i}}{p_{i}}}={\frac {1}{4}}\sum _{i}p_{i}\;d(\log p_{i})\;d(\log p_{i})\end{aligned}}}
6436:{\displaystyle ds^{2}={\frac {\langle \delta \psi \mid \delta \psi \rangle }{\langle \psi \mid \psi \rangle }}-{\frac {\langle \delta \psi \mid \psi \rangle \;\langle \psi \mid \delta \psi \rangle }{{\langle \psi \mid \psi \rangle }^{2}}}.}
5742:
2257:
707:
6716:
4210:
5944:
380:
3537:
3306:
7367:
185:
4222:
1369:{\displaystyle g_{jk}(\theta )={\frac {\partial ^{2}A(\theta )}{\partial \theta _{j}\,\partial \theta _{k}}}-{\frac {\partial ^{2}\eta (\theta )}{\partial \theta _{j}\,\partial \theta _{k}}}\cdot \mathrm {E} }
6732:
2609:
2461:
1865:
6239:
4127:
1873:
1501:
6497:
7127:
1058:
695:
2136:
2252:
8505:
5246:
4909:
4556:
1189:
4043:
369:
2520:
2055:
1789:
1497:
6162:
8275:
8025:
6634:
4438:
2909:
3235:
4829:
The last can be recognized as one-fourth of the Fisher information metric. To complete the process, recall that the probabilities are parametric functions of the manifold variables
3969:
3264:. This can be thought of intuitively as: "The distance between two infinitesimally close points on a statistical differential manifold is the informational difference between them."
8088:
1448:
1413:
7334:{\displaystyle {\begin{aligned}h={}&{\frac {1}{4}}\mathrm {E} \left+\mathrm {E} \left-\left(\mathrm {E} \left\right)^{2}\\{}&+{\frac {i}{2}}\mathrm {E} \left\end{aligned}}}
8710:
6097:
4896:
2942:
1016:
6056:
2762:
1726:
4078:
3498:
2858:
7975:
7043:
6277:
5982:
4540:
2195:
220:
8748:
4494:
2601:
1684:
3846:{\displaystyle \int _{a}^{b}{\sqrt {{\frac {\partial \theta ^{j}}{\partial t}}g_{jk}{\frac {\partial \theta ^{k}}{\partial t}}}}\,dt={\sqrt {8}}\int _{a}^{b}{\sqrt {dJSD}}}
3149:{\displaystyle f_{\theta _{0}}(\theta ):=D_{\mathrm {KL} }={\frac {1}{2}}\sum _{jk}\Delta \theta ^{j}\Delta \theta ^{k}g_{jk}(\theta _{0})+\mathrm {O} (\Delta \theta ^{3})}
8226:
7078:
6021:
8449:
8422:
7114:
3262:
2565:
8842:
8783:
8259:
8055:
7918:
6509:
8667:
8647:
8627:
8176:
5765:
5668:
4847:
2782:
2163:
601:
558:
311:
240:
8114:
3188:
3901:
8862:
8807:
8607:
8529:
8152:
919:{\displaystyle g_{jk}(\theta )=\int _{R}{\frac {\partial ^{2}i(x\mid \theta )}{\partial \theta _{j}\,\partial \theta _{k}}}p(x\mid \theta )\,dx=\mathrm {E} \left.}
5799:
approach; that is, to note that the above manipulations remain valid in the category of probabilities. Here, one should note that such a category would have the
5788:
4365:
291:
260:
6650:
4139:
2392:{\displaystyle \phi \mapsto {\mathcal {N}}\left({\frac {\mu _{0}+\mu _{1}}{2}}+{\frac {\mu _{1}-\mu _{0}}{2}}\cos \phi ,\sigma ^{2}\sin ^{2}\phi \right)}
522:{\displaystyle g_{jk}(\theta )=-\int _{R}{\frac {\partial ^{2}\log p(x\mid \theta )}{\partial \theta _{j}\,\partial \theta _{k}}}p(x\mid \theta )\,dx.}
5860:
5811:; these are square-integrable, and in the manipulations above, this is sufficient to safely replace the sum over squares by an integral over squares.
3681:{\displaystyle (b-a)\int _{a}^{b}{\frac {\partial \theta ^{j}}{\partial t}}g_{jk}{\frac {\partial \theta ^{k}}{\partial t}}\,dt=8\int _{a}^{b}dJSD}
3426:{\displaystyle A={\frac {1}{2}}\int _{a}^{b}{\frac {\partial \theta ^{j}}{\partial t}}g_{jk}(\theta ){\frac {\partial \theta ^{k}}{\partial t}}dt}
104:. Considered as a measurement technique, where it is used to estimate hidden parameters in terms of observed random variables, it is known as the
5843:. By setting the phase of the complex coordinate to zero, one obtains exactly one-fourth of the Fisher information metric, exactly as above.
4337:{\displaystyle h_{jk}^{\mathrm {flat} }=h\left({\frac {\partial }{\partial y_{j}}},{\frac {\partial }{\partial y_{k}}}\right)=\delta _{jk}}
119:
4347:
The superscript 'flat' is there to remind that, when written in coordinate form, this metric is with respect to the flat-space coordinate
2724:{\displaystyle P(\theta )=P(\theta _{0})+\sum _{j}\Delta \theta ^{j}\left.{\frac {\partial P}{\partial \theta ^{j}}}\right|_{\theta _{0}}}
5827:. This should perhaps be no surprise, as the Fubini–Study metric provides the means of measuring information in quantum mechanics. The
2402:
2000:{\displaystyle g={\begin{bmatrix}t&0\\0&(2t^{2})^{-1}\end{bmatrix}}=\sigma ^{-2}{\begin{bmatrix}1&0\\0&2\end{bmatrix}}}
1794:
1075:
6173:
8864:; for details, refer to original sources). Thus, one has the appearance of logarithms in the simpler definition, previously given.
4091:
9230:
Facchi, Paolo; et al. (2010). "Classical and
Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics".
6449:
1024:
633:
58:, the Fisher information metric on statistical models is the only Riemannian metric (up to rescaling) that is invariant under
9203:
8673:
8844:, the logarithm gives a point in the tangent space (roughly speaking, as again, one must transport from the origin to point
3856:
That is, the square root of the Jensen–Shannon divergence is just the Fisher metric (divided by the square root of 8).
2060:
2203:
8454:
3695:
is understood to be the infinitesimal change in the Jensen–Shannon divergence along the path taken. Similarly, for the
1643:{\displaystyle -\ln p(x|\mu ,\Sigma )={\frac {1}{2}}(x-\mu )^{T}\Sigma ^{-1}(x-\mu )+{\frac {1}{2}}\ln \det(\Sigma )+C}
9060:
8990:
3977:
316:
8577:
This definition of the metric can be seen to be equivalent to the previous, in several steps. First, one selects a
2466:
8387:{\displaystyle g(\sigma _{1},\sigma _{2})=\int _{X}{\frac {d\sigma _{1}}{d\mu }}{\frac {d\sigma _{2}}{d\mu }}d\mu }
1458:
624:
8507:. The abuse of notation is to write the tangent vectors as if they are derivatives, and to insert the extraneous
2010:
1731:
1463:
6109:
2535:
70:
7988:
6610:
4384:
3868:, that is, a probability space on a finite set of objects, the Fisher metric can be understood to simply be the
9430:
5792:
2863:
3197:
9440:
9435:
3916:
3528:
3516:
8063:
1418:
1383:
8540:
8679:
7894:
Again, the first term can be clearly seen to be (one fourth of) the Fisher information metric, by setting
6061:
4852:
2139:
8649:
is finite-dimensional, then so is the submanifold; likewise, the tangent space has the same dimension as
2914:
935:
623:
dimensions smaller than the original space. In this case, the metric can be explicitly derived from the
6026:
5819:
The above manipulations deriving the Fisher metric from the
Euclidean metric can be extended to complex
2737:
1692:
9342:
Feng, Edward H.; Crooks, Gavin E. (2009). "Far-from-equilibrium measurements of thermodynamic length".
8555:
5800:
4051:
3865:
3458:
2791:
1377:
7942:
7019:
6247:
5952:
4502:
2168:
2145:
The shortest paths (geodesics) between two univariate normal distributions are either parallel to the
190:
8910:"A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions"
8715:
5804:
5747:
The superscript 'fisher' is present to remind that this expression is applicable for the coordinates
4454:
101:
2570:
1653:
5820:
82:
8196:
7057:
6597:{\displaystyle \langle \phi \mid \psi \rangle =\int _{X}\phi ^{*}(x;\theta )\psi (x;\theta )\,dx.}
5991:
3515:
path between the desired endpoints of the process. The geodesic minimizes the entropy, due to the
9284:
8427:
8400:
7083:
6283:
5824:
3240:
86:
8873:
5737:{\displaystyle d\theta _{j}\left({\frac {\partial }{\partial \theta _{k}}}\right)=\delta _{jk}.}
2541:
8812:
8753:
8234:
8034:
7897:
6636:
can be understood to be an infinitesimal variation; equivalently, it can be understood to be a
55:
9183:
5835:, is identical to the Fubini–Study metric, although the latter is usually written in terms of
77:
of the divergence. Alternately, it can be understood as the metric induced by the flat space
8978:
8676:
provides a map from vectors in a tangent space to points in an underlying manifold. Thus, if
8652:
8632:
8612:
8161:
5985:
5847:
5840:
5750:
4832:
3511:: in order to minimize the change in free entropy of a system, one should follow the minimum
2767:
2148:
586:
543:
296:
225:
90:
8093:
3163:
9361:
9249:
9213:
8931:
8888:
8120:
6287:
3879:
3503:
as the change in free entropy. This observation has resulted in practical applications in
105:
59:
32:
20:
8847:
8792:
8592:
8514:
8178:
held fixed). Note that this space is infinite-dimensional, and is commonly taken to be a
8137:
8:
7936:
4445:
3508:
3297:
604:
561:
44:
9365:
9253:
9099:
9076:
Brody, Dorje; Hook, Daniel (2008). "Information geometry in vapour-liquid equilibrium".
8954:
8935:
8909:
9396:
9385:
9351:
9265:
9239:
9161:
9135:
9103:
9085:
9033:
9015:
8921:
8883:
8878:
8786:
6644:. Using the infinitesimal notation, the polar form of the probability above is simply
5796:
5773:
4350:
1069:
698:
276:
245:
8511:
in writing the integral: the integration is meant to be carried out using the measure
9377:
9199:
9153:
9107:
9056:
8986:
8959:
8559:
8266:
8028:
6711:{\displaystyle \delta \psi =\left({\frac {\delta p}{2p}}+i\delta \alpha \right)\psi }
3519:, which states that the action is bounded below by the length of the curve, squared.
3504:
608:
580:
48:
28:
9269:
9389:
9369:
9307:
9299:
9261:
9257:
9191:
9165:
9145:
9095:
9037:
9025:
8949:
8939:
8558:, and more specifically, the tangent space is restricted to those vectors that are
5851:
4205:{\displaystyle dy_{j}\left({\frac {\partial }{\partial y_{k}}}\right)=\delta _{jk}}
3869:
3527:
The Fisher metric also allows the action and the curve length to be related to the
3293:
3273:
78:
9149:
8179:
9410:, Translations of mathematical monographs; v. 191, American Mathematical Society.
9303:
9209:
8563:
7353:
7345:
6641:
5832:
4085:
3277:
267:
51:. It can be used to calculate the informational difference between measurements.
40:
9413:
Paolo
Gibilisco, Eva Riccomagno, Maria Piera Rogantin and Henry P. Wynn, (2009)
5939:{\displaystyle \psi (x;\theta )={\sqrt {p(x;\theta )}}\;e^{i\alpha (x;\theta )}}
65:
It can also be understood to be the infinitesimal form of the relative entropy (
9373:
8536:
3191:
74:
9029:
9424:
8571:
8567:
8262:
8229:
8058:
7978:
7051:
5808:
4130:
3281:
619:
different expectation values constant, then the dimension of the manifold is
576:
4496:. The sphere condition now becomes the probability normalization condition
607:; Lagrange multipliers are used to enforce constraints, such as holding the
9381:
9157:
8963:
5828:
3696:
222:
for the likelihood, that is the probability density of x as a function of
94:
6167:
is equivalently expressed by the idea the square amplitude be normalized:
2530:
Alternatively, the metric can be obtained as the second derivative of the
8578:
7349:
1689:
The metric splits to a mean part and a precision/variance part, because
9312:
9006:
Dowty, James G. (2018). "Chentsov's theorem for exponential families".
6058:
are strictly real. The previous calculations are obtained by setting
5836:
929:
To show that the equivalent form equals the above definition note that
9334:
9195:
8944:
8535:. This abuse of notation is, in fact, taken to be perfectly normal in
7928:
A slightly more formal, abstract definition can be given, as follows.
180:{\displaystyle \theta =(\theta _{1},\theta _{2},\ldots ,\theta _{n})}
9020:
8926:
3512:
81:, after appropriate changes of variable. When extended to complex
9356:
9244:
9140:
9090:
6100:
3873:
3441:
2456:{\displaystyle \sigma ={\frac {\mu _{1}-\mu _{0}}{2{\sqrt {2}}}}}
1860:{\displaystyle g_{T,T}=-{\frac {1}{2}}\nabla _{T}^{2}\ln \det T}
7116:
and noting that the integrals are just expectation values; so:
7047:
6637:
6234:{\displaystyle \int _{X}\vert \psi (x;\theta )\vert ^{2}\,dx=1}
4898:. Thus, the above induces a metric on the parameter manifold:
4081:
4378: + 1)-dimensional Euclidean space may be defined as
4122:{\displaystyle \textstyle {\frac {\partial }{\partial y_{j}}}}
2538:. To obtain this, one considers two probability distributions
2525:
1376:
The metric has a particularly simple form if we are using the
9229:
1068:
The Fisher information metric is particularly simple for the
6492:{\displaystyle \langle x\mid \psi \rangle =\psi (x;\theta )}
3522:
9126:
Crooks, Gavin E. (2009). "Measuring thermodynamic length".
8672:
With some additional abuse of language, one notes that the
5791:
well-defined, convergent, etc. The other way, as noted by
3903:) of a unit sphere, after appropriate changes of variable.
2675:
8609:
that are parameterized by some smoothly varying parameter
6286:, written in infinitesimal form, using quantum-mechanical
5235:
or, in coordinate form, the Fisher information metric is:
2603:, which are infinitesimally close to one another, so that
9190:. Zürich: European Mathematical Society. pp. 51–78.
1053:{\displaystyle {\frac {\partial }{\partial \theta _{k}}}}
690:{\displaystyle i(x\mid \theta )=-\log {}p(x\mid \theta )}
8546:
In order for the integral to be well-defined, the space
6721:
Inserting the above into the Fubini–Study metric gives:
3284:
as the ones found in equilibrium statistical mechanics.
2788:
direction. Then, since the
Kullback–Leibler divergence
1870:
In particular, for single variable normal distribution,
9403:, Lecture Notes in Statistics, Springer-Verlag, Berlin.
8750:
is the corresponding probability associated with point
6099:. The usual condition that probabilities lie within a
8562:. Square integrability is equivalent to saying that a
4095:
4055:
3440:; this action can be understood to give the change in
2131:{\displaystyle ds^{2}=2{\frac {dx^{2}+dy^{2}}{y^{2}}}}
1966:
1888:
8985:. New York: Oxford University Press. pp. 37–40.
8850:
8815:
8795:
8756:
8718:
8682:
8655:
8635:
8615:
8595:
8517:
8457:
8430:
8403:
8278:
8237:
8199:
8164:
8140:
8096:
8066:
8037:
7991:
7945:
7900:
7365:
7125:
7086:
7060:
7022:
6730:
6653:
6613:
6512:
6452:
6299:
6250:
6176:
6112:
6064:
6029:
5994:
5955:
5863:
5776:
5753:
5671:
5244:
4907:
4855:
4835:
4554:
4505:
4457:
4387:
4353:
4225:
4142:
4094:
4054:
3980:
3919:
3882:
3708:
3540:
3461:
3309:
3243:
3200:
3166:
2953:
2917:
2866:
2794:
2770:
2740:
2612:
2573:
2544:
2469:
2405:
2260:
2247:{\displaystyle \delta _{\mu _{0}},\delta _{\mu _{1}}}
2206:
2171:
2151:
2063:
2013:
1876:
1797:
1734:
1695:
1656:
1504:
1466:
1421:
1386:
1197:
1078:
1027:
938:
710:
636:
589:
546:
383:
319:
299:
279:
248:
228:
193:
122:
8500:{\displaystyle \sigma _{1},\sigma _{2}\in T_{\mu }S}
7923:
5839:, as below, whereas the Bures metric is written for
374:The Fisher information metric then takes the form:
8856:
8836:
8801:
8777:
8742:
8704:
8661:
8641:
8621:
8601:
8523:
8499:
8443:
8416:
8386:
8253:
8220:
8170:
8146:
8108:
8082:
8049:
8019:
7969:
7912:
7883:
7333:
7108:
7072:
7054:. The change of notation is done simply replacing
7037:
7005:
6710:
6628:
6596:
6491:
6435:
6271:
6233:
6156:
6091:
6050:
6015:
5976:
5938:
5846:One begins with the same trick, of constructing a
5782:
5759:
5736:
5651:
5224:
4890:
4841:
4818:
4534:
4488:
4432:
4359:
4336:
4204:
4121:
4072:
4037:
3963:
3895:
3845:
3680:
3492:
3425:
3256:
3229:
3182:
3148:
2936:
2903:
2852:
2776:
2756:
2723:
2595:
2559:
2514:
2455:
2391:
2246:
2189:
2157:
2130:
2049:
1999:
1859:
1783:
1720:
1678:
1642:
1491:
1442:
1407:
1368:
1183:
1052:
1010:
918:
689:
627:; a derivation and discussion is presented there.
595:
552:
521:
363:
305:
285:
254:
234:
214:
179:
100:Considered purely as a matrix, it is known as the
9184:"In a search for a structure, Part 1: On entropy"
8570:: the space contains its limit points. Note that
3280:are the Fisher information metric calculated for
3267:
1103:
701:, an equivalent form of the above definition is:
572:index the local coordinate axes on the manifold.
54:The metric is interesting in several aspects. By
9422:
9055:(2nd ed.). Hoboken: John Wiley & Sons.
8979:"Chentsov's theorem and some historical remarks"
1851:
1622:
1184:{\displaystyle p(x\mid \theta )=\exp \!{\bigl }}
9285:"Fisher information metric and Poisson kernels"
4038:{\displaystyle h=\sum _{i=0}^{N}dy_{i}\;dy_{i}}
3906:Consider a flat, Euclidean space, of dimension
364:{\displaystyle \int _{R}p(x\mid \theta )\,dx=1}
9401:Differential-geometrical methods in statistics
6499:and integration over the entire measure space
3971:. The metric for Euclidean space is given by
2515:{\displaystyle s={\sqrt {2}}\ln \tan(\phi /2)}
116:Given a statistical manifold with coordinates
9415:Algebraic and Geometric Methods in Statistics
2911:, one has an expansion up to second order in
1176:
1106:
9282:
8976:
8451:are vectors in the tangent space; that is,
6623:
6614:
6525:
6513:
6465:
6453:
6418:
6406:
6400:
6385:
6381:
6366:
6354:
6342:
6337:
6319:
6209:
6187:
3020:
2832:
2165:axis, or half circular arcs centered on the
2050:{\displaystyle x=\mu /{\sqrt {2}},y=\sigma }
1784:{\displaystyle g_{\mu _{i},\mu _{j}}=T_{ij}}
1492:{\displaystyle {\mathcal {N}}(\mu ,\Sigma )}
9050:
8977:Amari, Shun-ichi; Nagaoka, Horishi (2000).
8261:. The Fisher information metric is then an
6157:{\displaystyle \int _{X}p(x;\theta )\,dx=1}
2526:Relation to the Kullback–Leibler divergence
9341:
9332:The information geometry of mirror descent
9330:Garvesh Raskutti Sayan Mukherjee, (2014).
9292:Differential Geometry and Its Applications
9283:Itoh, Mitsuhiro; Shishido, Yuichi (2008).
8020:{\displaystyle (\Omega ,{\mathcal {F}},P)}
6988:
6889:
6853:
6800:
6629:{\displaystyle \vert \delta \psi \rangle }
6384:
6279:is real, this is the surface of a sphere.
5907:
5807:holds in this category. This includes the
5590:
5471:
5421:
5093:
4996:
4961:
4786:
4760:
4700:
4636:
4592:
4433:{\displaystyle \sum _{i=0}^{N}y_{i}^{2}=1}
4021:
532:The integral is performed over all values
9355:
9311:
9243:
9139:
9089:
9075:
9051:Cover, Thomas M.; Thomas, Joy A. (2006).
9019:
8953:
8943:
8925:
6992:
6893:
6857:
6804:
6584:
6218:
6141:
5814:
3795:
3638:
3523:Relation to the Jensen–Shannon divergence
3489:
3287:
2904:{\displaystyle P(\theta )=P(\theta _{0})}
1728:. The mean part is the precision matrix:
1323:
1263:
892:
827:
792:
611:of some quantity constant. If there are
575:When the probability is derived from the
509:
474:
348:
9406:Shun'ichi Amari, Hiroshi Nagaoka (2000)
9225:
9223:
9177:
9175:
9121:
9119:
9117:
8134:is defined as the space of all measures
3230:{\displaystyle f_{\theta _{0}}(\theta )}
2463:, and the arc-length parametrization is
8907:
8712:is a vector in the tangent space, then
4216:the Euclidean metric may be written as
3964:{\displaystyle y=(y_{0},\cdots ,y_{n})}
3190:is positive (semi) definite and is the
16:Metric on a smooth statistical manifold
9423:
9181:
9125:
8566:converges to a finite value under the
8539:; it is the standard notation for the
8083:{\displaystyle {\mathcal {F}}=\Sigma }
4374:-dimensional unit sphere embedded in (
3859:
1453:
1443:{\displaystyle \nabla _{\theta }^{2}A}
1408:{\displaystyle \eta (\theta )=\theta }
9220:
9172:
9114:
9005:
8589:) by considering only those measures
7356:. In index notation, the metric is:
4084:; they are the basis vectors for the
3444:of a system as it is moved from time
273:. The likelihood is normalized over
9069:
8705:{\displaystyle \sigma \in T_{\mu }S}
6092:{\displaystyle \alpha (x;\theta )=0}
4891:{\displaystyle p_{i}=p_{i}(\theta )}
4451:Consider now the change of variable
9335:https://arxiv.org/pdf/1310.7780.pdf
2937:{\displaystyle \theta =\theta _{0}}
2764:an infinitesimally small change of
1011:{\displaystyle \mathrm {E} \left=0}
13:
8165:
8077:
8069:
8038:
8003:
7995:
7955:
7856:
7842:
7823:
7815:
7793:
7785:
7766:
7752:
7740:
7697:
7689:
7678:
7657:
7649:
7638:
7613:
7605:
7586:
7578:
7566:
7531:
7517:
7498:
7484:
7472:
7429:
7425:
7404:
7400:
7292:
7237:
7194:
7151:
6051:{\displaystyle \alpha (x;\theta )}
5695:
5691:
5624:
5594:
5574:
5544:
5532:
5505:
5475:
5455:
5425:
5361:
5357:
5336:
5332:
5306:
5303:
5300:
5297:
5294:
5291:
5176:
5146:
5127:
5097:
4297:
4293:
4272:
4268:
4249:
4246:
4243:
4240:
4166:
4162:
4102:
4098:
3784:
3769:
3744:
3729:
3629:
3614:
3589:
3574:
3462:
3408:
3393:
3359:
3344:
3130:
3123:
3077:
3064:
2992:
2989:
2860:has an absolute minimum of 0 when
2804:
2801:
2757:{\displaystyle \Delta \theta ^{j}}
2741:
2688:
2680:
2660:
2269:
1831:
1721:{\displaystyle g_{\mu ,\Sigma }=0}
1707:
1664:
1628:
1576:
1534:
1483:
1469:
1423:
1344:
1324:
1310:
1287:
1264:
1250:
1227:
1034:
1030:
982:
951:
940:
893:
879:
850:
838:
793:
779:
750:
475:
461:
426:
14:
9452:
8265:on the tangent space. With some
7924:Continuously-valued probabilities
4073:{\displaystyle \textstyle dy_{i}}
3493:{\displaystyle \Delta S=(b-a)A\,}
2853:{\displaystyle D_{\mathrm {KL} }}
31:which can be defined on a smooth
9417:, Cambridge U. Press, Cambridge.
9188:European Congress of Mathematics
7970:{\displaystyle (X,\Sigma ,\mu )}
7038:{\displaystyle \delta \alpha =0}
6272:{\displaystyle \psi (x;\theta )}
5977:{\displaystyle \psi (x;\theta )}
5823:. In this case, one obtains the
4535:{\displaystyle \sum _{i}p_{i}=1}
3436:The path parameter here is time
2190:{\displaystyle \mu /{\sqrt {2}}}
1459:Multivariate normal distribution
215:{\displaystyle p(x\mid \theta )}
9408:Methods of information geometry
8983:Methods of Information Geometry
8743:{\displaystyle p=\exp(\sigma )}
6446:In this notation, one has that
4489:{\displaystyle p_{i}=y_{i}^{2}}
603:can also be understood to be a
266:for a (discrete or continuous)
9276:
9262:10.1016/j.physleta.2010.10.005
9053:Elements of Information Theory
9044:
8999:
8970:
8901:
8831:
8825:
8772:
8766:
8737:
8731:
8308:
8282:
8215:
8209:
8014:
7992:
7964:
7946:
7213:
7203:
7176:
7160:
7100:
7064:
6985:
6943:
6844:
6834:
6791:
6775:
6581:
6569:
6563:
6551:
6486:
6474:
6266:
6254:
6205:
6193:
6138:
6126:
6080:
6068:
6045:
6033:
6010:
5998:
5971:
5959:
5931:
5919:
5902:
5890:
5879:
5867:
5619:
5613:
5569:
5563:
5500:
5494:
5450:
5444:
5418:
5412:
5268:
5262:
5171:
5165:
5122:
5116:
5090:
5084:
5028:
5025:
5019:
5000:
4993:
4990:
4984:
4965:
4958:
4952:
4885:
4879:
4809:
4790:
4783:
4764:
3958:
3926:
3553:
3541:
3483:
3471:
3387:
3381:
3268:Relation to Ruppeiner geometry
3224:
3218:
3143:
3127:
3116:
3103:
3035:
3032:
3026:
3017:
3004:
2998:
2977:
2971:
2898:
2885:
2876:
2870:
2847:
2844:
2838:
2829:
2816:
2810:
2644:
2631:
2622:
2616:
2596:{\displaystyle P(\theta _{0})}
2590:
2577:
2554:
2548:
2509:
2495:
2264:
1925:
1908:
1679:{\displaystyle T=\Sigma ^{-1}}
1631:
1625:
1600:
1588:
1566:
1553:
1537:
1524:
1517:
1486:
1474:
1396:
1390:
1363:
1360:
1354:
1348:
1305:
1299:
1245:
1239:
1217:
1211:
1168:
1162:
1153:
1147:
1138:
1132:
1123:
1117:
1094:
1082:
977:
965:
874:
862:
824:
812:
774:
762:
730:
724:
684:
672:
652:
640:
506:
494:
456:
444:
403:
397:
345:
333:
262:is drawn from the value space
209:
197:
174:
129:
1:
9324:
9150:10.1103/PhysRevLett.99.100602
9100:10.1088/1751-8113/42/2/023001
8809:.) Conversely, given a point
4129:as the basis vectors for the
111:
9304:10.1016/j.difgeo.2007.11.027
8221:{\displaystyle \mu \in S(X)}
7073:{\displaystyle \delta \to d}
6016:{\displaystyle p(x;\theta )}
7:
8867:
8444:{\displaystyle \sigma _{2}}
8417:{\displaystyle \sigma _{1}}
7109:{\displaystyle ds^{2}\to h}
3257:{\displaystyle \theta _{0}}
2536:Kullback–Leibler divergence
1063:
89:; when written in terms of
73:); specifically, it is the
71:Kullback–Leibler divergence
10:
9457:
9374:10.1103/PhysRevE.79.012104
8789:of the exponential map to
3866:discrete probability space
2560:{\displaystyle P(\theta )}
9030:10.1007/s41884-018-0006-4
8837:{\displaystyle p\in S(X)}
8778:{\displaystyle p\in S(X)}
8254:{\displaystyle T_{\mu }S}
8050:{\displaystyle \Omega =X}
7913:{\displaystyle \alpha =0}
5821:projective Hilbert spaces
5770:When the random variable
4545:while the metric becomes
3913:, parametrized by points
3872:restricted to a positive
3529:Jensen–Shannon divergence
3517:Cauchy–Schwarz inequality
2140:Poincaré half-plane model
1686:be the precision matrix.
579:, as it would be for any
560:is now a coordinate on a
102:Fisher information matrix
25:Fisher information metric
8894:
8541:Radon–Nikodym derivative
8269:, one may write this as
8158:(with the sigma-algebra
7344:The imaginary term is a
3531:. Specifically, one has
3452:. Specifically, one has
2200:The geodesic connecting
1791:. The precision part is
1415:, so the metric is just
83:projective Hilbert space
9128:Physical Review Letters
8908:Nielsen, Frank (2023).
8662:{\displaystyle \theta }
8642:{\displaystyle \theta }
8622:{\displaystyle \theta }
8574:possess this property.
8171:{\displaystyle \Sigma }
5760:{\displaystyle \theta }
4842:{\displaystyle \theta }
2777:{\displaystyle \theta }
2158:{\displaystyle \sigma }
596:{\displaystyle \theta }
553:{\displaystyle \theta }
306:{\displaystyle \theta }
235:{\displaystyle \theta }
9182:Gromov, Misha (2013).
8858:
8838:
8803:
8779:
8744:
8706:
8663:
8643:
8623:
8603:
8556:Radon–Nikodym property
8525:
8501:
8445:
8418:
8388:
8255:
8222:
8172:
8148:
8110:
8109:{\displaystyle P=\mu }
8084:
8051:
8021:
7971:
7914:
7885:
7335:
7110:
7074:
7039:
7007:
6712:
6630:
6598:
6493:
6437:
6273:
6235:
6158:
6093:
6052:
6017:
5978:
5940:
5815:As Fubini–Study metric
5801:Radon–Nikodym property
5784:
5761:
5738:
5653:
5226:
4892:
4843:
4820:
4536:
4490:
4434:
4408:
4361:
4338:
4206:
4123:
4074:
4039:
4007:
3965:
3897:
3847:
3682:
3494:
3427:
3288:Change in free entropy
3258:
3237:at the extremum point
3231:
3184:
3183:{\displaystyle g_{jk}}
3150:
2938:
2905:
2854:
2778:
2758:
2725:
2597:
2561:
2516:
2457:
2393:
2248:
2191:
2159:
2132:
2051:
2001:
1861:
1785:
1722:
1680:
1644:
1493:
1444:
1409:
1370:
1185:
1054:
1012:
920:
691:
597:
554:
523:
365:
307:
287:
256:
236:
216:
181:
9431:Differential geometry
8859:
8839:
8804:
8780:
8745:
8707:
8664:
8644:
8624:
8604:
8531:over the whole space
8526:
8502:
8446:
8419:
8389:
8256:
8223:
8173:
8149:
8111:
8085:
8052:
8022:
7972:
7915:
7886:
7336:
7111:
7075:
7040:
7008:
6713:
6631:
6599:
6494:
6438:
6274:
6236:
6159:
6094:
6053:
6018:
5986:probability amplitude
5979:
5941:
5848:probability amplitude
5805:Radon–Nikodym theorem
5785:
5762:
5739:
5654:
5227:
4893:
4844:
4821:
4537:
4491:
4435:
4388:
4362:
4339:
4207:
4124:
4075:
4040:
3987:
3966:
3898:
3896:{\displaystyle R^{2}}
3848:
3683:
3495:
3428:
3259:
3232:
3185:
3160:The symmetric matrix
3151:
2939:
2906:
2855:
2779:
2759:
2726:
2598:
2562:
2517:
2458:
2394:
2249:
2192:
2160:
2133:
2052:
2002:
1862:
1786:
1723:
1681:
1645:
1494:
1445:
1410:
1371:
1186:
1055:
1013:
921:
692:
598:
555:
524:
366:
308:
288:
257:
237:
217:
182:
60:sufficient statistics
9441:Statistical distance
9436:Information geometry
9078:Journal of Physics A
9008:Information Geometry
8889:Information geometry
8857:{\displaystyle \mu }
8848:
8813:
8802:{\displaystyle \mu }
8793:
8754:
8716:
8680:
8653:
8633:
8613:
8602:{\displaystyle \mu }
8593:
8524:{\displaystyle \mu }
8515:
8455:
8428:
8401:
8276:
8235:
8197:
8162:
8147:{\displaystyle \mu }
8138:
8121:statistical manifold
8094:
8064:
8035:
7989:
7985:. Equivalently, let
7943:
7898:
7363:
7123:
7084:
7058:
7020:
6728:
6651:
6611:
6510:
6450:
6297:
6248:
6174:
6110:
6062:
6027:
5992:
5984:is a complex-valued
5953:
5861:
5831:, also known as the
5774:
5751:
5669:
5242:
4905:
4853:
4833:
4552:
4503:
4455:
4446:Lagrange multipliers
4385:
4351:
4223:
4140:
4092:
4052:
3978:
3917:
3880:
3876:(e.g. "quadrant" in
3706:
3691:where the integrand
3538:
3459:
3307:
3241:
3198:
3164:
2951:
2915:
2864:
2792:
2768:
2738:
2610:
2571:
2542:
2467:
2403:
2258:
2204:
2169:
2149:
2061:
2011:
1874:
1795:
1732:
1693:
1654:
1502:
1464:
1419:
1384:
1195:
1076:
1025:
936:
708:
634:
615:constraints holding
587:
544:
381:
317:
297:
277:
246:
226:
191:
120:
106:observed information
93:, it is the quantum
47:defined on a common
45:probability measures
33:statistical manifold
21:information geometry
9366:2009PhRvE..79a2104F
9254:2010PhLA..374.4801F
8936:2023Entrp..25..654N
7937:orientable manifold
6284:Fubini–Study metric
5825:Fubini–Study metric
5311:
4849:, that is, one has
4485:
4423:
4254:
3860:As Euclidean metric
3826:
3723:
3665:
3570:
3509:processing industry
3340:
3298:Riemannian manifold
3282:Gibbs distributions
1844:
1454:Normal distribution
1436:
605:Lagrange multiplier
87:Fubini–Study metric
9350:(1 Pt 1): 012104.
8884:Hellinger distance
8879:Fisher information
8854:
8834:
8799:
8787:parallel transport
8775:
8740:
8702:
8659:
8639:
8619:
8599:
8521:
8497:
8441:
8414:
8384:
8251:
8218:
8168:
8144:
8106:
8080:
8047:
8017:
7967:
7910:
7881:
7879:
7331:
7329:
7106:
7070:
7035:
7003:
7001:
6708:
6626:
6594:
6489:
6433:
6269:
6231:
6154:
6089:
6048:
6013:
5974:
5936:
5797:category-theoretic
5780:
5757:
5734:
5662:where, as before,
5649:
5647:
5401:
5277:
5222:
5220:
5073:
5063:
4941:
4888:
4839:
4816:
4814:
4749:
4683:
4618:
4578:
4532:
4515:
4486:
4471:
4430:
4409:
4357:
4334:
4226:
4202:
4119:
4118:
4070:
4069:
4035:
3961:
3893:
3843:
3812:
3709:
3678:
3651:
3556:
3490:
3423:
3326:
3254:
3227:
3180:
3146:
3063:
2934:
2901:
2850:
2774:
2754:
2721:
2659:
2593:
2557:
2512:
2453:
2389:
2244:
2187:
2155:
2128:
2047:
1997:
1991:
1939:
1857:
1830:
1781:
1718:
1676:
1640:
1489:
1440:
1422:
1405:
1378:natural parameters
1366:
1181:
1070:exponential family
1050:
1008:
916:
699:information theory
687:
625:partition function
593:
550:
519:
361:
303:
283:
252:
232:
212:
177:
56:Chentsov’s theorem
9344:Physical Review E
9238:(48): 4801–4803.
9232:Physics Letters A
9205:978-3-03719-120-0
8945:10.3390/e25040654
8560:square-integrable
8376:
8349:
8267:abuse of notation
8228:and consider the
8182:. The points of
8029:probability space
7870:
7837:
7807:
7780:
7737:
7711:
7671:
7627:
7600:
7545:
7512:
7469:
7443:
7418:
7289:
7148:
6931:
6763:
6686:
6428:
6358:
5905:
5852:polar coordinates
5783:{\displaystyle p}
5709:
5638:
5588:
5519:
5469:
5392:
5375:
5350:
5190:
5141:
5064:
5051:
5049:
4932:
4930:
4740:
4738:
4725:
4674:
4672:
4652:
4634:
4609:
4569:
4506:
4360:{\displaystyle y}
4311:
4286:
4180:
4116:
3841:
3810:
3793:
3791:
3751:
3636:
3596:
3415:
3366:
3324:
3051:
3049:
2702:
2650:
2481:
2451:
2448:
2344:
2309:
2185:
2126:
2033:
1828:
1614:
1551:
1338:
1278:
1173:
1113:
1048:
996:
907:
807:
609:expectation value
581:Markovian process
540:. The variable
489:
286:{\displaystyle x}
255:{\displaystyle x}
85:, it becomes the
49:probability space
43:whose points are
29:Riemannian metric
9448:
9393:
9359:
9318:
9317:
9315:
9289:
9280:
9274:
9273:
9247:
9227:
9218:
9217:
9179:
9170:
9169:
9143:
9123:
9112:
9111:
9093:
9073:
9067:
9066:
9048:
9042:
9041:
9023:
9003:
8997:
8996:
8974:
8968:
8967:
8957:
8947:
8929:
8905:
8874:Cramér–Rao bound
8863:
8861:
8860:
8855:
8843:
8841:
8840:
8835:
8808:
8806:
8805:
8800:
8784:
8782:
8781:
8776:
8749:
8747:
8746:
8741:
8711:
8709:
8708:
8703:
8698:
8697:
8668:
8666:
8665:
8660:
8648:
8646:
8645:
8640:
8628:
8626:
8625:
8620:
8608:
8606:
8605:
8600:
8554:) must have the
8530:
8528:
8527:
8522:
8506:
8504:
8503:
8498:
8493:
8492:
8480:
8479:
8467:
8466:
8450:
8448:
8447:
8442:
8440:
8439:
8423:
8421:
8420:
8415:
8413:
8412:
8393:
8391:
8390:
8385:
8377:
8375:
8367:
8366:
8365:
8352:
8350:
8348:
8340:
8339:
8338:
8325:
8323:
8322:
8307:
8306:
8294:
8293:
8260:
8258:
8257:
8252:
8247:
8246:
8227:
8225:
8224:
8219:
8190:) are measures.
8177:
8175:
8174:
8169:
8153:
8151:
8150:
8145:
8115:
8113:
8112:
8107:
8090:and probability
8089:
8087:
8086:
8081:
8073:
8072:
8056:
8054:
8053:
8048:
8026:
8024:
8023:
8018:
8007:
8006:
7976:
7974:
7973:
7968:
7919:
7917:
7916:
7911:
7890:
7888:
7887:
7882:
7880:
7876:
7872:
7871:
7869:
7868:
7867:
7854:
7840:
7838:
7836:
7835:
7834:
7821:
7813:
7808:
7806:
7805:
7804:
7791:
7783:
7781:
7779:
7778:
7777:
7764:
7750:
7743:
7738:
7730:
7725:
7720:
7716:
7712:
7710:
7709:
7708:
7695:
7687:
7681:
7676:
7672:
7670:
7669:
7668:
7655:
7647:
7641:
7633:
7629:
7628:
7626:
7625:
7624:
7611:
7603:
7601:
7599:
7598:
7597:
7584:
7576:
7569:
7557:
7551:
7547:
7546:
7544:
7543:
7542:
7529:
7515:
7513:
7511:
7510:
7509:
7496:
7482:
7475:
7470:
7462:
7458:
7449:
7445:
7444:
7442:
7441:
7440:
7424:
7419:
7417:
7416:
7415:
7399:
7387:
7382:
7381:
7340:
7338:
7337:
7332:
7330:
7326:
7322:
7295:
7290:
7282:
7273:
7267:
7266:
7261:
7257:
7256:
7252:
7240:
7226:
7222:
7221:
7220:
7197:
7189:
7185:
7184:
7183:
7154:
7149:
7141:
7137:
7115:
7113:
7112:
7107:
7099:
7098:
7079:
7077:
7076:
7071:
7044:
7042:
7041:
7036:
7012:
7010:
7009:
7004:
7002:
6942:
6941:
6932:
6924:
6919:
6914:
6910:
6909:
6904:
6900:
6882:
6881:
6852:
6851:
6833:
6832:
6816:
6799:
6798:
6774:
6773:
6764:
6756:
6752:
6747:
6746:
6717:
6715:
6714:
6709:
6704:
6700:
6687:
6685:
6677:
6669:
6635:
6633:
6632:
6627:
6603:
6601:
6600:
6595:
6550:
6549:
6540:
6539:
6498:
6496:
6495:
6490:
6442:
6440:
6439:
6434:
6429:
6427:
6426:
6421:
6403:
6364:
6359:
6357:
6340:
6317:
6312:
6311:
6288:bra–ket notation
6278:
6276:
6275:
6270:
6240:
6238:
6237:
6232:
6217:
6216:
6186:
6185:
6163:
6161:
6160:
6155:
6122:
6121:
6098:
6096:
6095:
6090:
6057:
6055:
6054:
6049:
6022:
6020:
6019:
6014:
5983:
5981:
5980:
5975:
5945:
5943:
5942:
5937:
5935:
5934:
5906:
5886:
5789:
5787:
5786:
5781:
5766:
5764:
5763:
5758:
5743:
5741:
5740:
5735:
5730:
5729:
5714:
5710:
5708:
5707:
5706:
5690:
5684:
5683:
5658:
5656:
5655:
5650:
5648:
5644:
5640:
5639:
5637:
5636:
5635:
5622:
5612:
5611:
5592:
5589:
5587:
5586:
5585:
5572:
5562:
5561:
5542:
5535:
5524:
5520:
5518:
5517:
5516:
5503:
5493:
5492:
5473:
5470:
5468:
5467:
5466:
5453:
5443:
5442:
5423:
5411:
5410:
5400:
5385:
5381:
5377:
5376:
5374:
5373:
5372:
5356:
5351:
5349:
5348:
5347:
5331:
5310:
5309:
5288:
5261:
5260:
5231:
5229:
5228:
5223:
5221:
5217:
5216:
5204:
5203:
5191:
5189:
5188:
5187:
5174:
5164:
5163:
5144:
5142:
5140:
5139:
5138:
5125:
5115:
5114:
5095:
5083:
5082:
5072:
5062:
5050:
5042:
5034:
5018:
5017:
4983:
4982:
4951:
4950:
4940:
4931:
4923:
4897:
4895:
4894:
4889:
4878:
4877:
4865:
4864:
4848:
4846:
4845:
4840:
4825:
4823:
4822:
4817:
4815:
4808:
4807:
4782:
4781:
4759:
4758:
4748:
4739:
4731:
4726:
4724:
4723:
4714:
4713:
4712:
4699:
4698:
4685:
4682:
4673:
4665:
4657:
4653:
4651:
4650:
4641:
4635:
4633:
4632:
4623:
4617:
4605:
4604:
4591:
4590:
4577:
4541:
4539:
4538:
4533:
4525:
4524:
4514:
4495:
4493:
4492:
4487:
4484:
4479:
4467:
4466:
4439:
4437:
4436:
4431:
4422:
4417:
4407:
4402:
4366:
4364:
4363:
4358:
4343:
4341:
4340:
4335:
4333:
4332:
4317:
4313:
4312:
4310:
4309:
4308:
4292:
4287:
4285:
4284:
4283:
4267:
4253:
4252:
4237:
4211:
4209:
4208:
4203:
4201:
4200:
4185:
4181:
4179:
4178:
4177:
4161:
4155:
4154:
4128:
4126:
4125:
4120:
4117:
4115:
4114:
4113:
4097:
4079:
4077:
4076:
4071:
4068:
4067:
4044:
4042:
4041:
4036:
4034:
4033:
4020:
4019:
4006:
4001:
3970:
3968:
3967:
3962:
3957:
3956:
3938:
3937:
3912:
3902:
3900:
3899:
3894:
3892:
3891:
3870:Euclidean metric
3852:
3850:
3849:
3844:
3842:
3828:
3825:
3820:
3811:
3806:
3794:
3792:
3790:
3782:
3781:
3780:
3767:
3765:
3764:
3752:
3750:
3742:
3741:
3740:
3727:
3725:
3722:
3717:
3687:
3685:
3684:
3679:
3664:
3659:
3637:
3635:
3627:
3626:
3625:
3612:
3610:
3609:
3597:
3595:
3587:
3586:
3585:
3572:
3569:
3564:
3499:
3497:
3496:
3491:
3432:
3430:
3429:
3424:
3416:
3414:
3406:
3405:
3404:
3391:
3380:
3379:
3367:
3365:
3357:
3356:
3355:
3342:
3339:
3334:
3325:
3317:
3296:of a curve on a
3274:Ruppeiner metric
3263:
3261:
3260:
3255:
3253:
3252:
3236:
3234:
3233:
3228:
3217:
3216:
3215:
3214:
3194:of the function
3189:
3187:
3186:
3181:
3179:
3178:
3155:
3153:
3152:
3147:
3142:
3141:
3126:
3115:
3114:
3102:
3101:
3089:
3088:
3076:
3075:
3062:
3050:
3042:
3016:
3015:
2997:
2996:
2995:
2970:
2969:
2968:
2967:
2943:
2941:
2940:
2935:
2933:
2932:
2910:
2908:
2907:
2902:
2897:
2896:
2859:
2857:
2856:
2851:
2828:
2827:
2809:
2808:
2807:
2783:
2781:
2780:
2775:
2763:
2761:
2760:
2755:
2753:
2752:
2730:
2728:
2727:
2722:
2720:
2719:
2718:
2717:
2707:
2703:
2701:
2700:
2699:
2686:
2678:
2672:
2671:
2658:
2643:
2642:
2602:
2600:
2599:
2594:
2589:
2588:
2566:
2564:
2563:
2558:
2532:relative entropy
2521:
2519:
2518:
2513:
2505:
2482:
2477:
2462:
2460:
2459:
2454:
2452:
2450:
2449:
2444:
2438:
2437:
2436:
2424:
2423:
2413:
2398:
2396:
2395:
2390:
2388:
2384:
2377:
2376:
2367:
2366:
2345:
2340:
2339:
2338:
2326:
2325:
2315:
2310:
2305:
2304:
2303:
2291:
2290:
2280:
2273:
2272:
2253:
2251:
2250:
2245:
2243:
2242:
2241:
2240:
2223:
2222:
2221:
2220:
2196:
2194:
2193:
2188:
2186:
2181:
2179:
2164:
2162:
2161:
2156:
2137:
2135:
2134:
2129:
2127:
2125:
2124:
2115:
2114:
2113:
2098:
2097:
2084:
2076:
2075:
2056:
2054:
2053:
2048:
2034:
2029:
2027:
2006:
2004:
2003:
1998:
1996:
1995:
1960:
1959:
1944:
1943:
1936:
1935:
1923:
1922:
1866:
1864:
1863:
1858:
1843:
1838:
1829:
1821:
1813:
1812:
1790:
1788:
1787:
1782:
1780:
1779:
1764:
1763:
1762:
1761:
1749:
1748:
1727:
1725:
1724:
1719:
1711:
1710:
1685:
1683:
1682:
1677:
1675:
1674:
1649:
1647:
1646:
1641:
1615:
1607:
1587:
1586:
1574:
1573:
1552:
1544:
1527:
1498:
1496:
1495:
1490:
1473:
1472:
1449:
1447:
1446:
1441:
1435:
1430:
1414:
1412:
1411:
1406:
1380:. In this case,
1375:
1373:
1372:
1367:
1347:
1339:
1337:
1336:
1335:
1322:
1321:
1308:
1295:
1294:
1284:
1279:
1277:
1276:
1275:
1262:
1261:
1248:
1235:
1234:
1224:
1210:
1209:
1190:
1188:
1187:
1182:
1180:
1179:
1171:
1111:
1110:
1109:
1059:
1057:
1056:
1051:
1049:
1047:
1046:
1045:
1029:
1017:
1015:
1014:
1009:
1001:
997:
995:
994:
993:
980:
961:
949:
943:
925:
923:
922:
917:
912:
908:
906:
905:
904:
891:
890:
877:
858:
857:
847:
841:
808:
806:
805:
804:
791:
790:
777:
758:
757:
747:
745:
744:
723:
722:
696:
694:
693:
688:
668:
602:
600:
599:
594:
562:Riemann manifold
559:
557:
556:
551:
528:
526:
525:
520:
490:
488:
487:
486:
473:
472:
459:
434:
433:
423:
421:
420:
396:
395:
370:
368:
367:
362:
329:
328:
312:
310:
309:
304:
292:
290:
289:
284:
261:
259:
258:
253:
241:
239:
238:
233:
221:
219:
218:
213:
186:
184:
183:
178:
173:
172:
154:
153:
141:
140:
79:Euclidean metric
27:is a particular
9456:
9455:
9451:
9450:
9449:
9447:
9446:
9445:
9421:
9420:
9397:Shun'ichi Amari
9327:
9322:
9321:
9287:
9281:
9277:
9228:
9221:
9206:
9196:10.4171/120-1/4
9180:
9173:
9124:
9115:
9074:
9070:
9063:
9049:
9045:
9004:
9000:
8993:
8975:
8971:
8906:
8902:
8897:
8870:
8849:
8846:
8845:
8814:
8811:
8810:
8794:
8791:
8790:
8755:
8752:
8751:
8717:
8714:
8713:
8693:
8689:
8681:
8678:
8677:
8674:exponential map
8654:
8651:
8650:
8634:
8631:
8630:
8614:
8611:
8610:
8594:
8591:
8590:
8564:Cauchy sequence
8516:
8513:
8512:
8488:
8484:
8475:
8471:
8462:
8458:
8456:
8453:
8452:
8435:
8431:
8429:
8426:
8425:
8408:
8404:
8402:
8399:
8398:
8368:
8361:
8357:
8353:
8351:
8341:
8334:
8330:
8326:
8324:
8318:
8314:
8302:
8298:
8289:
8285:
8277:
8274:
8273:
8242:
8238:
8236:
8233:
8232:
8198:
8195:
8194:
8163:
8160:
8159:
8139:
8136:
8135:
8095:
8092:
8091:
8068:
8067:
8065:
8062:
8061:
8036:
8033:
8032:
8002:
8001:
7990:
7987:
7986:
7944:
7941:
7940:
7926:
7899:
7896:
7895:
7878:
7877:
7863:
7859:
7855:
7841:
7839:
7830:
7826:
7822:
7814:
7812:
7800:
7796:
7792:
7784:
7782:
7773:
7769:
7765:
7751:
7749:
7748:
7744:
7739:
7729:
7724:
7718:
7717:
7704:
7700:
7696:
7688:
7686:
7682:
7677:
7664:
7660:
7656:
7648:
7646:
7642:
7637:
7620:
7616:
7612:
7604:
7602:
7593:
7589:
7585:
7577:
7575:
7574:
7570:
7565:
7558:
7556:
7553:
7552:
7538:
7534:
7530:
7516:
7514:
7505:
7501:
7497:
7483:
7481:
7480:
7476:
7471:
7461:
7459:
7457:
7451:
7450:
7436:
7432:
7428:
7423:
7411:
7407:
7403:
7398:
7397:
7393:
7388:
7386:
7374:
7370:
7366:
7364:
7361:
7360:
7354:geometric phase
7346:symplectic form
7328:
7327:
7300:
7296:
7291:
7281:
7274:
7272:
7269:
7268:
7262:
7245:
7241:
7236:
7235:
7231:
7230:
7216:
7212:
7202:
7198:
7193:
7179:
7175:
7159:
7155:
7150:
7140:
7138:
7136:
7126:
7124:
7121:
7120:
7094:
7090:
7085:
7082:
7081:
7059:
7056:
7055:
7021:
7018:
7017:
7000:
6999:
6937:
6933:
6923:
6918:
6912:
6911:
6905:
6877:
6873:
6872:
6868:
6867:
6847:
6843:
6828:
6824:
6817:
6815:
6812:
6811:
6794:
6790:
6769:
6765:
6755:
6753:
6751:
6742:
6738:
6731:
6729:
6726:
6725:
6678:
6670:
6668:
6667:
6663:
6652:
6649:
6648:
6642:cotangent space
6612:
6609:
6608:
6607:The expression
6545:
6541:
6535:
6531:
6511:
6508:
6507:
6451:
6448:
6447:
6422:
6405:
6404:
6365:
6363:
6341:
6318:
6316:
6307:
6303:
6298:
6295:
6294:
6249:
6246:
6245:
6212:
6208:
6181:
6177:
6175:
6172:
6171:
6117:
6113:
6111:
6108:
6107:
6063:
6060:
6059:
6028:
6025:
6024:
5993:
5990:
5989:
5954:
5951:
5950:
5912:
5908:
5885:
5862:
5859:
5858:
5833:Helstrom metric
5817:
5803:, that is, the
5775:
5772:
5771:
5752:
5749:
5748:
5722:
5718:
5702:
5698:
5694:
5689:
5685:
5679:
5675:
5670:
5667:
5666:
5646:
5645:
5631:
5627:
5623:
5607:
5603:
5593:
5591:
5581:
5577:
5573:
5557:
5553:
5543:
5541:
5540:
5536:
5531:
5522:
5521:
5512:
5508:
5504:
5488:
5484:
5474:
5472:
5462:
5458:
5454:
5438:
5434:
5424:
5422:
5406:
5402:
5396:
5383:
5382:
5368:
5364:
5360:
5355:
5343:
5339:
5335:
5330:
5329:
5325:
5312:
5290:
5289:
5281:
5253:
5249:
5245:
5243:
5240:
5239:
5219:
5218:
5212:
5208:
5199:
5195:
5183:
5179:
5175:
5159:
5155:
5145:
5143:
5134:
5130:
5126:
5110:
5106:
5096:
5094:
5078:
5074:
5068:
5055:
5041:
5032:
5031:
5013:
5009:
4978:
4974:
4946:
4942:
4936:
4922:
4915:
4908:
4906:
4903:
4902:
4873:
4869:
4860:
4856:
4854:
4851:
4850:
4834:
4831:
4830:
4813:
4812:
4803:
4799:
4777:
4773:
4754:
4750:
4744:
4730:
4719:
4715:
4708:
4704:
4694:
4690:
4686:
4684:
4678:
4664:
4655:
4654:
4646:
4642:
4640:
4628:
4624:
4622:
4613:
4600:
4596:
4586:
4582:
4573:
4562:
4555:
4553:
4550:
4549:
4520:
4516:
4510:
4504:
4501:
4500:
4480:
4475:
4462:
4458:
4456:
4453:
4452:
4418:
4413:
4403:
4392:
4386:
4383:
4382:
4352:
4349:
4348:
4325:
4321:
4304:
4300:
4296:
4291:
4279:
4275:
4271:
4266:
4265:
4261:
4239:
4238:
4230:
4224:
4221:
4220:
4193:
4189:
4173:
4169:
4165:
4160:
4156:
4150:
4146:
4141:
4138:
4137:
4109:
4105:
4101:
4096:
4093:
4090:
4089:
4086:cotangent space
4063:
4059:
4053:
4050:
4049:
4029:
4025:
4015:
4011:
4002:
3991:
3979:
3976:
3975:
3952:
3948:
3933:
3929:
3918:
3915:
3914:
3907:
3887:
3883:
3881:
3878:
3877:
3862:
3827:
3821:
3816:
3805:
3783:
3776:
3772:
3768:
3766:
3757:
3753:
3743:
3736:
3732:
3728:
3726:
3724:
3718:
3713:
3707:
3704:
3703:
3660:
3655:
3628:
3621:
3617:
3613:
3611:
3602:
3598:
3588:
3581:
3577:
3573:
3571:
3565:
3560:
3539:
3536:
3535:
3525:
3460:
3457:
3456:
3407:
3400:
3396:
3392:
3390:
3372:
3368:
3358:
3351:
3347:
3343:
3341:
3335:
3330:
3316:
3308:
3305:
3304:
3290:
3278:Weinhold metric
3270:
3248:
3244:
3242:
3239:
3238:
3210:
3206:
3205:
3201:
3199:
3196:
3195:
3171:
3167:
3165:
3162:
3161:
3137:
3133:
3122:
3110:
3106:
3094:
3090:
3084:
3080:
3071:
3067:
3055:
3041:
3011:
3007:
2988:
2987:
2983:
2963:
2959:
2958:
2954:
2952:
2949:
2948:
2928:
2924:
2916:
2913:
2912:
2892:
2888:
2865:
2862:
2861:
2823:
2819:
2800:
2799:
2795:
2793:
2790:
2789:
2769:
2766:
2765:
2748:
2744:
2739:
2736:
2735:
2713:
2709:
2708:
2695:
2691:
2687:
2679:
2677:
2674:
2673:
2667:
2663:
2654:
2638:
2634:
2611:
2608:
2607:
2584:
2580:
2572:
2569:
2568:
2543:
2540:
2539:
2528:
2501:
2476:
2468:
2465:
2464:
2443:
2439:
2432:
2428:
2419:
2415:
2414:
2412:
2404:
2401:
2400:
2372:
2368:
2362:
2358:
2334:
2330:
2321:
2317:
2316:
2314:
2299:
2295:
2286:
2282:
2281:
2279:
2278:
2274:
2268:
2267:
2259:
2256:
2255:
2236:
2232:
2231:
2227:
2216:
2212:
2211:
2207:
2205:
2202:
2201:
2180:
2175:
2170:
2167:
2166:
2150:
2147:
2146:
2120:
2116:
2109:
2105:
2093:
2089:
2085:
2083:
2071:
2067:
2062:
2059:
2058:
2028:
2023:
2012:
2009:
2008:
1990:
1989:
1984:
1978:
1977:
1972:
1962:
1961:
1952:
1948:
1938:
1937:
1928:
1924:
1918:
1914:
1906:
1900:
1899:
1894:
1884:
1883:
1875:
1872:
1871:
1839:
1834:
1820:
1802:
1798:
1796:
1793:
1792:
1772:
1768:
1757:
1753:
1744:
1740:
1739:
1735:
1733:
1730:
1729:
1700:
1696:
1694:
1691:
1690:
1667:
1663:
1655:
1652:
1651:
1606:
1579:
1575:
1569:
1565:
1543:
1523:
1503:
1500:
1499:
1468:
1467:
1465:
1462:
1461:
1456:
1431:
1426:
1420:
1417:
1416:
1385:
1382:
1381:
1343:
1331:
1327:
1317:
1313:
1309:
1290:
1286:
1285:
1283:
1271:
1267:
1257:
1253:
1249:
1230:
1226:
1225:
1223:
1202:
1198:
1196:
1193:
1192:
1175:
1174:
1105:
1104:
1077:
1074:
1073:
1066:
1060:on both sides.
1041:
1037:
1033:
1028:
1026:
1023:
1022:
989:
985:
981:
960:
950:
948:
944:
939:
937:
934:
933:
900:
896:
886:
882:
878:
853:
849:
848:
846:
842:
837:
800:
796:
786:
782:
778:
753:
749:
748:
746:
740:
736:
715:
711:
709:
706:
705:
667:
635:
632:
631:
588:
585:
584:
545:
542:
541:
482:
478:
468:
464:
460:
429:
425:
424:
422:
416:
412:
388:
384:
382:
379:
378:
324:
320:
318:
315:
314:
298:
295:
294:
278:
275:
274:
268:random variable
247:
244:
243:
227:
224:
223:
192:
189:
188:
168:
164:
149:
145:
136:
132:
121:
118:
117:
114:
41:smooth manifold
17:
12:
11:
5:
9454:
9444:
9443:
9438:
9433:
9419:
9418:
9411:
9404:
9394:
9338:
9337:
9326:
9323:
9320:
9319:
9298:(4): 347–356.
9275:
9219:
9204:
9171:
9134:(10): 100602.
9113:
9068:
9061:
9043:
9014:(1): 117–135.
8998:
8991:
8969:
8899:
8898:
8896:
8893:
8892:
8891:
8886:
8881:
8876:
8869:
8866:
8853:
8833:
8830:
8827:
8824:
8821:
8818:
8798:
8774:
8771:
8768:
8765:
8762:
8759:
8739:
8736:
8733:
8730:
8727:
8724:
8721:
8701:
8696:
8692:
8688:
8685:
8658:
8638:
8618:
8598:
8572:Hilbert spaces
8537:measure theory
8520:
8496:
8491:
8487:
8483:
8478:
8474:
8470:
8465:
8461:
8438:
8434:
8411:
8407:
8395:
8394:
8383:
8380:
8374:
8371:
8364:
8360:
8356:
8347:
8344:
8337:
8333:
8329:
8321:
8317:
8313:
8310:
8305:
8301:
8297:
8292:
8288:
8284:
8281:
8250:
8245:
8241:
8217:
8214:
8211:
8208:
8205:
8202:
8167:
8143:
8105:
8102:
8099:
8079:
8076:
8071:
8046:
8043:
8040:
8016:
8013:
8010:
8005:
8000:
7997:
7994:
7966:
7963:
7960:
7957:
7954:
7951:
7948:
7925:
7922:
7909:
7906:
7903:
7892:
7891:
7875:
7866:
7862:
7858:
7853:
7850:
7847:
7844:
7833:
7829:
7825:
7820:
7817:
7811:
7803:
7799:
7795:
7790:
7787:
7776:
7772:
7768:
7763:
7760:
7757:
7754:
7747:
7742:
7736:
7733:
7728:
7723:
7721:
7719:
7715:
7707:
7703:
7699:
7694:
7691:
7685:
7680:
7675:
7667:
7663:
7659:
7654:
7651:
7645:
7640:
7636:
7632:
7623:
7619:
7615:
7610:
7607:
7596:
7592:
7588:
7583:
7580:
7573:
7568:
7564:
7561:
7559:
7555:
7554:
7550:
7541:
7537:
7533:
7528:
7525:
7522:
7519:
7508:
7504:
7500:
7495:
7492:
7489:
7486:
7479:
7474:
7468:
7465:
7460:
7456:
7453:
7452:
7448:
7439:
7435:
7431:
7427:
7422:
7414:
7410:
7406:
7402:
7396:
7392:
7389:
7385:
7380:
7377:
7373:
7369:
7368:
7342:
7341:
7325:
7321:
7318:
7315:
7312:
7309:
7306:
7303:
7299:
7294:
7288:
7285:
7280:
7277:
7275:
7271:
7270:
7265:
7260:
7255:
7251:
7248:
7244:
7239:
7234:
7229:
7225:
7219:
7215:
7211:
7208:
7205:
7201:
7196:
7192:
7188:
7182:
7178:
7174:
7171:
7168:
7165:
7162:
7158:
7153:
7147:
7144:
7139:
7135:
7132:
7129:
7128:
7105:
7102:
7097:
7093:
7089:
7069:
7066:
7063:
7050:acting on the
7034:
7031:
7028:
7025:
7014:
7013:
6998:
6995:
6991:
6987:
6984:
6981:
6978:
6975:
6972:
6969:
6966:
6963:
6960:
6957:
6954:
6951:
6948:
6945:
6940:
6936:
6930:
6927:
6922:
6917:
6915:
6913:
6908:
6903:
6899:
6896:
6892:
6888:
6885:
6880:
6876:
6871:
6866:
6863:
6860:
6856:
6850:
6846:
6842:
6839:
6836:
6831:
6827:
6823:
6820:
6818:
6814:
6813:
6810:
6807:
6803:
6797:
6793:
6789:
6786:
6783:
6780:
6777:
6772:
6768:
6762:
6759:
6754:
6750:
6745:
6741:
6737:
6734:
6733:
6719:
6718:
6707:
6703:
6699:
6696:
6693:
6690:
6684:
6681:
6676:
6673:
6666:
6662:
6659:
6656:
6625:
6622:
6619:
6616:
6605:
6604:
6593:
6590:
6587:
6583:
6580:
6577:
6574:
6571:
6568:
6565:
6562:
6559:
6556:
6553:
6548:
6544:
6538:
6534:
6530:
6527:
6524:
6521:
6518:
6515:
6503:is written as
6488:
6485:
6482:
6479:
6476:
6473:
6470:
6467:
6464:
6461:
6458:
6455:
6444:
6443:
6432:
6425:
6420:
6417:
6414:
6411:
6408:
6402:
6399:
6396:
6393:
6390:
6387:
6383:
6380:
6377:
6374:
6371:
6368:
6362:
6356:
6353:
6350:
6347:
6344:
6339:
6336:
6333:
6330:
6327:
6324:
6321:
6315:
6310:
6306:
6302:
6268:
6265:
6262:
6259:
6256:
6253:
6242:
6241:
6230:
6227:
6224:
6221:
6215:
6211:
6207:
6204:
6201:
6198:
6195:
6192:
6189:
6184:
6180:
6165:
6164:
6153:
6150:
6147:
6144:
6140:
6137:
6134:
6131:
6128:
6125:
6120:
6116:
6103:, namely that
6088:
6085:
6082:
6079:
6076:
6073:
6070:
6067:
6047:
6044:
6041:
6038:
6035:
6032:
6012:
6009:
6006:
6003:
6000:
5997:
5973:
5970:
5967:
5964:
5961:
5958:
5947:
5946:
5933:
5930:
5927:
5924:
5921:
5918:
5915:
5911:
5904:
5901:
5898:
5895:
5892:
5889:
5884:
5881:
5878:
5875:
5872:
5869:
5866:
5816:
5813:
5809:Hilbert spaces
5795:, is to use a
5779:
5756:
5745:
5744:
5733:
5728:
5725:
5721:
5717:
5713:
5705:
5701:
5697:
5693:
5688:
5682:
5678:
5674:
5660:
5659:
5643:
5634:
5630:
5626:
5621:
5618:
5615:
5610:
5606:
5602:
5599:
5596:
5584:
5580:
5576:
5571:
5568:
5565:
5560:
5556:
5552:
5549:
5546:
5539:
5534:
5530:
5527:
5525:
5523:
5515:
5511:
5507:
5502:
5499:
5496:
5491:
5487:
5483:
5480:
5477:
5465:
5461:
5457:
5452:
5449:
5446:
5441:
5437:
5433:
5430:
5427:
5420:
5417:
5414:
5409:
5405:
5399:
5395:
5391:
5388:
5386:
5384:
5380:
5371:
5367:
5363:
5359:
5354:
5346:
5342:
5338:
5334:
5328:
5324:
5321:
5318:
5315:
5313:
5308:
5305:
5302:
5299:
5296:
5293:
5287:
5284:
5280:
5276:
5273:
5270:
5267:
5264:
5259:
5256:
5252:
5248:
5247:
5233:
5232:
5215:
5211:
5207:
5202:
5198:
5194:
5186:
5182:
5178:
5173:
5170:
5167:
5162:
5158:
5154:
5151:
5148:
5137:
5133:
5129:
5124:
5121:
5118:
5113:
5109:
5105:
5102:
5099:
5092:
5089:
5086:
5081:
5077:
5071:
5067:
5061:
5058:
5054:
5048:
5045:
5040:
5037:
5035:
5033:
5030:
5027:
5024:
5021:
5016:
5012:
5008:
5005:
5002:
4999:
4995:
4992:
4989:
4986:
4981:
4977:
4973:
4970:
4967:
4964:
4960:
4957:
4954:
4949:
4945:
4939:
4935:
4929:
4926:
4921:
4918:
4916:
4914:
4911:
4910:
4887:
4884:
4881:
4876:
4872:
4868:
4863:
4859:
4838:
4827:
4826:
4811:
4806:
4802:
4798:
4795:
4792:
4789:
4785:
4780:
4776:
4772:
4769:
4766:
4763:
4757:
4753:
4747:
4743:
4737:
4734:
4729:
4722:
4718:
4711:
4707:
4703:
4697:
4693:
4689:
4681:
4677:
4671:
4668:
4663:
4660:
4658:
4656:
4649:
4645:
4639:
4631:
4627:
4621:
4616:
4612:
4608:
4603:
4599:
4595:
4589:
4585:
4581:
4576:
4572:
4568:
4565:
4563:
4561:
4558:
4557:
4543:
4542:
4531:
4528:
4523:
4519:
4513:
4509:
4483:
4478:
4474:
4470:
4465:
4461:
4441:
4440:
4429:
4426:
4421:
4416:
4412:
4406:
4401:
4398:
4395:
4391:
4356:
4345:
4344:
4331:
4328:
4324:
4320:
4316:
4307:
4303:
4299:
4295:
4290:
4282:
4278:
4274:
4270:
4264:
4260:
4257:
4251:
4248:
4245:
4242:
4236:
4233:
4229:
4214:
4213:
4199:
4196:
4192:
4188:
4184:
4176:
4172:
4168:
4164:
4159:
4153:
4149:
4145:
4112:
4108:
4104:
4100:
4066:
4062:
4058:
4046:
4045:
4032:
4028:
4024:
4018:
4014:
4010:
4005:
4000:
3997:
3994:
3990:
3986:
3983:
3960:
3955:
3951:
3947:
3944:
3941:
3936:
3932:
3928:
3925:
3922:
3890:
3886:
3861:
3858:
3854:
3853:
3840:
3837:
3834:
3831:
3824:
3819:
3815:
3809:
3804:
3801:
3798:
3789:
3786:
3779:
3775:
3771:
3763:
3760:
3756:
3749:
3746:
3739:
3735:
3731:
3721:
3716:
3712:
3689:
3688:
3677:
3674:
3671:
3668:
3663:
3658:
3654:
3650:
3647:
3644:
3641:
3634:
3631:
3624:
3620:
3616:
3608:
3605:
3601:
3594:
3591:
3584:
3580:
3576:
3568:
3563:
3559:
3555:
3552:
3549:
3546:
3543:
3524:
3521:
3501:
3500:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3434:
3433:
3422:
3419:
3413:
3410:
3403:
3399:
3395:
3389:
3386:
3383:
3378:
3375:
3371:
3364:
3361:
3354:
3350:
3346:
3338:
3333:
3329:
3323:
3320:
3315:
3312:
3289:
3286:
3269:
3266:
3251:
3247:
3226:
3223:
3220:
3213:
3209:
3204:
3192:Hessian matrix
3177:
3174:
3170:
3158:
3157:
3145:
3140:
3136:
3132:
3129:
3125:
3121:
3118:
3113:
3109:
3105:
3100:
3097:
3093:
3087:
3083:
3079:
3074:
3070:
3066:
3061:
3058:
3054:
3048:
3045:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3014:
3010:
3006:
3003:
3000:
2994:
2991:
2986:
2982:
2979:
2976:
2973:
2966:
2962:
2957:
2931:
2927:
2923:
2920:
2900:
2895:
2891:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2826:
2822:
2818:
2815:
2812:
2806:
2803:
2798:
2773:
2751:
2747:
2743:
2732:
2731:
2716:
2712:
2706:
2698:
2694:
2690:
2685:
2682:
2676:
2670:
2666:
2662:
2657:
2653:
2649:
2646:
2641:
2637:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2592:
2587:
2583:
2579:
2576:
2556:
2553:
2550:
2547:
2527:
2524:
2511:
2508:
2504:
2500:
2497:
2494:
2491:
2488:
2485:
2480:
2475:
2472:
2447:
2442:
2435:
2431:
2427:
2422:
2418:
2411:
2408:
2387:
2383:
2380:
2375:
2371:
2365:
2361:
2357:
2354:
2351:
2348:
2343:
2337:
2333:
2329:
2324:
2320:
2313:
2308:
2302:
2298:
2294:
2289:
2285:
2277:
2271:
2266:
2263:
2239:
2235:
2230:
2226:
2219:
2215:
2210:
2184:
2178:
2174:
2154:
2138:. This is the
2123:
2119:
2112:
2108:
2104:
2101:
2096:
2092:
2088:
2082:
2079:
2074:
2070:
2066:
2046:
2043:
2040:
2037:
2032:
2026:
2022:
2019:
2016:
1994:
1988:
1985:
1983:
1980:
1979:
1976:
1973:
1971:
1968:
1967:
1965:
1958:
1955:
1951:
1947:
1942:
1934:
1931:
1927:
1921:
1917:
1913:
1910:
1907:
1905:
1902:
1901:
1898:
1895:
1893:
1890:
1889:
1887:
1882:
1879:
1856:
1853:
1850:
1847:
1842:
1837:
1833:
1827:
1824:
1819:
1816:
1811:
1808:
1805:
1801:
1778:
1775:
1771:
1767:
1760:
1756:
1752:
1747:
1743:
1738:
1717:
1714:
1709:
1706:
1703:
1699:
1673:
1670:
1666:
1662:
1659:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1613:
1610:
1605:
1602:
1599:
1596:
1593:
1590:
1585:
1582:
1578:
1572:
1568:
1564:
1561:
1558:
1555:
1550:
1547:
1542:
1539:
1536:
1533:
1530:
1526:
1522:
1519:
1516:
1513:
1510:
1507:
1488:
1485:
1482:
1479:
1476:
1471:
1455:
1452:
1439:
1434:
1429:
1425:
1404:
1401:
1398:
1395:
1392:
1389:
1365:
1362:
1359:
1356:
1353:
1350:
1346:
1342:
1334:
1330:
1326:
1320:
1316:
1312:
1307:
1304:
1301:
1298:
1293:
1289:
1282:
1274:
1270:
1266:
1260:
1256:
1252:
1247:
1244:
1241:
1238:
1233:
1229:
1222:
1219:
1216:
1213:
1208:
1205:
1201:
1191:The metric is
1178:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1108:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1065:
1062:
1044:
1040:
1036:
1032:
1019:
1018:
1007:
1004:
1000:
992:
988:
984:
979:
976:
973:
970:
967:
964:
959:
956:
953:
947:
942:
927:
926:
915:
911:
903:
899:
895:
889:
885:
881:
876:
873:
870:
867:
864:
861:
856:
852:
845:
840:
836:
833:
830:
826:
823:
820:
817:
814:
811:
803:
799:
795:
789:
785:
781:
776:
773:
770:
767:
764:
761:
756:
752:
743:
739:
735:
732:
729:
726:
721:
718:
714:
686:
683:
680:
677:
674:
671:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
592:
564:. The labels
549:
530:
529:
518:
515:
512:
508:
505:
502:
499:
496:
493:
485:
481:
477:
471:
467:
463:
458:
455:
452:
449:
446:
443:
440:
437:
432:
428:
419:
415:
411:
408:
405:
402:
399:
394:
391:
387:
360:
357:
354:
351:
347:
344:
341:
338:
335:
332:
327:
323:
302:
282:
251:
231:
211:
208:
205:
202:
199:
196:
176:
171:
167:
163:
160:
157:
152:
148:
144:
139:
135:
131:
128:
125:
113:
110:
15:
9:
6:
4:
3:
2:
9453:
9442:
9439:
9437:
9434:
9432:
9429:
9428:
9426:
9416:
9412:
9409:
9405:
9402:
9398:
9395:
9391:
9387:
9383:
9379:
9375:
9371:
9367:
9363:
9358:
9353:
9349:
9345:
9340:
9339:
9336:
9333:
9329:
9328:
9314:
9309:
9305:
9301:
9297:
9293:
9286:
9279:
9271:
9267:
9263:
9259:
9255:
9251:
9246:
9241:
9237:
9233:
9226:
9224:
9215:
9211:
9207:
9201:
9197:
9193:
9189:
9185:
9178:
9176:
9167:
9163:
9159:
9155:
9151:
9147:
9142:
9137:
9133:
9129:
9122:
9120:
9118:
9109:
9105:
9101:
9097:
9092:
9087:
9084:(2): 023001.
9083:
9079:
9072:
9064:
9062:0-471-24195-4
9058:
9054:
9047:
9039:
9035:
9031:
9027:
9022:
9017:
9013:
9009:
9002:
8994:
8992:0-8218-0531-2
8988:
8984:
8980:
8973:
8965:
8961:
8956:
8951:
8946:
8941:
8937:
8933:
8928:
8923:
8919:
8915:
8911:
8904:
8900:
8890:
8887:
8885:
8882:
8880:
8877:
8875:
8872:
8871:
8865:
8851:
8828:
8822:
8819:
8816:
8796:
8788:
8769:
8763:
8760:
8757:
8734:
8728:
8725:
8722:
8719:
8699:
8694:
8690:
8686:
8683:
8675:
8670:
8656:
8636:
8616:
8596:
8588:
8584:
8580:
8575:
8573:
8569:
8568:weak topology
8565:
8561:
8557:
8553:
8549:
8544:
8542:
8538:
8534:
8518:
8510:
8494:
8489:
8485:
8481:
8476:
8472:
8468:
8463:
8459:
8436:
8432:
8409:
8405:
8381:
8378:
8372:
8369:
8362:
8358:
8354:
8345:
8342:
8335:
8331:
8327:
8319:
8315:
8311:
8303:
8299:
8295:
8290:
8286:
8279:
8272:
8271:
8270:
8268:
8264:
8263:inner product
8248:
8243:
8239:
8231:
8230:tangent space
8212:
8206:
8203:
8200:
8193:Pick a point
8191:
8189:
8185:
8181:
8180:Fréchet space
8157:
8141:
8133:
8129:
8125:
8122:
8117:
8103:
8100:
8097:
8074:
8060:
8059:sigma algebra
8044:
8041:
8030:
8011:
8008:
7998:
7984:
7980:
7961:
7958:
7952:
7949:
7938:
7934:
7929:
7921:
7907:
7904:
7901:
7873:
7864:
7860:
7851:
7848:
7845:
7831:
7827:
7818:
7809:
7801:
7797:
7788:
7774:
7770:
7761:
7758:
7755:
7745:
7734:
7731:
7726:
7722:
7713:
7705:
7701:
7692:
7683:
7673:
7665:
7661:
7652:
7643:
7634:
7630:
7621:
7617:
7608:
7594:
7590:
7581:
7571:
7562:
7560:
7548:
7539:
7535:
7526:
7523:
7520:
7506:
7502:
7493:
7490:
7487:
7477:
7466:
7463:
7454:
7446:
7437:
7433:
7420:
7412:
7408:
7394:
7390:
7383:
7378:
7375:
7371:
7359:
7358:
7357:
7355:
7351:
7347:
7323:
7319:
7316:
7313:
7310:
7307:
7304:
7301:
7297:
7286:
7283:
7278:
7276:
7263:
7258:
7253:
7249:
7246:
7242:
7232:
7227:
7223:
7217:
7209:
7206:
7199:
7190:
7186:
7180:
7172:
7169:
7166:
7163:
7156:
7145:
7142:
7133:
7130:
7119:
7118:
7117:
7103:
7095:
7091:
7087:
7067:
7061:
7053:
7052:tangent space
7049:
7032:
7029:
7026:
7023:
6996:
6993:
6989:
6982:
6979:
6976:
6973:
6970:
6967:
6964:
6961:
6958:
6955:
6952:
6949:
6946:
6938:
6934:
6928:
6925:
6920:
6916:
6906:
6901:
6897:
6894:
6890:
6886:
6883:
6878:
6874:
6869:
6864:
6861:
6858:
6854:
6848:
6840:
6837:
6829:
6825:
6821:
6819:
6808:
6805:
6801:
6795:
6787:
6784:
6781:
6778:
6770:
6766:
6760:
6757:
6748:
6743:
6739:
6735:
6724:
6723:
6722:
6705:
6701:
6697:
6694:
6691:
6688:
6682:
6679:
6674:
6671:
6664:
6660:
6657:
6654:
6647:
6646:
6645:
6643:
6639:
6620:
6617:
6591:
6588:
6585:
6578:
6575:
6572:
6566:
6560:
6557:
6554:
6546:
6542:
6536:
6532:
6528:
6522:
6519:
6516:
6506:
6505:
6504:
6502:
6483:
6480:
6477:
6471:
6468:
6462:
6459:
6456:
6430:
6423:
6415:
6412:
6409:
6397:
6394:
6391:
6388:
6378:
6375:
6372:
6369:
6360:
6351:
6348:
6345:
6334:
6331:
6328:
6325:
6322:
6313:
6308:
6304:
6300:
6293:
6292:
6291:
6289:
6285:
6280:
6263:
6260:
6257:
6251:
6228:
6225:
6222:
6219:
6213:
6202:
6199:
6196:
6190:
6182:
6178:
6170:
6169:
6168:
6151:
6148:
6145:
6142:
6135:
6132:
6129:
6123:
6118:
6114:
6106:
6105:
6104:
6102:
6086:
6083:
6077:
6074:
6071:
6065:
6042:
6039:
6036:
6030:
6007:
6004:
6001:
5995:
5987:
5968:
5965:
5962:
5956:
5928:
5925:
5922:
5916:
5913:
5909:
5899:
5896:
5893:
5887:
5882:
5876:
5873:
5870:
5864:
5857:
5856:
5855:
5853:
5850:, written in
5849:
5844:
5842:
5838:
5834:
5830:
5826:
5822:
5812:
5810:
5806:
5802:
5798:
5794:
5777:
5768:
5754:
5731:
5726:
5723:
5719:
5715:
5711:
5703:
5699:
5686:
5680:
5676:
5672:
5665:
5664:
5663:
5641:
5632:
5628:
5616:
5608:
5604:
5600:
5597:
5582:
5578:
5566:
5558:
5554:
5550:
5547:
5537:
5528:
5526:
5513:
5509:
5497:
5489:
5485:
5481:
5478:
5463:
5459:
5447:
5439:
5435:
5431:
5428:
5415:
5407:
5403:
5397:
5393:
5389:
5387:
5378:
5369:
5365:
5352:
5344:
5340:
5326:
5322:
5319:
5316:
5314:
5285:
5282:
5278:
5274:
5271:
5265:
5257:
5254:
5250:
5238:
5237:
5236:
5213:
5209:
5205:
5200:
5196:
5192:
5184:
5180:
5168:
5160:
5156:
5152:
5149:
5135:
5131:
5119:
5111:
5107:
5103:
5100:
5087:
5079:
5075:
5069:
5065:
5059:
5056:
5052:
5046:
5043:
5038:
5036:
5022:
5014:
5010:
5006:
5003:
4997:
4987:
4979:
4975:
4971:
4968:
4962:
4955:
4947:
4943:
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4927:
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4900:
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4874:
4870:
4866:
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4857:
4836:
4804:
4800:
4796:
4793:
4787:
4778:
4774:
4770:
4767:
4761:
4755:
4751:
4745:
4741:
4735:
4732:
4727:
4720:
4716:
4709:
4705:
4701:
4695:
4691:
4687:
4679:
4675:
4669:
4666:
4661:
4659:
4647:
4643:
4637:
4629:
4625:
4619:
4614:
4610:
4606:
4601:
4597:
4593:
4587:
4583:
4579:
4574:
4570:
4566:
4564:
4559:
4548:
4547:
4546:
4529:
4526:
4521:
4517:
4511:
4507:
4499:
4498:
4497:
4481:
4476:
4472:
4468:
4463:
4459:
4449:
4447:
4444:technique of
4427:
4424:
4419:
4414:
4410:
4404:
4399:
4396:
4393:
4389:
4381:
4380:
4379:
4377:
4373:
4368:
4354:
4329:
4326:
4322:
4318:
4314:
4305:
4301:
4288:
4280:
4276:
4262:
4258:
4255:
4234:
4231:
4227:
4219:
4218:
4217:
4197:
4194:
4190:
4186:
4182:
4174:
4170:
4157:
4151:
4147:
4143:
4136:
4135:
4134:
4132:
4131:tangent space
4110:
4106:
4087:
4083:
4064:
4060:
4056:
4030:
4026:
4022:
4016:
4012:
4008:
4003:
3998:
3995:
3992:
3988:
3984:
3981:
3974:
3973:
3972:
3953:
3949:
3945:
3942:
3939:
3934:
3930:
3923:
3920:
3910:
3904:
3888:
3884:
3875:
3871:
3867:
3857:
3838:
3835:
3832:
3829:
3822:
3817:
3813:
3807:
3802:
3799:
3796:
3787:
3777:
3773:
3761:
3758:
3754:
3747:
3737:
3733:
3719:
3714:
3710:
3702:
3701:
3700:
3698:
3694:
3675:
3672:
3669:
3666:
3661:
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3648:
3645:
3642:
3639:
3632:
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3618:
3606:
3603:
3599:
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3582:
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3530:
3520:
3518:
3514:
3510:
3506:
3486:
3480:
3477:
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3454:
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3451:
3447:
3443:
3439:
3420:
3417:
3411:
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3397:
3384:
3376:
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3369:
3362:
3352:
3348:
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3331:
3327:
3321:
3318:
3313:
3310:
3303:
3302:
3301:
3299:
3295:
3285:
3283:
3279:
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3265:
3249:
3245:
3221:
3211:
3207:
3202:
3193:
3175:
3172:
3168:
3138:
3134:
3119:
3111:
3107:
3098:
3095:
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3085:
3081:
3072:
3068:
3059:
3056:
3052:
3046:
3043:
3038:
3029:
3023:
3012:
3008:
3001:
2984:
2980:
2974:
2964:
2960:
2955:
2947:
2946:
2945:
2929:
2925:
2921:
2918:
2893:
2889:
2882:
2879:
2873:
2867:
2841:
2835:
2824:
2820:
2813:
2796:
2787:
2771:
2749:
2745:
2714:
2710:
2704:
2696:
2692:
2683:
2668:
2664:
2655:
2651:
2647:
2639:
2635:
2628:
2625:
2619:
2613:
2606:
2605:
2604:
2585:
2581:
2574:
2551:
2545:
2537:
2533:
2523:
2506:
2502:
2498:
2492:
2489:
2486:
2483:
2478:
2473:
2470:
2445:
2440:
2433:
2429:
2425:
2420:
2416:
2409:
2406:
2385:
2381:
2378:
2373:
2369:
2363:
2359:
2355:
2352:
2349:
2346:
2341:
2335:
2331:
2327:
2322:
2318:
2311:
2306:
2300:
2296:
2292:
2287:
2283:
2275:
2261:
2237:
2233:
2228:
2224:
2217:
2213:
2208:
2198:
2182:
2176:
2172:
2152:
2143:
2141:
2121:
2117:
2110:
2106:
2102:
2099:
2094:
2090:
2086:
2080:
2077:
2072:
2068:
2064:
2044:
2041:
2038:
2035:
2030:
2024:
2020:
2017:
2014:
1992:
1986:
1981:
1974:
1969:
1963:
1956:
1953:
1949:
1945:
1940:
1932:
1929:
1919:
1915:
1911:
1903:
1896:
1891:
1885:
1880:
1877:
1868:
1854:
1848:
1845:
1840:
1835:
1825:
1822:
1817:
1814:
1809:
1806:
1803:
1799:
1776:
1773:
1769:
1765:
1758:
1754:
1750:
1745:
1741:
1736:
1715:
1712:
1704:
1701:
1697:
1687:
1671:
1668:
1660:
1657:
1637:
1634:
1619:
1616:
1611:
1608:
1603:
1597:
1594:
1591:
1583:
1580:
1570:
1562:
1559:
1556:
1548:
1545:
1540:
1531:
1528:
1520:
1514:
1511:
1508:
1505:
1480:
1477:
1460:
1451:
1437:
1432:
1427:
1402:
1399:
1393:
1387:
1379:
1357:
1351:
1340:
1332:
1328:
1318:
1314:
1302:
1296:
1291:
1280:
1272:
1268:
1258:
1254:
1242:
1236:
1231:
1220:
1214:
1206:
1203:
1199:
1165:
1159:
1156:
1150:
1144:
1141:
1135:
1129:
1126:
1120:
1114:
1100:
1097:
1091:
1088:
1085:
1079:
1071:
1061:
1042:
1038:
1005:
1002:
998:
990:
986:
974:
971:
968:
962:
957:
954:
945:
932:
931:
930:
913:
909:
901:
897:
887:
883:
871:
868:
865:
859:
854:
843:
834:
831:
828:
821:
818:
815:
809:
801:
797:
787:
783:
771:
768:
765:
759:
754:
741:
737:
733:
727:
719:
716:
712:
704:
703:
702:
700:
681:
678:
675:
669:
664:
661:
658:
655:
649:
646:
643:
637:
630:Substituting
628:
626:
622:
618:
614:
610:
606:
590:
582:
578:
577:Gibbs measure
573:
571:
567:
563:
547:
539:
535:
516:
513:
510:
503:
500:
497:
491:
483:
479:
469:
465:
453:
450:
447:
441:
438:
435:
430:
417:
413:
409:
406:
400:
392:
389:
385:
377:
376:
375:
372:
358:
355:
352:
349:
342:
339:
336:
330:
325:
321:
300:
280:
272:
269:
265:
249:
229:
206:
203:
200:
194:
187:, one writes
169:
165:
161:
158:
155:
150:
146:
142:
137:
133:
126:
123:
109:
107:
103:
98:
96:
92:
88:
84:
80:
76:
72:
68:
63:
61:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
9414:
9407:
9400:
9347:
9343:
9331:
9295:
9291:
9278:
9235:
9231:
9187:
9131:
9127:
9081:
9077:
9071:
9052:
9046:
9011:
9007:
9001:
8982:
8972:
8917:
8913:
8903:
8671:
8629:. Then, if
8586:
8582:
8576:
8551:
8547:
8545:
8532:
8508:
8396:
8192:
8187:
8183:
8155:
8131:
8127:
8123:
8118:
7982:
7932:
7930:
7927:
7893:
7348:, it is the
7343:
7015:
6720:
6606:
6500:
6445:
6281:
6243:
6166:
5948:
5845:
5841:mixed states
5829:Bures metric
5818:
5769:
5746:
5661:
5234:
4828:
4544:
4450:
4442:
4375:
4371:
4369:
4346:
4215:
4047:
3908:
3905:
3863:
3855:
3697:curve length
3692:
3690:
3526:
3502:
3449:
3445:
3442:free entropy
3437:
3435:
3300:is given by
3291:
3271:
3159:
2944:of the form
2785:
2733:
2531:
2529:
2254:has formula
2199:
2144:
1869:
1688:
1457:
1072:, which has
1067:
1020:
928:
629:
620:
616:
612:
574:
569:
565:
537:
533:
531:
373:
270:
263:
115:
99:
95:Bures metric
91:mixed states
66:
64:
53:
36:
24:
18:
9313:2241/100265
8785:(after the
8579:submanifold
7350:Berry phase
5837:pure states
4088:. Writing
9425:Categories
9325:References
9021:1701.08895
8927:2302.08175
8920:(4): 654.
7939:, and let
4133:, so that
4048:where the
3699:, one has
1021:and apply
112:Definition
9357:0807.0621
9245:1009.5219
9141:0706.0559
9108:118311636
9091:0809.1166
8852:μ
8820:∈
8797:μ
8761:∈
8735:σ
8729:
8695:μ
8687:∈
8684:σ
8657:θ
8637:θ
8617:θ
8597:μ
8519:μ
8490:μ
8482:∈
8473:σ
8460:σ
8433:σ
8406:σ
8382:μ
8373:μ
8359:σ
8346:μ
8332:σ
8316:∫
8300:σ
8287:σ
8244:μ
8204:∈
8201:μ
8166:Σ
8142:μ
8104:μ
8078:Σ
8039:Ω
7996:Ω
7962:μ
7956:Σ
7902:α
7861:θ
7857:∂
7849:
7843:∂
7828:θ
7824:∂
7819:α
7816:∂
7810:−
7798:θ
7794:∂
7789:α
7786:∂
7771:θ
7767:∂
7759:
7753:∂
7702:θ
7698:∂
7693:α
7690:∂
7662:θ
7658:∂
7653:α
7650:∂
7635:−
7618:θ
7614:∂
7609:α
7606:∂
7591:θ
7587:∂
7582:α
7579:∂
7536:θ
7532:∂
7524:
7518:∂
7503:θ
7499:∂
7491:
7485:∂
7434:θ
7430:∂
7426:∂
7409:θ
7405:∂
7401:∂
7320:α
7314:∧
7308:
7250:α
7228:−
7210:α
7170:
7101:→
7065:→
7062:δ
7027:α
7024:δ
6980:
6974:δ
6971:α
6968:δ
6965:−
6962:α
6959:δ
6953:
6947:δ
6935:∫
6887:α
6884:δ
6875:∫
6865:−
6841:α
6838:δ
6826:∫
6785:
6779:δ
6767:∫
6706:ψ
6698:α
6695:δ
6672:δ
6658:ψ
6655:δ
6624:⟩
6621:ψ
6618:δ
6579:θ
6567:ψ
6561:θ
6547:∗
6543:ϕ
6533:∫
6526:⟩
6523:ψ
6520:∣
6517:ϕ
6514:⟨
6484:θ
6472:ψ
6466:⟩
6463:ψ
6460:∣
6454:⟨
6419:⟩
6416:ψ
6413:∣
6410:ψ
6407:⟨
6401:⟩
6398:ψ
6395:δ
6392:∣
6389:ψ
6386:⟨
6382:⟩
6379:ψ
6376:∣
6373:ψ
6370:δ
6367:⟨
6361:−
6355:⟩
6352:ψ
6349:∣
6346:ψ
6343:⟨
6338:⟩
6335:ψ
6332:δ
6329:∣
6326:ψ
6323:δ
6320:⟨
6264:θ
6252:ψ
6203:θ
6191:ψ
6179:∫
6136:θ
6115:∫
6078:θ
6066:α
6043:θ
6031:α
6008:θ
5969:θ
5957:ψ
5929:θ
5917:α
5900:θ
5877:θ
5865:ψ
5755:θ
5720:δ
5700:θ
5696:∂
5692:∂
5677:θ
5629:θ
5625:∂
5617:θ
5601:
5595:∂
5579:θ
5575:∂
5567:θ
5551:
5545:∂
5510:θ
5506:∂
5498:θ
5482:
5476:∂
5460:θ
5456:∂
5448:θ
5432:
5426:∂
5416:θ
5394:∑
5366:θ
5362:∂
5358:∂
5341:θ
5337:∂
5333:∂
5266:θ
5210:θ
5197:θ
5181:θ
5177:∂
5169:θ
5153:
5147:∂
5132:θ
5128:∂
5120:θ
5104:
5098:∂
5088:θ
5066:∑
5053:∑
5023:θ
5007:
4988:θ
4972:
4956:θ
4934:∑
4883:θ
4837:θ
4797:
4771:
4742:∑
4676:∑
4611:∑
4571:∑
4508:∑
4390:∑
4323:δ
4298:∂
4294:∂
4273:∂
4269:∂
4191:δ
4167:∂
4163:∂
4103:∂
4099:∂
3989:∑
3943:⋯
3814:∫
3785:∂
3774:θ
3770:∂
3745:∂
3734:θ
3730:∂
3711:∫
3653:∫
3630:∂
3619:θ
3615:∂
3590:∂
3579:θ
3575:∂
3558:∫
3548:−
3478:−
3463:Δ
3409:∂
3398:θ
3394:∂
3385:θ
3360:∂
3349:θ
3345:∂
3328:∫
3246:θ
3222:θ
3208:θ
3135:θ
3131:Δ
3108:θ
3082:θ
3078:Δ
3069:θ
3065:Δ
3053:∑
3030:θ
3021:‖
3009:θ
2975:θ
2961:θ
2926:θ
2919:θ
2890:θ
2874:θ
2842:θ
2833:‖
2821:θ
2772:θ
2746:θ
2742:Δ
2711:θ
2693:θ
2689:∂
2681:∂
2665:θ
2661:Δ
2652:∑
2636:θ
2620:θ
2582:θ
2552:θ
2499:ϕ
2493:
2487:
2430:μ
2426:−
2417:μ
2407:σ
2382:ϕ
2379:
2360:σ
2353:ϕ
2350:
2332:μ
2328:−
2319:μ
2297:μ
2284:μ
2265:↦
2262:ϕ
2234:μ
2229:δ
2214:μ
2209:δ
2173:μ
2153:σ
2045:σ
2021:μ
1954:−
1950:σ
1930:−
1849:
1832:∇
1818:−
1755:μ
1742:μ
1708:Σ
1702:μ
1669:−
1665:Σ
1629:Σ
1620:
1598:μ
1595:−
1581:−
1577:Σ
1563:μ
1560:−
1535:Σ
1529:μ
1512:
1506:−
1484:Σ
1478:μ
1428:θ
1424:∇
1403:θ
1394:θ
1388:η
1341:⋅
1329:θ
1325:∂
1315:θ
1311:∂
1303:θ
1297:η
1288:∂
1281:−
1269:θ
1265:∂
1255:θ
1251:∂
1243:θ
1228:∂
1215:θ
1151:θ
1142:−
1127:⋅
1121:θ
1115:η
1092:θ
1089:∣
1039:θ
1035:∂
1031:∂
987:θ
983:∂
975:θ
972:∣
958:
952:∂
898:θ
894:∂
884:θ
880:∂
872:θ
869:∣
851:∂
822:θ
819:∣
798:θ
794:∂
784:θ
780:∂
772:θ
769:∣
751:∂
738:∫
728:θ
682:θ
679:∣
665:
659:−
650:θ
647:∣
591:θ
548:θ
504:θ
501:∣
480:θ
476:∂
466:θ
462:∂
454:θ
451:∣
439:
427:∂
414:∫
410:−
401:θ
343:θ
340:∣
322:∫
301:θ
230:θ
207:θ
204:∣
166:θ
159:…
147:θ
134:θ
124:θ
9382:19257090
9270:55558124
9158:17930381
8964:37190442
8955:10137715
8868:See also
7016:Setting
3513:geodesic
3505:chemical
3448:to time
1064:Examples
293:but not
242:. Here
9399:(1985)
9390:8210246
9362:Bibcode
9250:Bibcode
9214:3469115
9166:7527491
9038:5954036
8932:Bibcode
8914:Entropy
8057:, with
7979:measure
6640:in the
6101:simplex
4082:1-forms
3874:orthant
2784:in the
2197:-axis.
2057:, then
583:, then
75:Hessian
9388:
9380:
9268:
9212:
9202:
9164:
9156:
9106:
9059:
9036:
8989:
8962:
8952:
8397:Here,
7935:be an
7048:2-form
6638:1-form
5949:Here,
5854:, so:
5793:Gromov
3864:For a
3294:action
2399:where
2007:. Let
1172:
1112:
69:, the
23:, the
9386:S2CID
9352:arXiv
9288:(PDF)
9266:S2CID
9240:arXiv
9162:S2CID
9136:arXiv
9104:S2CID
9086:arXiv
9034:S2CID
9016:arXiv
8922:arXiv
8895:Notes
8130:) of
8027:be a
7977:be a
6290:, is
6244:When
2734:with
697:from
9378:PMID
9200:ISBN
9154:PMID
9057:ISBN
8987:ISBN
8960:PMID
8424:and
8119:The
7931:Let
7080:and
6282:The
6023:and
4080:are
3693:dJSD
3507:and
3292:The
3276:and
3272:The
2567:and
1650:Let
568:and
67:i.e.
39:, a
37:i.e.
9370:doi
9308:hdl
9300:doi
9258:doi
9236:374
9192:doi
9146:doi
9096:doi
9026:doi
8950:PMC
8940:doi
8726:exp
8581:of
8154:on
8031:on
7981:on
7846:log
7756:log
7521:log
7488:log
7352:or
7305:log
7167:log
6977:log
6950:log
6782:log
5598:log
5548:log
5479:log
5429:log
5150:log
5101:log
5004:log
4969:log
4794:log
4768:log
4370:An
2534:or
2490:tan
2370:sin
2347:cos
1852:det
1623:det
1101:exp
955:log
662:log
536:in
436:log
313::
19:In
9427::
9384:.
9376:.
9368:.
9360:.
9348:79
9346:.
9306:.
9296:26
9294:.
9290:.
9264:.
9256:.
9248:.
9234:.
9222:^
9210:MR
9208:.
9198:.
9186:.
9174:^
9160:.
9152:.
9144:.
9132:99
9130:.
9116:^
9102:.
9094:.
9082:42
9080:.
9032:.
9024:.
9010:.
8981:.
8958:.
8948:.
8938:.
8930:.
8918:25
8916:.
8912:.
8669:.
8543:.
8116:.
5988:;
4448:.
4367:.
3911:+1
2981::=
2522:.
2484:ln
2142:.
1867:.
1846:ln
1617:ln
1509:ln
1450:.
371:.
108:.
97:.
62:.
35:,
9392:.
9372::
9364::
9354::
9316:.
9310::
9302::
9272:.
9260::
9252::
9242::
9216:.
9194::
9168:.
9148::
9138::
9110:.
9098::
9088::
9065:.
9040:.
9028::
9018::
9012:1
8995:.
8966:.
8942::
8934::
8924::
8832:)
8829:X
8826:(
8823:S
8817:p
8773:)
8770:X
8767:(
8764:S
8758:p
8738:)
8732:(
8723:=
8720:p
8700:S
8691:T
8587:X
8585:(
8583:S
8552:X
8550:(
8548:S
8533:X
8509:d
8495:S
8486:T
8477:2
8469:,
8464:1
8437:2
8410:1
8379:d
8370:d
8363:2
8355:d
8343:d
8336:1
8328:d
8320:X
8312:=
8309:)
8304:2
8296:,
8291:1
8283:(
8280:g
8249:S
8240:T
8216:)
8213:X
8210:(
8207:S
8188:X
8186:(
8184:S
8156:X
8132:X
8128:X
8126:(
8124:S
8101:=
8098:P
8075:=
8070:F
8045:X
8042:=
8015:)
8012:P
8009:,
8004:F
7999:,
7993:(
7983:X
7965:)
7959:,
7953:,
7950:X
7947:(
7933:X
7908:0
7905:=
7874:]
7865:k
7852:p
7832:j
7802:k
7775:j
7762:p
7746:[
7741:E
7735:2
7732:i
7727:+
7714:]
7706:k
7684:[
7679:E
7674:]
7666:j
7644:[
7639:E
7631:]
7622:k
7595:j
7572:[
7567:E
7563:+
7549:]
7540:k
7527:p
7507:j
7494:p
7478:[
7473:E
7467:4
7464:1
7455:=
7447:)
7438:k
7421:,
7413:j
7395:(
7391:h
7384:=
7379:k
7376:j
7372:h
7324:]
7317:d
7311:p
7302:d
7298:[
7293:E
7287:2
7284:i
7279:+
7264:2
7259:)
7254:]
7247:d
7243:[
7238:E
7233:(
7224:]
7218:2
7214:)
7207:d
7204:(
7200:[
7195:E
7191:+
7187:]
7181:2
7177:)
7173:p
7164:d
7161:(
7157:[
7152:E
7146:4
7143:1
7134:=
7131:h
7104:h
7096:2
7092:s
7088:d
7068:d
7033:0
7030:=
6997:x
6994:d
6990:p
6986:)
6983:p
6956:p
6944:(
6939:X
6929:2
6926:i
6921:+
6907:2
6902:)
6898:x
6895:d
6891:p
6879:X
6870:(
6862:x
6859:d
6855:p
6849:2
6845:)
6835:(
6830:X
6822:+
6809:x
6806:d
6802:p
6796:2
6792:)
6788:p
6776:(
6771:X
6761:4
6758:1
6749:=
6744:2
6740:s
6736:d
6702:)
6692:i
6689:+
6683:p
6680:2
6675:p
6665:(
6661:=
6615:|
6592:.
6589:x
6586:d
6582:)
6576:;
6573:x
6570:(
6564:)
6558:;
6555:x
6552:(
6537:X
6529:=
6501:X
6487:)
6481:;
6478:x
6475:(
6469:=
6457:x
6431:.
6424:2
6314:=
6309:2
6305:s
6301:d
6267:)
6261:;
6258:x
6255:(
6229:1
6226:=
6223:x
6220:d
6214:2
6210:|
6206:)
6200:;
6197:x
6194:(
6188:|
6183:X
6152:1
6149:=
6146:x
6143:d
6139:)
6133:;
6130:x
6127:(
6124:p
6119:X
6087:0
6084:=
6081:)
6075:;
6072:x
6069:(
6046:)
6040:;
6037:x
6034:(
6011:)
6005:;
6002:x
5999:(
5996:p
5972:)
5966:;
5963:x
5960:(
5932:)
5926:;
5923:x
5920:(
5914:i
5910:e
5903:)
5897:;
5894:x
5891:(
5888:p
5883:=
5880:)
5874:;
5871:x
5868:(
5778:p
5732:.
5727:k
5724:j
5716:=
5712:)
5704:k
5687:(
5681:j
5673:d
5642:]
5633:k
5620:)
5614:(
5609:i
5605:p
5583:j
5570:)
5564:(
5559:i
5555:p
5538:[
5533:E
5529:=
5514:k
5501:)
5495:(
5490:i
5486:p
5464:j
5451:)
5445:(
5440:i
5436:p
5419:)
5413:(
5408:i
5404:p
5398:i
5390:=
5379:)
5370:k
5353:,
5345:j
5327:(
5323:h
5320:4
5317:=
5307:r
5304:e
5301:h
5298:s
5295:i
5292:f
5286:k
5283:j
5279:h
5275:4
5272:=
5269:)
5263:(
5258:k
5255:j
5251:g
5214:k
5206:d
5201:j
5193:d
5185:k
5172:)
5166:(
5161:i
5157:p
5136:j
5123:)
5117:(
5112:i
5108:p
5091:)
5085:(
5080:i
5076:p
5070:i
5060:k
5057:j
5047:4
5044:1
5039:=
5029:)
5026:)
5020:(
5015:i
5011:p
5001:(
4998:d
4994:)
4991:)
4985:(
4980:i
4976:p
4966:(
4963:d
4959:)
4953:(
4948:i
4944:p
4938:i
4928:4
4925:1
4920:=
4913:h
4886:)
4880:(
4875:i
4871:p
4867:=
4862:i
4858:p
4810:)
4805:i
4801:p
4791:(
4788:d
4784:)
4779:i
4775:p
4765:(
4762:d
4756:i
4752:p
4746:i
4736:4
4733:1
4728:=
4721:i
4717:p
4710:i
4706:p
4702:d
4696:i
4692:p
4688:d
4680:i
4670:4
4667:1
4662:=
4648:i
4644:p
4638:d
4630:i
4626:p
4620:d
4615:i
4607:=
4602:i
4598:y
4594:d
4588:i
4584:y
4580:d
4575:i
4567:=
4560:h
4530:1
4527:=
4522:i
4518:p
4512:i
4482:2
4477:i
4473:y
4469:=
4464:i
4460:p
4428:1
4425:=
4420:2
4415:i
4411:y
4405:N
4400:0
4397:=
4394:i
4376:N
4372:N
4355:y
4330:k
4327:j
4319:=
4315:)
4306:k
4302:y
4289:,
4281:j
4277:y
4263:(
4259:h
4256:=
4250:t
4247:a
4244:l
4241:f
4235:k
4232:j
4228:h
4212:,
4198:k
4195:j
4187:=
4183:)
4175:k
4171:y
4158:(
4152:j
4148:y
4144:d
4111:j
4107:y
4065:i
4061:y
4057:d
4031:i
4027:y
4023:d
4017:i
4013:y
4009:d
4004:N
3999:0
3996:=
3993:i
3985:=
3982:h
3959:)
3954:n
3950:y
3946:,
3940:,
3935:0
3931:y
3927:(
3924:=
3921:y
3909:N
3889:2
3885:R
3839:D
3836:S
3833:J
3830:d
3823:b
3818:a
3808:8
3803:=
3800:t
3797:d
3788:t
3778:k
3762:k
3759:j
3755:g
3748:t
3738:j
3720:b
3715:a
3676:D
3673:S
3670:J
3667:d
3662:b
3657:a
3649:8
3646:=
3643:t
3640:d
3633:t
3623:k
3607:k
3604:j
3600:g
3593:t
3583:j
3567:b
3562:a
3554:)
3551:a
3545:b
3542:(
3487:A
3484:)
3481:a
3475:b
3472:(
3469:=
3466:S
3450:b
3446:a
3438:t
3421:t
3418:d
3412:t
3402:k
3388:)
3382:(
3377:k
3374:j
3370:g
3363:t
3353:j
3337:b
3332:a
3322:2
3319:1
3314:=
3311:A
3250:0
3225:)
3219:(
3212:0
3203:f
3176:k
3173:j
3169:g
3156:.
3144:)
3139:3
3128:(
3124:O
3120:+
3117:)
3112:0
3104:(
3099:k
3096:j
3092:g
3086:k
3073:j
3060:k
3057:j
3047:2
3044:1
3039:=
3036:]
3033:)
3027:(
3024:P
3018:)
3013:0
3005:(
3002:P
2999:[
2993:L
2990:K
2985:D
2978:)
2972:(
2965:0
2956:f
2930:0
2922:=
2899:)
2894:0
2886:(
2883:P
2880:=
2877:)
2871:(
2868:P
2848:]
2845:)
2839:(
2836:P
2830:)
2825:0
2817:(
2814:P
2811:[
2805:L
2802:K
2797:D
2786:j
2750:j
2715:0
2705:|
2697:j
2684:P
2669:j
2656:j
2648:+
2645:)
2640:0
2632:(
2629:P
2626:=
2623:)
2617:(
2614:P
2591:)
2586:0
2578:(
2575:P
2555:)
2549:(
2546:P
2510:)
2507:2
2503:/
2496:(
2479:2
2474:=
2471:s
2446:2
2441:2
2434:0
2421:1
2410:=
2386:)
2374:2
2364:2
2356:,
2342:2
2336:0
2323:1
2312:+
2307:2
2301:1
2293:+
2288:0
2276:(
2270:N
2238:1
2225:,
2218:0
2183:2
2177:/
2122:2
2118:y
2111:2
2107:y
2103:d
2100:+
2095:2
2091:x
2087:d
2081:2
2078:=
2073:2
2069:s
2065:d
2042:=
2039:y
2036:,
2031:2
2025:/
2018:=
2015:x
1993:]
1987:2
1982:0
1975:0
1970:1
1964:[
1957:2
1946:=
1941:]
1933:1
1926:)
1920:2
1916:t
1912:2
1909:(
1904:0
1897:0
1892:t
1886:[
1881:=
1878:g
1855:T
1841:2
1836:T
1826:2
1823:1
1815:=
1810:T
1807:,
1804:T
1800:g
1777:j
1774:i
1770:T
1766:=
1759:j
1751:,
1746:i
1737:g
1716:0
1713:=
1705:,
1698:g
1672:1
1661:=
1658:T
1638:C
1635:+
1632:)
1626:(
1612:2
1609:1
1604:+
1601:)
1592:x
1589:(
1584:1
1571:T
1567:)
1557:x
1554:(
1549:2
1546:1
1541:=
1538:)
1532:,
1525:|
1521:x
1518:(
1515:p
1487:)
1481:,
1475:(
1470:N
1438:A
1433:2
1400:=
1397:)
1391:(
1364:]
1361:)
1358:x
1355:(
1352:T
1349:[
1345:E
1333:k
1319:j
1306:)
1300:(
1292:2
1273:k
1259:j
1246:)
1240:(
1237:A
1232:2
1221:=
1218:)
1212:(
1207:k
1204:j
1200:g
1177:]
1169:)
1166:x
1163:(
1160:B
1157:+
1154:)
1148:(
1145:A
1139:)
1136:x
1133:(
1130:T
1124:)
1118:(
1107:[
1098:=
1095:)
1086:x
1083:(
1080:p
1043:k
1006:0
1003:=
999:]
991:j
978:)
969:x
966:(
963:p
946:[
941:E
914:.
910:]
902:k
888:j
875:)
866:x
863:(
860:i
855:2
844:[
839:E
835:=
832:x
829:d
825:)
816:x
813:(
810:p
802:k
788:j
775:)
766:x
763:(
760:i
755:2
742:R
734:=
731:)
725:(
720:k
717:j
713:g
685:)
676:x
673:(
670:p
656:=
653:)
644:x
641:(
638:i
621:n
617:n
613:n
570:k
566:j
538:R
534:x
517:.
514:x
511:d
507:)
498:x
495:(
492:p
484:k
470:j
457:)
448:x
445:(
442:p
431:2
418:R
407:=
404:)
398:(
393:k
390:j
386:g
359:1
356:=
353:x
350:d
346:)
337:x
334:(
331:p
326:R
281:x
271:X
264:R
250:x
210:)
201:x
198:(
195:p
175:)
170:n
162:,
156:,
151:2
143:,
138:1
130:(
127:=
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