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In this paper, we showed how the framework from {{cite:00ebb13a41f057f181337e6534e55ce0061c872b}} allows us to recover the individual sample and subset bounds from {{cite:18fa3ffc7b92bf5185861df29a08789371a85376}} and {{cite:986de244cbbb2d601c17429c1fa3e5ebd79aae80}}, as well as to generate their parallels in the randomized subsample setting from {{cite:be7220ae071474abc131a234b5c6e451ffcafd45}}.
Moreover, we showed why the framework does not allow us to obtain bounds with all the expectations outside the square root, such as the tightest bounds from {{cite:986de244cbbb2d601c17429c1fa3e5ebd79aae80}} and {{cite:7b15974fb96436e91df99fe5e09d4b6aee9bccc1}}.
Finally, we extended the LD bounds from {{cite:7b15974fb96436e91df99fe5e09d4b6aee9bccc1}} to SGLD and we refined them to loss functions with potentially large gradient norms.
| d | 78613c18420bbed98d460a58eb6fbdfd |
In the limit of MPS = 1, our model transforms to an active lattice gas model without any on-site alignment interaction and we observe a variety of self-organized patterns which show the coexistence between dilute and dense phases typical of MIPS {{cite:7c479e3dec0cf8b17fafcf2886e912712d9a19f2}}, {{cite:cdd81b4a382593f6dec5faf821c01c420b66ef9f}}, {{cite:ecb5c860ec6c0e9a8d31135ddf20407d951d4d91}}. At very small activity, however, the system remains in a gaseous phase at low densities and exhibits a homogeneous aggregate phase at higher densities. The emergence of such diverse jamming patterns results from the interplay between the local density, local orientation and particle speed although we do not observe traffic jams, gliders, or bands reported for a similar kind of model {{cite:29ed3a138ad881bdb080dbc0019c28a0a1f9c993}} due to the absence of nearest neighbor interaction. We further characterize the system by the liquid and gas binodals with Pe = 8 as the critical Péclet value, and the resulting MIPS phase diagram agrees qualitatively with the MIPS phase diagram obtained for the active lattice gas {{cite:541e10a9bdb9c692341ab10167b61115f744bcbc}}.
For hardcore and softcore repulsions, the APM local alignment interaction is present and apart from the three prevalent phases of the APM (disordered gas, polar liquid, and liquid-gas coexistence region), the system exhibits diverse self-organized patterns as functions of different control parameters and volume exclusion effects. These patterns include locally ordered high-density traffic jams, MIPS, high-density homogeneous aggregate phases bounded by well-defined domains, and self-segregated clusters. Through several phase diagrams, we provide a complete description of the system and demonstrate its large-scale behavior where the jamming of particles plays a predominant role. We reach the following conclusions from the phase diagrams that connect to the jamming transitions in several other systems {{cite:8567f7e39f54d01acb196c45d774ea66d49875c5}}, {{cite:b015feace9fe9df45a4561e3ed10836a8bf51d0f}}, {{cite:6650d62cb5f1ab49e18f5f5c8e5f7ceeefe0759b}}, {{cite:a120460567220de37661c7622c14352b3813690a}}, {{cite:8a0b093c36dd35e5af6cd4802a6e8e41d95d0a1c}}, {{cite:efb7221a921f47682920f4682637bbf182eb473a}}, {{cite:68fcd5fce9e6cf239995427218c35030f2488ad8}}, {{cite:541e10a9bdb9c692341ab10167b61115f744bcbc}}: (a) increasing average particle density leads to overcrowding, promoting a jammed phase, (b) increasing particle self-propulsion velocity also enhances the possibility of jamming due to small hopping rates in the non-preferred directions of motion, and (c) thermal fluctuation facilitates dissolving a jamming phase via more flipping of the active particles.
| d | 47198631bd22684704483e9f692aa858 |
paragraph4
.5em plus1ex minus.2ex-.5emMini-batch statistics in inference can help reduce
train-test inconsistency.
To our knowledge, applying this technique in R-CNN's head has not
appeared in previous works,
but {{cite:3ff430ad04a76ec2bf374a80ef77d562a54be80f}} also observes that population statistics perform poorly
in R-CNN's head.
| d | 4864d8d671251ed7471a67157cb61e26 |
Acknowledgments. We thank Joel Primack and Matthew Reece for useful discussions. We thank Vivian Poulin for permission to reprint figure REF (left) from {{cite:cd8eeca7bc7ddb3a5f0ceaefccc0e38a7d00c898}}. We gratefully acknowledge the hospitality of the University of California, Santa Cruz, where a portion of this work was completed. This work was supported in part by the Berkeley Center for Theoretical Physics; by the Department of Energy, Office of Science, Office of High Energy Physics under QuantISED Award DE-SC0019380 and under contract DE-AC02-05CH11231; and by the National Science Foundation under Award Number 2112880.
| d | 108eb2f1efc287a7b5616ad2fe095226 |
Let us record here an important corollary for the theory of quantum information. The partial transposition criterion {{cite:8a6967c3b32ee3b065bf3bc9c2612b4d99544886}}, {{cite:35a046778a830786fe08145635152808340178a5}} states that a separable (i.e. non-entangled) density matrix {{formula:89f81f14-014e-4741-982e-f8944343741e}} has a positive semidefinite partial transposition: {{formula:7fa86009-0c1a-4e65-aa11-7bcdb80665ba}} . This fact is mostly used in the reverse direction, as an entanglement criterion: given a density matrix {{formula:c9f7791f-f909-4331-91b8-59f21fb3dbc9}} having a partial transposition with at least one negative eigenvalue ({{formula:ba415032-5031-4aa1-9c03-6a2b524b00c2}} ) is entangled. We have thus the following result, regarding the bosonic induced ensemble defined in Definition REF .
| r | 0ef42eeea646eba5da161fe4c446da2f |
As shown in the experimental results, the distributed control algorithm is able to accomplish formation tasks, even when these
are not considered when generating the training data. Similarly, the trajectories used in the training data were generated using systems with different agent dynamics and communication graphs, and not specifically the ones used in the tests. The aforementioned attributes signify the ability of the proposed algorithm to generalize to different tasks and systems with different dynamic models and communication topologies.
Nevertheless, the proposed control policy is currently restricted to systems of the form () satisfying Assumption REF and is not able to take into account under-actuated systems (e.g., the acrobot or cart-pole system) or systems with non-holonomic constraints; the analysis of such systems consists part of our future work. Finally, the discontinuities of (REF ) might be problematic and create
chattering when implemented in real actuators. A continuous approximation that has shown to yield satisfying performance is the boundary-layer technique ({{cite:76f6f894b9e35c7867ad9151807954d16fb32487}}).
| d | 76ca0fb5f1faec48162f314a550b831c |
Based on the Chapman-Enskog expansion of the LBE, an evolution rule is applied to every distribution function {{cite:94d491e55811e0099d710c08585cdf16452a916f}}:
{{formula:63b108d5-4340-436f-b842-546f73a4de68}}
| m | 9223a02f1690fa3a87c3a5313567d14e |
Deep neural networks have achieved remarkable achievements in various machine learning fields, including image classification {{cite:ceb46b6b4cd3c9d16b9beb4fe5a5f9c8bfdf0299}}, {{cite:04d11713d459cd954fe713ee7138e042ac671c92}}, semantic segmentation {{cite:4422e1234dbc0ab19e112fb06e0d828f0f80fe78}}, and object recognition {{cite:c4152b8da2724469709607f99235e480f238c76a}}. However, recent achievements heavily relied on large-scale datasets under the major assumption that the training and testing data have a consistent feature distribution. While in practice, the data collected from various scenes may suffer from different lighting conditions, color saturation, hue, visual angles, and background environments, resulting in domain gaps between training and testing data. Aiming at the above problems, domain adaptation can map data from different distributions of source and target domains into a feature space, and optimize the corresponding model parameters to make the distance of features from different domains as close as possible in this space.
According to the existence of labels in the target domain, domain adaptation can be divided into supervised domain adaptation, semi-supervised domain adaptation and unsupervised domain adaptation (UDA) {{cite:dd15c3839975ffb27a9273095f7db3e9def0fe51}}. Since data labeling is often time-consuming, labor-intensive, and expensive, obtaining high-quality labels has always been an important factor limiting performance improvement. Compared with supervised and semi-supervised methods, UDA does not require any labeled target domain data and has more potential application scenarios, which deserves further attention. Therefore, in this paper, we focus on exploring UDA method aiming to transfer knowledge from a labeled source domain to a different unlabeled target domain.
{{figure:20da0326-e167-4a8b-8ab6-c3416b4374f0}} | i | 16490110311673fd61184e6da443023c |
In this section, we validate the performance of the proposed denoising algorithm and compare it with several state-of-the-art denoising methods, including BM3D {{cite:4c43d77d0690ddda0c74e45b3eb11afd913d4aca}}, EPLL {{cite:e9854319a1fef0d69f0f835ddaf44938d559e51f}}, NCSR {{cite:ccec717c4ab3f96e66f522138f8e387cf3910a83}}, GID {{cite:09f692b8c6f13570cecb12a2ca8cb794093c9fb9}}, LINE {{cite:72954b85230b67bb6bd268cb2957c68f9afc4003}}, PGPD {{cite:80dfac5e285c27cd255b68a2a62ef6f64f539a11}} and aGMM {{cite:0a378e514f95a615edb417dc728f29ff05efe123}}. We evaluate the competing methods on 14 typical natural images, whose scene are displayed in Fig. REF . The training groups used in our experiments were sampled based on NSS scheme from the Kodak photoCD datasethttp://r0k.us/graphics/kodak/.. The detailed setting of parameters are shown in Table REF , including the number of Gaussian components {{formula:86ac39ba-9782-44b5-950b-288666b54390}} Since possible patterns and variables to learn can be reduced, the number of Gaussian component is not necessarily large., search window {{formula:551d3e15-fc0a-4a21-871c-e32294b913fb}} , the number of similar patches in a group {{formula:fbeed7c6-c905-48fd-929b-7fd701657e72}} , patch size {{formula:2a1014b3-441c-43cb-a698-7e540646d934}} and {{formula:05af35b1-4aae-4e83-916f-15055f12e27e}} , {{formula:473dda4f-b3c8-493c-8d11-17796c8a6b1b}} , {{formula:91b22040-cbb2-4e76-892b-8b4ca4a87863}} .
{{table:d8754bdd-ff4e-4f7a-bd80-65c9fb29c730}} | r | 0ee8833065a5ade581a67ea6c64569f6 |
Fig. REF shows signal-to-distortion ratio (SDR) improvements of all methods, which were computed using the BSSEval toolbox {{cite:dd316d6938a16cf24fc9f0370ee58d8877a72c34}} and averaged over the 50 test mixtures for each instrument pair.
Although the SDR improvements of FSCM+DNN greatly varied with the instrument pairs, Gauss- and EB-IDLMAs worked robustly against the instrument variations.
The proposed EB-IDLMA provided higher SDR improvements than Gauss-IDLMA for all instrument pairs, showing the effectiveness of considering the reliability of the estimated source model.
As shown in Fig. REF , the estimated degree of freedom parameter {{formula:bdfb2596-98ea-4893-a702-98790d5c8881}} varied with frequency, which means that the estimated source model {{formula:22f6a390-490e-4496-bfbf-24a30cf2f878}} was less reliable in the lower frequency band.
This should be due to the fact that the spectrograms of Ba. and Vo. markedly overlapped in the lower frequency band.
| r | fbcca07dd08d80d717a3ad455eb86537 |
Table REF shows the percentage of analogies rated as meaningful, based on majority vote, for the various models and the human references from saqa. There were <2% ties or cases with `Can't decide' as the majority, which were discarded. The Fleiss' kappa {{cite:70a1780265b82d163fd8387cb18cc806f5090b1c}} inter-annotator agreement was 0.347 in case of wsrc (plus human references for the selected concepts for wsrc concepts), indicating fair agreement and 0.553 in case of no_src (plus human references for the selected concepts for no_src concepts) indicating moderate agreement.
| r | 96f8678aebc823e66b0112ae144ee540 |
Decreasing the T-count of addition reduces the T-count of any construction based on addition.
For example, in {{cite:6bf8125029c82fdff3a493c9f68602f487cf2982}} it is estimated that factoring a 2048-bit number on a surface-code-based quantum computer would take {{formula:8d24cb2e-89c0-4bf3-a6a0-14689ef2dead}} distilled {{formula:b12b7aaf-aac6-4b62-8c3c-ec8d6399fff4}} states and have a measurement-depth spanning 27 hours (though the actual computation would likely be bottlenecked on distillation rather than measurement).
The time estimate assumes Toffolis have a measurement-depth of 3, and the T-count estimate assumes Toffolis have a T-count of 7.
Shor's algorithm is dominated by the cost of controlled additions.
With our techniques, the average T-count and measurement-depth of the relevant Toffolis is {{formula:343f5a4b-6395-4ed4-8609-7bf03626705a}} and 1 respectively.
With these numbers, the measurement-depth estimate is reduced to 9 hours and the T-count estimate is reduced to {{formula:432b57c5-a53d-4599-a421-718b9dd88e12}} distilled {{formula:8c7ea903-5f47-41dc-ba0d-7d15486c4261}} states.
| r | 414fd09b31c7ab929c2fbc853d7b075e |
A perfect example of this blending is represented by CNN-NLM {{cite:abda0503f02e75ac1c64d347c3e55099e14f2856}}, {{cite:4a9d0cb8eb823a363d9e8294cf028ff91e978fe5}}, where despeckling is carried out by nonlocal means, a simple and well-understood linear filtering algorithm.
The clean target pixel is estimated as a weighted average of neighboring noisy pixels, with weights that depend on the similarity between target and estimator.
In CNN-NLM the similarity metric is replaced by a suitably trained CNN.
The network takes as input a patch extracted from the original-domain image, and outputs a set of filter weights, adapted to the local image content.
In {{cite:abda0503f02e75ac1c64d347c3e55099e14f2856}} a rather standard CNN is used with 12 convolutional layers,
while in {{cite:4a9d0cb8eb823a363d9e8294cf028ff91e978fe5}} a 20-layer CNN is proposed which includes also two {{formula:26fbf86a-1c8b-450c-9142-c677e8117567}} layers proposed in {{cite:8ec3fd0577f8d016b019246a7d9ea4c62e94303e}} to exploit image self-similarities.
These layers associate the set of its {{formula:d7acd66c-8fd1-4271-9adc-796354a14511}} nearest neighbors with each input feature, which can be exploited for subsequent nonlocal processing steps.
Training is both on synthetic data and on real multilooked SAR images, like in {{cite:6ab483242cd9ab14adf854d92bb61584e2369ceb}}.
Results are much better than those of conventional nonlocal methods, like PPB {{cite:2771a139d4155952d5a7dccd344873eb5d168c62}},
which provides some hints on how the filter weights should be chosen given the underlying signal and the noise strength.
Moreover, the performance matches that of state-of-the-art CNN-based methods, which is quite interesting, considering that the filtering engine is fully linear.
The fact that, despite the non-additive nature of the noise, a linear filtering method can be competitive with highly nonlinear deep networks may deserve further studies.
| m | d3fe9c16823218da5532b589bd66aef2 |
A major challenge and limitation for utilizing deep learning in FM has been the difficulty in establishing standardized staining protocols that would enable more homogeneous marker combinations to train supervised models.
With our methods proposed herein, a single model is shown to perform comparably or superior to a number of individual problem-specific models that would be infeasible in practice due to the exponential growth in model parameters and training time with an increasing number of markers.
In addition, the versatility of our methods enables them to be easily applied to different network architectures for tasks beyond semantic segmentation, such as classification {{cite:e98593f7a7981d678052f7b7041079a221469401}}, {{cite:aa41b33016061681b12a6d1ba1c36b5095ac0127}}, detection {{cite:799bcc945580f3f0ece341d2dae8afeb423f602c}}, {{cite:9a8d3792cbca3cf301298420b36a4929e638dff9}}, or instance segmentation {{cite:6b09052fb7738925fb5841f2ce0a9d911359de97}}, {{cite:b132139ff8370c1eea229f8ebaf43e13f650123c}}.
These contributions can facilitate the sharing and exchange of trained CNNs across labs in the field as well as a faster adoption of neural network solutions in routine lab work at, e.g., microscopy facilities.
| d | a645e66a5dacabc1eec400df63720726 |
To evaluate the performance of the designed network, we resort to the DeepMIMO dataset {{cite:ac4284c3d6a16cae3fd9337f40abc74fa68ae787}}, which is widely used in DL applications for massive MIMO systems.
The outdoor ray-tracing scenario `O1' of the DeepMIMO dataset is adopted and the BS 1 in the `O1' scenario is set as the ULA with 64 antennas.
We summarize the simulation parameters in TABLE REF .
To generate the channel dataset, we first fix the user velocity {{formula:a6e257e3-8711-416c-9ff2-49967379cdf5}} .
The Doppler shift is {{formula:1f16adb9-b04d-4e60-9b49-7f81a6a032ba}} , where {{formula:0be7fc0d-15bc-4b7b-b8a3-2464a790b1dc}} denotes the angle between the user movement direction and the {{formula:ed4e8d5d-b91d-4fd5-8e49-bf034c35f24f}} -th path.
To reduce the size of the dataset and save the memory, we select {{formula:d5af020b-5ce2-4ed4-80fa-8c376ac76422}} .
Then, the parameters {{formula:05f10cac-47d1-485b-8e37-24008b49a3c0}} , {{formula:d40f3e32-773d-431e-93c9-424e0b1d4fda}} , {{formula:5c409b11-7e46-41a5-a59e-de6a1d7bea9b}} and {{formula:ca4bf564-055e-4465-94bc-a3515e4cc844}} of each user can be obtained form the DeepMIMO dataset in the `O1' scenario.
With the parameters {{formula:1e09c559-0228-439e-8eba-b67bcde51d05}} , {{formula:95917d0b-04c0-4299-ad4c-f114a03fe62b}} , {{formula:5f091afb-d808-4569-8cd6-076c4d1ede00}} and {{formula:2f85f453-4639-47bb-a21d-fc4a98be5aaa}} , we can generate the time-varying channel sample of each user according to (REF ) for {{formula:0bebc89c-e95f-42e4-9db2-f5fe905337bd}} .
We select the users located in the region from the 401-th row to the 510-th row.
Since each row in the user's region contains 181 users in the `O1' scenario, the total number of the channel sample is 19910.
We employ 80% of the dataset for training and the rest for validation.
Specifically, the dimension of the hidden state is the same as that of the latent state and is set as {{formula:50a1e21d-89cf-497f-9bcf-f51d9b906f9c}} .
The total epochs for training is 1000.
We set the initial learning rate as 0.004 and halve it every 50 epochs.
The AdaMax optimizer is adopted for network training with batch size {{formula:fc21052d-6177-4390-8c9e-7ec29ec06d94}} .
To evaluate the channel extrapolation performance, we use the normalized MSE (NMSE) {{formula:dcfc9327-e242-4603-a0c8-4aacb2279779}} as the metric.
| r | 798de5103e8515f02695b134da49be0c |
Uniform rectifiability is a stronger quantitative analogue of rectifiability (every UR set is rectifiable but not every rectifiable set is UR); it has found applications to singular integrals {{cite:d29559d57530e9f9e324ab9756aedd77934b9000}}, {{cite:431eff7d225ad2c99c62cd62a797b107ab15bac4}}, {{cite:65f9be55e02a7aef728b15f06945526dce258ff2}} and harmonic measure, see {{cite:e0b2e0f0ec99de22893f1ff2eb888a153c4dc87f}} and the references therein. When David and Semmes defined uniform rectifiability in {{cite:d29559d57530e9f9e324ab9756aedd77934b9000}}, they proved many equivalent conditions. One important result (now a black-box in the theory of UR) is a characterization of UR sets by the Bilateral Weak Geometric Lemma.
| i | 300dd4c50946318cbbb2eb686c7a1719 |
Let {{formula:b4c48f4c-c825-4cfd-89cb-16491783dd86}} be an input space, e.g. images, and {{formula:4236fb80-c080-46bc-a41c-470963d64fb1}} a label space, e.g. classification labels. Let {{formula:5640c949-cbce-47c8-926c-9c88978ecc51}} be the data distribution over {{formula:9afb01a3-636a-49cb-a86e-05c7e758e3ed}} . A model, {{formula:3cb99fb7-4949-4737-89b1-ceb79e887b5d}} , is called a prediction function, {{formula:10bafe87-4cbe-4b8c-aa7e-5e3e48e93556}} is a given loss function. Given a labeled set {{formula:bb8a8cb0-f9dc-4b5a-8684-7f8465a34870}} sampled i.i.d. from {{formula:5ef99c70-a679-46d9-a7fe-53bfd548da60}} , where {{formula:2c1e58ab-4281-42a7-bc63-883b72b3fedc}} is the number of training samples. According to {{cite:2296c2e224b95470ddb7c2308833358de52706b4}}, the true risk of the prediction function {{formula:8a35b4ed-4b61-4c26-8db2-63b68ffa0a69}} w.r.t. {{formula:1771f058-22ab-4521-9d69-82e8be1159e8}} is {{formula:9c0e71c3-d35f-41c0-b972-7c3a635f19ef}} while the empirical risk of the prediction function {{formula:e4e687e3-093d-46a6-818d-2c8423f7a06b}} is {{formula:c95ab80a-632c-437c-aa8c-16ecb32d38cf}} .
| m | fee8da38f5f8ea376e61d461259a585b |
Instant social video sharing based on the combination of online social networks and user-generated video streaming, has rapidly emerged as one of the most important social media services for users to access contents online {{cite:5927c6b7801ac74a5827e9b1d87ca085e44a2b30}}. A fundamental reason for the popularity of social video sharing is that it satisfies the users' inherent interests in sharing video contents which are generated and uploaded by users themselves {{cite:9afc80a0335d24cf63cf5e59c38e4c2846a619d1}}, with their friends {{cite:baad4cc9b988f9279417234f4e578e883f6facea}}. When viewing such user-generated videos, other users need to download the media files from servers over the Internet.
| i | effdc88c6b0a51fe3e0533d9633a4a75 |
The ImageNet dataset contains about 1.2 million high-resolution natural images for training that spans 1000 categories of objects. The validation set contains 50k images. We use Resnet ({{cite:7cdfe60c5732e5c8465ebc25a1b19a3b187a1926}}) as network topology. The images are resized to 224x224 before fed into the network. We report our classification performance using Top-1 and Top-5 accuracies.
| r | 09923f78903ed0ae8965aabaccee4e2b |
Foreground and Background Evaluations:
We compare the prediction performance separately in the foreground and the background regions of the scene in Table REF and Table REF on KITTI and Cityscapes, respectively.
We use off-the-shelf semantic segmentation models to extract the objects in the scene and assume some of the semantic classes as the foreground. Specifically, we use the pre-trained model by {{cite:c0a7ea678dbca6c05557ceb1fe970459c88a99fe}}, {{cite:42f19a13c47023aea3b761d1f17554676edbbbd0}}, which is the best model on KITTI semantic segmentation leaderboard with code available (the second-best overall), to obtain the masks on this dataset. On Cityscapes, we use the pre-trained model by {{cite:d1aae5b5481f7f64d2ee85198a430c912a426e07}} for similar reasons. We choose the following classes as the foreground objects: “person, rider, car, truck, bus, train, motorcycle, bicycle" and consider the remaining pixels as the background.
For foreground results, we extract the foreground regions based on the segmentation result and ignore all the other pixels in the background by assigning the mean color, gray.
We do the opposite masking for the background region by setting all of the foreground regions to gray, and calculate the metrics again.
| r | 8db1a1a89428c243ef54aa85a1034851 |
Machine learning (ML) has become an extremely prevalent tool in the classification of hadronically decaying highly Lorentz-boosted objects ("boosted jets") and to study the internal structure of hadronic jets ("jet substructure") {{cite:3e44d393dbe684ce4af6b33373c88d6bdfb6c4a1}}, {{cite:3c0e81c0d477ddb7873d6a0fa911a99f52dd542a}}, {{cite:217141d8be5008f49fa7b380a2503fd1d125b899}}, {{cite:ca778a3e5b2d0e62cd206fa43d7c6bdaa6789fd0}}, {{cite:4f4517e0b64a4b1ffe39c1d694fb483067e02910}}, {{cite:68ea49efa2c8312c97861a63ff617639c72e5a50}}, {{cite:33036f361be1876991b4baab91a1a8de025fc53e}}. In these areas, classification tasks are common, for which artificial neural networks (ANNs) are well suited. Recent work in deep neural networks (DNNs) has shown tremendous improvements in identification of boosted jets over selections based on expert variables (see Ref. {{cite:4f4517e0b64a4b1ffe39c1d694fb483067e02910}} and references therein). These algorithms typically make use of classifiers based on convolutional neural networks (CNNs) and recurrent neural networks (RNNs).
| i | a30a9857a3172b8965eafb9a3d4c8757 |
The two-level optimization scheme makes the dimensionless learning very flexible. The first-level scheme guarantees the dimensional invariance (or called dimensional homogeneity), and thus many representation learning methods can be used for the second-level scheme to capture scale-free relationships. We demonstrate polynomial and tree-based XGBoost {{cite:457c4b7c840d0488ab2cdf97fc3c4784b70f5f91}} methods in the previous sections. However, the capability of the dimensionless learning can be improved by leveraging more methods, including deep neural networks {{cite:ed5b74ebb091e0680348fcbddd9ff28196ff9988}}, symbolic regression {{cite:76c5a2d225b768d2f9bba5e1fcabe72f86177601}}, and Bayesian machine learning {{cite:8dccab33b51e5df2555e17e8a94c0ddb79602a7b}}. Another highly promising candidate is the sparse identification of nonlinear dynamics (SINDy) {{cite:6c5cf6eff6f93decd9ef2b5537017f0c3c00b4a9}}, which can identify ordinary differential equation (ODE) or partial differential equation (PDE) from data. By integrating the proposed dimensionless learning with SINDy, the forms of dimensionally-homogeneous differential equations can be discovered directly from data, which provides even more insights and interpretations of the physical system. We demonstrate this using a simple example of a spring-mass-damper system in Supplementary Information Section 5. Using the proposed two-level optimization with SINDy to analysis the time series of the spring-mass-damper system, we can obtain not only the governing equation, but several key physical parameters such as the natural time scale (i.e., the inverse of natural frequency), natural length scale and dimensionless damping coefficient (Fig. REF ). The dimensionless form of the governing equation involves the reduced number of variables from six to three, which is the minimalistic representation of the system.
{{figure:7cd2d536-d856-4240-9058-eb7ab132fad3}} | d | 3c37506f1c34c2ecff34c37960cdc958 |
For the sake of completeness, we also present two simpler but more frequently used quantities in the supporting information (SI): the isotropic and anisotropic components of the interaction polarizability in the {{formula:52c62b5c-7873-4dfd-a9ba-9e9bdd42a226}} frame.
It has been shown that fourth order perturbation theory effects are important in the accurate description of field induced electric properties.{{cite:ee6d6da6fcf31cb3edc2c3d9220bee86cbcdf054}}
We therefore choose the coupled cluster singles, doubles, and perturbative triples [CCSD(T)]{{cite:95bd3e1d75979eeb8336325e629629517e4c9230}}
method for reference. The coupled cluster reference calculations are performed with the Molpro quantum chemistry software{{cite:2bf0278bd452788cbf5cadcddeec819fd69d3b82}}).
Since the basis set convergence of CCSD(T) is fairly slow and comes with high computational overhead even for relatively small systems, we use the explicitly correlated F12b method{{cite:75cc7ab3b5bcb2fb4ca41cf954658a3dc3eb6f2d}} [3*A(fix,nox) variant as implemented in Molpro]
with the aug-cc-pVDZ-F12{{cite:5ccfe5181d4bc24335e43a4a84dcf4f1701b94ba}}, {{cite:0d046ba7da5f6fff1bffba982d3f5e9a3a4ed5e0}}, {{cite:fbc5c35f2ce515f2902abc0a6194ff3f0969cc86}}
basis set and the aug-cc-pVTZ-F12 auxiliary basis set.{{cite:fbc5c35f2ce515f2902abc0a6194ff3f0969cc86}}, {{cite:366ec2fb29f009bbc108921739c7f45fc229d32b}}, {{cite:89d3d45fc3a6277b276433a42e2ba6055ec70722}}
This level of theory has shown to yield a reasonably small interaction energy error (0.048 kcal mol{{formula:9ff6ea64-9e7d-4e0e-8b33-a007ad86a6f1}} on average) at zero field strength for the benchmark S66 energies.{{cite:739efba77994272e07d2637ed67513d4253f255b}}
| m | a36d40f8a463ae8f4b9b1155acb8246f |
Thus, FWHM(H{{formula:6d447668-b5f8-4859-87e1-275bda8c3aeb}}){{formula:ddf6b1bb-c1d5-4092-bbaf-1718aec77c12}} km s{{formula:208ca03b-2c10-4022-ab84-9c7c21eca3a5}} .
Finally, the calculated BH mass is {{formula:66437c66-799d-4ad6-81cf-9f0c08d4f508}} {{formula:2b187e4f-2996-408f-866e-96a9604f46d7}} . I take this value as {{formula:04be03ec-a83d-401a-909a-80c5e85df6de}} (Tab. ). Additionally, I have calculated BH mass based on the Mg2 line following the similar procedure. I have used equation (7.28) from {{cite:02696db9bc38ce9d3a72f3c49612dc15ba593053}}:
{{formula:3225bc9a-5a3c-4a98-810e-a54dc3eab6a6}}
| r | ecfdb903be0d3125b4c39458771d9b64 |
Clustered FL: If one uses a discrete mixture
model for the population distribution then the iterative algorithm
suggested by our framework connects to
{{cite:ac3b6d67cefb96e5a613e426c4256cca8caad7c8}}, {{cite:51352704ce18a3f82e0da72419f9b777f5a7a91d}}, {{cite:973f479d11068493fc195fe40d3a0ed282a78732}}, {{cite:901bf0bad67dc5824d219bb29669c1a98583fcd7}}, {{cite:6f9875d42c95476c6892f500e3d166bed9da5d3a}}. In
particular, consider a population model with parameters in the {{formula:c44a4f30-a2db-4f4a-a25e-78ee971899c9}} -dimensional probability simplex
{{formula:5905ad5d-5c63-4fea-be00-684924bd845d}} which describing a distribution. If
there are {{formula:49a4c0d1-d148-4d43-b5f7-f6e87371de98}} (unknown) discrete distributions
{{formula:1a7f0350-2a4a-44ea-bd70-e78149af626c}} , one can consider these as
the unknown description of the population model in addition to
{{formula:04bda094-eb66-48c5-8946-1e6d9ea55bcb}} . Therefore, each local data are generated either as a
mixture as in {{cite:6f9875d42c95476c6892f500e3d166bed9da5d3a}} or by choosing one of the
unknown discrete distributions with probability {{formula:50c8bd5b-4006-4829-bce1-c1e1c413ebac}}
dictating the probability of choosing {{formula:0732687b-530e-4383-ba66-23c8aba1c11c}} , when hard
clustering is used (e.g., {{cite:51352704ce18a3f82e0da72419f9b777f5a7a91d}}). Each
node {{formula:ba12237c-fed3-47eb-a2da-412a47f3bada}} chooses a mixture probability {{formula:85dd6238-c8da-4607-93c2-acd3f53fffe5}} uniformly
from the {{formula:85c7aa1b-fde3-43c2-80b2-c79f0595a449}} -dimensional probability simplex. In the former case, it
uses this mixture probability to generate a local mixture
distribution. In the latter it chooses {{formula:d64a72be-aa4f-415b-b449-c787296eda67}} with
probability {{formula:24736a8c-664b-45df-8934-e66d59878cb8}} .
| m | 16c2d41df23ef7d414c97fd4c96ba0f7 |
is the evolutionary family
{{formula:066b961a-1556-4a59-9b1c-da8fa14923ab}}
of bounded linear operators
{{formula:70cf9f6c-7b2a-4352-bbcc-973d25bbf08a}}
that yield the unique classical solution
{{formula:506889f7-3d2b-4e02-a178-8e4a8ada8e55}} in {{formula:b9f74eab-b596-4c32-877e-8228666d1397}}
to the homogeneous abstract differential equation
(REF )
(with {{formula:4f0c17bb-7e21-4233-8272-04a3f49ed04f}} ) on the interval {{formula:8677314c-47f2-48a4-b398-9b94fa569873}} for each {{formula:6c2936c0-3167-4f4e-a2bb-d7c333bb039b}} ,
with the initial condition {{formula:4823166a-d5bc-4922-a19d-adcc525354e8}} .
We recall that the auxiliary functions
{{formula:7f5a36e9-a417-453c-8166-dfbd17b7f9c5}}
have been defined in eq. (REF ).
We refer the reader to the general results in
A. Pazy {{cite:5cfdc8209353bd417b9e64ce2a488e682f022985}},
Theorem 6.8 in §5.6 on p. 164 and
Theorem 7.1 in §5.7 on p. 168.
It is easy to see that
{{formula:7037293b-b16c-49f2-85e1-c4c875942fa1}}
| m | 8a6e9145c3e9e8f3232bbd9346dc637d |
The latest round of PLMs such as GPT-3 {{cite:22008b16c0569a8ff5edc51aed43f8151f290945}}, Megatron-Turing NLG {{cite:b912b9b35817d86e3b6009ad3bde6e7906e8051c}}, the Switch Transformer {{cite:2cf14ae2b4734f94b62a87de16522ccb1976bc2d}}, Gopher {{cite:8b5476770d216458353a4c2ec218d0b57abd3873}}, among others, feature hundreds of billions of parameters and have achieved impressive performance in many NLP tasks using in-context learning — a new few-shot learning paradigm first introduced by {{cite:22008b16c0569a8ff5edc51aed43f8151f290945}}. In-context learning allows PLMs to use their natural language generation capabilities to solve any task almost like how humans would — by completing a piece of text or prompt. As described in Figure REF (a), this allows these large PLMs to solve various NLP problems without updating their parameters, potentially resulting in massive savings in both data annotation and engineering overhead compared with standard model training. Even more impressively, GPT-3 in-context learning yields competitive performance against fully supervised baselines in many NLP tasks by adding only a handful of demonstrative examples in the prompt {{cite:22008b16c0569a8ff5edc51aed43f8151f290945}}.
| i | c73b49ad1ce9e34827e5460491372884 |
A Feynman-type path integral approach has been used to determine a Fokker-Plank type of equation which reflects the entire pandemic scenario. Feynman path integral is a quantization method which uses the quantum Lagrangian function, while Schrödinger's quantization uses the Hamiltonian function {{cite:ac8d243dc4041b6f0af3ed1cb8e1463cdfd5c1ac}}. As this path integral approach provides a different view point from Schrödinger's quantization,it is very useful tool not only in quantum physics but also in engineering, biophysics, economics and finance {{cite:e7b6f64cd725ed54abd23f43ca5199bcafae943d}}, {{cite:ed61595485952a13aa264d322a203146ec371e19}}, {{cite:f70f01edf4569034f7041ec950db52f5cc2e2392}}, {{cite:ac8d243dc4041b6f0af3ed1cb8e1463cdfd5c1ac}}. These two methods are believed to be equivalent but, this equivalence has not fully proved mathematically as the mathematical difficulties lie in the fact that the Feynman path integral is not an integral by means of a countably additive measure {{cite:9d8332020016ca577820839ac25a52be8cff86fe}}, {{cite:ac8d243dc4041b6f0af3ed1cb8e1463cdfd5c1ac}}. As the complexity and memory requirements of grid-based partial differential equation (PDE) solvers increase exponentially as the dimension of the system increases, this method becomes impractical in the case with high dimensions {{cite:f70f01edf4569034f7041ec950db52f5cc2e2392}}. As an alternative one can use a Monte Carlo scheme and this is the main idea of path integral control {{cite:e7b6f64cd725ed54abd23f43ca5199bcafae943d}}, {{cite:3484121d52440634019d1cf0106bede4483e062a}}, {{cite:48a36d532a5ac232eba1e1ba281b2b5ff57d3501}}, {{cite:5be429544c6cb1bbeb598cb11808a0807238ba2c}}. This path integral control solves a class a stochastic control problems with a Monte Carlo method for a HJB equation and this approach avoids the need of a global grid of the domain of HJB equation {{cite:f70f01edf4569034f7041ec950db52f5cc2e2392}}. In future research I want to use this approach under {{formula:6abd877d-7b96-454e-bce5-85b74ef747fe}} Liouville-like quantum gravity surface {{cite:4c941969ae6f512e71b4a01819a96c99bdfb1275}}.
| d | e236c5d1604ceafa1aea786ef8f6e378 |
Adversarial Knowledge Disentanglement.
For the visual representation, we consider disentangling distribution-invariant knowledge from the task and domain-specific information. Suppose we have an image {{formula:7b9c3be2-2c7d-408e-bf95-2e87d57a2c1e}} from domain {{formula:4c0b7d4e-14f9-429a-b3dc-e9936ea10136}} and at task {{formula:a761acc5-9d3c-4ce5-9d57-d82e6cfe6064}} with label {{formula:8ea753ab-7598-494f-a79b-4c9646ea03fa}} (see Fig. REF ), this image is processed through Local Nets {{formula:6fdc9214-7cca-4e7d-a050-ce105395c01a}} and Global Net {{formula:d3256371-2295-406c-99ff-9d5bfb9d057d}} . {{formula:3f046716-f20f-48a9-8b5f-e7fa53f29f22}} is instantiated for each task {{formula:2d963a7d-f889-4436-ac76-2bd492a34e52}} and is aimed at learning task-variant information. Moreover, we encourage the Global Net {{formula:65aa1544-6e8e-4d2c-b89c-a6bf8fe51b86}} to learn task and domain-invariant knowledge. In practice, we may optionally store a small set of samples from previous seen tasksalso known as episodic memory in continual learning literature., denoted as {{formula:cb8a0cc9-2978-414e-9340-60fba256d38c}} , {{formula:852a9fcf-89e2-45ca-86dd-92247231a9d9}} represents source domains. Inspired by {{cite:6009094015ea8390b27cc876245bc739832703de}}, we design a disentangled loss to encourage the features in the global network to be independent with every task-variant local net; see Eqn. REF .
{{formula:018c5ec0-9047-4af8-8651-8323ba366ae0}}
| m | 5d0f9133e9c2dae7c7bf854fc33515af |
Table REF compares the SwinV2-G model with previous best results on ADE20K semantic segmentation benchmark. It achieves 59.9 mIoU on ADE20K val set, which is +1.5 higher than the previous best number (58.4 by {{cite:f4d9e4244a9c32fc43a1ac58cf0c6e26f00ff827}}). This indicates scaling up vision model is beneficial for pixel-level vision recognition tasks. Using a larger window size at test time can additionally bring +0.2 gains, probably attributed to the effective Log-spaced CPB approach.
{{table:51af2748-bc6c-4cc9-9f29-315660528552}} | r | 8d3cd648d0d085807f403c9beffe8df7 |
We evaluated the performances of SGD, Momentum, and Adam with different batch sizes.
The metrics were the number of steps {{formula:b1e0769d-114b-4ed4-a7d0-18adf5f041a3}} and the SFO complexity {{formula:1e6f79ee-05c2-4c7e-a3fb-d5ee1106a921}} satisfying {{formula:ce2956f6-5c19-4853-b8a4-530ae91fb47e}} , where {{formula:978deca7-f452-475d-abff-278af44aeb08}} is generated for each of SGD, Momentum, and Adam using batch size {{formula:4fdce47a-c6bb-4e37-92eb-aa74ba046dc4}} .
The stopping condition was 200 epochs.
The experimental environment consisted of two Intel(R) Xeon(R) Gold 6148 2.4-GHz CPUs with 20 cores each, a 16-GB NVIDIA Tesla V100 900-Gbps GPU, and the Red Hat Enterprise Linux 7.6 OS.
The code was written in Python 3.8.2 using the NumPy 1.17.3 and PyTorch 1.3.0 packages.
A constant learning rate {{formula:eef9b21f-bc79-435a-af97-b1c59c38dcf2}} was commonly used.
Momentum used {{formula:b0a54768-cd0c-46b2-a9ac-8395323e8a53}} .
Adam used {{formula:cda999fd-951f-4e46-a26a-fbaac52058d8}} and {{formula:fd2002fe-99c4-49eb-b5e5-4e051a83bd07}} {{cite:d7b3c90df08b4b4315958e228da32aca068cdc09}}.
| r | b896824cf164f7ac6c59a84959044f6f |
Fig. REF demonstrates the restoration of order in a KPZ like system with the inclusion of activity. It is well known that the solutions to the 2d isotropic KPZ equation are known not to be smooth or regular but rather `fractal' or `rough' and physically could be thought as a growing surface due to random deposition of particles on it. Fig. REF (a) and Fig. REF (c) represent such rough surfaces for {{formula:00064ea2-7c05-4f0f-b02b-0210546b456d}} which signifies {{formula:01cd6423-a748-4645-a83a-9dd2f97edeff}} in Eq. REF or {{formula:cd6e8261-0316-4011-91cf-2149094a42bc}} (conditions at which Eq. REF reduces to the KPZ equation). The {{formula:2996caa0-02d0-464d-8379-2454de1127fc}} KPZ distabilizes and shows a rough phase for any nonzero {{formula:ef729090-9479-49ef-9cc8-bac4351dcecb}} {{cite:bbac481745a050d9ed9ad85c10766a5ed628552e}}. This however is true in the thermodynamic limit. For a finite system, it exhibits a rough phase when {{formula:46f4348f-f4cb-498c-ac3c-585954fb18d0}} crosses a threshold that depends on {{formula:f84c3f20-203c-459c-ae1f-6e65e93b714f}} , the system size. This threshold is expected to be a decreasing function of {{formula:a89b9796-bb06-4828-96a5-3a79db5ad1ef}} , ultimately vanishing in the thermodynamic limit. This is consistent with what we observe in our simulations: we find that the instability threshold {{formula:139fdd4d-ecf9-4481-83b5-6f4214939eed}} decreases to {{formula:bf0837ce-0334-4816-8a0a-c4c1ff114385}} , in agreement with the discussions above. We also find that the “rough” or disordered phase of the immobile oscillators (with {{formula:01912a77-c1a2-4997-9cac-7971ff7201b1}} ) do get ordered upon making {{formula:12c509ac-43a5-46a3-9e50-a6929c28299d}} and {{formula:7372d7f8-aea3-442d-bbcc-e3e70bbaa116}} non-zero; see snapshots in Fig. REF (b) and Fig. REF (d) corresponding to ordered phases obtained by switching on the active parameters {{formula:28048213-cfc4-43f0-80ba-800a8dfd3ca1}} , {{formula:c1071a8d-190e-4196-bb38-6e69f269c4be}} and {{formula:f8bc24e1-63fd-471c-b09b-fe141e666a91}} but for same values of {{formula:0f2e260e-dc2d-48b6-a57e-b93ede9a7322}} .
{{figure:701da337-60d2-4373-97a2-d324752a9639}} | r | c7d9e465922ebabc5f3c2daa5742f787 |
Meanwhile, our method shows substantial improvements over all implemented baselines.
Our method reports roughly half the Median Error aggregated over all categories, and further demonstrates a roughly six-fold increase at Acc30.
We also note that this improvement cannot solely be attributed to the scale of DINO's ImageNet pre-training: the SWaV-based baseline uses self-supervised features also trained on ImageNet {{cite:7deef9b5801a10ba1f1c021908449fe96231b9f2}}, and PoseContrast is initialised with MoCo-v2 {{cite:429e7c0d3b04dedcc25f403c304c6ab8673569d4}} weights, again from self-supervision on ImageNet.
| r | 424f9c1f010a09df858d362bb880cad9 |
Computer-readable expressions for all partonic beam functions and matching coefficients can be found in an ancillary file provided with this submission.
We check the results for all matching coefficients
against the {{formula:2019e37a-669e-45d9-af72-ec0a14d1cd78}} results in Refs. {{cite:1378359ee8d80189d6789b748de4ea149c5ecb48}}, {{cite:c8077b9d8f69ead77dad3243300745ab26d5f88b}} and find full agreement. We discuss the calculation of the soft function in the next section.
| r | c7061457d030e42ddf69b49534359b1e |
Ethical Considerations.
A technique to synthesize photorealistic images of people from a monocular video could be misused to create manipulated imagery of real people. Such misuse of media synthesis techniques pose a societal threat. Viable countermeasure solutions include watermarking the model data or output {{cite:86bbead122305629a8f4023d9ebb6db044bb49bc}}, {{cite:0aa2c080c1a8ed9119a6bd4cc3f24169c8802002}}.
| d | 22610fd44c3cb64db422a05ebd1ea3de |
Information Extraction (IE) is the task of turning the unstructured information expressed in natural language (NL) text into a structured representation in the form of relational tuples consisting of a set of arguments and a phrase denoting a semantic relation between them: {{formula:f54da454-487c-46fd-8be7-e95af94965df}}arg1; rel; arg2{{formula:a5488e55-94ca-4e36-aa74-76112c823829}} . Unlike traditional IE methods, Open IE is not limited to a small set of target relations known in advance, but rather extracts all types of relations found in a text. In that way, it facilitates the domain-independent discovery of relations extracted from text and scales to large, heterogeneous corpora such as the Web. Since its introduction by Banko07, a large body of work on the task of Open IE has been described. By analyzing the output of state-of-the-art systems (e.g., {{cite:aaefd0d540bf6ba778e9334c13ea67e8f86a4ffa}}, {{cite:a8210ed034e319d9f7481a341ee98c7e29fb68e7}}, {{cite:12b4b0800ea7a2993e35bd35c7d0076bca81a657}}), we observed three common shortcomings.
| i | 94a1c36499d843c9f25cebbf01853890 |
In particular, Neural Radiance Fields {{cite:e9f800826446c4d9a63ada2fdba86bc92b6fa031}} (NeRF) and many follow-up works {{cite:a0e93da46099b6c8989b23f1700cd1f1f973ee0d}}, {{cite:e1e75dc779644963afd718a7d6109a4c8bbd7294}}, {{cite:3e9cb7fc46606783b6d141bd369361c6d49a5bd8}}, {{cite:6dcc9ca9b73700bed26dac9863a468242746d701}}, {{cite:fef6ea682040599ab580154ea6cdf756c431759d}}, {{cite:bb225275dda3f7d37e0610ffa0c803a4e4e74d13}} have shown that implicit networks can capture accurate geometry and render photorealistic images by representing a static scene as an implicit 5D function which outputs view-dependent radiance fields.
They use differentiable volumetric rendering, a scene geometry, and the view-dependent radiance that can be encoded into an implicit network using only image supervisions.
These components allow the networks to capture high fidelity photometric features, such as reflection and refraction in a differentiable manner unlike the conventional explicit 3D representations.
| i | 9e088e111fe11b1055defb759ef37777 |
Multi-task learning (MTL) aims to solve multiple learning tasks simultaneously while exploiting similarities/differences across tasks {{cite:f5c0d84e742362af0c6f71876b1c2751f038bc2c}}. Multi-task learning is commonly used in applications that warrant strong privacy guarantees. For example, MTL has been explored in healthcare applications, as a way to learn over diverse populations or between multiple institutions {{cite:4f391946a407b3ca83d1baaaa5238718ee99826b}}, {{cite:c9e8e22c6a2e9b19e841445fd80a174bee2dc2fb}}, {{cite:23fbaeb1e025b8c77aef2f389dd13c2218e43c2e}}; in financial forecasting, to combine knowledge from multiple indicators or across organizations {{cite:aa47f3329e9ab37faeac7088be224ba60e8c0af7}}, {{cite:e0de6f1c1aede63623fbf15e83faedc18fb0d73a}}; and in IoT computing, as an approach for learning in federated networks of heterogeneous devices {{cite:20a607d894428003d16a522bc7835cd78025058f}}, {{cite:a1bb7b069e3595854ab50082accc386e6567f968}}, {{cite:2215e6bbba30e9385a83df4fc6e83f2270363fc4}}, {{cite:2fb9f8d2873a9f11a7089c1ed16b798d899fe053}}, {{cite:90d059abad7738c80c0e74f285dffe19fd6ddc4f}}, {{cite:9ba545c805fcd4c354bcc6593eeae3cbc3e6c2a7}}, {{cite:56dfdc5cbb96958dc8699d6c0e7de12e1e97abd9}}. While MTL can significantly improve accuracy when learning in these applications, there is a dearth of work studying the privacy implications of multi-task learning.
| i | 63e0e1cfdb47e86c705db11ed8aa6651 |
The general results presented in this paper can also be used for statistical inference for the log-contrast model considered in {{cite:378755357f3db91a76cf2a185e9d076ee04f1f33}}. This type of de-biased estimates were also proposed in
{{cite:13644db5eb1d08dc5349c4ec25e7475ab29ac401}} and {{cite:19f9a275fe1fb6f8a29393439cf1aec4beffa2c6}}. {{cite:acec9e7a93b6396ca44ffa9064b0a668c1e8d4a6}} proposed an exact inference procedure for lasso by characterizing the distribution of a post-selection estimator conditioned on the selection event. It is interesting to extend their approach to the high-dimensional regression problems with constraints.
{{cite:64ffe6b519e83a96f87409440ed56f7f5c488090}} developed a bootstrap smoothing procedure for computing the standard errors and confidence intervals for predictions, which is different from what was considered in this paper. Efron's procedure can be applied directly to make inferences on predictions using the methods developed here.
| d | d0f6b24fcfa7924385df875bf177f51e |
As the usage and complexity of industrial robots increase, they take on unfamiliar shapes and thus, complicate the interaction and establishment of trust in shared environments with humans. Robot-related factors were shown to be the most relevant for the development of trust in these interactions {{cite:eb782ad2011d2c04047360d3cbd9f7c7fc1c7346}}. Among these factors, the design of the robot is particularly important to get the human interaction partners to trust the robot appropriately {{cite:bbe5a0ee768bd01f15c1fcd9a6f7872829673a19}}. Anthropomorphic features may aid the trust, but they are not often present in industrial robots. As the complexity of the new systems increase, the perception of these systems as collaborators rather than machines have been deemed as positive {{cite:ba54a564cd83f52c36f75791615b312b8f7bb31f}}. For example, the addition of a pair of sunglasses to a industrial robotic hand and gripper, along with a set of breathing-like movements and gaze behavior, improved metrics from participants such as the perceived sociability and likeability of the system {{cite:5fcfb74afa64481439a708bc80b5fc45e4296381}}. However, in their study, the authors did not find differences in trust. One possible reason for this is that they included a scale that was not originally designed for industrial collaborations {{cite:79fa51ec186497c0a86d320ac88ffd7810d6a748}}. Another study showed that trust does not seem to be affected as a result of anthropomorphism in industrial settings {{cite:15011bebddb48fca9c2c406bccfaca290db0e78a}}. In this case the authors used a validated scale to trust in industrial collaboration, developed by Charalambous et al.{{cite:e9c4e41be41627bacb39ce5f7e69095caebba1bb}}. Nevertheless, the study employed a limited form of anthropomorphism in which a face appeared on a screen attached to a robotic arm and gripper. In contrast to previous research, our study does not include a tactile interaction. Our research explores a new perceived modality of navigation for a non-humanoid robot that can potentially improve trust on the system by using a humanoid robot, NAO, in industrial settings.
| i | b353cad1ea7860f6a5376046391d2591 |
We address the following problem : is it possible to generate a state maximizing {{formula:a353772f-ed15-4003-b050-8a15e8a1f65b}} by applying a {{formula:cfaa8710-a367-4061-af6b-aa641e4e04fe}} gate circuit on a state of the LU orbit of {{formula:e574a6aa-e475-4879-916e-e798999d3171}} ?
We use the classical Z-Y decomposition of a single qubit unitary operator in the form {{formula:0e5cdda5-7d76-4e65-822c-27830082d36f}} (see e.g. {{cite:b6e9cd9edcd21dd4ca7a20370fdd6abc27e39030}}).
Using this decomposition, any fully factorized unitary operator {{formula:1f2c082b-98d5-454b-9e03-5dc81fc940c8}} depends, up to a global phase, on the 12 real parameters of the matrix
{{formula:a23f0c7c-837e-41df-9399-75037a10c18b}}
| m | 1ce9e3cafe20580e77517625012335c9 |
Following {{cite:73655c494023d9f617a46091aa8217517958c539}}, we can also select the fitted model based
on a tiered approach, first performing the F-test on the Ser and DevExp fits.
Galaxies for which the DevExp fit gives a statistically significant improvement
are then tested again to determine whether the SerExp fit is preferable to the
DevExp fit. The preferred fit (either Ser, DevExp, or SerExp) of the SerExp mocks is then used to assess the bias in the halflight radius and magnitude. We
tested this approach and found that it did not significantly alter the results.
| d | 2ae49e3e44b2a78273dd320ada5b5bed |
The fundamental invariant in General Relativity is the
Ricciscalar {{formula:980f2412-46c1-457f-950a-12ebb3ec46ae}} of the Levi-Civita symmetric connection, but that is not the
case the Teleparallel Equivalent of General Relativity (TEGR), where the
fundamental geometric invariant used for the definition of the gravitational
Action Integral is the torsion scalar {{formula:b961ffa1-b3c0-4800-a05d-adccaf0fb80e}} . It is defined by the antisymmetric
connection of the nonholonomic basis {{cite:cd9e444e362bce2de7d741de52de94bf5e6e04bb}}, {{cite:d8f5a9b0d8274bf77217268c60f18f36acfc730c}}, {{cite:916a5bbe9f669c4c1e258de7074b4dff912ef928}}, {{cite:4c42793ae3d9967e914a0275cd9100cb8a89dea6}},
the so-called curvature-less Weitzenböck connection {{cite:2e601c5cb8cb7758cd5d6011c20d2c747fa70ad9}}. In
{{cite:c1c4e4e67fcf6d59c5343b7423d0b8565f710f17}} it was found that in the Teleparallel Equivalence of General
Relativity (TEGR) {{cite:cd9e444e362bce2de7d741de52de94bf5e6e04bb}}, {{cite:d8f5a9b0d8274bf77217268c60f18f36acfc730c}}, {{cite:916a5bbe9f669c4c1e258de7074b4dff912ef928}}, {{cite:4c42793ae3d9967e914a0275cd9100cb8a89dea6}}, the energy
distribution using Einstein, Bergmann-Thomson and Landau-Lifshitz
energy-momentum complexes are the same either in general relativity or
teleparallel gravity.
| i | b399e1f9eb91ca2b9cac76053e02f7ec |
The developed framework employs phase field modeling approach, a popular continuous fracture modeling technique. VE-XPINNs are trained on the governing coupled PDE (variational form) of the phase field approach to encode the vector valued elastic field and the scalar valued phase field based on the initial crack location, material properties and the characteristic width of the crack. One significant advantage of the proposed variational energy formulation is that it requires derivatives one order lower than the conventional residual minimization approach, which results in better computational efficiency. Additionally, motivated by the previous works of V-PINNs {{cite:2d4c4895dcaba2ffbb7069a812025cb3199485d2}}, we use Gauss quadrature points to evaluate the integrals, over a domain. To begin with, the computational domain is divided into a number of elements and then, the quadrature points are generated within each element, for an efficient integration of non-smooth functions like fracture.
| i | f78d5e61276d7556db26b8ee7ad7eb9a |
Recently, the area of computational topology has grown rapidly and provides plenty of computational tools for data analysis ({{cite:f0b7ed31802be41742799c2eb115b44009f6671f}}). One of the common tools is persistent homology from topological data analysis, which observes the dynamics of topological features in a sequence of nested topological spaces ({{cite:25fc06a12d77e5563cecbe16d20f5817286edeeb}}). This information can be visualized as a persistence diagram (PD), a set of points in the two-dimensional plane where a new topological feature appears at {{formula:a7c5f662-0a35-40a0-a18b-71200e2a94de}} -coordinate and disappears at {{formula:307f3c16-5673-4511-bf15-01eed81cb6f8}} -coordinate. This compact description of topological features is more useful for the analysis of complex and high-dimensional data because many important topological features are invariant under dimensionality reduction in some sense. It is noteworthy that the spaces of PDs can be equipped with a metric function and it is known that these functions are stable with respect to small perturbations of given data. See, for example {{cite:3f994390427a3e36c367642d633f493cd02ade7e}}, {{cite:64ca16a5b53a92989b4538f5eeb8ff0f9f9b7ba8}}, {{cite:e87d06ec928f43e520e907f6c1ceb66c2930ab79}}, {{cite:54ca934cdebc2f4cb4a5e0b414981675593c01c1}}, {{cite:dda1e44fd57ad82f3984968b62fac97d8427adea}} where bottleneck and Wasserstein metrics were adopted and the stability issue was addressed.
| i | 5946012b096b5347cc345966586a6688 |
Inspired by perturbation approaches to generate saliency maps for image-based black-box models {{cite:e32c1bf96d4ba06d8c81a494037790d270c79d2d}}, {{cite:07bae6d9b51a7711392d532595d66a645fe5ad3c}}, {{cite:cf1df42148a68b8744c3d09b5e752b5c6d6ec890}}, we leverage the principle of analysis by occlusion.
We propose OccAM: Occlusion-based Attribution Maps for 3D object detectors on LiDAR data.
We estimate the importance of points by testing the model with randomly generated subsets of the input point cloud.
The underlying assumption is that an object will be detected less accurately or not at all if areas of the input that are important for the detection have been removed or perturbed.
However, the special characteristics of point clouds, like the unstructured nature and the depth-dependent point density in LiDAR data, poses new challenges compared to the analysis of image-based models.
Thus, we propose a voxel-based sub-sampling strategy in which the sampling probability is adapted to the point density to challenge the detector appropriately.
To evaluate the influence on the detections as precisely as possible, we further use a similarity metric that is optimized for the properties of 3D bounding boxes.
| i | 664e4b4fe5e12f72954bc31da194978e |
A classic topic in commutative algebra is the study of determinantal ideals. These are the ideals generated by the {{formula:04d35ffc-bd84-4972-8a66-accbb6549b82}} -minors of any matrix, and a special attention received the case of the minors of a generic matrix, whose entries are indeterminates, see for instance {{cite:7743d64c1146bf26d0b6ca2ea44b79a188022ede}} and {{cite:f289d50475484e64dda80d223fca3f9d156a9e4c}}. More generally, ideals of {{formula:eea2e319-b2d2-458e-b996-244ac6c08d54}} -minors of 2-sided ladders were studied, see {{cite:b0a9287671fda4d6b35148a46c05583d188bbe98}}, {{cite:3ac4bbd84cabcd30c447d20acacf22f1458f0cbc}}, {{cite:2fd062fa2c50010122936349e7adc4b819457437}}, {{cite:2dcfe9def025c481c27c4f518931970354e30427}}. When considering the case of 2-minors, these classes of ideals are special cases of the ideal {{formula:6cfee6ed-d4cd-41ea-b236-be424739ad4a}} of inner 2-minors of a polyomino {{formula:138e141c-cd30-44f1-b8be-47b5814edcf5}} in the polynomial ring over a field {{formula:a8658d41-bcb1-4235-aac4-9bf29da768d3}} in the variables {{formula:5274f6d8-e4f5-4bd1-a3e5-500baa2ed6be}} where {{formula:f7f61481-31a4-4899-a19a-65a06ef833d2}} is a vertex of {{formula:9adbfcc2-138e-4231-aaa5-cf520c8de9b7}} . This type of ideal, called polyomino ideal, was introduced in 2012 by Qureshi {{cite:a23affa19d22abbb7652734476e4dd0c9b50da7a}}. Since then, the study of the main algebraic properties of the polyomino ideal and of its quotient ring {{formula:b1efa97d-ccf9-4ea5-bbaf-bc2f9c6f4be3}} in terms of the shape of {{formula:a34c0034-c5ce-446e-ab0c-792941d90749}} has become an exciting area of research. For instance, several mathematicians have studied the primality of {{formula:a42793b1-3bf9-4786-9ace-820378e0d788}} , see {{cite:02b64877316367500aef5b4be5fe8ffd5122cfd4}}, {{cite:484276bbb7faa3b4ded1550e665a299601cc1d37}}, {{cite:68591980e70de89f9e6d37b27cede1dfa584fa54}}, {{cite:531f6a27a793f1ab60601e5ea9f75403edad2bb4}}, {{cite:07268741df9614086c6e86dea0deedaa8e314121}}, {{cite:4fbfa75990f11ef94e3465951799b09f2963e6bc}}, {{cite:11cc8b41f1e997011179b99ee7d43f2bba5afd96}}, {{cite:a418fc08d244a2076b68b3900de5c4551259b638}}. Moreover,in {{cite:17a904b5b96043ba2155fc9ca75d86a3e30f72ff}} and {{cite:5d650c7c0b96088561bd0fafa8cd8bd85a0497ae}}, the authors showed that {{formula:5dd75112-f30f-4cfd-9cc4-ca883f69ca90}} is a normal Cohen-Macaulay domain if {{formula:9be7468b-ae5f-435e-9467-5977d9f232af}} is a simple polyomino, i.e. a polyomino without holes; a precise definition will be given in Section . See also the references {{cite:9ca214814be9ae9399a595bea55d08ff375bafdd}}, {{cite:f3e7c440c59be416a928fa704c853934c1006b49}}, {{cite:f28569c7696ba69b5ef94cff9c228cae8c7f53b6}}, {{cite:4ff78f2850dee769acb50d92e2051ea481f18763}}, {{cite:6bfc31ef7166ea77aa6e8006dc2c80c9342d71f8}}.
| i | b6c7501beb5c8cdad87ee91648c2e5cd |
We employ transfer learning by fine-tuning Transformer based language model, RoBERTa {{cite:b648b51ac90387a2d0a21769855df5f016d8dd3c}} on the propaganda dataset.
We address the issue of minority class classification by designing ensemble of one-versus-one (OVO) classifiers, that vote for the presence/absence of the specific minority class.
We handle the intricacies in the Repetition class by employing a novel algorithm based on dynamic least common sub-sequence approach.
| i | 20103429907b9f05338edf61c25b00c9 |
In machine learning we often encounter minimax optimization problems, where the decision variables are partitioned into two groups: one for minimization and one for maximization. This framework covers many important problems as specific instantiations, including adversarial learning {{cite:1f3fd2c947b0a865b0fe2dda47ebf2a6ff4b899f}}, robust optimization {{cite:beddb75e7cab7ef4c932b251c0b004d0ef75fa85}}, {{cite:0b979db62cdc445f54b05a8d9a414d40f10d7c1d}}, reinforcement learning {{cite:cd4a2a315789a0640c8b4259432abffa3a0f9719}}, {{cite:0797c23754ddfa7228009b9f60b1e715fe2ae024}} and AUC maximization {{cite:973cf87289309dd9c76aecedae7a4b1cad3aa9f7}}, {{cite:932004a34de12533514c5d9dc9915215d313eda2}}, {{cite:7575d08f03a427e2074e64f5d7dead06ba638785}}, {{cite:23afa472c6dc702978518943414e6880a06d32bc}}, {{cite:f2d5e956cc34b8e53b74f5b4f0d6613eaefccf5c}}. To solve these problems, researchers have proposed various efficient optimization algorithms, for which a representative algorithm is the stochastic gradient descent ascent (SGDA) due to its simplicity and widespread use in real-world applications.
| i | 50a7185a1bce9e285ccef9ee38de87ce |
[noitemsep,topsep=0pt]
CMB: the Planck 2018 legacy release temperature and polarization CMB measurements {{cite:2c207c030786b052a7185d3217e89d7351d69dfc}}, {{cite:f999ee371aa6b86b5a2dd912e3f02970142274c7}}.
lensing: the Planck 2018 CMB lensing reconstruction likelihood {{cite:bee597bce20b0a26331e3d1acc8f19e4163e0d91}}.
BAO: the BAO measurements from 6dFGS {{cite:195b1c64b1f3e619ad101734667ca4e06a519da6}}, SDSS-MGS {{cite:06cfb7a3c9d7a1ba335f4d60b1d254cc0f0ae469}}, and BOSS DR12 {{cite:53e4caa99e4ac555ec74f06b93b2643bc61c4f06}}.
Pantheon: the Pantheon sample of 1048 Supernovae Type Ia distributed in the redshift interval {{formula:abb3cb64-b257-4c03-afbd-b2ea3510e41f}} {{cite:c671233c4f99977ada3b0f519dbb081d2c88f3c0}}.
R20: a gaussian prior on the Hubble constant in agreement with the SH0ES collaboration measurement in {{cite:3222b850d9c8c63891b8eff018a05fea424b9bcb}}.
| m | b163687163279106718e8312b5c747e6 |
The
underlying abstract
formulations of quantum
mechanics using any nontrivial
inner-product metric {{formula:9c16d412-a9d1-4248-9a82-4ba0cea9f300}}
are known as quasi-Hermitian quantum mechanics
{{cite:043d328e9bcb32ea0933f1c5594739b3875c22c9}},
or as pseudo-Hermitian quantum mechanics
{{cite:f03ef3e6fb681c8907384c186b3ec9b56d15080f}}.
Once one adds the specific factorization ansatz (REF ),
operators {{formula:2f3c149c-3300-4e23-acb4-ee7003df6f0d}} and {{formula:08696eaa-6eb9-4e27-bb94-a7df8924e48c}}
are interpreted,
most often, as parity
and charge, respectively. Still,
also certain less specific forms
of the factors {{formula:b3e22a1d-4cf9-4a02-b15a-02eff7315799}} and {{formula:7d7b60e2-e461-4ad1-a876-fe2533928843}} forming the metric
can be found discussed in the literature {{cite:45a3ce59ce8fa8c8452ab4f946b1b0da54b55fa0}}, {{cite:715304ae7fc35ee529838c6b3f796eb58b83ed04}}, {{cite:12f3fd8f486f981e96ed3dc26fd46d9e6e9b8cf6}}, {{cite:837a773886982c8de1ea6d15aae710f9e8f8558c}}.
In our present letter we intend to
reveal and describe an unexpected new connection between the
abstract
mathematics of the Hilbert-space-metric
factorization as sampled by Eq. (REF )
and the realistic requirements in
physics where one often has to demand the generic
consistent coexistence of several non-commutative
non-Hermitian quantum
observables (say, {{formula:6002712e-488c-4a98-a977-2360c0eac5a1}} ).
| i | dbb9122c30170aeb5e274e807455673a |
The out-of-domain training data consists of about {{formula:1af2dbb1-69d3-4ee0-a523-aaa710385034}} sentence pairs for English-to-German and {{formula:7a8522a8-9665-4f4e-8cf2-9ab586444f00}} sentence pairs for English-to-Russian. In-domain training data is about {{formula:4b61ad94-5a54-4ad7-b6cf-2b6acf08a8db}} sentence pairs for English-to-German and {{formula:6c76f817-0823-4e7a-9f49-39cb78ddaa09}} sentence pairs for English-to-Russian.
Training data is tokenized, truecased and segmented into subword units using byte-pair encoding (BPE) {{cite:938a92c14b0f2e0b84be77f80a63d8a4f1b7e072}}.
| m | 5ade665449a4a1d5fb548743d2f8ed4e |
Firstly we note that with similar hyperparameter settings, we obtain an equivalent performance corresponding to the results given in base paper {{cite:bdb088c79bf3838353c6f85b5bd90dd19bc08435}}.
We obtain major improvements in word similarity tasks (columns 1-3 in table REF ) when implementing retrofitting with all the top-down knowledge databases with the exception of FrameNet. In fact, with the use of FrameNet on any evaluation task, we seldom observe any improvements (in some cases, we observe decrease in performance) with retrofitting. Faruqui et al. cite the reason being that the nature of top-down knowledge captured by FrameNet between words is very abstract and distantly related, (for example, prominent and eye-catching).
In the case of TOEFL task, we observe very high improvements in accuracy, the highest increment {{formula:fc3ca47b-f2ad-4f24-ae71-f3a38d2be894}} occurring in the case of GC vectors with PPDB retrofit.
We observe modest improvements in the case of Sentiment Analysis task (SA), the highest increment {{formula:ef68eee4-0f94-478e-8fa4-5f3161e8e028}} occurring in the case of GC vectors with PPDB retrofit.
For the SYN-REL task, we observe improvement in case of GC {{formula:cb41992b-06b8-4a2a-9a56-18403da31d76}} and SG {{formula:cc4b9d5f-5cda-458a-8a6a-821ccafd6aa4}} with PPDB retrofit. For all other cases, we always observe drop in accuracy, the highest reduction occurring in the case of SG vectors with {{formula:7a0bd6e9-1f76-48c4-9e36-5c9904303c36}} retrofit {{formula:4c77ae0f-099b-48ef-970f-62a19c6c4287}} . As stated by Faruqui et al. in the base paper {{cite:bdb088c79bf3838353c6f85b5bd90dd19bc08435}}, the poor performance can be attributed to the fact that SYN-REL is inherently a syntactic task and by retrofitting, we only incorporate semantic information in the vector representations.
Finally, we note that retrofitting with PPDB database gives the best improvement across a majority of the combinations (Word vector set {{formula:2e96096d-d85c-431c-a364-3a564794248c}} Evaluation task), followed by {{formula:80e5eaa0-891c-403a-8406-cdc1015cb833}} , same as in the case of results provided in the base paper {{cite:bdb088c79bf3838353c6f85b5bd90dd19bc08435}}.
| r | 754603d70ba6c7e1f189cb50624c5adc |
According to Verlinde's argument, the total number of bits on the
holographic screen is proportional to the area, {{formula:b880ebc6-60b7-489c-9bfd-474b938185b6}} , and can be
specified as {{formula:08ec2d5e-2b46-48ec-8c9e-d3bef75f27b2}} . Indeed, the derivation of
Newton's law of gravity as well as Friedmann equations, in Verlide
formalism, depend on the entropy-area relationship {{formula:fda34b10-32fa-4d45-9cf6-3ab3cc96ea7f}} , where {{formula:b2e30337-e920-479e-863b-d743a30f4e38}} represents the area of the
horizon {{cite:a20d440c3b98315a946fde2291be41347cd46d7e}}. However, it is well known that the area formula
of black hole entropy no longer holds in higher derivative
gravities. So it would be interesting to see whether one can derive
Newton's law of gravity as well as the corresponding Friedmann
equations in these gravities in the framework of entropic force
perspective developed by Verlinde {{cite:a20d440c3b98315a946fde2291be41347cd46d7e}}.
| d | ce54075a2298563aea62765896146da2 |
The above is slightly stronger than the assumptions on second order moments typically found in the stochastic gradient literature, e.g., {{cite:fdc248e3d518f9419d34ed8134731249669d3e5b}}, as we require a uniform bound on the gradient noise. This condition is common for the algorithms using Markovian samples {{cite:92d165d655d8f85c36677ac2ca7582004bd48149}}, {{cite:8ee4ca78d744357207bb3dc6c27c18b707382fa5}}, {{cite:ec0b2c8fc8438997e1527287f2fcd6616fbc6c3f}}, which requires that the oscillation of stochastic gradient is controlled. For strategic classicaition problems, it is satisfied for the finite dataset setting in ex:br.
Moreover, similar to {{cite:5c7dc32fc111d0efb646417dc95a8b89d3a9a223}}, this bound is adapted to the growth of {{formula:035d5af6-feee-4066-8a94-4560f14dde98}} which is compatible with the strong convexity of the loss function {{formula:9f6d384f-1bf3-48b0-8137-c6a07e439bcc}} .
| r | aaefbce9867d70c7f4179ff95088ef71 |
Symbolic music generation aims at generating music scores automatically and has drawn more and more attention in recent years {{cite:4128e1e5d9fc53d6fd3aef94e46d3ff03dd856e8}}, {{cite:39299226e540c51ecb39fac9006490f7cbb6b7ea}}, {{cite:b9887316a1aa16c598d3611838945ea6c900deff}}. Since music can be represented in organized sequences of discrete tokens just like text, Transformer-based models, which have been demonstrated to work well on text generation {{cite:5be58b8302abd3355e81a2ae9e4a66711041f348}}, {{cite:90b6d9ac5243d8ac26591733bce01959ecdfca65}}, are increasingly applied in music generation {{cite:b9887316a1aa16c598d3611838945ea6c900deff}}, {{cite:4ea24d5620b89749607e02310e528eb3160ea282}}, {{cite:2ee9c1d9513596117678906bd0838e00f785770a}}, {{cite:b69f6363d87d936fcde1388020da3512bd4d1047}}, {{cite:ede5c15d13e4d9683bc81bcb3229a571fa065248}}, {{cite:30329af2dce74b4ff56423a61df4b8012ec0fcdc}}, {{cite:2c859b54287bca0577c93bbb1063f7ec45617e60}} and have made great success.
While the self-attention mechanism empowers Transformer to capture the complex correlations in music, there are two ubiquitous challenges to solve for this task: 1) Long sequence modeling. Music sequences are typically very long, especially for multi-instrument polyphonic music where the lengths can usually exceed 10000. The quadratic complexity of full attention limits its scalability to that length. 2) Music structure modeling. Music has its unique structures, where a piece can usually repeat some patterns of a previous piece, occasionally with some variations, after either a short or a long distance (see fig:score for an example). Successfully generating reasonable structures would make the music more realistic just like human-made music.
{{figure:e785f6e0-ae6d-42d7-9a2e-0250614a6529}} | i | 35ac35f5faa9868f6485b31f951e9904 |
The effect of re-activation and generalization ability of CREAM. Following {{cite:3cdca09812d47cf516cf5675efcb8078f94d8857}}, we re-implement CAM, HAS {{cite:15d5dfcaa8bef5e39eccd4faafd1321d9dcb4f39}} and SPG {{cite:d0c439fad602bbd565adf4fa0d1ca1a685cbaaf8}} on BoxAccV2 {{cite:3cdca09812d47cf516cf5675efcb8078f94d8857}} using VGG16. As illustrated in Figure REF (left), HAS and SPG perform better than CAM only when {{formula:876fdbbb-470e-4cee-ba2d-90aa81aae119}} is small (e.g. {{formula:219173d6-f81b-4bfd-b145-5d944561dddd}} <0.1). When {{formula:905878df-a0a6-414f-af59-2eaf94bd5a93}} is large (e.g. {{formula:547d76df-b6a7-44f8-b547-8f7b2ce3a96f}} >0.2), however, there is no apparent improvement by CAM. In comparison, {{formula:7bc06116-d2a0-41ff-8cd3-23bd1d1d0eb8}} curve is above CAM curve for most thresholds. Besides, it is flatter around the peak, showing that {{formula:e5b3b92d-7d5d-44df-84ae-e32531c7b35f}} is more robust to the choice of threshold.
| d | 0ae1bf90f624fa39f20f2e67f6e00577 |
Recently, computation-heavy applications are experiencing a dramatic increasing over the Fifth Generation (5G) and future wireless networks.
There is evidence that such applications, including mining process for Proof-of-Work (PoW) in blockchain, interactive gaming, virtual reality, video services, etc., have become premier drivers of the exponential computing task growth {{cite:8176c500779e3b7362073e882a1168acd6d9511d}}, {{cite:d37a6e197e731ec76c04c40433b1bb48650c9185}}, {{cite:6cd4a16c2d218d16243c3848d22bb19f284b2673}}.
To handle such increasing computing requirements, hybrid edge and cloud computing (ECC) systems have been expected to provide low-latency and on-demand computing services to users {{cite:b17e27828ac43915a5bb2700960af1c1a82be9c7}}, {{cite:1e891dbc08f73938116ba9b8b5629777f515b3e4}}, {{cite:284cc964e03480852b9c00c714df3ea41dda2f35}}.
In ECC systems, cloud computing, as the traditional solution of computation offloading for user devices, is usually implemented at cloud nodes physically located far from users, which results in a long latency service response.
Aiming at this problem, edge computing has been proposed as the complement of cloud computing by enabling users to upload computational tasks to the edge of networks {{cite:74ec3f61dad0560eb27d77c2d8addcde9d50afa7}}, {{cite:e03f641ee76311886fd104c09317602ff9d424da}}, which can eliminate the latency and enhance the reliability of services.
However, with the growing amount of computational task requirements, computational power limited edge servers might be overwhelmed with severe performance degradation. A feasible solution for this problem is forwarding these tasks at edge nodes to the remote cloud center {{cite:3dcffc3e2e2a05b403837fc5670f29fdb8451794}}, {{cite:240e594885cbabae7f527c2939a1675501c9a762}}, which can be considered as computation offloading between edge computing service providers (ECPs) and the cloud computing service provider (CCP).
Therefore, to achieve the optimal and stable performance of CCP systems, an efficient cloud computing resource sharing mechanism plays an important role resulting from the constrained resource equipped by the CCP and time-varying user requirements among the CCP system.
In addition, such mechanism is more challenging when the dynamic service subscription of users is taken into account {{cite:4399b1224663571be7dfc6c8937cd60862427299}}.
This work will establish a hybrid ECC system, in which users can upload their computational tasks to nearby ECPs or the remote CCP dynamically.
In addition, by considering the dynamic service subscription of users among the CCP and ECPs, this work will focus on the computing resource sharing and computation offloading mechanism design in the ECC system to realize an efficient utilization of computing resource and satisfy the service requirements of users.
| i | f94e7e3b6d5a015e3bfcae40193a4e22 |
Having identified the relevant Regge trajectories, we followed them to evaluate the contributions they make to the ICS of the chosen transition. These contributions may take various forms depending on the proximity of the trajectory to the real {{formula:8c90df23-4809-487d-b2ba-08454b62622a}} -axis, and the magnitude of its residue. In the higher energy range a single trajectory corresponding to the resonance {{formula:9a796583-6bf7-4971-930e-5ba544d93b3e}} is responsible for modulated sinusoidal oscillations [see Eq.(REF )] for all transitions considered (see Fig.). The oscillations for different transitions are in phase, and the resonance pattern survives summation over helicities and rotational quantum numbers {{cite:1f6e63bec729fee217430d3692de5ecdc69ddd68}}. This allows these oscillations to be visible in recent molecular beams experiments {{cite:a3991ffcf8499b31de58946ac97be6e2a67c0ac0}}. In the lower energy range, for the {{formula:754737d5-88f4-40cc-8880-694014d2c04e}} transition, we assign the first two, and the subsequent three peaks, to the trajectories {{formula:9bc2a9d8-3448-45f2-b5fc-0993ee9a2fb3}} B{{formula:defc9cce-cc99-4307-b7f4-ea5a1d34dc7e}} (3,1,1)(0,0,0){{formula:c1936761-831f-436d-ab46-4d4211fd675f}}{{formula:b348957e-f040-4a43-b6a7-d44746c3a0f3}} E1{{formula:b5acf966-3c4d-441a-bc6b-8fd596a80b97}} (3,0,0)(0,0,0)(E){{formula:612957b5-b920-4faa-9544-43a214ff9138}} (3,1,1)(0,0,0)(E){{formula:c31d26b2-b4a3-4a5c-bbc3-2d710de6ab02}} E1{{formula:3ab694d8-5652-4a60-9422-ba92278bddc7}} (3,2,2)(0,0,0){{formula:afe67b6d-ddc1-49a6-a4c7-9e35841b416f}} (3,2,2)(0,0,0){{formula:395dc5db-f24e-4f5f-8edb-a2cb7f9f017d}} E2{{formula:10445be6-1c54-4301-ad37-51830f24ac49}} res(3,2,2)(0,0,0)(E){{formula:824c5c98-f641-47d3-86aa-d49611a5467e}} Finally, the accuracy of a Padé reconstruction depends on the accuracy of the input data and on the numerical format (in this work, exponential with four significant digits) they are stored in. In general, the quality of the input data was found sufficient to accurately reproduce a Regge trajectory, with its different parts consistently reproduced in the analysis of different transitions. Determination of weak (small residue) trajectories passing close to the real axis proved to be most difficult. Such are the trajectories {{formula:a4508781-754a-47fc-86aa-80fdd4d87f46}} and {{formula:ba030766-d41a-4c5b-ba2b-80464a05ca8e}} , which make a notable contribution only at {{formula:2b048ac6-6521-423e-b449-ab59b64052be}} when {{formula:b30276df-7d2f-453f-baf6-b224f3f3d693}} is close to an integer {{formula:a31dea3b-1851-43b9-8094-35054dd3a13d}} , where {{formula:e73f83dc-8d67-4e01-8564-e047bfaa6432}} can be determined with a good accuracy. For energies between {{formula:7d338e38-0ea6-40cc-b65c-3b9fc6e1e7ca}} , {{formula:5507a8fc-31bf-4377-a5cd-951c4b7965c8}} found by Padé reconstruction fluctuates. Fortunately, this is not important, since there the contribution of such a trajectory vanishes. Thus, we can connect {{formula:615f605d-7219-491d-a913-7b15fc950268}} by a smooth line, as was done for the resonances {{formula:156866f9-b8b7-46f4-b9a5-be0972ca499f}} and {{formula:a5eafc75-b8c4-4a30-bb86-5064b91213c6}} in Fig. ).
| d | 88f9e8010be435077bd1810df6641fa7 |
In this paper, we proposed a Gaussian sampling-based optimization algorithm to generate approximate solutions to Max-k-Cut, and the Max-Agree variant of correlation clustering using {{formula:247cfaaa-49f0-4853-abb7-d0ad2d44be35}} memory. The approximation guarantees given in {{cite:0d01f6eced0c0deff61f990caf54532d5bd0b5db}}, {{cite:44579efd459c287c3e6b11c12348e65c1ad6fea3}}, {{cite:97965633d16fca1256ce2bb3d852e68972e07f94}} for these problems are based on solving SDP relaxations of these problems that have {{formula:ea58903f-631e-4a22-a915-c75155260e1e}} constraints.
The key observation that led to the low-memory method proposed in this paper was that the approximation guarantees from literature are preserved for both problems even if we solve their weaker SDP relaxations with only {{formula:ad19ff5f-23ba-45b9-b6f6-6e22d50f982f}} constraints. We showed that for Max-k-Cut, and the Max-Agree variant of correlation clustering, our approach nearly preserves the quality of the solution as given in {{cite:0d01f6eced0c0deff61f990caf54532d5bd0b5db}}, {{cite:44579efd459c287c3e6b11c12348e65c1ad6fea3}}. We also implemented the method outlined in Section to generate approximate clustering for random graphs with provable guarantees. The numerical experiments showed that while the method was simple to implement, it was slow in practice. However, there is scope for improving the convergence rate of our method so that it can potentially be applied to the large-scale instances of various real-life applications of clustering.
| d | 0664a1858b652f8d706d36280d652c65 |
where {{formula:e67a5c45-c6da-4d35-be2e-b6a8f8204950}} refers to the Rician factor, {{formula:1b0511da-e06b-4328-be9d-746f0465c144}} and {{formula:08cc4f14-9a70-417f-961e-22ee0e3a689b}} denote the LoS deterministic components and the non-LOS Rayleigh fading components, respectively. {{formula:b56ecc87-3d07-4108-b8a4-3975e61fc125}} is equivalent to a Rayleigh fading channel when {{formula:011348be-3716-494a-a363-7be2402ba458}} and a LoS channel when {{formula:635f67ad-e221-4d6d-8241-c7bc57b4ee69}} . Note that for the WD-BS channel model, we need to multiply the square root of the large-scale fading coefficient by the elements of the small scale fading coefficient. Similarly, the WD-IRS and IRS-BS channels can also be generated by following the above procedure with {{formula:a312b8c9-34db-498a-a391-9272e3ddfe49}} and {{formula:ffd4c867-b3bc-4638-898e-a95d6a94625b}} denoting the Rician factors of them. Furthermore, we set {{formula:73a9df74-052b-4db6-ba78-9affd8852a39}} , {{formula:ecfa8c95-d960-44c1-bfca-067bcdae0d7d}} , and {{formula:28733ab2-b0d0-4aca-b975-48d42f9ba878}} as {{cite:06b5e7bfacee5491d1367d4aaeae0397dd567a0a}}. The default settings of these parameters are specified in the “Communications model” block of Table II. Besides, the computing settings follow {{cite:0198502f2b27bb1b99bd897b9a4ed99d5b387b15}}, and are specified in the “Computing model” block of Table II.
{{figure:c65a8514-fdd8-4ce6-a756-f864402184d1}} | r | 79dec750984cde7d807b1a2c5fb971b2 |
For the operator {{formula:bd0386b6-052e-4d68-8d16-61c449195637}} on {{formula:e71f8c08-6be8-4eb3-a980-5190c6a97432}} we note that {{formula:d2e0f181-e0e4-46cc-97e1-934c013cac00}} Since the set of all {{formula:fae20151-6e13-4474-adf6-c1224696af9b}} -valued polynomials is dense in {{formula:b4763418-1d6a-42e7-9ba2-db0e993524fc}} it follows that the operator {{formula:3b530018-f3a0-495d-95ca-7b40cb26fabd}} on {{formula:79c18f60-af5b-445c-a00d-efeb8b7f01bb}} has the wandering subspace property. Following {{cite:db2d77ed7059851a857ea7bedad50133834736f0}}, we say that an operator {{formula:82d3373b-02dc-4e8f-8e98-80a90b93a425}} in {{formula:81e50538-6250-45a6-b344-73a6a9cd5389}} has the wandering subspace property if it satisfies
{{formula:8d3f6720-edb6-4476-8c0d-a85d82b4b0d5}}
| i | c3a8c69f22964fa03e987489f5837e83 |
The decay widths of the {{formula:da3e0f3c-6fce-4ef4-819e-4d10c2570628}} as the {{formula:9fbc9dea-5ab1-4343-8eaf-e1b72005cc32}} is shown in Table REF , and the predicted total decay width is 109.5 MeV, in well agreement with the experimental measurement {{formula:51401006-575e-4429-af94-eab6815ef831}} MeV {{cite:ab94a2fe56d1687f8f288c8e075ad247c30e8797}}. In this case, the dominant decay mode is {{formula:67c2fac7-f282-4679-96a5-cf3e63872b3c}} , and the decay mode {{formula:b11fdc2a-03bc-43a5-b36b-8958fb15b8a1}} is forbidden for the {{formula:3ad1ad04-3c91-466f-8cbd-1981ea369644}} assignment. It should be pointed out that the decay mode {{formula:0dfa0855-1857-43ec-8d0d-bd5754765c80}} has not yet finally been confirmed by the LHCb {{cite:a711b599ffb96b0de8aff67348bf49307639e257}}, which implies that assignments of the {{formula:54b845b0-76b7-43d9-aa82-72171a808673}} as the {{formula:b9d80ab4-4dc8-4448-aa35-ee1674c7abf3}} is acceptable.
| r | 57bee0a63df041ce0443658b82adcf97 |
Next, the feature generations of protein-protein interaction complexes is performed. The element-specific algebraic topological analysis on complex structures is implemented to generate topological bar codes {{cite:07964b51652f0f9b1b78bf207de136564b9fd67a}}, {{cite:5f439d4a043fea6356b5d4ad09c7f3883ab05e4a}}, {{cite:a1374ea7767918a8211578c3171bca949275d31a}}, {{cite:3b73d7d55a1a819c6b7293d98179e98a4df71c9a}}. In addition, biochemistry and biophysics features such as Coulomb interactions, surface areas, electrostatics, et al., are combined with topological features {{cite:d53703ed2c451367ff1bcd0702b01ecf5ec8ed07}}. The detailed information about the topology-based models will be demonstrated in sec:topology. Lastly, deep neural networks for SARS-CoV-2 are constructed for the BFE change prediction of protein-protein interactions {{cite:07964b51652f0f9b1b78bf207de136564b9fd67a}}. The detailed descriptions of dataset and machine learning model are found in the literature {{cite:97a04936cd5474ea36fa45c00d18161e40b68a9b}}, {{cite:5d6b7bbcd4eb27fad2375fae2cbc2b23ade9fc67}}, {{cite:07964b51652f0f9b1b78bf207de136564b9fd67a}} and are available at TopNetmAb.
| m | e05d56216d4e44ee36aac3b343da2935 |
Apart from this first motivation, the use of Wasserstein distances, and more generally of the theory of optimal transport, has shown to be an efficient tool in widely different recent problems of machine learning, with fast implementations and sound theoretical results (see e.g. {{cite:93666d4621e2c8820849ad86875334b73cda07c1}} for a survey). From a statistical perspective, most of the attention has been dedicated to studying rates of convergence between a probability distribution {{formula:9bf336da-e166-4c1b-bf7d-ac032feeaaa6}} and its empirical counterpart {{formula:fded145f-7f03-43e6-8bb0-49057fd558f7}} {{cite:aabcbdb0e41f48bb5ecf868f51db16cb2d89a055}}, {{cite:0e459530648e770e2fe54f45489e96a01cab7614}}, {{cite:d2cb0b8d7027fa1c61f72a3528089929964fa417}}, {{cite:4f245ddc8c0002f683c1d2e85dd36a255dad5644}}, {{cite:d461a58ca63383ffa0cdc2f421e52e69a8965ee5}}, {{cite:b33d289056e4a0ecdb77a133805981cfc2b2b916}}. Unsurprisingly, if more regularity is assumed on {{formula:884a3d9c-1bc3-4b37-896a-a7b5a0426dba}} , then it is possible to build estimators with smaller risks than the empirical measure {{formula:905fd80b-681c-457b-973b-fa8776fe117c}} . Assume for instance that {{formula:16476753-d45c-48ba-b890-55f85a5691d0}} is a probability distribution on the cube {{formula:beb2ebd1-6ef4-4480-b6e4-b7397b32b97d}} , with density {{formula:96862652-6322-46c9-89db-750a7ff61bea}} of regularity {{formula:676f5e6b-1f61-4228-bc76-337f74869d47}} (measured through the Besov scale {{formula:6e9d15e7-ac59-4d4b-804c-94ae6403d64d}} ). In this setting, it has been shown in {{cite:641c6bc9f3fdff56695133df58d8174a16e330ad}} that, given {{formula:633fbe64-08ad-47ba-a5d2-c53296e03ed3}} i.i.d. points of law {{formula:192f8dd0-d89c-45be-9c94-e258db602fb9}} , the minimax rate (up to logarithmic factors) for the estimation of {{formula:532583d8-a051-4379-b40e-22e523ffac30}} with respect to the Wasserstein distance {{formula:d744b3ef-46b1-4898-a0d3-dcd10c060cbf}} is of order
{{formula:8262f3c7-58bd-4ab5-90cb-94face80dd99}}
| i | 58408290996d43a18a26ece475c5912f |
Lastly, we present the results of GTZAN in Table REF . In addition to the models presented above, we also include the results from a regular Transformer {{cite:9096f366d6c5a3ecbe1f1617ec0851560ae58026}}, which contains only the temporal encoder {{cite:3b6a80c47065d9c64483e983fbfd39dfbde67316}}. As it is non-hierarchical and without the spectral encoder, we treat the comparison as the ablation study. It is clear that the regular Transformer does not work well in this case, most likely because the training data is insufficient. Whereas SpecTNT's design can handle this well by leveraging spectral encoders in a stacked architecture. Moreover, we once again see similar results to previous experiments when comparing among SpecTNT, SpecTNT-TCN, and TCN.
| r | cffa1cf2fedde58f4e62c890a2a91b4d |
We also know that material must travel from the CGM to the disk. Long term star formation rates in galaxies are inconsistent with instantaneous censuses of available stellar fuel in galaxies - there does not appear to be enough fuel in a galaxy at any given time to cover its entire star formation history {{cite:533dde04e72dc280a356786f9339b5c92ccea553}}. This means additional material must be entering galactic disks in order for star formation to continue at the observed rates {{cite:fe358759e4506984c625b83e6f7589abbb38b8dc}}. This balance between in-going and outgoing material is crucial to understanding the CGM.
| i | f8beaa6bbe4bad8648b94d882618c063 |
So far, all of our bounds have been based on cosmological data alone, including the one on {{formula:d9a147b0-b0df-43de-91a8-a2c867b924e4}} . On the other hand, oscillation experiments show that there is a lower limit on the sum of neutrino masses of {{formula:35dd08d7-3faa-4537-8ff8-1a7f093add90}} {{cite:cfc4267ae2effa2ff88174e9fefa6e529d44b466}}, {{cite:29616bd83970656fd317fd3c792150d5c9383078}}, {{cite:2c3898300900832ea8e40db1709b1c637c392c19}}. As such, by fixing the sum of neutrino masses to this minimum value, the bound on {{formula:51a2501a-8aaf-443f-87c0-f5d19e6f829d}} automatically translates into an upper limit on the neutrino number density of {{formula:72f2a512-4b47-439d-966c-bf6d5b20254b}} for a single neutrino species. Importantly, this constraint can be considered as a robust upper bound on the number density, irrespective of the value of the neutrino mass, or the form of the distribution function.
| r | 3d20f775500d0fd903f27b0e49320d57 |
which is called Riesz potential (see {{cite:88903784175bf808c06979e931f4fd1c6d5df81a}}), we will omit the constant {{formula:6330eb49-6b4a-49a6-8ea9-f189ccc34b78}} in the following. It is clear that {{formula:94c81ded-0486-4eb8-9c62-bae4a829d9c9}} for all {{formula:839e441a-09ab-4222-82dd-43e6c7c2b6a0}} . Then the system (REF ) can be reduced to the Schrödinger equation with nonlocal term:
{{formula:bc8e29c0-7e10-4eb1-a67c-84382e73aba0}}
| r | 745bc2a2beed72e45d9aecba6b588800 |
Results: Other Baselines.
We also compared our method with eight other OSR methods from the literature. With respect to standalone classifiers, we evaluated SVM {{cite:4a4384dc935f006a665880920a0744f439931083}} with thresholding, {{formula:c5c31b3c-f020-4bab-9c46-d33eeecb349d}} -SVM {{cite:1fa3ee5e2709a768ecbe0ed4b567d416d96a81cd}}, W-SVM {{cite:0b6af8823a248af348555e11367b034656a946bc}}, and the EVM {{cite:856badb8dd61ca127c774fc3c26e4496335647f3}} with MSD-Net features. With respect to deep learning-based methods, we evaluated OpenMax {{cite:b420887b259dc32ac45ff4b1a30a3daa3b3bdc57}}, OSRCI {{cite:e1543ed09eca2dc84a0d7001db756c794e7d5796}}, CROSR {{cite:cc8f3d8c7496f664256189207273174e2f34959b}}, and CAC-OSR {{cite:73d850ade423e7ad740ed3a7a93e280e53f40489}}. See Supp. Mat. Sec. 2 for descriptions of these approaches. For these experiments, we used the same dataset used to evaluate MSD-Net training with different loss configurations. As we did not heavily tune hyperparameters in our proposed approach to avoid overfitting, for a fair comparison we also refrained from doing so for all baseline methods.
Table REF shows results on known samples and unknown samples from the test set for all methods, including the different configurations of MSD-Net training.
| r | 2348b017b2326503f7daa757d295152d |
Furthermore, we observe that the SSIM loss term improves the stability of the adversarial training process and prevents GANs-induced up-sampling artifacts such as checkerboard patterns {{cite:7c0de51978872e3b964ff644af83ea14a5be7305}}. It encourages the discriminator to instill realistic contrast levels and image structures (e.g., fields boundaries and forested/urban areas), which have finer details. However, we notice a subsequent forcing in luminance that pushes generated crops toward lighter tones, while an {{formula:dd9b86e1-936f-4226-98e2-a2611077b9f4}} -only supervision renders more faithful colors.
| r | 2e848fa46bdbd95f53854af997d2095e |
As we explain in the following subsections, conventional methods such as ILRMA {{cite:db6c17e62d374b752e23fc402158be68fee806b1}}, IDLMA{{cite:6db86eb3e9de47be90909c8fe4e354bdcf9b6c12}}, and IPSDTA{{cite:c287f354a746902ed33ce22e5ec6e9c34476e161}} only differ in the way FCMs are modeled.
| m | 261f61ec84be23b5fc2fd0f204768ce5 |
The OpenMX code {{cite:f0a2925c3bab02f3d31542eeb6c9f4569a330055}}, {{cite:8b19668942e6a2c05062da1423a4ad1d19187132}} was used for structural optimization and electronic structure calculations based on the density functional theory (DFT). The exchange-correlation functional was considered within the generalized gradient approximation (GGA) proposed by Perdew-Burke-Ernzerhof (PBE) {{cite:5a0f903798f1507273cd7048ddeecf4d39a4d1ea}}. For the calculations of thin films, we prepared the slabs with the bulk in-plane lattice parameter ({{formula:231f69e1-a420-4e59-aa9f-d9d456058496}} {{cite:40d6fcd05cb5359a95187b59e20a80230245c0a6}}) and inserted the vacuum space along the {{formula:6777e8db-5c30-4f85-bc42-00b7a3631ed2}} axis which is larger than {{formula:48f19456-dfec-4148-9f30-daf0b1e35288}} . The quasi-Newton method was used for the relaxation of the atomic positions and the lattice vectors until the residual force becomes lesser than {{formula:46df56dd-38bb-47ab-a5c9-eb2631b338d1}} per atom. The structural optimization was performed by nonrelativistic calculations for the ferromagnetic state, except for the S-end bilayer and trilayer systems, where we took interlayer antiferromagnetic and ferrimagnetic states, respectively. Then, the electronic structure calculations were performed by relativistic calculations for various magnetic states (ferromagnetic, collinear antiferromagnetic and 120-degree noncollinear antiferromagnetic for the intraplane, and ferromagnetic and antiferromagnetic for the interplane),
and employ the lowest-energy solution. We set the cutoff energy for the FFT grid to 1800 Ry, and sampled the Brillouin zone with {{formula:3e18e593-9481-4382-a5b8-44ba82a0da6c}} and {{formula:6f34fefc-fbb0-4cf2-9bd3-dac018385420}} {{formula:9968b041-4ae4-4e82-b1f2-86ef4713d99e}} points for the thin films and the bulk, respectively. The local magnetic moments are calculated from the Mulliken population analysis.
| m | 4828da8bb39f91c9ed10d6490488da2d |
We begin with sufficiency. In a directed acyclic graph {{formula:2f170a75-388d-4078-8b60-deb3a8856f1e}} , suppose there are two directed paths {{formula:cb9d2851-1f71-42a4-a226-eb86fa8ee98a}} between {{formula:696d25ac-48d8-4126-b6f8-dab37754c806}} and {{formula:19c84813-dc52-4a03-a240-6a617b707e8c}} . We refer to these paths as {{formula:fcb18f55-dc3f-4665-8115-3b5fd0c850e5}} and {{formula:1dee86e7-2c29-4614-ac42-a18166eb53e7}} , where {{formula:40bafc4f-937e-4a6a-bf38-26ac26a5dd49}} of length {{formula:21193d57-8822-4cda-9463-b6bf1d4bffc8}} , for some set of {{formula:e882e710-221b-4133-acdd-fd1b1747e203}} intermediate nodes {{formula:51e02ee7-f07a-4eb4-a681-d4c03eea037c}} .
We use a result by {{cite:32b1a0845512757a8b1cc6b109b3a19bd34446da}} that characterises a correlation as the sum of all the paths between {{formula:203c2b7e-fc34-45fc-ac0b-e58c41f1ebd5}} and {{formula:56d74ad3-40bf-4247-8078-c7fcb597c62a}} where each directed path between {{formula:5bf5b0c5-c43c-4c0d-a892-55a889f47435}} and {{formula:d231cdbc-d6c3-4538-a5f0-9c73291bb6d6}} equals the product of path coefficients {{formula:0c19d3a5-febd-4de8-b052-d774b4301aba}} on that path, in this case of two paths {{formula:601696b4-8be7-4016-99ba-f2e30584cbd7}} , where {{formula:e5f7df8f-0a2b-4d49-8d3d-1ac340c0ae76}} is defined by (REF ). By Lemma REF only a directed path from {{formula:244c49e5-aca9-4b4a-aabf-712b2eb175fd}} to {{formula:627f362e-9780-4839-9945-e6fdf43879c9}} will contribute to the marginal correlation, and any common cause {{formula:f2df9676-2d76-4269-abc1-c696e8a6d63e}} or common effect {{formula:68b7ee63-2dfe-478d-9890-6f1407e58b3a}} will not.
Suppose additionally that there is a direct connection {{formula:9b90f61b-35c7-4203-b87b-ddad11f37f92}} with coefficient {{formula:bca3c2f8-56b1-4ce0-b53c-11473b34e049}} . Then the correlation between {{formula:97f6c74d-433f-4a42-ae77-354f055df164}} and {{formula:bc27bc6a-5b17-4636-9d61-1ad6b03a9525}} is {{formula:8a8d0e15-c342-4ad7-a5f4-a967d219eedb}} . It is now easily seen that whenever
{{formula:1c00b7ee-b454-4a10-8444-12f0222c0b03}}
where {{formula:50f849e6-0f33-49fd-8f46-ceb4e46a6f3c}} is the set of edges of all directed paths from {{formula:ad8f4424-36a0-4cfe-872e-bf6c7b865c65}} to {{formula:4682a2aa-0a9e-4921-b4ee-5da6ca0a9c50}} , the direct connection {{formula:2792cf3c-a8b4-4a04-8ea0-cee8d8b3f537}} with path coefficient {{formula:cfe40909-3c63-483e-9e87-3cc2b52a651c}} remains undetected by criterion (REF ). Only for values where the reverse holds will criterion (REF ) work. It is obvious that this argument remains valid if the number of paths between {{formula:f5246f95-d539-43e6-8cc8-5ccd72700246}} and {{formula:40fd9578-5364-459c-87a4-a9dbd812c75e}} is increased to any finite {{formula:6531a770-7884-4df7-b30a-68bf43cd665f}} .
We continue with necessity. Suppose there is no direct connection {{formula:385d54ce-5275-4234-9be1-70eed43bebb3}} but there is a directed path from {{formula:6f7a3e78-8291-452b-9e01-d809a89567f5}} to {{formula:29957866-18e8-4383-94d2-df3fa1ec4194}} , {{formula:96e39c12-b303-4d69-886e-4a5c1bd8f24b}} of length {{formula:3ff7eacb-1991-4dc3-8320-cc29e251efcf}} . Without loss of generality we can assume the variables to be standardised so that {{formula:1ef602f3-d9e5-4acb-b454-65a68b545290}} . Then from (REF ) it follows that {{formula:2cc38283-188f-43ee-9c87-baae34c19418}} , and so {{formula:17b0bdd4-9096-4dbe-b03e-48717aa7f944}} for all {{formula:a1fb8bb2-c76a-4b53-b036-6f2b83671b76}} on the path, giving criterion (REF ).
{{formula:950a538f-f397-4408-8fb7-5688a9d4c897}}
Appendix: Path analysis by Wright
We provide some details here on the calculations of the path contributions to the correlation between nodes {{formula:2bd56dbe-7948-4824-b0a4-944754277a61}} and {{formula:e6b172cd-ce5f-41bf-9109-6bb17bb37552}} for the small graphs in Figure REF . We have the system of equations
{{formula:56ec3654-9ffb-4142-9f8d-b68890c3d915}}
where the {{formula:19cf91f9-c0de-4892-9e49-4425e0db98ae}} are independent and identical normally distributed with mean 0 and variance 1. Then, each variable {{formula:5ca37e61-4a6c-45d7-b854-5872cfdfe843}} has mean 0, and
{{formula:585e2183-3824-44c2-914c-8f167e1efc53}}
as {{formula:2706abc0-2276-4d22-b632-258b62430c20}} and {{formula:28f95f8a-4862-442c-a515-99ec4bf57a3e}} for both {{formula:dcbfec47-32cb-4ca9-ba92-106e8a8f9113}} and {{formula:0f972108-5f86-46f8-84c2-4b1a3dbeecbd}} . So, for the correlation between nodes {{formula:85199fa2-1079-452e-8388-cdd0eba17edf}} and {{formula:728baa4f-50c3-40b6-8e66-9c4e76a40e21}} (see also Figure REF (a))
{{formula:47b7574a-5704-4662-bf23-00dea55fad27}}
by the assumptions that {{formula:771d3c35-1565-4885-bee2-3c1454322c77}} and {{formula:771651e0-fee0-4abe-9534-169ae7f3881f}} .
Then for the correlation between {{formula:a42298c4-edab-4b5c-b40d-25db575acb2b}} and {{formula:43fd6c05-0891-4c2b-8f52-c4f23a7ae7d4}} we have
{{formula:19806e39-6d09-44c5-bd89-eff6f5e98db3}}
Because the residuals {{formula:e7b263fa-1b07-4684-996c-a800385de65f}} are independent normal with mean 0 we obtain {{formula:e7fd457a-2533-463c-a9f2-9ee44d6ab49d}} for any {{formula:8964847c-1426-4175-85f1-91a3af5eceac}} and {{formula:8d9f61ab-2e8b-4cb5-b081-9988f6153963}} (also {{formula:6659f278-e823-47d0-b0a5-a833b92ba765}} ). Subsequently, we fill in for {{formula:ff9d181b-1614-4f9f-b480-2ed71fa6bbe7}} and {{formula:e4b9cea0-8c59-43fd-892c-8420248d9968}} . Then we obtain
{{formula:6ef41879-d83b-4141-84d4-0d41465c65fb}}
Noting that {{formula:3679dfc5-f3c4-4d70-b3c7-b5580e0280fe}} by assumption, we obtain the result. (See also {{cite:f47a7c40a24d1f82ba7d36bd42f81fdee9ee3602}} for a convenient alternative formulation.)
Appendix: Calculations of example in Section REF
Here are several details of the analysis for the graphs in Figure REF (a). We use the system of equations corresponding to Figure REF (a), which for the observational context {{formula:dad0a108-4648-425a-983e-3a9ac17af4b1}} is
{{formula:b7580e88-d7ce-4ee3-b7db-ec960e6a9d7e}}
And {{formula:360cc9bf-d122-43df-b39a-1bd3d8c704ee}} for a hard intervention and {{formula:19eaa58b-6672-4631-ae37-1a6929281a69}} for a soft intervention, where {{formula:13bc8044-f0b6-4688-bcbb-0d4dc6d24505}} is a random variable with mean {{formula:f40c5c57-e6cb-479a-a682-0b0ea14dfc8b}} and variance {{formula:f3209e31-5ce1-492a-99e8-4f0deeb90158}} . We limit ourselves here to covariance analysis since we need to show that in certain cases the covariance (and hence the correlation) is 0. In the observational context {{formula:bbc2b911-9411-4bf8-969b-26238395d1d2}} the covariance between {{formula:bd6c4b30-2587-4df0-be77-d77d4b37e583}} and {{formula:acab81a5-88ed-454a-9941-bd0da7adc4d2}} is
{{formula:56c86b7a-9868-4acf-9d5b-ec271a2426fd}}
where we assumed that the variance of all {{formula:2dec258a-166d-404c-9965-d0cefb94c58a}} is 1. When conditioning on {{formula:98d7fe0f-480c-4c2f-aec4-6b5376817dcf}} we have that the covariance between {{formula:8a84304e-9f0d-4fe9-b639-79ad5dec3aa1}} and {{formula:6e516e56-921b-4a21-89c8-32e156020ac1}} in context {{formula:e8d1260f-1762-4404-b43f-7fd9f39b7769}} is
{{formula:41d34983-6499-415b-8f15-b6a11ed26320}}
which corresponds to the graph in Figure REF (a) where there is no edge {{formula:d9718667-03c5-4f6c-9465-2401defd61b6}} . In the hard intervention, including conditioning on {{formula:b36f6830-8c5f-4fbc-be9b-734209e79678}} we find
{{formula:c4b363ca-9f5c-4765-9fc5-f61c815db27e}}
since {{formula:a325453b-5521-48d2-bbdb-3b57de1750a6}} is independent of {{formula:efb54756-478e-4f0a-8d01-46a8db7d570f}} . Similarly, we find for the soft intervention when conditioning on {{formula:23ec7b0f-c74b-4096-9032-66e9b9cbc8b5}}
{{formula:74925a72-f30c-433c-bf3a-f9c85cf90e4c}}
Without conditioning on {{formula:76f39c3a-c13a-4e3f-871f-7e106a6011e4}} we find for the hard intervention
{{formula:6ff92b56-bc39-4f65-ae4e-46e87228d35c}}
And for the soft intervention
{{formula:457935f3-ca42-4106-a257-b7fe9653a026}}
This shows that that the correct inference is obtained only when conditioning on {{formula:5d749f7c-dcfd-41f0-b37e-376eba5ce80c}} .
Appendix: Conditional invariant prediction with unobserved confounders
In causal analyses it is often assumed that all relevant variables are observed and included in the analysis (causal sufficiency). If causal sufficiency holds, then the assumption that the residual {{formula:ec0a6aa9-7511-4116-8444-499dff4d9a25}} is independent of the support set {{formula:d756c541-f379-459e-b178-5755256adeaa}} should hold. However, to assume causal sufficiency is rather presumptuous {{cite:a9709be02277a36b5c4d38bc0a691b9103c7eb85}}. For example, suppose that we have the graph of Figure REF (a). Here node {{formula:cacdac08-1563-4950-ba2f-fea03c285bb3}} is unobserved (hidden) and so we work with the marginal distribution over the nodes {{formula:1ad03793-36c8-4baa-8210-452f7d5897a2}} and a binary variable {{formula:90218dd5-01f4-4715-b624-b52d24c9a43c}} that induces a soft intervention such that {{formula:b47a0f12-d3e8-4061-a7c3-82f41c5dff19}} .
{{figure:a143b02b-f203-467c-87a0-13065580d65e}}The system of linear equations corresponding to the graph in Figure REF (a) is as follows.
{{formula:e7fc42f7-4f69-4867-b734-6f7cfaf8bac6}}
If we know node {{formula:a3bfa9d7-fb08-4b63-a0c2-23d0a5d73e30}} then we have that the residual of {{formula:f2e8562b-52d1-45dd-997e-eaf31a5b8c15}} and its support {{formula:f34c2112-6472-4eb4-81ab-f34266b17d3a}} are independent, because with the correct coefficient {{formula:de4c0d8e-1e45-4ef5-8a47-378dc8865751}}
{{formula:d6f9da99-a2e1-408d-a504-ca5fdd397a1a}}
where {{formula:98e5cedf-391d-4419-ad01-bbb92739e3f2}} is the projection matrix on the space of nodes {{formula:b7a3b7fc-7d08-4c6a-80a4-38d2a210c685}} and {{formula:ab9dfc72-6fb6-482d-832b-44ff0d44e439}} , and {{formula:906aca0e-45ef-4306-ad9e-5390a9a7b85c}} .
However, when {{formula:e04424c4-989c-4ef2-9b77-3134b9af6523}} is unobserved we make the residual orthogonal to {{formula:817b8177-1b61-492b-9d5b-fbf8b72f63ec}} only, whence {{formula:10808d4c-7f1b-4a4c-9717-cd5eda2ddc8e}} and we obtain
{{formula:f8423c48-2c76-4d46-8509-c2d2e7beaaaa}}
wher {{formula:1d3c8011-9744-46bb-9d24-1d04c6a90e27}} is the regression coefficient with only {{formula:8e752b06-5953-4bad-b24f-9ba40a29350c}} as predictor. This is 0 only if {{formula:7d15d09e-d2d5-406a-8e97-7c0c14303417}} or if {{formula:e7595e4d-1ab9-4c5f-b3b9-d5d57908f35a}} . It follows that when there are unobserved confounders, it is not possible to demand that the residual {{formula:2666b9e8-9c37-47e8-b1f4-7f08303ca1d4}} and its support {{formula:3291ff03-5829-4a8a-a567-948bba0c5f01}} are independent.
We can then take two differente routes to infer causal relations in the setting of unobserved confounders. (i) We can weaken Definition REF and allow for dependence between the residual and predictors as we saw above; or (ii) we can include an additional variable for which we know the relation to the target and source variable (instrumental variable).
To begin with (i) where we weaken Definition REF , the idea is that this weakened definition represents causal effects that may have been caused by unobserved variables {{cite:3823c723b1169cecea0de91f34c446411c722b60}}, and so the conclusion is weakened to ancestors as possible causes. The representation that is obtained from the measured variables is simply an incomplete picture but it is known that some predictors are ancestors of the target variable.
The second route is to use an observed variable to remove the correlation between the residual and the predictors. We therefore have to invoke an experiment or known environment with known causal relation, such that (1) for any soft univariate intervention from {{formula:4ad4831a-9bb1-4b50-bdd8-fb8f5ceb02db}} to {{formula:dd12cb8d-6a51-44a0-9d80-d5ed6d2e1eca}} the effect cannot be constant, i.e., {{formula:8a4f483a-5a6b-4adc-b64d-41030281d56c}} , (2) the variable {{formula:643d8a35-a795-4e65-b602-25f33cb5a52a}} ({{formula:d225a06c-a268-423b-9103-5575f1397e8d}} or 1) is connected to a single predictor, and (3) there is no feedback from the target variable {{formula:c073eb99-3a55-4078-bf1e-3db41cba5993}} to the other nodes {{cite:3823c723b1169cecea0de91f34c446411c722b60}}. Figure REF (b) shows two (dotted) edges that violate the criteria. The dotted edge {{formula:11fadf64-06ed-4332-a912-a5ed88781ffc}} violates (2) that there be no feedback from the target variable, and the edge {{formula:202df6a0-8945-4120-8e00-a4662594408a}} violates (3) that the instrumental variable only affects a single node. In practice, we can choose an environment, some intervention on a particular (single) node {{formula:f19c3ce9-add4-45d1-823b-eb17a604f109}} as the instrumental variable {{formula:cace6416-b82a-4f8d-8a5f-82206db0f8ce}} . Considering the assumptions (1)-(3) we must demand that the intervention has no feedback from the target node to any of the other ndoes (criterion (3)). So, this is basically a stronger version of conditional invariant prediction, where we assume a more fine-grained intervention of the environment (the experiment nudges a particular variable), and an assumption on no feedback.
Appendix: Additional lemmas
Lemma 13.1 (Reichenbach's principle)
Let {{formula:f2039243-eaf6-4156-aa23-a37a6e27048f}} be a directed acyclic graph with probability distribution {{formula:eced7c17-fa71-4506-8c81-5a1c033b3c59}} that is Markov and faithful to {{formula:7ffbf5e1-5b74-4166-9241-5176d5ceb49e}} . Suppose that for nodes {{formula:880c36cb-cb8f-4c32-8749-ae97c9eed63d}} and {{formula:bd9d1aa7-8b1c-4a12-ba6e-b39adf54e151}} in {{formula:2ce31b79-a096-4d48-a8a4-d8c2fcaf3a25}} there is a non-zero correlation {{formula:59b890b7-4db5-4d34-88b9-489b8eecfdde}} . Then there is a directed path {{formula:712ab135-1385-4589-8b9f-15d7786dd8bf}} from {{formula:4cc54826-a95c-47b5-9ca3-053e1c83d8bd}} to {{formula:d1dcea00-30db-4cd8-ac4e-b431d5b8b5fd}} , or {{formula:c2e7e047-5699-4194-9038-af16ffa053da}} from {{formula:18c7a525-1f9f-491d-96a1-445d7b874dfc}} to {{formula:ca98babb-9a22-4870-a340-0db48c14811e}} , where the intermediate node set could be empty (a direct connection), or there is a common cause such that {{formula:7a5a91f4-4042-4dab-b4bd-278feaa9a2f7}} .
Proof
Assume we obtain a correlation {{formula:686fa821-7ab9-44b1-94ca-75e3a97c6d89}} in the directed acyclic graph {{formula:866396fc-2ea5-4536-b5c4-347bded6ab1e}} . Because we assume faithfulness (Assumption REF ) this implies that there must be some path between {{formula:decee389-31a3-45b3-ba84-2e2fa86bc22f}} and {{formula:f6ce8e22-e3d9-4143-a6d8-34c82d8d285d}} . If there is a direct connection {{formula:cedb8e6c-180d-429a-a61a-845c95125c6a}} or {{formula:a6714f9e-31c5-4e4f-ab55-f0193061c8da}} , then we are done. Suppose then that there is some path between {{formula:bae588ec-2dad-4c20-865d-20ae6c443318}} and {{formula:723a0783-fc98-41a6-9e6b-5094b4afb89a}} with at least one intermediate node. Towards a contradiction, suppose that on this path between {{formula:07dce85e-cccd-4ace-840e-0908fbb52e0c}} and {{formula:5f8a1bdf-a02f-4b53-858a-21f28097c774}} there is a collider {{formula:acde483d-be8b-4529-ad5e-ccb01ea80760}} . By the Markov condition (Assumption REF ), this implies the independence relation {{formula:0f6439be-f7cf-4312-9190-f71f508ba24b}} . But this contradicts that we already have {{formula:0907870f-68ea-40db-a3e6-ca5e3082ebc7}} . The only remaining possibilities for a DAG are
{{formula:f55f50f4-f668-4fa3-ae32-af72264cd18f}} or {{formula:84f6d248-53c7-4494-936b-a00dc2872b13}} or {{formula:03de3b11-57d8-4360-a885-0e157dc1fee6}} .
{{formula:94435ec1-aef4-4928-97b9-b93e5b7ca4b3}}
Lemma 13.2
Let {{formula:c3b63c15-8690-453f-bdef-b8101b8447fa}} be a directed acyclic graph with probability distribution {{formula:31d6e265-1eaf-43a1-abff-8ce399de2e6b}} that is Markov and faithful. If there is a non-zero correlation {{formula:95311aa3-d48c-49be-9c5f-ec04d67b30f1}} between {{formula:6df3ace7-0e91-4b48-b980-d71b146840de}} and {{formula:7c2b264f-6ca8-4ec5-b538-b09fb0ceb043}} , and if we intervene on node {{formula:868f688b-6ad5-4b22-ba97-1056a3b21ac7}} in {{formula:77cfcd7b-2edd-49f8-acda-22f28894d32c}} , then we observe a change in the distribution of {{formula:39afa500-10ad-4573-a1cb-c68e2faef142}} only if there is some directed path {{formula:d813162f-9cb9-4d8a-8e72-7053770a49ef}} .
Proof
By Lemma REF , the faithfulness and Markov conditions imply that are three options upon the finding of a non-zero correlation {{formula:7a258105-b280-408b-a335-2af2561fd4b4}} . A hard (or soft) intervention will result in a change in the conditional distribution such that {{formula:02aeb183-0728-454e-a629-a56c4b1b385e}} . By the faithfulness condition, this implies that there must exist a directed path {{formula:79add44e-263c-4236-a80c-a3d46d260bad}} .
{{formula:8090b405-56e9-414f-8532-650580beb07c}}
Appendix: Simulation examples R code
# libraries
library(Rgraphiz)
library(pcalg)
library(qgraph)
# general settings
n <- 100 # nr of samples
p <- 3 # nr of variables
set.seed(123)
# model s -> u -> t
graph <- matrix(c(
0, 1, 0,
0, 0, 1,
0, 0, 0
),ncol=p,byrow=TRUE)
labels <- c("s","u","t")
rownames(graph) <- colnames(graph) <- labels
qgraph(graph,labels=labels)
# model parameters, beta != 1
betaUS <- 1.8
betaTU <- 0.9
# model
B <- matrix(c(
0, betaUS, 0,
0, 0, betaTU,
0, 0, 0
),ncol=p,byrow=TRUE)
IminB <- diag(1,p) - B
# data generation according to graph
Z <- matrix(rnorm(n*p),ncol=p,nrow=n) # n x p matrix of standard normal variables
X <- Z
## intervene on s by stochastic perfect intervention
# model s -> u -> t
Zi <- Z
Zi[,1] <- rnorm(n) + 1 # intervention on node s
Xi <- Zi
# plot scaled regressions
Xs <- scale(X)
Xis <- scale(Xi)
plot(Xs[,1],Xs[,3],pch=16,col='#FF000050',bty='n',xlab='node s',
ylab='node t',xlim=c(-3,3),ylim=c(-3,3))
points(Xis[,1],Xis[,3],pch=16,col='#0000FF50')
## intervene on s by stochastic perfect intervention
# model s -> u -> t
Zi <- Z
Zi[,1] <- rnorm(n)*0.3 + 1 # Z[,1]*0.3 + 1 #+ 1.5*rnorm(n)
Xi <- Zi
# plot scaled regressions
Xs <- scale(X)
Xis <- scale(Xi)
plot(Xs[,1],Xs[,3],pch=16,col='#FF000050',bty='n',xlab='node s',
ylab='node t',xlim=c(-3,3),ylim=c(-3,3))
points(Xis[,1],Xis[,3],pch=16,col='#0000FF50')
# plot conditional (on u) scaled regressions
Xs <- scale(X)
Xis <- scale(Xi)
Xis.ip <- scale(residuals(lm(Xi[,1]~Xi[,2])))
Xs.ip <- scale(residuals(lm(X[,1]~X[,2])))
Xit.ip <- scale(residuals(lm(Xi[,3]~Xi[,2])))
Xt.ip <- scale(residuals(lm(X[,3]~X[,2])))
plot(Xs.ip,Xt.ip,pch=16,col='#FF000050',bty='n',xlab='node s',
ylab='node t',xlim=c(-3,3),ylim=c(-3,3))
points(Xis.ip,Xit.ip,pch=16,col='#0000FF50')
############# common cause model #############
set.seed(123)
# model s <- u -> t
graph2 <- matrix(c(
0, 0, 0,
1, 0, 1,
0, 0, 0
),ncol=p,byrow=TRUE)
labels <- c("s","u","t")
rownames(graph2) <- colnames(graph) <- labels
qgraph(graph2,labels=labels)
# model parameters, beta != 1
betaSU <- 1.8
betaTU <- 0.9
# model
B2 <- matrix(c(
0, 0, 0,
betaSU, 0, betaTU,
0, 0, 0
),ncol=p,byrow=TRUE)
IminB2 <- diag(1,p) - B2
# data generation according to graph
Z2 <- matrix(rnorm(n*p),ncol=p,nrow=n) # n x p matrix of standard normal variables
X2 <- Z
## intervene on s by stochastic perfect intervention
# model s <- u -> t
Z2i <- Z2
Z2i[,1] <- rnorm(n)*0.3 + 1
X2i <- Z2i
# plot scaled regressions
X2s <- scale(X2)
X2is <- scale(X2i)
plot(X2s[,1],X2s[,3],pch=16,col='#FF000050',bty='n',xlab='node s',
ylab='node t',xlim=c(-3,3),ylim=c(-3,3))
points(X2is[,1],X2is[,3],pch=16,col='#0000FF50')
## intervene on s by stochastic perfect intervention on s
# model s <- u -> t
Z2i <- Z2
Z2i[,1] <- rnorm(n) + 1
X2i <- Z2i
# plot scaled regressions
X2s <- scale(X2)
X2is <- scale(X2i)
plot(X2s[,1],X2s[,3],pch=16,col='#FF000050',bty='n',xlab='node s',
ylab='node t',xlim=c(-3,3),ylim=c(-3,3))
points(X2is[,1],X2is[,3],pch=16,col='#0000FF50')
| r | 7aea455d2f9dca71bea5d1b23395dc89 |
Lamprinakou et al. {{cite:ef09236cf9b41af0ae7395ea3d0488d6acd7b840}} have introduced a novel epidemic model using a latent Hawkes process with temporal covariates for the infections and a probability distribution with a mean driven by the underlying Hawkes process for the reported infection cases. A Kernel Density Particle Filter (KDPF) {{cite:6e4bd7083ec81f219ef79492e348e2bc35144e1a}}, {{cite:9408083950372939a5685d03b4c0f4be56b6e73d}} is proposed for inference of both latent cases and instantaneous reproduction number and for predicting the new infections over short time horizons. Modelling the infections via a Hawkes process allows us to estimate by whom an infected individual was infected {{cite:ed39349db683c847940006f470a1c556631d8089}}.
| i | 0ba645c52bae87ca8a939951995e525d |
In this work, we check the necessity of the sophisticated operation, shifted window partitioning, for bringing back the global receptive field proposed in Swin {{cite:a6c5a0d2a0f0916ac4ec62316358e19cb251cdd3}}.
To differentiate from the shifted window-based local attention, we term the vanilla window-based local attention without shifting as plain window-based local attention.
Surprisingly, we discover that, after pairing a plain window-based local attention layer with a simple depthwise convolution, it has consistently better performance than Swin Transformer in image recognition, semantic segmentation, and object detection. Meanwhile, with the existence of depthwise convolutions, the shifted window partitioning in Swin can not bring additional performance improvement. Thus, we degenerate Swin to a plain Window-attention Transformer (Win), which is conceptually simpler than Swin, Twins, and Shuffle Transformer. Specifically, we have removed quite a few lines of code in Swin Transformer. Meanwhile, our Win Transformer achieves competitive performance in image recognition, object detection, and semantic segmentation. Compared with previous Vision Transformers, the proposed Win Transformer does contain any sophisticated operations. On the contrary, it removes some redundant operations. Taking the simplicity, the efficiency, and the effectiveness into consideration, our Win Transformer is a good choice for real applications.
| i | a455ba858bd002ece145f50a86ee1c25 |
Next we record the following interpolation theorem from {{cite:dbf134fa459141eca85cc211579dfbcdc71c9b8d}} for further use.
| r | 1ad8b6c705304b47ab4e18cad662a70c |
From Tables 1 and 2, it is observed that approaches specifically targeted towards mitigating distribution in a partial domain adaptation setup yield better accuracy than standard domain adaptation methods like DAN {{cite:f5940196d4bfc86887a0fd7f5cb76cc8e5793d73}}, DANN {{cite:0e34b58d0c926dd0075a46b46324c05ffc6df574}}, ADDA {{cite:fce2c17f7ecd67461dfce1986b738a53dcc93a81}} and RTN {{cite:1be28753eab87b4d7cff803d01a32d6cc9d625bf}}. When tested on the Office-31 dataset, the proposed model achieves best performance in two out of six tasks. It produces the second-best average accuracy value (trailing by 0.12%) when compared to other baselines. During evaluation on a much larger and complex dataset (Office-Home), it is observed that our model outperforms the rest in seven out of fourteen tasks, in addition to achieving the best average performance.
| r | a460a5ac50d200b683841070d2fa4e9f |
From this structured prediction perspective, we see a considerable limitation of existing works on uncertainty-aware learning for GNNs, namely that they only employ nodewise metrics as evaluation criteria. Given the abundant existing work on uncertainty-aware learning for standard multi-class classification {{cite:362072c641adf4369acbf05ac09bff133de8e827}}, {{cite:0812e7a3f893323f5f271154118c36730c6cfabd}}, {{cite:ee1c9ea19859d508e8d53c797f399c426a0540c0}}, many ideas have been adapted to the GNN case. This includes approaches based on Bayesian formulations {{cite:98419236a815eecdd938bb1619d2a3f4d68565fb}}, {{cite:dad666efaea932b5c0ae5c23dfaa58bf5928a4f2}}, evidential learning {{cite:104af32953202086620b238ee625ff24fdaab5e9}}, {{cite:488235c1109d63453f0e6a449b7a63426622f0d5}}, as well as post-hoc calibration methods {{cite:8230b97b940c081e256a50628ff485b12e5d69db}}, {{cite:e429078c835207d04d1c17113c72c0d3926375fb}}, {{cite:979d2cdd1bf444d769b6139228ca1df37ccd2f9f}}. However, for estimating the quality of predictive uncertainty, these approaches directly use the nodewise metrics from the multi-class classification literature, which are intended for i.i.d. test samples and ignore the graph structure.
| i | 777a80f60904f795dbf1aed9431b050f |
The {{formula:8a6582a8-d3f7-45f5-84cf-f4f69ad85584}} -means clustering algorithm{{cite:2a71bc5d97de1125969d2ef6d49f0d2469fa256a}}, on the other hand, has the advantages of being simple to implement, with a single intuitive choice of parameter (number of clusters, {{formula:1c0d930f-ded8-4d17-920e-4eb311c95071}} ), as well as being scalable to large datasets. The {{formula:fb8ee8b5-afed-485e-916a-3f55dd763a42}} -means clustering algorithm has been extensively studied in cases with outliers as well, and it has been shown in works such as those by Im et al {{cite:d54895b3b0cd184f4aa62e3fa11ea44af3e9ef2a}} and Bhaskara et al {{cite:53f19564800800b55ec7d72d4f68de1cd4dbbe8e}} that even in datasets with noise, a high-quality clustering can be obtained after removal of outliers, while using the {{formula:1a2d5857-911c-4051-ab8f-5b1899587551}} -means algorithm. Other literature on using k-means on a noisy dataset can also be found in the references {{cite:1b41037edc768e019299bcdc03ea6c130c6d50af}}, {{cite:b4316dabb184af96ffdea2a995be04d4a8483e6b}}, {{cite:839c43b7fdbbe0332b296f064848e38aac9a14eb}}, {{cite:5681402701ddacc5a88ac0bd2f521911f6a45e12}}, {{cite:5b26364d760b4060698eb7f4b26648b2bcf31a81}}. In addition to {{formula:8444720f-d3ff-4263-9573-245c137413bb}} -means, hierarchical agglomerative clustering (HAC){{cite:4a6777f4114271dcb6b0b2a4def95962e3aa1f17}} is also a possible choice of algorithm, as it is also scalable to large datasets and has an intuitive visualization in terms of the dendrogram. Users are able to easily view the distance between each level of agglomerated clusters and decide on an appropriate distance threshold. Thus, for this paper, we will be using the {{formula:bc0f7605-af57-4180-ba6b-1bede491ba1a}} -means clustering algorithm as well as the HAC algorithm for the purposes of clustering.
| m | 8842c674e1435992f1c05c39889e42d1 |
{{formula:493420f1-f51b-4d72-a4a6-601a99d3dd39}} is the density operator of chaotic field. This equation
enlightens us to have a new approach for deriving {{formula:41ed5804-5d81-4861-85da-78f2ade98ef9}} {{formula:b4a59e6d-8608-4b50-8d52-12fd87411d1b}} in doubled Hilbert space should be such constructed
that its partial trace over the tilde freedom may lead to density operator {{formula:f2cacbcc-477d-4935-82c4-7b51620fca26}} of the system. In the following we shall employ the
technique of integration within an ordered product (IWOP) of operators {{cite:e42fa86f5a46a80b6ed380ed7d3332ffd0892047}}, {{cite:3f4c0c11015a7efcd34699c07e1a1b1d3458d924}}, {{cite:61bb5bd8c81e52d34c455a2c6a918d769fd9d623}} to realize this goal.
| m | 6a5f66ae36299be4bef0ec049ae61033 |
Another important problem is to clarify to what extent
the giant graviton expansion works.
In this work we focused only on the maximally supersymmetric theories:
4d {{formula:4b016c0c-a7b9-4405-b607-656296506ab6}} SYM on D3-branes,
3d {{formula:23d3e908-453f-4757-b560-c4d12661301b}} SCFT on M2-branes,
and 6d {{formula:9be2ea41-a06a-49f7-b25d-70ceb77249cf}} SCFT on M5-branes.
The analysis on the gauge theory side {{cite:da42cba28b3a6648594f3f4ed38c4f52b86c3d55}}, {{cite:928f5c0aa6714ce3206c905ef5a1b291a4584846}}
found
the structure of the giant graviton expansions
in variety of theories.
It would be interesting to study the applicability of our method
to more general class of theories
which have holographic dual description.
| d | 610540fcdad72749950c709df966be46 |
In this section, we introduce our method via three subsections.
In our study, we focus on two crucial semantic labels when describing human faces, age and gender, and generate images directly from these labels, rather than editing randomly generated images.
To do this, we take advantage of the closed-form factorisation algorithm proposed by{{cite:314e49b71f9a225f44710b2314eda5f11b0f191c}} to project the latent vectors of our latent space into the scaling factors of the linear space and then use a pre-trained classifier which can output a continuous outcome to classify the randomly generated images according to their pre-defined semantic rules. This approach classifies the corresponding scaling factors at the same time with a different value range.
As a next step to achieve better diversity of the generated images, we propose to uniformly sample the scaling factors in each channel within the corresponding value range.
In the following subsections, we will elaborate on the methodology.
| m | 0f55dba7d5d7bedffd525eace3b7012e |
We separate the oscillations {{formula:9adf26f8-157f-4bde-b717-20d9b3e2987b}} and {{formula:1d651ad2-b3f7-4737-aa11-bc9abb71e3ff}} in the sum {{formula:96761508-b9c7-4cad-9b70-07ca44e851db}} by using the delta symbol {{formula:2659632a-2c4e-4218-8479-6197b5fc5be9}} which is defined on the set of integers by {{formula:758fc2ff-3fd1-43a7-bcbb-636121b8b113}} and {{formula:5f9c4500-94c9-45a7-8674-964f405ea487}} if {{formula:87b7ac33-9cd7-407a-af52-166bc1e512ff}} . We have the following expression for {{formula:486182ee-f839-480e-84a6-c2c8a28cc1eb}} which is due to Duke, Friedlander and Iwaniec {{cite:8c908b0d4172a37e8622f1ed01df794920b82249}}. Let {{formula:25d4bc76-c95c-4f99-8f5f-e0a5e4d0460c}} be a large number. For {{formula:f9c84cf4-21b5-4ad2-ba44-05ea5e85646b}} , we have
{{formula:b2fb5c2a-6a88-481a-af10-d6a22e17bff4}}
| m | 539216d50962ef7aef833327d67e1b19 |
There are many scenarios, however, where it is natural to transition away
from this strongly supervised setting with fully labeled examples. Above we
note ranking: individuals are very unlikely to provide full
feedback {{cite:7e10f8c62f1bd7e3c264dc8ca8d4ab027214cb00}}, {{cite:b3280e99ccc160e90bd47e3d976e1cf731071dfc}}, {{cite:d0a61d1b8d4e216dc0a4e4cea1c352804d8d58b9}}. In multi-label
image classification {{cite:53eddcb31e8ec834b742881644b9c94cf59e2555}}, {{cite:0131c9ce8dc0be27318bf7bd12e8a6b2194ff46c}}, a labeler may
identify a few items in a given scene but not all, leading to partial
labeled feedback. A major challenge in industrial machine learning
deployment is to monitor models once they are in production, where it
may be challenging to collect high-quality labels, but weak supervision—in
the form of clicks on a recommended website, or agreeing to a suggested text
message completion—is relatively easy and cheap to collect.
In all of these, developing valid confidence sets and measures for
our predictions is of growing importance, as we wish for models
to be trustable, usable, verifiable.
| i | 97e213f4b35a26d0fd3ac6fd7ece75ea |
Having derived scaling laws from our experimental observations, we are able to make predictions for both smaller and larger scales. Extrapolation has its limits, as saturation effects at both lower and higher scale ranges have been previously observed. We can however extrapolate to scales close to the ones we have already measured. A prediction for larger ViT-g/14 trained on LAION-2B with 34B samples delivers an estimate of 79.1% ImageNet top-1 accuracy. This may appear at first sight modest compared to results reported by BASIC (85.7% {{cite:64a63c2e73a95aa731215c11ca9ec7ef7fbdcf46}}), LiT (85.2% {{cite:130eb06f1fcce012e96559bf0c20678d41c20d7a}}) or CoCA (86.1% {{cite:996f7fbf8f0f5e014f587c8e48a621c4e593ca8f}}).
However, these works leverage an internal JFT dataset with labels which can be used for supervised pre-training. Moreover, for 973/1000 ImageNet classes, researchers were able to manually identify a correspondance from a JFT class {{cite:f134939d28dcfbbb02cdd694fc783734b3b6727b}}.
These works also use larger encoders, larger private data, and pre-train the encoders in multiple stages. Nonetheless, we estimate based on our empirical findings that further increasing model and data scale could result in competitive models even without using labeled data, additional supervised pre-training stages or additional losses.
Finally, we observe that the improvement of zero-shot ImageNet accuracy due to scaling up is accompanied by closely aligned improvements on robustness benchmarks.
| d | 63992fb14dbcb82e57404ff7222ccb38 |
Dual Learnable Prompts. Instead of learning a single prompt for a class {{cite:35f7d0567f7e9e724622185caec420823b16bbc7}}, we propose Dual Context Optimization (DualCoOp) which learns two contrastive prompts' contexts for each class. The learnable part in dual prompts carries positive and negative contextual surroundings individually and can be optimized end-to-end from data via binary classification loss. Specifically, we define the pair of prompts given to the text encoder as follows:
{{formula:c63ac30f-a8e7-42a6-8c01-75291f87a2e6}}
| m | 654368e36eb297960acc670f3dcde126 |
Rethinking ImageNet pre-training FGVC datasets remain significantly smaller than modern counterparts on generic classification {{cite:ceed32a9deaa19b215cb4642def49eee89d14757}}, {{cite:ed9e73177ad7bbbdc4e3e9f71370014040c60289}}. This is a direct result of the bottleneck on acquiring expert labels. Consequently, almost all contemporary competitive FGVC models rely heavily on pre-training: the model must be fine-tuned upon the pre-trained weights of an ImageNet classifier. While useful in ameliorating the otherwise fatal lack of data, such practice comes with a cost of potential mismatch to the FGVC task – model capacity for distinguishing between “dog”' and “cat” is of little relevance with that for differentiating “Giant Ibis” and “flamingo”. In fact, our paper argues otherwise – that coarse-level feature learning is best disentangled from that of fine-grained. Recent advances on self-supervised representation learning provide a promising label-efficient way to tailor pre-training approaches for downstream tasks {{cite:8b0f2af93dab646b4209f0878d8b0b268d510831}}, {{cite:9540acd3b733ada696f833e12d011288a7700d18}}. However, its efficacy remains unknown for FGVC.
| d | abbebe8e3740bda76cc8ecec680ca04e |
While such advances are a great scientific achievement, they come at non-negligible costs in terms of resource consumption {{cite:a48b289db3766e31e796a43b9bfee5fad8f2e5fa}}, {{cite:0bcfa155573d07ba1e1c114f5765677ae7667ca1}}, {{cite:3c8a86a3d9fe7f3453683995f201fdb7cabc2a22}}.
For instance the largest model in {{cite:89c6fb133f51e20e9d2a25c103b4e7e76989efae}} is trained during two weeks on 64 GPUs, and this is even not that expensive compared to larger models or models trained on far more data {{cite:ff479d2349d15bb674cc1a9bdb1f2ff121ea3fa9}}. We can argue this is still a limited cost as such models are trained once and used for many applications.
However, SSL models are often a component of the whole system, and they need to be fine-tuned on the downstream task to achieve good results.
This practice multiplies costly training phases, which may lead to huge resource consumption.
| i | 5f38ffde47d4d7ff9c66aea9b5b6dbb9 |
Many new problems in machine learning and signal processing require
the robust estimation of geodesic distances between nodes of a nearest
neighbors (NN) graph. For example, when the nodes represent points
sampled from a manifold, estimating feature space distances between
these points can be an important step in unsupervised
{{cite:c057c9c1bf9bfbaee96bc68e35f56b3b183b65c7}} and semi-supervised {{cite:e20833ea94b424d4d384b7b7b5807d9ff8433fca}} learning.
This problem often reduces to that of having accurate estimates of
each point's neighbors, as described below.
| i | 1b824f541b00a962cef416776cc54cb9 |
PWC-Net {{formula:b5548344-b785-4af4-a019-ec6e25b5bafc}} GRU We can apply a similar strategy to PWC-Net {{cite:b28952c0bf7e1acc88ed2a39f084eade54c35c15}}, a recently introduced network that achieves excellent performance
for two-frame optical flow prediction tasks. The network first feeds two images into separate siamese networks,
which consist of a series of convolutional structures. Then it decodes features and learns
abstract representations at different levels. Similarly to FlowNetS {{formula:2339ba14-1554-42fe-9585-5e03eeaa9557}} GRU, we can also feed encoded features at
different levels to GRU-RCN units and we call it PWC-Net {{formula:9a823d0c-ecbc-4dcf-ae83-3c8b08e3f36c}} GRU. A depiction of this second network structure is omitted for clarity.
| m | 907c5b37d5503f851fa5a7451ec91522 |
The final performance of the RR+RA model with optimised hyperparameters was compared to that of the state-of-the-art BI-R "baseline" model, using the optimised hyperparameters as stated by van de Ven et al {{cite:d23fbc4b5c4f8f3a51e6ef1d5c424f132f79ad13}}. Given that CF directly reduces the final classification precision of previously seen classes, this was main metric employed as an indicator of continual learning performance. All results stated relate to the average test precision across 5 runs of the experiment with optimal hyperparameters. Due to the nature of CF, it will tend to have a greater detrimental effect on the learned representations of earlier-seen classes, whose internal representations are more likely to be 'over-written'. Thus, the average precision on the first 50% of classes seen will tend to be lower than the average over the last 50% and any successful method will likely lead to an improvement, not necessarily in the overall average precision, but in the average precision of the first 50% and most likely the first 20% of classes.
| r | 4e5e4a4fe49aaa1725ad7215e5a5f972 |
Much of our predictive performance evaluation is based on the empirical probability that credible and/or prediction intervals cover the true value, which inherently conflates a frequentist property (empirical coverage probability) with Bayesian modeling frameworks. This type of assessment is in line with the notion of calibrated Bayes {{cite:350e1aa86fa8a2ada25f97bd232bfcc1ec15f064}} and recommended in a predictive context {{cite:cabe8f6d8df9195b4828cd44a2a7cc33ca559d58}}; moreover, it is the authors' experience that coverage probabilities are frequently used in assessing Bayesian models, particularly in a spatial context (see {{cite:b4534f8d9c2b560a3f0c4b693809c60b3f1bbfd8}}, {{cite:cbf97a59eb6f27dfefc5917ec932bc7cf4a776a7}}, {{cite:1e62e32e7c5be42b8c26a9cb6374de02e5d825a5}} as example), where prediction is the main goal.
| d | 708f9df0b386337405b6524c558446b2 |
Given the above, future research into team verification and design should perhaps
more closely incorporate computational complexity analyses like those given here.
Such research could initially focus on more fully characterizing those combinations
of restrictions that do and do not render team verification and design tractable.
Such work has already been started {{cite:f73e2b39961f958e5fcfc2df578c2c508a60368b}}, {{cite:72db63bdea849d120b8a63d85f0e559e5a77ee92}}, {{cite:ec48d4ad0dafad8a19b1c7c16c8744008e6ecd53}}, {{cite:ee57eee0f07c1001731943660fe4c8284cb7a143}}, {{cite:abde1976b17bc0e961260250e2acde3829fe31fe}}, {{cite:c8a4aeb395d03c55683d5ac7352c689d44b89000}}
for team verification and design in swarm robotics relative a variety of restrictions
using more advanced analysis techniques (e.g., parameterized
complexity analysis {{cite:30d5797b88e8ff107878d7094dde3c8d7fb877b5}}). Additional restrictions of particular interest here
are those that “break” the reductions underlying our intractability results, e.g.,
restrictions on the degree and type of structure encountered by agents in their
environments (including the presences of other agents). Once these initial intractability
maps have been derived for simplified team operation models, they should be extended
to more realistic models incorporating stochasticity and uncertainty. Part of this
can be done by using complexity analysis techniques that explicitly incorporate
stochasticity
{{cite:297907640c967322b1d2656423ce7e352de903b5}}, {{cite:18aac5b7f750b71287e54214e3fd6c5a381e6daa}}, {{cite:5c09206f9c4e1fba67da93a352c505ba6abad5b6}}. Complexity-based frameworks that incrementally build on
simplified operation models in a systematic and principled manner to create results
applicable to more realistic models (analogous to those developed in linguistics
{{cite:00f1c5d9fce0cf7cb2d7623aec69e2201f02e1d5}} and cognitive science {{cite:ec91c1a6e529e1e05ec0e896056e339d7d1479e5}}) may also be of use in this
endeavour.
| d | d3af836b4b89d7f7cac99604056198c6 |
paragraph41.75ex plus1ex minus.2ex-1emConfirmation of landscape independence.
As a consequence of our proof techniques, we also confirm a prediction of {{cite:22da7123926dc3f0b5779a90043db98db6ff4333}} in the {{formula:174eafd4-9379-49ec-b8cd-98233b0eff61}} -depth regime for {{formula:de98072f-2a2a-48de-a394-d8e6ec9704b1}} by showing that the output values of {{formula:b665d5d4-6a18-4b99-9f04-ae44225b5147}} on a random {{formula:02a0cd57-dbbc-4f9b-ba5d-e430538e0e22}} -{{formula:0ee86800-1e42-4e37-a938-ac2620cfdf9e}} instance (with depth {{formula:f7c5fe7c-d43e-4e6c-963c-abc4228774a9}} as stated in thm:vanish-nghbd) concentrate very heavily around the expected value. Once again, the expectation here is with respect to the input distribution as well as the internal randomness of the algorithm.
| r | 855174d6d67db4447b8a72a65db4d6b5 |
The QCD sum rules is a powerful theoretical approach in exploring the masses and decay widths of the {{formula:64b73951-a0d1-4d88-866b-f425932df244}} , {{formula:2621243c-25ee-4601-ab10-91c4a3d867be}} and {{formula:332f31d7-c9cb-462f-aaf8-334e320201ab}} states, and has achieved many successful descriptions in the scenario of tetraquark states {{cite:974881aba42e3529d33ba0b7e1c070abd80eba6b}}, {{cite:3dd6d77f9cfc8b0202277c9d2c15137a52d9dae7}}, {{cite:41bda13fa2fbe71a2bf691b6d4dd703e0348d394}}, {{cite:37ac80da5fcc09e11a7e10872682281d9d55815c}}, {{cite:fe2c7d7e1aa0a96c77e3dbd1a1e5f081a0c9668a}}, {{cite:12b5f76a256c296538d5c71e95fb3a5a9cc8f363}}, {{cite:c93ea303416f364345b4c958fefc8f71748615fd}}, {{cite:c529343eaaa05a8f91097eeb8cff3469d77e2b6d}} or tetraquark molecular states {{cite:06a6d54117ffe2a112fd1b6d83764d485782d43c}}, {{cite:94299b4ea129407ce24ca66c2e194ce1483d2a75}}, {{cite:b762c88cc17b45feea303a3c40c72791932275ec}}, {{cite:b1a299f025d112005016808c870ef3915ce641a8}}, {{cite:d7c9f9c4825e36f48f64eb661c107dfcfcc9377f}}.
| i | 78695d57d3e70dba2b66b10758510ccb |
Further, as a first demonstration, currently we use a stochastic gradient-based method to solve the inverse problem. While it is easy to implement, it can also result in slow convergence. As the propagation through imaging system part of the forward model can be efficiently inverted {{cite:f83205a34480ac5688bb6e2d572ee6b89eafbb37}}, we anticipate deploying variable splitting methods {{cite:ff5aee8491cb2523b811bb62ae479b43e73aef36}} can significantly improve the convergence speed, as well as allows flexible use of regularization methods may not have an explicit form {{cite:7daab9861a29653977bd5dfd378e728dd7ad6764}}, for example. We hope to explore in these directions to ensure successful translation of T{{formula:0cbfb3a4-80b0-4632-b391-c6b7b04ef4c9}} DPC to future clinical and scientific studies.
| d | 8e58f46aba248426733351add0e30ca7 |
As shown in Section REF , we focus on the score-based attacks. The analysis and evaluation of RND against decision-based attacks are shown in Section C of supplementary materials.
Besides, as we mainly focus on practical query-based black-box attacks where the models and the training datasets are unknown to the attackers, we don’t cover the attacks utilizing the transferability from surrogate to target models.
These methods usually assume that the surrogate models are trained on the same training set with the target model.
It is difficult to obtain the training set behind the target model in real scenarios. Meanwhile, there have been some defense strategies developed for transfer-based attacks {{cite:55b63fb73f62f61122afc2a80777bb242e2b4542}}, {{cite:dcfe922764af8bcc950f30430ce9f4c62623b162}}, {{cite:412d18160502454af84060147f6b107f16fcc3d2}}.
It is interesting to explore the combination of RND and these works towards better defense performance against transfer-based attacks. We leave it to future works.
| d | 162c7f9cb4fe8899a4e683377d17544c |
2) Class Balanced Re-weighting Loss {{cite:4603179d089fd7a13ff24513ccd6677a5645b0f2}}: This method uses the inversed effective number of samples to re-balance the loss. The loss is calculated as
{{formula:9b71bfd3-b032-42e0-986a-976af64f2dc0}}
| m | 10fb33f38d06883aaaf48f584121a94b |
For the classical Newton's method, the step size is fixed to {{formula:11e8efc4-84b8-4f3b-ae5f-7f9a70030214}} . However, many modern implementations implement Newton's method as line-search using 1 as default value and adapting the step size if necessary {{cite:52d25b8eb245f9752313b91a28b831e4e3eb57fa}}.
| m | 474bc3b53fbacbc6c84859c701290b4d |
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