1003:
6141:
6545:
5172:
1674:, and are trained to minimize the negative log-likelihood of data samples from the target distribution. These architectures are usually designed such that only the forward pass of the neural network is required in both the inverse and the Jacobian determinant calculations. Examples of such architectures include NICE, RealNVP, and Glow.
5855:
4713:
7361:, this allows "free-form" Jacobian. Here, "free-form" means that there is no restriction on the Jacobian's form. It is contrasted with previous discrete models of normalizing flow, where the Jacobian is carefully designed to be only upper- or lower-diagonal, so that the Jacobian can be evaluated efficiently.
6275:
8073:
Despite normalizing flows success in estimating high-dimensional densities, some downsides still exist in their designs. First of all, their latent space where input data is projected onto is not a lower-dimensional space and therefore, flow-based models do not allow for compression of data by
8016:
4961:
8064:
are hyperparameters. The first term punishes the model for oscillating the flow field over time, and the second term punishes it for oscillating the flow field over space. Both terms together guide the model into a flow that is smooth (not "bumpy") over space and time.
8077:
Flow-based models are also notorious for failing in estimating the likelihood of out-of-distribution samples (i.e.: samples that were not drawn from the same distribution as the training set). Some hypotheses were formulated to explain this phenomenon, among which the
3035:
1561:
7731:
By adding extra dimensions, the CNF gains enough freedom to reverse orientation and go beyond ambient isotopy (just like how one can pick up a polygon from a desk and flip it around in 3-space, or unknot a knot in 4-space), yielding the "augmented neural ODE".
4411:
4538:
2265:
2666:
3849:
3615:
7154:
2504:
1978:
5314:
7840:
6136:{\displaystyle {\begin{aligned}x_{1}\sim &N(\mu _{1},\sigma _{1}^{2})\\x_{2}\sim &N(\mu _{2}(x_{1}),\sigma _{2}(x_{1})^{2})\\&\cdots \\x_{n}\sim &N(\mu _{n}(x_{1:n-1}),\sigma _{n}(x_{1:n-1})^{2})\\\end{aligned}}}
8093:
map. This property is given by constraints in the design of the models (cf.: RealNVP, Glow) which guarantee theoretical invertibility. The integrity of the inverse is important in order to ensure the applicability of the
6693:
7319:
3736:
3327:
6540:{\displaystyle {\begin{aligned}x_{1}=&\mu _{1}+\sigma _{1}z_{1}\\x_{2}=&\mu _{2}(x_{1})+\sigma _{2}(x_{1})z_{2}\\&\cdots \\x_{n}=&\mu _{n}(x_{1:n-1})+\sigma _{n}(x_{1:n-1})z_{n}\\\end{aligned}}}
6949:
3352:
we want the model to learn, which only leaves the expectation of the negative log-likelihood to minimize under the target distribution. This intractable term can be approximated with a Monte-Carlo method by
4021:
6263:
4214:
8151:
6197:
5745:
4892:
2059:
5388:
5167:{\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}=f_{\theta }(z)={\begin{bmatrix}z_{1}\\e^{s_{\theta }(z_{1})}\odot z_{2}\end{bmatrix}}+{\begin{bmatrix}0\\m_{\theta }(z_{1})\end{bmatrix}}}
8949:
Kumar, Manoj; Babaeizadeh, Mohammad; Erhan, Dumitru; Finn, Chelsea; Levine, Sergey; Dinh, Laurent; Kingma, Durk (2019). "VideoFlow: A Conditional Flow-Based Model for
Stochastic Video Generation".
2851:
1377:
6280:
5860:
5519:
8062:
7407:
866:
5612:
8569:
Grathwohl, Will; Chen, Ricky T. Q.; Bettencourt, Jesse; Sutskever, Ilya; Duvenaud, David (2018-10-22). "FFJORD: Free-form
Continuous Dynamics for Scalable Reversible Generative Models".
904:
4481:
3949:
2332:
4533:
1798:
8527:
Grathwohl, Will; Chen, Ricky T. Q.; Bettencourt, Jesse; Sutskever, Ilya; Duvenaud, David (2018). "FFJORD: Free-form
Continuous Dynamics for Scalable Reversible Generative Models".
7442:
2839:
7640:
5657:
1191:
7800:
2117:
7359:
4257:
3407:
2778:
2515:
861:
7762:
6758:
6593:
5848:
4797:
4249:
4053:
3747:
4130:
3451:
1668:
1616:
1270:
851:
8865:
Behrmann, Jens; Vicol, Paul; Wang, Kuan-Chieh; Grosse, Roger; Jacobsen, Jörn-Henrik (2020). "Understanding and
Mitigating Exploding Inverses in Invertible Neural Networks".
5421:
692:
8928:
Yang, Guandao; Huang, Xun; Hao, Zekun; Liu, Ming-Yu; Belongie, Serge; Hariharan, Bharath (2019). "PointFlow: 3D Point Cloud
Generation with Continuous Normalizing Flows".
5784:
1082:
6723:
4740:
4708:{\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}=f_{\theta }(z)={\begin{bmatrix}z_{1}\\z_{2}\end{bmatrix}}+{\begin{bmatrix}0\\m_{\theta }(z_{1})\end{bmatrix}}}
3074:
2102:
1308:
1129:
6829:
4440:
3443:
7547:
4760:
3900:
3350:
2729:
8907:
Shi, Chence; Xu, Minkai; Zhu, Zhaocheng; Zhang, Weinan; Zhang, Ming; Tang, Jian (2020). "GraphAF: A Flow-based
Autoregressive Model for Molecular Graph Generation".
7586:
6983:
2364:
1840:
6793:
4948:
3101:
2696:
1832:
1734:
1707:
1365:
1335:
1218:
1035:
8102:
of the map as well as sampling with the model. However, in practice this invertibility is violated and the inverse map explodes because of numerical imprecision.
7487:
4921:
899:
7831:
7726:
7706:
7667:
7177:
6972:
5804:
5441:
4086:
2352:
5179:
5813:
The idea of using the invertible 1x1 convolution is to permute all layers in general, instead of merely permuting the first and second half, as in Real NVP.
856:
707:
438:
939:
742:
8782:
Nalisnick, Eric; Matsukawa, Teh; Zhao, Yee Whye; Song, Zhao (2019). "Detecting Out-of-Distribution Inputs to Deep
Generative Models Using Typicality".
979:
The direct modeling of likelihood provides many advantages. For example, the negative log-likelihood can be directly computed and minimized as the
4135:
818:
3332:
The second term on the right-hand side of the equation corresponds to the entropy of the target distribution and is independent of the parameter
367:
8011:{\displaystyle \lambda _{K}\int _{0}^{T}\left\|f(z_{t},t)\right\|^{2}dt+\lambda _{J}\int _{0}^{T}\left\|\nabla _{z}f(z_{t},t)\right\|_{F}^{2}dt}
6604:
7185:
3626:
8716:
Zhang, Han; Gao, Xi; Unterman, Jacob; Arodz, Tom (2019-07-30). "Approximation
Capabilities of Neural ODEs and Invertible Residual Networks".
3109:
8970:
Rudolph, Marco; Wandt, Bastian; Rosenhahn, Bodo (2021). "Same Same But DifferNet: Semi-Supervised Defect
Detection with Normalizing Flows".
876:
639:
174:
8740:
Helminger, Leonhard; Djelouah, Abdelaziz; Gross, Markus; Schroers, Christopher (2020). "Lossy Image
Compression with Normalizing Flows".
8662:
894:
8548:
Lipman, Yaron; Chen, Ricky T. Q.; Ben-Hamu, Heli; Nickel, Maximilian; Le, Matt (2022-10-01). "Flow Matching for Generative Modeling".
6837:
5665:
727:
702:
651:
3957:
775:
770:
423:
6202:
6768:
Instead of constructing flow by function composition, another approach is to formulate the flow as a continuous-time dynamic. Let
5536:
6146:
433:
71:
8694:
8490:
8457:
4802:
3855:
3046:
983:. Additionally, novel samples can be generated by sampling from the initial distribution, and applying the flow transformation.
8099:
7807:
2066:
1986:
8886:
Ping, Wei; Peng, Kainan; Gorur, Dilan; Lakshminarayanan, Balaji (2019). "WaveFlow: A Compact Flow-based Model for Raw Audio".
828:
5319:
3030:{\displaystyle \log p_{K}(z_{K})=\log p_{0}(z_{0})-\sum _{i=1}^{K}\log \left|\det {\frac {df_{i}(z_{i-1})}{dz_{i-1}}}\right|}
1556:{\displaystyle \log p_{K}(z_{K})=\log p_{0}(z_{0})-\sum _{i=1}^{K}\log \left|\det {\frac {df_{i}(z_{i-1})}{dz_{i-1}}}\right|}
991:
932:
592:
413:
8761:
Nalisnick, Eric; Matsukawa, Teh; Zhao, Yee Whye; Song, Zhao (2018). "Do Deep Generative Models Know What They Don't Know?".
7684:(for example, it's impossible to flip a left-hand to a right-hand by continuous deforming of space, and it's impossible to
9024:
3859:
803:
505:
281:
8994:
8844:
Caterini, Anthony L.; Loaiza-Ganem, Gabriel (2022). "Entropic Issues in Likelihood-Based OOD Detection". pp. 21–26.
8337:
Papamakarios, George; Nalisnick, Eric; Rezende, Danilo Jimenez; Shakir, Mohamed; Balaji, Lakshminarayanan (March 2021).
8095:
8082:
hypothesis, estimation issues when training models, or fundamental issues due to the entropy of the data distributions.
5446:
1804:
973:
760:
697:
607:
585:
428:
418:
9019:
911:
823:
808:
269:
91:
798:
8021:
7371:
871:
548:
443:
231:
164:
124:
8230:
Papamakarios, George; Nalisnick, Eric; Jimenez Rezende, Danilo; Mohamed, Shakir; Bakshminarayanan, Balaji (2021).
9014:
7803:
6598:
The forward mapping is slow (because it's sequential), but the backward mapping is fast (because it's parallel).
925:
531:
299:
169:
8074:
default and require a lot of computation. However, it is still possible to perform image compression with them.
1618:
should be 1. easy to invert, and 2. easy to compute the determinant of its Jacobian. In practice, the functions
6760:
of MAF results in Inverse Autoregressive Flow (IAF), which has fast forward mapping and slow backward mapping.
4445:
3905:
2273:
2108:
965:
553:
473:
396:
314:
144:
106:
101:
61:
56:
4494:
6269:
3045:
As is generally done when training a deep learning model, the goal with normalizing flows is to minimize the
1739:
8266:
Dinh, Laurent; Krueger, David; Bengio, Yoshua (2014). "NICE: Non-linear Independent Components Estimation".
7643:
2260:{\displaystyle p_{1}(z_{1})=p_{0}(z_{0})\left|\det \left({\frac {df_{1}(z_{0})}{dz_{0}}}\right)^{-1}\right|}
500:
349:
249:
76:
7412:
2783:
8316:
Kingma, Diederik P.; Dhariwal, Prafulla (2018). "Glow: Generative Flow with Invertible 1x1 Convolutions".
7591:
5617:
4406:{\displaystyle |\det(I+h'(\langle w,z\rangle +b)uw^{T})|=|1+h'(\langle w,z\rangle +b)\langle u,w\rangle |}
2842:
2661:{\displaystyle \log p_{1}(z_{1})=\log p_{0}(z_{0})-\log \left|\det {\frac {df_{1}(z_{0})}{dz_{0}}}\right|}
1134:
680:
656:
558:
319:
294:
254:
66:
7767:
3844:{\displaystyle {\underset {\theta }{\operatorname {arg\,max} }}\ \sum _{i=0}^{N}\log(p_{\theta }(x_{i}))}
7327:
3610:{\displaystyle -{\hat {\mathbb {E} }}_{p^{*}(x)}=-{\frac {1}{N}}\sum _{i=0}^{N}\log(p_{\theta }(x_{i}))}
634:
456:
408:
264:
179:
51:
7833:, one can impose regularization losses. The paper proposed the following regularization loss based on
3360:
2734:
7738:
6728:
6550:
5824:
4767:
4219:
4029:
4091:
1621:
1569:
1223:
563:
513:
7673:
methods would be needed. Indeed, CNF was first proposed in the same paper that proposed neural ODE.
5393:
3354:
666:
602:
573:
478:
304:
237:
223:
209:
184:
134:
86:
46:
8489:
Kingma, Durk P; Salimans, Tim; Jozefowicz, Rafal; Chen, Xi; Sutskever, Ilya; Welling, Max (2016).
5750:
1044:
8971:
8950:
8929:
8908:
8887:
8866:
8845:
8783:
8762:
8741:
8717:
8673:
8668:. In Bengio, S.; Wallach, H.; Larochelle, H.; Grauman, K.; Cesa-Bianchi, N.; Garnett, R. (eds.).
8602:
8570:
8549:
8528:
8502:
8469:
8436:
8417:
8383:
8350:
8317:
8291:
8290:
Dinh, Laurent; Sohl-Dickstein, Jascha; Bengio, Samy (2016). "Density estimation using Real NVP".
8267:
8243:
8212:
8079:
6701:
4718:
3052:
644:
568:
354:
149:
8435:
Danilo Jimenez Rezende; Mohamed, Shakir (2015). "Variational Inference with Normalizing Flows".
2072:
1278:
8623:
7149:{\displaystyle z_{0}=F^{-1}(x)=z_{T}+\int _{T}^{0}f(z_{t},t)dt=z_{T}-\int _{0}^{T}f(z_{t},t)dt}
2499:{\displaystyle p_{1}(z_{1})=p_{0}(z_{0})\left|\det {\frac {df_{1}(z_{0})}{dz_{0}}}\right|^{-1}}
1973:{\displaystyle p_{1}(z_{1})=p_{0}(z_{0})\left|\det {\frac {df_{1}^{-1}(z_{1})}{dz_{1}}}\right|}
1090:
8826:
8643:
8409:
8401:
7834:
6798:
4419:
3412:
2355:
737:
580:
493:
289:
259:
204:
199:
154:
96:
8624:"A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines"
7492:
4745:
3872:
3335:
2701:
8816:
8635:
8393:
8202:
8194:
8182:
8163:
8086:
7556:
1273:
961:
957:
765:
518:
468:
378:
362:
332:
194:
189:
139:
129:
27:
7708:
might be ill-behaved, due to degeneracy (that is, there are an infinite number of possible
6771:
4926:
3079:
2674:
1810:
1712:
1685:
1343:
1313:
1196:
1013:
8110:
Flow-based generative models have been applied on a variety of modeling tasks, including:
7685:
7681:
4535:
be even-dimensional, and split them in the middle. Then the normalizing flow functions are
1038:
793:
597:
463:
403:
7447:
5309:{\displaystyle z_{1}=x_{1},z_{2}=e^{-s_{\theta }(x_{1})}\odot (x_{2}-m_{\theta }(x_{1}))}
4900:
8821:
7816:
7711:
7691:
7652:
7162:
6957:
5789:
5426:
4071:
2337:
813:
344:
81:
9008:
8999:
8805:"Understanding Failures in Out-of-Distribution Detection with Deep Generative Models"
8593:
Finlay, Chris; Jacobsen, Joern-Henrik; Nurbekyan, Levon; Oberman, Adam (2020-11-21).
8421:
8231:
7677:
3049:
between the model's likelihood and the target distribution to be estimated. Denoting
1671:
980:
732:
661:
543:
274:
159:
8594:
8216:
8804:
8661:
Chen, Ricky T. Q.; Rubanova, Yulia; Bettencourt, Jesse; Duvenaud, David K. (2018).
8595:"How to Train Your Neural ODE: the World of Jacobian and Kinetic Regularization"
7676:
There are two main deficiencies of CNF, one is that a continuous flow must be a
4416:
For it to be invertible everywhere, it must be nonzero everywhere. For example,
2062:
987:
538:
32:
8397:
8371:
8167:
1002:
8639:
7670:
687:
383:
309:
8647:
8405:
8090:
3858:
between the model's likelihood and the target distribution is equivalent to
976:
law of probabilities to transform a simple distribution into a complex one.
846:
627:
8830:
8413:
8338:
6831:. Map this latent variable to data space with the following flow function:
6688:{\displaystyle \sigma _{1}\sigma _{2}(x_{1})\cdots \sigma _{n}(x_{1:n-1})}
7314:{\displaystyle \log(p(x))=\log(p(z_{0}))-\int _{0}^{T}{\text{Tr}}\leftdt}
5423:. Since the Real NVP map keeps the first and second halves of the vector
8370:
Kobyzev, Ivan; Prince, Simon J.D.; Brubaker, Marcus A. (November 2021).
3731:{\displaystyle {\underset {\theta }{\operatorname {arg\,min} }}\ D_{KL}}
6974:
is an arbitrary function and can be modeled with e.g. neural networks.
3322:{\displaystyle D_{KL}=-\mathbb {E} _{p^{*}(x)}+\mathbb {E} _{p^{*}(x)}}
622:
8735:
8733:
8207:
8198:
4894:, and the Jacobian is just 1, that is, the flow is volume-preserving.
8767:
373:
8976:
8955:
8934:
8913:
8892:
8871:
8850:
8788:
8746:
8722:
8678:
8607:
8575:
8554:
8533:
8507:
8474:
8441:
8388:
8355:
8322:
8296:
8248:
8085:
One of the most interesting properties of normalizing flows is the
8372:"Normalizing Flows: An Introduction and Review of Current Methods"
8272:
3103:
the target distribution to learn, the (forward) KL-divergence is:
1001:
986:
In contrast, many alternative generative modeling methods such as
617:
612:
339:
8491:"Improved Variational Inference with Inverse Autoregressive Flow"
3409:
of samples each independently drawn from the target distribution
6944:{\displaystyle x=F(z_{0})=z_{T}=z_{0}+\int _{0}^{T}f(z_{t},t)dt}
6272:, the autoregressive model is generalized to a normalizing flow:
4958:
The Real Non-Volume Preserving model generalizes NICE model by:
8376:
IEEE Transactions on Pattern Analysis and Machine Intelligence
4016:{\displaystyle \max _{\theta }\sum _{j}\ln p_{\theta }(x_{j})}
7324:
Since the trace depends only on the diagonal of the Jacobian
6258:{\displaystyle \sigma _{i}:\mathbb {R} ^{i-1}\to (0,\infty )}
8339:"Normalizing Flows for Probabilistic Modeling and Inference"
8232:"Normalizing flows for probabilistic modeling and inference"
4209:{\displaystyle x=f_{\theta }(z)=z+uh(\langle w,z\rangle +b)}
8456:
Papamakarios, George; Pavlakou, Theo; Murray, Iain (2017).
6192:{\displaystyle \mu _{i}:\mathbb {R} ^{i-1}\to \mathbb {R} }
3865:
A pseudocode for training normalizing flows is as follows:
7688:, or undo a knot), and the other is that the learned flow
6601:
The Jacobian matrix is lower-diagonal, so the Jacobian is
6265:
are fixed functions that define the autoregressive model.
4887:{\displaystyle z_{1}=x_{1},z_{2}=x_{2}-m_{\theta }(x_{1})}
8628:
Communications in Statistics - Simulation and Computation
8183:"A family of nonparametric density estimation algorithms"
8152:"Density estimation by dual ascent of the log-likelihood"
2054:{\displaystyle \det {\frac {df_{1}^{-1}(z_{1})}{dz_{1}}}}
1566:
To efficiently compute the log likelihood, the functions
8803:
Zhang, Lily; Goldstein, Mark; Ranganath, Rajesh (2021).
905:
List of datasets in computer vision and image processing
8693:
Dupont, Emilien; Doucet, Arnaud; Teh, Yee Whye (2019).
5383:{\displaystyle \prod _{i=1}^{n}e^{s_{\theta }(z_{1,})}}
5120:
5041:
4976:
4661:
4618:
4553:
8024:
7843:
7819:
7770:
7741:
7714:
7694:
7655:
7594:
7559:
7495:
7450:
7415:
7374:
7330:
7188:
7165:
6986:
6960:
6840:
6801:
6774:
6731:
6704:
6607:
6553:
6278:
6205:
6149:
5858:
5827:
5792:
5753:
5668:
5620:
5539:
5449:
5443:
separate, it's usually required to add a permutation
5429:
5396:
5322:
5182:
4964:
4929:
4903:
4805:
4770:
4748:
4721:
4541:
4497:
4448:
4422:
4260:
4222:
4138:
4094:
4074:
4032:
3960:
3908:
3875:
3750:
3629:
3454:
3415:
3363:
3338:
3112:
3082:
3055:
2854:
2786:
2737:
2704:
2677:
2518:
2367:
2340:
2276:
2120:
2075:
1989:
1843:
1813:
1742:
1715:
1688:
1624:
1572:
1380:
1346:
1316:
1281:
1226:
1199:
1137:
1093:
1047:
1016:
994:
do not explicitly represent the likelihood function.
2841:
subtracted by a non-recursive term, we can infer by
8588:
8586:
8458:"Masked Autoregressive Flow for Density Estimation"
7364:
The trace can be estimated by "Hutchinson's trick":
4068:The earliest example. Fix some activation function
3862:under observed samples of the target distribution.
1193:be a sequence of random variables transformed from
8056:
8010:
7825:
7794:
7756:
7720:
7700:
7661:
7634:
7580:
7541:
7481:
7436:
7401:
7353:
7313:
7171:
7148:
6966:
6943:
6823:
6787:
6752:
6717:
6687:
6587:
6539:
6257:
6191:
6135:
5842:
5798:
5778:
5739:
5651:
5606:
5529:In generative flow model, each layer has 3 parts:
5514:{\displaystyle (x_{1},x_{2})\mapsto (x_{2},x_{1})}
5513:
5435:
5415:
5382:
5308:
5166:
4942:
4915:
4886:
4791:
4754:
4734:
4707:
4527:
4487:Nonlinear Independent Components Estimation (NICE)
4475:
4434:
4405:
4243:
4208:
4124:
4080:
4047:
4015:
3943:
3894:
3843:
3730:
3609:
3437:
3401:
3344:
3321:
3095:
3068:
3029:
2833:
2772:
2723:
2690:
2660:
2498:
2346:
2326:
2259:
2096:
2053:
1972:
1826:
1792:
1728:
1701:
1662:
1610:
1555:
1359:
1329:
1302:
1264:
1212:
1185:
1123:
1076:
1029:
8699:Advances in Neural Information Processing Systems
8670:Advances in Neural Information Processing Systems
8495:Advances in Neural Information Processing Systems
8462:Advances in Neural Information Processing Systems
7764:can be approximated by a neural ODE operating on
6547:The autoregressive model is recovered by setting
5809:Real NVP, with Jacobian as described in Real NVP.
1368:
8150:Tabak, Esteban G.; Vanden-Eijnden, Eric (2010).
5850:is defined as the following stochastic process:
5754:
5740:{\displaystyle z_{cij}=\sum _{c'}K_{cc'}y_{cij}}
4266:
3962:
2957:
2600:
2429:
2302:
2277:
2181:
1990:
1904:
1483:
8181:Tabak, Esteban G.; Turner, Cristina V. (2012).
8187:Communications on Pure and Applied Mathematics
8057:{\displaystyle \lambda _{K},\lambda _{J}>0}
900:List of datasets for machine-learning research
7402:{\displaystyle W\in \mathbb {R} ^{n\times n}}
5821:An autoregressive model of a distribution on
4923:, this is seen as a curvy shearing along the
933:
8:
8599:International Conference on Machine Learning
7623:
7595:
5607:{\displaystyle y_{cij}=s_{c}(x_{cij}+b_{c})}
4461:
4449:
4395:
4383:
4371:
4359:
4301:
4289:
4194:
4182:
3378:
3364:
7553:Usually, the random vector is sampled from
7549:. (Proof: expand the expectation directly.)
972:, which is a statistical method using the
940:
926:
18:
16:Statistical model used in machine learning
8975:
8954:
8933:
8912:
8891:
8870:
8849:
8820:
8787:
8766:
8745:
8721:
8677:
8606:
8574:
8553:
8532:
8522:
8520:
8518:
8506:
8473:
8440:
8387:
8354:
8321:
8295:
8271:
8247:
8206:
8042:
8029:
8023:
7996:
7991:
7971:
7955:
7939:
7934:
7924:
7905:
7885:
7863:
7858:
7848:
7842:
7818:
7777:
7773:
7772:
7769:
7748:
7744:
7743:
7740:
7713:
7693:
7654:
7626:
7612:
7605:
7593:
7558:
7506:
7494:
7464:
7449:
7428:
7424:
7423:
7414:
7387:
7383:
7382:
7373:
7340:
7335:
7329:
7292:
7274:
7265:
7259:
7254:
7235:
7187:
7164:
7125:
7109:
7104:
7091:
7063:
7047:
7042:
7029:
7004:
6991:
6985:
6959:
6920:
6904:
6899:
6886:
6873:
6857:
6839:
6812:
6800:
6795:be the latent variable with distribution
6779:
6773:
6741:
6736:
6730:
6709:
6703:
6664:
6651:
6635:
6622:
6612:
6606:
6576:
6552:
6527:
6502:
6489:
6461:
6448:
6433:
6409:
6396:
6383:
6367:
6354:
6339:
6325:
6315:
6302:
6287:
6279:
6277:
6225:
6221:
6220:
6210:
6204:
6185:
6184:
6169:
6165:
6164:
6154:
6148:
6120:
6098:
6085:
6057:
6044:
6023:
5996:
5986:
5973:
5957:
5944:
5923:
5906:
5901:
5888:
5867:
5859:
5857:
5834:
5830:
5829:
5826:
5791:
5767:
5752:
5725:
5707:
5692:
5673:
5667:
5640:
5635:
5625:
5619:
5595:
5576:
5563:
5544:
5538:
5502:
5489:
5470:
5457:
5448:
5428:
5401:
5395:
5390:. The NICE model is recovered by setting
5366:
5353:
5348:
5338:
5327:
5321:
5294:
5281:
5268:
5247:
5234:
5226:
5213:
5200:
5187:
5181:
5147:
5134:
5115:
5098:
5080:
5067:
5062:
5048:
5036:
5018:
4997:
4983:
4971:
4963:
4934:
4928:
4902:
4875:
4862:
4849:
4836:
4823:
4810:
4804:
4780:
4775:
4769:
4747:
4726:
4720:
4688:
4675:
4656:
4639:
4625:
4613:
4595:
4574:
4560:
4548:
4540:
4516:
4512:
4511:
4496:
4476:{\displaystyle \langle u,w\rangle >-1}
4447:
4421:
4398:
4337:
4329:
4320:
4261:
4259:
4232:
4227:
4221:
4149:
4137:
4093:
4073:
4034:
4033:
4031:
4004:
3991:
3975:
3965:
3959:
3944:{\displaystyle f_{\theta }(\cdot ),p_{0}}
3935:
3913:
3907:
3880:
3874:
3829:
3816:
3797:
3786:
3763:
3753:
3751:
3749:
3710:
3701:
3696:
3681:
3665:
3642:
3632:
3630:
3628:
3595:
3582:
3563:
3552:
3538:
3511:
3478:
3473:
3463:
3462:
3460:
3459:
3453:
3420:
3414:
3381:
3371:
3362:
3337:
3298:
3265:
3260:
3256:
3255:
3230:
3197:
3192:
3188:
3187:
3162:
3153:
3148:
3133:
3117:
3111:
3087:
3081:
3060:
3054:
3007:
2983:
2970:
2960:
2940:
2929:
2913:
2900:
2878:
2865:
2853:
2816:
2797:
2785:
2761:
2748:
2736:
2709:
2703:
2682:
2676:
2644:
2626:
2613:
2603:
2577:
2564:
2542:
2529:
2517:
2487:
2473:
2455:
2442:
2432:
2414:
2401:
2385:
2372:
2366:
2339:
2327:{\displaystyle \det(A^{-1})=\det(A)^{-1}}
2315:
2287:
2275:
2243:
2230:
2212:
2199:
2189:
2167:
2154:
2138:
2125:
2119:
2085:
2080:
2074:
2042:
2024:
2008:
2003:
1993:
1988:
1956:
1938:
1922:
1917:
1907:
1890:
1877:
1861:
1848:
1842:
1818:
1812:
1781:
1765:
1760:
1747:
1741:
1720:
1714:
1693:
1687:
1654:
1629:
1623:
1602:
1577:
1571:
1533:
1509:
1496:
1486:
1466:
1455:
1439:
1426:
1404:
1391:
1379:
1351:
1345:
1321:
1315:
1291:
1286:
1280:
1256:
1231:
1225:
1204:
1198:
1168:
1155:
1142:
1136:
1092:
1065:
1052:
1046:
1021:
1015:
8809:Proceedings of Machine Learning Research
8672:. Vol. 31. Curran Associates, Inc.
8663:"Neural Ordinary Differential Equations"
6977:The inverse function is then naturally:
4528:{\displaystyle x,z\in \mathbb {R} ^{2n}}
4251:has no closed-form solution in general.
8156:Communications in Mathematical Sciences
8142:
1793:{\displaystyle z_{0}=f_{1}^{-1}(z_{1})}
26:
8311:
8309:
8307:
3445:, then this term can be estimated as:
7437:{\displaystyle u\in \mathbb {R} ^{n}}
4954:Real Non-Volume Preserving (Real NVP)
4132:with the appropriate dimensions, then
2834:{\displaystyle \log p_{i-1}(z_{i-1})}
2671:In general, the above applies to any
7:
8343:Journal of Machine Learning Research
8285:
8283:
8261:
8259:
8236:Journal of Machine Learning Research
7669:is implemented as a neural network,
7635:{\displaystyle \{\pm n^{-1/2}\}^{n}}
5652:{\displaystyle \prod _{c}s_{c}^{HW}}
1186:{\displaystyle z_{i}=f_{i}(z_{i-1})}
7795:{\displaystyle \mathbb {R} ^{2n+1}}
4742:is any neural network with weights
895:Glossary of artificial intelligence
7952:
7728:that all solve the same problem).
7354:{\displaystyle \partial _{z_{t}}f}
7332:
7285:
7277:
6249:
3770:
3767:
3764:
3760:
3757:
3754:
3649:
3646:
3643:
3639:
3636:
3633:
3620:Therefore, the learning objective
14:
8995:Flow-based Deep Generative Models
8622:Hutchinson, M.F. (January 1989).
6764:Continuous Normalizing Flow (CNF)
3402:{\displaystyle \{x_{i}\}_{i=1:N}}
2773:{\displaystyle \log p_{i}(z_{i})}
7757:{\displaystyle \mathbb {R} ^{n}}
7680:, thus preserve orientation and
6753:{\displaystyle f_{\theta }^{-1}}
6588:{\displaystyle z\sim N(0,I_{n})}
5843:{\displaystyle \mathbb {R} ^{n}}
5817:Masked autoregressive flow (MAF)
4792:{\displaystyle f_{\theta }^{-1}}
4244:{\displaystyle f_{\theta }^{-1}}
4048:{\displaystyle {\hat {\theta }}}
1337:models the target distribution.
7808:universal approximation theorem
4125:{\displaystyle \theta =(u,w,b)}
3860:maximizing the model likelihood
3854:In other words, minimizing the
3357:. Indeed, if we have a dataset
1663:{\displaystyle f_{1},...,f_{K}}
1611:{\displaystyle f_{1},...,f_{K}}
1272:should be invertible, i.e. the
1265:{\displaystyle f_{1},...,f_{K}}
7987:
7983:
7964:
7947:
7901:
7897:
7878:
7871:
7575:
7563:
7536:
7530:
7518:
7499:
7470:
7454:
7244:
7241:
7228:
7222:
7210:
7207:
7201:
7195:
7137:
7118:
7075:
7056:
7019:
7013:
6932:
6913:
6863:
6850:
6818:
6805:
6682:
6657:
6641:
6628:
6582:
6563:
6520:
6495:
6479:
6454:
6402:
6389:
6373:
6360:
6252:
6240:
6237:
6181:
6126:
6117:
6091:
6075:
6050:
6037:
6002:
5993:
5979:
5963:
5950:
5937:
5912:
5881:
5764:
5757:
5601:
5569:
5508:
5482:
5479:
5476:
5450:
5375:
5359:
5303:
5300:
5287:
5261:
5253:
5240:
5153:
5140:
5086:
5073:
5030:
5024:
4881:
4868:
4694:
4681:
4607:
4601:
4399:
4380:
4356:
4338:
4330:
4326:
4310:
4286:
4269:
4262:
4203:
4179:
4161:
4155:
4119:
4101:
4039:
4010:
3997:
3925:
3919:
3838:
3835:
3822:
3809:
3725:
3722:
3716:
3702:
3697:
3693:
3687:
3674:
3604:
3601:
3588:
3575:
3529:
3526:
3523:
3517:
3504:
3495:
3490:
3484:
3467:
3432:
3426:
3316:
3313:
3310:
3304:
3291:
3282:
3277:
3271:
3248:
3245:
3242:
3236:
3223:
3214:
3209:
3203:
3177:
3174:
3168:
3154:
3149:
3145:
3139:
3126:
2995:
2976:
2919:
2906:
2884:
2871:
2828:
2809:
2767:
2754:
2632:
2619:
2583:
2570:
2548:
2535:
2461:
2448:
2420:
2407:
2391:
2378:
2312:
2305:
2296:
2280:
2218:
2205:
2173:
2160:
2144:
2131:
2030:
2017:
1944:
1931:
1896:
1883:
1867:
1854:
1787:
1774:
1521:
1502:
1445:
1432:
1410:
1397:
1180:
1161:
1071:
1058:
992:generative adversarial network
315:Relevance vector machine (RVM)
1:
5533:channel-wise affine transform
5416:{\displaystyle s_{\theta }=0}
1807:formula, the distribution of
1037:be a (possibly multivariate)
988:variational autoencoder (VAE)
804:Computational learning theory
368:Expectation–maximization (EM)
5779:{\displaystyle \det(K)^{HW}}
5521:after every Real NVP layer.
2509:The log likelihood is thus:
1678:Derivation of log likelihood
1077:{\displaystyle p_{0}(z_{0})}
1006:Scheme for normalizing flows
761:Coefficient of determination
608:Convolutional neural network
320:Support vector machine (SVM)
6718:{\displaystyle f_{\theta }}
4735:{\displaystyle m_{\theta }}
4483:satisfies the requirement.
3856:Kullback–Leibler divergence
3076:the model's likelihood and
3069:{\displaystyle p_{\theta }}
3047:Kullback–Leibler divergence
954:flow-based generative model
912:Outline of machine learning
809:Empirical risk minimization
9041:
8501:. Curran Associates, Inc.
8468:. Curran Associates, Inc.
8398:10.1109/TPAMI.2020.2992934
8168:10.4310/CMS.2010.v8.n1.a11
8120:Molecular graph generation
8096:change-of-variable theorem
7159:And the log-likelihood of
5662:invertible 1x1 convolution
2097:{\displaystyle f_{1}^{-1}}
1303:{\displaystyle f_{i}^{-1}}
549:Feedforward neural network
300:Artificial neural networks
8705:. Curran Associates, Inc.
8640:10.1080/03610918908812806
8098:, the computation of the
7804:Whitney embedding theorem
7588:(normal distribution) or
5806:is any invertible matrix.
3902:, normalizing flow model
1310:exists. The final output
1124:{\displaystyle i=1,...,K}
964:that explicitly models a
532:Artificial neural network
7835:optimal transport theory
7686:turn a sphere inside out
6824:{\displaystyle p(z_{0})}
4435:{\displaystyle h=\tanh }
3438:{\displaystyle p^{*}(x)}
2109:inverse function theorem
966:probability distribution
841:Journals and conferences
788:Mathematical foundations
698:Temporal difference (TD)
554:Recurrent neural network
474:Conditional random field
397:Dimensionality reduction
145:Dimensionality reduction
107:Quantum machine learning
102:Neuromorphic engineering
62:Self-supervised learning
57:Semi-supervised learning
9000:Normalizing flow models
8695:"Augmented Neural ODEs"
8129:Lossy image compression
7813:To regularize the flow
7644:Radamacher distribution
7542:{\displaystyle E=tr(W)}
6698:Reversing the two maps
6270:reparametrization trick
4755:{\displaystyle \theta }
3895:{\displaystyle x_{1:n}}
3345:{\displaystyle \theta }
2724:{\displaystyle z_{i-1}}
250:Apprenticeship learning
8058:
8012:
7827:
7806:for manifolds and the
7802:, proved by combining
7796:
7758:
7722:
7702:
7663:
7636:
7582:
7581:{\displaystyle N(0,I)}
7551:
7543:
7483:
7438:
7403:
7355:
7315:
7173:
7150:
6968:
6945:
6825:
6789:
6754:
6719:
6689:
6589:
6541:
6259:
6193:
6137:
5844:
5800:
5780:
5741:
5653:
5608:
5525:Generative Flow (Glow)
5515:
5437:
5417:
5384:
5343:
5316:, and its Jacobian is
5310:
5168:
4944:
4917:
4888:
4793:
4756:
4736:
4709:
4529:
4477:
4436:
4407:
4245:
4210:
4126:
4082:
4049:
4017:
3945:
3896:
3845:
3802:
3732:
3611:
3568:
3439:
3403:
3346:
3323:
3097:
3070:
3031:
2945:
2835:
2774:
2725:
2692:
2662:
2500:
2348:
2328:
2261:
2098:
2055:
1974:
1828:
1794:
1730:
1703:
1664:
1612:
1557:
1471:
1361:
1340:The log likelihood of
1331:
1304:
1266:
1214:
1187:
1125:
1078:
1031:
1007:
799:Bias–variance tradeoff
681:Reinforcement learning
657:Spiking neural network
67:Reinforcement learning
8059:
8013:
7828:
7810:for neural networks.
7797:
7759:
7735:Any homeomorphism of
7723:
7703:
7664:
7637:
7583:
7544:
7484:
7439:
7404:
7366:
7356:
7316:
7174:
7151:
6969:
6946:
6826:
6790:
6788:{\displaystyle z_{0}}
6755:
6720:
6690:
6590:
6542:
6260:
6194:
6138:
5845:
5801:
5781:
5742:
5654:
5609:
5516:
5438:
5418:
5385:
5323:
5311:
5169:
4945:
4943:{\displaystyle x_{2}}
4918:
4889:
4794:
4757:
4737:
4710:
4530:
4478:
4437:
4408:
4246:
4211:
4127:
4083:
4050:
4018:
3946:
3897:
3846:
3782:
3733:
3612:
3548:
3440:
3404:
3347:
3324:
3098:
3096:{\displaystyle p^{*}}
3071:
3032:
2925:
2836:
2775:
2726:
2693:
2691:{\displaystyle z_{i}}
2663:
2501:
2349:
2329:
2262:
2099:
2056:
1975:
1829:
1827:{\displaystyle z_{1}}
1795:
1731:
1729:{\displaystyle z_{0}}
1704:
1702:{\displaystyle z_{1}}
1665:
1613:
1558:
1451:
1362:
1360:{\displaystyle z_{K}}
1332:
1330:{\displaystyle z_{K}}
1305:
1267:
1215:
1213:{\displaystyle z_{0}}
1188:
1126:
1079:
1032:
1030:{\displaystyle z_{0}}
1005:
635:Neural radiance field
457:Structured prediction
180:Structured prediction
52:Unsupervised learning
9025:Probabilistic models
8123:Point-cloud modeling
8022:
7841:
7817:
7768:
7739:
7712:
7692:
7653:
7592:
7557:
7493:
7448:
7413:
7372:
7328:
7186:
7163:
6984:
6958:
6838:
6799:
6772:
6729:
6702:
6605:
6551:
6276:
6203:
6147:
5856:
5825:
5790:
5751:
5666:
5618:
5537:
5447:
5427:
5394:
5320:
5180:
4962:
4927:
4901:
4803:
4768:
4746:
4719:
4539:
4495:
4446:
4420:
4258:
4220:
4136:
4092:
4072:
4030:
3958:
3906:
3873:
3748:
3627:
3452:
3413:
3361:
3336:
3110:
3080:
3053:
2852:
2784:
2735:
2702:
2675:
2516:
2365:
2338:
2274:
2118:
2073:
1987:
1841:
1811:
1740:
1713:
1686:
1672:deep neural networks
1622:
1570:
1378:
1344:
1314:
1279:
1224:
1197:
1135:
1091:
1045:
1014:
824:Statistical learning
722:Learning with humans
514:Local outlier factor
8601:. PMLR: 3154–3164.
8001:
7944:
7868:
7482:{\displaystyle E=I}
7264:
7114:
7052:
6909:
6749:
5911:
5648:
4916:{\displaystyle n=1}
4788:
4240:
4023:by gradient descent
3355:importance sampling
2093:
2016:
1930:
1773:
1299:
667:Electrochemical RAM
574:reservoir computing
305:Logistic regression
224:Supervised learning
210:Multimodal learning
185:Feature engineering
130:Generative modeling
92:Rule-based learning
87:Curriculum learning
47:Supervised learning
22:Part of a series on
9020:Statistical models
8054:
8008:
7945:
7930:
7854:
7823:
7792:
7754:
7718:
7698:
7659:
7632:
7578:
7539:
7479:
7434:
7399:
7351:
7311:
7250:
7169:
7146:
7100:
7038:
6964:
6941:
6895:
6821:
6785:
6750:
6732:
6715:
6685:
6585:
6537:
6535:
6255:
6189:
6133:
6131:
5897:
5840:
5796:
5776:
5737:
5702:
5649:
5631:
5630:
5604:
5511:
5433:
5413:
5380:
5306:
5164:
5158:
5106:
5005:
4940:
4913:
4884:
4789:
4771:
4752:
4732:
4705:
4699:
4647:
4582:
4525:
4473:
4432:
4403:
4241:
4223:
4206:
4122:
4078:
4045:
4013:
3980:
3970:
3941:
3892:
3841:
3777:
3728:
3656:
3607:
3435:
3399:
3342:
3319:
3093:
3066:
3027:
2831:
2770:
2721:
2688:
2658:
2496:
2344:
2324:
2257:
2094:
2076:
2051:
1999:
1970:
1913:
1824:
1805:change of variable
1790:
1756:
1726:
1699:
1670:are modeled using
1660:
1608:
1553:
1357:
1327:
1300:
1282:
1262:
1210:
1183:
1121:
1074:
1041:with distribution
1027:
1008:
974:change-of-variable
235: •
150:Density estimation
8382:(11): 3964–3979.
8199:10.1002/cpa.21423
8132:Anomaly detection
8089:of their learned
7826:{\displaystyle f}
7721:{\displaystyle f}
7701:{\displaystyle f}
7662:{\displaystyle f}
7409:, and any random
7368:Given any matrix
7299:
7268:
7179:can be found as:
7172:{\displaystyle x}
6967:{\displaystyle f}
5799:{\displaystyle K}
5688:
5621:
5436:{\displaystyle x}
4081:{\displaystyle h}
4042:
3971:
3961:
3781:
3752:
3660:
3631:
3546:
3470:
3020:
2651:
2480:
2356:invertible matrix
2347:{\displaystyle A}
2237:
2049:
1963:
1546:
950:
949:
755:Model diagnostics
738:Human-in-the-loop
581:Boltzmann machine
494:Anomaly detection
290:Linear regression
205:Ontology learning
200:Grammar induction
175:Semantic analysis
170:Association rules
155:Anomaly detection
97:Neuro-symbolic AI
9032:
9015:Machine learning
8982:
8981:
8979:
8967:
8961:
8960:
8958:
8946:
8940:
8939:
8937:
8925:
8919:
8918:
8916:
8904:
8898:
8897:
8895:
8883:
8877:
8876:
8874:
8862:
8856:
8855:
8853:
8841:
8835:
8834:
8824:
8800:
8794:
8793:
8791:
8779:
8773:
8772:
8770:
8758:
8752:
8751:
8749:
8737:
8728:
8727:
8725:
8713:
8707:
8706:
8690:
8684:
8683:
8681:
8667:
8658:
8652:
8651:
8634:(3): 1059–1076.
8619:
8613:
8612:
8610:
8590:
8581:
8580:
8578:
8566:
8560:
8559:
8557:
8545:
8539:
8538:
8536:
8524:
8513:
8512:
8510:
8486:
8480:
8479:
8477:
8453:
8447:
8446:
8444:
8432:
8426:
8425:
8391:
8367:
8361:
8360:
8358:
8334:
8328:
8327:
8325:
8313:
8302:
8301:
8299:
8287:
8278:
8277:
8275:
8263:
8254:
8253:
8251:
8242:(1): 2617–2680.
8227:
8221:
8220:
8210:
8178:
8172:
8171:
8147:
8126:Video generation
8117:Image generation
8114:Audio generation
8063:
8061:
8060:
8055:
8047:
8046:
8034:
8033:
8017:
8015:
8014:
8009:
8000:
7995:
7990:
7986:
7976:
7975:
7960:
7959:
7943:
7938:
7929:
7928:
7910:
7909:
7904:
7900:
7890:
7889:
7867:
7862:
7853:
7852:
7832:
7830:
7829:
7824:
7801:
7799:
7798:
7793:
7791:
7790:
7776:
7763:
7761:
7760:
7755:
7753:
7752:
7747:
7727:
7725:
7724:
7719:
7707:
7705:
7704:
7699:
7668:
7666:
7665:
7660:
7641:
7639:
7638:
7633:
7631:
7630:
7621:
7620:
7616:
7587:
7585:
7584:
7579:
7548:
7546:
7545:
7540:
7511:
7510:
7488:
7486:
7485:
7480:
7469:
7468:
7443:
7441:
7440:
7435:
7433:
7432:
7427:
7408:
7406:
7405:
7400:
7398:
7397:
7386:
7360:
7358:
7357:
7352:
7347:
7346:
7345:
7344:
7320:
7318:
7317:
7312:
7304:
7300:
7298:
7297:
7296:
7283:
7275:
7269:
7266:
7263:
7258:
7240:
7239:
7178:
7176:
7175:
7170:
7155:
7153:
7152:
7147:
7130:
7129:
7113:
7108:
7096:
7095:
7068:
7067:
7051:
7046:
7034:
7033:
7012:
7011:
6996:
6995:
6973:
6971:
6970:
6965:
6950:
6948:
6947:
6942:
6925:
6924:
6908:
6903:
6891:
6890:
6878:
6877:
6862:
6861:
6830:
6828:
6827:
6822:
6817:
6816:
6794:
6792:
6791:
6786:
6784:
6783:
6759:
6757:
6756:
6751:
6748:
6740:
6724:
6722:
6721:
6716:
6714:
6713:
6694:
6692:
6691:
6686:
6681:
6680:
6656:
6655:
6640:
6639:
6627:
6626:
6617:
6616:
6594:
6592:
6591:
6586:
6581:
6580:
6546:
6544:
6543:
6538:
6536:
6532:
6531:
6519:
6518:
6494:
6493:
6478:
6477:
6453:
6452:
6438:
6437:
6418:
6414:
6413:
6401:
6400:
6388:
6387:
6372:
6371:
6359:
6358:
6344:
6343:
6330:
6329:
6320:
6319:
6307:
6306:
6292:
6291:
6264:
6262:
6261:
6256:
6236:
6235:
6224:
6215:
6214:
6198:
6196:
6195:
6190:
6188:
6180:
6179:
6168:
6159:
6158:
6142:
6140:
6139:
6134:
6132:
6125:
6124:
6115:
6114:
6090:
6089:
6074:
6073:
6049:
6048:
6028:
6027:
6008:
6001:
6000:
5991:
5990:
5978:
5977:
5962:
5961:
5949:
5948:
5928:
5927:
5910:
5905:
5893:
5892:
5872:
5871:
5849:
5847:
5846:
5841:
5839:
5838:
5833:
5805:
5803:
5802:
5797:
5785:
5783:
5782:
5777:
5775:
5774:
5746:
5744:
5743:
5738:
5736:
5735:
5720:
5719:
5718:
5701:
5700:
5684:
5683:
5658:
5656:
5655:
5650:
5647:
5639:
5629:
5613:
5611:
5610:
5605:
5600:
5599:
5587:
5586:
5568:
5567:
5555:
5554:
5520:
5518:
5517:
5512:
5507:
5506:
5494:
5493:
5475:
5474:
5462:
5461:
5442:
5440:
5439:
5434:
5422:
5420:
5419:
5414:
5406:
5405:
5389:
5387:
5386:
5381:
5379:
5378:
5374:
5373:
5358:
5357:
5342:
5337:
5315:
5313:
5312:
5307:
5299:
5298:
5286:
5285:
5273:
5272:
5257:
5256:
5252:
5251:
5239:
5238:
5218:
5217:
5205:
5204:
5192:
5191:
5173:
5171:
5170:
5165:
5163:
5162:
5152:
5151:
5139:
5138:
5111:
5110:
5103:
5102:
5090:
5089:
5085:
5084:
5072:
5071:
5053:
5052:
5023:
5022:
5010:
5009:
5002:
5001:
4988:
4987:
4949:
4947:
4946:
4941:
4939:
4938:
4922:
4920:
4919:
4914:
4893:
4891:
4890:
4885:
4880:
4879:
4867:
4866:
4854:
4853:
4841:
4840:
4828:
4827:
4815:
4814:
4798:
4796:
4795:
4790:
4787:
4779:
4761:
4759:
4758:
4753:
4741:
4739:
4738:
4733:
4731:
4730:
4714:
4712:
4711:
4706:
4704:
4703:
4693:
4692:
4680:
4679:
4652:
4651:
4644:
4643:
4630:
4629:
4600:
4599:
4587:
4586:
4579:
4578:
4565:
4564:
4534:
4532:
4531:
4526:
4524:
4523:
4515:
4482:
4480:
4479:
4474:
4441:
4439:
4438:
4433:
4412:
4410:
4409:
4404:
4402:
4355:
4341:
4333:
4325:
4324:
4285:
4265:
4254:The Jacobian is
4250:
4248:
4247:
4242:
4239:
4231:
4215:
4213:
4212:
4207:
4154:
4153:
4131:
4129:
4128:
4123:
4087:
4085:
4084:
4079:
4054:
4052:
4051:
4046:
4044:
4043:
4035:
4022:
4020:
4019:
4014:
4009:
4008:
3996:
3995:
3979:
3969:
3950:
3948:
3947:
3942:
3940:
3939:
3918:
3917:
3901:
3899:
3898:
3893:
3891:
3890:
3850:
3848:
3847:
3842:
3834:
3833:
3821:
3820:
3801:
3796:
3779:
3778:
3773:
3741:is replaced by
3737:
3735:
3734:
3729:
3715:
3714:
3705:
3700:
3686:
3685:
3673:
3672:
3658:
3657:
3652:
3616:
3614:
3613:
3608:
3600:
3599:
3587:
3586:
3567:
3562:
3547:
3539:
3516:
3515:
3494:
3493:
3483:
3482:
3472:
3471:
3466:
3461:
3444:
3442:
3441:
3436:
3425:
3424:
3408:
3406:
3405:
3400:
3398:
3397:
3376:
3375:
3351:
3349:
3348:
3343:
3328:
3326:
3325:
3320:
3303:
3302:
3281:
3280:
3270:
3269:
3259:
3235:
3234:
3213:
3212:
3202:
3201:
3191:
3167:
3166:
3157:
3152:
3138:
3137:
3125:
3124:
3102:
3100:
3099:
3094:
3092:
3091:
3075:
3073:
3072:
3067:
3065:
3064:
3036:
3034:
3033:
3028:
3026:
3022:
3021:
3019:
3018:
3017:
2998:
2994:
2993:
2975:
2974:
2961:
2944:
2939:
2918:
2917:
2905:
2904:
2883:
2882:
2870:
2869:
2840:
2838:
2837:
2832:
2827:
2826:
2808:
2807:
2779:
2777:
2776:
2771:
2766:
2765:
2753:
2752:
2730:
2728:
2727:
2722:
2720:
2719:
2697:
2695:
2694:
2689:
2687:
2686:
2667:
2665:
2664:
2659:
2657:
2653:
2652:
2650:
2649:
2648:
2635:
2631:
2630:
2618:
2617:
2604:
2582:
2581:
2569:
2568:
2547:
2546:
2534:
2533:
2505:
2503:
2502:
2497:
2495:
2494:
2486:
2482:
2481:
2479:
2478:
2477:
2464:
2460:
2459:
2447:
2446:
2433:
2419:
2418:
2406:
2405:
2390:
2389:
2377:
2376:
2353:
2351:
2350:
2345:
2333:
2331:
2330:
2325:
2323:
2322:
2295:
2294:
2270:By the identity
2266:
2264:
2263:
2258:
2256:
2252:
2251:
2250:
2242:
2238:
2236:
2235:
2234:
2221:
2217:
2216:
2204:
2203:
2190:
2172:
2171:
2159:
2158:
2143:
2142:
2130:
2129:
2103:
2101:
2100:
2095:
2092:
2084:
2060:
2058:
2057:
2052:
2050:
2048:
2047:
2046:
2033:
2029:
2028:
2015:
2007:
1994:
1979:
1977:
1976:
1971:
1969:
1965:
1964:
1962:
1961:
1960:
1947:
1943:
1942:
1929:
1921:
1908:
1895:
1894:
1882:
1881:
1866:
1865:
1853:
1852:
1833:
1831:
1830:
1825:
1823:
1822:
1799:
1797:
1796:
1791:
1786:
1785:
1772:
1764:
1752:
1751:
1735:
1733:
1732:
1727:
1725:
1724:
1708:
1706:
1705:
1700:
1698:
1697:
1669:
1667:
1666:
1661:
1659:
1658:
1634:
1633:
1617:
1615:
1614:
1609:
1607:
1606:
1582:
1581:
1562:
1560:
1559:
1554:
1552:
1548:
1547:
1545:
1544:
1543:
1524:
1520:
1519:
1501:
1500:
1487:
1470:
1465:
1444:
1443:
1431:
1430:
1409:
1408:
1396:
1395:
1366:
1364:
1363:
1358:
1356:
1355:
1336:
1334:
1333:
1328:
1326:
1325:
1309:
1307:
1306:
1301:
1298:
1290:
1274:inverse function
1271:
1269:
1268:
1263:
1261:
1260:
1236:
1235:
1220:. The functions
1219:
1217:
1216:
1211:
1209:
1208:
1192:
1190:
1189:
1184:
1179:
1178:
1160:
1159:
1147:
1146:
1130:
1128:
1127:
1122:
1083:
1081:
1080:
1075:
1070:
1069:
1057:
1056:
1036:
1034:
1033:
1028:
1026:
1025:
970:normalizing flow
962:machine learning
958:generative model
942:
935:
928:
889:Related articles
766:Confusion matrix
519:Isolation forest
464:Graphical models
243:
242:
195:Learning to rank
190:Feature learning
28:Machine learning
19:
9040:
9039:
9035:
9034:
9033:
9031:
9030:
9029:
9005:
9004:
8991:
8986:
8985:
8969:
8968:
8964:
8948:
8947:
8943:
8927:
8926:
8922:
8906:
8905:
8901:
8885:
8884:
8880:
8864:
8863:
8859:
8843:
8842:
8838:
8815:: 12427–12436.
8802:
8801:
8797:
8781:
8780:
8776:
8760:
8759:
8755:
8739:
8738:
8731:
8715:
8714:
8710:
8692:
8691:
8687:
8665:
8660:
8659:
8655:
8621:
8620:
8616:
8592:
8591:
8584:
8568:
8567:
8563:
8547:
8546:
8542:
8526:
8525:
8516:
8488:
8487:
8483:
8455:
8454:
8450:
8434:
8433:
8429:
8369:
8368:
8364:
8336:
8335:
8331:
8315:
8314:
8305:
8289:
8288:
8281:
8265:
8264:
8257:
8229:
8228:
8224:
8180:
8179:
8175:
8149:
8148:
8144:
8139:
8108:
8071:
8038:
8025:
8020:
8019:
7967:
7951:
7950:
7946:
7920:
7881:
7874:
7870:
7869:
7844:
7839:
7838:
7815:
7814:
7771:
7766:
7765:
7742:
7737:
7736:
7710:
7709:
7690:
7689:
7682:ambient isotopy
7651:
7650:
7622:
7601:
7590:
7589:
7555:
7554:
7502:
7491:
7490:
7460:
7446:
7445:
7422:
7411:
7410:
7381:
7370:
7369:
7336:
7331:
7326:
7325:
7288:
7284:
7276:
7270:
7231:
7184:
7183:
7161:
7160:
7121:
7087:
7059:
7025:
7000:
6987:
6982:
6981:
6956:
6955:
6916:
6882:
6869:
6853:
6836:
6835:
6808:
6797:
6796:
6775:
6770:
6769:
6766:
6727:
6726:
6705:
6700:
6699:
6660:
6647:
6631:
6618:
6608:
6603:
6602:
6572:
6549:
6548:
6534:
6533:
6523:
6498:
6485:
6457:
6444:
6442:
6429:
6426:
6425:
6416:
6415:
6405:
6392:
6379:
6363:
6350:
6348:
6335:
6332:
6331:
6321:
6311:
6298:
6296:
6283:
6274:
6273:
6219:
6206:
6201:
6200:
6163:
6150:
6145:
6144:
6130:
6129:
6116:
6094:
6081:
6053:
6040:
6032:
6019:
6016:
6015:
6006:
6005:
5992:
5982:
5969:
5953:
5940:
5932:
5919:
5916:
5915:
5884:
5876:
5863:
5854:
5853:
5828:
5823:
5822:
5819:
5788:
5787:
5763:
5749:
5748:
5721:
5711:
5703:
5693:
5669:
5664:
5663:
5616:
5615:
5591:
5572:
5559:
5540:
5535:
5534:
5527:
5498:
5485:
5466:
5453:
5445:
5444:
5425:
5424:
5397:
5392:
5391:
5362:
5349:
5344:
5318:
5317:
5290:
5277:
5264:
5243:
5230:
5222:
5209:
5196:
5183:
5178:
5177:
5176:Its inverse is
5157:
5156:
5143:
5130:
5127:
5126:
5116:
5105:
5104:
5094:
5076:
5063:
5058:
5055:
5054:
5044:
5037:
5014:
5004:
5003:
4993:
4990:
4989:
4979:
4972:
4960:
4959:
4956:
4930:
4925:
4924:
4899:
4898:
4871:
4858:
4845:
4832:
4819:
4806:
4801:
4800:
4766:
4765:
4744:
4743:
4722:
4717:
4716:
4698:
4697:
4684:
4671:
4668:
4667:
4657:
4646:
4645:
4635:
4632:
4631:
4621:
4614:
4591:
4581:
4580:
4570:
4567:
4566:
4556:
4549:
4537:
4536:
4510:
4493:
4492:
4489:
4444:
4443:
4418:
4417:
4348:
4316:
4278:
4256:
4255:
4218:
4217:
4145:
4134:
4133:
4090:
4089:
4070:
4069:
4066:
4061:
4028:
4027:
4000:
3987:
3956:
3955:
3931:
3909:
3904:
3903:
3876:
3871:
3870:
3869:INPUT. dataset
3825:
3812:
3746:
3745:
3706:
3677:
3661:
3625:
3624:
3591:
3578:
3507:
3474:
3458:
3450:
3449:
3416:
3411:
3410:
3377:
3367:
3359:
3358:
3334:
3333:
3294:
3261:
3254:
3226:
3193:
3186:
3158:
3129:
3113:
3108:
3107:
3083:
3078:
3077:
3056:
3051:
3050:
3043:
3041:Training method
3003:
2999:
2979:
2966:
2962:
2956:
2952:
2909:
2896:
2874:
2861:
2850:
2849:
2812:
2793:
2782:
2781:
2757:
2744:
2733:
2732:
2705:
2700:
2699:
2678:
2673:
2672:
2640:
2636:
2622:
2609:
2605:
2599:
2595:
2573:
2560:
2538:
2525:
2514:
2513:
2469:
2465:
2451:
2438:
2434:
2428:
2424:
2423:
2410:
2397:
2381:
2368:
2363:
2362:
2336:
2335:
2311:
2283:
2272:
2271:
2226:
2222:
2208:
2195:
2191:
2185:
2184:
2180:
2176:
2163:
2150:
2134:
2121:
2116:
2115:
2071:
2070:
2067:Jacobian matrix
2038:
2034:
2020:
1995:
1985:
1984:
1952:
1948:
1934:
1909:
1903:
1899:
1886:
1873:
1857:
1844:
1839:
1838:
1814:
1809:
1808:
1777:
1743:
1738:
1737:
1716:
1711:
1710:
1689:
1684:
1683:
1680:
1650:
1625:
1620:
1619:
1598:
1573:
1568:
1567:
1529:
1525:
1505:
1492:
1488:
1482:
1478:
1435:
1422:
1400:
1387:
1376:
1375:
1347:
1342:
1341:
1317:
1312:
1311:
1277:
1276:
1252:
1227:
1222:
1221:
1200:
1195:
1194:
1164:
1151:
1138:
1133:
1132:
1089:
1088:
1061:
1048:
1043:
1042:
1039:random variable
1017:
1012:
1011:
1000:
946:
917:
916:
890:
882:
881:
842:
834:
833:
794:Kernel machines
789:
781:
780:
756:
748:
747:
728:Active learning
723:
715:
714:
683:
673:
672:
598:Diffusion model
534:
524:
523:
496:
486:
485:
459:
449:
448:
404:Factor analysis
399:
389:
388:
372:
335:
325:
324:
245:
244:
228:
227:
226:
215:
214:
120:
112:
111:
77:Online learning
42:
30:
17:
12:
11:
5:
9038:
9036:
9028:
9027:
9022:
9017:
9007:
9006:
9003:
9002:
8997:
8990:
8989:External links
8987:
8984:
8983:
8962:
8941:
8920:
8899:
8878:
8857:
8836:
8795:
8774:
8753:
8729:
8708:
8685:
8653:
8614:
8582:
8561:
8540:
8514:
8481:
8448:
8427:
8362:
8329:
8303:
8279:
8255:
8222:
8193:(2): 145–164.
8173:
8162:(1): 217–233.
8141:
8140:
8138:
8135:
8134:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8107:
8104:
8070:
8067:
8053:
8050:
8045:
8041:
8037:
8032:
8028:
8007:
8004:
7999:
7994:
7989:
7985:
7982:
7979:
7974:
7970:
7966:
7963:
7958:
7954:
7949:
7942:
7937:
7933:
7927:
7923:
7919:
7916:
7913:
7908:
7903:
7899:
7896:
7893:
7888:
7884:
7880:
7877:
7873:
7866:
7861:
7857:
7851:
7847:
7822:
7789:
7786:
7783:
7780:
7775:
7751:
7746:
7717:
7697:
7658:
7629:
7625:
7619:
7615:
7611:
7608:
7604:
7600:
7597:
7577:
7574:
7571:
7568:
7565:
7562:
7538:
7535:
7532:
7529:
7526:
7523:
7520:
7517:
7514:
7509:
7505:
7501:
7498:
7478:
7475:
7472:
7467:
7463:
7459:
7456:
7453:
7431:
7426:
7421:
7418:
7396:
7393:
7390:
7385:
7380:
7377:
7350:
7343:
7339:
7334:
7322:
7321:
7310:
7307:
7303:
7295:
7291:
7287:
7282:
7279:
7273:
7262:
7257:
7253:
7249:
7246:
7243:
7238:
7234:
7230:
7227:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7194:
7191:
7168:
7157:
7156:
7145:
7142:
7139:
7136:
7133:
7128:
7124:
7120:
7117:
7112:
7107:
7103:
7099:
7094:
7090:
7086:
7083:
7080:
7077:
7074:
7071:
7066:
7062:
7058:
7055:
7050:
7045:
7041:
7037:
7032:
7028:
7024:
7021:
7018:
7015:
7010:
7007:
7003:
6999:
6994:
6990:
6963:
6952:
6951:
6940:
6937:
6934:
6931:
6928:
6923:
6919:
6915:
6912:
6907:
6902:
6898:
6894:
6889:
6885:
6881:
6876:
6872:
6868:
6865:
6860:
6856:
6852:
6849:
6846:
6843:
6820:
6815:
6811:
6807:
6804:
6782:
6778:
6765:
6762:
6747:
6744:
6739:
6735:
6712:
6708:
6684:
6679:
6676:
6673:
6670:
6667:
6663:
6659:
6654:
6650:
6646:
6643:
6638:
6634:
6630:
6625:
6621:
6615:
6611:
6584:
6579:
6575:
6571:
6568:
6565:
6562:
6559:
6556:
6530:
6526:
6522:
6517:
6514:
6511:
6508:
6505:
6501:
6497:
6492:
6488:
6484:
6481:
6476:
6473:
6470:
6467:
6464:
6460:
6456:
6451:
6447:
6443:
6441:
6436:
6432:
6428:
6427:
6424:
6421:
6419:
6417:
6412:
6408:
6404:
6399:
6395:
6391:
6386:
6382:
6378:
6375:
6370:
6366:
6362:
6357:
6353:
6349:
6347:
6342:
6338:
6334:
6333:
6328:
6324:
6318:
6314:
6310:
6305:
6301:
6297:
6295:
6290:
6286:
6282:
6281:
6254:
6251:
6248:
6245:
6242:
6239:
6234:
6231:
6228:
6223:
6218:
6213:
6209:
6187:
6183:
6178:
6175:
6172:
6167:
6162:
6157:
6153:
6128:
6123:
6119:
6113:
6110:
6107:
6104:
6101:
6097:
6093:
6088:
6084:
6080:
6077:
6072:
6069:
6066:
6063:
6060:
6056:
6052:
6047:
6043:
6039:
6036:
6033:
6031:
6026:
6022:
6018:
6017:
6014:
6011:
6009:
6007:
6004:
5999:
5995:
5989:
5985:
5981:
5976:
5972:
5968:
5965:
5960:
5956:
5952:
5947:
5943:
5939:
5936:
5933:
5931:
5926:
5922:
5918:
5917:
5914:
5909:
5904:
5900:
5896:
5891:
5887:
5883:
5880:
5877:
5875:
5870:
5866:
5862:
5861:
5837:
5832:
5818:
5815:
5811:
5810:
5807:
5795:
5773:
5770:
5766:
5762:
5759:
5756:
5747:with Jacobian
5734:
5731:
5728:
5724:
5717:
5714:
5710:
5706:
5699:
5696:
5691:
5687:
5682:
5679:
5676:
5672:
5660:
5646:
5643:
5638:
5634:
5628:
5624:
5614:with Jacobian
5603:
5598:
5594:
5590:
5585:
5582:
5579:
5575:
5571:
5566:
5562:
5558:
5553:
5550:
5547:
5543:
5526:
5523:
5510:
5505:
5501:
5497:
5492:
5488:
5484:
5481:
5478:
5473:
5469:
5465:
5460:
5456:
5452:
5432:
5412:
5409:
5404:
5400:
5377:
5372:
5369:
5365:
5361:
5356:
5352:
5347:
5341:
5336:
5333:
5330:
5326:
5305:
5302:
5297:
5293:
5289:
5284:
5280:
5276:
5271:
5267:
5263:
5260:
5255:
5250:
5246:
5242:
5237:
5233:
5229:
5225:
5221:
5216:
5212:
5208:
5203:
5199:
5195:
5190:
5186:
5161:
5155:
5150:
5146:
5142:
5137:
5133:
5129:
5128:
5125:
5122:
5121:
5119:
5114:
5109:
5101:
5097:
5093:
5088:
5083:
5079:
5075:
5070:
5066:
5061:
5057:
5056:
5051:
5047:
5043:
5042:
5040:
5035:
5032:
5029:
5026:
5021:
5017:
5013:
5008:
5000:
4996:
4992:
4991:
4986:
4982:
4978:
4977:
4975:
4970:
4967:
4955:
4952:
4937:
4933:
4912:
4909:
4906:
4883:
4878:
4874:
4870:
4865:
4861:
4857:
4852:
4848:
4844:
4839:
4835:
4831:
4826:
4822:
4818:
4813:
4809:
4786:
4783:
4778:
4774:
4751:
4729:
4725:
4702:
4696:
4691:
4687:
4683:
4678:
4674:
4670:
4669:
4666:
4663:
4662:
4660:
4655:
4650:
4642:
4638:
4634:
4633:
4628:
4624:
4620:
4619:
4617:
4612:
4609:
4606:
4603:
4598:
4594:
4590:
4585:
4577:
4573:
4569:
4568:
4563:
4559:
4555:
4554:
4552:
4547:
4544:
4522:
4519:
4514:
4509:
4506:
4503:
4500:
4488:
4485:
4472:
4469:
4466:
4463:
4460:
4457:
4454:
4451:
4431:
4428:
4425:
4401:
4397:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4373:
4370:
4367:
4364:
4361:
4358:
4354:
4351:
4347:
4344:
4340:
4336:
4332:
4328:
4323:
4319:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4284:
4281:
4277:
4274:
4271:
4268:
4264:
4238:
4235:
4230:
4226:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4152:
4148:
4144:
4141:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4077:
4065:
4062:
4060:
4057:
4056:
4055:
4041:
4038:
4024:
4012:
4007:
4003:
3999:
3994:
3990:
3986:
3983:
3978:
3974:
3968:
3964:
3952:
3938:
3934:
3930:
3927:
3924:
3921:
3916:
3912:
3889:
3886:
3883:
3879:
3852:
3851:
3840:
3837:
3832:
3828:
3824:
3819:
3815:
3811:
3808:
3805:
3800:
3795:
3792:
3789:
3785:
3776:
3772:
3769:
3766:
3762:
3759:
3756:
3739:
3738:
3727:
3724:
3721:
3718:
3713:
3709:
3704:
3699:
3695:
3692:
3689:
3684:
3680:
3676:
3671:
3668:
3664:
3655:
3651:
3648:
3645:
3641:
3638:
3635:
3618:
3617:
3606:
3603:
3598:
3594:
3590:
3585:
3581:
3577:
3574:
3571:
3566:
3561:
3558:
3555:
3551:
3545:
3542:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3514:
3510:
3506:
3503:
3500:
3497:
3492:
3489:
3486:
3481:
3477:
3469:
3465:
3457:
3434:
3431:
3428:
3423:
3419:
3396:
3393:
3390:
3387:
3384:
3380:
3374:
3370:
3366:
3341:
3330:
3329:
3318:
3315:
3312:
3309:
3306:
3301:
3297:
3293:
3290:
3287:
3284:
3279:
3276:
3273:
3268:
3264:
3258:
3253:
3250:
3247:
3244:
3241:
3238:
3233:
3229:
3225:
3222:
3219:
3216:
3211:
3208:
3205:
3200:
3196:
3190:
3185:
3182:
3179:
3176:
3173:
3170:
3165:
3161:
3156:
3151:
3147:
3144:
3141:
3136:
3132:
3128:
3123:
3120:
3116:
3090:
3086:
3063:
3059:
3042:
3039:
3038:
3037:
3025:
3016:
3013:
3010:
3006:
3002:
2997:
2992:
2989:
2986:
2982:
2978:
2973:
2969:
2965:
2959:
2955:
2951:
2948:
2943:
2938:
2935:
2932:
2928:
2924:
2921:
2916:
2912:
2908:
2903:
2899:
2895:
2892:
2889:
2886:
2881:
2877:
2873:
2868:
2864:
2860:
2857:
2830:
2825:
2822:
2819:
2815:
2811:
2806:
2803:
2800:
2796:
2792:
2789:
2769:
2764:
2760:
2756:
2751:
2747:
2743:
2740:
2718:
2715:
2712:
2708:
2685:
2681:
2669:
2668:
2656:
2647:
2643:
2639:
2634:
2629:
2625:
2621:
2616:
2612:
2608:
2602:
2598:
2594:
2591:
2588:
2585:
2580:
2576:
2572:
2567:
2563:
2559:
2556:
2553:
2550:
2545:
2541:
2537:
2532:
2528:
2524:
2521:
2507:
2506:
2493:
2490:
2485:
2476:
2472:
2468:
2463:
2458:
2454:
2450:
2445:
2441:
2437:
2431:
2427:
2422:
2417:
2413:
2409:
2404:
2400:
2396:
2393:
2388:
2384:
2380:
2375:
2371:
2343:
2321:
2318:
2314:
2310:
2307:
2304:
2301:
2298:
2293:
2290:
2286:
2282:
2279:
2268:
2267:
2255:
2249:
2246:
2241:
2233:
2229:
2225:
2220:
2215:
2211:
2207:
2202:
2198:
2194:
2188:
2183:
2179:
2175:
2170:
2166:
2162:
2157:
2153:
2149:
2146:
2141:
2137:
2133:
2128:
2124:
2091:
2088:
2083:
2079:
2045:
2041:
2037:
2032:
2027:
2023:
2019:
2014:
2011:
2006:
2002:
1998:
1992:
1981:
1980:
1968:
1959:
1955:
1951:
1946:
1941:
1937:
1933:
1928:
1925:
1920:
1916:
1912:
1906:
1902:
1898:
1893:
1889:
1885:
1880:
1876:
1872:
1869:
1864:
1860:
1856:
1851:
1847:
1821:
1817:
1789:
1784:
1780:
1776:
1771:
1768:
1763:
1759:
1755:
1750:
1746:
1723:
1719:
1696:
1692:
1679:
1676:
1657:
1653:
1649:
1646:
1643:
1640:
1637:
1632:
1628:
1605:
1601:
1597:
1594:
1591:
1588:
1585:
1580:
1576:
1564:
1563:
1551:
1542:
1539:
1536:
1532:
1528:
1523:
1518:
1515:
1512:
1508:
1504:
1499:
1495:
1491:
1485:
1481:
1477:
1474:
1469:
1464:
1461:
1458:
1454:
1450:
1447:
1442:
1438:
1434:
1429:
1425:
1421:
1418:
1415:
1412:
1407:
1403:
1399:
1394:
1390:
1386:
1383:
1354:
1350:
1324:
1320:
1297:
1294:
1289:
1285:
1259:
1255:
1251:
1248:
1245:
1242:
1239:
1234:
1230:
1207:
1203:
1182:
1177:
1174:
1171:
1167:
1163:
1158:
1154:
1150:
1145:
1141:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1073:
1068:
1064:
1060:
1055:
1051:
1024:
1020:
999:
996:
968:by leveraging
948:
947:
945:
944:
937:
930:
922:
919:
918:
915:
914:
909:
908:
907:
897:
891:
888:
887:
884:
883:
880:
879:
874:
869:
864:
859:
854:
849:
843:
840:
839:
836:
835:
832:
831:
826:
821:
816:
814:Occam learning
811:
806:
801:
796:
790:
787:
786:
783:
782:
779:
778:
773:
771:Learning curve
768:
763:
757:
754:
753:
750:
749:
746:
745:
740:
735:
730:
724:
721:
720:
717:
716:
713:
712:
711:
710:
700:
695:
690:
684:
679:
678:
675:
674:
671:
670:
664:
659:
654:
649:
648:
647:
637:
632:
631:
630:
625:
620:
615:
605:
600:
595:
590:
589:
588:
578:
577:
576:
571:
566:
561:
551:
546:
541:
535:
530:
529:
526:
525:
522:
521:
516:
511:
503:
497:
492:
491:
488:
487:
484:
483:
482:
481:
476:
471:
460:
455:
454:
451:
450:
447:
446:
441:
436:
431:
426:
421:
416:
411:
406:
400:
395:
394:
391:
390:
387:
386:
381:
376:
370:
365:
360:
352:
347:
342:
336:
331:
330:
327:
326:
323:
322:
317:
312:
307:
302:
297:
292:
287:
279:
278:
277:
272:
267:
257:
255:Decision trees
252:
246:
232:classification
222:
221:
220:
217:
216:
213:
212:
207:
202:
197:
192:
187:
182:
177:
172:
167:
162:
157:
152:
147:
142:
137:
132:
127:
125:Classification
121:
118:
117:
114:
113:
110:
109:
104:
99:
94:
89:
84:
82:Batch learning
79:
74:
69:
64:
59:
54:
49:
43:
40:
39:
36:
35:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
9037:
9026:
9023:
9021:
9018:
9016:
9013:
9012:
9010:
9001:
8998:
8996:
8993:
8992:
8988:
8978:
8973:
8966:
8963:
8957:
8952:
8945:
8942:
8936:
8931:
8924:
8921:
8915:
8910:
8903:
8900:
8894:
8889:
8882:
8879:
8873:
8868:
8861:
8858:
8852:
8847:
8840:
8837:
8832:
8828:
8823:
8818:
8814:
8810:
8806:
8799:
8796:
8790:
8785:
8778:
8775:
8769:
8764:
8757:
8754:
8748:
8743:
8736:
8734:
8730:
8724:
8719:
8712:
8709:
8704:
8700:
8696:
8689:
8686:
8680:
8675:
8671:
8664:
8657:
8654:
8649:
8645:
8641:
8637:
8633:
8629:
8625:
8618:
8615:
8609:
8604:
8600:
8596:
8589:
8587:
8583:
8577:
8572:
8565:
8562:
8556:
8551:
8544:
8541:
8535:
8530:
8523:
8521:
8519:
8515:
8509:
8504:
8500:
8496:
8492:
8485:
8482:
8476:
8471:
8467:
8463:
8459:
8452:
8449:
8443:
8438:
8431:
8428:
8423:
8419:
8415:
8411:
8407:
8403:
8399:
8395:
8390:
8385:
8381:
8377:
8373:
8366:
8363:
8357:
8352:
8348:
8344:
8340:
8333:
8330:
8324:
8319:
8312:
8310:
8308:
8304:
8298:
8293:
8286:
8284:
8280:
8274:
8269:
8262:
8260:
8256:
8250:
8245:
8241:
8237:
8233:
8226:
8223:
8218:
8214:
8209:
8204:
8200:
8196:
8192:
8188:
8184:
8177:
8174:
8169:
8165:
8161:
8157:
8153:
8146:
8143:
8136:
8131:
8128:
8125:
8122:
8119:
8116:
8113:
8112:
8111:
8105:
8103:
8101:
8097:
8092:
8088:
8087:invertibility
8083:
8081:
8075:
8068:
8066:
8051:
8048:
8043:
8039:
8035:
8030:
8026:
8005:
8002:
7997:
7992:
7980:
7977:
7972:
7968:
7961:
7956:
7940:
7935:
7931:
7925:
7921:
7917:
7914:
7911:
7906:
7894:
7891:
7886:
7882:
7875:
7864:
7859:
7855:
7849:
7845:
7836:
7820:
7811:
7809:
7805:
7787:
7784:
7781:
7778:
7749:
7733:
7729:
7715:
7695:
7687:
7683:
7679:
7678:homeomorphism
7674:
7672:
7656:
7647:
7645:
7627:
7617:
7613:
7609:
7606:
7602:
7598:
7572:
7569:
7566:
7560:
7550:
7533:
7527:
7524:
7521:
7515:
7512:
7507:
7503:
7496:
7476:
7473:
7465:
7461:
7457:
7451:
7429:
7419:
7416:
7394:
7391:
7388:
7378:
7375:
7365:
7362:
7348:
7341:
7337:
7308:
7305:
7301:
7293:
7289:
7280:
7271:
7260:
7255:
7251:
7247:
7236:
7232:
7225:
7219:
7216:
7213:
7204:
7198:
7192:
7189:
7182:
7181:
7180:
7166:
7143:
7140:
7134:
7131:
7126:
7122:
7115:
7110:
7105:
7101:
7097:
7092:
7088:
7084:
7081:
7078:
7072:
7069:
7064:
7060:
7053:
7048:
7043:
7039:
7035:
7030:
7026:
7022:
7016:
7008:
7005:
7001:
6997:
6992:
6988:
6980:
6979:
6978:
6975:
6961:
6938:
6935:
6929:
6926:
6921:
6917:
6910:
6905:
6900:
6896:
6892:
6887:
6883:
6879:
6874:
6870:
6866:
6858:
6854:
6847:
6844:
6841:
6834:
6833:
6832:
6813:
6809:
6802:
6780:
6776:
6763:
6761:
6745:
6742:
6737:
6733:
6710:
6706:
6696:
6677:
6674:
6671:
6668:
6665:
6661:
6652:
6648:
6644:
6636:
6632:
6623:
6619:
6613:
6609:
6599:
6596:
6577:
6573:
6569:
6566:
6560:
6557:
6554:
6528:
6524:
6515:
6512:
6509:
6506:
6503:
6499:
6490:
6486:
6482:
6474:
6471:
6468:
6465:
6462:
6458:
6449:
6445:
6439:
6434:
6430:
6422:
6420:
6410:
6406:
6397:
6393:
6384:
6380:
6376:
6368:
6364:
6355:
6351:
6345:
6340:
6336:
6326:
6322:
6316:
6312:
6308:
6303:
6299:
6293:
6288:
6284:
6271:
6266:
6246:
6243:
6232:
6229:
6226:
6216:
6211:
6207:
6176:
6173:
6170:
6160:
6155:
6151:
6121:
6111:
6108:
6105:
6102:
6099:
6095:
6086:
6082:
6078:
6070:
6067:
6064:
6061:
6058:
6054:
6045:
6041:
6034:
6029:
6024:
6020:
6012:
6010:
5997:
5987:
5983:
5974:
5970:
5966:
5958:
5954:
5945:
5941:
5934:
5929:
5924:
5920:
5907:
5902:
5898:
5894:
5889:
5885:
5878:
5873:
5868:
5864:
5851:
5835:
5816:
5814:
5808:
5793:
5771:
5768:
5760:
5732:
5729:
5726:
5722:
5715:
5712:
5708:
5704:
5697:
5694:
5689:
5685:
5680:
5677:
5674:
5670:
5661:
5644:
5641:
5636:
5632:
5626:
5622:
5596:
5592:
5588:
5583:
5580:
5577:
5573:
5564:
5560:
5556:
5551:
5548:
5545:
5541:
5532:
5531:
5530:
5524:
5522:
5503:
5499:
5495:
5490:
5486:
5471:
5467:
5463:
5458:
5454:
5430:
5410:
5407:
5402:
5398:
5370:
5367:
5363:
5354:
5350:
5345:
5339:
5334:
5331:
5328:
5324:
5295:
5291:
5282:
5278:
5274:
5269:
5265:
5258:
5248:
5244:
5235:
5231:
5227:
5223:
5219:
5214:
5210:
5206:
5201:
5197:
5193:
5188:
5184:
5174:
5159:
5148:
5144:
5135:
5131:
5123:
5117:
5112:
5107:
5099:
5095:
5091:
5081:
5077:
5068:
5064:
5059:
5049:
5045:
5038:
5033:
5027:
5019:
5015:
5011:
5006:
4998:
4994:
4984:
4980:
4973:
4968:
4965:
4953:
4951:
4935:
4931:
4910:
4907:
4904:
4895:
4876:
4872:
4863:
4859:
4855:
4850:
4846:
4842:
4837:
4833:
4829:
4824:
4820:
4816:
4811:
4807:
4784:
4781:
4776:
4772:
4763:
4749:
4727:
4723:
4700:
4689:
4685:
4676:
4672:
4664:
4658:
4653:
4648:
4640:
4636:
4626:
4622:
4615:
4610:
4604:
4596:
4592:
4588:
4583:
4575:
4571:
4561:
4557:
4550:
4545:
4542:
4520:
4517:
4507:
4504:
4501:
4498:
4486:
4484:
4470:
4467:
4464:
4458:
4455:
4452:
4429:
4426:
4423:
4414:
4392:
4389:
4386:
4377:
4374:
4368:
4365:
4362:
4352:
4349:
4345:
4342:
4334:
4321:
4317:
4313:
4307:
4304:
4298:
4295:
4292:
4282:
4279:
4275:
4272:
4252:
4236:
4233:
4228:
4224:
4200:
4197:
4191:
4188:
4185:
4176:
4173:
4170:
4167:
4164:
4158:
4150:
4146:
4142:
4139:
4116:
4113:
4110:
4107:
4104:
4098:
4095:
4075:
4063:
4058:
4036:
4025:
4005:
4001:
3992:
3988:
3984:
3981:
3976:
3972:
3966:
3953:
3936:
3932:
3928:
3922:
3914:
3910:
3887:
3884:
3881:
3877:
3868:
3867:
3866:
3863:
3861:
3857:
3830:
3826:
3817:
3813:
3806:
3803:
3798:
3793:
3790:
3787:
3783:
3774:
3744:
3743:
3742:
3719:
3711:
3707:
3690:
3682:
3678:
3669:
3666:
3662:
3653:
3623:
3622:
3621:
3596:
3592:
3583:
3579:
3572:
3569:
3564:
3559:
3556:
3553:
3549:
3543:
3540:
3535:
3532:
3520:
3512:
3508:
3501:
3498:
3487:
3479:
3475:
3455:
3448:
3447:
3446:
3429:
3421:
3417:
3394:
3391:
3388:
3385:
3382:
3372:
3368:
3356:
3339:
3307:
3299:
3295:
3288:
3285:
3274:
3266:
3262:
3251:
3239:
3231:
3227:
3220:
3217:
3206:
3198:
3194:
3183:
3180:
3171:
3163:
3159:
3142:
3134:
3130:
3121:
3118:
3114:
3106:
3105:
3104:
3088:
3084:
3061:
3057:
3048:
3040:
3023:
3014:
3011:
3008:
3004:
3000:
2990:
2987:
2984:
2980:
2971:
2967:
2963:
2953:
2949:
2946:
2941:
2936:
2933:
2930:
2926:
2922:
2914:
2910:
2901:
2897:
2893:
2890:
2887:
2879:
2875:
2866:
2862:
2858:
2855:
2848:
2847:
2846:
2844:
2823:
2820:
2817:
2813:
2804:
2801:
2798:
2794:
2790:
2787:
2762:
2758:
2749:
2745:
2741:
2738:
2716:
2713:
2710:
2706:
2683:
2679:
2654:
2645:
2641:
2637:
2627:
2623:
2614:
2610:
2606:
2596:
2592:
2589:
2586:
2578:
2574:
2565:
2561:
2557:
2554:
2551:
2543:
2539:
2530:
2526:
2522:
2519:
2512:
2511:
2510:
2491:
2488:
2483:
2474:
2470:
2466:
2456:
2452:
2443:
2439:
2435:
2425:
2415:
2411:
2402:
2398:
2394:
2386:
2382:
2373:
2369:
2361:
2360:
2359:
2357:
2341:
2319:
2316:
2308:
2299:
2291:
2288:
2284:
2253:
2247:
2244:
2239:
2231:
2227:
2223:
2213:
2209:
2200:
2196:
2192:
2186:
2177:
2168:
2164:
2155:
2151:
2147:
2139:
2135:
2126:
2122:
2114:
2113:
2112:
2110:
2105:
2089:
2086:
2081:
2077:
2068:
2064:
2043:
2039:
2035:
2025:
2021:
2012:
2009:
2004:
2000:
1996:
1966:
1957:
1953:
1949:
1939:
1935:
1926:
1923:
1918:
1914:
1910:
1900:
1891:
1887:
1878:
1874:
1870:
1862:
1858:
1849:
1845:
1837:
1836:
1835:
1819:
1815:
1806:
1801:
1782:
1778:
1769:
1766:
1761:
1757:
1753:
1748:
1744:
1721:
1717:
1694:
1690:
1677:
1675:
1673:
1655:
1651:
1647:
1644:
1641:
1638:
1635:
1630:
1626:
1603:
1599:
1595:
1592:
1589:
1586:
1583:
1578:
1574:
1549:
1540:
1537:
1534:
1530:
1526:
1516:
1513:
1510:
1506:
1497:
1493:
1489:
1479:
1475:
1472:
1467:
1462:
1459:
1456:
1452:
1448:
1440:
1436:
1427:
1423:
1419:
1416:
1413:
1405:
1401:
1392:
1388:
1384:
1381:
1374:
1373:
1372:
1370:
1352:
1348:
1338:
1322:
1318:
1295:
1292:
1287:
1283:
1275:
1257:
1253:
1249:
1246:
1243:
1240:
1237:
1232:
1228:
1205:
1201:
1175:
1172:
1169:
1165:
1156:
1152:
1148:
1143:
1139:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1085:
1066:
1062:
1053:
1049:
1040:
1022:
1018:
1004:
997:
995:
993:
989:
984:
982:
981:loss function
977:
975:
971:
967:
963:
959:
955:
943:
938:
936:
931:
929:
924:
923:
921:
920:
913:
910:
906:
903:
902:
901:
898:
896:
893:
892:
886:
885:
878:
875:
873:
870:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
844:
838:
837:
830:
827:
825:
822:
820:
817:
815:
812:
810:
807:
805:
802:
800:
797:
795:
792:
791:
785:
784:
777:
774:
772:
769:
767:
764:
762:
759:
758:
752:
751:
744:
741:
739:
736:
734:
733:Crowdsourcing
731:
729:
726:
725:
719:
718:
709:
706:
705:
704:
701:
699:
696:
694:
691:
689:
686:
685:
682:
677:
676:
668:
665:
663:
662:Memtransistor
660:
658:
655:
653:
650:
646:
643:
642:
641:
638:
636:
633:
629:
626:
624:
621:
619:
616:
614:
611:
610:
609:
606:
604:
601:
599:
596:
594:
591:
587:
584:
583:
582:
579:
575:
572:
570:
567:
565:
562:
560:
557:
556:
555:
552:
550:
547:
545:
544:Deep learning
542:
540:
537:
536:
533:
528:
527:
520:
517:
515:
512:
510:
508:
504:
502:
499:
498:
495:
490:
489:
480:
479:Hidden Markov
477:
475:
472:
470:
467:
466:
465:
462:
461:
458:
453:
452:
445:
442:
440:
437:
435:
432:
430:
427:
425:
422:
420:
417:
415:
412:
410:
407:
405:
402:
401:
398:
393:
392:
385:
382:
380:
377:
375:
371:
369:
366:
364:
361:
359:
357:
353:
351:
348:
346:
343:
341:
338:
337:
334:
329:
328:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
296:
293:
291:
288:
286:
284:
280:
276:
275:Random forest
273:
271:
268:
266:
263:
262:
261:
258:
256:
253:
251:
248:
247:
240:
239:
234:
233:
225:
219:
218:
211:
208:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
160:Data cleaning
158:
156:
153:
151:
148:
146:
143:
141:
138:
136:
133:
131:
128:
126:
123:
122:
116:
115:
108:
105:
103:
100:
98:
95:
93:
90:
88:
85:
83:
80:
78:
75:
73:
72:Meta-learning
70:
68:
65:
63:
60:
58:
55:
53:
50:
48:
45:
44:
38:
37:
34:
29:
25:
21:
20:
8965:
8944:
8923:
8902:
8881:
8860:
8839:
8812:
8808:
8798:
8777:
8768:1810.09136v3
8756:
8711:
8702:
8698:
8688:
8669:
8656:
8631:
8627:
8617:
8598:
8564:
8543:
8498:
8494:
8484:
8465:
8461:
8451:
8430:
8379:
8375:
8365:
8349:(57): 1–64.
8346:
8342:
8332:
8239:
8235:
8225:
8190:
8186:
8176:
8159:
8155:
8145:
8109:
8106:Applications
8084:
8076:
8072:
7812:
7734:
7730:
7675:
7648:
7552:
7367:
7363:
7323:
7158:
6976:
6953:
6767:
6697:
6600:
6597:
6267:
5852:
5820:
5812:
5528:
5175:
4957:
4896:
4764:
4490:
4415:
4253:
4216:The inverse
4067:
3864:
3853:
3740:
3619:
3331:
3044:
2780:is equal to
2670:
2508:
2358:), we have:
2269:
2106:
1982:
1802:
1736:. Note that
1681:
1565:
1339:
1086:
1009:
985:
978:
969:
953:
951:
819:PAC learning
506:
355:
350:Hierarchical
282:
236:
230:
8080:typical set
4950:direction.
4064:Planar Flow
2063:determinant
703:Multi-agent
640:Transformer
539:Autoencoder
295:Naive Bayes
33:data mining
9009:Categories
8977:2008.12577
8956:1903.01434
8935:1906.12320
8914:2001.09382
8893:1912.01219
8872:2006.09347
8851:2109.10794
8789:1906.02994
8747:2008.10486
8723:1907.12998
8679:1806.07366
8608:2002.02798
8576:1810.01367
8555:2210.02747
8534:1810.01367
8508:1606.04934
8475:1705.07057
8442:1505.05770
8389:1908.09257
8356:1912.02762
8323:1807.03039
8297:1605.08803
8249:1912.02762
8208:11336/8930
8137:References
7671:neural ODE
7489:, we have
4088:, and let
1369:derivation
688:Q-learning
586:Restricted
384:Mean shift
333:Clustering
310:Perceptron
238:regression
140:Clustering
135:Regression
8648:0361-0918
8422:208910764
8406:1939-3539
8273:1410.8516
8091:bijective
8069:Downsides
8040:λ
8027:λ
7953:∇
7932:∫
7922:λ
7856:∫
7846:λ
7607:−
7599:±
7420:∈
7392:×
7379:∈
7333:∂
7286:∂
7278:∂
7252:∫
7248:−
7220:
7193:
7102:∫
7098:−
7040:∫
7006:−
6897:∫
6743:−
6738:θ
6711:θ
6675:−
6649:σ
6645:⋯
6620:σ
6610:σ
6558:∼
6513:−
6487:σ
6472:−
6446:μ
6423:⋯
6381:σ
6352:μ
6313:σ
6300:μ
6250:∞
6238:→
6230:−
6208:σ
6182:→
6174:−
6152:μ
6109:−
6083:σ
6068:−
6042:μ
6030:∼
6013:⋯
5971:σ
5942:μ
5930:∼
5899:σ
5886:μ
5874:∼
5690:∑
5623:∏
5480:↦
5403:θ
5355:θ
5325:∏
5283:θ
5275:−
5259:⊙
5236:θ
5228:−
5136:θ
5092:⊙
5069:θ
5020:θ
4864:θ
4856:−
4782:−
4777:θ
4750:θ
4728:θ
4677:θ
4597:θ
4508:∈
4468:−
4462:⟩
4450:⟨
4396:⟩
4384:⟨
4372:⟩
4360:⟨
4302:⟩
4290:⟨
4234:−
4229:θ
4195:⟩
4183:⟨
4151:θ
4096:θ
4040:^
4037:θ
3993:θ
3985:
3973:∑
3967:θ
3923:⋅
3915:θ
3818:θ
3807:
3784:∑
3775:θ
3712:θ
3683:∗
3654:θ
3584:θ
3573:
3550:∑
3536:−
3513:θ
3502:
3480:∗
3468:^
3456:−
3422:∗
3340:θ
3300:∗
3289:
3267:∗
3232:θ
3221:
3199:∗
3184:−
3164:θ
3135:∗
3089:∗
3062:θ
3012:−
2988:−
2950:
2927:∑
2923:−
2894:
2859:
2843:induction
2821:−
2802:−
2791:
2742:
2714:−
2593:
2587:−
2558:
2523:
2489:−
2317:−
2289:−
2245:−
2087:−
2010:−
1924:−
1767:−
1682:Consider
1538:−
1514:−
1476:
1453:∑
1449:−
1420:
1385:
1293:−
1173:−
847:ECML PKDD
829:VC theory
776:ROC curve
708:Self-play
628:DeepDream
469:Bayes net
260:Ensembles
41:Paradigms
8831:35860036
8414:32396070
8217:17820269
8100:Jacobian
7988:‖
7948:‖
7902:‖
7872:‖
5716:′
5698:′
4799:is just
4353:′
4283:′
4059:Variants
4026:RETURN.
2731:. Since
1367:is (see
960:used in
270:Boosting
119:Problems
8822:9295254
6268:By the
5786:. Here
3954:SOLVE.
2334:(where
2107:By the
2065:of the
2061:is the
1803:By the
852:NeurIPS
669:(ECRAM)
623:AlexNet
265:Bagging
8829:
8819:
8646:
8420:
8412:
8404:
8215:
8018:where
6954:where
6143:where
4715:where
3780:
3659:
2845:that:
2354:is an
1983:Where
1131:, let
998:Method
645:Vision
501:RANSAC
379:OPTICS
374:DBSCAN
358:-means
165:AutoML
8972:arXiv
8951:arXiv
8930:arXiv
8909:arXiv
8888:arXiv
8867:arXiv
8846:arXiv
8784:arXiv
8763:arXiv
8742:arXiv
8718:arXiv
8674:arXiv
8666:(PDF)
8603:arXiv
8571:arXiv
8550:arXiv
8529:arXiv
8503:arXiv
8470:arXiv
8437:arXiv
8418:S2CID
8384:arXiv
8351:arXiv
8318:arXiv
8292:arXiv
8268:arXiv
8244:arXiv
8213:S2CID
7649:When
7444:with
4897:When
956:is a
867:IJCAI
693:SARSA
652:Mamba
618:LeNet
613:U-Net
439:t-SNE
363:Fuzzy
340:BIRCH
8827:PMID
8644:ISSN
8410:PMID
8402:ISSN
8049:>
6725:and
6199:and
4491:Let
4465:>
4442:and
4430:tanh
2698:and
1834:is:
1709:and
1087:For
1010:Let
990:and
877:JMLR
862:ICLR
857:ICML
743:RLHF
559:LSTM
345:CURE
31:and
8817:PMC
8813:139
8636:doi
8394:doi
8203:hdl
8195:doi
8164:doi
7646:).
7217:log
7190:log
5755:det
4267:det
3963:max
3804:log
3570:log
3499:log
3286:log
3218:log
2958:det
2947:log
2891:log
2856:log
2788:log
2739:log
2601:det
2590:log
2555:log
2520:log
2430:det
2303:det
2278:det
2182:det
2069:of
1991:det
1905:det
1484:det
1473:log
1417:log
1382:log
1371:):
603:SOM
593:GAN
569:ESN
564:GRU
509:-NN
444:SDL
434:PGD
429:PCA
424:NMF
419:LDA
414:ICA
409:CCA
285:-NN
9011::
8825:.
8811:.
8807:.
8732:^
8703:32
8701:.
8697:.
8642:.
8632:18
8630:.
8626:.
8597:.
8585:^
8517:^
8499:29
8497:.
8493:.
8466:30
8464:.
8460:.
8416:.
8408:.
8400:.
8392:.
8380:43
8378:.
8374:.
8347:22
8345:.
8341:.
8306:^
8282:^
8258:^
8240:22
8238:.
8234:.
8211:.
8201:.
8191:66
8189:.
8185:.
8158:.
8154:.
7267:Tr
6695:.
6595:.
4762:.
4413:.
3982:ln
2111::
2104:.
1800:.
1084:.
952:A
872:ML
8980:.
8974::
8959:.
8953::
8938:.
8932::
8917:.
8911::
8896:.
8890::
8875:.
8869::
8854:.
8848::
8833:.
8792:.
8786::
8771:.
8765::
8750:.
8744::
8726:.
8720::
8682:.
8676::
8650:.
8638::
8611:.
8605::
8579:.
8573::
8558:.
8552::
8537:.
8531::
8511:.
8505::
8478:.
8472::
8445:.
8439::
8424:.
8396::
8386::
8359:.
8353::
8326:.
8320::
8300:.
8294::
8276:.
8270::
8252:.
8246::
8219:.
8205::
8197::
8170:.
8166::
8160:8
8052:0
8044:J
8036:,
8031:K
8006:t
8003:d
7998:2
7993:F
7984:)
7981:t
7978:,
7973:t
7969:z
7965:(
7962:f
7957:z
7941:T
7936:0
7926:J
7918:+
7915:t
7912:d
7907:2
7898:)
7895:t
7892:,
7887:t
7883:z
7879:(
7876:f
7865:T
7860:0
7850:K
7837::
7821:f
7788:1
7785:+
7782:n
7779:2
7774:R
7750:n
7745:R
7716:f
7696:f
7657:f
7642:(
7628:n
7624:}
7618:2
7614:/
7610:1
7603:n
7596:{
7576:)
7573:I
7570:,
7567:0
7564:(
7561:N
7537:)
7534:W
7531:(
7528:r
7525:t
7522:=
7519:]
7516:u
7513:W
7508:T
7504:u
7500:[
7497:E
7477:I
7474:=
7471:]
7466:T
7462:u
7458:u
7455:[
7452:E
7430:n
7425:R
7417:u
7395:n
7389:n
7384:R
7376:W
7349:f
7342:t
7338:z
7309:t
7306:d
7302:]
7294:t
7290:z
7281:f
7272:[
7261:T
7256:0
7245:)
7242:)
7237:0
7233:z
7229:(
7226:p
7223:(
7214:=
7211:)
7208:)
7205:x
7202:(
7199:p
7196:(
7167:x
7144:t
7141:d
7138:)
7135:t
7132:,
7127:t
7123:z
7119:(
7116:f
7111:T
7106:0
7093:T
7089:z
7085:=
7082:t
7079:d
7076:)
7073:t
7070:,
7065:t
7061:z
7057:(
7054:f
7049:0
7044:T
7036:+
7031:T
7027:z
7023:=
7020:)
7017:x
7014:(
7009:1
7002:F
6998:=
6993:0
6989:z
6962:f
6939:t
6936:d
6933:)
6930:t
6927:,
6922:t
6918:z
6914:(
6911:f
6906:T
6901:0
6893:+
6888:0
6884:z
6880:=
6875:T
6871:z
6867:=
6864:)
6859:0
6855:z
6851:(
6848:F
6845:=
6842:x
6819:)
6814:0
6810:z
6806:(
6803:p
6781:0
6777:z
6746:1
6734:f
6707:f
6683:)
6678:1
6672:n
6669::
6666:1
6662:x
6658:(
6653:n
6642:)
6637:1
6633:x
6629:(
6624:2
6614:1
6583:)
6578:n
6574:I
6570:,
6567:0
6564:(
6561:N
6555:z
6529:n
6525:z
6521:)
6516:1
6510:n
6507::
6504:1
6500:x
6496:(
6491:n
6483:+
6480:)
6475:1
6469:n
6466::
6463:1
6459:x
6455:(
6450:n
6440:=
6435:n
6431:x
6411:2
6407:z
6403:)
6398:1
6394:x
6390:(
6385:2
6377:+
6374:)
6369:1
6365:x
6361:(
6356:2
6346:=
6341:2
6337:x
6327:1
6323:z
6317:1
6309:+
6304:1
6294:=
6289:1
6285:x
6253:)
6247:,
6244:0
6241:(
6233:1
6227:i
6222:R
6217::
6212:i
6186:R
6177:1
6171:i
6166:R
6161::
6156:i
6127:)
6122:2
6118:)
6112:1
6106:n
6103::
6100:1
6096:x
6092:(
6087:n
6079:,
6076:)
6071:1
6065:n
6062::
6059:1
6055:x
6051:(
6046:n
6038:(
6035:N
6025:n
6021:x
6003:)
5998:2
5994:)
5988:1
5984:x
5980:(
5975:2
5967:,
5964:)
5959:1
5955:x
5951:(
5946:2
5938:(
5935:N
5925:2
5921:x
5913:)
5908:2
5903:1
5895:,
5890:1
5882:(
5879:N
5869:1
5865:x
5836:n
5831:R
5794:K
5772:W
5769:H
5765:)
5761:K
5758:(
5733:j
5730:i
5727:c
5723:y
5713:c
5709:c
5705:K
5695:c
5686:=
5681:j
5678:i
5675:c
5671:z
5659:.
5645:W
5642:H
5637:c
5633:s
5627:c
5602:)
5597:c
5593:b
5589:+
5584:j
5581:i
5578:c
5574:x
5570:(
5565:c
5561:s
5557:=
5552:j
5549:i
5546:c
5542:y
5509:)
5504:1
5500:x
5496:,
5491:2
5487:x
5483:(
5477:)
5472:2
5468:x
5464:,
5459:1
5455:x
5451:(
5431:x
5411:0
5408:=
5399:s
5376:)
5371:,
5368:1
5364:z
5360:(
5351:s
5346:e
5340:n
5335:1
5332:=
5329:i
5304:)
5301:)
5296:1
5292:x
5288:(
5279:m
5270:2
5266:x
5262:(
5254:)
5249:1
5245:x
5241:(
5232:s
5224:e
5220:=
5215:2
5211:z
5207:,
5202:1
5198:x
5194:=
5189:1
5185:z
5160:]
5154:)
5149:1
5145:z
5141:(
5132:m
5124:0
5118:[
5113:+
5108:]
5100:2
5096:z
5087:)
5082:1
5078:z
5074:(
5065:s
5060:e
5050:1
5046:z
5039:[
5034:=
5031:)
5028:z
5025:(
5016:f
5012:=
5007:]
4999:2
4995:x
4985:1
4981:x
4974:[
4969:=
4966:x
4936:2
4932:x
4911:1
4908:=
4905:n
4882:)
4877:1
4873:x
4869:(
4860:m
4851:2
4847:x
4843:=
4838:2
4834:z
4830:,
4825:1
4821:x
4817:=
4812:1
4808:z
4785:1
4773:f
4724:m
4701:]
4695:)
4690:1
4686:z
4682:(
4673:m
4665:0
4659:[
4654:+
4649:]
4641:2
4637:z
4627:1
4623:z
4616:[
4611:=
4608:)
4605:z
4602:(
4593:f
4589:=
4584:]
4576:2
4572:x
4562:1
4558:x
4551:[
4546:=
4543:x
4521:n
4518:2
4513:R
4505:z
4502:,
4499:x
4471:1
4459:w
4456:,
4453:u
4427:=
4424:h
4400:|
4393:w
4390:,
4387:u
4381:)
4378:b
4375:+
4369:z
4366:,
4363:w
4357:(
4350:h
4346:+
4343:1
4339:|
4335:=
4331:|
4327:)
4322:T
4318:w
4314:u
4311:)
4308:b
4305:+
4299:z
4296:,
4293:w
4287:(
4280:h
4276:+
4273:I
4270:(
4263:|
4237:1
4225:f
4204:)
4201:b
4198:+
4192:z
4189:,
4186:w
4180:(
4177:h
4174:u
4171:+
4168:z
4165:=
4162:)
4159:z
4156:(
4147:f
4143:=
4140:x
4120:)
4117:b
4114:,
4111:w
4108:,
4105:u
4102:(
4099:=
4076:h
4011:)
4006:j
4002:x
3998:(
3989:p
3977:j
3951:.
3937:0
3933:p
3929:,
3926:)
3920:(
3911:f
3888:n
3885::
3882:1
3878:x
3839:)
3836:)
3831:i
3827:x
3823:(
3814:p
3810:(
3799:N
3794:0
3791:=
3788:i
3771:x
3768:a
3765:m
3761:g
3758:r
3755:a
3726:]
3723:)
3720:x
3717:(
3708:p
3703:|
3698:|
3694:)
3691:x
3688:(
3679:p
3675:[
3670:L
3667:K
3663:D
3650:n
3647:i
3644:m
3640:g
3637:r
3634:a
3605:)
3602:)
3597:i
3593:x
3589:(
3580:p
3576:(
3565:N
3560:0
3557:=
3554:i
3544:N
3541:1
3533:=
3530:]
3527:)
3524:)
3521:x
3518:(
3509:p
3505:(
3496:[
3491:)
3488:x
3485:(
3476:p
3464:E
3433:)
3430:x
3427:(
3418:p
3395:N
3392::
3389:1
3386:=
3383:i
3379:}
3373:i
3369:x
3365:{
3317:]
3314:)
3311:)
3308:x
3305:(
3296:p
3292:(
3283:[
3278:)
3275:x
3272:(
3263:p
3257:E
3252:+
3249:]
3246:)
3243:)
3240:x
3237:(
3228:p
3224:(
3215:[
3210:)
3207:x
3204:(
3195:p
3189:E
3181:=
3178:]
3175:)
3172:x
3169:(
3160:p
3155:|
3150:|
3146:)
3143:x
3140:(
3131:p
3127:[
3122:L
3119:K
3115:D
3085:p
3058:p
3024:|
3015:1
3009:i
3005:z
3001:d
2996:)
2991:1
2985:i
2981:z
2977:(
2972:i
2968:f
2964:d
2954:|
2942:K
2937:1
2934:=
2931:i
2920:)
2915:0
2911:z
2907:(
2902:0
2898:p
2888:=
2885:)
2880:K
2876:z
2872:(
2867:K
2863:p
2829:)
2824:1
2818:i
2814:z
2810:(
2805:1
2799:i
2795:p
2768:)
2763:i
2759:z
2755:(
2750:i
2746:p
2717:1
2711:i
2707:z
2684:i
2680:z
2655:|
2646:0
2642:z
2638:d
2633:)
2628:0
2624:z
2620:(
2615:1
2611:f
2607:d
2597:|
2584:)
2579:0
2575:z
2571:(
2566:0
2562:p
2552:=
2549:)
2544:1
2540:z
2536:(
2531:1
2527:p
2492:1
2484:|
2475:0
2471:z
2467:d
2462:)
2457:0
2453:z
2449:(
2444:1
2440:f
2436:d
2426:|
2421:)
2416:0
2412:z
2408:(
2403:0
2399:p
2395:=
2392:)
2387:1
2383:z
2379:(
2374:1
2370:p
2342:A
2320:1
2313:)
2309:A
2306:(
2300:=
2297:)
2292:1
2285:A
2281:(
2254:|
2248:1
2240:)
2232:0
2228:z
2224:d
2219:)
2214:0
2210:z
2206:(
2201:1
2197:f
2193:d
2187:(
2178:|
2174:)
2169:0
2165:z
2161:(
2156:0
2152:p
2148:=
2145:)
2140:1
2136:z
2132:(
2127:1
2123:p
2090:1
2082:1
2078:f
2044:1
2040:z
2036:d
2031:)
2026:1
2022:z
2018:(
2013:1
2005:1
2001:f
1997:d
1967:|
1958:1
1954:z
1950:d
1945:)
1940:1
1936:z
1932:(
1927:1
1919:1
1915:f
1911:d
1901:|
1897:)
1892:0
1888:z
1884:(
1879:0
1875:p
1871:=
1868:)
1863:1
1859:z
1855:(
1850:1
1846:p
1820:1
1816:z
1788:)
1783:1
1779:z
1775:(
1770:1
1762:1
1758:f
1754:=
1749:0
1745:z
1722:0
1718:z
1695:1
1691:z
1656:K
1652:f
1648:,
1645:.
1642:.
1639:.
1636:,
1631:1
1627:f
1604:K
1600:f
1596:,
1593:.
1590:.
1587:.
1584:,
1579:1
1575:f
1550:|
1541:1
1535:i
1531:z
1527:d
1522:)
1517:1
1511:i
1507:z
1503:(
1498:i
1494:f
1490:d
1480:|
1468:K
1463:1
1460:=
1457:i
1446:)
1441:0
1437:z
1433:(
1428:0
1424:p
1414:=
1411:)
1406:K
1402:z
1398:(
1393:K
1389:p
1353:K
1349:z
1323:K
1319:z
1296:1
1288:i
1284:f
1258:K
1254:f
1250:,
1247:.
1244:.
1241:.
1238:,
1233:1
1229:f
1206:0
1202:z
1181:)
1176:1
1170:i
1166:z
1162:(
1157:i
1153:f
1149:=
1144:i
1140:z
1119:K
1116:,
1113:.
1110:.
1107:.
1104:,
1101:1
1098:=
1095:i
1072:)
1067:0
1063:z
1059:(
1054:0
1050:p
1023:0
1019:z
941:e
934:t
927:v
507:k
356:k
283:k
241:)
229:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.