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Polymeric systems exhibit a wide range of characteristic time and length scales.
This is readily illustrated for the example of natural rubber, i.e. melts of cis-PI chains with a typical length of {{formula:67ba0215-4e3d-4641-bf1f-ece78a024d2c}} Kuhn segments.
Important characteristic length scales comprise (i) the Kuhn length, {{formula:8b779ed3-0318-40c0-8a1b-9bdb7fa2718e}} , (ii) the tube diameter, {{formula:91920d59-e932-4a01-9531-249360d99375}} , (iii) the coil diameter, {{formula:a1d912a7-4d0e-4c71-b4c8-fec35b81551e}} , and (iv) the contour length, {{formula:ba613425-cc0a-4dd1-a7da-13f2de34dcf5}} .
The spread is even larger between the characteristic time scales.
There are already almost three orders of magnitude between the Kuhn time, {{formula:2e2b2068-6292-46b6-b398-b3c7d3c57b70}} , and the entanglement time, {{formula:2e60ef22-8fcb-4b0b-b051-6fb1baba7c47}} . The Rouse time of {{formula:8713024c-f812-4da0-ae38-1eb144362c9e}} governs fast processes such as the tension equilibration inside the tube {{cite:db833dc088c704887024fb140b44bbfc252f6c24}}, while the estimated disentanglement time is {{formula:e841b019-22e8-4efa-9397-669cf324f7f6}} .
| d | 8957b18e9d114967e2dd5b2e91ce79f3 |
Current experimental techniques have allowed the manipulation of atomic systems to previously unthinkable degrees, paving the way to the development of new technologies and the observation of very small quantum effects. One such technologies is the quantum computer; trapped ions systems have been implemented successfully to perform logical operation {{cite:15ba362e3759653ed3c37f7672b7c3f5a5245cb1}}, {{cite:e829ea253986922641091d0fad68fcfcd745eb67}}, {{cite:8563a526491f180d7cd1bcf27c5dfebfd0d80460}} making atomic systems a strong candidate for scalable quantum bits. Furthermore, the implementation of highly excited states (Rydberg atoms) has been proposed for quantum computing because of the length of their interaction and their long coherence times {{cite:8563a526491f180d7cd1bcf27c5dfebfd0d80460}}, {{cite:2dc7343a0d2ffe6b5700d123f874fb5f162541cd}}, {{cite:a1375f19f179c1d425dfa0f088c8e0880bb2a094}}, {{cite:e9d2959cf60e133d8965ef3901367d25496c89c7}}. In order for atoms be suitable for quantum technologies it is necessary for them to have long coherence times, which is ultimately limited by their interaction with environmental particles {{cite:da68b3b3139f1c03367307fb839b8a8794ee5c9a}}. In this paper we propose a new decoherence mechanism of excited atomic systems induced by particles through short-distance interactions. This would not only presents a fundamental limit to the stability of atomic systems, but also can be applied to the detection of weakly-interacting particles by means of analyzing the evolution of the atomic state. A search for exotic interactions using the decoherence of a Ramsey interferometer has been suggested previously {{cite:6792b7b685952c8dcf4d1e62e2426b937a7902af}}, {{cite:963d27f111167fb660072d3c62b71c97484a802c}}, however the mechanism for linking the decoherence of internal degrees of freedom of the atom (energy levels) with momentum transfer was left open.
In our work, we propose that when a particle is scattered by the atomic nucleus it will produce an almost instant change in the position of the nucleus which will be perceived by the electron in the atom as a sudden change in the electrostatic potential, projecting the wave function of the atom into the eigenstates of the new potential.
First we calculate the change in the state of the atom as a result of the nuclear displacement, obtaining a projection over lower energy levels only. We then express the order of the displacement in terms of the properties of the nucleus and the scattered particle, observing that the effect is more prominent for Rydberg atoms. Finally, we extend the analysis to multiple scattering events and analyze the evolution of the state of an atom scattering photons and massive particles.
| i | 5fbf23a8db1b682667be99bb68887862 |
where {{formula:d11af20e-787c-48d1-9185-e38285733f4c}} and {{formula:a88ce351-125a-470a-8ff2-281009bda008}} . The primal-dual hybrid gradient (PDHG) method (see, for example, {{cite:751e6e614401be402e6aa0c246f8f2dbf6e41f01}}, {{cite:10948d814c1bf0c46b37ad09237374fbeffd7141}}) is one of the popular methods for solving problem (REF ) and it is a preconditioned proximal point algorithm with {{formula:449c3d38-2fc3-4a0b-8a59-aa10874de0dd}} for the saddle subdifferential operator of {{formula:8f341bc5-d7a8-44cc-bd1f-d103951c7587}} given in (REF ) (see, {{cite:46b9812ab2ae26a875f55614fc8dfdc960355849}}, {{cite:0ab65289bd0dca83ac30611be39a18a054a4c5d4}}). Given the associated preconditioner as
{{formula:c1e93225-c705-4485-a73a-81d13a480e7b}}
| m | ac1909a0a2ec398d09b1f31ad3ec9269 |
However, retrieving information from the electron spectrum beyond {{formula:01651ede-abdd-4ea9-b229-98140232a14a}} in general requires an automatized, data-driven approach.
This is all the more the case because PIC simulations are computationally (potentially very) expensive, which motivates researchers to this day to improve PIC simulation codes, for example, algorithmically or by improved hardware utilization {{cite:0ce97fd7d16da952cfb6b011dc1a6aa79f249102}}, {{cite:951d3ca2cdebb45d0c3fb165467411d4e16352b2}}.
Therefore, employing machine learning (ML) algorithms and ML-based surrogate models is essential to decrease the overall computational effort which would otherwise be needed due to the necessity of performing an excessive amount of simulations.
For example, Djordjević et al. used deep learning to predict the time evolution of ion cutoff energies and electron mean kinetic energies in overdense laser-ion acceleration {{cite:5132d091f9494e2d757f8ab49ee2a1924e2524bd}}.
| i | fe0ca64f9291dd0b6350930678652ae0 |
For all neural network models, we choose a validation split of 10%, a maximum number of epochs of 200, and an Adam optimiser. We also investigate the effect of different learning rates on the training of the models, with eleven logarithmically spaced values in the range [0.1,0.000001]. We make sure that the chosen batch size is not larger than the tenth part of the training set. We assign a batch size of 8 to the original (non-augmented) datasets and experiment with sizes of 128, 512, and 1024 for augmented datasets. We choose cross-entropy as the loss function for both the MLP and the CNN (TCNN already includes the triplet loss). We choose all these parameters and parameter ranges in light of good practices for CNNs {{cite:24ab5c02c67c0a5337ea02b53cdcc6b9d361819c}}.
| m | 87271f75d20c5d390fcd74c7bff5c2e4 |
3D molecular structures contain spatial information important to molecular property prediction, such as bond angles and lengths.
Many recent methods try to answer how to utilize this 3D information in the GNN model.
For instance, Klicpera et al.{{cite:6efcc961bc45fb22c6f0382cc3ef8274c133ff0d}} proposed to let message embeddings interact based on the distance between atoms and the angle between directions to improve quantum mechanical property prediction.
Lu et al.{{cite:93ecfa9871e98bc65b468a5ef5d7be342fe1a5bd}} proposed a Multilevel Graph Convolutional neural Network that extracts features from the conformation and spatial information with multilevel interactions.
Liu et al.{{cite:e2a2d2263208169a9c85618c5a38d0077f757180}} proposed SphereNet, a 3D graph network framework with spherical message passing.
3D-based methods achieve significant performance improvements. But obtaining 3D structures is time-costing, which reduces the value of 3D-based methods in large-scale applications.
| m | b2c64f97ce258061e8b327354f898969 |
Finally, Credence's evaluation is conditional on the assumptions encoded by the users in learning the DGP, and therefore its diagnostics are as good as these assumptions. For instance, Credence assumes no interference, and no measurement error; if this were not true, the expected performance of candidate estimators of the ATE provided by Credence may be invalid. However, Credence is sufficiently flexible that such evaluation can in principle be conducted under identifying conditions that may be varied in a form of sensitivity analysis.
Background on Variational Autoencoders (VAE)
Autoencoders refer to a particular machine learning estimator that aims to learn a lower dimensional representation of the data which provides a one-to-one mapping between the original data and the lower-dimensional representation {{cite:4b73ce217076ff26646e5d6b4904f14ac7137df3}}. Variational autoencoders (VAE) extend this idea to learn a representation of a high-dimensional complex data set as a standard normal distribution in a lower dimensional latent space. This allows sampling from standard normal distributions in the latent space and projecting it to the original high-dimensional space while ensuring that the distributional properties of the projection and the original data are virtually identical {{cite:c75e852d716d2df965c98867c7f9026fb89dfbb6}}. Typically, a VAE constitutes two parts – an encoder and a decoder. The encoder takes in the data
in the original high dimensional space ({{formula:248251fa-1fb5-495b-8de4-80cb67cb15dc}} ) and maps it to {{formula:b625ebc0-28e6-40b6-9715-7d184228df7b}} , a lower dimensional latent space. A decoder performs the opposite operation to that of the encoder, mapping a vector {{formula:6ea3f9d1-f63f-4213-9098-734e38550124}} in lower dimensional latent space to vector {{formula:9d2e3c59-0c44-471a-b397-623b436e1317}} in higher dimensional space of the original data. The learning algorithm includes (1) encoding the data {{formula:8c6686d4-ec34-4cc7-8a54-ba473ab42e66}} as a vector in low-dimensional latent space {{formula:acbb133b-f61e-4756-8c60-68731ba5c60a}} , (2) sampling random vector {{formula:c861431d-0398-4060-98f7-b6ba215557a4}} in the latent space from a multivariate normal distribution with mean {{formula:1dafaaf7-f7b0-4278-9c6c-2ae677032b43}} and covariance matrix {{formula:1977e31e-7009-409c-a054-bcbbc358ff29}} derived from the encoded vector, and (3) decoding the sample vector {{formula:96e80703-33ba-4248-99f3-a1c87af234d6}} from the latent space by projecting it into the space of original data. The VAE loss function has two parts: (i) reconstruction loss (similarity of the input and the decoded output), and (ii) KL divergence between the normal distribution {{formula:440d921f-edba-4871-a2c3-f5d466c2cb38}} and standard normal distribution {{formula:7abef5c8-e468-485c-b954-9e8b403db1db}} . The reconstruction loss enforces similarity between the empirical distributions of the original and generated samples, while KL divergence ensures that the distribution of latent vectors is as close to a standard normal distribution as possible. This ensures that sampling from standard normal distribution and decoding will have distributional congruence with the original data.
Data Generative Procedure for Synthetic Data Experiments
Quadratic DGP:
The pre-treatment covariates {{formula:4e391b3c-526d-4d52-b628-c9eb2736f001}} are sampled from a multivariate normal distribution {{formula:65618026-c60f-4813-8199-694c22f0beaa}} , and the potential outcome function and treatment selection function are defined as follows:
{{formula:fe16b03b-1571-4afc-8a1f-a937d250626f}}
Friedman's DGP:
This DGP {{cite:7f1f85d3ebff16357057700873dfb79a74738b3a}} was first proposed to assess the performance of prediction methods. We augment Friedman's simulation setup to evaluate causal inference methods. The pre-treatment covariates are sampled from the standard uniform distribution. The potential outcome {{formula:a9cd2bdf-8ebc-477a-9ca3-d3732e544359}} is defined by Friedman's function {{cite:7f1f85d3ebff16357057700873dfb79a74738b3a}}, {{cite:aaf41942e9f1d41bdfa9e9a23a0dd665167a24ea}}. The expected treatment effect we study is equal to the cosine of the product of the first two covariates scaled by the third covariate.
{{formula:1acf33db-03f0-42be-a50c-184b4ced2480}}
{{figure:d3a467fd-964d-4def-96ee-b9a7b93a408d}}
Credence's Goodness of Fit
In this section, we discuss if the Credence data generated using the learned DGP for Lalonde data and Project STAR data is comparable to the observed real data. We have argued in the paper that the comparing first and second moment is not a sufficient metric of similarity. However, in this section, we compare the correlation matrix of the generated data with the real data because it is easy to visualize and communicate. As we standardize the data removing the mean from each covariate and scaling it by the variance of the same, the vector of means for covariates in the observed and generated data is anchored at 0. In Figure REF (a) and (b), we show that the correlation matrices (a proxy measure for second moment of the distribution) of the Credence generated data under constraints {{formula:b6b33f98-092c-4351-a117-c895c8c62212}} and {{formula:ae07bb42-a71e-4f1a-9ee2-8343e9275337}} is visually extremely similar to the correlation matrices of the observables in the real data. This provides a convincing evidence that Credence's generated data has similar distributional properties.
Implementation of Causal Methods
In this section, we discuss our implementation of causal inference methods studied in Section such as double machine learning (DML), doubly robust estimation, propensity score matching, causal BART, causal forest, metalearners and TMLE. We used the existing libraries and packages
We used the commonly used MatchIt's implementation of propensity score matching {{cite:5b34dbed3a23ed92f990c0db996c9ff7aa8bded7}}. We chose the method to match with replacement for estimating ATE.
Our paper uses causal forest implementation from the `grf' R package {{cite:c1d89fa52ff58e6232dcc4457ef5625e3758fab1}}. We used the default setting designed by the developer with 2000 number of trees and {{formula:4dd42094-52d5-48c7-b7e1-9d6f873b9eb5}} variables tried per split.
For causal BART, we use R implementation of BART by Vincent Dorie {{cite:f5984bed9847aed5013a763884d0c567d7809eb9}}. We only use the method with default hyperparameters.
Our implementation of GBT DML and linear DML used EconML's implementation of these methods {{cite:54f2fc3630c9dc69e7c1c98472bb7d06833efdad}}. We used the scikit-learn's machine learning API for the same {{cite:f257218d28b04e1e792f6012f6d3ab8e8c34cdb0}}. For GBT DML, we used the method with 100 trees, and the linear DML used ridge regression.
Further, we used EconML's implementation of metalearners such as S learner, T learner and X learner {{cite:54f2fc3630c9dc69e7c1c98472bb7d06833efdad}}. Similar to DML, we used scikit-learn's ML API to for gradient boosting trees and ridge regression {{cite:f257218d28b04e1e792f6012f6d3ab8e8c34cdb0}}.
We used Paul Zivich's implementation of TMLE {{cite:3af8c34f538ef5c01db95c4529d3f55ed785dffc}}.
| d | 515e01f6b523137035bf2ab60e0b91c8 |
Fairness in artificial intelligence (AI) is a relatively new but fast-growing research field which deals with assessing and addressing potential bias in AI models. For example, an early landmark paper {{cite:d0e7a152db66eb78a5932dc200bc586c62610859}} found differences in performance of a video-based gender classification model for different racial groups. With AI models starting to be deployed in the real world it is essential that the benefits of AI are shared equitably according to race, gender and other demographic characteristics, and so efforts to ensure the fairness of deployed models have generated much interest. Most work so far has focused on computer vision problems but some applications in healthcare are starting to emerge {{cite:057b07681e07106793c7e4bfd74db1f3555176a4}}, {{cite:2cee1e6970e72c8f3e75f570b49d64aba13aa6b7}}.
| i | c86bf2251676a0b7398529f6d408416e |
We detected molecular emission from multiple strong CH{{formula:7ea42551-8fb9-4d8f-973c-1b0689394b87}} OH transitions near 241.7 GHz as well as continuum emission from dust in the coma of Wirtanen. Molecular line emission was modeled using a three-dimensional radiative transfer method based on the Line Modeling Engine {{cite:0bfc80f2521d87b479f39e3680de3ff73b65fb3a}} adapted for cometary atmospheres, including a full non-LTE treatment of coma gases, collisions with H{{formula:d784a6d5-9f51-4332-9a16-b6d5190b4a5b}} O and electrons, and pumping by solar radiation {{cite:d26e5546cf51c671b8fc73a0892b708236566f18}}. We calculated the number density of molecules released from the nucleus as a function of distance following the Haser formalism {{cite:94fa4171003ea8073deee9c10af030ad55ee107c}} as
{{formula:548d1234-13ab-4209-bbac-e2d6d70668c0}}
| r | e4e88621c8081161bb6a7f1b43d57849 |
Based on this approach, we need to model the networks' global context issue as an optimization problem. Take the convolutional neural networks (CNN) as an example for further discussion. The networks output a tensor {{formula:fa188106-2ece-42a0-b2a7-e6e055e440c1}} after we feed into an image. Since the tensor can be seen as a set of {{formula:c1a8dcd8-2eae-4846-837e-fe657db2f486}} {{formula:43672ac4-4d57-4e70-824f-d5983cef3f8c}} -dimensional hyper-pixels, we unfold the tensor into a matrix {{formula:509fd958-d267-4bf6-8140-ff22f3d8db73}} . When the module learns the long-range dependencies or the global context, the hidden assumption is that the hyper-pixels are inherently correlated. For the sake of simplicity, we assume that hyper-pixels are linearly dependent, which means that each hyper-pixel in {{formula:3975e3c7-1aca-42f9-85e0-fa226f296c01}} can be expressed as the linear combination of bases whose elements are typically much less than {{formula:85d3141b-c790-4a3d-b83c-1149db5c6091}} . In the ideal situation, the global information hidden in {{formula:9264fe54-abc1-4ae9-a93c-7a6ffa3f52a7}} can be low-rank. However, due to vanilla CNN's poor ability to model the global context {{cite:996cebd0aad6bf99889993b43915da1cfc435157}}, {{cite:de0d3c9b391dd2f3cce1e8f693ecf6e959499d38}}, the learned {{formula:17a06bc8-1e1c-4d6a-a3c1-3edeba1f0f7b}} is usually corrupted with redundant information or incompleteness.
The above analysis suggests a potential method to model the global context, i.e., by completing the low-rank part {{formula:62cdeeed-e11d-435f-9800-9bf949074cc1}} in the unfolded matrix {{formula:f7a2addd-eed6-4815-b60e-aa734bddfbab}} and discarding the noise part {{formula:d002b96a-9a2f-4792-83e0-2566d775f76a}} , using the classic matrix decomposition models described in eq.md.model, which filters out the redundancy and incompleteness at the same time.
We thus model learning the global context as a low-rank recovery problem with matrix decomposition as its solution.
Using the notion of sec.warm up, the general objective function of matrix decomposition is
{{formula:53648a17-305a-4290-83a6-a1115d6e9392}}
| m | ac8ed2254cbe7e19e6b72ce63886a5cc |
Generalization to different net architectures. We transfer the learned LR schedules for different net architectures training. All the methods are trained on CIFAR-10 with different net architectures. The hyper-parameters of all methods are the same with the setting of CIFAR-10 with ResNet-18.
We test the learned LR schedule to different configurations of DenseNet {{cite:315d06a6883f7d9a3ed2ffc6d2ac150a2e9645a0}}. As shown in Fig. REF , our method perform slightly stable than MultiStep strategy at about 75-125 epochs. This tends to show the superiority of adaptive LR to train the DenseNets. Also, we transfer the LR schedules to several novel networks, the results are presented in Fig. 8 in the main paper.
| r | c613636e65f26ea3c9aece21daa729c7 |
In order to accomplish our goal, we are going to apply the algebraic formalism developed by Ruzhansky and Turunen {{cite:c266a138b00291ad6583b200d0e528a773d26fa9}}, where the notion of a global symbol has been introduced for describing the Hörmander classes of pseudo-differential operators, see Hörmander {{cite:b4e72bc5042edcd6df0a3c09ce984123b5c27724}}. In such a formalism, the symbol of an operator is globally defined on the non-commutative phase space {{formula:66365ee5-6efe-481e-8bff-707c6edcd4e0}} where {{formula:f7d0e5b7-6401-4469-bd56-8dd41276ea40}} is the unitary dual of {{formula:2c6a0a6c-8f1f-4444-81cc-61dffe46d614}} instead of the classical local notion of a symbol defined by charts, which is defined on the cotangent space {{formula:faf1c613-89e8-4e67-9834-d9bb886d1d47}} In particular the paper {{cite:e43904da54061a635ed43b803cd064f7b5643fe5}} by Ruzhansky, Turunen and Wirth is a source of many open problems, between them, the problem of computing geometric invariants of {{formula:07b2fa07-dfd9-424e-a056-f37e199d737b}} in terms of the matrix-valued symbols, where the Wodzicki residue is a fundamental one on the list.
| i | f43c906b668feee1cd50f9b4e36c248c |
We extend the universal approximation theorem of {{cite:9b4dec58093331e11410f07c454cea484e34af42}} to Theorem REF , where we show that given any measurable operator {{formula:0aa13cbb-e2cb-4850-871b-ec9278a0fa04}} , for {{formula:fdd8e24f-e10f-40c5-b941-23f4f7e91044}} , {{formula:c3e653e1-1142-4a8a-ae98-6003a0e12623}} , with respect to an underlying measure {{formula:3b7d8aae-1254-4057-89a0-b24ef084d5ed}} , there exists a DeepOnet of the form (REF ), which can approximate it to arbitrary accuracy. In particular, we remove the continuity (of {{formula:d03c0041-e906-4d3d-869c-785680af3b7e}} ) and compactness (of subsets of {{formula:0b8423ef-68e4-47c4-91e2-7abbe4452108}} ) assumptions of {{cite:9b4dec58093331e11410f07c454cea484e34af42}} and pave the way for the application of DeepOnets to approximate operators that arise in applications of PDEs to fields such as hypersonics {{cite:cf055b97a1553465881a89d4c4261cb7761b1f89}}.
We provide an upper bound (REF ) on the DeepOnet error (REF ) by decomposing it into three parts, i.e., an encoding error (REF ) stemming from the encoder {{formula:2c3dd7d5-4ff5-46b5-a2d1-d44d317e59fc}} , an approximation error that arises from the approximator neural network {{formula:fa95897a-b50c-4115-b03a-2bdf586a9b1f}} that maps between finite-dimensional spaces and a reconstruction error (), corresponding to the trunk net induced affine reconstructor {{formula:3e243956-c9aa-4f18-a3fd-23819f2c7f6b}} (REF ).
In Theorem REF , we prove lower bounds on the reconstruction error () by utilizing optimal errors for projections on finite dimensional affine subspaces of separable Hilbert spaces (Theorem REF ). This allows us in Theorem REF to prove two-sided bounds on the DeepOnet error (REF ). In particular, the lower bound is explicitly given in terms of the decay of the eigenvalues of the covariance operator (REF ), associated with the push-forward measure {{formula:6f3f7c66-aff3-44cc-a39c-ba1c57e54ae6}} (REF ). Moreover, this construction also allows us to infer the number of trunk nets {{formula:1edf170e-6187-4b34-8cce-9196994e8c41}} and that these trunk nets should approximate the eigenfunctions of the covariance operator in-order to obtain optimal reconstruction errors. Furthermore, we also provide bounds (REF ) on the reconstruction error that leverage the Sobolev regularity of the image of the nonlinear operator {{formula:4b30814c-8554-4394-8674-681316219117}} .
To control the encoding error (REF ) corresponding to the encoder {{formula:07d446f4-9d07-407c-898c-ba202ea3346e}} , which is a pointwise evaluation of the input at {{formula:178e87fe-b56d-4535-8abd-08eb7023d71b}} sensor locations, we construct a decoder {{formula:245c8525-a630-410c-bd4e-2a142cd1e284}} (approximate inverse of the encoder) (REF ). We show in Corollary REF that sensors chosen at random on the underlying domain {{formula:956f8c9d-e8f4-4ecf-814d-bc494288f0fc}} suffice to provide an almost optimal (optimal modulo a {{formula:cefc0300-83af-4f2c-ba6e-160f819c19a3}} ) bound on the encoding error. This further highlights the fact that DeepOnets allow for a general approximation framework , i.e., no explicit information is needed about the location of sensor points and they can be chosen randomly.
Finally, estimating the approximation error () reduces to deriving bounds on a neural network {{formula:1d4270fd-6f28-4886-8585-523e6e8b9bd8}} that maps one finite (but possibly very-high) dimensional space to another. Hence, standard neural network approximation results such as from {{cite:12ee14511bdbef98452a1ad8c72c1e7ae3834a50}} can be applied. In particular, approximation results for holomorphic maps, such as those derived in {{cite:48a5bcefba389a9c0dad3cf31e1c84205a4aee41}}, {{cite:10cc69ea3b728f7df447be8873cb535ebf59191b}}, {{cite:6f8937c39a1a26636dc9d5f65aae2d97ebf87b9b}} are important in this context.
| d | 3c660a55bf72426286aee72817fce2ce |
Remark 1 Here we provide a sufficient condition on which {{formula:11a8db31-497e-4f49-abc2-6dc551f4ac21}} implies {{formula:b9abc043-bc21-4270-976d-8f2bbf76a7e8}} weakly, this condition can be found in {{cite:e4020d225829479f9a3e18666999a089637347fc}}. Since Mirrored Stein Fisher information depends on the target distribution {{formula:443fa977-32b7-43b2-a101-f6ad635de119}} , mirror function {{formula:068dd1e1-2ade-4a43-ba24-c9d8aeee9160}} and kernel {{formula:5c1f24af-ddc4-494a-b614-c3df5efe5c4c}} , we need the following two properties on {{formula:4314c7e1-9fa9-415a-ae2f-c93e36bb0705}} and {{formula:1c6c008f-66af-4c92-a5c6-edd64b2dfcc4}} respectively:
| r | 321f061358fc702b0bc17b045bd17eeb |
The methodology we present is rather general. We find that classification trees and random forests work very well in conjunction with the 0–1 loss. They are less suitable for loss functions that directly depend on the posterior model probability such as the multinomial deviance loss. However, one may use any other classification method that is quick and leads to accurate predictions for the application at hand. For example, logistic regression provides natural and smooth estimates for the posterior model probabilities, but it is also less flexible due to the linear form of the predictor. Generalised additive models may improve the accuracy of logistic regression at the expense of a higher computing time. Other fast classification methods include linear discriminant analysis and its extensions like mixture and flexible discriminant analysis. If a higher computing time for the classifier is acceptable and a high predictive power is desired, more elaborate methods such as neural networks may be applied.
In general, for most applications it will be preferable to use a classification method where the optimal choice of the tuning parameters is insensitive to the selected design or where standard settings are available that work reasonably well in most circumstances. Otherwise the optimal tuning parameters have to be determined for each new design, for example via cross-validation. Apart from choosing different classification methods, one may also consider different loss functions. The choice of the loss function determines the functional form of the penalty for not correctly estimating the true class. Alternatives to the 0–1 loss and multinomial deviance loss include the exponential, logit, and hinge loss functions. For an overview of all the aforementioned methods and loss functions, see {{cite:7fbc48ff04f300fbf095833333b8380491d2b9cb}}.
| d | 94e88f2ad0b83c304293f1cd2e9a1c12 |
(1) Datasets. Due to the relative shortage of professional sketch artists, achieving large numbers of images remains an open problem, impeding the development of FSS.
Furthermore, more diversified sketch (or drawing) styles are needed for building more attractive models and achieving better synthesis results.
To address these issues, we believe novel data augmentation techniques {{cite:42de6fc9f3c2f9dd0263998dc27beaf0eb86021a}}, {{cite:d3531e96dd68fd3be332653f3462f841a07d89df}}, {{cite:1f14b7862876249fddf15d630a98a84d235127aa}} and transfer learning strategies {{cite:5a7c7ebee2ab08a2f101d5f4e23966c161880a26}}, {{cite:0425f6d215f61a38ad6b9595b01bbc5e73fad7c6}}, {{cite:c7e40d2d3b099e0ea14bf02b92aa82cffe57a715}} specifically designed for FSS are promising directions of study.
| d | afc48740e1cdd7fbf2becd7d48a9f099 |
A central quest in the field of opinion dynamics is to understand how opinion exchange among individuals in a social network can give rise to emergent social phenomena such as consensus formation, polarization, and fragmentation of opinions.
Due to advances in information technology and accessibility to massive social data, empirical studies of opinion dynamics have received increasing attention and provide insights into developing quantitative models of opinion dynamics {{cite:a924a73bff214b3609d49da8c8634c3915ab83f5}}, {{cite:9a6d4656748a56254f49c842d84c59cfda4f8dd0}}, {{cite:93fdbe2ba651633e0b51bd98f36abe5182c2cc9e}}, {{cite:94fd9cf540d70e37cf7ccba0828b640a568da910}}, {{cite:9075fe909dc510f13ec108f3cc26e2e4165ff3b1}}, {{cite:87a9ab5697da6a8819513dcd655dbf86c8622cb6}}, {{cite:de19af13954e83c8969c82f0e37ed6af61c5a4a4}}.
Even before the social media age, many theoretical models had been proposed to explain social phenomena.
Among the most studied ones are the Voter model {{cite:d63cff05fdece0336e72144e949bd61e88144ba3}}, {{cite:d899284c2ccfc5565628690bdc231e366a53330d}} and its variants {{cite:7b12f2429d5648a63437ee89ecc4e9a312942c57}}, {{cite:1c6ab457327557bcb8b1d02a701d91bfe05d3ed0}}, which describe dynamics of discrete opinions in a population, such as electoral votes for political parties, through the lens of interacting particle systems {{cite:b23dd3c60e6bac7cfb1f5f3ec0076cfc53ebeea1}}.
Although these models are highly idealized, tools from discrete classical spin models in statistical mechanics can be employed to elucidate that consensus formation is a collective phenomenon, similar to how a magnetic ordered phase can emerge from microscopic interactions among discrete-valued spins in magnetic systems {{cite:7b12f2429d5648a63437ee89ecc4e9a312942c57}}, {{cite:27301477ffd511f165c72b21151c7927f890ad5d}}. Despite a clear connection to discrete-spin systems, Ising-like models of opinion dynamics are not appropriate when opinions are real valued and continuous, especially in financial markets. There, information is also exchanged simultaneously, not just pairwise.
| i | ea0e48a9e4dc203dd24d07cbf850b8dc |
Bin Center Predictor (BCP):
Previous works {{cite:4d93a246d8b0b1eec16277242ba8c480d3ff53e2}} have used a vision transformer (ViT) to predict bin centers that discretize the image depth into a fixed number of intervals. ViT divides the image feature map into {{formula:556dda8d-c195-4d40-940e-20e2e7e9bb70}} patches and uses self-attention layers to exchange information among the patches. The first embedding is passed through an MLP head to predict the bin centers. Instead of decoding the feature map to high resolution and then using ViT, we propose to use the initial pixel queries to predict the bin centers. Apart from being more efficient, the proposed design helps in embedding the depth information into the pixel queries via direct ground truth supervision.
Our BCP module consists of a simple Global Average Pooling followed by an MLP layer to predict the bin widths {{formula:8d162a59-8f8c-45f9-ab9b-a9bbc94af51a}} of dimension {{formula:0bddb867-3d45-416a-b61a-4ca8ce1fb093}} . Here, {{formula:a843d75a-f9ce-47ce-9634-13e1d6a9b23a}} denotes the number of adaptive bins per image. We use {{formula:449f4f4b-3751-4dbf-ae76-1b73c7e12732}} for our model as suggested in {{cite:4d93a246d8b0b1eec16277242ba8c480d3ff53e2}}.
Given the pixel queries {{formula:41e06bd3-54ce-4072-9df6-119d2d1cc4bb}} of size {{formula:ad911d4d-d8b0-4ab7-a26f-6480f758ef76}} , we predict:
{{formula:992e2df3-3943-4e12-9544-384399d63b13}}
| m | d5e2c3b1d1b858d07dbce786811ed194 |
Limitations. For compute reasons we have only studied the effects of networks on the SAC algorithm — which is a state-of-the-art algorithm for continuous control underlying many popular algorithms – nonetheless, there are plenty of other algorithms. The idea of obtaining gradients through subnetworks is common and algorithms such as DDPG {{cite:9a435125d09d30c012ca2d0eeb424886bb8fde42}} and Dreamer {{cite:b886cb90676db6d747669dd53f5ab0fcddeab813}} might also benefit from smoothing. Spectral normalization {{cite:b4562fb2cdf60bf060b586054c3e18f958a62c42}} is a relatively straightforward smoothing strategy and many alternatives which might perform better are known. We emphasize that the goal of our paper is to demonstrate that enabling large networks is an important and feasible problem, to not provide a solution for every conceivable RL algorithm. Finally, there are also further environments to try out. We have focused on continuous control from pixels as it is a setting relevant for real-world applications, but other common benchmarks such as Atari games {{cite:ee4984b34db0c1e083558bb1dcfc7beb2eb82f3d}} and board games {{cite:0f6d17679cc3c9daccb95332897f6f9d88abf82f}} are also important.
| d | 3c954497b9dd0df23817983561115e25 |
Since every FNE is NE (see Lemma REF (i)), we apply,
for every {{formula:409006fb-b8b9-46e7-a4f2-b79ca1d42c7a}} {{cite:6f03e7606cb80d19e09346aa7c9d357fe5d33686}} to
the family {{formula:6b381b8b-1ec5-4a2f-8b39-df5632db6c31}} and conclude that the string
operator {{formula:94f137e0-6b08-41da-afce-c41842c1ccd1}} is NE. Thus,
from (REF ) which guarantees the non-emptiness of
{{formula:50b9840b-8526-42b8-b791-756a9891f4d4}} and due to the fact that every NE with a fixed point
is QNE, the family of operators {{formula:7c261f81-cd13-4d19-bb49-e7cbb07286a4}} is a family
of QNEs.
| m | 7ae4aaac5cc551ee02b1566a5bdd9086 |
Note that it is impossible to decide NWT in time {{formula:442eedb9-ea84-425f-8837-6101dc37d45f}} , so when {{formula:a46bc525-5584-40ab-ac0a-27ddc561832b}} is polynomially bounded, our algorithm has only polylogarithmic overhead over decision. Note also that {{cite:22173d1e4c2925c2cf6c8ee2911fcec5e01102bc}} provides a subcubic reduction from listing negative-weight triangles to NWT, although it has polynomial overhead and so does not imply our result. Together with an algorithm of Williams {{cite:09283ac4fe5caa39b39cf05890c3c578d0804a24}}, Theorem REF implies the following.
| r | 4d858324ad3d9b652cacabb12b376c85 |
The standard object detection frameworks {{cite:2c482ba339d6cb02e6bbbc107ab56b5cea5cf9a9}}, {{cite:410881a7c394643029ca1b5dc11971afa8b60d7a}}, {{cite:894d514cfa9c610ca8718ff416432f8434d2435b}}, {{cite:549c3a88c00466495fe8d9b618c491bbd185aa36}} can be characterised as a function ({{formula:b06c8d41-548f-40e7-84b3-0c5dae9b19ed}} ), that takes an input image and transforms it into a set of bounding boxes enclosing objects, each of which is classified into one of the classes, known a priori. {{formula:fcafddc5-2491-4c9e-a595-fbd026a3e9c1}} is trained on large amounts of annotated data corresponding to each class, using variants of stochastic gradient descent.
A class-incremental object detector relaxes the constraint that all the class data is available beforehand. As and when new class information is available, the detector should modify itself to be competent on detecting the new classes along with the old classes
by combating itself against catastrophic forgetting {{cite:f48cbc83cc70a0ca1772e4289169a237d3de0bce}}, {{cite:82ae6234fed7f76a1ff280323609c82653919eb8}}.
| m | bbd1f0e9cc93329b0ff46418640a1910 |
We list the BLEU scores of our proposed model in Table REF .
Moses-1 {{cite:8f687545b4e934a72708c41d8a7e3b7eab4729d5}} is the state-of-the-art phrase-based SMT system with the default configuration and a 4-gram language model trained on the target portion of training data.
Moses-2 is the same as Moses-1 except that the language model is trained using the target data plus 10M Xinhua portion of Gigaword corpus.
The BLEU score of our NMT baseline, which is an attention-based NMT as introduced in Section , is about 4.5 higher than the state-of-the-art SMT system Moses-2.
{{table:0eb23239-cfd7-487a-bdde-9f74efeecff8}} | r | bfeb05cca8df5d1e4ed73962dce5b911 |
HA historical average {{cite:cdb2b32f88c839b3d563586c334f3de7ecac2a5e}}. HA models the data as a seasonal process and computes the predictions as a weighted average of the previous seasons.
ARIMA integrated moving average model with Kalman filter which is widely used for time series forecasting.
VAR {{cite:f43cd870500e7a552abdcc78d5f1221bfa6866d6}} vector auto-regression.
SVR support vector regression.
FC-LSTM {{cite:02148cbc6fbe849a3addf9a44d0e7b10279e1880}} recurrent neural network with fully connected LSTM units.
WaveNet {{cite:d44b6c8f2a1c0013030e23ef6fdab16b3565660b}} a dilated causal convolution network.
STGCN {{cite:30516a74fc4a6cd1e6d7b23274b736d711099c92}} spatial-temporal GCN that combines 1D convolution with graph convolution.
GCRNN {{cite:cdb2b32f88c839b3d563586c334f3de7ecac2a5e}} graph convolutional recurrent neural network.
DCRNN {{cite:cdb2b32f88c839b3d563586c334f3de7ecac2a5e}} diffusion convolutional recurrent neural network.
| r | 4f86bdf1975a01f8d322e616acf85414 |
It is also potentially useful to use information about the r-band to inform the
fits of neighboring bands. Indeed {{cite:73655c494023d9f617a46091aa8217517958c539}} attempted this by requiring
many parameters (i. e., Sérsic index, radius, ellipticity) of the fitting
model to be identical across the g and r bands, essentially using the two bands
as a form of coadded data to increase the S/N. This increase of S/N comes at
the expense of dis-allowing variation in the matched parameters, which may or
may not be an appropriate assumption (i. e., in a two-component fit, we might
expect the bulge size to change across bands, which is dis-allowed).
Additionally, {{cite:c56d1eaeb019aee0a054dbebe68c955c4a44357d}} enforced simple polynomial relationships in
parameters across bands, using the neighboring bands to further constrain the
acceptable parameter space to be searched by the fitting algorithm. The
most flexible method is to fit each band independently and examine the
systematic effects of each band as necessary, making additional
cross-band comparisons including color {{cite:b04fff922a1a401100e6aa69a98fe6b873347788}}.
This is our preferred method for the data presented here and in M2013.
| d | 8670bdf4d73c8a9dc40e9438ca54e5be |
Several results for control sets near local bifurcations are available. For
transcritical and pitchfork bifurcations in the one-dimensional case and for
Hopf bifurcations, cf. {{cite:f1ca894046553a0e72a30bc722f29df5aba0291e}}, also for
applications to physically relevant systems and further references. Lamb,
Rasmussen, and Rodrigues {{cite:d1ecc25b28e66dd9874f866b85b77e769c0d0d01}} develop a topological bifurcation
theory for minimal invariant sets (which coincide with invariant control sets)
of set-valued dynamical systems. The only contribution for control sets near a
homoclinic bifurcation is due to Häckl and Schneider {{cite:bb5dd7ff653f982d084287455d8b66c3cc508325}} who study
systems when the uncontrolled two-dimensional system is obtained by the
universal unfolding of a Takens-Bogdanov singularity. The relation of our
results to {{cite:bb5dd7ff653f982d084287455d8b66c3cc508325}} is discussed in more detail in Remark REF
and Remark REF . Control sets near homoclinic and heteroclinic
orbits are also of relevance in the study of models for ship roll motion, cf.
Gayer {{cite:f4f31f20fb1632bd989ee33782973c09ad242cec}}, {{cite:b81a52dfc354edeee5e73d3949ba52b8dd9cd6f5}} and Colonius, Kreuzer, Marquardt, and Sichermann
{{cite:f49c8fc9aa9e44ef9a930a89260ca67dcdbc1ed3}}. While in the latter references the uncontrolled and
unperturbed system is Hamiltonian, the present paper considers non-Hamiltonian
cases. We use the monograph Kuznetsov {{cite:b6d1aeb45a37e6bf2bc2ac25ffd4cebf2b71a7b7}} as a basic reference for
homoclinic bifurcations, cp. also Guckenheimer and Holmes {{cite:293bb60c6777211fc17b68859ede05d9bb5c835b}} and
Wiggins {{cite:8a615f41eed098ce1f688d3854417b75d43dfee0}}.
| i | b27df8920721c5738b6c001c7ee1bc2f |
In contact mechanics, the first and the third terms on the right-hand side of (REF ) usually can be formulated as the integrals. Then the objective function in (REF ) can be viewed as a sum of expectations of several random variables, which can be solved by stochastic gradient descent methods or its variants (cf. {{cite:75659788d5b8cb3fa998a8632b553d24d7ec4fb4}}). We refer to Subsection 4.1 for details along this line.
| m | 4861a7858a373c08368f795b3803231d |
There are a few research directions which are worthy of further study. {{cite:1d963746329150ec90ee5a043a99d48e67cb8ef6}} and {{cite:83d87a66b81391eaccdfae58c835469f9a6f3fa1}} have studied posterior consistency under model misspecification. In the same spirit, theoretical properties of the proposed model under misspecification of the degrees of freedom can be potential avenues of exploration. {{cite:8bdf9302b989475e50339157bc6a93ec1811d142}} has introduced the infinite mixture of infinite factor analysers (IMIFA) model, which is a Pitman-Yor mixture of the model of {{cite:06de19444c2bb13a17d83e8ffd1f30f42faba718}}. The same extension of the proposed model from normal likelihood to Student's {{formula:dcbea486-061c-413b-ac68-ae5c7fede165}} -likelihood can also be made when some or all of the mixture components are suspected to follow heavy-tailed distribution. Finally, the proposed model is not completely choice-free, due to step size parameter {{formula:3ac1306e-ee14-4f5f-994f-e8091fa65fb3}} used in No-U-Turn sampler update for {{formula:79a95ab7-e071-4310-a4e4-be250a412588}} . {{cite:af5d2ab8900df594d7ebf88d03194e8c60a42556}} suggested a method of adaptive setting for the value of {{formula:b50ace25-4eff-4715-a3f0-c79118851038}} . This, however, is not directly applicable in our settings, because we are using a single iteration of No-U-Turn sampler whose target function changes as estimates of the other parameters change. Devising a method of tuning {{formula:5c4ab60b-ddcc-429b-a89e-f6ac627ca86e}} would be an improvement on our work.
| d | 607bcaad7e32c7a6e7c009f6eb434504 |
According to the definition of the preceding statement, our method belongs to a typical two-stage method. In the front-end, projection heatmaps of 3D BBCs are predicted by the tutorial described in DeepHMap {{cite:be5a108a513b8079242b646ebc45f68ef2e3f391}}. The core task of the paper is to design a comprehensive postprocessing in the back-end. Our proposed postprocessing consists of two modules: projection grouping and correspondence learning based hypothesis selection. We describe each necessary step in this section.
| m | 2b612dbc3b87051b5ae8dec7562d3196 |
Sensitivity analysis of misclassified confounding variables is one example of sensitivity analyses for observational epidemiologic studies. Several approaches exist to assess sensitivity of causal conclusions to unmeasured confounding {{cite:c2103b357effdc4029b51432830b51eea4660a9a}}, {{cite:9c416dc426d250c5099ad08f8008fb6a606da5b3}}, {{cite:d152846c757809f2b5113d935bfeec503591d53b}}, that aim to quantify the impact of violations of the assumption of no unmeasured confounding, while our approach aims to quantify the impact of violations of the assumption that all confounding variables are measured without error. Although this paper discusses classification error in a dichotomous confounding variable, the same principles apply to measurement error in a categorical or continuous confounding variable or when multiple confounding variables are considered. Clearly, in such more complex situations, more elaborate assumptions about the structure of measurement error should be made {{cite:f92c7da5c285a58162091690381ee07cd3c9d97d}}.
| d | 4fb7875740db11b35dc9e17b859a6b0c |
Prototypical methods {{cite:bb6e73b332c95042416f9420ed9c3dd190f639de}}, {{cite:1ea4743188f41ddba9b8fd320c297765ef874fce}}, {{cite:9f442184a7217086eccdd17bfe04a40e9620a629}} learn a tuple of prototypes {{formula:35ec6539-0b3f-4926-9393-40a8761ca2fa}} , where {{formula:429d2bdb-6301-4ec7-b96c-0e00ad06e273}} , which are representations of the most important regions in the training dataset. They facilitate a refined recognition of images. These methods usually contain four modules: backbone, add-on, comparison and decision module. The input image {{formula:57dab19d-8b4a-4e8f-aa55-044d4565ec32}} is first passed on to the backbone module to obtain a feature map {{formula:fd176f07-0f29-4a04-85c7-5c551baa4c63}} which is then passed on to an add-on module to obtain a feature map {{formula:52f6c56f-e333-4929-a4e7-5c5ee55e9867}} . The add-on module outputs {{formula:4dc96bd1-8412-4ace-8b22-598a7c7609e9}} local representations or patches of dimension {{formula:7379fa71-a507-4771-8200-0c2514d7cf8d}} . We denote them by a set {{formula:787eb066-0410-46af-89f5-1924c60e5415}} , where {{formula:7158c17e-b0eb-4da0-9b22-5bf94b4127ec}} . Note that the dimensionality of prototypes is same as local representations, i.e., {{formula:074306f5-fef1-4720-835e-6f3fd19dfe46}} .
The comparison module then compares {{formula:89252f61-1d20-4210-8f9a-426c4272af64}} with {{formula:66e72e36-fadf-4754-ba89-9ada558bea55}} using similarity scores. The scores between the prototypes and their nearest patch are then passed on to the decision module. In ProtoPNet {{cite:bb6e73b332c95042416f9420ed9c3dd190f639de}}, the decision module is a fully connected layer with dimensionality {{formula:b8a395a8-99e1-479f-85c4-9af855be9983}} , where {{formula:0cbe0b3b-be8e-4759-8516-cd33e449bb28}} is the number of classes. Also, each prototype belongs to a particular class which makes the weights of the decision module sparse.
ProtoPShare {{cite:1ea4743188f41ddba9b8fd320c297765ef874fce}} improves on ProtoPNet and additionally merges semantically similar prototypes using novel data-dependent merge-pruning algorithm. In ProtoTree {{cite:9f442184a7217086eccdd17bfe04a40e9620a629}}, the decision module is a decision tree where the prototypes are arranged as nodes of the decision tree. For clarity, the notations with descriptions are mentioned in tab:notation.
| m | 9d8665d156d80ea767a4742e25169907 |
Other Tasks.
In this paper, we worked on the super-resolution with the bicubic LR images.
Recently, some researchers studied the blind SR {{cite:b2b9be297a9141084ffa28ee1abbe73e8ca78be5}}, where the input LR images are from unknown degradation.
In addition, other low-level vision tasks, such as deblurring, denoising, deraining, and JPEG artifact reduction, were developed by attention mechanisms {{cite:5dc2ac82a7f3250a25d18c383758b3cc28da0f87}}, {{cite:92524d735de9a3d0587be280a4932736a6520aa8}}, {{cite:348f8b9c84ecd696e8b828ced8afb26046135b8c}}, {{cite:421e5db3aaa2247e0338c78222d4f2619d95cf3f}}.
Since these image restoration tasks also need the contextual information of distorted regions like the bicubic SR, our model introducing N-Gram to image would be helpful.
Secondarily, we visualize how our work can boost high-level vision tasks, such as classification (CIFAR10 {{cite:69d609a9c4cff7bb5c5a28c4ce2a135efb94fa41}}) and ST-VQA (Scene Text Visual Question Answering) {{cite:587c5a51b1c94a9243a24e0c8ef43a6e92f6644a}} in figcifar1,,figcifar2,,figstvqa.
| d | 9c917c9ac3ae42464a8a6c0881235fc6 |
Regret and Equilibration. First, we benchmark their performance against the optimal choice distribution in a cost model that internalizes agents' utilities from exploring the space (Lemma REF ), and show that in this context, the SQL dynamics enjoy a constant total regret bound in arbitrary games that depends logarithmically in the number of actions (Theorem REF ). Second, we show that they converge to Quantal Response Equilibria (QRE)The prototypical extension of NE for games with bounded rationality {{cite:871a73dfccabae328cf82bed16adc0c7b47514fe}}. in weighted potential games with heterogeneous agents of arbitrary size (Theorem REF ).Apart from their standard applications, see {{cite:80e09797039883c3c18d16f1ec42135ab4bc76b3}}, {{cite:3f521d1781045b612f9bef239f83becb766bf1ec}}, {{cite:7cc3e391b9d9abf9e52186c5df2575686df23044}} and references therein, weighted potential games naturally emerge in distributed settings such as recommendation systems {{cite:1281cb2b9b3c54fb9762d6692fad1c3c84231ca3}}. The underpinning intuition is that agents' deviations from pure exploitation are not a result of their bounded rationality but rather a perfectly rational action in the quest for more information about unexplored choices which creates value on its own. This is explicitly captured by a correspondingly modified Lyapunov function (potential) which combines the original potential with the entropy of each agent's choice distribution (Lemma REF ).
| i | b98338f6f00887991bf01809f848e04f |
finally, we investigate the limit {{formula:431fd023-1c58-4472-85e3-929ff4b7be3d}} within the so-called constant-roll inflation {{cite:199f6f6693ca16d1607f6653533e055038ade4ed}}, {{cite:ce161cc1d641a5bc8c867fcc0cc28946c3a323f8}}, {{cite:bf79bd36e3eed0b29fa84910c5e4d001844e027f}}, {{cite:4610505ad538438e3527026b90d1d34305c2ca35}}, {{cite:87cd5bb7e99e1448cc7cb0604844f844fd27100a}}, {{cite:a52aff79ba23d494d66b8ead974adc9aa65f01e2}}, {{cite:d7c7f35212330eda5ccd9229e5ba3504cbed7c09}}, {{cite:a89da8aa30585285bf778f670985c347096234e9}}, {{cite:17c0da68cfc464b4ca52d4f65aaadd567c034f33}}, {{cite:a38e7ecdd8272a872bfc307299134269617e28c4}}, {{cite:0e36837c1ea7f58858a194e4d8c95359b7cd241c}}, {{cite:50fbbadbc60954037104b81d8eacac29e25a1fef}}, {{cite:78b8d180ec398ad4ebb7690ef5f263d27e7fd66e}}, {{cite:24f702f3bbab4697c251657f119d3e17d3e77902}}. Different types of inflation re-examined in the context of a constant roll scenario. Tachyon inflation is studied with a constant rate of rolling {{cite:86690a8e073905a73ca07e19af7e5e4ca24e6cb7}}, {{cite:33492d0339977d00805b00cae74982ac9d8d5a35}}. The authors derive the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts, and the tensor to scalar ratio to the first order by using the method of Bessel function. A constant roll scenario within modified gravity is investigated {{cite:640a7b322970df09056db77ff415a58738fdfbb6}}, {{cite:bc806a90d29ea0dfffd2a1ed8e4040b2d51d3e96}}, {{cite:60f2672edda62bf26eca54c320caa763dbbbfbcc}}, {{cite:3050f5336f941cce2a36021943df6e4ada3c3e16}}.
| i | 2772963c6f7a75342c7f362be66cac6b |
Next, we showcase the universality and reconfigurability of the MPLC platform by programming Haar random unitary matrices, which are of high interest for applications such as boson sampling{{cite:1b21f12ff0b9ec128743a53200c3c1638106027f}}, {{cite:83062ebb80206d0e343818b7796c135c4b776d03}} and quantum cryptography{{cite:b34d27c67672d06f4e29d97d29793556b31b67be}}. We experimentally implement 400 random unitary transformations on four spatial modes, sampled according to the Haar measure{{cite:458f47cdad2ac8049efc9afdfcfc3efec8e52dcd}}. For each random unitary, the coincidence rates between all pairs of output modes are measured. The probability distribution of the coincidence rates for different unitary transformations is presented in the inset of fig. REF a. For random transformations, according to the Porter-Thomas distribution, a negative exponential distribution is theoretically expected (black line){{cite:2728537e132aa8325e07dfcf6c3535484fc0ea1b}}, {{cite:868443693a4de35c97812cf90f60f423c867a027}}, with good agreement with the experimental results (blue dots).
| r | 2d30cc1d3932ce4636505b1856cf23bd |
It would be interesting to see whether the bosonization formalism
{{cite:4c3a686c966e26b82489b36b203f2b5400faacb6}}, eventually combined with the approximations
discussed in this paper, can also be employed to study twisted bilayer
graphene near a magic angle
{{cite:69ef1d96b3312a572c4a07e5c2619ff5f13fc9ce}}, {{cite:c8d575f1628a5e6a8f80fb2dbf397eed1a077c77}}, {{cite:fff5e22634fde81da12721acd90db0efd0042601}}, {{cite:57adffadb434023449ebe80798ed2dff51902ce1}}, {{cite:dd47109c77b526efe393d2ae30f91090ee3589c6}}, {{cite:5ab1c7b0e52c7e24c976057cf3c3f7133c4119ca}}, {{cite:3fd8a005ddc7df66916df492e3e4d4f10c773bd9}}, {{cite:9d28189ff0895220c43a986e9c6221b8455682d9}}.
Here the resulting moiré pattern induces an effective
superlattice and a set of flat-minibands in the moiré Brillouin zone.
In addition to a superconducting phase {{cite:69ef1d96b3312a572c4a07e5c2619ff5f13fc9ce}}, {{cite:c8d575f1628a5e6a8f80fb2dbf397eed1a077c77}},
evidences for a ferromagnetic phase at {{formula:b80846e1-0216-4d0d-926a-90cf6789ca2e}} -filling of the conduction
miniband are also found {{cite:fff5e22634fde81da12721acd90db0efd0042601}}.
In principle, a possible flat-band ferromagnetic phase of the
effective lattice model introduced in Ref. {{cite:3fd8a005ddc7df66916df492e3e4d4f10c773bd9}} for twisted
bilayer graphene could be studied within the bosonization scheme.
| d | 4036b9a3666d34c98cab04ee306363da |
Recently, in {{cite:4854c075c7334eb1ab243b1b4fb3a83e78fbaaf6}}, methods for estimating coherence witness and dimension witness have been proposed that would directly benefit from the approach presented herein. Generalizations of quantum fingerprinting {{cite:29558f94c50a02687f6e6998b382201ddda9b77a}} and photon distinguishability {{cite:df7321fefc0003a5310e4c6559973f66982efd4d}} to higher dimensions are natural and can be explored in this context as well. Recent work on determining quantum entanglement presents another interesting research direction {{cite:6b8c1f6e0be79cf27c7b200936298ce9e6a0f338}}, as does the possibility to extend the method of inner-product based genomic classifiers given in {{cite:707b20a3350ca9e42d40dda60dc95104a5204e12}}.
In all of the above mentioned applications, processing all states in the same circuit is beneficial due to its ability to generate true randomness in state sampling, which offers the possibility to manage the measurement and reduce the sampling complexity. Obtaining all state overlaps by pairwise comparison based on 2-state SWAP test requires additional circuits to randomly select the pair to be compared. In contrast, in our approach, we do not need any additional quantum circuits to achieve this randomness.
| d | 7b715d472745a3c12d9db431ca78d608 |
We believe hybridizing ML techniques with optimization algorithms is a very effective approach to solve combinatorial optimization problems. This is evidenced by the recent advance in the ML community, e.g., leveraging ML techniques to improve Branch-and-Bound algorithms for solving Mixed Integer Linear Programs {{cite:9d7f698e4d5232af5dd38e9ce101d5fb9b29389b}}, {{cite:bf2849b2529dfaa94d0488b237896457548744f3}}, {{cite:88bfcff2acdeb09a4054cc36defea4a001c71762}}, {{cite:8b4383a68fdaabe2b5bdd20b1c0134f60ba56e60}}, {{cite:5d4e55c81be1536309b0ada304980a4227662f14}}, or to boost formal reasoning algorithms for solving Boolean Formulas {{cite:caa1afb7e800353d41bc4033dbfe48223b94b73e}}, {{cite:8849501da75581f2a0572d1d0b4c1a498c4ab161}}, {{cite:d6c4152ea0e617359177514fdc773a44f4e56601}}. Our work introduces this idea to the meta-heuristic community, by showing that a widely-used meta-heuristic, ACO, can benefit a lot from ML and solution prediction.
| d | 63a91645bb3a5ce2065da04e2508899f |
We mention that Corlette {{cite:e9bb360f5866dd7811f501455ed5c8db252266d6}} studied the following heat flow
{{formula:e8825a11-4b13-49ba-b994-44b1355fb96e}}
| r | d5f6d93e1eee03129d710f959162cd7b |
In the Bullet cluster,
the gravitational potential as traced by the weak-lensing shows peaks that are separated from the central region traced by the hot gas. In MOND, these
two would be expected to coincideThe relativistic MOND theory {{cite:3c6a57af326b0439c3facc90ca0422fc7a2091e3}} proposed by Bekenstein could be used to explain the Bullet Cluster {{cite:770e606f8f57d3f1996d47412e140ff7dab18d2a}}.,
since the gravitational potential would trace the dominant visible component namely the hot gas, while if there is dark matter it would be expected to peak at the
location of the stellar component in the galaxies. The latter case is what has been observed as can been seen in Fig.1 of {{cite:f8bf447b8aad5b3455dc179b43eb261028967293}}.
For the rest of the article, we will not consider the MOND explanation, but instead take the view point that the flat rotation curves of galaxies and
clusters at large radii as an evidence for the
existence of dark matter. Furthermore, we believe that the dark matter explanation is much simpler and more natural compared to the MOND explanation.
| i | 9889efc0756bfc7ca63c94be0cfc3ef5 |
In the last twenty years, a lot of theoretical and numerical progress appeared in lattice QCD.
M. Luscher introduced the gradient flow to lattice gauge theory away from differential geometry, which is a fictitious euclidean time evolution based on a gradient of a field theoretic action
intended to build a trivializing map of gauge theory
{{cite:67c382d9cde60cebe3dbe130c16b0492ddf23fbf}}, {{cite:9815c329cc75b0286ca412110e111aa988c1a8bc}}.
Later, he pointed out the gradient flow can be regarded as a continuous version of stout smearing {{cite:4b8af123eeab210676d3809a18abaf7f3890e71d}}.
Gradient flow is not only useful for lattice gauge theory but also general field theory.
For example,
applying the gradient flow to the non-linear sigma model, one can obtain AdS geometry
{{cite:a8524c8630f2f7646e0b0a01a0e5e66b912ff069}}, {{cite:b455497106123d673a3ccf4f72a9c22d72db8de8}}, {{cite:6d3b8f6c2201d6580dfdb3dec23b096554be8813}}, {{cite:918669cb5c1b8c06ba4442bafe21de5265967ba7}}.
Another example is relation to the renormalization group.
Relations of the gradient flow and the renormalization group have been discussed {{cite:d9ca952ac7d925956de48caa927f8ba9cdd68421}}, {{cite:343be4247445819fb572059350cb365259c86d87}}, {{cite:f6f9d60f4b6a61e9d08023ece548c9de5b1c7189}}, {{cite:bb13a26a9873a5ed40299f01533441f92f04de65}}.
Recently, Sonoda et al. found that
flowed field can be regarded as smoothly regulated field respecting gauge symmetry, and one can define Polchinski-type
renormalization group equation with keeping gauge invariance {{cite:f6f9d60f4b6a61e9d08023ece548c9de5b1c7189}}, {{cite:bb13a26a9873a5ed40299f01533441f92f04de65}}.
| i | 635dcdaf31b7bf976ba9fa413ed658ba |
Online update plays a very vital role to maintain tracking performance, especially for offline deep trackers {{cite:86bdb10d606f5d7d5a95eabfecbed62b19b2fa26}}, {{cite:419213682fb8368259d355b13b91ca4fd4b9cf06}}, {{cite:2b6d34135b3b92ac99ab37681a54fb747dc6e2ee}}. As a result, a great number of approaches are presented to complete the issue. Deep discriminated trackers {{cite:2b6d34135b3b92ac99ab37681a54fb747dc6e2ee}}, {{cite:19fe72b2b6a25fa7062c707439f6d0960efd9d60}}, {{cite:a32de2f98029be7dd567fd97bfc23a5915f74721}} were usually updated by fine-tuning model online, which are extremely time-consuming and easy to overfit due to limited training samples, even if reinforcement learning {{cite:4c5abd6e9a83ac2e8b0db65b8e862d96f6fdeb00}} or adversarial learning {{cite:9d281850ef79e300b3b7209ea34c1b34ebc1886d}} has been employed. For Siamese trackers, some methods updated the initial exemplar through running moving {{cite:ce18d5de8db51dacdb535e88bc6d76438c071e2e}} or feature fast transformation [15]. Whereas, they often failed to satisfy complex updating demands with the impact of irregular appearance variations and sample noises. Nowadays, neural networks gradually became an alternative answer to update deep trackers. Among these, UpdateNet {{cite:5793266921d33c40547d237d806768f89b5c5a3f}} designed a powerful deep updating module which could benefit from the training on large-scale video datasets, and Meta-updater {{cite:8182475f067a945f577624f896eb5d39c0b206e4}} attempted to update long-term trackers with the LSTM {{cite:b10359f58cd3cdd02808af24ea27803498eaf18d}} based on meta learning. Deep networks are more adaptive and reliable for update, but they are still difficult to update Siamese trackers in an optimal way since they are completely separated with tracking models.
| m | 61bc7f0a22f7da79f61c8c0f505d8465 |
It follows from {{formula:3845da39-5579-4b72-9c0e-786ebe82962d}} that a well known theorem of classical mechanics {{cite:afe3d9b4d815c0b8eb5def81900ad089c82f72f1}}, {{cite:b1467a076ede01a6bcdfb63b21ef10fe13bb692b}}, {{cite:c0e1ffcb1891131c885be3f7b2152b46853bbca6}} is generalized to arbitrary dimension:
tK = =1N q f ,
where {{formula:41d1ac3f-9c2c-417a-930a-0e081be7cfc6}} is the total force acting on particle {{formula:2f2eaba9-033a-4542-b1b2-6b659678a06a}} . Following Goldstein {{cite:afe3d9b4d815c0b8eb5def81900ad089c82f72f1}} we decompose {{formula:ba0c791d-e908-4174-bfc5-8c382adce5b1}} as the sum of an external force {{formula:2fd87e60-222e-462a-bdae-ad16368221cc}} and the sum of the pair-wise additive forces due to the other particles {{formula:2afc0525-1ef9-42c0-82ed-a9d49882d862}} .
If Newton's weak law of action and reaction holds, i.e., if
{{formula:252f5296-b26d-4446-9438-5cc89be21dd6}} ,
then we have
tK = =1N q fe .
| d | 52a9f9ec00f3e640be683e8db221c624 |
Precise boundary conditions that fulfill these requirements were given in four spacetime dimensions in the papers {{cite:636253d966cb0f0e0687571f5d90c906015c270d}}, {{cite:cf8a7de42f5119d3421d149e9dddc173e0eb1161}}, {{cite:bbf5b380b2c31993ccd74d1f3f28fb2ed41a2d86}}. The boundary conditions of {{cite:636253d966cb0f0e0687571f5d90c906015c270d}} and those of {{cite:cf8a7de42f5119d3421d149e9dddc173e0eb1161}}, {{cite:bbf5b380b2c31993ccd74d1f3f28fb2ed41a2d86}} are actually distinct, but both differ from those of {{cite:fe463b495144ab80b0b84115e2d58cff7be5e623}} by a twist (that takes a different form in each case) in the parity conditions on the leading term in the expansion of the gravitational variables near spatial infinity. The effect of this twist is to elevate the BMS transformations to the status of non trivial gauge symmetries, i.e., gauge symmetries with non-vanishing charges – while the original boundary conditions of {{cite:fe463b495144ab80b0b84115e2d58cff7be5e623}} effectively factored out the BMS supertranslations, which had zero charge. A gauge transformation with non-vanishing generator is called “improper” {{cite:f17f336e48d7fa7e5d62dbb0abeb1a828c9e700d}} (the terminology “large gauge transformations” is sometimes also used).
| r | f03d8d7ba9fc89e3192b398b04e277b5 |
Fix {{formula:d351735f-1e1a-454d-b6a9-2bd1855fa9db}} , and let {{formula:b1828abc-28dc-4664-a79e-b66703107c08}} . It is well known that {{formula:9e33b3f1-0549-46dd-b752-21c9a3e32e9d}} is the circle {{formula:244dd698-78ad-47dc-8833-db7c56cd3651}} ; see for example {{cite:5b1aa1411c991c45ed9656024a5266bf1c9deeb4}}. By conformal invariance, we must have {{formula:1d993a7d-a70e-443f-9b01-1cefab2dce9a}} , where {{formula:98c09af6-4269-4323-aac2-5ef0507f821a}} . Rearranging and dividing yields
{{formula:072068bd-c964-49af-bd13-e8168883e4f7}}
| r | 792e7c0cd4bc5e94372ee22d304350f1 |
Finally, we execute gradual pruning with the sparsity schedule, augmentation choices, and cyclic linear learning-rate scheduler discussed above. The whole gradual pruning procedure lasts for 300 epochs, as in {{cite:d52505e31f4598aef014c6d902970d8a5fec1ae6}}.
We aim to obtain accurate sparse checkpoints for {{formula:16fabaa3-bd77-4b4f-8566-1580859ce000}} , {{formula:155c2934-aa4c-4c35-8cd5-9a0e5d48d51c}} , {{formula:09027f08-7435-497f-b93d-941226afb35a}} , {{formula:87d228e7-85f0-404d-927c-70bd9cdaf913}} , and {{formula:87953136-8cff-4b0d-b16a-2d03d0fd4fec}} sparsity.
For this, we prune to {{formula:56823f6a-5edc-44eb-8946-c7a1ae26ec19}} in the initial step, and increment sparsity every 20 epochs, until reaching {{formula:663f66f1-b829-4f1a-b86e-9ed2494b9a3f}} , with fine-tuning in between. (See Figure REF .) We select the accuracy of intermediate models which match the target sparsities; to examine the impact of fine-tuning, we trained each of the resulting sparse checkpoints for an additional 100 epochs, marked with ({{formula:14d2427b-3ed3-4a8a-8493-958326b3722d}} ).
We compare with global magnitude (GM) following the same schedule as oViT, as well as the state-of-the-art SViTE {{cite:401a56c900868d3b7f9db648722881b89b6f3701}} paper, which trains the sparse model from scratch using a variant of RigL {{cite:ff2fd1877c5730d058dd0c172445789da74dd772}}, but for a total of 600 epochs.
The results are in Table REF .
| r | 8a29c36ac135113f598f91b8ac879faf |
where {{formula:c568687a-aca1-496d-b378-abf69fbbf64a}} might be non-differentiable.
It is well known that the
optimal value and an optimal solution of problem (REF ) can be approximated as closely as desired by using subgradient methods.
In some cases we can use interior point or Newton methods to solve problem (REF ).
Although developed very early by Shor in the Soviet Union (see {{cite:e5f799e58b4d7956e8af7f98ed816d54d822ff3c}}), subgradient methods is still highly competitive with state-of-the-art methods such as interior point and Newton methods.
They can be applied to problem (REF ) with very large number of variables {{formula:e51bdbb1-db73-4f61-ba0b-6a398c1216bc}} because of their little memory requirement.
| i | ae08402994b24bc845dee82067bfd129 |
The above fields will be the main objects of interest throughout the paper. Their {{formula:da09ba12-a56d-4d3d-bcc1-6a61434e0293}} -adic (resp. {{formula:8dbd5388-5b46-4785-ad4d-18bd014a3fcf}} -adic) completions are typical examples of perfectoid fields in the sense of Scholze {{cite:4f16b711bac96c7972feb8e53227a58a115c0d68}} (see §). For any such field {{formula:be36f967-1934-4d49-91e7-39beb2e24431}} , one can define its tilt {{formula:bca856e0-4918-45b6-8e64-c9fab0f24082}} (see §REF ), which intuitively is its local function field analogue and serves as a good characteristic {{formula:dd1df562-2454-430a-ba10-3a1b0381f151}} approximation of {{formula:0ca78393-71a1-44c5-a12b-d2eb2fac5da1}} . In practice, this means that one can often reduce arithmetic problems about {{formula:c653fa0d-1679-4b8d-869c-d78658eb17a9}} to arithmetic problems about {{formula:84436a70-7f73-4f97-9028-1bb5495d2b26}} . This kind of transfer principle, which works for a fixed {{formula:2dd05661-5131-4072-848f-f9267e4dc5bd}} , should be contrasted with the Ax-Kochen principle which achieves such a reduction only asymptotically, i.e. with the residue characteristic {{formula:25df2939-19aa-4891-8594-5dae5134827e}} . This is explained in detail in Scholze's ICM report (see pg.2 {{cite:7d907995fb4bb13c69aa6fd71f7fbc734e6d3b18}}).
| i | 910e77964b0b4b8caca2d542b933c500 |
where {{formula:933331f5-26f7-4494-9a82-8b3eca75ea61}} will be a weighting coefficient of the Dyson series. Constructing such {{formula:e7c63560-fe47-4ce8-abc5-8fdcdc97ae3c}} operators is explained in the appendix B {{cite:2fd60f257d212f3caa50237a2decd689e565db14}}.
| m | 89944ec41712f083798b7a9f343e40ad |
Due to its intriguing nature, NGC 6946 has been mapped at multiple wavelengths. For example, in the 21 cm atomic hydrogen emission line {{cite:cde684b57b6a204b9b67d260379b23d2ebd31463}}, {{cite:473e47ea1f80a819a6890ff7e803dbbdba4a0cf8}}, {{cite:48fde37c7930be381d9600bf1299ae6a663cadcc}}, {{cite:327537ff3094e635f228add9d0ba8c670cd77471}}, in H{{formula:ba3969d2-fe70-4357-b2de-bfa97ff0fb72}} emission
of the ionized hydrogen {{cite:a7941d283e0f3cff0b497445567d08c1e0fe5bcb}}, {{cite:48fd2289173afdca8561e99b3d6338fe4225b967}}, {{cite:1f5af9b27a874151d8d5bc679ab950b8add61746}}, in the molecular (CO) emission lines {{cite:44781732dc15b00195dfe3d6250485726c5e0bc9}}, {{cite:bf849b072ee56261751d718b0dc08be82e06d6e6}}, {{cite:376262bc7a78614dd3c1ea0987ce21b49176bb9b}}, {{cite:49106d8d5f87ef2d842236e522e2adb69935078e}}, with the Hubble Space Telescope (HST) {{cite:57e010d2bc28dbfbbb569577df9204745d4315aa}}, and as part of large surveys, such as the Key Insights on Nearby Galaxies: a Far-Infrared Survey with Herschel {{cite:5b5af857114af994425cf2932104033d76b8a2b1}}, and
the Spitzer Infrared Nearby Galaxies Survey {{cite:769df2ecb3b33c95a65c4c21f161ebd53bcead94}}, among others.
| i | 6a08e6aae6e44655d42934d1cfdda625 |
A possibility that was not accounted for in this study is letting the nodes use different stepsizes to update each {{formula:8ae26785-6cd6-4832-a16c-9d54c5ec39d4}} {{cite:a9d7403ab381455245168d7f2fd31ad46eb91b67}}.
Indeed, function {{formula:69f35d72-c529-4cc9-bd7f-7c7b16beba7e}} is coordinate-wise smooth with constant {{formula:ac497efd-b770-4fa1-9d71-4d3787e09145}} for {{formula:71013baf-07f7-4645-9d59-96a8329de792}} ; this could be used by each node to choose a different stepsize {{formula:ba5e2d7f-1c8a-4ad1-aa7f-746bd8d7670e}} for each {{formula:79b94a4f-ef89-47a1-9fda-19430ea3f889}} , which would make convergence faster.
Methods to estimate the per-coordinate smoothness when it is not known a priori were discussed in {{cite:c81a3166b29a4cdbc75c741c5c4aa6b851aa9f6c}}, {{cite:a9d7403ab381455245168d7f2fd31ad46eb91b67}}.
| d | fb003c708700e42a90cba2cfe8b545d0 |
Neural ordinary differential equations (NODEs) have recently shown tremendous promise as a means to learn unknown continuous-time dynamical systems from trajectory data {{cite:997bf2746778e8f39cf712c4d22ba5da996dad61}}.
By parametrizing differential equations as neural networks, as opposed to directly fitting the available trajectory data, NODEs provide compact representations of continuous-time systems that are memory-efficient and that are well understood; they allow the user to harness an existing wealth of knowledge from applied mathematics, physics, and engineering.
For example, recent works have used NODEs as a means to incorporate physics-based knowledge into the learning of dynamical systems {{cite:9d11607a50fe3612b36c51b3d68b86e365a4dba1}}, {{cite:3ed39ec5490744e64d10be1f4c2f2144796e05c6}}, {{cite:7eb14362f7cfe2e5059a680ddf176c007335d7ac}}, {{cite:bbd3044f74bb50586ed8f5fa8bf31ade2ca953a9}}, {{cite:ab781f37db3fd8b89f6b7b3541a5b917a42a9933}}, {{cite:d6c3ecbb6b4b44899f4669d0b9c159e8883cca5e}}, {{cite:7bf589d446ae95b9768ac945c82031e587dd7c8c}}.
Furthermore, NODEs have been used to define continuous normalizing flows – a class of invertible density models – to learn complex probability distributions over data {{cite:997bf2746778e8f39cf712c4d22ba5da996dad61}}, {{cite:01187e9a3e254ec1af2162b3e97150b66c1868e8}}, {{cite:d063a0efa0be72986f40c784dffa2339e05eeaf4}}, {{cite:8f2fd428599d2f490f1ed90231ba03bc1f71fb6d}}.
{{figure:3e5df1c5-a5f9-4267-b4a7-d35754644c26}}{{figure:e86e75ed-75ae-4c77-9a53-51a5f7009796}} | i | 1b5e3743a24aecfb500fdcc56d06e2a4 |
The general position number of a graph was introduced in {{cite:e5780be6738d71b30846ca68169cd8654530b80f}}. A couple of years earlier, however, the invariant was in different terminology considered in {{cite:5ff25a2519c306d8a27bfcf31b05b3263741556d}}. Moreover, in the special case of hypercubes it was much earlier studied in {{cite:6cf37eb6b2d35daf117a10b2edcc91891947add3}}. In {{cite:57cfe5212c220497c72b4a68b10904504b3e50ef}}, general position sets in graphs were characterized. Several additional papers on the concept followed, many of them dealing with bounds on the general position number and exact results in product graphs, Kneser graphs, and more, see {{cite:e9cdd1b464bbb0858ecd777f057c0ae96c717955}}, {{cite:7ab6e09b07d8b76915a0c5b8b6f5328a28b178fd}}, {{cite:39d1e68e09d1d22e629c8a34141339ff05ca3af8}}, {{cite:3a1e692faa7fc030c6f47c2b0c82d3cdc151d169}}, {{cite:5bc978ff4720da3df14c95e4df71cf8aa98a35df}}, {{cite:79a5329a9ab3fd0c6e200fe499292f4bef268104}}, {{cite:a8dd91d5ca8969381d31ad6ccc7abfd59ef4ea5a}}, {{cite:4181b9a1f003554a514dc7c90560394a1345114e}}. In addition, the concept was very recently extended to the Steiner general position number {{cite:bad428536f3f9539d1caf2fbcc32e87ed455e7b0}}.
| i | 596ef0d20716ce5be9b0ec99258bc9ad |
We compare our model's performance with published image captioning models. The compared models include the top performing single-stage attention model, Att2all {{cite:181cc2f507fb0ac405308a634995957ea399906b}}; two-stages attention based models, n-babytalk {{cite:6f32d2f578a1b7850d9e4b283b65ea86c50ededa}} and up-down {{cite:e32626419d809d67148a9085cf86962561ff0cc3}}; visual scene graph based models, GCN-LSTM {{cite:2951cc47358aeda9f9b1b2c512c9118712f33022}}, AUTO-ENC {{cite:3ab5364cef975f546face4fe4328f61c6984cade}}, ALV {{cite:710369a75f94942f676ae47c39405401f7b72e2f}}, GCN-LSTM-HIP {{cite:196533cfa56ee2e868f714cb25e1f59bb205e8ba}}; and transformer based models Entangle-T {{cite:9f7d782e494683bb874584d4c014162a78047efe}}, AoA {{cite:a2afeb394d81c6e35164d03fc9db81940b8861b6}}, VORN {{cite:15eaf1368d4e1b0311f88b428adde72bf2478836}}. The comparison on the MSCOCO Karpathy offline test set is illustrated in Table REF . Our model achieves new state-of-the-art on the CIDEr and SPICE score, while other evaluation scores are comparable to the previous top performing models. Note that because most visual scene graph based models fused semantic and spatial scene graph, and require the auxiliary models to build the scene graph at first, our model is more computationally efficient. VORN {{cite:15eaf1368d4e1b0311f88b428adde72bf2478836}} also integrated spatial attention in their model, and our model performs better than them among all kinds of evaluation metrics, which shows the superiority of our spatial graph transformer layer.
The MSCOCO online testing results are listed in Tab. REF , our model outperforms previous transformer based model on several evaluation metrics.
{{table:d8a92786-5dd0-4685-a5cc-240d406f2fef}} | r | 6e9bfe6a46044f5d4622080a7b19cf0b |
In this paper, we explore the sequential order within the dialogue as the self-supervised signal to guide meaningful and coherent dialogue learning. We introduce a self-supervised learning task, inconsistent order detection, to explicitly capture the order signal of the dialogue. The task is defined as, given a target utterance pair triple, the model is required to predict whether the triple is correctly ordered or not.
For instance, the utterance pair triple {{formula:951341ea-5238-433d-b6e2-c052c6e26ecd}} is misordered.
The key to solving this task is to model the utterance order based on the dialogue context effectively.
But when directly encoding the full dialogue history along the temporal order, the model actually only focuses on the ending utterances, and earlier information is largely discarded {{cite:0b6d42cacf2eb73277fb30e0e89e4834a028114a}}.
Thus, we propose a sampling-based self-supervised network ({{formula:7ea16132-2645-40ef-8d30-00a1df3b867d}} ) to account for the forgetfulness problem and solve the inconsistent order detection task. In order to accurately predict if a target utterance triple is ordered or not, we randomly sample utterance triples from the dialogue history as the reference to incorporate the dialogue context.
Since for the same target utterance triple, the sampled triple references are different at different iterations during training. It essentially approximates the full dialogue history without suffering from the forgetfulness issue.
| i | bbdef66d5ed483b8a1d6ca591ed2a038 |
Read Mappers. We evaluate five read mapping systems, each of which adopts different optimization techniques to accelerate read mapping:
1) Base uses Minimap2 {{cite:bc6f87d020c653c85947c911657c2f1461b78f2b}}, a state-of-the-art software tool for read mapping.
2) SW-filter extends Minimap2 to filter out exactly-matching reads (i.e., reads that exactly match subsequences in one or more locations in the reference genome) using simple single-instruction-multiple-data (SIMD) operations, without requiring costly ASM operations.
3) Ideal-ISF uses an ideal In-Storage Filter (ISF) that can concurrently filter out exact-matching reads inside the SSD while the host CPU performs read mapping for non-filtered reads.
4) ACC uses a state-of-the-art hardware accelerator for short read mapping, GenCache {{cite:0d892bb6d56b93b852bfbaef6c0d30f35b478422}}.
5) Ideal-ISF+ACC uses an ideal in-storage filter (ISF) that can concurrently filter out exactly-matching reads inside the SSD while a hardware accelerator (ACC) performs read mapping for non-filtered reads.
| m | 7da8fad4f47da7b9b127ae47a352548c |
In the gaslike state, the quantity {{formula:b2c72a48-8164-4d0b-a2e6-75657f718a32}} increases towards the ideal limit of {{formula:8ddfd600-a8e8-49a1-a1c4-24e0a1157a2b}} in Fig. REF a. In the gaslike state, {{formula:0171942a-e7da-4e63-83d5-6791d206f5dd}} is related to the particle mean free path since the speed of sound {{formula:62b35f6d-3ac8-4ee1-9bcb-429d46475c81}} is proportional to the thermal velocity {{formula:50d13ff7-744f-46f1-b0a7-d62921b28534}} and {{formula:30f2a9f4-0f64-42b7-b023-4b1ed1f4d67d}} is related to the average time between particle collisions {{cite:206896a3061b9949220a68626de86adc4831707e}}. The decrease of heat capacity from 2 to 3/2 in the gaslike regime of fluid dynamics can be explained in terms of longitudinal modes at short distance in the fluid vanishing as {{formula:433ab890-583b-4f19-aebb-0cb8aa7e929b}} exceeds their wavelength {{cite:32f63b3103dbc1158f82ffc502206e611181fb1c}}. The collapse of all curves in the gaslike regime in Fig. REF a implies that {{formula:989006a0-070c-4b89-9160-60f54977a4df}} is the only parameter necessary to characterise the loss of these degrees of freedom in this regime.
| r | 065f0fe37d532c077443ea51a35d4812 |
Replace {{formula:c25a11f7-3f9b-4e93-b7df-11ba47edabdf}} of (REF ) with {{formula:48d2ec24-f79e-42a1-a937-5583214f5863}} , we achieve the update rule of {{formula:3929363c-6791-48c2-b2ed-0e07e2951b4a}} {{cite:76ba4768256fceadd7828a93fc0ddcedab57d008}}:
{{formula:3599398f-3473-4a75-b21f-c98dee5fd714}}
| m | b8669eb9732b471a6fa699c250e40664 |
If the aspiration of loop neurons is to constitute systems of exceptional scale and complexity, might the simplicity of the dendritic building block limit the dynamical repertoire? We hope this model helps answer this question. It is known that even simple systems with simple rules for propagation, such as cellular automata, can give rise to behavior of great sophistication {{cite:e1398feea563e032da991cdca29283f743a46080}}, {{cite:7f6847e2c0453f97a5f900600ef6c6a7572592c3}}. It has been argued that similar concepts can be applied as a starting point for physics, with the rich, natural world emerging based on the interactions of fundamental nodes {{cite:b87c770bdc055dda4b42215cc5091ec506f329ed}}. Similarly, elements as simple as two-state spins interacting with nearest neighbors (Ising models) can give rise to phase transitions and critical phenomena, including crucial long-range correlations {{cite:fb42a55782bbb27b27e90ea073b0b38ff91df13b}}, {{cite:d82ed624aa9d466ef242a9468b1eb2eb25fd4b12}} as well as attractor dynamics for associative memory storage and retrieval {{cite:f7d526f79fd3f83312263f8fa6af88ecfa57ef31}}, {{cite:8d61b73a19ce65fa5aead802d756dc224af12e97}}, {{cite:ffc1e096c7520f450babc6b6cafdfe053b939624}}, {{cite:607e3d718b5b7075499dbf24bc57ce82287868f7}}. By comparison, the dendrites studied here are multi-dimensional and nuanced. The nonlinear spatio-temporal convolutions occurring in each neuron's synapto-dendritic tree provide a deep repository that can inform the neuron's behavior. Transmission of action potentials to many destinations at light speed enable complex network topologies far beyond nearest-neighbor interactions. The use of superconducting circuits allows signal retention across a broad range of time scales through the choice of the {{formula:03312226-a11f-4e0d-87a8-94f13b7e3b6c}} parameters (the {{formula:a6952fb9-0b47-4c44-a8cf-fa59b79f2472}} distribution) that specify the leak rates in Eq. REF , as has been demonstrated experimentally {{cite:7755168319e0fe14dfd249d7325d5f4a1d578bae}}. The range of integration loop inductances (the {{formula:ba1621ac-efa0-4ebd-98e7-1ec2de009d3d}} distribution) in conjunction with the bias currents will establish the dynamic range of the network. With near and distant connectivity as well as a wide spectrum of dissipation times and an enormous dynamic range, SOENs are likely to achieve the long-range spatiotemporal correlations present in the critical states that optimize neural information processing {{cite:43f13c2e750937db158261e9469a80ebb9a9acb1}}, {{cite:3e4f40a9663505e5e0e0a622c9eced0157423520}}, {{cite:f0b15891affcafae717f01caf542ef07f716c715}}, {{cite:a68443fab0ab908fb96236e157d6acb2435e8332}}. The hierarchy of spatial connectivity in conjunction with the hierarchy of information retention times and response magnitudes appears excellent for enabling the fractal use of space and time that supports information integration and cognition {{cite:432f25ae59d286bd5f9c8c40301db96536463071}}, {{cite:48c960c2ab2165fd67f69fcd92f64169c2e0b7d4}}, {{cite:169102058e63c05002eb3938cf1bb8b488fa4dfb}}, {{cite:cc2f833a3a9c5b3e8a0657ae5d123db62ddc4817}}, {{cite:00b91892c5392988e88bb077cd90425e9fc54ab1}}, {{cite:f0fde49d6f4b7f87f1f314496a8de5effcdb9dc8}}. Yet the simplicity of constructing the majority of computational grey matter from similar building-block components brings an advantage in modeling and technological implementation.
| d | 2a7b326211c35fb7b67ee8392c05258e |
It is worth mentioning that the Hubble constant tension has been interpreted as an early/late universe tension, while it shall maybe be seen as a thin/large beam tension {{cite:a3702b23164b38cdaec818f8762f049531b33c56}}, {{cite:f6c4f540920c41cd9e69d2cd1f502c80bc65f13d}}, {{cite:58bce5f9c1ba7931773a1a06edd4c033a6db54a2}}. This new emerging tension, to be confirmed by more BBN and large scale estimations of the baryonic density, is a primordial/late time tension, so that the CMB would be tied between two lever arms at redshifts of order {{formula:7ebc45c7-9593-498c-aacb-5d63dbfb9bce}} and {{formula:ccf353b6-53cd-4ede-b526-16ab8a201672}} . If confirmed, the status of BBN, with the lithium-problem and a mildly-constraining helium-4, would have to be reconsidered. Note also that unlike the cosmological lithium problem, this deuterium tension can be mitigated easily by invoking a small contribution from most models developed to solve the lithium problem as they overproduce deuterium {{cite:9f1f52825acb8b249f7278c6b1e04868118c48ef}}, {{cite:0a161a797cd197132643609e3b3ed8e8da7b7553}}, {{cite:98b06a9cbec1d6910366508eec0c1df2af6c9448}}, {{cite:6e17cd507002f8ad04406fa144b9d192835f21c9}}, {{cite:4443d68d508fd670a8dd947de86198fdc98070ad}}. To finish, note
also that since BBN theory assumes a perfect FL geometry and since the spectroscopic data are located on our past light-cone at low redshift – and thus well inside the CMB sky – the Copernican principle could be at stake {{cite:82f2b9530ba601f9ff6067ceda1dfdc0b79da29d}}, {{cite:393a7213fe653a741d8acf598f28565e709e5bbd}}.
| d | b8668163bc6384562cf88fb494e6d242 |
In (REF ), we have an under-determined set of equations with {{formula:dfa8ead4-e157-4edb-af85-45e936201504}} observations and {{formula:8bd18b45-848b-4adb-b842-e9c97f33952b}} unknowns. While this problem has infinite number of solutions, it could be transformed to a tractable problem by assuming that {{formula:49d61994-d397-4da1-be7c-b2b571125b40}} which is reasonable in massive MIMO systems. This strategy is built upon well-known continuous CS approaches {{cite:136f2d3b5e495173d160e9cc5ca7c1057b5c90b6}}, {{cite:ca3dea882ff8ddd1fca9d0e958ad7983198aa849}}, {{cite:7dbc272e987854b27e7363727690c5c3871a976d}} and provides a unique optimal set of solutions for matrices {{formula:b600094d-a5f1-48f6-a4c1-59b0a796111d}} s in (REF ) leading to the least number of atoms under the affine constraints of (REF ). Thus, we form the following optimization problem to reflect the structure of {{formula:76f730e5-8aef-468c-89e6-87908db9511a}} s:
{{formula:d409d8fc-8785-4809-a73f-2f8017362aff}}
| m | 7751650439c73ae21690578cad676f1c |
We propose that the increase in coronal height inferred from our reverberation analysis coincides with the launch of the blob that {{formula:f02ef691-7fb9-4d8e-9f53-b79bbbe18144}} days later causes a radio flare when it reaches a large enough scale to become radio bright (Fig. REF b, c). We note that due to the low-cadence radio monitoring between MJD 58302 and 58304, we cannot rule out that there was another earlier flare that was missed, and thus the time delay between corona expansion and radio flare would be less than 5 days. If the blob is launched when the corona is most compact (MJD {{formula:ac6559b4-4893-4134-acb4-a3b42d831743}} ) and travels at {{formula:70256ca5-e555-4082-a1f3-d9a7f18cfae1}} {{cite:5afdb03ee7469cac836de966bf36d3b62d5cf0e1}}, then a delay of {{formula:65df1b55-82fc-40f5-a4a9-f8a187819154}} days until a radio flare
would suggest that the radio-emitting blob is at {{formula:c5f0f44d-61b2-4757-b867-9dbfda168d2e}} {{formula:be0eb824-2459-4dcb-9f54-22ceae87e03e}} . Unfortunately, there have been no published radio measurements of the synchrotron break to independently estimate the radius and distance of the jet ejecta, but this distance is reasonable, when compared to the BHB MAXI J1535–571 where the emission height above the BH is {{formula:43838d15-a83b-4d7c-8c28-c216d98fdf20}} {{formula:f58310f4-a763-42ea-a60e-ac7a9dade5d6}} {{cite:ae3002d4d3fb42807a5544d646e7f9526b0ec0a1}}. We also note that recent Chandra imaging showed a transient X-ray jet ejecta at the time of the radio flare, inferring that the ejecta was a cone of {{formula:b3578789-dc35-479f-849d-7e20d9c490ee}} AU by {{formula:ba2645b2-ed44-4e4a-8ea2-cc8ab3e4b0b3}} AU {{cite:d796af44941f3721bf962ac23cf5020916512319}}. As 1 AU{{formula:e5319d1f-6341-43a4-9358-5550e24595ce}} {{formula:fa24f59f-bf79-4407-ab17-db3fd2016176}} for a 10 {{formula:2cee3b18-3502-4aa9-879a-3ebaba417c22}} BH, this would lead to a vertical extent of the transient jet emitting in X-ray of {{formula:98d19f4e-bbb5-4f46-b6b4-5d0ae3d68a73}} {{formula:3fc9c8e4-8b3a-469b-8938-d029df43162a}} , and the light travel distance of several days is always below this upper limit. Therefore, it is plausible that the expansion of X-ray corona is accompanied with a jet knot which propagates downstream (moves higher up) with the jet material when approaching the time when the radio flare was observed. This interpretation is also suggested to explain the increased hard lag amplitude during the transition in the high mass X-ray binary Cyg X-1 {{cite:42f55545b1630ed1cb89acd73ae6b00487a42411}}. It is consistent the physical picture presented in {{cite:cad7f946e966553997b188248a696c4b49751ec0}}, where the soft X-ray flux is observed to increase just before the radio flare in another LMXB MAXI J1535–571, because the X-ray flare could result from inverse Compton scattering of the synchrotron radiation from the knot after it was ejected {{cite:cad7f946e966553997b188248a696c4b49751ec0}}. Finally, this behavior may be analogous to observations in some high luminosity AGN, where X-ray flares and iron line narrowing precede radio ejection events ({{cite:3d69056083c6b2ed7c4bdcd6ffb4924bdb9d1122}}, {{cite:029a9de98c9137e7f4b4415845c45d88ca0e09a3}}, {{cite:6460cfee9beb9a0839db1510314662bc68ce8478}}). In this scenario, the source of the hard power-law emission is more likely to arise from synchrotron emission in addition to Thermal Comptonization because the optical depth along the jet axis ({{formula:a7db2629-4e00-47d3-9997-f5e60e41d8ea}} where {{formula:f080c06f-7e0a-4ce4-8091-2864a84abca9}} is the power injected in the jet, and {{formula:1d075dc3-2b15-4fe5-878e-f715f9c0e675}} is the radius of the jet) drops quickly {{cite:c5827b34a31c8cfb6094fa40df961f54e7c8834c}}, while efficient thermal Comptonization requires a mildly optically thin medium ({{formula:45139ebe-f98a-4351-a977-67ac93392f89}} ).
| d | d96bbc3df73033816f1c3900aed2217b |
in the sense that {{formula:dad1c250-8629-43cb-bb25-1b644fad17e7}} can be computed by applying {{formula:b4df6bcd-2ed1-4139-9481-dfa9c5473b93}} on certain iterates of {{formula:0e11bf0e-4358-4288-964e-f25dd1e52a06}} map. Here, {{formula:09d94d8e-ca16-49dd-8bb9-34939707426c}} indicates the classes of {{formula:bfb50f15-64bd-4c57-9c71-91c0fbb17508}} . Hence, {{cite:88a70da85987c38bf1e8e4d7f5cec9c8e001e824}} identified a reduction process as well as a factorization process which supply the {{formula:568dd82b-67a7-4ab5-ad77-ab3d44b613e5}} -reduced and factorized data {{formula:81b54f4d-017e-4714-bc52-948e6e3b46bc}} .
| i | 089c8f7fc4b769710af061cc7cbf76af |
The results in Section REF suggest that the unlabeled PowderFaces dataset only provides a marginal benefit in performance, if any at all. This ran contrary to our expectations: we expected unlabeled data to help in this context, given a similar approach achieved state-of-the-art results on Imagenet ({{cite:578ebcde923114c8b11c35a1b9e2242258699b6a}}). In that work however, the unlabeled dataset consisted of 300 million images – 300 times more than the Imagenet dataset itself. In our case, our unlabeled dataset of one million images is only about twice the size of the number of faces in AffectNet and Google FEC combined. Thus, to truly conclude whether the benefits of unlabeled data as shown on Imagenet transfer well to facial expressions, many more unlabeled images are needed. We plan to address this point in future work.
| d | 94409cd42fe3acaa3248fbd2fe85cb74 |
D-time: the running time (sec.) to execute the dynamic programming algorithm
in Stage 5 to compute a lower bound on the number
of all polymers {{formula:85d90103-16e3-404b-ba0e-fc5077657fe2}} of {{formula:e6267bd5-ec7d-40ae-bf6c-64cf1a6b94c2}}
and generate all (or up to 100) chemical isomers {{formula:827fc17d-5da9-44b8-91d2-95f2e58f6bca}} ;
{{formula:6af9a02a-23b5-4779-a658-251ca25247dd}} *{{formula:eff94dd1-8c2c-44e1-9fa6-c738db101444}}{{formula:0490578f-3e42-47e8-b5a8-b00ddcd88536}} aEb{{formula:bd61d3ee-179d-485a-8d30-a02db7ec4a64}} a10b{{formula:53647d3b-eb8f-475b-bf5f-8eafb3e11a97}}
{{formula:455dec24-c052-4dde-a77c-147b4b56c19b}} *{{formula:423c6b27-7c55-4589-ad90-7a7ccbc62c80}}{{formula:e614420f-2fac-4cfe-a3f1-6756e3714d22}}
From Table REF , we observe that
the number of isomers {{formula:be9024bc-6a51-4e15-8076-9377f15541c8}} of an output polymer {{formula:209e8e56-af0c-4ffc-ab83-6db97c974559}}
varies on each case, where
the polymer {{formula:d2e71f96-d13e-4d80-9317-59cb15a261fd}} admits
only 24 isomers {{formula:52f015b9-baf1-4929-b149-6cf645d5336d}} for instance {{formula:fbcb68bc-2d26-4740-89d8-901172d44ff3}} and {{formula:fb79f514-77fa-41d5-bd4d-073cd617943c}}Prm
and over {{formula:4d5d3b77-5c5c-43d5-8990-8776ac2cefa4}} for instance {{formula:0cb48d63-3444-4a23-b308-8f4e3f90a806}} and {{formula:5335bb43-0877-4242-a96f-4b7f3fe2af47}}Tg.
The computation time for generating at most 100 isomers {{formula:93fad278-11f8-4347-904b-1e3f322e64fe}} and
estimating a lower bound {{formula:2882840a-fa4c-4648-b03d-9f492ecbea20}}
Concluding Remarks
In this paper, we designed a method for inferring polymers
based on the framework for monomers proposed by
Akutsu and Nagamochi {{cite:f44329971d83b91d3e147dde6ec4784a524f5904}}.
To treat a polymer as a form of monomers with no connecting-edges,
we introduce a new way of representing a polymer with a monomer form
by distinguishing link-edges from other edges in polymers.
Since the link-edges of a polymer are characteristic to the polymer,
we included new descriptors that feature the link-edges of a polymer
into our feature vector.
We constructed prediction functions by linear regression
for eight chemical properties on polymers in Phase 1 of the framework.
We inferred polymers for the first time in Phase 2 of the framework.
The results of our computational experiments suggest that
the method still can infer a polymer with 50 non-hydrogen atoms
in the monomer form in a reasonable running time.
There are some chemical properties on polymers to which linear regression
did not provide a good prediction function.
It is left as a future work to use other learning methods such as
decision trees and neural networks
and find new effective descriptors in order
to construct a prediction function with a better performance
for these chemical properties on polymers.
Appendix
Linear Regressions
This section reviews the method for linear regression used
by Zhu et al. {{cite:74356bbe8d662afddc80670b4a4eb8fa955d9b16}} in the framework of inferring chemical graphs.
For an integer {{formula:66e99f2e-73df-4376-94d6-05337411bb94}} and a vector {{formula:89c1326d-225b-4ed6-a104-0685a7884f42}} , the {{formula:56eb925d-a4c1-4d2a-b387-201b8385f479}} -th entry of {{formula:eabc902c-d41d-48e6-a1da-c08143c78410}}
is denoted by {{formula:1a90a51e-1a75-4e7b-a9f8-19658138a303}} .
Let {{formula:21d57387-4694-4383-b346-24424b0b0eb0}} be a data set of chemical graphs {{formula:52bf9546-3aa8-4508-95a0-e8fac7a1988a}} a(R{{formula:6ce9fa0e-22ac-4c62-8c83-3c154b44d81a}} ai=a(i){{formula:c36a6c3d-99c3-47ef-9546-c12762dda963}} i{{formula:535fc225-a786-4994-8618-41f6cbfaa099}} Let {{formula:a49b2b01-4760-4a20-ad27-cf3d4b07d101}} be a feature function that maps a chemical graph {{formula:b118531a-990d-44f1-8f45-f65e1e930901}} f(RK{{formula:bee8722e-bfab-4479-baee-6612baa6a12b}} xi= f(i){{formula:f459f0ca-aa81-4e0f-b1fb-6e68e8ee3c56}} i{{formula:23ac7e86-8757-45ca-b115-6bf93dd4e316}} : RKR{{formula:05db0eaa-36a5-43e8-9b37-bc77cc864e10}}{{formula:99cff058-ffcd-40d3-82ca-6227f49bbebf}}{{formula:88f2e730-0efb-430a-ad64-e47818bff86e}} R2(,D){{formula:60bdeb08-b8af-44c7-916c-2227152071ff}}{{formula:09926bcc-5b20-4737-85f3-068fbe44ba70}}{{formula:4962ed3b-cede-47c1-96f8-4ce858d928f5}}
For a feature space {{formula:0826322a-b34f-4cbc-90f1-2dc34ddc02eb}} , a hyperplane is defined to be
a pair {{formula:47bb9d9c-813f-4548-8d5c-d744390e08b2}} of a vector {{formula:b6a1bd6a-c8c4-42b6-a877-3d9d1bdf6171}} and a real {{formula:385d1f3c-5d3b-4493-a0e0-63ae2bc73025}} .
Given a hyperplane {{formula:d207d51d-a2ce-4753-a73e-d8dd235ebfb6}} ,
a prediction function {{formula:0d83a36b-cd92-4431-ae7a-755991123cd9}} is defined by setting
{{formula:46054fac-bd64-4fa4-9b8c-b783621806aa}}
We observe that such a prediction function can be represented as an ANN
with an input layer with {{formula:c52f3501-1dbc-4c23-a6f8-5006368a4dc9}} nodes {{formula:3ad84806-9595-4665-87e2-d6909b7acaff}}
and an output layer with a single node {{formula:89449e7e-4d99-459b-9a52-aa1c76aebc9d}}
such that the weight of edge arc {{formula:61042bb9-ee33-40e8-af73-9273a073145f}} is set to be {{formula:53503286-9316-4c05-9b71-ceaa760a7db2}} ,
the bias of node {{formula:c9de85bd-72ba-48d5-a8fd-8957a337c8d4}} is set to be {{formula:a6416327-be05-4c52-ba86-9e6c850b491d}}
and the activation function at node {{formula:f77f5b51-bd90-4847-8fa5-c030d9ebbec4}} is set to be a linear function.
However, a learning algorithm for an ANN may not find a set of weights
{{formula:c616b7ed-cf70-4336-b6ca-dcdb5f6fdee7}} and {{formula:8946315b-277b-4499-8f53-c965512fe4dc}} that minimizes the error function, since
the algorithm simply iterates modification of the current weights and biases
until it terminates at a local optima in the minimization.
We wish to find a hyperplane {{formula:a4787b01-6894-4f3f-8d45-75899a146fe1}} that minimizes the error function
{{formula:c5c10f84-2133-4a83-8e04-c68adb19c260}} .
In many cases, a feature vector {{formula:9f9b48fc-d75e-47ca-9663-fc2feb23a29e}} contains descriptors that do not play
an essential role in constructing a good prediction function.
When we solve the minimization problem, the entries {{formula:cfc7267e-e53a-4ead-849a-8779a649dd85}} for some descriptors {{formula:32708264-3ca3-414f-92a0-ae20c989d7a7}}
in the resulting hyperplane {{formula:5f9ca4e0-c941-41c2-8d66-600baf62ec91}} become zero, which means that these descriptors were
not necessarily important for finding a prediction function {{formula:af90bc5e-991e-4823-a240-2e9d9c904fe1}} .
It is proposed that solving the minimization with an additional penalty term {{formula:0b7ec2cd-b3c5-47a8-b6f5-3009f3d47068}} to the error function
often results in a more number of entries {{formula:45a1b45b-49be-4935-9c84-81583503c1e8}} , reducing a set of descriptors
necessary for defining a prediction function {{formula:980a3a5f-3741-48a9-9113-6a74efecf9ef}} .
For an error function with such a penalty term,
a Ridge function
{{formula:c8717bd9-bde6-4d94-831b-8777b1223a45}} {{cite:cc098553c53b37f0239900c581ab165867fdcfba}}
and a Lasso function
{{formula:ea4f3183-4cbb-4ace-ac29-550c09d72d96}} {{cite:b7ee8cd6ccb14dfccd7686af537de9d71bbbb5b8}}
are known, where {{formula:fc6f1521-6bfa-45ec-9b1d-ab6212d2a726}} is a given real number.
Given a prediction function {{formula:7b8955d5-af51-4166-90bf-520f46bca005}} ,
we can simulate a process of computing
the output {{formula:448333bd-f08e-46e0-9efe-bac751b6e886}} for an input {{formula:8eae70d3-c8fb-42ea-8041-8946dc984671}}
as an MILP {{formula:467993d1-f14d-4df2-afa0-ee973bb3252f}} in the framework.
By solving such an MILP for a specified target value {{formula:88f6403e-d938-4ace-a50e-791d1427c236}} ,
we can find a vector {{formula:e3848f9e-64b9-4e59-915b-bed88358691b}} such that {{formula:7520c72a-a0b0-40ec-b995-510db7226d79}} .
Instead of specifying a single target value {{formula:f12e5529-bc06-47b4-9a8f-a64620683c05}} , we use
lower and upper bounds {{formula:300ac975-61fc-4fef-89b0-8ef7b7b80e47}}
on the value {{formula:d58e05f1-2e3c-4582-b661-773c9f4b6ecd}} of a chemical graph {{formula:f0b4aaf4-dc9f-4fac-a674-77b396ef51df}}y*{{formula:b2c85410-5ca5-47f3-8d56-fcb79be86d37}}y*{{formula:add8ad5b-57f5-45dd-aab0-98b2f54662a1}}
by setting {{formula:0492ee30-766e-4f88-8b2e-0f806932aff5}} and {{formula:06e5f55f-c811-4cab-a5ae-ddf12a6efe94}} to be close or different values.
A desired MILP is formulated as follows.
{{formula:09138c50-b4fe-4877-8502-fca27fd93a7b}} :
An MILP formulation for the inverse problem to prediction function
constants:
A hyperplane {{formula:fd5caa73-2de1-4bb7-92d3-ecb3c8958bc4}} with {{formula:88357d86-55c8-45b3-bf59-cc0bca1d2469}} and {{formula:b21c59bd-ee55-4e0b-ae5a-ab0dc8beaf2f}} ;
Real values {{formula:8260941b-6f05-43b4-bce0-2d6aef4f4ef2}} such that {{formula:eb66b500-5139-4cf4-9ecb-452b30c39908}} ;
A set {{formula:3d374b04-7c9f-4b84-b901-857199a37a7c}} of indices {{formula:4f88d672-9b1c-4178-b45d-de62919b709a}} such that
the {{formula:bfd83d41-a221-498c-a794-d56d4dc40019}} -th descriptor {{formula:1338ba85-ed09-46db-bbaf-c6524ebb10b4}} is always an integer;
A set {{formula:96b0daf0-c5eb-461a-a28e-fd477741bec4}} of indices {{formula:0c3d59b4-6125-40a5-be20-b99084e2e142}} such that
the {{formula:a176483e-5118-42d5-be5f-5c854494cacc}} -th descriptor {{formula:ce8186df-f537-4243-96d9-c33e7ccd1f8b}} is always non-negative;
{{formula:a02d0670-472d-4381-b0d8-1233474e2d3e}} : lower and upper bounds
on the {{formula:312c8a02-28f6-4b31-b10c-82003c52dfab}} th-descriptor;
variables:
Non-negative integer variable {{formula:1d48fc85-1af3-468b-a92c-0369d1575a4c}} ;
Integer variable {{formula:5bef72cc-ab9d-415e-82dd-40f5ed5937b3}} ;
Non-negative real variable {{formula:474d326d-6a0b-41a1-83c9-ef79cbfdf2a9}} ;
Real variable {{formula:ffba1e36-77cc-415b-a22f-f8b2d8b54c7c}} ;
constraints:
{{formula:c6bee0e4-5d29-4aa0-9377-ae00fa95b18e}}
{{formula:e7c491de-f8e8-44a4-a414-9c987304befa}}
objective function:
none.
The number of variables and constraints in the above MILP formulation is {{formula:c1ac5f37-d1e8-4a41-adb8-f3a7910b5cf0}} .
It is not difficult to see that the above MILP is an NP-hard problem.
The entire MILP for Stage 4 consists of the two MILPs
{{formula:ce252ca3-5e04-4ab4-b785-03d1a81d6f19}} and {{formula:d58ff3b3-18c8-4996-9a2c-5445b28b3762}}
with no objective function.
The latter represents the computation process of our feature function {{formula:bd59fad4-d5fc-455d-be97-df605aecf7dc}} and
a given topological specification.
See Appendix for the details of MILP {{formula:4569e791-234e-4605-8de4-b3bc7d3669a7}} .
A Full Description of Descriptors
Our definition of feature function is analogous with the one by
Zhu et al. {{cite:74356bbe8d662afddc80670b4a4eb8fa955d9b16}}
except for a necessary modification due to our polymer model
with link-edges.
Associated with the two functions
{{formula:01efa66a-e947-4820-8701-535895c7c92f}} and {{formula:3855705c-98de-4dae-8bb6-93af83f0979a}} in a chemical graph {{formula:e7d364d5-f05f-4aa2-94a0-dfe84ff4cce3}} ,
we introduce functions
{{formula:d8897c04-13fa-42a5-99f0-086bc3aea00f}} ,
{{formula:7d43e1f5-887c-4aca-b73d-12146737a014}} and
{{formula:effd16fa-1bdd-4638-ac6c-f6279b862a32}}
in the following.
To represent a feature of the exterior of {{formula:546e8e3f-a213-4bf4-9d0f-25c25e6a16c4}} ,
a chemical rooted tree in {{formula:cb887374-e706-4b84-bb5c-d270845afcd0}} is
called a fringe-configuration of {{formula:3ce2d413-60ad-40aa-9671-0787e7ade0eb}} .
We also represent leaf-edges in the exterior of {{formula:1289c6a0-9736-401d-8588-1224d28eb83d}} .
For a leaf-edge {{formula:c2251e9e-7238-4c9e-a78f-5ac5c2db6f4b}} with {{formula:f8d51a00-b167-434d-b71d-13f147cb011f}} , we define
the adjacency-configuration of {{formula:f1501968-6b6b-4860-a5ac-fb39d48573a6}} to be an ordered tuple
{{formula:43a1ea16-9c96-4204-aa8a-76b0b990ad8a}} .
Define
{{formula:b3fc60f9-1468-4802-b5f6-ab28a4e1d9df}}
as a set of possible adjacency-configurations for leaf-edges.
To represent a feature of an interior-vertex {{formula:44b89f91-ead1-4e02-8186-1eeb33ccc70d}} such that
{{formula:0ea4a0cb-b25c-49c6-aca9-60730d968ae9}} and {{formula:8a2e73a5-a1ca-48a4-b7a8-0e63937a25ad}}
(i.e., the number of non-hydrogen atoms adjacent to {{formula:b749e8b6-c79e-4a42-981a-db3c6df1bc8e}} is {{formula:d2351bb7-505a-40fd-a2d6-7bf6f5e4b023}} )
in a chemical graph {{formula:fc2650ab-cb2e-4dee-87a8-6d06a66c040e}} ,
we use a pair {{formula:35b7f0db-bb9f-4d0d-9e0e-a200b25b406e}} ,
which we call the chemical symbol {{formula:ae8f1984-b1e4-4b57-8e3e-f5ae64f1ce29}} of the vertex {{formula:382ca2e3-dd6e-4b38-92e9-3f9fe3a91688}} .
We treat {{formula:0118b08e-9e4b-4197-a4a1-a51e854ce1c3}} as a single symbol {{formula:eb6c96fb-12fe-4e7e-885b-107858a0ec49}} , and
define {{formula:0855fb4b-fb4f-4b3b-91bd-0a4a7741438f}} to be the set of all chemical symbols
{{formula:3a609459-8399-47ed-898c-90af49aebbc7}} .
We define a method for featuring interior-edges as follows.
Let {{formula:86da20bc-968f-4100-a8a1-85e7b9aa5b19}} be
an interior-edge {{formula:e656e1e9-e5d9-41d1-8904-a40e0bff8dfc}}
such that {{formula:6fea4a44-3659-488e-ba50-3ba75b19515b}} , {{formula:0dd90536-05dc-4179-96a7-44d6d39eab6f}} and {{formula:e01745cf-99c8-48eb-8e42-9fd30aaeca4d}}
in a chemical graph {{formula:517a3927-53f9-4bd0-8812-5cd45078f118}} .
To feature this edge {{formula:7fdaab30-ea98-4dd1-8d4c-ce553d3dba5a}} ,
we use a tuple {{formula:a13d007e-5f79-4243-bff0-539785aaa753}} ,
which we call the adjacency-configuration {{formula:e2d37042-7749-4505-990e-6035da44598f}} of the edge {{formula:d6f03772-e368-4a58-b016-8082af522a2d}} .
We introduce a total order {{formula:4a16fbbe-9f71-4934-a93c-f9cb5d7f4385}} over the elements in {{formula:0bb57d97-7170-4579-befa-59cf6875a7bb}}
to distinguish between {{formula:a2956fc8-c39d-472b-8bed-1082e824159a}} and {{formula:aee7d0f6-fc43-4c4e-ac00-cff3284ee86c}}
{{formula:08353433-87b8-4221-b190-b0131ffdb509}} notationally.
For a tuple {{formula:430bcd32-2fa4-4a27-b302-942b3558d129}} ,
let {{formula:2f1f9ac3-0708-45fe-b92f-7748876bdfb4}} denote the tuple {{formula:04ba8482-686b-497f-bbe9-c36c178ecf0a}} .
Let {{formula:6bec974f-2476-4a38-86c2-14b789e43082}} be
an interior-edge {{formula:43f49310-7f48-494f-b668-0a8baba64a98}}
such that {{formula:1c09548e-8b51-4f29-aa1c-01a32b6be3bf}} , {{formula:5b6d8318-ef9b-4664-a08f-960ec8cbea83}} and {{formula:d5f34aa8-b208-42a8-bc51-fa6510c5db61}}
in a chemical graph {{formula:466ae302-4669-46cb-9556-becf1d5ed425}} .
To feature this edge {{formula:f4da3a19-681c-46bc-aa9c-783607f6cb66}} ,
we use a tuple {{formula:2403aa79-df85-4f2b-9a0f-c6731df01283}} ,
which we call the edge-configuration {{formula:81c53307-5bf9-421b-bee5-329b800353da}} of the edge {{formula:c0507ecd-3e73-46ba-9424-b5329efb1ed9}} .
We introduce a total order {{formula:59437698-0ac6-4b3f-8937-8157476f162e}} over the elements in {{formula:c61ef60d-8785-4c00-9bff-41ce0dc01559}}
to distinguish between {{formula:77ed6be1-2e6b-41b8-b99c-83d6444dea5d}} and {{formula:20d6d7bc-186d-4d4e-b1f9-947bbb9635d8}}
{{formula:d8ca3023-56c8-4239-bcf8-461189c25bfe}} notationally.
For a tuple {{formula:5823b016-01a2-4305-8450-2f939137a24f}} ,
let {{formula:c5ef79ed-b03c-4047-b455-b65a6221f098}} denote the tuple {{formula:a525dcf9-1e51-4706-90a0-669dbe7ea148}} .
Let {{formula:705f302b-0764-4109-bcfc-256f1a9f1a3f}} be a chemical property for which we will construct
a prediction function {{formula:55f54395-3411-42e4-9356-050c767dcdca}} from a feature
vector {{formula:cad69373-a877-4fc2-91ca-46f6f8e6d3b6}} of a chemical graph {{formula:004374de-9db5-4d47-8148-ff7d03c2ddeb}} yR{{formula:7f176acb-fb13-4df5-af50-91ed785f9dd4}} .
We first choose a set {{formula:a33c887a-a753-4522-9183-af338654f52c}} of chemical elements
and then collect a data set {{formula:d82a19d4-ca70-434e-8827-9d4727aad816}} of
chemical compounds {{formula:c701965a-c73c-461b-a546-6f083daac019}} whose
chemical elements belong to {{formula:da75ab60-c7a7-4ca2-a19b-850c42b39a4e}} ,
where we regard {{formula:9851fda3-c3e1-4767-a490-80edc73fb38a}} as a set of chemical graphs {{formula:7b35d6bc-2056-463f-8401-23c64513b657}} C{{formula:552d5ee4-eac3-4a92-8bbc-5116fd91f2c4}} D{{formula:a5f02c0e-0159-429e-a229-5bf92ee0d379}} D{{formula:73b89dca-a211-4e96-929d-a1bec5fdff0d}}{{formula:e1bd5b75-d2bc-4d6c-bb98-38b9f23783e8}} =2{{formula:f24ac299-02d7-4169-9038-f3c0ca2ff1b5}} Let {{formula:6872855e-3034-4a46-95b9-79b05206bdf5}}
(resp.,
{{formula:debb96fc-a9b5-4571-a898-f30e78183b3e}} )
denote the set of chemical elements used in
the set {{formula:a8c05fc1-273e-458e-9090-5614b91bbb51}} of interior-vertices
(resp., the set {{formula:15d65c4e-8ac6-4079-a48b-17d5907c5673}} of exterior-vertices) of {{formula:9438edf6-8b0c-4ed9-ab9f-c6dfdadaf803}} D{{formula:20430098-f04a-4c00-82be-d08c8af13589}} int(D){{formula:1109d1b8-5b8e-429e-892c-651cef75ddee}} lnk(D){{formula:5f3cf326-80e3-44d4-b87e-d2f82c44bd93}} Eint({{formula:30690f85-d9d6-417c-b54b-818c7979860c}} Elnk({{formula:7d0bc1b3-72ad-4af6-925f-c4ef0653be62}}
over all chemical graphs {{formula:53c6875e-086b-4067-9ff3-2e017cbb1796}} .
Let {{formula:7f71725a-b7eb-4205-9e50-4e8e61a67095}} denote the set of
chemical rooted trees {{formula:ec586950-ae8b-4187-a482-003b2dee5aa2}}
r-isomorphic to a chemical rooted tree in {{formula:d08f4e39-d5b8-46d1-b15e-9da294b586aa}}
over all chemical graphs {{formula:80ba3cd8-2661-422d-942b-b591ffa9e6c5}} ,
where possibly a chemical rooted tree {{formula:6d0bd6aa-8102-4402-9309-635a11424e8e}}
consists of a single chemical element {{formula:22a80f92-3464-49e2-832d-8702fa51f4ec}} .
We define an integer encoding of a finite set {{formula:894fb569-d080-4c9e-b4b7-97867e659a8e}} of elements
to be a bijection {{formula:e28b7131-c21f-46e3-a02a-c1480c0f417a}} ,
where we denote by {{formula:71d139f4-931f-476b-8d1e-20ad930d9a95}} the set {{formula:dfbd08e0-2eb6-4404-ac23-9f9bf9c8b55f}} of integers.
Introduce an integer coding of each of the sets
{{formula:d3c92083-58f2-419f-aed9-873b043bb74e}} , {{formula:031c2702-059c-4fc2-80f6-7e7f96ff98bd}} ,
{{formula:20d785e5-1ce0-4937-9f98-6d38eb834da9}} and {{formula:424c9396-e4e3-46ee-87fc-ddfe65c63b35}} .
Let {{formula:4a984fd8-616c-4b15-a1ad-61b5faf32299}}
(resp., {{formula:e713ea90-585a-42c3-972d-c95bc984c7ef}} ) denote
the coded integer of an element {{formula:842abef6-245c-4e3f-9807-23c172c7a22b}}
(resp., {{formula:e41bd851-1e23-464f-803a-505b5c0b40e8}} ),
{{formula:1cd7b85b-19d5-405f-8c31-37085a6e5807}} denote
the coded integer of an element {{formula:8f93871b-d56c-46fc-a171-e182d64a8630}} in {{formula:86878f96-2a8e-443d-8aff-2b65b10464f2}}
and
{{formula:9a405187-e2ad-47de-8476-5899cf905618}} denote an element {{formula:cb8b76ce-5f3e-424d-9461-846a15f811d9}} in {{formula:af453e43-f4cb-41c3-889b-7b5f5092475b}} .
We assume that a chemical graph {{formula:9130411e-2976-4018-9ac9-9652eb08632a}} C (v)4{{formula:18fc478c-2581-4fa8-9a35-25156c641093}} C {{formula:f68dd237-2941-4235-972b-c83be051b18a}} In our model, we use an integer
{{formula:e2c8794d-f1d7-4226-a34d-3de5615f4bdd}} ,
for each {{formula:a1492932-82f1-42f3-b02d-2c7d1259148d}} .
We define the feature vector {{formula:c14171cf-b3e4-4541-a9ac-22e671bfa484}}
of a polymer {{formula:95c9907b-1e1d-4f0e-8f42-d7ca1ee5cd0d}}
to be a vector that consists of the following
non-negative integer descriptors {{formula:4b5d3e83-de5e-4236-9b56-7b0ed846ca1f}} , {{formula:85114a11-7fea-4ab0-88b7-71897316202c}} , where
{{formula:5151f623-78b4-48e9-b03c-799c1859320f}} .
{{formula:16481b37-e5c4-40ff-ac0c-ac1d3ae04cd5}} : the number {{formula:0006334f-c408-484d-ac55-dc23b20201f1}} of non-hydrogen atoms in {{formula:65c1873a-f057-4438-9be4-7a25cc009482}}
{{formula:7acb3cc9-7d67-4a37-99bc-f24a285dba18}} : the number {{formula:d912278f-02ac-41e3-aa12-5f0ebdcabadc}} of interior-vertices in {{formula:fff128a0-b9ba-404f-be78-fd834e1d7713}}
{{formula:ff39f01d-125f-4793-ab7e-328192586ff2}} : the number {{formula:46c517f3-39e8-4c26-aa21-71d57d3bca56}} of link-edges in {{formula:1e42fcbd-824f-4c72-a320-cebdad0b4767}}
{{formula:433ae038-6844-4e4f-976c-ed5e0152a3e2}} :
the average {{formula:d4fd18b2-67ea-4cc7-a446-1287766d3cf1}} of mass{{formula:8cebcb2e-e036-4ed5-8d9c-e936d63f70d3}}
over all atoms in {{formula:da662249-854f-45f6-8278-b1105e828238}}ms(1|V(H)|vV(H)mass*((v)){{formula:ea776edc-5758-4842-8b52-40d5544581cc}}
{{formula:aa6d16dc-db6d-466c-8a2b-0aa8bb6a44b5}} , {{formula:17ad8e61-b472-4368-9a8c-a8d98b13fbdf}} :
the number {{formula:a981b791-a3f0-40ec-b006-39f45920ae1c}} of non-hydrogen vertices {{formula:c3ab8ab7-c9fd-41bb-bec5-3d928e561553}}
of degree {{formula:15d928a4-3804-4a5f-a545-9bb66f471e85}}
in the hydrogen-suppressed chemical graph {{formula:af9e1516-193a-4295-9db2-083ac29cd023}} .
{{formula:407621c0-b032-4e05-9741-b2f46728bae7}} , {{formula:efc2970f-cfb6-4b1c-821e-239384e3f0de}} :
the number {{formula:de5460bb-9a30-431c-9ba3-4358309f9440}}
of interior-vertices of interior-degree {{formula:844cc226-be2f-434c-824a-531224636e11}}
in the interior {{formula:ad7fc9f1-186e-42c9-a9bd-b92c815bbb85}} of {{formula:3e918122-8fad-4de5-920b-5eb6fcf6bafc}}
{{formula:f95f4d1b-267f-47c2-bf71-a984b118b549}} , {{formula:d81a30fe-59f5-4c89-8644-ff46673f0015}} , {{formula:bbcd8742-0073-4dce-a39f-f3f7784548d8}} :
the number {{formula:c434df7f-9d3a-4301-89b9-0096c0f0a2c1}}
of interior-edges with bond multiplicity {{formula:bb8f200f-5484-45bc-821e-095ca73472f1}} in {{formula:3e86e1af-5ec4-4626-9ba6-4a8e16b8fe35}} bdmint({eEint((e)=m}{{formula:7028c396-9869-4386-ba31-fd306c43cc19}}
{{formula:a40c086d-81a1-446e-b165-086e78f9bc66}} , {{formula:ecf9725d-5298-4d69-a385-0734795d9b59}} ,
{{formula:9b3ef463-0fa1-4998-8ee2-5c9de1a45b32}} :
the frequency {{formula:ed33c25b-b9bd-43fe-8e42-bb428229a849}}
of chemical element {{formula:26cfc64d-2690-4442-b139-76756d1802c4}} in
the set {{formula:3034f0b2-e64a-4119-8ad6-1deeda734e72}} of interior-vertices in {{formula:b9a833f7-05b8-4dcc-9e9d-3e7897f145ec}}
{{formula:11109ae6-fd59-45fc-9b2e-087e1e595103}} ,
{{formula:799b9f84-57cc-4d5d-afa0-da76999a50b5}} ,
{{formula:bca5ac9d-5a92-455e-99a5-55807819d04d}} :
the frequency {{formula:219d6eb0-c02c-46ca-b927-1e53d46495de}}
of chemical element {{formula:27b9542b-6cfc-4a30-8e48-0d6b0cdce873}} in
the set {{formula:825d7d16-72c7-4b7b-ab88-09b43acf30bd}} of exterior-vertices in {{formula:4c816d2a-a266-45f6-afa1-6ca51ef5c2d0}}
{{formula:fbf6a906-4fc1-4fc7-95ee-9280f68f0980}} ,
{{formula:ba528610-e2ca-469b-8981-1733b9a01f8c}} ,
{{formula:52354082-8757-4d29-85bc-eb01e1044b6a}} :
the frequency {{formula:ea58554e-ba39-46ad-97a8-abd9a407ebd1}} of edge-configuration {{formula:9940eeb0-f2b1-48b5-be07-1d5b7a501b02}}
in the set {{formula:c5d7f895-2d4e-401a-bebd-d7655122329a}} of interior-edges in {{formula:1901b0e1-2e77-4c47-a802-97b9c0441b5b}}
{{formula:1f576ba0-7ed6-4fb1-a252-ef7699651413}} ,
{{formula:8bb1acaa-9731-4aa1-8119-a1e90e1514c4}} ,
{{formula:198e749f-3e6e-431c-b341-69207b7db2f0}} :
the frequency {{formula:950e69ad-8c72-49d0-a932-bf5f26445bcc}} of edge-configuration {{formula:926032e8-e8a1-4e47-a0ef-840720d43ba3}}
in the set {{formula:75b57096-7b9d-4406-babf-394f97358a59}} of link-edges in {{formula:8f49da73-9c41-4aef-b6db-cdd53b4b008e}}
{{formula:a3b3b138-19f2-45c6-a49d-0ea2e80f8aca}} ,
{{formula:c7f834b6-c090-4a55-ba2a-f1b7629ff04f}} ,
{{formula:30650ff1-394d-441b-94e5-f6a47c72aed7}} :
the frequency of chemical symbols {{formula:302fffec-4119-4af9-8138-dde19b3fa600}}
of connecting-vertices {{formula:bf3c2aa5-217e-4f37-a1e8-fde1886f3172}} in {{formula:35b93ed9-eb3d-472f-839b-309915ae53c6}}
{{formula:53f245df-e415-4a93-8a1f-299ac0f2f6cc}} ,
{{formula:b87ac112-2660-4f4e-a461-c5a3316a1ec9}} ,
{{formula:1a1a7c27-18ca-47f1-8e11-81a940a9b5d8}} :
the frequency {{formula:85ffa08d-5b5c-4700-810a-6adcad0a631b}} of fringe-configuration {{formula:ddaba060-1f89-454b-8d54-bd28a3839bee}}
in the set of {{formula:c6714385-ca4e-4e66-9c47-c0f199756297}} -fringe-trees in {{formula:58bdbfb7-99e1-4605-b83d-e505925904c0}}
{{formula:e24bb9bd-1c9a-414e-aaa0-79a53bea9a08}} ,
{{formula:5b6c8b3c-e1ac-4141-8c75-0ce92f00c77e}} ,
{{formula:caf41fbc-be9c-4a06-947a-12d3d10b8b03}} :
the frequency {{formula:673f75d9-1ac0-42f5-9aee-6238287d4432}} of adjacency-configuration {{formula:63ea1eff-6f07-4699-9cf9-56902f81087d}}
in the set of leaf-edges in {{formula:28fd2d67-a735-40ee-be91-e32b2c8b580b}} .
Specifying Target Chemical Graphs
Our definition of topological specification is analogous with the one by
Zhu et al. {{cite:74356bbe8d662afddc80670b4a4eb8fa955d9b16}}
except for a necessary modification due to our polymer model
with link-edges.
Seed Graph
A seed graph for a polymer is defined
to be a graph {{formula:992bcd3f-efe3-4a2b-b2bd-27c2236be9a9}} with a specified edge subset {{formula:37d587af-fa5b-4071-b812-96022b1f19ed}}
such that
the edge set {{formula:0367bcff-338a-4641-aed4-4bd6d61f95e8}} consists of four sets
{{formula:6905986e-634a-4e02-b5ae-dbf6871ba892}} , {{formula:99a6bb8e-8ff1-422c-a337-f4cc5caa7ebb}} , {{formula:858c425e-f4e6-4253-8ade-725daac71782}} and {{formula:065b7221-8efc-4dee-9128-48e407c7fbe9}} ,
where each of them can be empty, and
{{formula:0f38be72-9392-40f3-a1e3-e22af7cd429c}} is a circular set in {{formula:1b66f9ac-9f33-4eb8-9af8-b3f1e25e7557}} such that
{{formula:36aba200-6171-4529-a5d6-2a8122929cd5}} .
Figure REF (a) illustrates an example of a seed graph,
where {{formula:75a0945c-bbdd-4c99-aacc-ede719565d48}} ,
{{formula:ea42d8fb-aab0-4aef-81a1-88d455fa3c20}} ,
{{formula:46783c6b-a9a7-41d4-a8e9-397a575ec8fb}} ,
{{formula:db299493-89f0-469a-9e28-cddd1b9b8f6b}} ,
{{formula:4904eb99-10e0-4849-88f4-8ef441e7ffd3}} and
{{formula:dc1e9f9a-7a83-4f6e-846f-675a3fb6c56f}} .
A subdivision {{formula:89fdaad7-017c-45a3-9ecf-53c1f67ce1e4}} of {{formula:664e2e0b-6bc6-4953-b0ab-f8ed4e23d7a6}}
is a graph constructed from a seed graph {{formula:dac174d4-7c60-4696-b81f-4bb595f3a315}}
according to the following rules:
Each edge {{formula:816ffcb0-e2e2-44bc-b076-04449bf8818f}} is replaced
with a {{formula:7681d797-5a24-4fd3-8d4c-285201aabef3}} -path {{formula:ab7da322-83c4-40a2-94d3-aeee145ed9a4}} of length at least 2;
Each edge {{formula:2def5efd-b9e9-47a8-959a-80a5d5ecee16}} is replaced
with a {{formula:ffd1bff3-acc2-4d50-858a-a2f5aba58a2e}} -path {{formula:cb12f89a-6a30-4280-8e8a-6b5c7d19c015}} of length at least 1
(equivalently {{formula:2fc7eb5b-40c6-41c4-8503-065ef610d87b}} is directly used or replaced with
a {{formula:5276f904-edb2-4283-8098-ae3225d88f0c}} -path {{formula:988051ee-ce63-48e5-b1b2-ea48e2537ac6}} of length at least 2);
Each edge {{formula:a2214471-876e-4227-8f9c-2ab42d4d1124}} is either used or discarded; and
Each edge {{formula:b088f8d4-b32f-4fec-a22f-66a3fd37bc37}} is always used directly.
The set of link-edges in the monomer representation {{formula:0e20a72e-70cb-4b36-85b3-257387389601}} EClnk( E(=1)E(1)){{formula:31cd6f1b-14ab-4ec5-8157-23a05fc98983}} Pe{{formula:309ca27e-6164-4889-9288-3d512feb6086}} e=uvEClnk(E(1)E(2)){{formula:da6edd8e-aaa5-4883-824b-8a0326d58779}} S{{formula:80b0636a-e907-49aa-93d7-55f506c98928}} GC{{formula:db46c991-c1e1-4add-ad8e-5ca39c180725}} A target chemical graph {{formula:919f0e4a-81c3-4def-8f91-f2c998ba2f11}} will contain {{formula:ea8b6c1a-ed19-4759-baf1-dce3628ed3fc}} as a subgraph
of the interior {{formula:32abd228-bab4-466f-8451-4dd3ddc16daf}} of {{formula:a47fda9e-e893-4aae-a02c-835f17e8967d}}
Interior-specification
A graph {{formula:2da78fb2-19fc-4f9d-a642-d748c6da528d}} that serves as the interior {{formula:beb8b97b-210e-45ee-98b6-1754061b1871}} of
a target chemical graph {{formula:3e9a95d1-ffd2-4a7f-80b9-1716e1e02ef1}} S{{formula:d9cda7e6-47ad-4c7a-8534-7cd875e1ce0a}} GC{{formula:f8d21eba-e781-4a93-b922-45c57726ffbe}} e=u u'E(2)E(1){{formula:36cd1d9c-97d7-4329-b2af-30c72adeefb9}} u,u'{{formula:4bdd5bd9-b3c7-4bc8-a5b3-31e4ef3cd9e4}} Pe{{formula:6ed5eaf6-0ad6-41e0-8212-68cffb61be59}} H*{{formula:fde64930-651c-49b1-bd21-028f6bb7f0b6}} S{{formula:f5b6f85d-d95a-4e67-9d10-0d59c41eb49d}} Qv{{formula:dc73b142-4a04-4522-bc59-ecfc8f03441c}} vVC{{formula:e50ab433-93c4-40ed-9c7e-a6716f59fffe}} vV(Pe){u,u'}{{formula:70b70e5d-847b-4964-be9c-15a02be2463f}} u,u'{{formula:27fd2e38-64ef-4a74-8a6d-68be107d35cd}} Pe{{formula:bf0abe3c-44cf-4956-85cf-f15d1bfd58c4}} e=uu'E(2)E(1){{formula:c4079f3d-eca3-48dc-8076-4a61c86eff0b}} Qv=(v), E(Qv)={{formula:9dc37ebb-6dca-48ca-bed5-40d5ec99e43e}} v{{formula:cc3173f1-1d19-41ea-a728-735e3ec540f7}} H*{{formula:6ebf9379-7831-4666-b826-a029cc3dd506}} |E(Pe)|{{formula:ce1216b0-ed74-4c60-a147-81eed63c729d}} Pe{{formula:2c4fae68-6ca8-465a-b9d1-21f04740ad1b}} |E(Qv)|{{formula:2ac3092a-1bf4-467f-892b-3cbb8c4aa75d}} Qv{{formula:8111ad19-56f5-4f5c-a151-19aaff8ad2f7}} Qv{{formula:20554796-550a-4dd6-b8d2-fee4ff3e3630}} int{{formula:e713b34e-8089-4285-b099-803d3baab3a4}} nlnkLB, nlnkUBZ+{{formula:4b39e650-8267-4bf8-bbe7-523c9515cfc1}} .
For each edge {{formula:9457cac6-c0e4-48f3-8ccd-89629a74ea1a}} ,
a lower bound {{formula:ddef5861-9014-481e-a672-8e7aa26dcc36}} and
an upper bound {{formula:0654c21b-8a88-4838-b1a0-c77ccbac8d91}} on the length {{formula:3c12eef1-78c0-464b-a6fe-806024bb6652}} of
a pure {{formula:7e9c8e6c-59e8-4e37-942c-0331b24a40de}} -path {{formula:688d6480-5b51-4393-9999-987a2d0496c6}} .
(For a notational convenience, set
{{formula:199f2e88-2049-49df-b8ef-63989901ecd8}} , {{formula:4853d00a-3e58-480a-a03b-c88d8d73563d}} , {{formula:868b57ad-fd1c-41e8-98a4-a1406cb43459}} and
{{formula:04120e07-01ff-42c6-b721-131de9ea65c8}} , {{formula:9d2abc36-c650-44de-a93d-d67a3d5371e4}} , {{formula:7ae52c95-83b9-475f-88ab-695a7dd5c0ae}} .)
a lower bound {{formula:dd16aa36-562a-427f-899e-8f25370dd8c8}} and
an upper bound {{formula:52765614-0d35-48d8-bd7f-e1c848aa0018}} on the number of leaf paths {{formula:511cf82d-64f7-4e6d-9c75-c19882255bd6}} attached
at internal vertices {{formula:18778fbc-88df-46e1-b998-716e72e63d11}} of a pure {{formula:e08f9fd6-7b17-46aa-908c-e3e25b8be366}} -path {{formula:ce36d7c2-5a9b-43a5-8881-1057987c2584}} .
a lower bound {{formula:d341c0f1-ec26-44a5-9f65-df9163837016}} and
an upper bound {{formula:2a4e9eb1-637f-469d-a501-b501b440bdc8}} on the maximum
length {{formula:180b1715-9f41-4c1e-a1ce-5578583f0d4e}} of a leaf path {{formula:c2ef1f4d-b71a-4b2c-bfeb-1ba11abd05a1}} attached
at an internal vertex {{formula:06010aea-e1c0-4017-9fe0-ebba7b724d02}}
of a pure {{formula:17eaac3b-dac3-4fe2-9147-68cb36c4c683}} -path {{formula:33ac0ea0-0d32-4cfa-853f-9aa45d220e64}} .
For each vertex {{formula:98809e32-f709-4df7-833a-f1e504f61e18}} ,
a lower bound {{formula:5cc81ba6-5655-4fea-a167-fdd9e810d0c4}} and
an upper bound {{formula:3b848a24-e55a-4cef-976d-7a50488a24cf}} on
the number of leaf paths {{formula:14ddba25-cb1a-45bc-a76a-061f04ada4f3}} attached to {{formula:82641bda-e7ce-4725-a3f5-ae8d93112dda}} ,
where {{formula:c94bb765-3a0b-4d56-9bd4-9656d63ea205}} .
a lower bound {{formula:a9c9b23a-1fac-4c55-8626-5a793fd56d14}} and
an upper bound {{formula:42d42530-ea3d-4aee-a2ce-404b6ffb1c35}} on the
length {{formula:fec4f60f-4a7f-4061-af1f-61a10cdbe49c}} of a leaf path {{formula:190372f3-205e-4a8d-9498-8b20bae20774}} attached to {{formula:c91a37bd-a00c-4518-8414-0825f0af44e8}} .
For each edge {{formula:bf18b3a3-b3a4-4312-baec-1cf5e3e5f84f}} ,
a lower bound {{formula:ae195623-d281-4c50-ab5a-29aa19a9e98b}}
and an upper bound {{formula:da927ddb-c94a-4926-8c61-cc7bc6925d07}} on
the number of edges with bond-multiplicity {{formula:90bcecfc-8464-4b7f-b1c0-f3096160b3c8}} in
{{formula:274aa51b-1f26-4a00-b18c-5f9c508ce06e}} -path {{formula:6e78031a-1066-4b6b-b566-b1302573dd03}} , where we regard {{formula:dc07e96b-87de-49d6-b4c7-28a508c69c4a}} , {{formula:c5fa8def-098b-4d38-8b2f-9119849d1b68}}
as single edge {{formula:2572f043-46bb-44be-9208-8d008406a703}} .
We call a graph {{formula:481594fc-ff19-4b57-876a-6b45b7ed19a9}} that satisfies an interior-specification {{formula:18c17b8c-dc5b-4b50-b431-fe2a66acef36}}
a {{formula:e719409d-7ceb-4e5d-a60d-b3bc62f5ad92}} -extension of {{formula:2e679182-438e-4a74-ad6b-df3f858642d0}},
where the bond-multiplicity of each edge has been determined.
Table REF shows an example of
an interior-specification {{formula:1bc2c782-2074-4f3c-8ad5-1a68c2e88190}} to the seed graph {{formula:704b1a91-c267-41a6-9ddb-81b215bfa9ae}} in
Figure REF .
{{table:e165987f-aa7d-4c62-b9f7-636fe18a206c}}Figure REF illustrates an example of
an {{formula:a2609bd5-b934-4571-965e-57d7011e59c2}} -extension {{formula:b6c9ce1d-6dd5-406f-bd1e-2333e71feb8f}} of seed graph {{formula:026519b1-383c-48af-9efd-0fb2e59cb51c}} in
Figure REF (a)
under the interior-specification {{formula:e421525c-2382-4c14-808c-21ffab73434c}} in
Table REF .
Chemical-specification
Let {{formula:e97a90e0-a880-46bf-ab6f-46a0d16890df}} be a graph that serves as
the interior {{formula:777e640f-a2a2-40ef-9f4c-92f963153096}} of a target chemical graph {{formula:043ef74d-dd28-4dc3-85f2-38ad65a5021c}} H*{{formula:fce5ddb8-ef42-4297-9036-2dc016e77324}} from {{formula:cacf5e52-fada-44e9-9de5-d2e4df2b09b8}}
by choosing a chemical element {{formula:137348ff-b353-4fb4-aee6-6cc7141ac476}}
and assigning a {{formula:06247ac5-b439-4f08-8fed-65959e50b181}} -fringe-tree {{formula:d23fd0a8-0457-4584-a3b5-814211c6696a}}
to each interior-vertex {{formula:599ffd5d-192a-4f51-a6ee-f17a96283ba8}} .
We introduce the following rules for specifying
the size of {{formula:e3382513-58b1-4579-9df2-0a46953d6e99}}{{formula:83822d08-2e0a-4efe-ad66-fcc5f53c2e17}} ce{{formula:99bdc46c-561a-4034-bf6a-4d6ff065f071}}
Lower and upper bounds {{formula:b9e319d6-321c-4bb2-ac01-c68ab2a2e7de}}
on the number of vertices, where {{formula:3976c6a9-744b-4eba-af0b-727b185fe1dd}} .
A subset {{formula:300c5407-4e29-48d6-8e39-716d54505491}}
of chemical rooted trees {{formula:1abf7c56-a9f2-4c09-ab5e-8139f32d1ea6}} with {{formula:b5cdde64-fc95-4476-bc24-c18f72df3f5d}} , where
we require that
every {{formula:05c47da3-db6e-4430-83d2-412c66621dd4}} -fringe-tree {{formula:93b24b7a-ca7b-4011-a684-37a44fe30c8c}} rooted at an interior-vertex {{formula:39507e6c-c70c-4f6e-8c58-045b78eb31e0}}
in {{formula:b0fd0529-ecbb-4afe-a9ec-35c55f161072}} F*{{formula:d754d6bd-a71e-42d2-bd7b-93882b6eb5c9}} ex{{formula:f3a12592-aed8-403b-be2f-c9a5894a73ff}} F*{{formula:b94e6039-14f2-4ed8-b6e2-7afbcf3ddebb}}
A subset {{formula:31c2a190-8022-4e1d-be6d-b8cab3b93f6f}} , where
we require that every chemical element {{formula:72e6b949-6845-4210-8c5e-525edc868916}}
assigned to an interior-vertex {{formula:f425ff9d-25b5-4237-bf57-2334eb42eb16}} in {{formula:a34e0669-b35f-49d2-8b2a-e6719bc929a2}} int{{formula:2e9edad2-4825-4e4b-8c67-0006100177d1}} := intex{{formula:2db6313a-2c87-4bd2-8678-14691cc7451f}} naa({{formula:666e9df3-5016-40df-b233-1c63c25eb79b}} naaint({{formula:9e698193-1dbf-445e-abb3-6bdec7f7e77e}} naaex({{formula:4ee88593-8839-4e5e-aa96-a570f5be3794}} v{{formula:f8ae2688-7373-4728-8e75-95fefdb47e7e}} (v)=a{{formula:18853421-4819-453c-a143-90a54abaf536}} .
A set {{formula:f346ef70-48f5-4e78-8c08-2fc044335f57}} of chemical symbols.
Subsets {{formula:93815436-3650-4e10-9813-645d0407f383}} of {{formula:288aa321-2ca1-482f-9728-f93cb464d10c}}
of edge-configurations {{formula:8571ce11-d1e0-4427-a55a-4296675bd217}} with {{formula:1df579d4-c9d7-4fab-b5d7-17e5d66bbae0}} , where
we require that the edge-configuration {{formula:ccbf99c2-a292-495c-bf89-288fdf06c5e7}} of an interior-edge (resp., a link-edge) {{formula:34de098b-c1b6-4d8b-a30e-1385416faa99}} in {{formula:48b4b160-6066-4f48-ac5e-c16376881927}} int{{formula:544a180a-63fd-4f45-bfdb-bb89b18aef33}} lnk{{formula:a947bf0e-0812-42a5-ac73-bcd41af5fbef}} (,' ,m){{formula:e90b5d1d-d9eb-4262-93cd-9b51b1dccada}} (' , ,m){{formula:39610bcc-3138-4b9c-afff-bfeed9a3c9e9}}
Define {{formula:0b834d99-ffb2-4749-be22-32c618aa9704}} (resp., {{formula:71802de4-f52e-472e-9c0e-854cc0db52d6}} ) to be the set of adjacency-configurations such that
{{formula:694534d8-a50d-4f41-a980-8238d25c6e04}} .
Let {{formula:76330b0d-944c-4ea6-a543-7b37c4f768ee}}
(resp., {{formula:0bd835a9-c79f-4243-9a87-c8d214078dfa}} )
denote the number of interior-edges (resp., link-edges) {{formula:b1cd427d-312b-410c-8962-8a1b74cfb25d}}
such that {{formula:2b9e644b-22ea-4076-b0d8-1fbfb219ffde}} in {{formula:9673553f-72e6-44ab-aed8-c592aa86df61}}
Subsets {{formula:5f6711e9-0cf7-4352-b416-62242df62874}} ,
{{formula:bdcb4532-f6bf-4c44-8176-9e59baa678bc}} ,
we require that every chemical element {{formula:221fa927-870e-49c9-ace7-501717233b5e}}
assigned to a vertex {{formula:6b9f8e50-b69f-445b-8c76-0bf85a2841e7}}
in the seed graph belongs to {{formula:fd45be4b-e611-4d2b-8a0c-3311d8e092c1}} .
Lower and upper bound functions
{{formula:cffd2ed3-f7b3-48da-9772-dd23e16b9a28}} and
{{formula:3c6655bd-8328-4bfa-9b18-087849e5cf01}}
on the number of interior-vertices {{formula:c9873d65-ec56-4b0d-a481-b357fea13906}} such that {{formula:9daf1570-49ae-4af0-b25b-e6ab88fbf769}} in {{formula:1d3eb940-57f6-415f-9daa-9cc2413a12ad}}
Lower and upper bound functions
{{formula:42d8d272-a94b-41b6-b03d-a5dfe8393e24}}
on the number of interior-vertices {{formula:99ab0480-8f6e-4b0d-99da-52ecab0ad5e8}} such that {{formula:7f4f9fd8-0086-43ff-a008-7b49482bbeef}} in {{formula:cac4a4fb-afaa-44a6-a3d5-2317598a5d8f}}
Lower and upper bound functions
{{formula:54c43db1-1348-44b1-a812-cc9f5f9d02d3}}
on the number of connecting-vertices {{formula:f5be1eb3-22ca-4232-8211-7c155501b985}} such that {{formula:e981f26e-12d4-428f-bef2-b77358427edb}} in {{formula:bad3ef31-ad99-40bd-ad05-10ab039b1d00}}
Lower and upper bound functions
{{formula:cbaff3c5-f4c2-4aae-b1b8-327f97f8375f}}
({{formula:bbd2ae0d-1e7a-4429-a6f2-a0d19e443f5c}} )
on the number of interior-edges (resp., link-edges) {{formula:310b96e1-44da-4277-baea-b5fdeadefc65}} such that {{formula:e3c3b21a-272b-4e7b-a846-3145a20da960}} in {{formula:80e50a15-c5e7-4e73-82ab-0012642ca179}}
Lower and upper bound functions
{{formula:1644a525-872d-49fb-b179-8c2f19572bd4}}
(resp., {{formula:3d783fcf-b8c1-48b8-8942-5f8155825413}} )
on the number of interior-edges (resp., link-edges) {{formula:5af228c6-f9e0-4dfe-807b-02309eabd9dc}} such that {{formula:15837e60-2d23-4850-9197-477e4b54c382}} in {{formula:d3a301ca-d01b-470e-bc6f-91f7141ae214}}
Lower and upper bound functions
{{formula:012c876d-8d27-4f3e-a8dd-27906b91a6bd}}
on the number of interior-vertices {{formula:62d82314-90c4-42c4-ab5f-37a49e753d06}}
such that {{formula:e6a536a8-6174-4252-9766-79d8efd90e2b}} is r-isomorphic to {{formula:8f72eddc-257d-450a-93ae-ea9c4b703007}} in {{formula:4e0f89e7-25a5-43ee-aa66-30f688dc30b3}}
Lower and upper bound functions
{{formula:dadd636d-cd6c-40e9-85d2-9cb4b22ae21f}}
on the number of leaf-edges {{formula:c913d7f3-bdb1-4e85-9874-41a5c6aa6db8}} in {{formula:53ab84e3-45df-4439-894c-3e0aea75defa}}
with adjacency-configuration {{formula:2083e617-31c3-47ac-a929-9a3c55ba6687}} .
We call a chemical graph {{formula:d67223f7-7cf4-4bc4-8e8c-34b98eb28b73}} ce{{formula:14134681-48a5-4b92-a713-0e56fa0fd8a3}} G(GC, int,ce){{formula:c63889fe-a7af-43be-a457-8a6a947992e5}} (int,ce){{formula:73982592-c945-438c-bf8c-e70f108a22fe}} GC{{formula:ff6217b4-5d48-43c8-b7d6-bcdbe06f7cb3}} Table REF shows an example of
a chemical-specification {{formula:734777c9-b4c8-473f-a290-db9379db620a}} to the seed graph {{formula:fa9a8a96-03fb-4e54-9809-b8816a654771}}
in Figure REF .
{{table:82feb155-a373-49e3-867a-282536e572d7}}Figure REF
illustrates an example of
a {{formula:a5978f41-f65b-41ce-a2c3-122524580684}} -extension of {{formula:0377d8cf-74b9-4507-9afb-52e3c9be6048}} obtained
from the {{formula:068ddf8e-f473-404a-aab3-fdba66b84ada}} -extension {{formula:2a3af0d3-25f9-41b1-b784-a096e39c12e2}}
in Figure REF
under the chemical-specification {{formula:285671ba-a13b-415c-beb1-e89303212964}} in Table REF .
Test Instances for Stages 4 and 5
We prepared the following instances {{formula:4a5ccdf4-927b-4abe-98e0-1250345f1fba}} and {{formula:e5582369-ef7d-4bd1-92ff-f619d5bd8823}}
for conducting experiments
of Stages 4 and 5 in Phase 2.
In Stages 4 and 5, we use four properties
{{formula:2c3e6d48-e27f-436a-9fd3-a689b945c2bf}}AmD, HcL, RfId, Tg{{formula:46ff5e81-c258-449e-ba74-5a50823c8eb6}}
and define a set {{formula:a84114ff-7709-40c5-bca5-2e3ed71518da}} of chemical elements as follows:
{{formula:15b3f296-bbea-4b7b-b815-88528af6db5a}}AmD{{formula:cb998695-a15a-4746-a75a-aba9a0b52dcb}} ,
{{formula:59dc35f5-81bd-49f6-8240-e6d2894b8388}}HcL{{formula:fa2d405e-6af2-4f5e-afe0-786348de9c2e}}Tg{{formula:21b91ade-6edc-4753-b3b9-d46e32b1f8ec}} ,
{{formula:8835e668-1dbc-4818-89ab-f0d17bfb3af3}}RfId{{formula:e475f65e-e7fc-4731-b66b-086daf9c7625}} and
{{formula:3211a85e-6955-40fe-8583-4665abd4abb6}}Prm{{formula:e13e6efc-6a3f-4968-a034-89a7edc9079b}} .
(a)
{{formula:3aeeedf6-9617-4d51-ba08-6325f76fbe29}} : The instance
used in Appendix to explain the target specification.
For each property {{formula:5ad44df7-bf3a-4731-acb4-235974c2debb}}AmD, HcL, RfId, Tg, Prm{{formula:93ce1d7e-965c-4cd5-a3bb-2ab21df29412}} , we replace
{{formula:e4ca5eea-7c77-41b8-a98c-7c7c6a0fa598}}
in Table REF
with {{formula:676a2de5-176d-4d96-acf7-2d8e68b5c0fe}}
and remove from the {{formula:345f3ae7-cb65-4d7e-b5d5-7fca0550b8ed}}
all chemical symbols, edge-configurations and fringe-configurations
that cannot be constructed from the replaced element set
(i.e., those containing a chemical element in
{{formula:5ea18feb-a1bd-40ba-a3be-623ed6cd6850}} ).
(b)
{{formula:3b8a60c9-e603-4673-bab5-28ebd1c0d0fd}} : An instance that
represents a set of polymers that includes the four examples of polymers in
Fig. REF .
We set a seed graph {{formula:1e625a1e-2956-4f97-b509-a15c638cc4b6}} to be the graph
with two cycles {{formula:55442c44-ef7e-483f-b172-d1231f99d0d6}} and {{formula:519618ce-7d46-4830-b237-ab021e3e446b}} in Fig. REF (a),
where we set
{{formula:c1d4dec6-6fb8-4535-8ca0-c22deee6878d}} and
{{formula:79494f35-c0b3-4521-a999-c849394dc1a3}} .
Set {{formula:e759d107-75cc-488a-ba6e-dc3ef4c20496}} for each property {{formula:c10fe854-f189-4eb9-9278-e5f755fdd419}}AmD, HcL, RfId, Tg{{formula:32ce96ac-33b8-41c5-9bfa-11370dea6d73}} ,
and set {{formula:ea664373-fa92-4c4d-a6ed-86873223c8ff}} to be
the set of all possible chemical symbols in {{formula:91f1330f-89b5-484d-bef1-e79c0d1c57b7}} .
Set
{{formula:21f549db-21cc-480c-9d1d-2236b21ed9bc}} (resp., {{formula:6fd6c01b-ca7d-4982-b3d5-a9984cfd7da6}} )
to be the set of edge-configurations of the interior-edges
(resp., the link-edges)
used in the four examples of polymers in Fig. REF .
Set
{{formula:eb4d75ac-4a83-47ab-917f-b301f72ce798}} (resp., {{formula:5ed5da11-8a4e-4a97-9bac-d1db060724c3}} ) to be
the set of the adjacency-configurations of the edge-configurations in
{{formula:6147a90b-f5ea-4c3e-b7c6-07b1a0c71e86}} (resp., {{formula:d3d57292-e708-4ec4-a736-29130337225b}} ).
We specify {{formula:c86a2c6d-6223-4df5-9f5d-6d484e4b695c}} for each property {{formula:1b1d1f66-f8bf-484d-8349-40788aaa8550}}
and
set
{{formula:cd4a4ece-3e15-40fc-98a7-b6270b255dc6}} , {{formula:90e0d80a-cca9-488b-83c2-9ab8e799a3d8}} ,
{{formula:7a14580b-0b60-4009-94c0-41ee18c8baaa}} , {{formula:0ccfb543-f7bb-491a-ac6a-a043d8718b92}} .
For each link-edge {{formula:7e83fbbc-8728-42cd-8a5a-68d42f3b41e8}} ,
set
{{formula:4e56c101-18b6-49fb-b357-88f2a4db2a24}} ,
{{formula:ad6e5e84-647d-4edf-958f-6634c681b846}} ,
{{formula:c497948d-9752-4260-8657-1adffb665328}} ,
{{formula:9f37e8d7-3f49-40f1-82f0-fb5a6049609c}} ,
{{formula:23401a65-a947-427f-b120-b6623dfa808e}} and {{formula:0a507e32-298e-466a-8fdd-6c653e015825}} .
To form two benzene rings from the two cycles {{formula:9ac42753-7950-4c1c-b84c-4d9d503bf849}} and {{formula:0478e15e-134f-41dc-887b-314bc1ebd292}} , set
{{formula:684912be-489b-4000-bb29-460f9e5a0bfe}} ,
{{formula:13e4796d-5583-4375-b256-6c47e3dc85d0}} , {{formula:01f6437f-6742-44d7-a76c-ed696c68c669}} ,
{{formula:236bb225-96dc-403d-b3af-22ce7f7bfb98}} ,
{{formula:0b2dab3f-d544-482e-b202-621cb3d9361a}} .
Not to include any triple-bond, set
{{formula:458c5b54-deff-4ef8-82c4-95ec43e8d752}} .
Set lower bounds
{{formula:743ee3ac-fa4f-40af-9664-e1d7ab4c44b5}} , {{formula:f4ef5516-61f6-41b0-b482-af6021282dcc}} , {{formula:5044f22d-d71e-4db5-a299-bb37951f0f19}} , {{formula:0077d75c-bd2e-46ed-b98e-82d53da78a71}} ,
{{formula:6066354b-7e4a-40d5-8d30-68d7e1b2448b}} , {{formula:01526aea-4b0d-4a9b-98cc-f3dcb38a28b9}} , {{formula:1229cda4-c73d-4c7f-9536-244587017f3b}} , {{formula:61065725-be16-453c-8b9a-c439d752214c}} and {{formula:6b41727b-e6b8-4532-97b3-9ff0a686693f}} to be 0.
Set upper bounds
{{formula:51ee6480-d6a2-4ad0-bdda-58299c1995a1}} ,
{{formula:3f01e536-e218-4c4d-a528-f1f950a64ef2}} ,
{{formula:d0e204bc-41cd-49b7-939d-c726a0e91032}} ,
{{formula:4f7c8429-fb27-420b-b457-9856918683a8}} , and {{formula:784a4b85-b09b-4df6-b804-374fe8006da0}} , {{formula:6b17e867-acef-41e5-84a1-7ef2920249be}} ,
{{formula:13aff91a-453f-4bf0-8a64-98895a26c467}} , {{formula:486e5969-d078-4f72-92cb-f10c8bb4d524}} , {{formula:fc943b30-45ec-4ed6-9de4-1e47d02bc62a}} , {{formula:14aac3da-0a46-4db6-a1de-82402a21f948}}
and {{formula:7dd2bb78-6d14-47f6-9a56-3615ccb1a68c}} to be {{formula:48f85208-f923-4b0e-9025-1c4d0248a33a}} .
Set {{formula:00a680e6-aa13-4968-bf32-4d864e8507e5}} to be the set of the 17 chemical rooted trees {{formula:b3988e93-d8ee-46b4-b515-def1e084294f}}
in Fig. REF (b).
Set {{formula:7c1001fc-6e2a-4412-841d-1b2922fc0d89}} , {{formula:2604e2a6-caa6-4be6-94c3-c3db96bd424f}} and
{{formula:65d27b6e-0462-47c5-9443-5ccb7d6951be}} ,
{{formula:06074f75-860e-46ec-8066-049db86f150c}} ,
{{formula:4bd19778-fea0-4043-891d-4c3ce5b9dca1}} and
{{formula:5d8e6a97-057b-4361-aa4a-1baff97afdf5}} .
All Constraints in an MILP Formulation for Chemical Graphs
Our definition of an MILP formulation MILP {{formula:aec60637-3925-4b36-92d0-6c149afd4658}}
is analogous with the one by Zhu et al. {{cite:74356bbe8d662afddc80670b4a4eb8fa955d9b16}}
except for a necessary modification due to our polymer model
with link-edges.
We define a standard encoding of a finite set {{formula:c0e0214d-ba78-46c4-86ad-2d0ddb15cbec}} of elements
to be a bijection {{formula:8cf6b288-da2d-4481-9b38-136de484a3ac}} ,
where we denote by {{formula:a708ae32-a8d1-4224-88ed-3a48462e2b92}} the set {{formula:ef1b0295-fbb7-445b-b89a-c2fbe689b85a}} of integers
and by {{formula:5f3ab283-8a36-41aa-b5fb-2228509752d9}} the encoded element {{formula:021a83e4-0986-4822-ac57-3a44f79cf53b}} .
Let {{formula:1f2c958d-a21a-4e91-a000-ea75126b0606}} denote null, a fictitious chemical element
that does not belong to any set of chemical elements,
chemical symbols, adjacency-configurations and
edge-configurations in the following formulation.
Given a finite set {{formula:d23142e8-5bd4-4087-9b7c-4bc43291516f}} , let {{formula:225e1b38-1ddd-4b72-b232-ea7a1e0dc457}} denote the set {{formula:34808dad-970b-4ade-bd41-41082aa9d8f2}}
and define a standard encoding of {{formula:cfe2217c-94d6-487e-a496-b8b0b3cb617b}}
to be a bijection {{formula:4c4f2a8d-00b5-46cb-bafc-f47112f1365a}} such that
{{formula:e498449a-333a-4431-824f-f928757e9742}} ,
where we denote by {{formula:9f6847cb-9d72-4755-8617-ca8f62840f5f}} the set {{formula:119ddc77-b0af-417c-a657-83e518f4cb75}} of integers
and by {{formula:97fb4922-6943-44d0-a2f0-a6c04ae14548}} the encoded element {{formula:a67098b4-bdbc-4137-9743-6c6188c70b31}} ,
where {{formula:77eda93f-6b41-4084-b27f-38892d20f61a}} .
Let {{formula:53034aed-205d-484e-a215-d05a1aad4e07}} be a target specification,
{{formula:a7f4f562-2bb1-497b-8b73-9c87f8b8a87e}} denote the branch-parameter in the specification {{formula:81132e3d-4b82-4e78-adc2-bd86cd6df9aa}}
and {{formula:b41e0e5b-93fa-496f-8dd8-fa90fe07ca3d}} G(GC, int,ce){{formula:fe6f20a4-a016-4d65-84af-badfc1e455ff}}
Selecting a Cyclical-base
Recall that
{{formula:746ca679-767c-43f0-8eff-ec9bda383c41}}
A subset {{formula:6b407b7a-770a-4fdc-9731-1698cc04e727}} is given
for introducing link-edges in the monomer representation {{formula:d202f080-f506-43c8-a5e3-85d226a9738b}} aiE(0/1){{formula:5e47905c-fb3d-4053-a450-c691e226785e}} C {{formula:8210106b-0a89-4351-bae6-f49dd0b202ce}}
-
For each edge {{formula:212bf0c6-752f-4ef1-b2ed-f2ab89277bd5}} , edge {{formula:e886ddde-6c03-458e-83e0-22f695ab379a}} is not included in {{formula:203fdfb7-c2ce-4dce-8321-042f471340e1}}
and instead a path
{{formula:3808165b-75c5-4b8d-a1dd-5fe09a4e4ebf}}
of length at least 2
from vertex {{formula:c8baf28c-2cf4-4a6e-8af7-f57ed57524ac}} to vertex {{formula:8c9a86e5-2763-49ea-9438-4e2e003ecc8b}}
visiting some vertices in {{formula:3353d7eb-e851-46e5-ab28-b63ed48d1e3e}} is constructed in {{formula:a3eb36f8-3283-4611-887e-76deb1d6a0bc}} ; and
-
For each edge {{formula:706f47ef-1df5-4ced-bf64-b30603865c16}} , either edge {{formula:1c322a45-df7e-483e-82e4-2450106b2c8d}} is directly used in {{formula:f3ba1f59-8566-4d4a-bcd9-b6f9e947335b}} or
the above path {{formula:0018ac5c-c969-413a-beef-ac0cce16f6dd}} of length at least 2 is constructed in {{formula:1272a7c6-ed2e-4afa-b3b9-aa1cf159998c}} .
Let {{formula:bb4b43f8-1ccf-403d-8d9b-671abb4c0468}} and denote {{formula:5ae08479-ae12-442c-ab0b-c52eab32511f}} by
{{formula:0a2f85e4-99bd-4015-8e3b-27c1a371aa72}} .
Regard the seed graph {{formula:01911c80-0810-47f3-b945-a3199f7a6a58}} as a digraph such that
each edge {{formula:2520b8da-791c-4c3a-8df4-ae3dd9d0a995}} with end-vertices {{formula:dd7a1c9b-3b16-42d4-a9d3-be06a14c9996}} and {{formula:4c62e8cb-1076-44b6-80ed-29c97843e136}}
is directed from {{formula:cd563100-6927-480d-aa29-61e18d764d22}} to {{formula:3ac520ce-fba6-41f4-8df3-be27e2df3d91}} when {{formula:94e79191-f05b-4b24-97e1-360a293ae84e}} .
For each directed edge {{formula:ea009577-3797-47dc-9bd1-90a4de827b97}} ,
let {{formula:fd414523-926d-47bc-9d56-000c5aa8a246}} and {{formula:1bccd283-2ab8-4757-b965-0e610625dfd9}} denote the head and tail of {{formula:726ae14f-0169-4aca-b7c6-913d162f295d}} ;
i.e., {{formula:4fb88064-151d-4399-9ece-58702bef2ef4}} .
Define
{{formula:ebfcbefe-3029-4f18-b457-db67e0b28e4b}}
and denote {{formula:04f42ba8-5a61-40dc-adf3-ac5b0613452b}} ,
{{formula:18cb2497-2aab-4245-b5dc-4bbc70ffeab9}}
{{formula:8814c7f1-9837-4917-9985-26315fffa7af}}
Let {{formula:49faa3e1-f21b-405f-95f2-7491336bbfd0}} denote the set of indices {{formula:027b7b33-eacb-4773-856a-9ae23b487fae}} of edges {{formula:b96a9be5-b272-4db9-8e3f-32b6339d51e2}} .
Similarly for {{formula:db6665fc-8c7c-4e03-affa-3aa7270630ff}} , {{formula:ffe34a94-8274-4ada-b9b3-c5c77407b36b}} and {{formula:33967358-5476-4a28-ac52-5eb83480a58e}} .
Let {{formula:b087eefa-9a33-4773-a519-13b39ac1045d}} denote the set of indices {{formula:03e51ddc-f301-447f-baf6-b256060b46a5}} of edges {{formula:5c1e8216-2e92-4a8e-b28c-898369f9196a}} .
To control the construction of such a path {{formula:be50eeea-4c0e-41d5-bb54-b80e8358db61}}
for each edge {{formula:18b87444-2966-4d28-838d-6dc44d575882}} ,
we regard the index {{formula:44735e7b-f881-4acb-bbaf-99d6140ea2e2}} of each edge {{formula:c4e9a76f-c4d6-40cf-ad3d-f554eee40465}}
as the “color” of the edge.
To introduce necessary linear constraints
that can construct such a path {{formula:bef884ab-c64e-4834-9b4e-9c7273f5bbe5}} properly in our MILP,
we assign the color {{formula:7d918ab7-2a39-4e7d-9aa6-03ee04518dac}} to the vertices {{formula:de2ea6de-0ace-4d23-bd08-5a3a771022a3}}
{{formula:b993bf94-ba0d-48c7-9ad3-e22fb045a9fd}} in {{formula:98e3cfbf-473e-459e-898a-067f05f8836b}}
when the above path {{formula:3ac351ae-cf49-4be1-b678-1733f81ee423}} is used in {{formula:daa15c6e-237b-4f8e-bee6-a7010897169b}} .
For each index {{formula:8894e70f-7f50-4517-9674-2a884020cfd9}} , let
{{formula:1ff00f8e-c7ff-4e8b-bd69-9c8dbab5ce5b}} denote the set of edges {{formula:26eb7c2a-3899-4b42-8d03-6c1b855c7eb6}} incident to vertex {{formula:3c7aaafe-e384-4cec-9aa3-14dadb97e2c9}} ,
and
{{formula:a47f2013-a661-4e6e-8d63-8ceab9ca5dc1}} (resp., {{formula:c610a8c9-23e4-4f08-8c8d-8c4e0b2d9fb3}} ) denote the set of
edges {{formula:332c8768-42c0-447b-b7f9-fcc047a01338}} such that
the tail (resp., head) of {{formula:b4923a40-9d3e-4275-bfc2-c71c41287d62}} is vertex {{formula:171125fa-4091-4ecd-a039-0d17e59b8a3f}} .
Similarly for
{{formula:8fbf3ec0-8bd9-4027-8c2b-04c90b94e020}} , {{formula:6ca53d9a-70f2-4160-a901-a3b6a3bbab96}} , {{formula:fd667373-c1df-4264-b8f0-2b84f530f419}} , {{formula:7cf1aaf2-3743-41cc-988c-ad02dbbb4d1f}} ,
{{formula:f43bbb68-a1d0-4721-9ac9-751d61525c9c}} and {{formula:99c464fe-54b6-4f6d-8fb2-0367d4f23450}} .
Let {{formula:69f91f47-6ee4-498a-9b2c-a925abf420c1}} denote the set of indices {{formula:c5d5af66-ad26-4258-ad7d-fb46ff63ada5}} of edges {{formula:5f3567f3-a147-4d93-908b-dc991522954a}} .
Similarly for
{{formula:8850f746-3a2b-4319-876c-c8a7373adce2}} , {{formula:f0d94c0e-657e-47c9-be21-1cdf09f2ca93}} ,
{{formula:bc39da83-0a59-460b-b73f-9e4efd07bf2d}} , {{formula:cdb33546-18b4-41a3-8e3e-f0347470b4bf}} ,
{{formula:65666f8f-71bc-4503-b2ae-c4c26c1cf558}} , {{formula:c6a2862d-6ed1-4390-b7e4-3410d38bbe23}} ,
{{formula:8cc48035-0c7d-484f-8bf8-43581d2f7db2}} and {{formula:4764e698-a23e-47cb-900c-8a7bad34bc33}} .
Note that {{formula:c4e23d1e-ce24-40ab-9e06-a1ce3233dbce}} and
{{formula:de1fd2ae-9341-4c0b-87e1-564e36bb2020}} .
constants:
{{formula:c9b9309a-61b3-4a0a-9ef4-cf76f2132b4c}} : an upper bound
on the number {{formula:258913df-d707-4add-997d-623086a801c5}} of non-hydrogen atoms in {{formula:30d3ea31-e722-4513-a292-91297f9e1f87}}
{{formula:5fb8fe79-c2c6-4fcf-b09b-8acd63872706}} , {{formula:cf13fc38-2028-48cc-8746-07039a5508ee}} , {{formula:9694fdca-2fdf-4b15-bcc2-5a117a715b28}} ,
{{formula:06c2ad1e-1a0b-4693-b1c3-896ce307fe03}} , {{formula:da5bc2a2-fb76-4170-b88a-a76730fbf8d5}} .
Note that
{{formula:a0cb9821-c207-462b-800e-dfb8f81e0c52}} holds {{formula:b1305164-14c4-4db0-a570-a21871fcbe00}} ;
{{formula:2f18769c-6fe0-4875-adf5-21159ca9b6ae}} , {{formula:205d041f-20d5-4493-b2d3-9d9b7f9682c6}} :
lower and upper bounds on the length of path {{formula:4238a452-2642-4b71-b86b-db62db2fa49b}} ;
{{formula:cb1c633b-ca0d-46e7-bad1-aaf3b98414b1}} :
the number of link-edges created from {{formula:059b9d9a-ac23-459f-8f4f-f92cf524a522}} ;
{{formula:1d05b6ab-9e9f-4975-a0f3-20fc85c612bd}} :
lower and upper bounds on the number of link-edges
of a target polymer {{formula:e579977f-be58-4bc2-ab63-27b0a1e856e6}}
variables:
{{formula:f8e9c8a4-7dd8-4f72-aa3b-02c242a1a114}} , {{formula:8a348e3f-f3d5-41ca-a1e7-1db63870c5c5}} :
{{formula:6ad69dfb-339b-4484-ba2d-d2e0ebd55d22}} represents edge {{formula:29da823e-d626-4f12-848b-cbd9ca0e153d}} , {{formula:f415da6b-86b5-4371-83d2-2b5fa051c8c5}}
({{formula:7da1e777-321b-4ceb-ae90-9d439e3ab804}} , {{formula:9d564865-c9eb-4b28-a5f6-c5a7001245d4}} ; {{formula:0f13ad7b-a0e9-4237-b616-33cc76844a3e}} , {{formula:163389f1-7617-4632-9d53-5250bbcc44b0}} )
({{formula:bf525067-bde6-407b-aa6e-cc2e740c20b6}} {{formula:41b5aa67-3f97-4a77-b0b3-83005a76fc4b}} edge {{formula:47a9abd5-76c0-426c-b3ab-b3ba4fc90bef}} is used in {{formula:b41a9ddf-09b4-4a29-936a-e63ccad62ab2}} );
{{formula:f4d18716-8759-428f-be69-24a83ccde4ef}} , {{formula:6b6485e6-b548-40d8-9e7b-9873225820f5}} :
{{formula:f96b848e-bc9b-4c99-9cdf-8808596c190c}} {{formula:a619617e-7835-481c-b3dc-32ae1fe59818}} vertex {{formula:c3060a5a-2ac6-4393-a48c-934bc5932dcc}} is used in {{formula:a87e8a16-4f93-47a8-9f83-61e50d7d3b5a}} ;
{{formula:ebf3e5bc-3b07-4493-927c-0050d7fd09cb}} , {{formula:fd262740-8efa-4b47-9797-05e0ffd7a536}} : {{formula:33d3082b-cfa7-4b7f-b08a-d6a31e7f0ec4}} represents edge
{{formula:0cd7b2ac-f078-4415-a902-6dc5a0975bdb}} ,
where {{formula:c967f811-eef4-4c6b-8aad-f602454f7fca}} and {{formula:ad4522c7-29cc-45e3-9ba5-36f0127afd62}} are fictitious edges
({{formula:56d1da8a-c477-468c-bcc3-b4269a1b0157}} {{formula:b5bf0e5f-3e11-41ff-9071-2a9e6ddaf8c5}} edge {{formula:8985dfce-b505-43fe-956f-a6be496b72c1}} is used in {{formula:dbfca9da-5754-46dc-9dbc-b733d394c58e}} );
{{formula:fcf4cefd-1e58-4666-85d3-5e58bb50dacf}} , {{formula:922e394d-cafa-43e2-99b1-61c93bcf1f9a}} : {{formula:e555c699-4133-48f3-9f92-2932a5ba2238}} represents
the color assigned to vertex {{formula:c376c4ef-ddf2-4542-a6dd-4bd52ddaf424}}
({{formula:66bc2ff6-79a4-450e-b9b3-2fe0efd7d731}}
{{formula:45bc66d1-a4c4-4973-8dc3-4435479630a3}} vertex {{formula:730940d8-f751-4951-89b1-d61d158839dc}} is assigned color {{formula:e10959d8-33b7-4576-868a-a26fb8244232}} ;
{{formula:45c8b888-7c00-4d2c-b416-c78416d7dfce}} means that vertex {{formula:49b09808-7d5b-47b6-b6a1-ead8ee4c6437}} is not used in {{formula:b73cad05-884f-4ae3-a061-a8cc0720015c}} );
{{formula:d7fa12a8-9f09-447e-92af-fe18bcb4c222}} , {{formula:de3f6ce0-e26f-4d97-a128-91b3749af755}} ,
{{formula:73a6fe3e-ddcc-40da-8d61-e8d30115731d}} : the number of vertices
{{formula:58d6b66c-0985-4320-a624-7cfa5f1f3325}} with color {{formula:dab88d19-806b-452d-a26f-f4a0c9da94ca}} ;
{{formula:4e010d2b-7aec-4d56-812a-6146bb8bca1f}} , {{formula:e136266e-e0b4-4f9d-85b7-af4360eb8459}} :
{{formula:52f72ca4-485c-4e38-b510-716d539fcd80}} {{formula:01fefa84-72f1-48c3-93b2-3a5afa9cf056}} {{formula:df553928-8239-4e1b-9c87-b4c6d94ea65b}}
for some {{formula:25990bee-c8fd-4878-be0d-78b0af49e568}} ;
{{formula:8215d2d2-ec2a-4d2b-935e-343cb6c649e2}} , {{formula:b089590d-1849-4650-9d71-46b06a66c2a5}} , {{formula:9fc2dc71-1910-4865-a55d-c46dded96bd6}}
({{formula:9c6f0352-8c12-4ac7-9d40-2af3d9a9d9bb}} {{formula:ca4abd50-b92a-483f-b1a5-9159b4d60fba}} {{formula:faa68eb1-e5f5-4a7b-b859-b14a42569101}} );
{{formula:e8cdf6d7-1b7c-429a-88c8-1d98147df214}} , {{formula:666a4546-4000-407a-a709-58db6d5fa1b6}} :
the out-degree of vertex {{formula:82f1c8de-1b94-4e8c-bad6-691e0d05cd95}} with the used edges {{formula:c24e41d1-1afc-4c1a-8d51-65efd5abe3ed}} in {{formula:47e0aa46-2e37-4887-98f7-5f8c6f5fd47f}} ;
{{formula:87aea7ef-c6d5-4812-a2d1-0d6212ad487b}} , {{formula:ed763419-b868-43b2-8077-3b7bba140c96}} :
the in-degree of vertex {{formula:6bab4bb5-3bb8-4f3d-98fc-cf381ca7a783}} with the used edges {{formula:24d3b613-4e1b-4b09-b81e-c5fbdcdf4fd7}} in {{formula:1c623dba-b26b-4dc8-bc45-0745adf1a5f6}} ;
{{formula:f381e363-bc9a-4896-a302-07fe9d817aef}} :
the number of link-edges in {{formula:e509d308-9c47-4433-997f-e4557f2e47d5}}
constraints:
{{formula:a42d1c51-ebcc-4505-bc53-ae01cb280c94}}
{{formula:f5680a3f-aff5-40b3-a5d2-0a111fd00e6d}}
{{formula:ac8b4d0d-d581-46e8-baa0-84e14f3cc928}}
{{formula:efe9aa10-b21c-490f-a7f0-e553ef97ded4}}
{{formula:c870f647-a5f4-47e1-84f5-af7f884cb553}}
{{formula:2c2f6500-dcc9-4dce-9188-1837c7a58799}}
Constraints for Including Leaf Paths
Let
{{formula:843c5d4e-cfee-47b0-b2b9-1a292b021113}} denote the number of vertices {{formula:94851cad-049d-438c-b199-e704e5db7217}} such that
{{formula:0f33124b-0ff6-4081-ae35-aa556115e6df}} and assume that
{{formula:fba6c0bc-194c-45f8-a907-5ac59413796f}} so that
{{formula:57fd1875-590f-4a81-972e-1cc900018e2c}}
Define the set of colors for the vertex set
{{formula:921d6ad6-b368-4e81-b74d-98956ea2a322}}
to be {{formula:854a124a-40fe-48a4-bc55-3ea5d41b52ea}} with
{{formula:39648235-5068-47ea-a191-1e2756cc32b5}}
Let each vertex {{formula:cb3c6039-952f-4381-87f7-ba862ce17d32}} , {{formula:72168b2e-42c5-4344-8557-bf85c34c7d89}}
(resp., {{formula:062a7e1d-25dc-44ca-9a36-11544e926d34}} )
correspond to
a color {{formula:2370b2b2-23f5-423d-a972-ed6a41d4a38b}} (resp., {{formula:87b28124-7b08-402a-a623-6f9f7428c94b}} ).
When a path {{formula:72aea25b-8f9b-4fbd-a4fa-97554c824dc3}}
from a vertex {{formula:5b66510b-5efb-49ae-9e80-7e1d3b4a05f6}}
is used in {{formula:e46874a1-1e8c-41b2-b8a0-2e229be34ad0}} , we assign the color {{formula:d6f5d60a-39c0-4c9d-913e-33297adaaf7e}} of the vertex {{formula:52dcebd8-4ac4-47e1-9bc3-17a00449f15b}}
to the vertices {{formula:24782a8a-e018-49c2-9ad6-4bec73782040}} .
constants:
{{formula:a7ba2fcc-7cbc-4a5c-8d2b-7768c707fe32}} : the maximum number of different colors
assigned to the vertices in {{formula:a3a6f4c9-d5ff-4d55-95f1-a4500d3e27ce}} ;
{{formula:bbfa4100-6cdb-4fd0-8161-492742b7e35b}} :
lower and upper bounds on
the number of interior-vertices in {{formula:b08b9c1f-85ad-48ad-b82d-aaddbecb6a98}}
{{formula:d3ec3c0d-be19-4ca8-a325-3f34fba40311}} , {{formula:6f372d59-417b-4ebe-893d-7c384f47180c}} :
a lower bound on the number of leaf {{formula:531ac2a9-4a82-4879-9d24-a46b57266022}} -branches in
the leaf path rooted at a vertex {{formula:365acf60-234f-4ae2-94bc-3d1b565c89e1}} ;
{{formula:6d6329a6-9762-40e3-a147-0b18fddef9e8}} ,
{{formula:8f814c24-54f3-4c44-8f5d-435d87bdd7d5}} :
lower and upper bounds on the number of
leaf {{formula:ee7e5523-1672-4476-9fcc-1fd39afad933}} -branches in the trees rooted at internal vertices
of a pure path {{formula:012728de-f4cd-457b-8a99-9a0203eaeb2b}} for an edge {{formula:5b649743-6aff-4901-bdd0-1f8bb73dc439}} ;
variables:
{{formula:7c143bba-4b1d-4234-950e-db8705ce0136}} :
the number of interior-vertices in {{formula:57a4148c-a9cf-43dd-bf6d-aa6d71dd3918}}
{{formula:31467946-fd01-4e0d-8462-d0e10ffae3d5}} , {{formula:6b3de091-9846-4826-84c1-57662a7c15c9}} :
{{formula:177197c2-ce89-4bf4-8433-1d6f4f86aaa7}} {{formula:5d99d189-3705-496b-a588-932273fa872c}} vertex {{formula:71ca1f6c-6525-40bb-9b8a-dc2b745adabc}} is used in {{formula:fe8b3c63-f29b-4de4-9df8-b9c4646a3bdc}}
{{formula:fa140fe2-2e1e-4939-8969-9d05a9019534}} , {{formula:5cabe705-d8b4-4645-8f5d-e2cfc0cee112}} : {{formula:8ebdada0-5665-40d6-9197-7f61abefe208}} represents edge
{{formula:e2d7c433-6883-46bc-982f-329f34c49dca}} ,
where {{formula:ccb71336-a241-4b65-b7bc-3ab68218b194}} and {{formula:eedf9afb-6ee5-4d20-9311-4b248e14cc44}} are fictitious edges
({{formula:22b33b4f-97c8-4607-9eee-e4f871c2a9bc}} {{formula:c2d42312-2bf8-4cf6-af2c-65871b49137d}} edge {{formula:05d91632-7380-48a3-a65e-5216dba3947e}} is used in {{formula:85ffd2f8-15c4-4a61-82b1-4471669bd171}} F(i)[0,cF]{{formula:ca7f5f95-ad3b-4f1c-ae16-53b2f48da707}} i[1,tF]{{formula:cc0be2ad-1aa8-4958-b01f-01984547c38d}} F(i){{formula:1a0c0d28-0249-4117-8acf-2079e0b3fa39}} vFi{{formula:06369061-cf14-473a-94f0-28fcacc20bac}} F(i)=c{{formula:079a297a-c430-43b3-8a30-609628e4e71c}}{{formula:3bfc5bed-2c1c-4a20-8769-0a5d32a8f69e}} vFi{{formula:4e53cbc5-87e1-4e7d-a754-424412fe4d8b}} c{{formula:10a6e610-e7f6-4cff-8d6f-72001f0949c8}}
{{formula:82d83657-40c2-41b8-a1c7-8e6353ffd4d0}} , {{formula:bc771e27-19f1-416d-9958-ba63025fe432}} : the number of vertices {{formula:a08d5d70-0f0c-4d9a-b64e-fd0d9db88480}}
with color {{formula:a020e454-dd75-427e-8c82-8a4b9b028992}} ;
{{formula:7f89d05c-0ba6-4a5a-9436-44511d5fee62}} , {{formula:d6aeaab7-502a-4471-a569-768f26cafc65}} :
{{formula:ce9f467f-063c-42aa-a875-b6d7eb7c64e0}} {{formula:f65ad5ad-b157-46dc-bb1b-5e3f3bdc939d}} {{formula:886925ad-4f08-43d1-9ca2-bc16e0edb970}} for some {{formula:1d26ef4a-b35f-4a47-9cc9-58309dde1ec8}} ;
{{formula:b9b82acb-fd1e-4099-a14b-feee02acdf3d}} , {{formula:80aa2cf2-521e-4361-b108-6f522b9c7978}} :
{{formula:32f15595-914a-4d8f-91c3-db6f04ac6a66}} {{formula:64867f8c-12ed-4d4f-a506-54d46f98e465}} {{formula:d57bab42-53a2-4408-8cc5-767cff71abb0}} for some {{formula:5744aad1-8242-43b4-9166-2ad322f4fc25}} ;
{{formula:f9886a1b-f96a-4d0b-ac3e-34bfbba0a7f9}} ,
{{formula:61b15ce2-94f9-417a-b08e-1d82ffe174a7}} , {{formula:81b71312-9ec1-4839-8a89-43ff8117dabc}} :
{{formula:16f316a8-78f8-4518-9652-f5cfb2906307}} {{formula:0e962441-c66b-496b-9ca6-55067d2787cc}} {{formula:07b29968-1cf2-452b-aedc-abae29a5fce7}} ;
{{formula:58c1901b-6250-46e1-a688-63c6a2e0103f}} , {{formula:442be19b-cf5c-4439-bd08-78951af2d482}} , {{formula:c6f12a81-5009-45cb-b3ff-83e544ec0cfb}} :
{{formula:15eb78b8-a8d7-4588-993a-c5565fedc9c2}} {{formula:017d6b0e-8ef8-44f8-b3ef-d54c36af2e10}} path {{formula:595d1c65-6c18-4d2f-99da-7902f21d4c34}} contains vertex {{formula:14dc1158-aaad-4901-bebe-53e7199094db}}
as an internal vertex
and the {{formula:83fddcdc-09fa-4822-ba98-9aa7ad8eaa38}} -fringe-tree rooted at {{formula:b62f704d-f86a-4b4c-9a7b-a8d496970e83}} contains a leaf {{formula:8e94ca59-be66-4bc9-9f9c-505d402a0368}} -branch;
constraints:
{{formula:35565c70-90f2-4aae-97e7-b80f5526359b}}
{{formula:c32a89ce-9440-4552-9584-45bc1dc9b8e9}}
{{formula:998f68ed-8ed4-4804-a38b-1222d2eae521}}
{{formula:734bb528-1785-41db-a8cf-6396b93c3372}}
{{formula:29a33742-5799-43c0-8c27-92ae97a0fb61}}
{{formula:4f431e50-3b67-4ded-b0c2-b4b3a5816e0c}}
{{formula:e5b05ecc-ca91-4fdf-a647-a30a07203209}}
{{formula:0d87b2d8-4e5c-455c-b8ad-fb48449e69b7}}
Constraints for Including Fringe-trees
Recall that {{formula:2b7e2550-6924-43c9-8bfd-07ce8a36d0be}} denotes the set of
chemical rooted trees {{formula:bbe35bfd-dc57-4e74-9636-e179337fa7fd}}
r-isomorphic to a chemical rooted tree in {{formula:ffd0d34a-8933-4fe8-8cb3-a878abdde661}}
over all chemical graphs {{formula:e6cea67e-8940-4d8a-8e59-aae1d3659b56}} ,
where possibly a chemical rooted tree {{formula:cd94155f-512f-440d-b495-3295b03bb98b}}
consists of a single chemical element {{formula:3bbca1a2-f25c-4499-ac5d-fa6e855c477f}} .
To express the condition that
the {{formula:69cc7877-6a9a-460b-b505-0d035b5bb708}} -fringe-tree is chosen from a rooted tree {{formula:cc6ddebc-50b0-4483-a99b-d1d26f7aa231}} , {{formula:546d63e6-f0b5-4e5f-bbcd-025fd743ba4d}} or {{formula:e2a4c09b-7969-4bf5-85e5-87c4fd94eab1}} ,
we introduce the following set of variables and constraints.
constants:
{{formula:edc16c4f-f267-46ef-a431-9aceda3b10a0}} : a lower bound
on the number {{formula:237f4969-a171-43e7-be85-17ce7f67a1bf}} of non-hydrogen atoms in {{formula:1b024901-476d-4a24-a6fb-12e6496c0564}} nLB, n*nintLB{{formula:2046e6cb-9162-47fa-ba10-a665e7946a34}} chLB(i),chUB(i)[0,n* ]{{formula:2fdef0dc-9a75-4d4d-a453-33c48399024d}} i[1,tT]{{formula:59abf221-7076-439c-8a70-b2397f14b9bd}} Ti){{formula:66225055-fece-4330-a567-d48efe370c5f}} Ti{{formula:f194bd53-b56d-4314-8950-0d1b18417437}} vCi{{formula:c599a562-a50b-4201-b848-282d1221e131}}
{{formula:8c62741f-74ac-4d3b-a362-8eb63ead5408}} , {{formula:7b8aefe5-e31e-4c33-819e-095420c02ae5}} :
lower and upper bounds on the maximum
height {{formula:117e3469-373a-46ad-8f95-7c1c3cc05da9}} of the tree {{formula:3430f453-3dc4-406f-ad63-10fdddbc429e}} rooted at
an internal vertex of a path {{formula:e080e573-ae30-4a27-8889-eb218deed294}} for an edge {{formula:14cd19c3-2ea7-4b41-97dc-324202519e62}} ;
Prepare a coding of the set {{formula:57c85a89-78d5-4de2-913c-44033a151374}} and let
{{formula:e5c5f384-1e30-4358-bf58-b2f5af7b5305}} denote the coded integer of
an element {{formula:6c1e4074-cccb-4dd3-8928-ef180d194c9e}} in {{formula:afe26c9b-90a8-469b-9ba1-58b2f88c0515}} ;
Sets {{formula:7f236161-0942-4a52-90e5-d81bd6805ef3}}
and {{formula:e380072f-13f4-416a-95bf-35e155b938f4}}
of chemical rooted trees {{formula:46cbda8a-762e-470d-a257-126dadb7853e}} with {{formula:e1597ef2-02b3-43d6-b34f-84735583d365}} ;
Define
{{formula:6b4680cc-f5ef-4cc1-8497-801bb5785ca5}} ,
{{formula:7a935018-e657-4d25-ac43-a1f77d61f7d6}} , {{formula:460c35ea-2339-4f7b-98ef-4aa2edb7ff79}} ,
{{formula:bfeb3427-7f93-4ec2-a70f-902bcefa827c}} , {{formula:f196f5d4-cfa5-4656-914a-8c70be6746b4}} and
{{formula:c759a79e-d4eb-4d0c-aa2d-e97979b92065}} , {{formula:f9fbc507-7690-43ff-8d92-4fcc79cd2f7e}} ;
{{formula:be9a4b9e-c01d-43e4-aa39-66b2f47ba5c4}} :
lower and upper bound functions
on the number of interior-vertices {{formula:90424861-1212-472e-b9a1-bb1886e56ce8}}
such that {{formula:6862b1c3-4fc2-4d0f-a567-f7c19d8c6f2e}} is r-isomorphic to {{formula:597aa67a-72c5-442b-a59d-95e5309b620b}} in {{formula:af2aa222-fd95-43c1-a07a-72f4a7c1c84f}}
{{formula:173ae8bb-1cc5-4806-9c43-9b827aabdc32}} :
the set of chemical rooted trees {{formula:5a960478-90c5-4b38-9951-aa476b73b420}}
with {{formula:d9e9a2fa-e5e6-42b6-a0c5-f9fa05e89b87}} ;
{{formula:107035a2-8cb3-4842-8c54-3fa09934fa04}} :
the number {{formula:7b6f5e23-3e90-4534-9a7b-7d7fecb6d363}}
of non-root hydrogen vertices in a chemical rooted tree {{formula:ce89787b-aa4e-4cb0-8bc2-6b28f9a2c545}} ;
{{formula:fa0f28ad-1603-455f-9fa8-58265b4eff11}} :
the height {{formula:8e60cc33-753d-477e-902f-bb1e4c3d89f8}} of the
hydrogen-suppressed chemical rooted tree {{formula:45a6cfe2-4d78-4df0-89b7-87f9bf77f1d3}} ;
{{formula:22b2bdf8-86e6-4e90-ba76-4c3cf875fdd2}} :
the number {{formula:6685c9c6-0218-4d4d-b3d9-3c53b77b5433}} of non-hydrogen children of the root {{formula:f54740a4-c76a-40c6-ad6f-b25430a99bdf}}
of a chemical rooted tree {{formula:32688566-dc2d-4b02-af86-432762d3008e}} ;
{{formula:3711cbd1-9985-4db5-9614-1401b8daea1f}} :
the number {{formula:31e26a55-9523-48c7-980b-bea27c6c1701}}
of hydrogen children of the root {{formula:0fa63998-6bbf-4e6b-ac0a-0f6ad32a4f04}} of a chemical rooted tree {{formula:c68969c1-1a25-4283-ab7d-416b13b48a3c}} ;
{{formula:9d7a1642-a309-4080-84e3-4cc67a8af436}} :
the ion-valence of the root in {{formula:6aa67ea8-0177-42b5-bf36-4de13257b1b1}} ;
{{formula:80875c34-c6b2-4e6b-9f86-bc131c661290}} :
the frequency of leaf-edges with adjacency-configuration {{formula:6c99477b-0c83-4a22-a51e-9c11b0670ac4}} in {{formula:4f62a021-5678-4d5a-bf85-6fccdd063803}} ;
{{formula:46df0688-1a58-4d0d-8b23-e0190206f59d}} :
lower and upper bound functions on the number of leaf-edges {{formula:1289001e-22cb-4235-8858-25e4a98aba28}} in {{formula:b9a294bd-3b16-4cdd-ade4-e4861b2dfa44}}
with adjacency-configuration {{formula:b20806d8-6586-4ab6-afe0-8430da82ee7e}} ;
variables:
{{formula:cae88987-e66d-4577-a524-2cbd6f73477d}} : the number {{formula:9cd57018-98e3-41be-9b1a-05c7a997d248}} of non-hydrogen atoms in {{formula:92f3e382-2ca8-411a-aed3-70d7eb03d73a}} vX(i)[0,1], i[1,tX]{{formula:7d6aa9b5-a8db-4864-a2c5-63f46ae46e23}} X{T,F}{{formula:4843f631-341b-4137-9609-5306f5f40173}} vX(i)=1{{formula:3aec02cd-052e-4e96-9b74-c3299a27fd18}}{{formula:6d58dcca-8b9e-4c54-ad6b-3e1228c228da}} vXi{{formula:249241aa-6905-424c-a0bd-3c942c5cd141}} ;
{{formula:fa1919a7-426f-4c41-8e57-ce1d0a6e5656}} :
{{formula:0d2cb670-d8e9-48a5-a64f-52ba11d16a2f}} {{formula:800b1da4-4ce9-4791-ae88-0fe2581082dc}}
{{formula:9fa8fede-2ed3-4b9e-a552-de25a3e9ad3f}} is the {{formula:b246656c-2ed8-46ed-9c53-535196483b52}} -fringe-tree rooted at vertex {{formula:780be91d-d4b4-407e-9357-99a6f50a203e}} in {{formula:8b040a6b-d9be-4d9f-b8fe-ade4c33929ba}}
{{formula:4f89e80d-31e6-4e15-a7b5-b4109e1fc6a8}} :
the number of interior-vertices {{formula:5b81ec94-072b-461f-ba96-2151b148f778}}
such that {{formula:7d1bdee3-7591-495f-9b0f-3bcdce4904cf}} is r-isomorphic to {{formula:9c1074f7-d37f-4919-be26-1479ac3e3d0f}} in {{formula:bf8587c9-cebe-4a5a-8f95-f0385a0539a7}}
{{formula:d2270fa6-95ce-493e-8584-25f8e699501f}} :
the number of leaf-edge with adjacency-configuration {{formula:b7bbc73b-6132-48ec-bacd-50ed69df832a}} in {{formula:770a1112-74c9-4cbd-bed0-974f6b709207}}
{{formula:8fa6a61c-ea58-439d-a82e-898df6a692ee}} :
the number of non-hydrogen children of the root
of the {{formula:599458d5-3273-4292-b0d1-71ef3b84a9f6}} -fringe-tree rooted at vertex {{formula:6f54bbb1-7164-480c-8db5-13a17e1117f7}} in {{formula:c8cfea7a-4ba2-4712-ab5f-10141af8438a}}
{{formula:6bfecd44-7f27-4805-a36b-e5f49d522324}} , {{formula:2360efbf-d6a4-4143-b41e-920c534d1dbd}} ,
{{formula:916ca944-9c09-4d3d-9fd2-cee61cdf2d7c}} :
the number of hydrogen atoms adjacent to vertex {{formula:b0e7fdb0-f93c-4ae8-a26e-dade2d4637d1}}
(i.e., {{formula:a8784e42-25d2-40a7-8faf-90c167637c3b}} ) in {{formula:9031651b-22c4-4b92-aa60-8c8083de45cb}} ;
{{formula:79f8b73f-2a65-4660-9356-7822c872fa1a}} , {{formula:3b0f29a1-8af6-44c3-85ef-3be269386be6}} ,
{{formula:e8b8d3be-5431-424f-a467-2eaf1a94f42e}} :
the ion-valence {{formula:bfb2ef98-ba13-4035-aef8-7a94e36bae34}} of vertex {{formula:1efaedb9-8ef6-4fd6-bc5d-992e5e18d3be}}
(i.e., {{formula:42b5416e-9d6d-47e4-8628-d67ff4072ae6}}
for the {{formula:715cef6a-b990-46f6-a60f-7195f182729c}} -fringe-tree {{formula:84876147-c446-4e01-b96b-617a79d12fe1}} rooted at {{formula:29a2b542-3022-4717-90b7-e81301a5c1fa}} ) in {{formula:eab813fd-4241-46fe-94fc-18a90e03963a}} ;
{{formula:0f0d6a7c-df5c-41d9-b87e-1d10507e6152}} , {{formula:09914fd7-9ea7-409f-9f9d-29ab7de6d495}} ,
{{formula:b425179a-3624-40ea-9490-16c2f488e29e}} : the height {{formula:6d76f803-a028-49ff-b1a5-7bf289fb42d8}} of
the hydrogen-suppressed chemical rooted tree {{formula:e2e52e7d-6d0f-4958-a80d-1db160ae69da}} of
the {{formula:d8bba615-cd8d-416b-acc6-4232b0ba6465}} -fringe-tree {{formula:7206ab6a-6985-4df7-8ed9-ba1149729c55}} rooted at vertex {{formula:6ef5fd96-84a2-4d2a-b0f6-f4f4b1d9d7f3}} in {{formula:aba49290-4e06-4274-aed3-b09f268e0527}} (k,i)[0,1]{{formula:efa2315b-6492-4827-ab7b-be312da5e44f}} k [1,kC]=I(2)I(1), i[1,tT]{{formula:f9a7a2f2-eb87-4f2c-9f04-a3541c6d9240}} (k,i)=1{{formula:044b93d8-99c5-4075-b8d4-7ab03d512852}}{{formula:f5a60154-3a54-4c00-b2c5-fd20c93c700b}}{{formula:2056c915-8093-4ead-9866-00c89aae5b15}} Tv{{formula:5c6e80dc-893c-4e65-a67e-7191a05d9e17}} v=vTi{{formula:8ac58e5b-4712-4d1c-a7e1-76d0b7361fdb}} k{{formula:c3348026-b935-4322-8a7f-e36d1fb7c780}} v ){{formula:24d02c9d-bdf4-4eb8-a11d-4a0f477a3934}} Tv, vVT{{formula:d97b5acc-a928-4f10-8ab8-f4fb53be3757}}
constraints:
{{formula:c0ef458e-a741-47ab-89c2-49a26f810256}}
{{formula:6aaac113-36cb-4aa7-adf9-17d50669a2d7}}
{{formula:a680ea4b-68c4-4e60-aa13-cf57bd933f49}}
{{formula:28ff76e5-65e2-4c93-875a-8356c285900a}}
{{formula:d520a864-c05e-4b4c-8bdf-7bd6c3c9b72b}}
{{formula:7420cdf6-aff3-4c9e-8edd-9dff241cb3af}}
{{formula:8e688d5b-44f5-4a7d-a776-a91242cc2498}}
{{formula:85fdcbee-f916-4f00-9cdb-a57573c9ed7b}}
{{formula:c2b7a0a7-627b-46ac-af9a-9594c1b6b9bb}}
{{formula:4ed5f5bc-c838-41d3-8108-16571beb2298}}
{{formula:e9d3cfb7-d4f5-4ec1-a670-07cfa778f6ef}}
{{formula:db8348f9-bf9a-4d2d-84dd-e1cb5506c793}}
Descriptor for the Number of Specified Degree
We include constraints to compute descriptors for degrees in {{formula:45d7a269-8f4a-44e8-88b4-07be7b9b0a67}}
variables:
{{formula:1a97e983-4e5f-4f63-bfbe-2184c82a35ca}} , {{formula:41c59362-2d0b-422e-86ad-d9973366883a}} ,
{{formula:f7736f71-277e-4fea-9f63-be4dfe5acac3}} :
the number of non-hydrogen atoms adjacent to vertex {{formula:460829d7-e8ba-4d9c-85ef-18f1cc7d7b36}}
(i.e., {{formula:a8340977-daa4-407f-8f5a-cd6db27a5d7b}} CT(i)[0,4]{{formula:fdf6f38f-2157-4a1d-992a-58c1c0c5fb19}} i[1, tC]{{formula:9f7f958d-edc2-4bc0-aaf7-2083ab665c87}} vCi{{formula:dd9bcb11-d4ba-47cc-a197-b465a4ec3e83}} vTj{{formula:f9ffea28-6300-4605-94ca-cf7799cf32c8}} j[1,tT]{{formula:99617ab9-5692-43a7-b2d2-2bc320c46e29}} TC(i)[0,4]{{formula:6a4b1871-7f77-44ae-bbb5-c032d5234d41}} i[1, tC]{{formula:3f617846-0540-4267-a3a8-a2ad203ab6fa}} vTj{{formula:d8883239-1873-448e-951f-e3816eb5b0b3}} j[1,tT]{{formula:e0668d7c-4a03-4412-9ec4-9e6c3e0bbc82}} vCi{{formula:e99f182a-a566-4a49-80b8-e85bda47f2e7}} dgC(i,d)[0,1]{{formula:ee964f7b-07c6-46f3-9c47-2821987ac6da}} i[1,tC]{{formula:6428639c-81e8-47d3-926c-6f306d45f763}} d[1,4]{{formula:8f7df710-e183-453c-a803-a95981675689}} dgX(i,d)[0,1]{{formula:52086a0d-5ffb-4008-b2b5-5f32771c66e6}} i[1,tX]{{formula:ca78c5f9-c359-44dd-a61f-51a87f211414}} d[0,4]{{formula:fe1eb3c6-04a0-4fa0-bb27-23624d7ff889}} X{T,F}{{formula:9bf1f3fc-8774-4753-ace5-90739da8a46d}} dgX(i,d)=1{{formula:74ccaa08-7115-4867-b026-124ad8518833}}{{formula:28b130bf-75ed-41e0-9e70-cb484084d114}} X(i)+hyddegX(i)=d{{formula:cfcc286c-81fa-4978-96b2-e82d1f54c7a1}}
{{formula:b2f726e0-e5c9-490b-9036-09f2c9db7b94}} , {{formula:17433e8b-0385-4a53-8cad-d9b5f4d70812}} :
the number of interior-vertices {{formula:18de4715-c2aa-421f-8626-94fe23ad3caa}} with
{{formula:e720ea96-e935-4ef9-836d-5b7ccb6b6d81}} in {{formula:c22b8924-916b-45e2-b277-bb20fc8276a7}} ;
{{formula:1552427e-4ab4-4603-85fe-2847b5c7213b}} , {{formula:5cf38a4f-62ac-4d2d-a6ac-e22ae54aa46d}} ,
{{formula:d64d5f09-69d6-4172-8757-a9634b8dca3d}} , {{formula:3a9d0160-7c43-4556-9d43-673506990e84}} :
the interior-degree {{formula:128a2135-708a-462a-9c80-63ec0df7adac}}
in the interior {{formula:d9a2f92d-5ee4-4429-8499-582e1d294158}} of {{formula:83fc2b08-5803-4bc0-86ab-8124f4a2b02c}} vXi{{formula:7dd620dd-1fad-426c-a85b-dced8f4ffb7e}}
{{formula:c045b92b-0c29-4df6-8813-b213f5d93190}} , {{formula:1de89dd3-ab6e-485d-8f31-64f8e9cdf48e}} , {{formula:f96f0fc1-ff2b-4170-a19f-b9f4879809a5}} ,
{{formula:af9251c6-d3cc-43e1-ad77-d7a162f3f63a}} , {{formula:f41e0e6c-09b9-4b15-960c-92e38f192dd4}} ,
{{formula:b0af9464-be4f-4609-9f5a-06df98da6324}} , {{formula:06c7719b-d068-492e-b7b1-4493f2dfe3d8}} :
{{formula:59379dbe-7b98-47e0-881d-ee0a242caf3e}} {{formula:c5b844d5-b031-43e6-802d-d62781f90814}} {{formula:0b8cff36-a750-4315-814f-175f0f1ef24e}} ;
{{formula:b7bb2bb6-ebdd-4d0e-8350-c4c7726990b1}} , {{formula:4eb03541-69ad-40db-bc58-2d2e202b1f2f}} :
the number of interior-vertices {{formula:d97f08d6-4c33-4d15-aa98-a6d2bcbffaa8}} with
the interior-degree {{formula:bfc86988-0902-4612-ace8-c04ff87c0366}}
in the interior {{formula:75cbc1aa-25db-40b4-bf5a-ddb6d1958167}} of {{formula:1ede2aa1-4cf4-4054-9477-c0432db18e7c}} .
constraints:
{{formula:b76b2ee4-9a38-48a6-bf6b-101feb76c92b}}
{{formula:c93b208a-620a-431c-8fe8-28bfa06184d6}}
{{formula:d383f80c-8a15-4ba7-bac4-50ec13b14fa9}}
{{formula:d2306870-21d4-426b-8391-9f3137b27684}}
{{formula:be8812f2-a3c6-454b-9aba-2fe02db6ad0a}}
{{formula:bbecb4db-a78d-4e7d-ad3c-088d32787eac}}
{{formula:8d1c60b1-5a89-4ea6-8199-b5e3c3ec26af}}
{{formula:3329cb56-15f6-4170-9c4a-f77cee384b08}}
{{formula:6cbeec06-7ff3-4a79-8b66-589b6c673c1f}}
Assigning Multiplicity
We prepare an integer variable {{formula:cec0d59e-9cdd-41b0-8680-035c0921e580}}
for each edge {{formula:50cbb987-f0d1-4959-afec-27aa0de23512}} in the scheme graph {{formula:fafb2bc1-d5d5-40a6-8cc5-6b38e2075ef3}}
to denote the bond-multiplicity of {{formula:7dfa0dd8-b183-42ec-8746-ce6030e81998}} in a selected graph {{formula:9b1fd822-a9d6-44ec-9af7-3f70c5617239}} and
include necessary constraints for the variables to satisfy in {{formula:fed7952b-ccc7-4a71-8f53-9dbba1fb61b7}} .
constants:
{{formula:e0cd9b5c-99d8-4bca-b363-186d2e5a34f2}} : the sum {{formula:67f44303-7851-437c-bac5-3ae0e23ef08c}} of bond-multiplicities of edges
incident to the root {{formula:3ba3d431-ba15-41d3-adf3-052cac797e67}} of a chemical rooted tree {{formula:28a21721-86e8-4dcf-8137-e5cb48c46946}} ;
variables:
{{formula:b2b92c10-7318-4a34-b4a5-817be105e049}} , {{formula:4b2601f4-000d-4b23-8524-5ab655fd0b2b}} , {{formula:0fd7e71b-ff51-4f94-8b38-c1e66b936a34}} :
the bond-multiplicity of edge {{formula:1b6b0e88-45cf-454f-9cdd-80efeb1980c1}} in {{formula:23d2b2d5-7eff-4b4d-b1c5-df835629b201}}
{{formula:e5573bb9-c07b-4c13-8121-8c4c18a50acd}} , {{formula:7758aedf-c02b-4a42-b41d-8098e5abd371}} :
the bond-multiplicity of
edge {{formula:266e57a0-22d7-403c-9c30-6e7651f43ad7}} in {{formula:0ae13c4f-d8fb-4f88-9137-fe6337d0570e}} CT(k), TC(k)[0,3]{{formula:44e2c48f-1c7f-4b82-95e9-fb0986fea6fd}} k[1, kC]=I(2)I(1){{formula:fa4b5cba-b5ec-40c5-b24d-19e12e173d21}} Pk{{formula:baea3d3e-20bd-40b2-b3e7-fd4a54b67642}} ;
{{formula:51d3a1af-36b9-4883-9600-23f5be92a1a5}} :
the bond-multiplicity of the first edge of the leaf path {{formula:7dd7a0f4-213a-4318-806d-9a23cd265525}}
rooted at vertex {{formula:006ac12c-5025-48d0-810e-6e6109b58805}}
or {{formula:1d3aaba0-fa4b-4a23-8a5f-e1ec37bf32a0}} in {{formula:5cebb681-3d22-4620-9fef-4a28ad24c6c3}}
{{formula:ec568a3c-e989-4922-be1b-fa2407e40e18}} :
the sum {{formula:a0dcf5d2-72b9-40e7-9af0-d3a526ffcc5d}} of bond-multiplicities of edges in the {{formula:c27a68dd-4696-46f8-8cae-42a4152a5788}} -fringe-tree
{{formula:d73ff0c8-dde0-4add-b290-cf5bfe6e6bec}} rooted at interior-vertex {{formula:0a54e5c9-8160-4c86-9398-741fc27f2078}} ;
{{formula:511cf517-a180-40e0-aafa-b0050c85be64}} , {{formula:2da4ad13-792a-4e84-9431-95ffee5d540a}} , {{formula:dcdcdbc5-c13f-4861-a9a9-868744514ada}} ,
{{formula:3e1386da-255c-4303-8e7e-ce2b03c6f388}} :
{{formula:7df7074d-3be9-4f4c-a4e5-94d92c362c6d}} {{formula:1a1654f4-2874-4e0b-b90e-0ed374f672d1}} {{formula:62224f5a-a35f-41f8-afbc-f870808b2bfc}} ;
{{formula:58b09ec3-fd0c-447e-b833-c316327d8db8}} ,
{{formula:987f1050-1a2f-4323-8c05-d81474c7633a}} , {{formula:0b5d34e9-c9f7-46aa-b0ae-32c7d8849c1a}} :
{{formula:d2be23a0-81de-44aa-b356-02899eda972b}} {{formula:62c093bf-5ef7-4ec9-8e57-ca601ebfe563}} {{formula:4b696881-7fc3-47f3-8217-dcfb5b245f6e}} ;
{{formula:4f929786-a163-4926-af78-4aaabee57b07}} , {{formula:3dea10d8-5aa8-43bd-8aa0-095fcf2644c7}} , {{formula:1ea2681b-9373-49c4-b035-08d5b40f039e}} :
{{formula:d2ca44ac-b559-4c01-a3eb-d1c2a39229e6}} (resp., {{formula:33bf81e6-5f05-4d15-9f1a-3975444215a0}} ) {{formula:48ccc955-3da5-40d1-85af-c752463cd945}}
{{formula:279044a5-b254-41d6-a0f6-5ff23be950b8}} (resp., {{formula:50436d67-6c1b-4de8-8ac7-7c1b4d0bbcb9}} );
{{formula:d8ae4759-8204-47b0-9375-9ebe6d2ca052}} , {{formula:fdf68ec0-0c3a-4b92-b49e-d4061a9fb623}} ,
{{formula:4901ab0c-24a8-43c8-9ca3-05ffa7c07476}} :
{{formula:0c3d0fde-5b41-4b09-bcd3-e8cba93d160c}} {{formula:27a1e9c5-d607-44d2-b95e-d73e75de617f}} {{formula:724c87bc-82d4-49fd-a63b-171523987ddf}} ;
{{formula:c4f10ad3-068f-416c-966b-c5a3bf18b074}} , {{formula:6e96fcb9-d08d-4edd-b770-ab9591bc5edb}} :
the number of interior-edges with bond-multiplicity {{formula:8398fb75-86f3-4bab-a47d-1c444eb87508}} in {{formula:b778ac75-7d43-40a0-b213-fbae483151c0}}
{{formula:3472caae-5484-4257-8e59-f78b7a640fcb}} ,
{{formula:59828e01-c1b2-4cff-994f-2f9c6602d537}} , {{formula:afd0fe82-afe4-4a4e-9bb2-1a802f9b4609}} :
the number of interior-edges {{formula:9731d427-12c0-4a93-abce-f0d274b5b3b2}} with bond-multiplicity {{formula:f7485b61-6143-4e75-be26-746d13bd04c0}} in {{formula:82244d6e-4d7b-4168-af1e-1483baefcc66}}
constraints:
{{formula:3b2472d1-5f27-4969-8d4a-75bef80ec772}}
{{formula:604dbaa2-e21c-4548-ab87-b63034c1372f}}
{{formula:5dc86b18-f1e6-46fb-89d2-16a38de7bdf0}}
{{formula:2d75f90c-e9b6-4671-89dd-052b18431196}}
{{formula:a667c8f9-6b33-4710-a676-43fef7744f38}}
{{formula:dba4adb2-cafa-4e00-adbf-94c451a970f1}}
{{formula:08977125-9131-4407-a676-461419ff8fbf}}
{{formula:d24ba9a5-9326-421d-9e0f-b3318163bd4d}}
Assigning Chemical Elements and Valence Condition
We include constraints so that each vertex {{formula:c8012ad8-eea0-4f32-91d4-7401fd2b3c15}} in a selected graph {{formula:8bfa3fdf-5e35-4f11-a3b2-33f9c596909d}}
satisfies the valence condition; i.e.,
{{formula:0ff1a0b5-e1d5-412d-9a28-f20d0b454dd7}} ,
where {{formula:2d1e3634-8361-4e8a-95e6-49125df6fb60}} for the {{formula:b8afa8a3-a385-44cd-bea2-f95351d91ce9}} -fringe-tree {{formula:343c98c4-8b55-46ee-91b8-7e976ef0e5fa}}
r-isomorphic to {{formula:07d55166-3540-420d-b443-76729e194eaf}} .
With these constraints, a chemical graph
{{formula:68d40d29-743e-4054-a6d9-109e0db2ef26}} on a selected subgraph {{formula:a2c9e750-e9b1-4661-9746-95aa6c03b052}}
will be constructed.
constants:
Subsets
{{formula:6202a61d-8393-42dc-88f4-a3d35dddbd47}} of chemical elements,
where we denote by {{formula:b24154ba-bbfa-45c2-b193-69f3bee7cbf5}} (resp., {{formula:5ec56113-7919-4495-ab60-baedf6961fe6}} and {{formula:5ae2e3d7-e9a1-4ed8-a4b7-f81867d47fde}} )
of a standard encoding of an element {{formula:a213dc77-8b63-4365-94f6-0f8b90b855bb}} in the set {{formula:194892f5-9a65-48d8-8c58-970f9b6be509}}
(resp., {{formula:67ae8b6f-d2e5-47a4-9c0e-61460d3ec155}} and {{formula:bccc391c-f383-474b-b873-6bc80873b16a}} );
A valence function: {{formula:bd7c9dfe-61be-42b6-a5e0-6f469b37a135}} ;
A function {{formula:bf2b85f5-a656-41a0-9041-911787502244}}
(we let {{formula:322686e2-8461-4e8b-afa9-34b176539a0e}} denote the observed mass of a chemical element
{{formula:59a75b8b-efad-4c91-907a-daa52c7bfa6b}} , and define
{{formula:63d10159-500b-44d2-a190-5e323e94fe4a}} );
Subsets {{formula:f718b6bf-a567-4ae0-8925-dddf58421cd9}} , {{formula:c4a9ad9f-2d5c-4be4-ab35-e7aba9e17756}} ;
{{formula:89474a40-8817-465c-9c86-98a522f4f638}} , {{formula:cf1bbdd0-b5ae-4e3d-b262-ad9c554719ae}} :
lower and upper bounds on the number of vertices {{formula:5645106f-0604-4df4-a845-5d246babe3bb}} with {{formula:ce1cbfae-2e27-4d7a-bfb7-161a63ab727f}} ;
{{formula:23be6be6-feeb-470a-ac8a-ca6a34aab871}} ,
{{formula:495bb436-ec6d-43c4-857f-bcd094b693ed}} :
lower and upper bounds on the number of interior-vertices
{{formula:ab4d1828-46eb-427d-956d-898a38562a01}} with {{formula:0453805e-c540-4184-8886-8a365ec433ba}} ;
{{formula:feb00bcc-3c75-43c6-9260-2928380313a7}} :
the chemical element {{formula:54edb93e-ffde-48bd-a569-e88d2dc1dbe4}} of the root {{formula:76f102a4-47cd-43bb-86fe-830d4b5cff2b}} of {{formula:16e3aa96-c2d3-4da7-9a04-2036ffd9cf3b}} ;
{{formula:b1b51be7-6128-49fe-a152-5cfc13b9e529}} ,
{{formula:c09ab380-9778-4384-93b3-f29795791d9e}} :
the frequency of chemical element {{formula:9c595ef5-85b1-4d74-855b-0ce27d49fe13}} in the set of
non-rooted vertices in {{formula:5681e44e-e7d0-4af8-a4a5-6c0c289ed32f}} , where possibly {{formula:a25d241c-4cc2-48e4-9a2a-d52d500b12fa}} ;
A positive integer {{formula:a6be3694-4018-4a9d-b19e-7acc49883672}} :
an upper bound for the average {{formula:ac652305-6190-48c1-b935-f84d3b530aae}} of mass{{formula:ead86c76-c3a2-445a-8914-8ba17dcbf219}}
over all atoms in {{formula:08a47b58-5aab-4433-a53d-9554675425d0}}
variables:
{{formula:a3b52664-acaa-47a7-80a9-ae42dc021eb8}} :
the bond-multiplicity of edge {{formula:4c2f2403-d35c-4efe-8f6d-fe0627f72d5a}} (resp., {{formula:699e73ef-b772-47b1-a93a-7570038d9dfb}} )
if one exists;
{{formula:67671f10-77b3-46d1-b2df-f3c8fb11d59d}} :
the bond-multiplicity of {{formula:e88430b1-3846-4754-8984-784049ab2e04}} (resp., {{formula:952069e2-e455-453e-8c5c-8674284c9e30}} )
if one exists;
{{formula:6eec8f0a-415e-4bd9-8ff0-677495d0513f}} :
{{formula:c8d329f3-9d6b-4158-845b-96fa43a67c90}} (resp., {{formula:dda74dde-b0c6-4f4d-abb7-b5c6d6bbb032}} )
{{formula:0ec552e0-0b40-4e89-ae2e-98b78c7aaf23}} {{formula:fe416b61-b0ef-4b04-ad47-b641998de7bd}} (resp., {{formula:a48675f9-a390-43a5-ae9c-84cf980e9509}} )
{{formula:8d1e0a67-f7ff-431c-bc5a-24c1677f3830}} {{formula:2d0464fa-6d80-4b62-b3ca-dcb210115c82}}
(resp., vertex {{formula:5e9b633f-c8e9-466b-bfe5-d1f203be7124}} is not used in {{formula:83b92401-75cd-4610-877b-61570963e12c}}
{{formula:90c0787c-ccdd-41d8-8182-4e604e4d2870}} :
{{formula:f867c7e1-ffef-466d-8c46-03b2218e61e3}} {{formula:d935bbbc-5bf5-460d-b01c-60900bccba2e}} {{formula:61550f6a-450f-49b8-aeda-9b08f1a37541}} ;
{{formula:ea107dac-8605-4a6a-b0e9-9bc817b08359}} :
{{formula:f9b26c86-9f27-475b-b3d2-d40cd43de5b3}} ;
{{formula:1f937de7-c4a9-412e-b2c1-01498067cc71}} :
{{formula:f59b4919-cd95-495d-9e0c-7172406c0776}} ;
{{formula:569eb873-b707-4728-8708-e0354cdbd131}} :
{{formula:0178bf66-f7fe-475b-a487-db5c390397f3}} {{formula:2445ad20-298c-4a74-aba6-12913436833e}} {{formula:ee69ecba-bb43-4200-8206-11c009c5a4be}} ;
{{formula:8fd6cbfb-d279-4ba4-8734-48ee0b63ca4a}} ,
{{formula:f4c7c8b7-5670-44eb-ace4-c47c4bd0b52e}} :
the number of vertices {{formula:d9691322-a52f-4d46-ad68-17be3112f095}}
with {{formula:ebc33cee-c7de-4e8d-aca8-7e16c696493e}} , where possibly {{formula:ca01495c-5494-4a44-9263-ec297eeba8db}} ;
{{formula:db616508-2fc3-4dab-aea9-f840d82f5663}} ,
{{formula:250eea5d-8d30-4fc1-ac8d-62d3940db8c1}} :
the number of interior-vertices {{formula:f24a87e2-6c10-4915-825e-dcabdedaabfa}}
with {{formula:15c61b26-837d-4641-b30b-7ef7d47f5256}} ;
{{formula:86f6239a-fbc2-4cd7-a27e-3fca4b27d94a}} ,
{{formula:1ea3a0e2-bdf9-4a3d-a3ad-ff6fbe27d620}} , {{formula:7b36b87c-6ed9-433a-911b-d98fa45a7bdd}} :
the number of exterior-vertices rooted at vertices {{formula:393fb433-03e5-46aa-ab27-d2bfa7073b32}}
and the number of exterior-vertices {{formula:2ecbbcff-d549-4a1b-8a98-cd40644f6f88}}
such that {{formula:554dbd50-c53c-4213-ab8f-dd0645a6d1ef}} ;
constraints:
{{formula:b8644970-3aa4-4252-9656-5f5ef817fbf3}}
{{formula:ecc888f2-df04-4e7f-a0cd-e0fab1503710}}
{{formula:b5ad9d5f-f1cf-4c4e-a06e-5687e533585c}}
{{formula:ae8c50da-d8a0-48d5-85eb-c5dba3c1b143}}
{{formula:53327dbb-7889-41f9-8629-39d7bd8fb53d}}
{{formula:4506102c-a89a-4fc3-a47e-fb1ff9beefff}}
{{formula:1f5dd9cc-bdf2-4ee4-bef4-f0fda704aafd}}
{{formula:b4b8cba4-e670-42a1-ad9e-88b8553c364b}}
{{formula:6dc08da3-aea4-483c-849e-97f105645d65}}
{{formula:1c1d1c83-fa12-4381-9992-93a0c1b779d5}}
{{formula:ebcbf729-23d8-4b41-bff9-684495338c02}}
{{formula:08262164-4f0b-4fbf-be63-407ad648b458}}
{{formula:bf6cbd39-b057-4fce-952d-a4cfabacde24}}
{{formula:8fd13559-dcfa-4e01-b928-60b2f21d83e0}}
Constraints for Bounds on the Number of Bonds
We include constraints for specification of lower and upper bounds
{{formula:d5a30897-3e84-4915-abd1-4122762d4d6e}} and {{formula:87f20247-8ed9-4b13-a098-42b298b2d565}} .
constants:
{{formula:98d29839-e181-4dad-a870-6208f1078c1d}} ,
{{formula:7daff12c-b5c8-422b-bd3c-4e84e9d13247}} , {{formula:5f8f4fcf-39db-4d2d-86c0-2a2ff119eada}} : lower and upper bounds
on the number of edges {{formula:7c3e39ec-897f-443c-b64e-d48d77e6bbd9}} with bond-multiplicity {{formula:0e094408-6e2d-4fa9-a687-5326ceb508fb}}
in the pure path {{formula:238f2ce3-c212-4e38-a9fd-58e0c25d610c}} for edge {{formula:15a9e246-b1b0-4d22-a3c8-b3dddbb77f1d}} ;
variables :
{{formula:135a21fa-8a59-43b6-b219-1fd2fcfbc336}} , {{formula:d8e0e0cf-435a-4567-a9a5-b0c67fc0c6cf}} , {{formula:622fcf11-b641-4e55-9a95-b80a043b8c5f}} , {{formula:7dcf79c4-19ba-4528-acb7-bb947834e132}} :
{{formula:2644489d-e471-4757-8e7e-6cdefacd1379}} {{formula:8372baf9-865a-43a2-8a4b-9c08aee066b5}} the pure path {{formula:6fd41c61-3922-4d57-ac79-17ab14a0e3f3}} for edge {{formula:8bda2d8d-ec0c-4957-895b-d4e3137d0c5d}}
contains edge {{formula:490e7e60-d097-4845-adf5-25dcc7cf6589}} with {{formula:50016266-0673-487c-bb75-db612cd5c9df}} ;
constraints:
{{formula:a9e5c614-3430-4677-80f8-822546fd43f2}}
{{formula:2a1e08b4-acee-49cb-b819-65cc1dd12914}}
{{formula:07e6d76e-21b9-40bf-86ce-9c0305b9ab9a}}
{{formula:ab0c9d89-4e2d-4896-9818-9ce3e97a0d0b}}
Descriptor for the Number of Adjacency-configurations
We call a tuple {{formula:3844345c-0ad9-416a-a805-6bc643d2168a}}
an adjacency-configuration.
The adjacency-configuration of an edge-configuration
{{formula:568b9eb5-3a3f-49fd-931b-3f022d3bb706}} is defined to be
{{formula:612402e4-8181-4077-a25d-5afada551aff}} .
We include constraints to compute the frequency of each adjacency-configuration
in an inferred chemical graph {{formula:563259e0-1bad-4678-9fad-e785fbd15306}}
constants:
A set {{formula:5dea24a8-6154-4804-9a4f-332de9fac555}} of edge-configurations {{formula:c2f6d9d5-d0e0-4829-90e8-e8c2a7dd6197}}
with {{formula:48ee04d0-4d8d-4635-977d-5a23873ac3a0}} ;
Let {{formula:f2248480-6d32-4049-ab29-c532ec72bfca}} of an edge-configuration {{formula:dc4cdb7a-2c30-4854-aed6-f81696e7683f}}
denote the edge-configuration {{formula:8aa1eb46-408b-4f47-b129-569e46bd26ea}} ;
Let {{formula:17db8685-4643-45a8-896f-822960d07e37}} ,
{{formula:892b2c93-c03b-4980-8ce8-c78071e4afff}}
and {{formula:5d0846a9-ee37-41ab-b6b3-2b58f3269dc9}} ;
Let {{formula:f9ab505d-be2c-40f9-8a41-23a7b6939063}} , {{formula:91d79cab-a9d2-4f04-a603-89d479cbb207}} and {{formula:79a72d92-9e79-4854-9b31-c155efbaf330}}
denote the sets of the adjacency-configurations of
edge-configurations in the sets
{{formula:537a4fe6-30cf-4740-8eb9-6b67215911d6}} , {{formula:95cac7cb-2701-4e2a-9479-66cecbae0fea}} and {{formula:bf5ffe42-4c4c-4020-a5fc-c014f1251aa3}} ,
respectively;
Let {{formula:13638900-9890-41da-bee9-2a08246e546c}} of an adjacency-configuration {{formula:c08498de-dd5f-456a-bf0c-4d6ce1e1aa40}}
denote the adjacency-configuration {{formula:ed5fa1a7-c10d-425c-b0a6-11520025c95c}} ;
Prepare a coding of the set
{{formula:9a3f361d-f1eb-488f-8e80-111050099db0}} and let
{{formula:4efb27a7-a81d-490b-8b1d-46fcb82347eb}} denote
the coded integer of an element {{formula:a2e6f0f1-a206-4292-8e2f-894a09b0f582}} in {{formula:1e02149f-5e9b-48c8-9b6d-4a1cca89dc33}} ;
Choose subsets {{formula:e0ac96e6-0f6a-466e-9f8f-bf554160616c}} ;
To compute the frequency of adjacency-configurations exactly, set
{{formula:46c9a538-351a-4fe8-9712-0fa7cf138b88}} ;
{{formula:3b8c5bec-9c8d-4065-937a-81049b8234c7}} :
lower and upper bounds on the number
of interior-edges {{formula:63cda010-011d-4945-af08-002ee414fb43}} with {{formula:800d8e92-44d6-4079-8a7d-3897ccc86fe0}} ,
{{formula:a04c0f86-b564-4d1a-80fa-f1a5be8dc1f0}} and {{formula:6aeeb851-f734-4857-85dc-b9fa6f283efd}} ;
A subset {{formula:1b948564-5dbb-45c5-819e-e44124b3b05e}} for adjacency-configurations of link-edges.
Let {{formula:c820e597-0aea-442f-be60-16dc76d68666}} ,
{{formula:e8609795-a741-4bb2-9c07-cb4f7139ae09}} and
{{formula:37fdfeca-5830-40bd-9f03-362eb811fcc7}} ;
{{formula:6399b911-05c8-41e1-b10c-4576d5a440df}} :
lower and upper bounds on the number
of link-edges {{formula:c47f1313-ba07-4c7a-a2a2-13f7f53d0832}} with {{formula:62b062b8-3c9d-431e-8fbc-6f8b7afc1685}} ,
{{formula:17371d87-86f9-4c2d-9f41-8cc8c3d372c0}} and {{formula:ba7c9029-8dc1-44f4-a720-4dd3191e3c98}} ;
variables:
{{formula:f446dbbd-7548-4c9a-adaa-5b2a4f20400f}} :
the number of interior-edges with adjacency-configuration {{formula:48b86903-435f-4f9b-8ea0-b16f37719efc}} ;
{{formula:a0dd4812-a424-4ce0-b795-9ab1ee1fb70b}} ,
{{formula:14728b1e-8343-4ba7-8c45-27338affede6}} ,
{{formula:a7a71857-4d0a-4f42-99c4-a5f2519fd100}} :
the number of edges {{formula:899bc929-478a-4bff-a278-9e13badbe2ee}} (resp., edges {{formula:c53b2e6a-a791-4e53-bae1-06e1a5dca963}}
and edges {{formula:83a2b7f8-8528-4d45-a3e8-fba90cc9ae06}} ) with adjacency-configuration {{formula:d51304b4-86ab-4630-b365-aef777bee8df}} ;
{{formula:d717162a-1118-48c6-819b-5e9ba3a103bc}} ,
{{formula:0c988d5e-02fb-4ce6-8313-c786220e3de0}} ,
{{formula:8c7b385d-b807-4df3-a660-721eff5e8db4}} ,
{{formula:9db43b08-181a-4b2f-9ba5-a7f7b9bfcec3}} :
the number of edges {{formula:84238bb6-0ccf-4019-85d2-db0141bf0c95}}
(resp., edges {{formula:5dfdc875-0938-46d8-a04f-c36cec00027e}}
and edges {{formula:39b6c42e-d73f-4dfa-a5ea-7f07e5dc335a}} and {{formula:4e71e7c9-f7d9-42de-bdc3-0cd7e6a8f210}} ) with adjacency-configuration {{formula:0c8198d7-af78-4893-b7cf-a9b7618184e3}} ;
{{formula:6a5ffc74-fd23-42d0-92f5-fd200258c70f}} ,
{{formula:e0f8f01b-1b8c-42ce-b844-b3a21bee3d9b}} ,
{{formula:295f78c6-6f5d-4890-a55e-b9ad3e9aca25}} :
{{formula:bcdedaa4-10a6-4dff-891f-3dc96f1d09d9}} {{formula:6b5bfd9d-1677-48ed-991d-7e2198efd2a4}}
edge {{formula:c64404c4-d633-47fe-a863-dc2ccadb0e43}} has adjacency-configuration {{formula:1086e28b-d2a9-4f3c-9fd6-cecded5a8028}} ;
{{formula:7dff5059-64e3-453e-8923-567f43b58708}} :
{{formula:5ff56daa-d910-455a-896c-2fdb847eb609}} (resp., {{formula:0421fffc-d493-4fe9-bbca-8b5d0716e750}} ) {{formula:102e8f23-3bec-4227-a5eb-34e5726a23a0}}
edge {{formula:3f439398-8a48-461c-9202-50984b12a5f3}} (resp., {{formula:87dd7c20-6e37-454d-afa3-76543cc3dce1}} )
for some {{formula:eb948373-f772-4d4d-8886-61713111c1ad}} has adjacency-configuration {{formula:78dffba9-87cd-46c1-98df-5f40aaa6e8fb}} ;
{{formula:fafc9717-1789-4d23-9eac-1738db52a93a}} :
{{formula:c5f40c3d-c000-4f8c-be8a-e6403553d5bd}} {{formula:3e427a47-f445-4a47-aa4a-141ba6c7fa99}}
edge {{formula:1bdd2d4b-36b0-4031-b05b-18b40be6a41f}} for some {{formula:7630fb8d-cb84-41f0-a4c4-61168c125cee}} has adjacency-configuration {{formula:bbf9b402-6a7f-4b8a-8838-a4d8f0e047bc}} ;
{{formula:7ea35af4-0e85-4e7e-beea-e61ce187e689}} :
{{formula:7cdd309f-b1d8-4fb9-a444-63a562604356}} {{formula:3a68f7a5-01e9-4c9d-9102-6966ad8becb5}}
edge {{formula:3c9e6263-0b19-47f1-9be8-61716fa7fcce}}
for some {{formula:9f9509ff-41b0-4c9e-9b68-400bab4ca6b4}} has adjacency-configuration {{formula:4a14968e-8e43-4fb9-b18b-37a2292e67c2}} ;
{{formula:7a52a5a0-194f-4523-b578-6306f436d37d}} :
{{formula:09e580a4-da71-4e5f-92cc-f6c59e6baf40}} of the edge {{formula:5a494dfb-bd71-4b57-b320-541d13eb15c6}}
(resp., {{formula:ff40be95-7dbf-4f38-b806-764692e1b262}} ) if any;
{{formula:340a9360-4409-4e2e-a02b-0d5500be8aec}} :
{{formula:8458e6e1-4dfd-4d4d-81dc-60be1aade886}} of the edge {{formula:100fcc2c-5983-48d2-93c7-352bbae99f1a}} if any;
{{formula:9ca023e8-953c-4543-923f-a78c7ba5778d}} :
{{formula:04124ed3-4964-4462-85e1-83bbca404a5c}} of the edge {{formula:76045253-e3be-456d-9afa-605739298227}} if any;
{{formula:b123b9eb-e654-42a4-965b-9bddf0a5cfe3}} ,
{{formula:77993be2-75c3-4dd7-94d7-9a6a26ed72ed}} ,
{{formula:b8a36272-4656-40e8-a6a5-0b42934c6968}} :
{{formula:f9b46c7b-4a70-4583-8448-dd0fe95c4185}} (resp.,
{{formula:35a30eec-76b6-4888-b34b-57a6b5914174}} and {{formula:ee81a9df-4503-4699-9e21-b309069d3592}} ) {{formula:d53ef829-a0ac-4e05-95f6-abc821579386}}
edge {{formula:80ee9c3b-a661-4499-9206-b40611b498db}} is used in {{formula:2274e7db-c483-4cc2-b606-612a56f91f67}} eXiE(G){{formula:903782c0-9f0c-4280-850d-a4529e468ac5}}
{{formula:1e1ec5ee-736d-4a13-857d-d3acc3dbd061}} :
{{formula:6dbc7559-611e-413e-b644-488c964eb3e1}}
(resp., {{formula:42044869-872f-45f5-8bb6-c5419ebb7250}} and {{formula:4957ad6a-4ad7-47a0-b417-09f0bb623cbd}} )
{{formula:aef86fc2-faaf-457e-a9f9-f09c25ef3fa0}}
edge {{formula:8df86287-9aa2-4056-832e-9b09b6d6f6b7}}
for some {{formula:127d92f1-0d58-4328-8d2e-f2aaaf9ff65f}} is used in {{formula:7644e293-50a5-4d27-82fc-922bfc60d214}}
{{formula:df16e392-d0f1-4ca7-9b72-06fc0dea8ee3}} :
Analogous with {{formula:6c2cb869-cc6f-451a-965c-d45cdd3f4e89}} and {{formula:8518a5b7-62d6-4edf-b6c0-fbf5299d13d1}} ;
{{formula:5c8798a4-b144-4094-8f74-25d8adff377f}} :
{{formula:8721ae1a-9461-4e3b-8698-09043658909c}} (resp.,
{{formula:a9d561be-12af-4c7d-88d7-4fdb9b279a5c}} and {{formula:fc74e6e0-6eaa-477b-a469-732ed7408e4d}} )
{{formula:ee6416d5-a27b-4275-ad35-060990b6f93b}}
edge {{formula:724ff7ca-7b00-4c0e-8ac2-1f58e8b70b7e}}
for some {{formula:1e2fc720-efc8-4920-9178-a76a708098ba}} is used in {{formula:46099af4-a67f-430c-b951-ad7bc1d18e53}} acTF+(i)[0,|int|],
acTF-(i)[0,|int|], i[1,tT]{{formula:21da5c88-97d4-4581-9a72-f085de77dd56}} acCF+(c){{formula:f532132f-2c4f-40ba-805a-1134fb9da6d1}} acCF-(c){{formula:11c44e73-c65a-4b3c-b433-c70d1a810eb4}}
{{formula:c4819a33-f9e2-4347-b0ac-e78764325cc3}} :
the number of link-edges with adjacency-configuration {{formula:ed308c08-f30e-4401-96cf-3f66669e269f}} ;
{{formula:3b5c37d3-3ce3-4358-81c8-200a57bec24e}} :
the number of link-edges {{formula:8db118c9-8fd9-44c5-9b21-eec4034cb428}} (resp., edges {{formula:5006b079-2c37-4567-82f1-3d08e5fc04e1}} )
with adjacency-configuration {{formula:f0d9e818-e559-4c52-be45-8116ec34fd26}} ;
{{formula:af071c90-4d73-40c1-9493-7a6b71c6576d}} ,
{{formula:71336af5-35a2-45c1-a62a-631c42298d5b}} :
the number of link-edges {{formula:4fe7d5dc-c332-4041-a028-dca3b3e658b1}}
(resp., link-edges {{formula:1b9498ee-d738-45b4-b6a2-b216f866dff9}} ) with adjacency-configuration {{formula:75f84f30-0d86-4695-b497-ddf247d8a42e}} ;
{{formula:9472576d-570a-4fa3-9f05-b2d62a266766}} :
{{formula:e21916e5-5203-43a7-abaa-4448a8de5bd2}} {{formula:764ebe2d-f823-4635-8bb1-d090b7a184b1}}
edge {{formula:0e143084-c46e-4f8b-94c2-6168c8f86434}} is a link-edge with adjacency-configuration {{formula:648df5c6-af39-41fd-8eb0-3007c44d2ebe}} ;
constraints:
{{formula:8a698f57-b7a9-473d-b706-adb2b0082abf}}
{{formula:47e0dd23-2968-4942-8f21-c283c1a33990}}
{{formula:12678c7b-2754-4ac1-a9b0-068c717a8dd7}}
{{formula:87877973-7836-41c3-81fa-95e2f797d3b5}}
{{formula:b4f647a1-31e2-4b8a-9de6-34182a498ff5}}
{{formula:afabf42f-2e30-4e7c-a9b7-cde26d62d3c5}}
{{formula:8f333bc9-488f-4d6e-8acf-2f4352b459e4}}
{{formula:1a5f7c44-46fd-41f2-aa96-1f7c6dc9651f}}
{{formula:5c97bf1f-3199-4315-8e36-9a474bb55bb8}}
{{formula:d52b60d9-d510-4ff4-84a6-2e2b3c81d009}}
{{formula:a6776b4a-2f7f-4f47-8a42-3f2b008fd9b3}}
{{formula:fe62ce25-995b-47ae-b98f-224b88372958}}
{{formula:07a14070-59da-41fb-8904-b0a6c5601346}}
{{formula:dd437d0c-eff3-4fa4-abb9-ba890e395172}}
{{formula:5e00cd95-ec6a-4e09-83e9-b6bba9bfa582}}
{{formula:625f648c-804d-4540-b091-4a220d770eaf}}
Descriptor for the Number of Chemical Symbols
We include constraints for computing
the frequency of each chemical symbol in {{formula:cdcf2296-0e87-4f3e-866a-2f6b023a23ee}} .
Let {{formula:025f49ae-c453-4aae-8044-37383c3c62f1}} denote the chemical symbol of an interior-vertex {{formula:96b0233f-6085-4080-888f-65ec391767e9}} in
a chemical graph {{formula:986f2bb5-403a-48c5-9f47-69eaca51d141}} cs(v)==addg{{formula:9fe79090-7144-41fc-8e6f-1dd70eb854f0}} (v)=a{{formula:7b67f8be-4a33-4a2f-b584-7bac26b47cb8}} C (v)=H(v)-hydv)=d{{formula:93c06fcc-b985-483c-8aa6-86504470c0cf}} (H,,){{formula:00eccc5f-a845-4c5f-a256-0ba47945238e}}
constants:
A set {{formula:10df3c39-19d6-49ae-9374-9ca49c00d52f}} of chemical symbols;
Prepare a coding of each of the two sets
{{formula:2d9c1716-390f-422b-94e4-340b2baebbbb}} and let {{formula:cb89087c-836e-4349-b4e0-b6f40695e12f}} denote
the coded integer of an element {{formula:6996904d-ee2d-45e8-85f3-5d80e52ada18}} ;
Choose subsets {{formula:7f28e0c5-c8b7-4968-8c02-90574008077a}} :
To compute the frequency of chemical symbols exactly, set
{{formula:52133e10-cb0c-4ce4-b958-ec31670d5f47}} ;
variables:
{{formula:5925125f-5f64-4cc0-b1c1-e97e491fd68b}} , {{formula:94077e1e-e850-4364-a711-78cc1a634040}} :
the number of interior-vertices {{formula:8b6b36d5-b32c-44dd-9a32-8465425ff3ed}} with {{formula:f82d155c-61a8-4db4-afa1-33b0fbb33b84}} ;
{{formula:32dd48a4-6224-437f-953a-365fe9a5230b}} , {{formula:54eecf41-aa4d-4364-9a71-8b237cddaf20}} ,
{{formula:e9c8b9c7-c75d-4c6e-b3e5-3196aa654128}} ;
constraints:
{{formula:59e9ef47-5ac2-49ff-b98a-55103e0e28e8}}
{{formula:728f98b3-ec10-41ea-86f4-1ef524cfe16c}}
Descriptor for the Number of Edge-configurations
We include constraints to compute the frequency of each edge-configuration
in an inferred chemical graph {{formula:1282ed13-db18-4519-88dd-60e992672eb8}}
constants:
A set {{formula:73a6a9f4-7b3a-4a22-8d92-72212c551a21}} of edge-configurations {{formula:266b08fc-8b4b-4112-8b25-e6de7874ba30}}
with {{formula:21473b8d-1ce0-49f3-b327-5935592e3f25}} , where we let {{formula:6cac4991-46f4-4cb6-9468-06abf180fc5a}} denote {{formula:10813e6d-72c2-4522-8edf-7ebdd6cb9196}} ;
Let {{formula:a70be55c-cb9c-4f2c-b6bc-adf3a1bb8a8d}} ,
{{formula:949a4362-0840-45b8-b390-2d1921cac779}}
and {{formula:6e8d5ca1-8190-4367-9332-bf5f2de6e696}} ;
Prepare a coding of the set
{{formula:25b4e06d-aafa-4dfc-955f-cd7e42a88f91}} and let
{{formula:6d5abcb9-e5f7-4d56-ab6a-601faf0b6f46}} denote
the coded integer of an element {{formula:878d9629-e6fd-438b-8739-0795509c24b1}} in {{formula:d5ec5193-abb5-4451-a114-98efca63336d}} ;
Choose subsets {{formula:05cc54ca-4918-4397-82c2-2649cbb933f2}} ;
To compute the frequency of edge-configurations exactly, set
{{formula:1b089a82-1c4e-4811-b888-a511f7efa8df}} ;
{{formula:627eeff9-1bf5-412b-96c7-158ab56ed3fc}} :
lower and upper bounds on the number of interior-edges {{formula:6f1045dc-b24a-4b8a-a6d8-fb84231d59d3}}
with {{formula:4f0be110-3a1c-4bdd-a5d4-2b223477a257}} , {{formula:f5e695bc-7523-4e8a-a10f-e4cee5d90cd9}} and {{formula:fa5ba226-2d3a-447f-a0ec-8295c5235893}} ;
A subset {{formula:9c79ed4f-c05e-40b6-b8bc-10862739f83b}}
for edge-configurations of link-edges.
Let {{formula:c19c8e63-76f3-4d30-bad5-33b22be329c0}} ,
{{formula:3d8040d5-d012-4c55-bcf9-33b0bfeba1b2}} and
{{formula:68d57b89-035e-45dd-82a4-f6656f18a2eb}} ;
{{formula:fb6d5694-c413-43f3-b435-bb5b9911373e}} :
lower and upper bounds on the number of link-edges {{formula:58aa856d-818f-44f3-a67d-529c92a27633}}
with {{formula:e8913e65-6300-4c4b-aa46-cb9afe368096}} , {{formula:bc16ada6-5b12-4959-920c-b10fe59c0131}} and {{formula:8b6d5939-3c26-4966-b0fb-e8a9a89d1631}} ;
{{formula:ac20f942-bee6-47cc-95a7-7c7e48529cc1}} :
lower and upper bounds on the number of connecting-vertices {{formula:ed83bc8e-d741-45c0-9ddf-1593f118a3f7}}
with {{formula:2f91d548-1404-4024-8eae-4036a87d9611}} ; Define
{{formula:394c6d0e-69e6-4c12-b30e-f6485137e045}} ;
{{formula:0eca80ab-4b25-47f6-8379-774aaee71e33}} ;
{{formula:9b41d7af-4fb6-4f67-b596-0a99d6efe815}} ;
variables:
{{formula:0d5e2eba-6c58-48db-babb-d04682b314f2}} :
the number of interior-edges with edge-configuration {{formula:8d6b9af8-edab-4d87-bd9e-ff0e672885d2}} ;
{{formula:75c5f3b2-0ccd-478e-a080-e656a450852f}} ,
{{formula:03ce70ff-77bf-4186-ab28-a933595f8a8f}} ,
{{formula:cb379516-4430-439d-ba93-0bd8ac2ee343}} :
the number of edges {{formula:a4c8fc4b-d0c2-4e7b-8c1f-29e61d349707}} (resp., edges {{formula:cfb81869-b81b-465f-8ba5-513d28042566}}
and edges {{formula:335ef749-b653-4ecf-ab69-6e581f033303}} ) with edge-configuration {{formula:d126b80f-304c-4b27-a8da-0e5eae3c28b7}} ;
{{formula:7a8a7f7c-3d58-4690-b347-2f9831620605}} ,
{{formula:4b8dc4ce-419b-4110-95a6-6d60f01988a3}} ,
{{formula:5fb9d2da-ed09-4b46-8218-fa34211f09cb}} ,
{{formula:5aa9c38f-a392-4cd8-a0b2-4a9870e343e1}} :
the number of edges {{formula:9581c90a-695b-420a-a9d7-77e2dcf3a40a}}
(resp., edges {{formula:f1212e1e-4677-41c4-aa80-7b443c959544}}
and edges {{formula:6ab99c2c-2c2d-4e18-b84d-ddfc8bdd2e9c}} and {{formula:029404be-b474-46de-bfca-32cef4e83b51}} ) with edge-configuration {{formula:eb323dc1-a7db-4163-a845-d8e2f19e541b}} ;
{{formula:da76d637-665f-4b8c-afa0-a920ea718675}} ,
{{formula:481355d6-dc85-4d43-9290-9778718d3f0b}} ,
{{formula:2bcfd5af-9a6a-4946-81d1-3d2567cbe200}} :
{{formula:f9f95295-31ba-4903-86c3-9af12431c7ec}} {{formula:9afdc1ad-76f7-42f6-bc38-d1cbd30296d0}}
edge {{formula:0a932b48-f895-47b5-8ae5-6c017f24dc58}} has edge-configuration {{formula:1b2a90b5-5c4b-4f43-8e0d-59863e4d4d40}} ;
{{formula:92425201-f555-43d8-94ee-a60e120ba595}} :
{{formula:9e97517d-6ed4-443f-b586-c8f90767c557}} (resp., {{formula:a63d442c-1501-46e1-a422-ecb05be3e9c3}} )
{{formula:b5f1ee5a-8588-44bb-a854-9f486b178ee2}}
edge {{formula:652cdb79-600b-4942-9a9a-3afb91789c5d}} (resp., {{formula:3bcfc991-4c8f-42fe-a802-aaee8ad6081e}} )
for some {{formula:49554841-d43b-4949-8785-c2e1ae86712b}} has edge-configuration {{formula:46875f20-992d-426f-b4bd-901bbe93688f}} ;
{{formula:de7ade13-9e56-4054-81fb-fb4385167108}} :
{{formula:a7e540ef-0d42-4ad5-8fcf-37c07e2ce36c}} {{formula:0fdaede9-255c-4944-a6fd-e5ac520e198b}}
edge {{formula:4e3b80b3-0941-406d-9598-5b88c85129aa}} for some {{formula:e2373c7d-fac5-4d04-a58a-ead376f6c67e}} has edge-configuration {{formula:e27a7cda-3da4-4dd9-8f63-c77915172f6c}} ;
{{formula:dda472b6-23f8-47fe-9401-f4a4e9d2f7de}} :
{{formula:1bfd97d3-def9-4563-bf9a-be915d0fdb9c}} {{formula:61cc6364-1487-4e9f-8745-cb967fb48263}}
edge {{formula:4ee48166-738c-4c44-8685-50b78b43ab78}} for some {{formula:2079bff3-2001-4a31-9bcc-b1a582fdf629}} has edge-configuration {{formula:72cf5d20-38c8-4c2c-9049-47f835df3132}} ;
{{formula:048ba500-4503-4d20-92c5-fdc4af542141}} :
{{formula:a8bed1b2-a3af-4348-878e-e00459202770}} of an end-vertex {{formula:a525afce-4fdd-48c9-ad10-098abe2cbc8e}} of
the edge {{formula:968a581e-0c15-4347-bb1c-97d6d2fbfdb6}}
(resp., {{formula:5eed50a5-87cf-432c-a364-fab6ccfe4073}} ) if any;
{{formula:1c848ff1-3a50-49ee-84a7-4cded97c30f4}} :
{{formula:93fd0c6b-9504-43a5-9e86-687e4639c985}} of an end-vertex {{formula:3a42581a-781e-457b-bf07-6342cf6f0f62}} of
the edge {{formula:0ef1c6b6-7676-47b5-9bd3-65466ce60061}} if any;
{{formula:80882cba-0a7a-4a56-88f2-42e0654ac101}} :
{{formula:13b1eba0-7e91-4a7c-b7ce-ae851b06adc0}} of an end-vertex {{formula:80c8ebd7-ac20-43fa-bf23-f05d7acc76c7}} of
the edge {{formula:5f0112e7-6c57-447b-a6d6-179a5536d018}} if any;
{{formula:90d90a3a-39bf-4cfc-a2ca-133b10842d74}} ,
{{formula:1466c9ef-36c0-4920-8bce-6c35d496a256}} ,
{{formula:2e18151d-603f-4282-bdff-5a5401d10055}} :
{{formula:c698b613-0710-4d5a-90e3-838cc482e8b9}} (resp.,
{{formula:ff8c8295-66da-4f37-98ad-ab33e0cc0ca0}}
and {{formula:a460763a-cd99-462c-909f-47fe7d859d92}} ) {{formula:232b1ca2-ae98-4011-aa4a-74b42651d06e}}
edge {{formula:2d89cd93-3930-4f34-ae85-139a7835f426}} is used in {{formula:45f7b249-a2b7-4c1b-9492-434b5c1d3e39}} (resp., {{formula:3a087cf1-600c-4f28-a3d8-0362cfc9398e}} );
{{formula:607978a8-37d2-470b-9aa8-35e1844d8697}} :
{{formula:7dc98a16-ed2b-47c7-a89c-9addf3811be4}}
(resp., {{formula:796f19df-4d3d-4750-a31f-2f501057f343}}
and {{formula:d26b455a-ab44-4975-9e44-03555b1c8123}} )
{{formula:dd9c4608-7d08-4812-a767-8de120bba3cf}}
edge {{formula:ff62116a-3c9e-4e61-ae2c-71670e241ad2}}
for some {{formula:fac9a3b2-78e1-4766-bc14-3c94e583e038}} is used in {{formula:3b1fb3b1-15bf-4275-a4cb-a0bbd0614b43}} (resp., otherwise);
{{formula:6f0ede52-6cea-463a-9770-0265aad13f04}} :
Analogous with {{formula:d10901bf-d498-4bf6-99a2-a4a3d7b114d2}} and {{formula:2c3fd289-95b1-49c3-baa1-b3d8ebe20327}} ;
{{formula:8dcaf93a-8bf0-49ae-be03-9550c338c17f}} :
{{formula:31f90191-f49f-4822-881a-4b3f158befc9}} (resp.,
{{formula:da09c928-9819-4823-aab7-bb0f7fe861ce}}
and {{formula:d9ef1ced-6bf8-4c2a-b1e4-0f1b0468183f}} )
{{formula:a5319a8f-26a7-4b98-a081-80ab146cb366}}
edge {{formula:77f37c4b-4402-457d-b4fe-3db861dcdb9b}}
for some {{formula:1fb6082e-8255-4fb0-bb0b-9f8fbdfd0cef}} is used in {{formula:1722abff-e860-4126-9f25-e390b7453859}} (resp., otherwise);
{{formula:1d7d82bb-50c9-4dd0-b92d-3c9a01053382}} :
Analogous with {{formula:3afa841f-5132-4d01-a436-f28b48858b20}} and {{formula:e2cba918-27a0-4aed-9db9-09fc749f76e5}} ;
{{formula:8975e136-b4ec-4dea-8ef6-5a8ac9cf5d7b}} :
the number of link-edges with edge-configuration {{formula:298aebe2-fac8-4d53-bfa9-35cc18883260}} ;
{{formula:1296e0c4-3982-42f8-8564-6e07c8476f2b}} :
the number of link-edges {{formula:e2378d41-5758-408e-aeb5-105003bfaa37}} (resp., edges {{formula:1310ef00-2fd3-41e1-a0aa-ec3fef371a8b}} )
with edge-configuration {{formula:855da38a-4d51-4c13-a9db-f1e32f96f452}} ;
{{formula:79501200-1be4-4136-9bf1-075b23e27b46}} ,
{{formula:169f604c-6c52-4436-9716-2b637f40b766}} :
the number of link-edges {{formula:be60e9a3-6c88-4d8b-ba73-a7f882a69ce7}}
(resp., link-edges {{formula:1565b658-a8ba-4b0d-9851-7a98acd503d9}} ) with adjacency-configuration {{formula:28d7691c-d0f2-4213-a1b4-873b6a9d374f}} ;
{{formula:5d42440e-11d5-41c7-8bae-96c055d0d7b5}} :
{{formula:2e0e797b-81c6-406d-8c57-5892fd9398e6}} {{formula:3d298cf0-2814-4356-8298-3a918bf0e293}}
edge {{formula:bcf4252e-1ae1-4518-9eb3-03d7f5eb97da}} is a link-edge with edge-configuration {{formula:f7379463-fae5-4faa-baa9-b12726088226}} ;
{{formula:db8d1e38-2a29-4f4f-844b-5cab397ab01f}} :
{{formula:aa1d6b83-3617-4f74-9451-d2d7fc92109f}} {{formula:c6875fb6-a44e-4249-8e0e-5809919b60db}}
{{formula:e33468be-821b-4522-8ee9-5fe5f1d2c5c2}} for the link-edge {{formula:e2691a0b-9bdf-4de4-ac53-b6c9d9cfb909}} joining connecting-vertices;
constraints:
{{formula:936440b7-4428-411a-962f-4ba893fdd2da}}
{{formula:95d25096-6e74-4329-b0e4-e1a7ec7bf039}}
{{formula:ef1da738-8660-4eb3-9dfa-c2cad165b9c6}}
{{formula:f67d3fe3-ebce-4da6-ba46-90344bc6f38a}}
{{formula:2e6f5bab-c8b5-4986-b5ee-4886f01fdcae}}
{{formula:400abc60-a8a5-4aba-83f0-3299770894d9}}
{{formula:08ad0a8a-d1e4-4240-94cd-7addfba3f035}}
{{formula:026becfc-fe07-40d3-8e0f-74ff6dee768a}}
{{formula:ccacd2c0-df88-45ce-9ee4-78fad4bfdb9d}}
{{formula:869b133b-fed0-4d49-a42b-af9d02b1af7c}}
{{formula:a74a471d-8e03-4c45-9d3e-fb6b492e75d4}}
{{formula:c4b801db-aa8c-4771-a67c-a88fb79ac7e0}}
{{formula:f9c15a8c-326a-4f42-a687-e5ef58883ec0}}
{{formula:649f726b-1f07-4bf2-a263-2620174d8a2f}}
{{formula:ca10472e-4ea8-45fb-a75b-a4779bb98ede}}
{{formula:251cf8db-f5ba-470e-90ba-94371ed5fd87}}
{{formula:9eb04463-44c1-404f-8a73-8587f863c93d}}
{{formula:d821f3f5-e128-4083-82d0-cd9e8ed9372e}}
Constraints for Normalization of Feature Vectors
By introducing a tolerance {{formula:e126d67b-2516-49f6-8494-a9fae4005056}} in the conversion
between integers and reals, we include the following constraints for
normalizing of a feature vector {{formula:cf01b48b-c4a8-4406-b476-f78969701782}} :
{{formula:481daba4-d29d-44b6-b932-a84e1167fd8d}}
An example of a tolerance is {{formula:6f197f3d-c57b-4e0e-97b7-6c9bad668baf}} .
We use the same conversion for descriptor {{formula:0c85b2e8-2614-412b-98f4-f924412b0dfb}} .
| r | 4eda9490648e73002fd3bbf43545debe |
Method overview.
Motivated by DETR {{cite:e72c2ceb74efcb34d87b44ff7cc762ea61de02e7}}, we approach the problem of TAD by an encoder-decoder framework based on the transformer network.
As Fig. REF shows, the overall architecture of ReAct contains three parts: a video feature extractor, an action encoder, and an action decoder.
First, video clip features are extracted from each RGB frame by using the widely-used 3D-CNN (e.g., TSN {{cite:4502fdf672b626fe141595d539cea25d66d1c916}} or I3D {{cite:29dabf262200128a927151b7f45c272b49efcfb7}}). The optical flow features are also extracted using TVL1 optical flow algorithm {{cite:884d0d276b5f73e401b77a88632deb2e089ddeb6}}.
Following that, a 1-D conv layer is used to modify the feature dimension of the clip features. The output features are then passed to the action encoder, which is a {{formula:f55b4cc4-eb30-464b-92c6-ba1bbf35b665}} -layer transformer network. The encoded clip features serve as one of the inputs to the action decoder.
The decoder is a {{formula:113b335c-910c-49b1-91c8-23097d8a02a8}} -layer transformer, and it differs from the encoder in two aspects. It has action queries (which are learnable embeddings) as inputs, and the queries attend the encoder outputs in each layer of the decoder, known as Cross-attention.
Essentially, ReAct maps action instances as a set of action queries. The action queries are transformed by the decoder into output embeddings which are used for both action classification and temporal localization by separate feed-forward neural nets.
The details of the encoder structure are provided in the appendix.
{{figure:129770e2-f015-441d-b822-40ba2d3c6ecc}} | m | 479e7273d65abaa99df2b98a63213ee9 |
In sharp contrast with DJSCC methods {{cite:3da1999921f77eb034be25e11386e28ea8486929}}, {{cite:89c22b2259e7866f613287b957b795c6a2eaa26b}}, {{cite:d03806f8438d82118e20b8bd4048bfc1b6337950}}, {{cite:e84d47207d8eb892f55063a964a575873ed2ee31}}, we require our DJESCC method to address two issues:
| m | 821fa6208c9c1915e78bbcded075df4c |
In fact: the bidirectional, double-branch network structure and feature pyramid {{cite:96f738a98d7b88ac661be795872e27d9a3871658}} has appeared in several previous significant works {{cite:4f74253deac7be2fdacbf430bed7dad5d5cb5b83}}, {{cite:80b72e2bfb79435972e7243fa07b853e3be0075f}}, and has been proved that can implement feature extraction and image fusion hierarchically and more effectively. In particular, it is necessary to emphasize the distinction between the proposed TDNet and BDPN {{cite:80b72e2bfb79435972e7243fa07b853e3be0075f}}. Aiming at making full use of the high-frequency information in PAN images, BDPN extracts the multilevel details from PAN images and directly injects them into the upsampled LRMS images. Differently, we focus more on the mapping relations among images. Specifically, the extracted high-frequency information is adopted as the input of the fusion branch in multiple stages, and the non-linear “pixel-to-pixel” mapping is learned in the designed MRAB, which ensures a reasonable fusion. Besides, we choose the multi-scale convolution module (MSCB) to be used as a component of the network, which could achieve the purpose of increasing the receptive field while avoiding deep convolution layers of the TDNet. This can also explain why the number of parameters of TDNet is much smaller than that of the BDPN.
| d | abb5aa35ce85401bc9a2c76aa7f1d0ab |
Theorem REF shows that the eigencomponents of the error decay faster for {{formula:c3f4d1ef-839a-47b3-b1f2-1ed0c09bd2bb}} corresponding to large {{formula:2785fd06-0ae3-44ee-8531-95154bbf2651}} . Moreover, the difference in the convergence rate of different eigencomponents depends upon the ratio {{formula:86f47479-c6a8-4f8f-972e-d84ee33101f5}} . Theorem REF shows that if the activation function is taken to be the ReLU, then this ratio grows rapidly with the width and the components of the error corresponding to large eigenfunctions decay much faster than the components for small eigenfunction. Further, Theorem REF shows that the large eigenfunction are smooth functions, while the small eigenfunctions are highly oscillatory functions. This means that the high frequency components of the solution are learned much more slowly than the low frequency components. We argue that this is basis behind the spectral bias observed in {{cite:6af581907cd6e84cb11c5b4d1ec7a20611c18d61}}.
| d | bb6ad69fa13279f7fa5e0ed58852a6b8 |
Although SPH has been successfully applied to some complex problems in solid mechanics, fracture is still a challenging and open research area within the SPH community, and has not been extensively studied. This is because SPH is a continuum-based method and cannot, by its own, model fracture and distinct crack surfaces. SPH researchers have so far relied on ad-hoc empirical local damage models {{cite:2bc28c29454c9525e6905ccff53135b01eeb6b14}}, pseudo-spring and virtual link approaches {{cite:8e6458836855cdc594fff9a37c7f1a27935eb515}}, {{cite:2c9e0c28c81b4ebef68e6e353c2d4579e1b2f69b}}, {{cite:9ed52c730fbc4013f459de23f7f9e0862a8eb8ac}}, cohesive zone models {{cite:d2ac16ff250e776614ccafdba0c9e4d69cf07c97}}, {{cite:1e749ba64b8d6b72b9a3cb16224388a5a6d366d6}}, {{cite:cc9cbea054cb871a33e8555e3385c58178a19013}}, and “cracking particles" approaches {{cite:ac09ee8cdcd29f2690a376349ae5d0f6dd41cf13}}. However, the above-mentioned techniques suffer from drawbacks and limitations. Local damage models are based on empirical damage laws rather than comprehensive fracture theories, and lead to mesh dependency and non-convergent results under refinement. The “cracking particles" approach resembles the extended finite element method (XFEM) {{cite:09b5aad3d60c7821ee576609f83c34d5fa078610}}, {{cite:f4cd9e075d714078a3e579ec6f69061bd59ff794}}, therefore the fracture paths need to be predefined. In the pseudo-spring approach the damage evolution is based on rather simple linear damage models, the softening curve of the damage law might lead to instabilities, and past research has shown that they are prone to spurious damage patterns. Finally, in cohesive zone models there are a few tunable parameters for which there is not systematic way of derivation, the crack paths usually need to be known a-priori, and the fracture surfaces need to be tracked explicitly. Even though the aforementioned crack simulation techniques proved sufficient for certain classes of problems, it is evident that there is significant room for improvement when it comes to modeling fracture within the SPH framework.
| i | 4bfe79de6acda7d7c67d2d3732eed137 |
The Quadratic Unconstrained Binary Optimisation problem (QUBO) has become a unifying formulation for a wide range of combinatorial optimisation problems {{cite:1cf92201b53d86375a94a05f4a5f2e191bf760d9}}, {{cite:ea64751776c08c29d50e58fa435c1123fb773db8}}. Its objective form is closely related to the Hamiltonian of the Ising model in statistical physics {{cite:141491e013140579d555aba68a220aed2513eefc}}, where a variety of sampling methods based on Markov Chain Monte-Carlo have been developed to probe the energy landscape and find its ground states {{cite:c4c930a6940eadca3944f8d6fbef92eceaf170e7}}. These form the basis of commercially available QUBO solvers, such as those implemented on quantum annealers {{cite:9fdab3ae7bc0c4a1c8ff73539af47dfcf68fbc97}} and quantum-inspired computers {{cite:11493da307eccee90edd4f7dacf1a18758c30a5a}}.
| i | 4a4cad098003ea29ebf2988890d6620b |
On the other hand, deep learning based reconstruction algorithms recently have led to state-of-the-art results in solving such ill-posed problems in computational imaging {{cite:ff7b024c13daefb0f4286140c2cd45c3a3a4d909}}, {{cite:26958590307fba32bb298b96257d83f41d872959}} {{cite:35edcf08aafa888ca201d5380fd46222bb55a016}} {{cite:69d08246aeaac22de94f60285b918bce43e5a117}}. These approaches typically learn an inverse mapping from measurements to the signal by minimizing reconstruction loss on a set of training examples. However, this kind of training, popularly known as discriminative learning, makes the network task specific.
Furthermore, we need to retrain the network for various parameter settings of the forward model. For example, for every new setting of measurement rate and sensing matrix in SPC, we need to relearn the network parameters. Instead of having to design/retrain a different network for each task and parameter setting, it would be more efficient to have a generalized framework which can be used for solving various inverse problems.
| i | 613c72fdfa14666e9868dc4b7f47d87d |
In this study, based on the dimensionality reduction results, we found there are obvious regional chemical characteristics, as shown in Figure REF .
The stars in zone B are mainly MRSH, the stellar metallicity in this
region lies between -1 dex and -0.6 dex. But it still has lower {{formula:1b0bd4ba-3028-47e2-b718-b80209b83596}} -element
abundances. The {{formula:1edf1d9a-746e-48e6-ac8b-fb11bd5d9ad2}} , {{formula:7b3e8247-8fd3-447d-a943-2ca097751ce2}} ,
[{{formula:6b54aaa3-fae2-419e-a0a1-a8a4a047be4e}} /Fe] here is obviously much smaller than the overlap area.
It is very similar to the chemical characteristics of dwarf galaxies.
{{cite:eb84d4ba257c72309838c294ffa3ab6e9c3f448d}} showed such element abundance characteristics for field stars of dwarf galaxies (Fornax).
{{cite:5ac64f88df1752fe7c4cbe087bf68dc8be9dfc43}} also showed in more details the abundance patterns of the elements as a function of metallicity
for the disks stars, halo stars and dwarf galaxy (Fornax), which clearly implied lower {{formula:cd51cf3d-d6b7-4412-94bb-800880d2d0ab}} -element with [Fe/H] from -1.0 dex to -0.5 dex for dwarf galaxies.
{{cite:fa949d904fadf5f319f271b745ef875a3f94d241}} revealed the origin of the metal-rich halo
stars ({{formula:12f5ec93-fc4a-448d-b16b-630015de3aa3}} ) with highly eccentric orbits (high orbital anisotropy, {{formula:c36113fa-3732-4925-894c-21dda5f2d404}} ), by tracing
their stars back to the epoch of accretion and showed these stars could come from
a single dwarf galaxy. Those sample stars in area B also have extremely high
eccentricities in the last row of the middle column of Figure REF .
Therefore, we conclude that those stars in area B for the MRSH are chemically and kinematic consistent with dwarf galaxies.
| d | 3c5b85f7051c238cef930815e8ac3b6b |
Importantly, our analysis differentiates between two properties that might hinder network performance: high-norm readouts and non-normality.
In a number of classic studies {{cite:251b317ccbafdc7124f8bcfa4c3be819e2a99a62}}, {{cite:7f6ee32db1d6946ae91a4a9734a4c3c8c52d49f1}}, {{cite:ab3b8a5eab4ff5836a3926552ba76db2500dd140}}, large norms of the readout vector have been associated with impaired performance.
In addition, recent observations indicate that training performance is low in parameter regions where the open-loop dynamics is highly non-normal {{cite:c02beb50ff7c1e5ea457d44de6749953409dd4d6}}, and link low performance to large readout vectors.
In our framework, the two properties can be analyzed separately.
Non-normality can be measured from the angle {{formula:0ffc0dd2-1165-4e6e-ac8c-8d508bdd72b0}} between the two activity eigenvectors {{formula:0647ca38-09b4-4358-a7a1-010fbf69b7fa}} ; Fig. REFC indicates that non-normality is minimal at the resonance frequency {{formula:e4cd5979-4e54-461c-a3ab-22b6f09c007b}} . The norm of the readout vector {{formula:f250c3b7-8c07-45be-8f0f-11bb7c3d3d11}} can be instead derived from Eq. (REF ) (see Appendix REF ); we show in Supp. Fig. REF that, for every value of connectivity {{formula:1e62af54-c7c0-4324-a1d8-fca86adad53a}} , the norm of the readout vector is monotonic in the target frequency {{formula:a5dcaa4e-72bc-4b1f-95bb-92ec387ec6f1}} .
We conclude that these two quantities are not equivalent predictors of learning performance; in our setting, training performance is optimal close to {{formula:cf8db1c7-cac3-4eae-879c-c3411dbedd00}} , so that non-normality is identified as the dominating factor in controlling performance.
| d | dbd01bdbe6172e84da7ff93df7e05f6b |
Blockchain technologies originated in Bitcoin by Nakamoto {{cite:11fc8292b67b20894a4d5ff693233b60c80a3c43}} in
2008. Since then, Blockchain has attracted tremendous attention from both
research communities and industrial applications. Furthermore, many real
applications of blockchain benefit from a number of salient and excellent
features, for example, decentralization, distributed structure, availability,
persistency, consistency, anonymity, immutability, auditability, and
accountability. So far, blockchain has been envisioned as a powerful
backbone/framework for decentralized data processing and data-driven
autonomous organization in a peer-to-peer and open-access network. Readers may
refer to books by Narayanan et al. {{cite:3edf605eb00ddcade14501bc08072b8588204702}}, Bashir {{cite:8a5745b4d49a0999a3307c9feeb81a901ff59b2d}},
Raj {{cite:6a3336a21c4e41c37b62bfefb688e257dee0c007}}, Maleh et al. {{cite:a636fd8ecc48e86def2a899d886ee072134458bd}}, Rehan and Rehmani
{{cite:dee892682faf1b9491f4bcf36f78bda153d95a35}} and Schar and Berentsen {{cite:e3f4e6447d270b878559d3bbb1ec8d67e45bda04}}; and survey papers by
Wang et al. {{cite:c1314f43d9b25d9b33b5c8df2a4c99df79a5beea}}, Gorkhali et al. {{cite:e0e516b4e296ed667bdcf4eb996263b993ed5429}}, Belchior et al.
{{cite:6b1e2cc056ef6221349464b6dc5c26e0c239f5c7}} and Huang et al. {{cite:3abfb39b99659e0d9625936ff4238f6702a723a1}}; and further survey papers
with serval real areas by Fauziah et al. {{cite:db8daa169a2dd077f7b0a90ed5c26938fb3e6ac0}} for smart contracts,
Dai et al. {{cite:7fb24ea1632975ef5ab2761aa608f9a1766033a2}} for Internet of Things (IoT), Sharma et al.
{{cite:d7db5d21e78b8c7af934126807627b8fc29d3d08}} for cloud computing, Gorbunova et al. {{cite:739fc60138259e2837fd1efc15c6eefb8cd08b04}} for
industrial applications, and Ekramifard et al. {{cite:db45b0341619b100456ea51315caf0091574b1f8}} for artificial
intelligence (AI).
| i | a48581f96a5d65f811a8b955048a7188 |
It is reasonable to hope that the four-dimensional maximally-supersymmetric Yang-Mills theory may be exactly solved. In the large {{formula:6f032c7a-737e-412d-b02e-1bdc7168edc7}} `t Hooft limit, the theory becomes famously integrable, and Yangian symmetries emerge, which has led to the exact determination of various planar observables. At finite {{formula:b3f5f7b6-3a34-4bbd-a5c7-665584463aa6}} , these special properties are obscured; however, the theory enjoys a non-perturbative S-duality symmetry {{cite:506661d366b70d9f39dc1c6cb44ba4ff7427d0c4}}, {{cite:8b61ae843592ee04d9906821170afc1bdfc50c76}}, {{cite:5409330d2d14aaeb3c7d37c7b8fb6defb467b24f}}, {{cite:74db2ecaf17b804e614a1c194ef97486c8499268}}, which for simply-laced gauge groups is an invariance (up to global identifications {{cite:9b3959123414bef65273498eb032ea4c31e55509}}) under {{formula:9c3f996b-34fb-4908-a3e6-4f1bd82ceb14}} transformations of the complexified gauge coupling,
{{formula:743a51fd-6701-4d95-879e-052053afd4de}}
| i | 06b4239a551f4080cdc8798d3ea7eb24 |
These data sets employed a triggering system based on signals from several STAR detectors to select ultra-peripheral collisions that may contain {{formula:6c786525-805e-4cde-b22c-9e8e27388acf}} pairs decayed from the photo-nuclear production of a {{formula:f735ffd4-757a-425b-b3f0-c5bcccfaefe2}} vector meson or from direct photo-nuclear production of {{formula:41a6c27a-8892-4064-a521-17dd89b377c4}} pairs {{cite:493d0fab0d1dc711c600d35d3ffe145f7fc88935}}. For the Au+Au and U+U datasets, the trigger accomplishes this by selecting events with neutron emission resulting from a mutual Coulomb dissociation process {{cite:853e8d44d2d022282f3f51dc1863a69f34856009}} using two Zero Degree Calorimeters (ZDCs) located about 18 meters along the beam line on either side of the interaction point. The trigger requires a coincidence between signals in the two ZDCs {{cite:493d0fab0d1dc711c600d35d3ffe145f7fc88935}} on either side of the interaction point, to reject beam-background interactions in which a single beam interacts with the beam pipe or other material. Requiring a coincidence in the ZDCs also helps select interactions occurring at the center of STAR where its detectors have roughly uniform acceptance. Since the interaction probabilities for mutual Coulomb excitation and for the diffractive photoproduction processes are approximately independent of one another for a given impact parameter, the selection of mutual Coulomb dissociation events provides an effective technique for selecting these exclusive processes. Mutual Coulomb dissociation {{cite:853e8d44d2d022282f3f51dc1863a69f34856009}}, {{cite:2700f5f6f3c541cb3096294c719fe3d4ede0dc6f}} at RHIC is well modeled {{cite:c75a86b0b844585185b797fb5a0a2f26954298fd}} with an uncertainty of {{formula:8dbc7398-9ca6-4453-9daa-00c9416832cd}} and the various ratios of the cross sections from the model are in agreement with the experimental results within {{formula:1444b592-4225-4ec7-b8b3-d0e79b5a6b22}} uncertainty {{cite:fab2641736dcd0bdebbc6646319c890ff1c913b2}}. In addition to the neutron emission from mutual Coulomb dissociation, the trigger system also requires signals in the mid-rapidity STAR detectors (at large angles with respect to the beam) to be consistent with the presence of at least 2, but not more than 6 charged particles in order to effectively select exclusive events. Finally, in order to ensure that the selected events are diffractive in nature, the trigger system includes a veto on activity in the forward/backward regions between mid-rapidity and the ZDCs.
Since a mutual Coulomb dissociation trigger cannot be used for p+Au collisions, a similar UPC trigger is implemented by employing Roman Pots to detect the minimally deflected proton {{cite:f2878e8b184dfad945093f0f56245503f7f4b05e}}. In this way candidate exclusive events, such as diffractive photo-nuclear interactions, can be efficiently triggered even in p+Au events that do not satisfy the mutual Coulomb dissociation trigger.
| m | f2937489628b17774e14802aa8833a7d |
DL pipelines are effective {{cite:02f5b2fcbc369c7f54e609cb5eb3fe9e596c7cb6}}, {{cite:2184883349e9135f498f07695da53127e20095fa}}, {{cite:5a519736a3f2edeb669873c9b0c957f699891ab7}} at decoding facial emotions, but it is unclear what neural information is more important in this process. To address this, we compared XAI methods to determine their trustworthiness in this application {{cite:3a091af353f2a9a8d3e1a592f8f3c20f2c7c7ea6}}. We compared XAI saliency-maps (Figure REF ) generated from LRP-B, PatternNet, and Pattern-Attribution methods and identified features that are consistent with known patterns observed in EEG during facial emotion decoding for individuals with and without ASD {{cite:5ecacbe9d0583e3e03048f5d97a621e71c40e770}}, {{cite:0668393e4730c6cbfbc5e19b4e1b32ffdcaaccf3}}.
Even though the previously-mentioned XAI methods identify features that are consistent with prior knowledge on EEG-based facial emotion decoding {{cite:2184883349e9135f498f07695da53127e20095fa}}, we observed some quantitative differences between the associated relevance patterns. Because of these differences, it is not clear which method best represents the neural-network facial emotion-decoding process. This motivates the usage of ROAR and random and method-slices baseline for quantifying the level of relevance of each of these XAI methods.ROAR yields a reliable evaluation of what features of the EEG input are used to train the CNN for FER decoding. We observed that LRP, PatternNet and Pattern-Attribution identify the late time-ranges, after 500ms relative to the stimulus onset, as essential for correct facial emotion decoding, which is consistent with previous literature {{cite:caff1e5332f50c076f94b304b40b07e33037c840}}, {{cite:c396c5dc6a9d669588d35a6adefcac4176c90f62}}, {{cite:17139383c7c4d4a7f3c8fbcf90eb6a8cbd8d2845}}.
| d | 5eb195264d962170d57efe174c22cc37 |
After the seminal work on ta {{cite:e8e77763256622761e7f1a8b55e902a800523863}},
many extensions have been proposed aiming at generalising ta in different directions.
One such direction is the introduction of additional discrete data structures, usually resulting in increased expressive power.
A prominent example is the study of ta extended with a stack, such as the tpda model that we study in this work.
This permits reasoning about real-time programs with procedure calls,
which exhibit a subtle interaction between recursion and timing constraints.
In this section,
we provide an extensive discussion on the relationship between tpda and other related models which have been proposed in the literature.
| d | 6cb2fc45fb16481069754db0f5e10c2b |
Recently, gauge field theory which contains several Higgs field became of
interest to solve the questions of the neutrino mass, the matter-antimatter
in the universe and the nature of the dark matter{{cite:c75aa03e90ee5ebb486514504afed0c722186786}}. In obtaining
the physical results from these models, the pole mass plays an important
role in the process where the characteristic scale is close to the mass shell{{cite:5763b1ad4e985e7b1362a535633275eddda8062f}}. It was shown that the pole mass is infrared finite{{cite:1af75c3c70fd8b8c35195518b0a57bb7a548ac7f}} and invariant under the renormalization group{{cite:7d0cd9e15b448843997a981661fc61178f1b7be0}}. In case
of the quantized gauge field theory, we need to add a gauge-fixing terms{{cite:ee8c62035dd7f23592dbcf64c3e825f339144722}} which contain a gauge parameters and the physical quantities such as
the pole mass should be independant of these gauge parameters. In case of the
standard model which have single Higgs boson, the gauge independence of the Higgs
pole mass for the Higgs boson was shown in Ref.{{cite:772bfdbfa787c00530b67be6f5c1e05fafe93ab0}}. In this paper,
we will investigate the gauge parameter dependence of the Higgs pole masses in
multi-Higgs models in two different types of the gauge fixing conditions.
| i | efe0b6b2b0c3922dd5c22162f333839e |
Yet another way to build off of our formulation is to generalize our analysis to multidimensional trajectories. Multidimensional data sets are common such as when combining single molecule measurements from force spectroscopy and Förster resonance energy transfer {{cite:4c3e1bc1a8b297559060e57eafdc57d627418935}}, {{cite:d7a397bddbd5a679d393c06a8257d39264f3eb0c}}, {{cite:b1742b602c53bb2237c937e44441457f5a63943e}}. To treat these data sets within our GP framework is straightforward as we would simply need to choose a kernel allowing two- or higher-dimensional inputs{{cite:4dc9a55e2b13f96b1cfe4af864cd05a1071fabff}}.
| d | 059c3d6e19c49252c6016ec258afeb6e |
Kernel methods, such as the Wasserstein Kernel or the Sliced Wasserstein Kernel {{cite:60fce4ddb2412b8a42a36437c4b6d72829ea5e30}}, {{cite:0ecc05958fd1606a91b1dedc6cf01693e10785fe}}, while theoretically robust, use computationally expensive metrics to compute the distance between PDs. Additionally, kernel methods for PDs treat each point on a PD as if it were alone and do not take into account potential relationships between points, especially the concurrence of topological features represented by these points. Historically, topological spaces were distinguished by their {{formula:5f882196-266e-4c94-9c6b-e4ad0cc5a4ea}} th Betti numbers, the rank of their {{formula:05aee5bb-b42b-43dc-83af-b5518b9fc769}} th homology groups (see Section 2), which roughly count the number of {{formula:d566d0a5-fe05-4a38-8b84-6dc636fd3762}} -dimensional holes in the space. Kernel methods using the Wasserstein or Sliced Wasserstein distances usually measure mainly more the physical differences between PDs rather than what those differences mean for the topology of the underlying space.
| i | 0cdf14cc819fcadc150bb81ad0d4a2a1 |
[noitemsep,nolistsep]
1NN, SVM, and PCA
Transfer Component Analysis (TCA) {{cite:92bef8a9370b5c51c2a038bcb26231261cb869ba}}, which performs marginal distribution alignment
Geodesic Flow Kernel (GFK) {{cite:503d37ec5fb6e793b39d1c49a08eca31c5b08fbf}}, which performs manifold feature learning
Joint distribution alignment (JDA) {{cite:94b5767ab8688115589b3da6b98d0b73bbbbdd65}}, which adapts both marginal and conditional distribution
Transfer Joint Matching (TJM) {{cite:6d82e8520edc191c1e2277a075a044e02b0e4d72}}, which adapts marginal distribution with source sample selection
Adaptation Regularization (ARTL) {{cite:6f8bf2d06a8f4ce5715906e44f8c11a3d27959f1}}, which learns domain classifier in original space
CORrelation Alignment (CORAL) {{cite:ee3e54f611dcdb725fe8c8e1cd82288c2064ff19}}, which performs second-order subspace alignment
Scatter Component Analysis (SCA) {{cite:c37cc73fd3e54362cab7c5da33c1963221b09f1a}}, which adapts scatters in subspace
Joint Geometrical and Statistical Alignment (JGSA) {{cite:305b3bf6a8754ceeb3805f5d987750aa8246e0f1}}, which aligns marginal & conditional distributions with label propagation
Distribution Matching Machine (DMM) {{cite:62da2d58e21be2e46e9c8d02a5ffd1b494512922}}, which learns a transfer SVM to align distributions
| m | a3229bf7282afa9cd1d6f98febd25b49 |
The dual objective can be given by {{cite:32959805305ac7435354c57ed82a66c0244f5c26}}:
{{formula:d86b1dd7-e1a0-4000-b454-72648ed6cf6c}}
| m | 707083097d0af57fa65acfb8e755134e |
The ISOC acts as strong effective Zeeman fields, which polarize
electron spins oppositely to the out-of-plane direction at opposite
valleys, that is, at the {{formula:0cff52c8-55bc-43f4-9c68-ba9db2002ebc}} and {{formula:f329efc3-3a30-4ab6-8cd4-8c5f87e2ea9a}} points in
Fig. REF (b). If we choose the out-of-plane direction as the
{{formula:73db0b93-b2e7-48bf-b1fd-08fdb83b6741}} -axis, the ISOC term has the form {{cite:89baf49fb264bca1fade98452a8c5d1a01745a5e}}
{{formula:aedd85e2-39f6-4340-b81b-0f081395313a}}
| m | 48ba2167f58b21f57829e0cf8b202e0a |
Among the possible extensions of SM, a curious choice are the
3-3-1 models which encompass a class of models based on the gauge
group {{formula:1774b85d-1341-4ded-94a9-3d6f794ca484}} {{cite:f36d2cc7a00744820608ab075f879a5ab5b0b961}}, {{cite:d6c68bf8ec8d3b54f976788dcf77a77ec1860432}}, {{cite:bdd94f27ec6454c4ee77271a06efce1d6edf5692}}, {{cite:abfb0d58cb2975ac879d75c14f63cd20e2ec0444}}, {{cite:ef93ea0fd8040b53afbd68b747f7f6f72cb0e646}}, {{cite:fbedeec0e759985adb849a5ba9109be0d0350efb}}, {{cite:65b7eb74560260b826ac248aaec37176ccb9701a}}, {{cite:ef93ea0fd8040b53afbd68b747f7f6f72cb0e646}}, {{cite:3a844463d6070e7ce3dfabaed963502c6ae87005}}, {{cite:7c8d818d0d71f423ad44ff8a99bf1db7926c7441}}, {{cite:06a7f1f06d1e6ce8d6ba23721dbc9f4a9deaf89d}}, {{cite:3d233ee766036e41fa46b10f71b5cd23b714297c}}, {{cite:6877cf3c3552cdfa14cbae2bcb6f7ce4061dbaa1}}, {{cite:5af7964d92cd9b8dc72f063167104aa592c104d0}}, {{cite:4607f758e138b22e6fd7fae4993ee0949411a49c}}, {{cite:da383ce43a766546ade1afa27fbbb3e58ecaac02}}, {{cite:2140cf0593b0bd29799134b05cb70122893a1148}}, that is at first spontaneously broken to the SM group
{{formula:516fbdc1-a00e-4756-8953-608788f2965f}} and
then undergoes the spontaneously broken to {{formula:9563eacc-082a-42e8-ab64-1aa97a4d9b3c}} . The extension of the gauge group with respect to
SM leads to interesting consequences.
The first one is that the requirement of
anomaly cancelation together with that of asymptotic freedom of
QCD implies that the number of generations must necessarily be
equal to the number of colors, hence giving an explanation for the
existence of three generations. Furthermore, quark generations
should transform differently under the action of {{formula:6e2178b1-738e-4ce1-9a58-c79071f242d2}} . In
particular, two quark generations should transform as triplets,
one as an antitriplet.
| i | 1132792b4b88acf37fc40408f7e11896 |
Additionally, we have investigated an important limitation of our method: accurate predictions require similar mesh connectivity, i.e. our method is sensitive to remeshing of the input surface. We hypothesise that this limitation can be alleviated by data augmentation. We find that PointNet++ is more robust to remeshing, so it can be an option if heterogeneous mesh size is more important than {{formula:fc6cf01e-ed59-4049-9a19-fa3c1722424f}} symmetry. Furthermore, we see this as an opportunity for discretisation-independent neural networks, e.g. {{cite:da218683d88044c0a830354ffa790ff81fe9300d}}.
| d | 7581f61244abd6eb10ca15c9afb8d589 |
Test suites. Our experiments were conducted on networks obtained from real-world data as well as on a set of synthetically
generated networks using the LFR model {{cite:49af3b25d2eed6ea1eef5d872f94003e99e69600}}. The set of real-world networks is obtained from the instances available at the
10th DIMACS
challenge website {{cite:fb2d1e3d745e4f0f23bffc2a0072dc620b23eca3}}. The networks, which are undirected and unweighted, include – Jazz (network of jazz musicians;
{{formula:a7f09ffa-08cc-4c4e-b997-7eb01acc5b90}} ) {{cite:e1d2c36b521290cadb64d6740ba79edd10b8f751}}, Polbooks (network of books on USA
politics; {{formula:8219972d-c2f8-4510-b6c8-005d4f1e18da}} ) {{cite:6b6f653459bef862ad79db51763e8810d90c5b92}}, Chesapeake (Chesapeake bay mesohaline network; {{formula:b7556334-b6b9-4033-bf4f-9cd7c3375740}} ) {{cite:6d3b4b0498b2825414830a11d528e4f0ec4e5326}},
Dolphin (Dolphin social network; {{formula:a754ee52-e57c-408f-a3dd-f0ab1822426b}} ) {{cite:39f2df9f6da538c84434b4b6075f0cf4f317ef0c}}, Football (American college football; {{formula:5a438bdf-1174-4d25-b4fc-68a609eca749}} ) {{cite:ea7b06430d88074778e42021130d3224f0301132}}, Celegans (Metabolic network of C. elegans; {{formula:35f9cf0a-2a23-4601-b74b-7c71098e8af4}} ) {{cite:789d256bc6414b47cfb22c6b7d0795298628ef39}}, Power (topology of the Western
States Power Grid of the USA; {{formula:bb4b452d-1bdd-417d-b9c0-f9003729e8f9}} ) {{cite:a212bdfa05b0ebd69cd661a67018b1a2db6e014d}} and Email (e-mail
interchanges between members of the Univeristy Rovira i Virgili; {{formula:9257861d-512e-43ef-b17a-e614a0483ae7}} ) {{cite:f373a48c6415da6c0ce315df016f2ac397bc35d0}} (note that {{formula:db6d51a5-ef30-41fc-be9a-869f1c0fe7da}} refers to the number
of vertices and {{formula:be5e7f91-b0cf-43e2-828f-cd82cb817692}} refers to the number of edges). All these networks
exhibit scale-free degree distribution (see Figure S1 in the supplementary information).
| r | b6e2b3077f0824a49024b955fedae638 |
Initially, when an image is fed into a capsule network, the image is convolved with a set of filters to produce a feature map, i.e., to produce a tensor representation of the image that is better suited for classification {{cite:f0af46a7806841c2c5717ef947af0118b5302332}}. The size of the feature map is determined by the length, {{formula:786cc9c4-6b54-4f13-8748-8c69cf34ed57}} , and stride, {{formula:90fed4ec-b0dd-4fd1-bd5c-3879111c45a1}} , of the filters, and the total number of filters used, {{formula:841c7fce-1a74-4537-b248-164db50c979a}} . Higher level feature maps can be produced from these feature maps, giving rise to {{formula:7dc097fe-d40a-4e8f-98ef-4cf1b69d5f77}} feature maps in total. The {{formula:8dcac146-4a1f-4eda-83df-1a77497a6e8c}} feature map is convolved with a set of filters to produce a convolutional capsule (ConvCaps) layer. The ConvCaps layer contains {{formula:eef154c4-2e84-4326-986a-94639a27f7ab}} capsule types each of dimension {{formula:7aadec97-b4e3-46d8-9d41-51df17aa8abb}} , where all capsules of a particular type detect the same entity. The elements of a capsule in the ConvCaps layer are calculated using local regions of the image, so ConvCaps capsules detect simpler entities. Succeeding layer capsules are calculated using preceding layer capsules and transformation weight matrices. Hence, higher-level capsules look at broader regions of the image, and are expected to detect more complex entities.
| m | a412cee342267adeb8fbff103bbc7fa6 |
as typical
for a process which can occur, for {{formula:e36cf8c6-7677-4ecb-a5ef-29870d9433d0}} , only via quantum tunnelling.
This so-called Schwinger effect and its analogues have been suggested to play a role in many
problems of phenomenological and cosmological interest, ranging from black hole quantum
evaporation {{cite:7063d0cf306456167020a07395e8f54cf5417ecb}}, {{cite:e4d1a4638105a86c1cc877f43cce1cd088ef940c}}, {{cite:be0d18291fe8c47c723674cf0e223671417724aa}}, {{cite:93a0dee0e1cd63dd967a8321cd5217ef9a55f21a}}
to particle production in hadronic
collisions {{cite:ea47ffa4004ca29176ab37685cf4c466712ceb7c}}, {{cite:dfb1d7f89b0ccb05a045eafccbbeadb5f6d0e1fb}}, {{cite:915744acd94f7af74eeee5c5459f4cb047b5df7e}}
and in the early
universe {{cite:9ba8a8cb2c40355474436ba9c62a35590e0417bf}}, {{cite:21f318ad83b786478571ec775b6190cbc6154117}}, {{cite:14de8306e897e2aec5bbc8a8ebca212ac34246da}}, to
mention only a few.
Unfortunately, there is no practical way to produce a static electric field
of this strength in the foreseeable futureOne possibility considered was the field at the crossing of two intense laser
beams {{cite:552d3b828c0c92d05098539c24233e3b96d358d6}}, {{cite:ad45e6399ec9055d469b6a898c3366bca61b9d23}}, {{cite:c6c31242088d247a085dfc3cdcbad10b27281218}}, {{cite:3bc8435c36bee685dafe70ea152b0e6daf1bcbf2}}. However, the required laser peak power is in the hundreds of exawatt range (for a laser operating in the optical range, focussed
to the diffraction limit) {{cite:629f2a0bfa1c19eaf06d697a55a72d4066e0f547}} and thus still far beyond the present technology. However, the pair production can be
strongly enhanced if one superimposes the intense field at the laser focus with a further weak high frequency electromagnetic field {{cite:4d60f56f2ee24012e1a9fa0be9c1325216f30ae3}}, {{cite:cb37daf5691072a56c87a469c8f4e77505536e3f}}, paving the way to a possible detectibility at the Extreme Light Infrastructure ELI {{cite:64adba0627fea94c9ac6f2686afc91fa5cc3c81e}}, {{cite:c9565af79452157201764e7e76e20d3b5b3c0118}}.. Therefore, a direct laboratory test
of prediction (REF )
seems utopic.
{{figure:5b59b972-ceb4-4017-8509-dd373274f576}} | i | 5f8db648e795685fbb3b376d4b5b5e67 |
A second issue in which we consider our XMM-OM survey to have an
advantage over the GALEX study of {{cite:6e631beb328acfb7846c13fa98b38194280f160d}}
is in the shape of the UVW1 bandpass compared to GALEX NUV for
measuring rest-frame 1500 Å photometry at {{formula:ef6d6011-d287-49e8-b2e8-39124d8bc0b1}} . In the construction of
luminosity functions, the bandpass determines the K-correction, and so
the suitability of the bandpass relates directly to the systematics
associated with K-correction. A UVW1-based survey has the advantage
that K-corrections for different spectral shapes converge at {{formula:d146f02f-cf7a-4f86-b714-3de335d9086d}} ,
and the range of K-corrections is small for the redshift range
{{formula:4cc63ce5-329c-4aa9-86f5-a944b11a6606}} ; see Fig REF .
| d | 179108b7a961d3c3587983ceb611a2ff |
Neural networks have shown an unprecedented success in the hands of practitioners in several fields. Examples of successful applications include classification {{cite:137043dc41b21f9decc7158028aaa5f14026d695}}, {{cite:24ffe66954da954be9ef94d14b644975d23d203b}} and generative modeling {{cite:e1417545a613bf0cac3c7866f62e4ff27b0365cb}}, {{cite:2891ecad021e748eb1d08823142e4d8033997729}}. This practical efficiency has accelerated developing analytical tools for understanding the properties of neural networks. A long standing dilemma in the analysis of neural networks is their favorable generalization error despite typically being extremely overparameterized. It is well known that the classical statistical learning methods for bounding the generalization error, such as Vapnik–Chervonenkis (VC) dimension {{cite:d6d44547223b8066d1d6ef4fd38e402f6ce37715}}, {{cite:48e7fea77085101c60ee31fc499c00d0229e8167}} and Rademacher complexity {{cite:e11fc29573cba58e560c8c60dc5c95a27a328c2c}}, {{cite:f1795a879dd1f969095a6efae9fb3802f9bd2ef5}}, often result in vacuous bounds in the case of neural networks {{cite:d3e5344549eb614a1ebc56c0f6a7b08da7280a74}}.
| i | a5ceba14de32a2ce8c5ca163030eb31f |
In most model-based and learning-based control designs, the contact sequence is fixed or predefined {{cite:6d8daed3fc72bcdc8847935fca2ec33cfcb259a8}}, {{cite:df346b5086b0886d48f44858142785935bfed3a3}}, {{cite:bf358658803473f8b191889de7ecdc920a87a531}}, {{cite:b3debbdaa7f031ff1fb1df3d425baff50ab764ab}}, {{cite:02127c1a7b2015912b782fe2db8e1a0cb2e02c58}}, {{cite:adf6b28edc1e94af090c3a6b24dd974e3df5b66b}}, {{cite:048330e2585b01a4a10d29dc084b810888e191f7}}. Dynamic adaptation of the contact sequence is challenging because of the hybrid nature of legged locomotion dynamics as well as the inherent instability of such systems. While it is possible to generate adaptive contact schemes via trajectory optimization {{cite:fa659bd4a722a388e54b352444af3b292aba684e}}, {{cite:902aca6fead39cd9286552718e2fbf87447721c5}}, {{cite:f3fce3d45cce90d67218967d0b4bdc7f56bdae52}}, such approaches are generally too compute-intensive for real-time use.
| i | e9773a93e41bd8be18f05ba80364c1cc |
To change the unconditional GAN into a conditional one, we introduce class specific parameters that result in a class-specific modulation of the forward pass through the generator. This allows it to generate similar distributions to each target class. To prevent learning of separate modulation parameters for all of the classes, we propose to use hypernetworks to efficiently compute these class-specific weights – and importantly share the knowledge required to generate them among the classes. This is motivated by the fact that hypernetworks have been shown to efficiently transfer knowledge from one task to another one in the context of continual learning {{cite:2f76cf84563c96b4a599c9c62c7e5d6a7a3a44bc}}.
| m | 1b2dfdd7c30a8502c770fbc504328c4b |
This work/research was supported by KUIS AI Center Research Award. C. Pehlevan acknowledges support from the Intel Corporation.
Checklist
For all authors...
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
We noticeably express the contributions and scope of the paper, i.e., providing a general framework for constructing biologically plausible neural networks for separating both independent and correlated sources while covering more generic source domains.
Did you describe the limitations of your work?
See Section
Did you discuss any potential negative societal impacts of your work?
We strongly believe that this work does not have any potential negative societal impacts.
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We have read the ethics review guidelines and our article conforms to them.
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Did you state the full set of assumptions of all theoretical results?
Section provides the underlying assumptions for our theoretical results.
Did you include complete proofs of all theoretical results?
We provide the complete proof of Theorem REF in the appendix. Moreover, we include the detailed derivations of the proposed network dynamics in the appendix.
If you ran experiments...
Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
Yes, we include the code in the supplemental material with tutorial Python notebooks and multiple-run simulation scripts for our experiments in addition to README.md file.
Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
In the appendix, we provide the hyperparameter selections and training details of the experiments. We also share our code for numerical experiments.
Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
In the appendix, we provide the SINR convergence behaviors for several experiments based on multiple realizations with corresponding percentile envelopes.
Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
We did not discuss the computational power and memory concerns since our method does not require an advanced computing system. An individual source separation experiment can run on a basic computer.
If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
If your work uses existing assets, did you cite the creators?
We refer to {{cite:a787e02979fd085c7be95db360129531f7269f62}} and link to the data in REF for image patches we used for sparse dictionary learning.
Did you mention the license of the assets?
We were unable to find the official license of the image patches we used for sparse dictionary learning. However, we cite the creators of the data {{cite:a787e02979fd085c7be95db360129531f7269f62}} and give the link for the image patches in REF .
Did you include any new assets either in the supplemental material or as a URL?
Our code is the only new asset which we include in the supplemental material.
Did you discuss whether and how consent was obtained from people whose data you're using/curating?
Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
If you used crowdsourcing or conducted research with human subjects...
Did you include the full text of instructions given to participants and screenshots, if applicable?
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Appendix
On sufficient scattering condition for source vectors
The determinant maximization criterion used in matrix factorization frameworks is based on the assumption that the latent factors are sufficiently scattered in their presumed domain to somewhat reflect its shape. Both simplex structure matrix factorization and polytopic matrix factorization frameworks propose precise sufficient scattering conditions for the latent vectors to guarantee their identifiability under the determinant maximization criterion. In this section, we briefly summarize these conditions.
{{figure:3ce62317-9a26-4032-988c-6e6202489118}}Sufficient scattering condition for unit simplex sources:
An earlier latent factor identfiability assumption for SSMF required the inclusion of the vertices of the unit simplex in the generative latent vector samples {{cite:526ec6c4173f8d887e02791b13474c84bcc60324}}. This, so-called separability or local dominance, assumption was later replaced by a weaker sufficiently scattered condition (SSC) in {{cite:41bfcbe8a2eea028df21b76666ac6ae84f011a77}}. This new condition uses the second order cone
{{formula:da6e123f-26bc-402d-b96d-2d7b1db4e9c0}}
which is illustrated in Figure REF .(a) together with the unit simplex {{formula:4d9a4225-7f5a-4b6d-b382-4518e083d786}} , as a reference object for defining SSC. The SSC proposed in {{cite:41bfcbe8a2eea028df21b76666ac6ae84f011a77}} for SSMF requires that conic hull of the simplex samples contains {{formula:5b502204-f887-46b3-911b-e70018c9bc0a}} , i.e.,
{{formula:ca751cb3-0e3b-4a56-9d8b-43df6de31b5f}}
Let {{formula:cff49542-83b0-4c0c-a798-ab71cbd529a3}} represent the affine hull of {{formula:6e184aa7-3b64-4e48-bb96-d65133104e51}} . Figure REF .(b) illustrates this requirement restricted to {{formula:639ed283-9131-48b3-aa34-d50565a1f901}} : the red triangle is the boundary of {{formula:676f43dd-51f9-4bdc-be36-540cbc42bbe9}} , the blue dots are sufficiently scattered samples from {{formula:65e3f21a-3f56-4753-baeb-07ccdf0bc545}} , the black circle and the blue polyhedral region are the boundary of {{formula:b0440810-352d-4736-9de6-a5112785782c}} and the conic hull of sufficiently scattered samples from {{formula:bf5835e0-39a7-48c5-9ca3-3489b9353682}} restricted to {{formula:68426ae1-0dcb-40ae-a01c-7ffb3e53bd0a}} , respectively. There is an additional requirement that the boundaries of {{formula:a39d3e22-4288-40cd-bf9e-1fb5617320d9}} and {{formula:9855840b-59b6-4d15-8ae0-66844f0aa2ea}} intersect the boundary of {{formula:8c265440-290e-4a80-bc87-6124dc1b1ce1}} at the identical points.
Sufficient scattering condition for polytopic sources: The reference {{cite:83acc3a1dbcc816d84c4ee93d843a7cbabb00a9a}} offers a similar SSC for polytopic sources for which the reference object for SSC is the maximum volume inscribed ellipsoid (MVIE), represented by {{formula:f1211c5c-1fa0-4a8b-a805-203f3bc617a7}} of the polytope {{formula:58279419-17d3-49e2-92cc-90481afda04d}} . Figure REF .(c) illustrates MVIE (the black ellipsoid) for the polytope selection {{formula:c28da582-9296-4387-b727-6a2d3a01d5d4}} whose edges are the red lines. The SSC for polytopic sources require that convex hull of the polytopic samples contain the polytope's MVIE,i.e.,
{{formula:dd8e94ab-30b3-4990-8352-cd8f85bb9623}}
This condition is illustrated in Figure REF .(d), where the dots represent sufficiently scattered samples and the blue polyhedral region is their convex hull. The SSC in {{cite:83acc3a1dbcc816d84c4ee93d843a7cbabb00a9a}} further require that the boundaries of {{formula:1fd852e8-9f95-41e0-8dba-de593d5a8134}} and {{formula:1016f29c-71b0-4cce-83f6-e377fabfed3f}} intersect {{formula:b10841b9-c2c9-4216-be38-6d51f0d1846b}} at the identical points.
Proof of theorem 1
The proof of Theorem REF relies on the following lemma, which follows from equality constraints () and ():
Lemma 1 Given the mixing model in Section REF , for sufficiently large sample sizes enabling full column-rank condition on {{formula:5aa869da-9abd-486d-a7e0-332ee264fb17}} , the constraints in () and () define an arbitrary linear mapping between input and output vectors in the form
{{formula:0d0b7d1a-c2d4-412a-8351-92a4e9a43d8a}}
where {{formula:89729808-803f-4efb-adf8-4fba59373db9}} is full-rank.
[Proof of Lemma REF ]
To see the relation between {{formula:6ff0cd28-3ddb-4c07-8dfa-602b5d974069}} and {{formula:470b7098-91ed-4dc3-a7d1-ecc1d6023181}} , note that mixing relation (REF ) and equality constraint () enforces {{formula:3b5eb308-9cf0-47bf-b6b5-b0e2302a019c}} .
Defining {{formula:6e71591c-b267-49c4-9d8c-cff936ccdf44}} as the reduced SVD decomposition for {{formula:5f845180-fa52-450a-9eba-ec3688cc4dcc}} matrix with {{formula:9a876714-9719-496e-9329-362328dd5404}} , we can write
{{formula:7bf432dd-9c51-41fb-8718-5d4cdb10b1a2}} .
For sufficiently large sample sizes enabling full rank {{formula:00b3ee9a-2473-400a-85f6-8c04d1878667}} , these imply {{formula:41dce221-cb46-4857-9729-9479ba4d3877}} , {{formula:37538f18-0180-41d7-94da-7d5e3eb3a266}} ,
for some real-orthogonal matrix {{formula:64949879-0169-47d0-8e10-922ceb4b82da}} . From this expression, we can also write {{formula:ad1847f4-4787-4173-9740-9d500dedb514}} , {{formula:bdde1982-61aa-4544-9c24-07394cbb20c1}} , where {{formula:72ffdff9-d294-4987-beb9-c6f098e9119c}} is a matrix with orthonormal columns.
The weighted inner product matching condition in () implies that the output {{formula:3da840a1-f5fd-4a56-9b31-86628dcbab0b}} 's are related to the the slack vectors {{formula:038ca2af-4ea9-4080-b06f-7da7b9c4233d}} 's through the relationship {{formula:a807241b-dff3-40a5-ade0-ceca6404aa12}} , {{formula:09a1ce7a-7675-4e5f-b58e-969f953b8b9a}} ,
where {{formula:c3e13bff-a579-4edd-bfb4-91d4ddaf6775}} is another real-orthogonal matrix. Consequently {{formula:73a09e24-e214-466b-af87-1a3ac76818c6}} , {{formula:371246f8-26e9-4726-895e-3f94659f7856}} . Here, the multiplier of {{formula:8e07cbb9-4ad8-4916-9bb9-b364314f253e}} is in the form of a Singular Value Decomposition of a full rank matrix, i.e., {{formula:020de640-a873-4f6a-ac49-3fe006bfffe9}} which is left multiplied by a full rank diagonal matrix, i.e., {{formula:ae840e09-ac9d-4042-b502-752f701d9866}} . This implies that the equality constraints () and () in conjunction define an arbitrary linear mapping between input and output vectors, through inner product matching.
Now we can prove Theorem REF .
[Proof of Theorem REF ]
By Lemma REF ,
{{formula:bd9c785b-05da-4d8f-be6e-acc4ae7e1c4d}} , {{formula:82acb2b5-b9ab-4452-8c9d-558d3156698f}} , where {{formula:d16e904c-8f83-4a8f-942f-d8a3b8da2cea}} admits the form {{formula:b8b07332-6886-4427-9886-42ed3eb25c89}} .
Therefore, using the mixture-source relationship {{formula:a1e93bac-9d66-42c7-9b2f-fa9708e95ce9}} , we can write {{formula:1ed7817b-ac3c-48cd-8082-0166e459333f}} ,
where {{formula:c2fc98bf-aa3d-4518-87c8-8c2e13388747}} is the linear mapping relationship between outputs and sources. Plugging this into maximization objective in (), we obtain
{{formula:1f3bc7d9-54c1-4aa6-bbc1-8c8a09ce1803}} .To proceed, we define {{formula:02e9c94c-dd80-4d36-887e-daeaed5b1df9}} as the reduced SVD for matrix {{formula:aa9712a7-8b2b-4f26-96bb-237843cb7f44}} to obtain
{{formula:60e4aa38-a93d-4183-bff2-fa511bf4c30e}} .
Consequently,
{{formula:bb65dcd1-9964-4a56-95ef-0f4aa1adfc7d}} .
As a result, maximizing the objective in () is equivalent to minimizing {{formula:af6df050-c398-4323-bdca-f601f974ba91}} with some additional constant terms. Since {{formula:bc08ca84-023c-4151-a914-8896616bbc05}} and {{formula:7b226941-a999-4b89-84e9-eb3f7601aae5}} are diagonal, the equivalent function can be written as
{{formula:e0ee0b95-ed46-4911-9181-00b772c17813}} , which is the objective function in ().
Derivations
The simplification of the similarity matching cost functions
In this section, we provide the simplification of the similarity matching cost functions {{formula:e2410ae5-9efd-466b-9ce2-3b128c7ea478}} and {{formula:04cad2d2-196c-41e0-b33c-c27d124aadec}} in Section REF , by preserving only the quadratic terms that are relevant to online optimization with respect to {{formula:b93ac17c-9126-4435-805e-a9ac565312c1}} and {{formula:4bfb3d38-2925-4f58-b2c1-d7e998249661}} .
Using the matrix partitions {{formula:d8b33c98-e685-4657-8d24-d34c21bed047}} and {{formula:4fb41da8-a85b-491a-a4a6-85939dc0867a}} , we can write {{formula:12da87c7-47e0-4158-8f5e-a4113e6470f1}} more explicitly as
{{formula:ac32db6a-8a40-4c99-832a-abb16dc9c451}}
By keeping only the relevant part of this cost function for online optimization with respect to {{formula:e9672e25-777d-4d61-8976-629ece242b96}} , by scaling with {{formula:a0b100d9-4f58-4c02-a4e0-27bd97eb78d6}} , and ignoring the small final term, we obtain the effective
online cost function corresponding to {{formula:5ebd5e2c-8d65-457c-80d5-4faec2dd1bd6}} as
{{formula:d2f191e1-f227-4d45-a9f0-ba001dddaff8}}
where
{{formula:bb8a309d-ea63-440e-b4f7-26bd14463b65}}
If we apply the same procedure to {{formula:a9fdc841-aa62-4eb2-93c0-1c59346abe77}} :
{{formula:7706ab09-5909-4b38-b519-fa582d82302f}}
Similar to {{formula:34410bae-c821-4215-a13a-18bce6ed4b95}} , we can simplify the part of the {{formula:55cabac0-cac6-487b-8088-b121ac21f0a2}} cost function that is dependent on {{formula:bc1f7e58-7ee5-459e-a125-2cc37ac5370b}} and {{formula:a086e85e-bb47-46a3-9195-f1ea37f228d8}} as
{{formula:f3e76134-478b-4fb5-a849-2fcd14b8145f}}
where
{{formula:25f82cba-02b6-4691-9c14-f63b301c9ba7}}
As a result, we can write the effective online cost function {{formula:d0640f49-3f21-4e65-922e-73ae15eec432}} , corresponding to {{formula:8080c416-b0bd-447a-8dec-4c288047168f}} and {{formula:aeaa62f4-6ef0-4ec5-b6fd-2aa59d0ed61d}} as
{{formula:0a4dfea3-6e06-4bbc-a75e-c8c02da77183}}
Derivatives of the WSM cost function
In this section, we provide the expressions for the gradients of the online
WSM cost function {{formula:92bc3326-8654-4a96-81f5-cb529658a2c1}} in (REF ) to be used in the descent algorithm formulation in Section REF and Appendix . For the gradients with respect to {{formula:da61c815-7d18-45c9-b0ad-780d90ab625e}} and {{formula:abcfdd38-517f-4966-8074-9e228254dff1}} , we use {{formula:798c75e7-a46d-49f9-8ea2-f8b95feefd05}} in (REF ), which is the simplified version of {{formula:885ac4c0-2737-4649-96e5-7b1bd56da983}} , as derived in Section REF .
The (scaled) gradient with respect to {{formula:00c958b9-0574-4f03-8a97-032d00977d6d}} :
{{formula:7e479c87-b989-4cc7-8bee-d562488075ab}}
By applying the decomposition {{formula:8624a783-0b83-40db-984c-a048061ef66a}} , where
{{formula:7715aaf7-188c-4b42-8dc5-18773325834d}}
we can rewrite the gradient expression in (REF ) as
{{formula:ad6a87c9-9202-4f5d-9d7a-d1374c01e945}}
In (REF ), we used the substitution,
{{formula:e5188b06-7b21-4fa7-85d4-ceb61c4a1655}}
The gradient with respect to {{formula:dcbe73a2-f32d-4585-9e8e-4773fdda6183}} :
{{formula:09b97088-26fc-4b31-8331-23a4ccc156b5}}
Note that, since {{formula:75f2f611-9f3c-469b-ad6d-fbb7717390d9}} is positive,
{{formula:ee510178-3328-486f-96e2-0e42a829f448}}
is a descent direction. Furthermore, by decomposing {{formula:bba20fbe-8590-4099-856c-3519850c81c4}} , where
{{formula:6f053510-adf8-4ee6-b353-ce8184a131c3}}
we can rewrite the the descent direction in (REF ) as
{{formula:6e3c78eb-d252-4931-8b2a-fff3c1568f08}}
where we substituted
{{formula:f0bc5975-0a44-4816-95e3-a048a28094d1}}
The derivative with respect to {{formula:689a9aa7-1d21-4b1d-8472-dfcc44d5dff0}} :
{{formula:83e2491d-7a1a-46b0-bd8c-9bff513df512}}
The derivative with respect to {{formula:ec7fd201-8e6a-4c3f-9f0f-8aa334d7fa53}} :
{{formula:a2d40ef6-2f49-407e-99e6-4ba261a065f7}}
Det-max WSM neural networks for example source domains
The proposed Det-Max WSM framework is applicable to infinitely many source domains corresponding to different assumptions on the sources.
In this section, we provide derivations and illustrations of WSM-based Det-Max neural networks for some selected source domains.
Anti-sparse sources
Section REF covers the derivation of the network dynamics and the learning rules for antisparse sources, i.e., the source domain selection of {{formula:ff541649-b659-4afa-9f36-5239473eaf2f}} . If we summarize the dynamics equations obtained:
Update dynamics for the hidden layer {{formula:e819c720-0f4f-4642-a6f9-f96a1dcbef79}} :
{{formula:1434eae8-bb39-4598-9feb-50e7715c7171}}
where {{formula:65ff5402-b8a4-467e-8b30-5c85d135d996}} is the clipping nonlinearity with level {{formula:3aa1e856-3ca7-4eb9-a7bc-ff5ed8adbaaf}} .
Update dynamics for the output {{formula:a1cc2917-0717-4ad3-ac3e-29ffe1775628}} :
{{formula:1cee5e71-72eb-4258-8f9d-f3cc392f954e}}
Figure REF shows the corresponding two-layer neural network.
{{figure:404c1524-670a-4883-a5a2-69352eac5c7b}}
Nonnegative anti-sparse sources
For the case of nonnegative anti-sparse sources, the corresponding network is essentially the same as the antisparse case in Appendix REF . The only difference is that the clipping activation functions at the output layer are replaced with nonnegative clipping function {{formula:d04deeac-7de9-442e-a3a4-910223edd2dd}} illustrated in Figure REF .
{{figure:e91aa062-74db-4b79-b39b-1dba82e1901f}}As a result, we can write the network dynamics corresponding to the nonnegative anti-sparse case as
Update dynamics for the hidden layer {{formula:d28b628c-916e-4215-8922-29c7f784e795}} :
{{formula:fb793b2f-ec1e-464e-ba22-bacae266512d}}
Update dynamics for the output {{formula:5eec2e5a-33ca-496a-b7b4-45fb0d32caec}} :
{{formula:f76bac70-ae9a-4b66-bffb-ec7e1cbcf00d}}
The network corresponding to nonnegative anti-sparse sources is shown in Figure REF .
{{figure:38d0ade1-a4b6-486a-ab6c-5a88cfc5792a}}
Nonnegative sparse sources
For nonnegative sparse sources, i.e., {{formula:d619564b-d614-4edd-8af9-a27c416798a3}} , we consider the following optimization setting:
{{formula:c9d8cecf-3f53-4d1c-b4b6-b193740efa87}}
for which the Lagrangian based reformulation can be written as
{{formula:8027fbec-aabe-4380-9a65-74279ef3d04f}}
The updates for {{formula:1f3018ad-1375-42bc-9d80-35dd87f99e84}} , gain variables {{formula:6e88a28b-7ff1-41fa-bede-36b2057c238c}} and the synaptic weights follow the equations provided in Section REF .
For the output component {{formula:75684206-5f05-4ef3-8bb2-5e633fa8c215}} , the corresponding cost function is an {{formula:05496e70-e80c-487c-8521-45c84c7ef410}} regularized quadratic cost function. Following the primal-dual approach in {{cite:c42cb7a49a884c3988fac603205c8f0029adf18d}}, we can obtain the dynamic equations for output update as
{{formula:afc3a3ae-4b7f-423e-b9a2-0730c67b6aa6}}
where
{{formula:24e13ab9-b8f4-43ec-8721-ed9dea5e8c85}} is the rectified-linear unit mapping defined by {{formula:96bcde8c-c7a9-483f-acf9-66e7a7272289}} .
Based on the dual maximization, the Lagrangian variable {{formula:88b05892-babc-4e6e-a334-77415af08031}} is updated by
{{formula:49d880d0-84e8-460f-9d43-1c948750078d}}
According to the expressions obtained above, in addition to the hidden layer and the output layer neurons, there is an additional neuron corresponding to the Lagrangian variable {{formula:08e8b421-563b-4aad-beec-d1cf53c6521c}} of whose dynamics is governed by (REF ). The corresponding neuron generates an inhibition signal for the output neurons, based on the total output activation. The corresponding network structure is shown in Figure REF .
{{figure:49266f29-1662-4bd1-a4ea-c4d82b5042c5}}
Sparse sources
In the sparse source setting where {{formula:846e2b42-fa6d-499f-9814-9013d21c0672}} , the only change relative to the nonnegative sparse case is the replacement of the ReLU output activation function with the soft thresholding function
{{formula:3edd861a-5666-4d61-8787-7e2ba06e522c}}
Therefore, we can rewrite the output dynamics for {{formula:671ec9c3-7101-4054-8d9e-30ccd1fce646}} as
{{formula:80854f1e-e735-4150-8391-d5e9048f8fe4}}
Figure REF illustrates the WSM based Det-Max neural network for sparse BSS.
{{figure:4025dc07-2212-4db6-ad7f-b0d7081c0450}}
Unit simplex sources
The unit simplex set {{formula:270e733d-4824-454f-9bd8-b71f6319f993}} is a face of the polytope {{formula:2ec7a594-d45d-4c97-9dd2-83d499f5327f}} which is the domain for nonnegative sparse sources. Therefore, we replace the {{formula:a5c062a3-905e-48bf-add0-c005ffaeb3e6}} -norm inequality constraint in (REF ) with the equality constraint to obtain the Det-Max WSM optimization problem for the unit simplex domain:
{{formula:125d4336-2e0b-448c-8903-66f52c425768}}
Therefore, for the Lagrangian based formulation
{{formula:a89475fd-c65b-4f66-885d-7777fde9b69f}}
we no longer require {{formula:7b6299db-8651-4a4a-824a-4d17e673dc0b}} to be nonnegative. Therefore, for {{formula:8df955ed-3c55-4173-8286-1c16d2542d4b}} only required change relative to {{formula:148a9cf0-82c4-4522-843e-66ae39ee17fc}} is the replacement of the ReLU activation function of the rightmost inhibition neuron in Figure REF with the linear activation. As a result, the output dynamics for the unit simplex sources can be written as
{{formula:b3478313-e3dd-4f8f-8b90-ba77ad81e185}}
Figure REF shows the WSM-based Det-Max neural network for the unit-simplex sources.
{{figure:7ada1692-f65a-450b-9bd4-283022ccc873}}
Sources with mixed attributes
We consider the following polytope example provided in {{cite:83acc3a1dbcc816d84c4ee93d843a7cbabb00a9a}}
{{formula:0778eba7-672a-43ee-b7ae-1be013f16f5c}}
which is an example of domains where source attributes such as nonnegativity and sparsity defined only at the subvector level.
The Det-Max WSM optimization setting for this case can be written as
{{formula:bf4692b0-aa6c-4173-b2bb-2c3d260fdea4}}
for which the Lagrangian based reformulation can be written as
{{formula:49c22be6-e060-45c9-a47f-6d059f1cceec}}
The proximal operator corresponding to the Lagrangian terms can be defined as
{{formula:a1342b12-bda7-40ed-b830-c03599ad5cb6}}
Let {{formula:21ddf8a0-67f3-4759-be64-ea707190a725}} the output of the proximal operator. From the subdifferential set based optimality condition
if {{formula:353e546d-b88b-4bb6-98c9-b113ac9bb735}} then {{formula:055157d9-bfaa-4758-ab95-6e47411e6f3d}} which implies {{formula:3342beb7-ecf2-4b6e-9aed-e3fd45609470}} ,
if {{formula:10fac813-5e00-4ca2-8d5b-a12ac73b7497}} then {{formula:7877156a-7139-415c-aec0-eb8cd5f3ff51}} ,
if {{formula:c50cd6d3-07ac-476d-b9bc-b5d00cad8eaf}} then {{formula:edf45f1d-4430-4d4e-9609-a32a99d5f5d3}} .
Therefore, we can write {{formula:7680fea7-bb58-4e4d-96b6-1653e4c1b7f3}} , {{formula:a154cbe5-8786-4029-b0e3-ea16de9ecf6e}} and {{formula:e38baeae-bb15-41db-934c-3bb41fce4b39}} .
As a result, we can write the corresponding output dynamics expressions in the form
{{formula:b46ae59f-51b6-40df-ba4a-36e2c0e33916}}
Figure REF shows the Det-Max WSM neural network for the source domain in (REF ).
{{figure:053a8c30-2ef9-41ba-a182-7a174b7d2b82}}
Supplementary on numerical experiments
Update dynamics for the hidden layer {{formula:82e0c7e5-0799-4329-91de-7cad3ffd81be}} and the output vector {{formula:80948367-98e3-4e9a-9372-47ddf971f678}} are defined by differential equations depending on the selection of the source domain which lead to recursive neural dynamic iterations. Algorithm summarizes the neural dynamic iterations for anti-sparse sources covered in Section REF . Very similar output dynamic calculations for each source assumption can be acquired based on the derivations in Section . We run the neural dynamic iterations until a convergence check is satisfied or a predetermined maximum number of iterations {{formula:d1a78d8a-530d-4fb0-92b9-65287af8651b}} is reached. In Algorithm , {{formula:475ef09a-7de4-49f8-845f-bd3775645031}} denotes the tolerance in the relative error check for the stopping condition, and {{formula:8285cc6d-4606-42b0-bad1-ac07f0ccdf93}} represents the learning rate at the iteration count {{formula:a51477d6-954a-41f5-aa86-9329cffdcf82}} . In the following subsections, we provide the experimental details and additional source separation examples for different assumptions on the sources.
[H]
[1]
Initialize {{formula:944fe2ca-8b47-41fd-b283-d68a7539d55d}} , {{formula:85f5e92f-6a8e-4b8d-8a93-3ecf7a19a41a}} , and {{formula:76a798ff-4ae1-4a39-b37d-b4d7060b0d08}}
({{formula:c9ddc07c-d3a9-46d4-8ed7-7fe40eb2e524}} or {{formula:6489dd28-3700-443c-b3d6-0a3ff5e2ac9f}} ) and {{formula:1c875821-ec51-48f6-b33d-d7e47a990c04}}
{{formula:712b5054-b8b6-420c-b6e4-7b82b9bd5fb7}}
Apply Equation for {{formula:c3318ad0-dfca-4e86-908c-f979658e4006}}
{{formula:c32ef9d0-b2e2-440d-a995-34b632f58357}}
Apply Equation for {{formula:285c9280-ad50-410a-b5bd-8dbc1d1780a6}}
{{formula:fc68394a-a375-4741-97ae-f39ecd543b0f}}
Neural dynamic iterations for anti-sparse sources
Batch algorithms with correlated source separation capability
In this section, we briefly discuss two batch learning algorithms for blind separation of correlated sources, which reflect the Det-Max problem : 1. Polytopic Matrix Factorization {{cite:83acc3a1dbcc816d84c4ee93d843a7cbabb00a9a}}, 2. Log-Det Mutual Information Maximization {{cite:168cfb07bdee0a51f1292da2a496fb6856750749}}.
Polytopic Matrix Factorization: {{cite:83acc3a1dbcc816d84c4ee93d843a7cbabb00a9a}} recently introduced the Polytopic Matrix Factorization (PMF) as a structured matrix factorization framework that models the columns of the input matrix, i.e., the mixture signals in our problem, as a linear transformation of source vectors from a polytope. The choice of the underlying polytope in the PMF framework reflects the attributes of the sources possibly in a heterogeneous perspective; e.g., the polytope discussed in Section REF provides an example of heterogeneous feature assumptions at the subvector level such as mutual sparsity. Taking into account the mixing model in Section REF , PMF uses the following optimization problem,
[l]<b>
Y(t)Rn t, HRm n ((Y(t)Y(t)T))
X(t) = HY(t)
yi P, i=1, ..., t,
where {{formula:a7f2f70d-1b6d-449b-8300-2ea066a44355}} and {{formula:4f647e1c-1f88-4813-be9c-a7f8d7ab43b8}} correspond to the unknown mixing matrix and the source estimates, respectively. The aim of PMF is to obtain the original factors of {{formula:2df23c12-60f2-4b54-b804-80c77989986e}} and {{formula:d462d2da-6a37-4781-8b17-9a83e6775233}} up to some acceptable sign and permutation ambiguities, i.e., {{formula:af3c9ca2-61f5-4239-b758-c46bdc976ff3}} and {{formula:610bf70d-17fe-4b13-b6e8-2261877baa29}} . The reference {{cite:83acc3a1dbcc816d84c4ee93d843a7cbabb00a9a}} provides the sufficient condition for the identifiability of the original factors of {{formula:cdbd5292-5831-4e67-b76f-e3910adfaf90}} and {{formula:03596c56-b16f-42b5-bf18-6ec3588c98cc}} based on the sufficiently scattering condition discussed in Section , i.e., if the source vectors are sufficiently scattered in a permutation-and/or sign only invariant polytope {{formula:d63c743e-703f-4a49-a763-210a07e15929}} , then all global optima of the problem REF lead to the ideal separation. For the corresponding algorithm to solve the problem REF , we refer to the pseudo-code in {{cite:83acc3a1dbcc816d84c4ee93d843a7cbabb00a9a}}, which is a batch algorithm with a projected gradient search.
Log-Det Mutual Information Maximization: The reference {{cite:168cfb07bdee0a51f1292da2a496fb6856750749}} brings a statistical interpretation to the PMF framework based on a log-determinant (LD) based mutual information measure. According to this approach, the LD-mutual information between the input and output is maximized, under the constraint that the outputs are in the presumed source domain. The corresponding optimization setting is given by
[l]<b>
Y(t)Rn t 12(Ry + I ) - 12(Ry - Ryx(I + Rx)-1 RyxT+ I )
yi P, i=1, ..., t,
where the objective REF is defined in terms of sample covariance matrices, i.e., {{formula:daa38a8d-1d95-4626-ab63-fa23096c3ac3}} , and {{formula:030d1661-894f-479b-970f-a52cf03c7148}} . Similar to the PMF framework, the LD-InfoMax approach assumes that the source vectors are drawn from a presumed polytope {{formula:fe2c57b0-5776-4357-936f-701efa1a3f69}} . The LD-InfoMax approach is capable of separating correlated sources, since it does not assume any statistical independence or uncorrelatedness on the source vectors. Reference {{cite:168cfb07bdee0a51f1292da2a496fb6856750749}} proposes a projected gradient ascent-based algorithm to solve the problem REF as a batch learning approach.
We compare our algorithm with the PMF and LD-InfoMax frameworks for correlated source separation experiments in Sections REF , REF , REF , and for sparse source separation experiment in Section REF .
Synthetically correlated source separation with nonnegative anti-sparse sources
In this section, we provide the training details and hyperparameter selections for the numerical experiment provided in Section REF . For this network, we used the following hyperparameter selections and variable initializations:
{{formula:091b0cab-b6d0-474b-99e4-b17de852bb4d}} , and {{formula:9430e43d-d5c0-48ce-b087-981f2aee7688}} , where {{formula:d723c9d9-b02f-43e3-bab6-4b8b3907a48f}} is the identity matrix.
{{formula:5783342a-6aee-48d9-94c8-a534ba8db86f}} , and {{formula:9e9b5a99-c15d-4fd2-bedd-baa0c41bc771}} .
{{formula:75f03699-1ed6-4d31-b795-fe0532f4d026}} .
{{formula:13730585-f3ab-490f-9fe2-ec41c15d17f6}} is dynamically adjusted using {{formula:c06ad54c-41ed-4129-8da9-3ceb5a4ef674}} , where {{formula:52e5e06f-8e93-4d21-8751-cad844070d1b}} is the data sample index, and {{formula:dc8ec42d-0892-4b6e-8c1c-4fbe31692404}} .
{{formula:0b8d71a8-33bd-4dbb-b46c-bb69dd552bac}} .
{{formula:900b0548-f1de-408b-a65a-5b1431984b24}} .
Learning rate for the neural dynamic iterations is adjusted using {{formula:a32311a6-8ac2-4f6c-8718-56dcb5b04860}} , where {{formula:f704ffb2-3b42-4115-b2a7-e391aa06d52b}} is the neural dynamic iteration count.
The maximum number of neural dynamic iterations is restricted to {{formula:12fe3fab-50cb-4b0c-928d-f5a131b9d754}} if the stopping condition is not satisfied.
For the stability of the learning process, we keep the diagonal weights of {{formula:268b0d3b-e6a1-4b80-9100-f81ff155eb58}} and {{formula:f78af5a0-c02d-4235-9aed-1dc0338b5a50}} in a predetermined range, i.e., {{formula:06acd065-93c0-4fd3-8f5e-13642aaef304}} and {{formula:ac36d4bd-c73e-447f-af23-9cec0379768b}} .
Synthetically correlated source separation with anti-sparse sources
To illustrate the correlated source separation of WSM neural networks with antisparse sources, we consider a numerical experiment with four copula-T distributed sources in the range {{formula:25bce1cd-3ff9-4a35-863f-e225da0a4fa5}} with a Toeplitz correlation calibration matrix whose first row is {{formula:56848c21-a901-49b7-9af0-06f97d0a5338}} . We consider the range {{formula:0ae5b1e5-ef55-4230-a25c-c02b811d7d8e}} for the correlation level. The sources are mixed with an {{formula:1ad353c2-6dbd-4943-b338-31becf0cde54}} random matrix with i.i.d. standard normal entries, and corrupted by i.i.d. standard normal noise corresponding to 30dB SNR level. Antisparse-WSM neural network is employed in this experiment, which is illustrated in Figure REF . We compare the SINR performance of WSM algorithm with the BSM algorithm {{cite:658e0223e6cf55fa7a0217b97a214fc2c7a89d5f}}, Infomax ICA algorithm {{cite:60ad1c65acfaf618a3c55164e236e4b43aa10128}}, PMF algorithm {{cite:83acc3a1dbcc816d84c4ee93d843a7cbabb00a9a}}, and LD-InfoMax algorithm {{cite:168cfb07bdee0a51f1292da2a496fb6856750749}}. Figure REF illustrates the SINR performances of these algorithms (averaged over 100 realizations) with respect to the correlation parameter {{formula:6e680ed6-1730-4f6e-b0eb-e8d5701179a7}} . Similar to the results for nonnegative antisparse source separation experiments provided in Section REF , the WSM approach maintains its immunity against source correlations, whereas the BSM and ICA algorithms, which assume uncorrelated sources, deteriorate with increasing source correlation. LD-InfoMax and PMF algorithms achieve relatively similar SINR behaviors while their performance remains comparatively steady with respect to increasing source correlation. We note that both PMF and LD-InfoMax typically achieve better performances compared to our proposed online algorithm since these approaches utilize batch algorithms.
{{figure:3987d51c-9ac0-49ae-baec-3c09f471097c}}For the antisparse source separation setting, we used the following hyperparameter selections and variable initializations:
{{formula:630c353c-3255-44f8-875d-0fafecf23bf8}} , and {{formula:35abdfbd-d48b-4472-84a0-53576c0b2b44}} ,
{{formula:a7b07cf4-5850-4a2c-8c9f-d4195d145913}} , and {{formula:55bf78fd-74cf-46fe-aeed-fd27fe2b1f26}} ,
{{formula:593193cb-0278-4491-97e1-a52eba8c4aa9}} ,
{{formula:91323e5f-90af-4e27-8773-c82d71df626b}} is dynamically adjusted using {{formula:d8113d59-84ae-4959-8735-86ff7c688ba9}} , where {{formula:d69d7d52-659a-42e5-b941-8334ea1faa1c}} is the data sample index, and {{formula:775e5c18-010b-4b68-8c99-f9a3ec7ab6f8}} .
{{formula:287ae761-51d2-474a-b412-a3645af2118a}} ,
{{formula:0843900f-9731-4e21-8167-5479c4779cfb}} .
Learning rate for the neural dynamic iterations is adjusted using {{formula:61de8640-df8b-40f0-b9fe-aea047290861}} , where {{formula:c8c954c1-8960-4c8d-a463-efc95f1b937d}} is the neural dynamic iteration count.
The maximum number of neural dynamic iterations is restricted to {{formula:f91cc285-67e4-4d34-82c9-6506fb987cfa}} if the stopping condition is not satisfied.
For the stability of the learning process, we keep the diagonal weights of {{formula:7e19d485-407c-4502-bb3d-6c10695917b1}} and {{formula:89ac17e5-aec0-42c3-9088-021666f5c615}} in a predetermined range, i.e., {{formula:21e1fff2-f72f-4117-aebd-f213a98ed757}} and {{formula:8ebb0c77-df83-4678-b54d-02cf86613ca9}} .
Image separation
For the image separation example provided in Section REF , the WSM Det-Max Neural Network illustrated in Figure REF is employed. For this network, we used the following hyperparameter selections and variable initializations:
{{formula:d601b423-0f7a-4559-b6bd-c230ef823ade}} , and {{formula:d8a32ae3-1278-46a0-927e-ef9deb7785c1}} .
{{formula:00e73d2a-fffa-4ee8-9d69-d9b6699116fc}} , and {{formula:4cc2379c-b661-405d-96e6-0fc36f917b73}} .
{{formula:dd523a25-8c11-4d7f-8676-cb181e45f9d8}} .
{{formula:41232b94-d51c-4fc7-8cd1-69523ffbbdab}} is dynamically adjusted using {{formula:bb118182-6b16-4d89-9683-77c9f9ad199b}} , where {{formula:93250172-74fb-44ca-a89f-b41116804238}} is the data sample index.
{{formula:b11038bc-badb-4eec-a507-bc0ad642e386}} .
{{formula:bdb007e2-5564-40f2-8465-38afc9cc3acf}} .
Learning rate for the neural dynamic iterations is adjusted using {{formula:27cd102f-9ab5-4785-8b02-ae6aab7087a3}} , where {{formula:fc14f2a2-f2ed-42ee-ae7e-f56eaabe9e9e}} is the neural dynamic iteration count.
Maximum number of neural dynamic iterations is restricted to be {{formula:368052f4-986c-4415-a459-767b464ab8fd}} if stopping condition is not satisfied.
For the stability of the learning process, we keep the diagonal weights of {{formula:f8302476-6ac3-4ae8-8b85-e1173ee537a8}} and {{formula:61e3d624-e0b5-487e-ae32-0687dd83f335}} in a predetermined range, i.e., {{formula:012fa8ef-49c0-46af-b57f-3f5e3d967f20}} and {{formula:af96e0d9-9b98-4fe5-ac72-0b20acd5dbf3}} .
In this section, we also include the results of batch algorithms PMF and LD-InfoMax as illustrated in Figure REF in addition to the source images, mixture images, and the outputs of the ICA, NSM, and WSM algorithms with better resolutions compared to Figure REF . Recall that our WSM-based network outputs illustrated in Figure REF achieves SINR level of 27.49 dB. LD-InfoMax algorihtm's outputs in Figure REF obtain SINR level of 28.65 dB, and the PMF algorithm's outputs in Figure REF obtain the SINR level of 31.92 dB. As expected, both PMF and LD-InfoMax algorithms achieve better performances due to their batch nature whereas our proposed approach's output is compatible with these frameworks.
{{figure:68907e49-d476-45bb-b6b6-d278aa9f9b9e}}{{figure:c995d6a9-9c57-42e3-b3fe-c32cfeb3da8b}}
Sparse source separation
In order to illustrate the use of the proposed framework for a different source domain, we consider sparse sources where {{formula:c4e6e038-562f-46c8-8048-bbcac995a824}} . We generate {{formula:090f0277-6a75-4a35-a0a9-99dda50223ae}} dimensional source vectors, by projecting i.i.d. uniform vectors in {{formula:255c6a83-f496-4e4c-8fe7-c6e8db0a0fd3}} to {{formula:ee6fe6b3-a1cd-4fe9-83f7-f8f3146d4092}} . The mixing matrix is a {{formula:0f3e42fe-0a12-446d-8d0c-ed8e40f52fce}} -matrix with i.i.d. standard normal entries. The mixtures are used to train the sparse-WSM Det-Max network in Figure REF introduced in Appendix REF . For this network, we used the following hyperparameter selections and variable initializations:
{{formula:3b4f8658-5ac8-4d21-84f5-738333cad669}} , and {{formula:9ab25c0f-e4cd-46be-ace4-dd82b639ad44}} .
{{formula:2dd323b2-8a79-4477-a2ae-26f1654749ac}} , and {{formula:08a1471d-4a66-49ec-8c0a-4e15b39174d4}} .
{{formula:ecce897d-37aa-420e-b3e0-270efe239e51}} .
{{formula:60944ca8-83b3-4759-9a8f-170f1a43013d}} is dynamically adjusted using {{formula:35aec4e8-ad5c-44a9-8ea2-71b58fed36e2}} , where {{formula:f2d78057-6118-45ca-80d2-2ad946395e2f}} is the data sample index.
{{formula:0813d6fc-290d-4f64-8615-2984e35936f8}} .
{{formula:570247dd-50ee-48db-8d92-3297df88eb40}} matrices are initialized first with i.i.d. standard normal random variables. Then, we normalized the Euclidean norm of all rows to {{formula:4a3a278c-e2e9-4193-82ea-c16789860444}} by proper scaling.
Learning rate for the neural dynamic iterations is determined to be {{formula:b2909c54-f619-40b3-9651-98c56417ea5d}} .
Maximum number of neural dynamic iterations is restricted to be {{formula:f7abb566-f0fa-45b0-b6c5-1f9917e51f1d}} if stopping condition is not satisfied.
For the stability of the learning process, we keep the diagonal weights of {{formula:ce29b75c-ee47-4fe4-bcd5-8a78805b3f36}} and {{formula:6483869f-dd29-4a51-9c33-b76e78b30478}} in a predetermined range, i.e., {{formula:2e973e63-3f2e-45da-8258-bf17e40b984d}} and {{formula:df4e6464-b957-4317-8fb9-52a0912da50c}} .
Figure REF illustrates the SINR convergence behavior for the sparse-WSM network, as a function of update iterations, for the input SNR level of 30dB (averaged over 200 realizations).
{{figure:744a9a48-4f40-443e-85fd-226b27e7076e}}Figure REF demonstrates the separation performance of the sparse-WSM network for different noise levels.
{{figure:686af39a-9b2e-4d5f-b970-51b993747fa8}}To compare our online approach with the batch algorithms LD-InfoMax and PMF, we also performed experiments with these algorithms for the input SNR level of 30 dB. Table REF summarizes the averaged SINR results of each algorithm over 200 realizations for 30 dB input SNR level. In these experiments, we observe that both PMF and LD-InfoMax obtain better SINR performances on average compared to our WSM Det-Max network. This condition is due to the batch nature of both PMF and LD-InfoMax as discussed earlier.
{{table:24de2857-9994-41aa-8dca-3df5535210cb}}
Sparse dictionary learning
Related to the previous example, we consider the well-known example of sparse coding, which is the dictionary learning for natural image patches {{cite:a787e02979fd085c7be95db360129531f7269f62}}.
For this experiment, we used {{formula:c366ddd9-bddb-4b7f-97ea-ced21e46a233}} prewhitened image patches obtained from the website, http://www.rctn.org/bruno/sparsenet. We used the vectorized versions of these patches to train the sparse Det-Max WSM neural network in Figure REF . Figure REF shows the receptive field images obtained from the columns of the inverse of the sparse-WSM separator, which correspond to localized Gabor-like edge features. This example confirms that the sparse WSM neural network with a local update rule successfully captures the behavior observed in primates' primary visual cortical neurons.
{{figure:e954cb2b-438e-4dc4-89f0-9beb18c34ea6}}
Source separation with mixed latent attributes
In this section, we illustrate the source separation setting with different identifiable-enabling polytopes similar to the given example in REF . These experiments demonstrate the capability of the proposed WSM Neural Network for general identifiable polytopes. The identifiability of the provided sets in this section are verified by the graph automorphism-based identifiability characterization algorithm presented in {{cite:8eb555239ef97cd4282fe7cd1b71bfba51cda3f9}}.
Special polytope example in appendix REF
We provide numerical experiment results for the WSM Det-Max network in Figure REF corresponding to the polytope in (REF ). To employ this WSM Det-Max Neural Network, we synthetically generated {{formula:54c6e3a7-017b-4561-98c7-321642b45c9f}} dimensional uniform vectors in this polytope and mixed them by a random {{formula:f168f104-0ab3-437c-8e58-de00aa791152}} -matrix with i.i.d. standard normal entries. Also, the mixtures are corrupted by i.i.d. standard normal noise corresponding to 30dB SNR level. Figure REF illustrates the behavior of the overall SINR and individual source SNRs in addition to the behavior of diagonal weight matrices ({{formula:908011c4-0372-4261-93e2-79748249069c}} and {{formula:1bbeaf33-4e4f-4448-98e4-c8571fcbd406}} ) with respect to the number of update iterations for a single experiment. To measure the average behavior of this neural network, we run experiments for 100 different source and mixing matrix generation, and Figure REF illustrates the averaged SINR convergence behavior with the {{formula:55626229-d3c7-4662-a307-c9f25d0f0e40}} -percentile envelope, as a function of update iterations.
{{figure:c9a4178d-8cf1-4f6a-84ff-fe3a970be7e2}}{{figure:1406d57c-3fed-410d-aed3-09ee1bad974c}}For this network, we used the following hyperparameter selections and variable initializations:
{{formula:43912d00-74db-41e7-90aa-bc8fde34d83d}} , and {{formula:1224bd28-7196-4015-b144-9c28f6eac23d}} .
{{formula:08c3ef1b-3b30-408f-a661-e65599c2f645}} , and {{formula:aff9d085-f5f6-4c8e-b201-aa8de536d3eb}} ({{formula:56c93272-f02a-49ff-b8bd-acbcd8e47f3f}} for the experiment visualized in Figure REF ).
{{formula:f58646f4-8a1a-4657-8a8e-a1202c13d5ff}} .
{{formula:50ff5701-b7c2-4e1c-97a7-e1439efb498d}} is dynamically adjusted using {{formula:113074bf-dabb-4f16-9d85-c0b5281b741a}} , where {{formula:718dd059-3c6c-4ff5-8934-b5792e82baa2}} is the data sample index.
{{formula:a6fb7d3e-643f-45f1-8523-c3e67ab3b021}} .
{{formula:7c8fd595-893e-4ae1-ac78-0a0ed1071d8e}} matrices are initialized first with i.i.d. standard normal random variables. Then, we normalized the Euclidean norm of all rows to {{formula:f1a55993-340a-45f5-88ed-4c6f905efc6c}} by proper scaling.
Learning rate for the neural dynamic iterations is determined to be {{formula:dfd0fa80-d684-4db0-b28c-182b5136084d}} .
Maximum number of neural dynamic iterations is restricted to be {{formula:53dbdfb7-c96d-4488-bc81-edb4c9602c60}} if stopping condition is not satisfied.
For the stability of the learning process, we keep the diagonal weights of {{formula:276b8bd2-9550-4d1e-9122-1fdc6f76d7b9}} and {{formula:401302a8-c299-47c2-95b9-ad15bd9ac574}} in a predetermined range, i.e., {{formula:bce98e1a-7fa2-480c-9823-990a23865bf3}} and {{formula:9844e73c-e0b2-4973-a71d-6c9f12d9ced5}} .
Mixed anti-sparse and nonnegative anti-sparse sources
As another identifiable polytope example, we consider the following set which assigns mixed antisparse attributes to the source components: signed or nonnegative. For this experiment, we randomly selected two components to be nonnegative whereas the remaining three components are antisparse. The mixing matrix is a {{formula:4953acd5-1309-4fae-b4fd-b72c732a7630}} matrix with i.i.d. standard normal entries. The mixtures are used to train the WSM Det-Max network similar to Figure REF where the clippings at the output layer corresponding to nonnegative sources are replaced with nonnegative clipping.
{{formula:be0a4f28-8c10-4ccf-b897-a76344601f21}}
To train the WSM Det-Max network in this scenario, we used the following hyperparameter selections and variable initializations:
{{formula:606128aa-b948-4795-9578-1be1511a50ed}} , and {{formula:5a950db3-8a55-4df8-a717-11c638fef850}} .
{{formula:2f6970e5-539b-4333-b4f1-5912ed9e99db}} , and {{formula:7792dadf-3034-486c-a4c3-c762d572cbc0}} .
{{formula:ca061514-bb7d-46fa-b7e6-5d34a46b813b}} .
{{formula:234f6c66-f8d1-435f-ac79-c9ed1f780682}} is dynamically adjusted using {{formula:ff363ac0-b925-4526-a754-3cb5f2b19de6}} , where {{formula:c61863be-2dc4-4b15-8f41-221bc22e6a0c}} is the data sample index.
{{formula:7e460f61-1feb-4212-9b0f-c5801c153c28}} .
{{formula:63f9aba0-669d-411b-970b-2423df1930b4}} .
Learning rate for the neural dynamic iterations is adjusted using {{formula:ca8ead4a-2700-4544-a657-4539532387d7}} , where {{formula:6775b26d-8b28-4973-8247-a38d0dec741d}} is the neural dynamic iteration count.
Maximum number of neural dynamic iterations is restricted to be {{formula:7990c858-2df4-47fa-bbe2-194c5b781a66}} if stopping condition is not satisfied.
For the stability of the learning process, we keep the diagonal weights of {{formula:f7788a0f-3d5c-4228-8e88-31ef95b4a1b3}} and {{formula:a5ccd193-887c-4b28-8ec3-f4782565df52}} in a predetermined range, i.e., {{formula:53f3b034-fc23-4693-a250-f9033c1c1150}} and {{formula:73fcf237-c3ba-4898-8bb1-e4fe3f3bdf17}} .
{{figure:a970c4b2-e58b-44cf-bf33-3209672f37a0}}
Mixed sparse and nonnegative anti-sparse sources
As the last illustration of source separation on identifiable domains, we consider the following polytope,
{{formula:ed40a12a-a878-4dfc-87a9-9d8bf90210bb}}
where only one component is nonnegative and the subvector containing the remaining components is sparse. To demonstrate the source separation ability of WSM Det-Max Neural Network for this underlying domain, we generated {{formula:4efa743d-5546-4cda-a587-d80c713ade0e}} dimensional uniform vectors in this polytope. The sources are mixed with a {{formula:d16cc73e-6859-40b7-beac-0c502de2f5ca}} random matrix with standard normal entries. To train the WSM Det-Max network in this setting, we used the following hyperparameter selections and variable initializations:
{{formula:87d56e3a-a2ba-49d3-97c8-3c466a3406b0}} , and {{formula:bc3ef5e2-9d59-4b78-b379-6b48be74b0c4}} .
{{formula:d7add070-cbbe-420b-84ea-441d53d18deb}} , and {{formula:9e9a84fd-65c0-46f2-bbd5-4097b784c942}} .
{{formula:42130b71-4b25-424a-bb42-6c8031caabff}} .
{{formula:5d0d83d8-db06-476e-9031-e6536600eeb1}} is dynamically adjusted using {{formula:5aa3eb9a-2f2a-49cb-a9b0-55c3209d7f91}} , where {{formula:dfcc0679-191d-4e99-ae7f-3bb388da289c}} is the data sample index.
{{formula:29dd6925-157d-457a-945b-a2e43f8e8811}} .
{{formula:171c0818-9d3c-4523-84aa-c691180c37d4}} matrices are initialized first with i.i.d. standard normal random variables. Then, we normalized the Euclidean norm of all rows to {{formula:32ce326d-0684-4833-849d-d290086bc543}} by proper scaling.
Learning rate for the neural dynamic iterations is adjusted using {{formula:60d0b3c6-40bb-4103-82d3-5c838c01a9c7}} , where {{formula:b9043f5c-dcf7-4215-affc-5edc2966c055}} is the neural dynamic iteration count.
Maximum number of neural dynamic iterations is restricted to be {{formula:d7bb7cfd-7591-44de-acff-46975aac847b}} if stopping condition is not satisfied.
For the stability of the learning process, we keep the diagonal weights of {{formula:c7405895-8d65-4207-960c-286f9e36c2c2}} and {{formula:2ffde243-8b4b-4547-8ae1-6ac1331a910a}} in a predetermined range, i.e., {{formula:caef5cd2-0e20-4137-8d8c-a7a10ca02cc2}} and {{formula:ab32226b-121e-471d-89d3-46e178941b7d}} .
Figure REF illustrates the SINR convergence behavior (averaged over 100 realizations) of the WSM Det-Max network for this scenario, as a function of update iterations.
{{figure:87e3e85a-868b-4e7f-84ef-29e7388195a7}}
Digital communication example: 4-PAM modulation scheme
We consider the 4 Pulse-amplitude modulation (4-PAM) scheme as a realistic application of blind separation of digital communication signals, with the symbols {{formula:c5bc203b-7d28-4e5c-9a13-634512c19041}} . We consider a uniform symbol distribution, i.e., {{formula:9b78f640-c418-4f44-83d8-16ac202c780a}} , where {{formula:8a4af1e7-5af0-4bcc-bce5-4eac7468267a}} represents the transmitted symbol. We assume that 5 sources are transmitted, with 400000 samples each, and mixed through a {{formula:218f9293-a5ab-4a50-95c0-309ecb3e4154}} random matrix with standard normal entries. Without loss of generality, we make use of {{formula:9b0dd04c-2587-4626-8353-c36063b29850}} polytope as the source domain assumption so that we feed the mixtures to the WSM Det-Max neural network for the antisparse sources. To train this network, we used the following hyperparameter selections and variable initializations:
{{formula:bb2af25e-52e0-46a6-9f71-8ac691b274fc}} , and {{formula:6057d5e8-2b26-497f-a736-39e858a6b798}} .
{{formula:52e1245c-aa66-4478-8dc2-d4f87078cb5e}} , and {{formula:56bb69e9-b4b2-4bdd-ae20-ef1c2f191dd1}} .
{{formula:9495b780-4dc1-43e0-98f6-f544971ada00}} , and {{formula:657acb43-2e11-4194-bedd-8058ce8d556d}} .
{{formula:e7c5a20f-74ab-4a95-94fc-a4e11f43af5c}} is dynamically adjusted using {{formula:72d41c83-8f97-45bf-a452-0f9f40f37bb4}} , where {{formula:81cdbaca-eb93-41b6-848a-beb4b7828785}} is the data sample index.
{{formula:a631c5f5-5c8e-4147-8df3-fab06befdd76}} .
{{formula:ed985255-c221-418b-8720-ee910f7c04d9}} matrices are initialized first with i.i.d. standard normal random variables. Then, we normalized the Euclidean norm of all rows to {{formula:7531ceae-f344-4e10-934a-b9c0734f8441}} by proper scaling.
Learning rate for the neural dynamic iterations is adjusted using {{formula:64e06411-5b74-48c9-94a3-93cefe057474}} , where {{formula:5481374c-b936-4bb8-8916-ad066711e033}} is the neural dynamic iteration count.
Maximum number of neural dynamic iterations is restricted to be {{formula:cf363c8c-7aa9-4222-aede-9a4c5d16aeb3}} if stopping condition is not satisfied.
For the stability of the learning process, we keep the diagonal weights of {{formula:ac5adca7-4d87-4de6-9ceb-34199336ffab}} and {{formula:03e886d3-3e06-49f6-8340-e4c9a41b49b6}} in a predetermined range, i.e., {{formula:a92a0176-58b1-4d83-8859-9948b21a90b7}} and {{formula:8178cc88-fe96-4934-a374-5d5436fdee64}} .
Figure REF illustrates the SINR convergence behavior (averaged over 20 realizations) of the WSM Det-Max network for this scenario, as a function of update iterations.We conclude that our proposed approach is able to separate the source symbols from their mixtures.
{{figure:1fc21097-7509-439a-a1fa-cfadc0ecfdbd}}
Ablation study on hyperparameter selection for nonnegative sparse sources
The proposed Det-Max WSM framework requires many hyperparameter selections. In Section , we discuss the selection of these hyperparameters for different source domains. Most of the time, we find these hyperparameters by trial error and sensitivity analysis. Several ablation studies similar to grid search are useful to find the optimal values for the hyperparameters. In this section, we provide such ablation studies on effects of the selection of {{formula:9ecd1784-e4f1-4e3c-af08-408c17adc790}} , {{formula:f8e75d99-5e5b-4b9d-92bb-dae2d528446a}} , {{formula:4a14498c-761b-4055-b63e-c28fd2cc15bb}} , and {{formula:faa6b9ff-e8d5-4e6b-a560-fc01fc54a5f5}} . We chose to focus on {{formula:64bb00cb-6d99-40c0-a0e9-e11e73b17397}} here because we observed that it is one of the most sensitive parameters. Although the other parameters appear to have less of an effect on the final result than {{formula:9710bbed-fd1e-46bc-90cf-fb84d3acb171}} , the cumulative impacts of the combined hyperparameter choices can substantially influence overall performance.
We consider nonnegative sparse source separation setup, i.e., {{formula:877ee3a5-eb17-4c09-b724-183793896778}} . We generate {{formula:23301bd7-3f41-452a-9082-d69fd6eba870}} dimensional source vectors uniformly in {{formula:68f07c89-3c3b-4a95-9d39-4f8d0869c8c9}} , and the mixing matrix is a {{formula:cf6ae805-dbd3-4712-8a49-f12eae5ff7bb}} -matrix with i.i.d. standard normal entries. The mixtures train the nonnegative sparse-WSM Det-Max network illustrated in Figure REF . In these ablation studies, we specifically consider the effect of hyperparameter selection for {{formula:b7c1bcc3-2d53-4d94-b27c-cb99c13399ed}} , initial {{formula:ef0c242a-996f-43d4-a9be-b221fe824222}} , {{formula:df5fc909-3749-4b00-b3bb-517acb8209e4}} , and initial {{formula:1817abf0-1017-47f3-bfc8-ffec63e77a0e}} . For each of the mentioned hyperparameters, we consider the following choices,
{{formula:96ed81f5-cd1f-4aad-98fa-c53e271304d5}} ,
{{formula:0b51cf2a-b9dc-4977-bc25-39787be44896}} ,
{{formula:83c7d7e9-a83e-4bf9-9ebc-2796d4e51730}} ,
{{formula:4ba98599-85a5-49dc-a670-cbc02ba5d183}}
While experimenting with one hyperparameter, we fixed the rest of them as given in the following list,
{{formula:39e0f450-d06e-48ee-942a-edef6c24475d}} , and {{formula:6c64cd9c-c949-485a-b83e-eb6f688b4238}} .
{{formula:3eb4c249-1afb-4bee-8426-008b6e424405}} , and {{formula:6ab4bec2-74ef-4dcd-bad4-835c3f6829b5}} .
{{formula:44259ffa-9849-4bd6-b107-3c1c04b59e37}} , and {{formula:e4872b0f-d367-43ce-8003-873b74d27f0f}} .
{{formula:f64f905c-96c8-4e3b-ac90-49df74db643f}} is dynamically adjusted using {{formula:42bbf53d-0c79-4773-8b5f-cda3035c7208}} , where {{formula:cb7b7244-95f5-4ff8-bc1e-4dca91cb5207}} is the data sample index.
{{formula:3dd99aaf-798e-468b-ad11-d15b77fd351b}} .
{{formula:13249425-9337-4c72-897b-a83c98194b5e}} matrices are first initialized with i.i.d. standard normal random variables. Then, we normalize the Euclidean norm of all rows to {{formula:4804201e-4e6e-42c6-aae7-e177667c99fd}} by proper scaling.
The learning rate for the neural dynamic iterations is adjusted using {{formula:57278e7d-0577-4444-a8f2-037e52f83eea}} , where {{formula:8370f6cb-635c-4b98-bd3b-e7d543506c4b}} is the neural dynamic iteration count.
Maximum number of neural dynamic iterations is restricted to be {{formula:e37a1b2f-5426-417b-b0f0-0290cd5629f0}} if stopping condition is not satisfied.
For the stability of the learning process, we keep the diagonal weights of {{formula:e47c06af-20f3-454e-8ea6-64d2d9d6986e}} and {{formula:7ce39112-5a10-46e4-b242-8b0ac5f11717}} in a predetermined range, i.e., {{formula:c7fc9921-2800-4ec0-9c41-95d9c2575b42}} and {{formula:e07da5a0-4a2d-4739-bc33-e052658a2ba4}} .
Figure REF illustrates the SINR performance of the WSM Det-Max network concerning {{formula:ce1e9cc8-e903-4a86-848f-96c93796e504}} , and it demonstrates that it significantly affects the final SINR behavior of the proposed approach. We argue that the selection {{formula:9f49e6ef-4537-4f4e-ab5d-7bff8a6885fe}} is a near-optimal for nonnegative sparse source separation with the WSM Det-Max network, whereas one can also implement a more detailed search based on possibly other hyperparameter dependencies. We also analyze the effect of initial {{formula:cf79c8d2-0366-4eb6-b572-6d23fe0d2c07}} on the final SINR, and Figure REF demonstrates the performance change with {{formula:8af8d077-bfb3-4a15-aebf-928c07e989ed}} gain initialization. We inspect that the WSM Det-Max network for nonnegative sparse sources relatively maintains its averaged performance against different initial gain parameters, whereas the selection of {{formula:ec879826-9cd3-4df9-914c-03962063bba6}} leads to best performance with a significantly lower variance compared to other initialization choices. In Figure REF , we visualize the effect of learning rate choice for {{formula:06f6c977-238b-420f-a9f4-a3a4cc71fc17}} . It is noticeable that {{formula:05126e09-d384-4ac4-b3b8-1856ea108c3b}} is less effective in SINR performance compared to other considered hyperparameters, but {{formula:81c94078-609e-43f2-adb4-22eb35a8eda3}} achieves the best average result with a lower variance. As the final ablation study on hyperparameter selection, we consider the initial value of {{formula:a87c7fa8-d8f1-459d-bd4e-94fe654675d2}} which we dynamically adjust using {{formula:64b0402a-dd44-40c6-ba54-20ee26078def}} , where {{formula:c7dfe84e-2c07-4167-84fc-5a39e394d334}} is the data sample index and {{formula:af92d712-3532-40f1-9a96-92b7e6f58151}} is the initial value. Figure REF illustrates the effect for the initial value {{formula:1f7cf406-9b60-4bb0-ab60-625b9f96bb29}} , and it is remarkable that an improved result is attained for {{formula:f95d3df8-290b-4dd5-aa3e-f9d4c8de584e}} .
{{figure:827cc53a-9679-4baf-9d7c-ffdee48f37bf}}{{figure:36f0284b-9500-4106-af38-2f0aa548d76f}}
Discussion on the complexity of the proposed approach
In this section, we discuss the computational complexity of the proposed WSM Det-Max neural network implementations. For simplicity, we consider the antisparse source separation cases discussed in Section REF . Remarkably, the overall complexity is due to the output computation complexities which are determined by (REF )-() and (REF )-(). Note that these differential equations are naturally solved in neuromorphic implementations. However, in digital computer simulations, we need to implement loops to obtain their iterative solutions, as summarized in Algorithm . As described in Section REF , assume that there are {{formula:495be4ac-9dd7-4947-b0b4-9ee9330cc6d3}} sources and {{formula:c71513d2-f64a-4a72-8605-e376979af7a0}} mixtures, i.e., {{formula:c67cd263-e6c0-48e3-a798-ba09c411622f}} , and {{formula:6f40851c-c8a7-4cb0-8b1a-8a06420988e4}} for all {{formula:b14ebafa-8e7f-4cf9-8bb2-2ad076d3e85b}} . Assuming that the factors {{formula:ec9979c8-1244-492b-8179-a25f42222d16}} , {{formula:d5cf335e-1212-42f0-9174-dfcef73cb4f3}} , {{formula:37c52ecd-48e3-4eab-99cd-26f60404bf78}} , and {{formula:249c9783-6cf8-4c64-8e3f-f53960b36e48}} are computed outside the iterative loop of Algorithm , the expressions in (REF ) and (REF ) require {{formula:6366a096-e151-48b3-b156-8cd89f2a6c65}} and {{formula:f1ca6a4d-3607-4b8c-8514-bf5efc5b819f}} multiplications, respectively. If we assume that the neural dynamic loop reaches to the pre-determined maximum number of iterations {{formula:786f1bab-90f9-4c44-b71c-248aced8ded5}} , i.e., the numerical relative error check for the convergence is not satisfied, then the total number of multiplication is dominated by the factor {{formula:fb15e822-da6f-419f-9f38-21b6c9de6e4b}} . If we analyze the computational requirements of the factors {{formula:33df761b-8c6e-4246-8b74-5e15cd22c842}} , {{formula:926f45fe-e406-4bad-9bed-c2e41a601366}} , {{formula:2b500215-57ee-4520-a733-7fd7dbdda7b5}} , and {{formula:fb036200-5585-43df-9382-931934e478c5}} , these calculations require multiplications of {{formula:c51eafde-8d02-41b6-95ab-eeb67bbbb074}} , {{formula:211e3b5b-7b28-4a7d-8e48-dd3c0dc9621a}} , {{formula:babf400f-fc1d-4774-b735-b46cd533c637}} , and {{formula:c8cb10f1-cdb6-440f-8efe-57fd226142df}} , respectively, since {{formula:7a3a31f6-68c1-4160-a00c-eb788ecf98c0}} and {{formula:65e3e871-e009-4070-91bf-c1b47bb44428}} are diagonal matrices and {{formula:63c7e34a-833a-4bae-a0ea-8031eafa31c2}} are symmetric matrices. Therefore, the complexity of the neural dynamics of our proposed approach is dominated by the factor of {{formula:333e5263-8184-49d4-b6c1-1e3236059efc}} . The complexity of the update rules of the gain variables expressed in equations (REF ) and (REF ) is dominated by the multiplication factor of {{formula:c6db1078-cca9-49f5-af02-92323e43f621}} for all {{formula:3753bc49-18c2-4fb5-a72c-da9f5a50a5cc}} variables, leading to the dominant multiplication factor of {{formula:b3e1bdbc-da84-4e3a-9fd1-fca305040ae9}} . Moreover, the update rules of the synaptic weight updates expressed in equation (REF ) are dominated by the multiplication factor of {{formula:f6eb604a-186a-4ba8-97ad-a560e306a7dc}} or {{formula:c9142274-75bb-4dde-9e3c-267dd56b0fd1}} , leading to {{formula:c417e8e7-9d0a-445b-acf3-c51469d1dd40}} number of multiplications. Therefore, the worst-case complexity of our proposed method per sample in terms of the big-O notation is
{{formula:58a3c921-0014-4c1c-ae7c-39769ac0e8ca}} .
We now compare this with the complexity of the NSM and BSM algorithms. We first consider the prewhitening layer introduced in {{cite:1b96180a12c0ce5e5f760fe5f8ee1bccc78470d3}}, as both algorithms require input to be prewhitened. Taking into account equations (28), (29), and (30) in {{cite:1b96180a12c0ce5e5f760fe5f8ee1bccc78470d3}} for output computation and synaptic weight updates of the prewhitening layer, the complexity can be expressed in terms of big-O notation as {{formula:bbd9979b-4dd1-4938-9674-e201a511322c}} , where {{formula:1b6c1bca-902e-44d0-a6a4-0f374bf2ec46}} is an integer introduced as a result of the Lagrangian multiplier in equation (12) in the reference {{cite:1b96180a12c0ce5e5f760fe5f8ee1bccc78470d3}}, and {{formula:cde05804-4327-44a7-be5a-f620fbfeea21}} is the maximum predetermined number of iterations for the neural dynamic loop of NSM (see equation (28) and (33) in the reference). The output dynamics and the synaptic weight updates of the second layer of the online NSM network is described by the equations (33), (34), and (35) in {{cite:1b96180a12c0ce5e5f760fe5f8ee1bccc78470d3}} which lead to the complexity in terms of big-O notation of {{formula:1d1b6e2c-1308-4ccd-91df-135a8b3e4e19}} . As a result, the overall complexity of the NSM algorithm per sample can be stated as {{formula:ebc62c91-bc90-4af3-ac2f-b01fec510861}} . Similar to the WSM network, the neural dynamic loop of BSM has the complexity of {{formula:5b2b46d6-e38c-42ad-9410-8ec59c600b15}} as a result of recursion defined by Equation (17) in {{cite:658e0223e6cf55fa7a0217b97a214fc2c7a89d5f}}, where {{formula:f5e004a7-404b-414c-b185-1e58b6c4f602}} is the predetermined maximum number of iterations for the neural dynamic loop of BSM. Furthermore, synaptic weight and gain updates introduce {{formula:daef205c-4518-4132-940f-2927f14b2553}} complexity similar to the WSM algorithm. Therefore, combining with prewhitening, the overall complexity of the BSM algorithm per sample becomes {{formula:22ead9b8-f2b5-4d56-a059-8f1bc357a47d}}
In conclusion, for the biologically plausible neural network solutions to the blind source separation problem, the overall complexity is determined by the recursive neural dynamic loops due to the implicit definition of the network output. Although this condition makes the implementation of such algorithms less feasible for digital hardware, they enable low-power implementations in future analog neuromorphic systems with local learning constraints.
| d | 8da6de058142baf7bef0382dffaf744b |
We show that this bound is tight. We further prove that for any number {{formula:990524cd-6183-4e0d-9fc1-7ea52fa5137c}} of ex post constraints, a stronger tight bound of {{formula:e63e5c4d-8573-490c-8909-4ffd2f0caa23}} holds, just as in the unconstrained setting of {{cite:d733cba82b23fcd81467fa0e884b9bced3d8bbee}}; the same proof outline yields an alternative proof to the result of {{cite:79c36abbe7f6437cdf5a6c751b989841020a8d79}} on ex ante constraints, as shown in Appendix .
| r | 824f4e79bc0b61d8884bb31a4ab2a6b6 |
In this work, we have assumed that the stoichiometric matrix {{formula:76a78d47-b500-4403-a0cf-a931425f1599}} is regular.
This implies that the system can always relax to a chemical equilibrium state when the volume is fixed, i.e., in the isochoric situation {{cite:4001823179a7b048a1972459e285db7638c5761a}}.
In other words, the system never reaches a state that continously produces entropy with constant volume, namely, the conventional nonequilibrium steady state (NESS) {{cite:e1a3a8b3f7a9b14a83c8f3cf486610e60d2cd591}}, {{cite:703519e1473c714b42552aa90671df71566c2afa}}, {{cite:f1c1d22335563f8e06556e5657fc8dbbc023ca36}}, {{cite:66d7280807c3d687aab1dda48b519a4d182d80aa}}, {{cite:1425079e73583697465b16f5ab1f15fb435e2f1a}}, {{cite:c72cb15f9ac88607b45d4c0e837706a977078469}}, {{cite:474cb47f91abaa4f15f83ccaa9927a0b5ccf3236}}, {{cite:dc6ee1a7ce6b87f543f0b2247e76e5aa956a4b84}}.
Accordingly, the nonequilibrium states treated here, notably the steady growing state, are realized due to the change of the volume.
This nonequilibrium state with changing volume originates in the extensivity of thermodynamics and should be distinct from the conventional NESS with constant volume.
| d | c18a3274220a52ee254378871e67e62b |
We now turn to results about automatic sequences.
This is a large and interesting class of sequences where the {{formula:4db8d052-56be-4180-bf74-e023e63c2938}} th term
is computed by a finite automaton taking as input the representation of {{formula:aac6f2e8-dd40-4dcf-9f86-96e4a7a80665}}
in some base (or generalizations, such as Fibonacci base).
For more information about automatic sequences, see {{cite:039802dad983d6870ce05db5cd784d787f9ad188}}.
| r | e2ef1705ce9ac6cbbe2d80ca4f6c2912 |
The current work reports for the first time the NuSTAR detection of a low-frequency QPO at {{formula:2d665e41-8577-4099-8d49-fcf0736fd751}} Hz in the LMXB GRS 1716–249. Earlier studies from the 1993 outbursts of GRS 1716–249 detected a QPOs at {{formula:a5ffe438-b64e-48ab-bb26-2b9d73d314da}} 0.04 Hz which slowly drifted to 0.3 Hz at the end of the observation {{cite:97c12ce1a45786851a6c20eae902033ddf0b1594}}, but there were no QPO detections at frequencies {{formula:8fdced9a-4d91-4784-80a4-ad88c244e53f}} 1 Hz.
| d | 915730398cedd9d67ccb7206e424bbfb |
All the code in these experiments was written in the Julia language {{cite:36a5b7fe2acbbaf44a1458a3b556c04c667d53ac}}, as part of the open source jInv framework {{cite:70d70fb8e257dc9dcaa0808908f7d78c012b69a5}}. Some critical parts of the code, like the LU solver (forward + backward substitution) and matrix-vector products for the forward modelling is written in C++ and is parallelized using shared memory OpenMP. The experiments were computed on a workstation with Intel
Xeon Gold 5117 2GHz X 2 (14 cores per socket) with 256 GB RAM, running on Centos 7 Linux distribution. Our code is available online at https://github.com/JuliaInv/jInvSeismic.jl.
| r | a25c1339f92e950a07d7fd31489339d9 |
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