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where {{formula:64820e0c-660a-49e2-b351-74ac01410643}} represents the fluid velocity, {{formula:4680ac25-6e8e-41b4-a471-8ecaf8cf8b0d}} the pressure, {{formula:cb00b14f-9c6e-49c3-b39e-7b57ce9c2471}} is a Cylindrical Brownian Motion, {{formula:d0dda1fa-30a7-4a1d-87d2-b6f3241519f4}} represents the nonlinear term and {{formula:fabdf921-b2f5-4e13-a187-61f1516c3309}} is a first order differential operator (the SALT Operator) formally addressed in Subsection REF . Intrinsic to this stochastic methodology is that {{formula:fd2cad67-fa6f-4284-a82f-1e1bd2690dff}} is defined relative to a collection of functions {{formula:572f28cb-e6e6-4154-81f8-0ba89ffc8d3c}} which physically represent spatial correlations. These {{formula:e754334e-434b-4a7b-9156-92fc740d44ec}} can be determined at coarse-grain resolutions from finely resolved numerical simulations, and mathematically are derived as eigenvectors of a velocity-velocity correlation matrix (see [{{cite:552fdaec9986fd29a8ad1d57ce4c251970d3fd39}}, {{cite:59cef20df83e1e21434c3ff88265b9b72a1e3001}}, {{cite:3a23384b409c59483ac44831b448c8a4216c0db0}}]). We pose the equation (REF ) in {{formula:034d25c1-8a5f-446c-bccf-b8ed7b5a2b92}} dimensions for {{formula:892e1eee-6fc7-4e15-a628-c71c3e4933cf}} and impose the divergence free constraint on {{formula:7e171c5a-a4e8-47d2-9b66-6113b584082a}} . We shall consider the problem both over the torus {{formula:3e326452-5f1c-4e3a-9d4c-a818c79deb87}} O RN{{formula:07f58f62-66b8-46ad-9f18-a13c784cd41e}} n{{formula:6bd457ed-48c4-4911-b782-da718009f553}} w{{formula:b496473a-5aaa-4b27-9c5f-a38697991a39}} 2D{{formula:cb19333c-2588-4a0f-b703-eb135db08c8d}} 3D{{formula:9990d39d-73c9-4686-93d8-f6227283d9aa}} 2D{{formula:05c8da06-c3d4-47da-9736-2b634ed2956f}} This work continues the theoretical development of fluid models perturbed by a transport type noise, which has seen significant developments since the seminal works [{{cite:587b83688fc2a47f74b1e748a63324192ac6c039}}, {{cite:1157024c7446915f4c18a18bb442362eac9fbe36}}]. This paper partners that of [{{cite:9fe6f01417f186f30309a318c20c7b47d0f34aef}}] where we showed the existence and uniqueness of maximal solutions to Stochastic Partial Differential Equations satisfying an abstract framework, built to cope with a general transport type noise as we see in (REF ). The significance of such equations in modelling, numerical schemes and data assimilation is reviewed there, along with the theoretical developments of these equations. We only draw particular attention here to the Navier-Stokes Equations, and results on a bounded domain. The Navier-Stokes Equations have been studied with transport type noise, for example in the works [{{cite:a752e5447586f7c4ed4f7ffc46f5a3bb52ad30ae}}, {{cite:5577fe38984c7567cde74c0980bb5fcff385dbb4}}, {{cite:07ac68a841f2fde3a3b64ca53d4b46799ca417e9}}], though typically solutions are analytically weak and where strong solutions are considered major concessions in the noise are made. In these cases a cancellation property is evident in the noise term, so that in energy methods the differential operator is not felt. These difficulties have been addressed on the torus, in the likes of the papers [{{cite:b7abe2d3e3baae3215537977adaf6ef4ef107143}}, {{cite:d89dfd0ce809b45c250834cbaa1bf7eac4ed9f65}}] and those further addressed in [{{cite:9fe6f01417f186f30309a318c20c7b47d0f34aef}}], but extending a control of this noise term to a bounded domain remains open. Indeed the situation becomes more complex in the presence of viscosity, as energy methods require non-standard Sobolev inner products to conduct the required integration by parts in the bounded domain, rendering control on the noise terms completely out of familiarity. The problem of analytically strong solutions to fluid equations perturbed by a transport type noise in the bounded domain has been considered in [{{cite:f62ef7f26ef58ba40785675c31c2a3e1b41e182e}}], though the authors assume that the gradient dependency is of a small enough size to be directly controlled and that the noise terms are traceless under Leray Projection; such assumptions are designed to circumvent the technical difficulties of a first order noise operator on a bounded domain. Our results of Section pertaining to a bounded domain thus represent a substantial improvement on the literature.
| i | f1d9f3ccb14952ef920f1f2ddf8f2669 |
First, the evaluation results of the proposed models regarding the deployment speed are provided. The performance is tested on a low-power NVIDIA Jetson TX2 module with 8GB of memory, which is a state of the art GPU used for on-board UAV perception. Additionally, in order to accelerate the deployment speed and achieve real-time deployment, TensorRThttps://developer.nvidia.com/tensorrt deep learning inference optimizer is utilized. TensorRT is a library that optimizes deep learning models providing FP32 (default) and FP16 optimizations for production deployments of various applications.
In Table REF we provide the detection speed in terms of fps for the two proposed models on the NVIDIA Jetson TX2 module without the utilization of the TensorRT optimizer, with the TensorRT on the default mode, and finally with TensorRT on the FP16 mode. As we can see TensorRT and in particular the FP16 mode significantly accelerates the proposed models, achieving detection in-real time for high-resolution images. To gain some intuition about the deployment speed, we note that state-of-the-art detectors run at notably fewer FPS on Jetson TX2, and also for lower resolution input images. For example, YOLO v.2 {{cite:847a365c7c0610584ad49ec6ec4f2780b24d2cc8}} runs at 3.1 fps for input of size {{formula:09c36e03-9b56-4ffc-b1a8-b352059132c0}} , while utilizing TensorRT (FP32) runs at 7.8 fps, and further speed up is achieved with the FP16 mode up to 14.4 fps, which remains far away from real-time even for lower input resolution. Finally, we should highlight that the deployment speed regards all the models, that is with and without the proposed regularizer, since the regularizer does not affect the deployment speed.
{{table:11136afc-5fdb-4c08-b625-a7a802c7aad8}} | r | 2f736fa9228cfc71117b0b10ad4aa589 |
One of the main advantages of using PINNs is to leverage Automatic Differentiation (AD) for approximating with high accuracy a derivative at an arbitrary point in the domain. This is a powerful and flexible mechanism for evaluating the residual function in PINN. However, we note that it is possible to calculate the residual function using a fixed number of collocation points, e.g., the nodes of a uniform grid, and approximate the derivative using a Finite Difference (FD) approximation. A similar approach is used in classical fractional PINN {{cite:0754e3f8378d95002345446229a43daabc37bc1c}}. This method comes with the disadvantage of using a fixed grid point, hurting the generalization of the solution to different collocation points and expressing the calculations on finite difference stencils.
{{figure:453eba26-3044-40fa-8f9c-dbc01de2809f}} | r | 9745d0be95c52e63ce5adc4cb7d08556 |
This generalizes the prediction of Vafa-Witten in {{cite:3f2d1d9f5678207bc9c03c678ea5188bdf73236d}} to the gauge group {{formula:65c66ec2-98b0-4ae8-9701-47f87b11106a}} and proves Formula
(4.11) in {{cite:2b02fb2eeacbc7cf6f62757fb621da46cece88ee}}.
| d | 106088b0e136bcd8a7556778d18fa4ce |
For the high-energy data set (Fig. REF ) we used a beam at {{formula:1320e29b-4848-4a16-b952-7c5f60aafcc3}} 590 from the accelerator, which was degraded by the Al foil down to 471{{formula:2122916f-4293-4953-b5c9-dd32d074be2a}} 25 . This energy value was chosen mainly for purposes of comparison with the previous measurements on the eROSITA mirror sample {{cite:614224adbb57771f7f81bebdfb207a019cd655c4}}, {{cite:764adb2094f4a9f96e0603fae31e1be5b74b2181}}. The rationale behind the low-energy value can be found in the work of {{cite:9d2c79796e4f32ed2e17840abe0fea0f6a9c57a9}}, who showed that the highest transmission efficiency of soft protons is observed for those protons impacting the mirrors with 40-60 keV, for both instruments on board of Athena. Hence, it is crucial to investigate the scattering of soft protons at energies around and below 100 keV. Unfortunately, the present setup could not reach such low energies, limiting us to a proton beam with an energy of {{formula:1e586fd7-ebc3-4ee5-8054-79e4a3d38d0e}} 340 at the exit of the accelerator, degraded to 172{{formula:f0c42dbd-5c51-46f3-ae98-19ab5df6d354}} 30 by the Al foil. In both cases, the values of the incident energies were determined by simulations with the software TRIMThe TRIM code is one of the SRIM (Stopping and Range of Ions in Solids) group of programs, available at http://www.srim.org/index.htm#HOMETOP. {{cite:cd469f6922aa4debbbb3ef3dce922af971748c78}}, already validated in {{cite:614224adbb57771f7f81bebdfb207a019cd655c4}}.
{{figure:08b5c842-55f3-49c6-9309-9b24747a70e2}}{{figure:7848c6af-cd75-48e9-a521-6f97a4eb0353}} | r | 1eff4bb2bf4f67cd8816c00a767360df |
1)
The metastable state in the probe computation of {{cite:1f2bcd1f36959356c85d96d346177f0d564816a4}} is a spherical NS5 state.
2)
A natural candidate for the resolution of the observed supergravity singularities involves the formation of a spherical NS5-brane state á la Polchinski-Strassler {{cite:1f982426cb26d424023358e817c4a01e50471107}}.
3)
The exact supergravity arguments of {{cite:5142cfcadc7b77a9b87d4a828e3647fc32101bb8}}, {{cite:b8b8825782f7bf8c5bfb2f7d7ea010eff07029ec}}, which are based on Smarr relations, provided a natural explanation for the observed singularities and indicated that the no-go theorems for point-like anti-D3 branes can be evaded for spherical NS5 branes in agreement with the probe computation. In particular, there are no regular extremal solutions of point-like anti-D3 branes. Here and in what follows, the terminology `point-like anti-D3' refers to solutions with vanishing NS5 brane dipole charge and spherical horizon topology (more precisely, {{formula:738ce94c-6ace-4c69-8892-724555a665a5}} horizon topology). Regular solutions of NS5 branes wrapping a 2-cycle, with horizon topology {{formula:3d6dd281-ed44-4afb-a1b1-dfaae5af5a06}} , can evade the no-go theorem. At finite temperature, point-like anti-D3 black brane solutions can in principle exist but they cannot have a regular extremal limit. Regular black NS5 brane solutions wrapping a 2-cycle are in principle allowed.
| d | d6d6cefab6cabd27798930a046fdf593 |
Some other works seek novel transformations besides geometric transformation. UVTON {{cite:c9d7845022c1627adc27b60ed5c1f1cc313c1025}} deforms on the 3D dense pose {{cite:f2fea94824ae4d3e858cde6bad5049363c86c48e}} and then fill the surface with clothing textures. Han et al. {{cite:16734f705d716c91a0a50d5110959f55131dff40}} propose to warp based on clothing flow.
| m | d3d23583e28eca6ef8fc3077fb1062fa |
In Figure REF , we consider a set of parameters satisfying the dispersive regime described in the previous section, {{formula:270c1cc3-837b-4a91-a975-b4806774bfe4}} , where the tripartite hybrid system reproduces a strongly coupled optomechanical system well, that is, {{formula:e24ece18-73c0-4941-9a19-18519334d3e3}} , {{formula:3627662d-5d93-44b2-924d-d66c8c2727f2}} , {{formula:6132e00a-5f6f-4a6d-9795-f307a164ade1}} , {{formula:a174b16e-6e72-49a4-892c-583987c66f82}} , {{formula:761d4526-dc04-40cf-83e8-b6e8f9660974}} . In this case, in fact, {{formula:51f14924-9737-4df3-b946-03e06d6343fc}} , and, as predicted, the two master equations, Equations (REF ) and (REF ), yield almost indistinguishable predictions. Moreover, the typical signatures of strong optomechanical coupling manifest themselves because we see that, in the weak excitation limit of small driving rate {{formula:55708302-26bb-494c-b331-46734248ef7e}} , and in the resolved sideband regime {{formula:118360c4-98aa-4748-b8b0-0d61b91e82d2}} (the blue and red curves of Figure REF ), the resonance peaks corresponding to the absorption of single mechanical quanta are clearly visible {{cite:0d92605a7b57953c0f01d31eca83e92ee271af32}}. At a finite thermal phonon number {{formula:85aa7b8e-4c2e-4e19-bf8f-63f7e1de3235}} (see the red curves and the caption of Figure REF ), additional peaks appear even though for increasing {{formula:d264a6aa-e9a5-49f4-9a5f-3b9e106b3fba}} they tend to blur into a broad thermal background {{cite:0d92605a7b57953c0f01d31eca83e92ee271af32}}. The various resonances overlap and vanish as soon as we move to the unresolved sideband regime {{formula:9dc09963-54bf-4faf-b36c-a305c7cc1bf0}} (dashed black lines in Figure REF ), and we get a broad peak, even larger than the standard Lorenztian response of the cavity, which tends to be reproduced at strong driving and not too strong coupling. We notice that thanks to the mediating action of the off-resonant qubit, the optomechanical coupling has been increased by three orders of magnitude. We also notice that in Figure REF a, we use the effective detuning {{formula:184e78f1-451f-4a04-baad-111d591f9f11}} rather than {{formula:b8535024-864b-435f-a16a-3b99e61fd51a}} , in order to take into account the cavity frequency shift associated with the mechanical displacement term, {{formula:3fdf7058-2396-4c32-964c-7f21ab37cf47}} in Equation (). All the other parameters are given in the figure caption.
{{figure:c2006968-7cc2-4719-920a-b2cbbdf236e1}} | r | a5b759208d57c415e406a60db13dbaaf |
Following and extending the renewal approach from {{cite:0dca8cf0ecaaf9c5a3713f6a6c179b323e8a8ff4}}
(see also {{cite:72add659191dc60a131769548563cfe31d9d8074}}, {{cite:726e1867b68831341e558cf7aeaea9038fd96dd8}}, {{cite:8962ccab10c3c64a5ee433d5a3df746548228dd4}}), we compute the full propagator
with resetting, denoted as {{formula:a79a4733-3d0b-4b03-9640-f4ef7b6af04e}} , by counting the
number of resettings up to time {{formula:bf2b8b5a-e1ab-4598-a47e-2a88e30bdab5}} and adding their contributions:
{{formula:adf955d6-a187-4364-92ce-6fc12eecd21e}}
| r | a6045ec8ce567ebe9dfb8add70369425 |
Using kernel methods, it is possible to model non-linear relationships between the data points with a low computational complexity, thanks to the so-called kernel trick.
For this reason, these have been widely used to extend many traditional algorithms to the non-linear framework, such as PCA {{cite:89f8f438533aac067c7ac1ea4305d693e1306e24}}, linear discriminant analysis {{cite:324ab9bcb5eccb51af275e9e9c7f9af4b82929d3}}, {{cite:e57861bff417f988954f6e40fadc782c63ea195f}}, {{cite:e9e133bf1198a5fc5706cc3e6b66c62402d6faf9}} and ridge regression {{cite:0a2096f7c7c83723b2a127a5d8fd9acc3fd8bcf3}}, {{cite:d7dd96504a3e6a21defeab6adb730a1df3288cf8}}.
| m | 5c6f64ea99c58c942efe529b94ad84d9 |
In order to study collective effects in a specific collision system, the anisotropic flow observable {{cite:8691ff76e78820a0fe491d14c68a8e6b575da93b}} is used. In a heavy-ion collision, the non-zero impact parameter makes the collision region anisotropic. This uneven geometry will result in anisotropies in the momentum distribution of the produced particles as the QGP expands. Measuring the angular distribution of the final state particles allows one to compute the flow coefficients ({{formula:1a31ffc6-f984-4dc2-8c9a-64722495d417}} ). Each of them is sensitive to different effects (initial geometrical anisotropies for {{formula:f76bf8dc-2290-42b6-a09c-0487d916a702}} , event-by-event fluctuations for {{formula:508777ef-3df2-4096-a4af-1b018c14a5c8}} , etc.) and can then be compared to theory.
{{figure:1a48ca2b-c051-4c67-811c-880e28f84be0}} | r | 0f17f20337faccd8d64272339156af97 |
Uncertainty {{cite:c6c7001be68c2b8fb2e3e30f1f2e8f2265dc8e8a}} assumes that the higher the uncertainty of task data is, the lower the weight of this task loss should be assigned. They design a learnable parameter {{formula:0193c349-52a6-44c1-a606-758ddc7211a7}} to model the uncertainty for each task. Specifically, they optimize the model parameters and {{formula:fafb5e33-dc8b-451e-a45f-1794acc9a472}} to minimize the following objective:
{{formula:d19ba247-f6b0-452d-9ee6-13569eaa1d70}}
| m | 5b1f91caa8235596292597519aec75cf |
As proved in {{cite:8fbb0351448e9d6b1bf3b33fb8d9679af6dcdcd1}}, current framework inevitably mines incorrect rules
with high confidences, i.e., if there are several rules sharing one or more predicates,
confidences of rules would be coupled mutually. Intuitively, this is because Eq. (REF )
distributes the score of a rule to the predicates that constitute the rule at different hops.
For instance, brotherOf {{formula:6aba1d05-5f3c-420c-ae14-3f49e7b63fc4}} sonOf and brotherOf
{{formula:1e021994-8d12-463c-804d-8561aec29b15}} sisterOf share brotherOf at first hop.
However, in case our query is sonOf, if brotherOf wins high confidence at first hop,
the score of the second rule may not be too low, which is absolutely an incorrect result.
This reduces the interpretability of the output rules.
Present models are faced with the dilemma where there would be invertible relation
pairs and rules of varied lengths mixing up. (i) A relation pair {{formula:ab183341-f396-4bd1-83dc-af7267fa55c4}}
is invertible if there simultaneously exist two triplets {{formula:31fa918e-61cd-43b1-80d3-6454f414ba3c}} and {{formula:45e782a5-ae49-49e2-9133-ff537cdb2117}}
in a KG. (ii) The KG example shown in Fig. REF consists of
candidate rules of length 2, 3 and 4 for query brotherOf({{formula:723fd0bf-63eb-4f22-8e4f-8d88188d7895}} , {{formula:17344e82-c3a4-49d7-9ef4-a61e590659c2}} ).
These two factors jointly may cause invalid induction results, under the condition that
we choose an improper hyperparameter {{formula:c028270c-0e8b-4d2b-8b58-d7d7d0f41619}} as the maximum length of rules, e.g., if we set {{formula:5de76844-66a1-4c2c-895b-4d67971b5bd3}} ,
the rule path sonOf({{formula:3c853705-c337-4c97-b133-d56559bb2a3f}} , {{formula:88c57dcb-caa7-463a-8ae8-96e8caab8f9a}} ) {{formula:c8278246-05bb-42b8-b7eb-776ed060063a}} motherOf({{formula:3fc001d2-219a-4afc-b570-f0139a859bfa}} , {{formula:7e758991-47e1-4766-ad83-f68d88820ed0}} ) {{formula:88e71768-ef97-45b1-902e-dc006c34eb22}}
sonOf({{formula:89423783-1179-4563-87f5-48dcdc474e6c}} , {{formula:af489910-9571-4570-a0f7-69a07d83f618}} ) {{formula:08bac9a5-ed98-4e66-9f03-23a52a64c74a}} motherOf({{formula:b02c74ac-c35c-4f10-bbb7-eac42a4fecf0}} , {{formula:ebb75d0f-84c8-46c0-ac04-14a594d5f274}} ) {{formula:e7facd84-4608-4f57-baac-a5ff727f6691}}
brotherOf({{formula:68efbd14-758d-463f-afb6-e7b6f2b32945}} , {{formula:7579e061-1a9f-4e06-b2eb-a8e65890da70}} ) is possible but meaningless. This is also
essentially due to the distribution of confidences brought by Eq. (REF ),
and thus impedes the way for multi-hop reasoning over long rules.
A high ranking of an entity results not solely from a top-scored rule,
but also a number of relatively low-scored rules. As formulated in Section REF ,
the product of vector-matrix multiplication is a scalar representing the number of unique paths,
and the final score of the entity is computed by summing up the confidences of all paths.
Again, metrics like MRR and Hit@{{formula:6afd8e61-d437-456b-a938-237032d04f0c}} only assess models in terms of the ranking of the desired entity,
rather than the quality of mined reasoning rules.
Thus, models following the Neural LP framework with high MRR and Hit@{{formula:4e6a58cd-1646-42f1-89ce-125f614936a8}} may be better suited to
tasks like question answering or relation completion {{cite:93bed5d9c44ace906e0fb7e64b2f68149d2c9791}},
but this may not be applied to rule mining.
{{figure:684afaae-392a-4239-823d-c484ced95c32}} | d | d411c9fe74bba8137b9ed8fd6eedcf8e |
The rapid pace of adoption of GNNs in neuroimaging applications urges us to pause and ponder about the rationale and success of this adoption. While we agree that the motivation is well-grounded in the field and in fact necessary for sample-efficient factual learning models, we question whether the current methods are living up to the promises of the motivation. We argue that it is difficult to answer this question without a systematic evaluation of the methods and their current alternatives under a uniform fair setup. In this study, our aim was to build such a framework by benchmarking popular classes of GNN architectures through the applications of brain disorder diagnosis and phenotype prediction. The choice of the application is motivated by the trend and interest of the researchers to use GNNs for such tasks and the open-source nature of the datasets.
In building our framework, we strive for a rigorous fair and setup through i) uniform model selection and assessment strategies across methods, ii) fair allocation of computational resources to all the methods by providing an exhaustive range of parameter search spaces for each architecture/task combination to obtain the best possible performance for each method iii) uniform training setup (e.g. loss functions, regularization, initialization schemes) for all deep learning methods to disentangle only architectural effects on the performance.
Our empirical results show that, on average, GNNs fail to outperform current learning alternatives, and that and no one GNN architecture was consistently superior over the others. A simple 2 layer 1D CNN where all the ROIs stacked as channels with no structural awareness of the graph consistently outperformed sophisticated structurally aware GNN methods. The results of AST-GCN – a dynamic model that learns the graph structure at the cost of more training parameters – showed an improved performance at the largest sample size over FC-driven graphs. This opens the question whether graph structures defined using thresholded functional connectivity, as is done throughout this work and as is the current practice in the field, is the most optimal strategy for constructing the graphs. We highlighted that thresholded FC is a major issue when computing graph theory measures in network neuroscience, yet, remains an unacknowledged problem in GNNs research. We proposed to integrate graph diffusion as a prepossessing step and show that it can partially alleviate the problem and consistently improve the results. While not the only solution, this calls for a more data-centeric approach when developing learning models from fMRI graphs over a model-centeric approach where we observed that architectural differences did not play an important role in the final outcome in the studied applications.
On a more general note, the empirical results open the debate of the utility of deep learning in neuroimaging applications {{cite:f2c7fd10e2aa8e45e0853370f66184681b36d082}}, {{cite:4bfce4227c7c2f4f37e18672e0942b9d31ffb78b}}. On the clinical datasets, the SVM model performed competitively with the DL models without the need for engineering complex neural network architectures. We argue that the competitiveness of DL methods at this scale is a positive rather than a criticism. On the UkBioBank, the dynamic DL models (1D CNN, AST-GCN) do show improved performance over the SVM and other static baselines. This guides towards developing DL models that utilize minimally pre-processed dynamic signal over static feature engineered summaries such as correlation values. Further, to objectively identify the merits of applying GNNs for single-subject prediction we have to look further than the test metrics and consider the advantage of enabling the application of deep learning methods in this domain. DL is a versatile class of algorithms with applications that extend beyond supervised classification and would potentially be of remarkable value when applied on fMRI data. For example, graph explainability techniques {{cite:864b68058322fa53b277f7b08e9d325cb691f626}}, {{cite:4504e40e1a01d13c9ec8a1a930aa8fcbafe19f38}} could be utilized to find potential functional connectivity biomarkers for phenotypes and psychiatric disorders. The application of graph self-supervised learning techniques {{cite:a5eddedd24d1609e07797b2c445b91e78ff0a26f}}, {{cite:02446877324b781649bfcdda97117e0da92ae18d}}, {{cite:bd40d9eec62df951cb962c0c9eb5105b8373e51e}} and graph normative modelling {{cite:f349d252d25347b748706e76362c0d0ff5fb60df}} on non-clinical datasets would assist in learning representations that could further be transferred to a downstream clinical tasks. Graph augmentation methods {{cite:dcc67940421ae761d65d22aceab31f14bb235864}} could boost the statistical power of the clinical datasets and assist in solving the problem of labeled data scarcity and regularize overfitting. These are some of the promising directions that support the utility of GNNs in neuroimaging. That said, the empirical results presented in this work advocate for more moderation and more rigorous validation when reporting GNN results that do not clearly outperform graph-agnostic baselines. Finally, we hope that framework developed throughout this work could serve as a baseline for future novel GNN architectures or applications on fMRI data and assist in identifying key components that could improve existing methods.
| d | c3093397f15476ed1a986ba3bddae70c |
Recently, the Transformer {{cite:608917206b56ca1b9a27d3c53ea9457caee3c28f}} architecture has shown state-of-the-art performance in a variety of tasks ranging from NLP {{cite:608917206b56ca1b9a27d3c53ea9457caee3c28f}}, to image classification {{cite:2f5d5ed4618233ff35d9b172401358ba2e63aeb5}} and video object tracking {{cite:8d1fd6430caee317c8ffd306441865acdb45b9d2}}, among others, and has been proposed as a replacement for both CNNs and RNNs, or combined with convolutional layers in a Conformer {{cite:15ab6e27412117262a5cb6f114bce603473be510}} architecture.
Transformers base their representational power on self-attention (SA) layers that can model longer temporal or spatial dependencies than typical convolutional layers, while, in contrast to RNNs, they can be efficiently parallelized making them significantly faster during inference.
Recently transformers have shown strong state-of-the-art performance in SED tasks {{cite:72d75bb0fab3bb71fee3f7227455452ed3c60ed1}}, while their use in SSL and SELD proposals has remained limited. Regarding source localization, Schymura et al. integrated self-attention into the outputs of the RNN layers in a CRNN model {{cite:b4f0e8352422c4ec62ce385fe0c597403dfa78cf}} showing performance gains over the standard CRNN.
In subsequent work {{cite:8333cb85a6910cb8690ca46ad4faaace766f6939}}, RNNs are dropped for transformer layers including linear positional encoding, bringing further performance improvements.
With regard to SELD, the first work using SA seems to be the DCASE2020 challenge submission of {{cite:af5eaa08240eccf62166b028195de40edb989c7f}} which follows a SELDnet-like CRNN architecture, augmented with SA layers following the bidirectional RNN layers.
The best performing team in DCASE2020 also seems to employ attention in the form of conformer blocks, as detailed in a later report {{cite:3241fa4ac19af5d2919ade7912c4e46bf8946b14}}. Following DCASE2020, Cao et al. {{cite:fbcd80f4e4c9f86af70a62786a16fe90d8f127ba}} proposed their Event Independent Network V2 (EINV2), realizing a track-based output format instead of the class-based one of standard SELDnet, using multi-head self-attention (MHSA) layers following convolutional feature extractors.
Sinusoidal positional encoding is used before the MHSA as in {{cite:608917206b56ca1b9a27d3c53ea9457caee3c28f}}.
Since the above SELD proposals include various other improvements and modifications over the basic SELDnet CRNN, such as modified loss functions {{cite:af5eaa08240eccf62166b028195de40edb989c7f}}, partially independent models for SED and SSL with parameter sharing {{cite:fbcd80f4e4c9f86af70a62786a16fe90d8f127ba}}, or various data augmentation strategies {{cite:3241fa4ac19af5d2919ade7912c4e46bf8946b14}}, the effect of adding self-attention in isolation to the result is not clear.
| i | 6d45e9fc8b5ee167f873f9d41e55fd84 |
In previous studies, the Boltzmann collision operator {{cite:0adaffe509d88899bb9b9f4e5448e60c2d024ccf}}, {{cite:91352240a67c9eb800ac834e157ff0d00298988f}} and the Ornstein–Uhlenbeck operator {{cite:cb35de12e1e8f639b77899628d38c9b4b9079a7c}}, {{cite:09f38da3d897e64d0c17f144d9b8ef28ce1ac65c}} have been used for a description of dissipative effects in the quantum Boltzmann equation and quantum Fokker-Planck equation (QFPE), respectively. The former one, however, is phenomenological {{cite:a8d43f0646a81e8b6e5846b3fe2c1c7a9e7c2f1e}}, whereas the latter one is valid only at high temperature {{cite:2a26b900b7de8313d354c3e951e95ce26c0f0cf5}} that leads to a breakdown of the positivity of population distributions at low temperature {{cite:389e12d5edfaf7b4646b43b9b50066df9d251202}}, {{cite:b2668f63b5adb2fcbcbf3f3b42cf6a5cf2282c14}}, {{cite:5a90b1ce19d4c4fec0b2ac909cbb8b05dbff37c4}}. This is because a Markovian assumption cannot take into account the effects of quantum noise, which is non-Markovian at low temperature. Thus, numerically “exact” approach, for example quantum hierarchical Fokker-Planck equations (QHFPE) {{cite:d1df5c8a0877b13bb6e9a3db3935096ce7cc8f03}} for a reduced WDF must be used as the rigorous quantum mechanical treatments. These equations are derived on the basis of the hierarchical equation of motion (HEOM) formalism {{cite:83cb0623adf641e5de2cd566ec2630da88c0d91c}}, {{cite:389e12d5edfaf7b4646b43b9b50066df9d251202}}, {{cite:b2668f63b5adb2fcbcbf3f3b42cf6a5cf2282c14}}. By using the QHFPE, for example, self-excited currentoscillations of the resonant tunneling diode (RTD) in the negative differential resistance region described by a Caldeira-Leggett model was discovered in a numerically rigorous manner {{cite:9249ff76491219f52b87a7f9ade63307351a53d7}}, {{cite:8f1c54a2c8ccf3ffa3858c767d9d9bd353558d3d}}, {{cite:465fdb3d17795c8215da3ae07361e96f3980a784}}.
| i | 27e70334c35020a9a2a07da4f4486247 |
To build a relationship between excess entropy and bulk rheology, we next investigate the connection of {{formula:33974be2-25e4-49b3-996d-a971c45d2157}} to the other dynamical metrics. For this comparison, we compute the ratio of the second to first harmonic amplitude, which we denote as {{formula:b54e55af-7f35-450e-b62d-8cf6116d3b72}} . We can relate {{formula:a2725e9e-8afd-400c-9b79-793a852f22a2}} to several quantities in our system (Fig. REF ). For example, {{formula:f1232ae4-0079-4bdc-b13f-6a958b28a507}} scales with the product of {{formula:2e6caa90-8144-4ed4-aca2-e68be42c5f7d}} and {{formula:654e1bca-0fb9-4d66-ab90-5d4334540e4b}} (Fig. REF a), where {{formula:235cb82c-cb28-4093-8e3a-b81e65781d8e}} is the amplitude of the prescribed shear force. This relationship between dimensionless parameters suggests that when the imposed force on the system grows larger than {{formula:2e311ebc-2975-4448-88a6-ac03b88ae982}} , the microstructure begins to permanently change, losing stored memory. Rapid variation of {{formula:051e19eb-2dad-4de5-a2fb-c21909a191cd}} also signifies the transition. These findings build on recent work that links excess entropy and non-affine particle dynamics {{cite:c7b76ababd2fbc4926cab073fe435c6cedb22b30}}, {{cite:4f2527dd964ad96c48dd90ed33d28d34ff7a941e}}. Note that the scaling in the present case is quadratic because {{formula:6c9d0236-5a03-4646-9ade-5d1248a5d24c}} varies nearly linearly with the imposed force, {{formula:ee95c846-f1e6-41ff-be1f-d79578f010cf}} (see Supplemental Materials). Finally, we find that the product of {{formula:44e065f6-290c-4453-8a2d-5b57de748cf5}} and {{formula:a5265896-cbc6-4cc2-816a-1ebcdbfdb286}} scales linearly with {{formula:f9efb237-0c33-40cd-809d-ba850c3302c3}} (Fig. REF c). The scaling factor for this linear relationship is {{formula:b70d109f-d1aa-436d-a00a-065249a7a9e4}} ; here {{formula:26026532-65f9-4aad-8372-d14870215209}} quantifies the particle spatial density, {{formula:efb600ea-915b-450f-9ed2-21da27283689}} is the average nearest neighbor distance derived from the first peak of {{formula:0b0ca962-145d-4787-9fb2-10a85145fa28}} (Fig. REF c: inset), and {{formula:ab63ad41-e29f-47b4-9d52-b1f1e5d05e0e}} is the total area of the observed sample or simulation.
| r | a07d86428af2b95b0eff369029551863 |
The pipeline of original NeRF {{cite:7c0a4dea25854b0319dc85becd48504774b3afe7}} consists of the coarse and fine stages. During training, the coarse stage obtains the density distribution over the whole scene. It uniformly and densely samples points and calculates corresponding densities by a coarse MLP. However, as will be shown in Table REF , for common scenes with uniformly sampling, there are only around {{formula:ddbdd549-536c-45f7-b4b4-c7ac5c98c938}} - {{formula:f0d8dbb7-4fdc-4a21-8384-4199c8128817}} of valid samples (in Eq. (REF )) – {{formula:36955cdd-bc89-4108-b6e2-bac404f174ad}} - {{formula:6737b909-3cb4-49d6-abfd-652f8ab9cb89}} are pivotal samples (in Eq. (REF )).
| i | 4e6c1faccde47367e10f87d3f0c74107 |
In broad terms, galaxies could form via either the monolithic formation or the hierarchical accumulation. In the monolithic formation mechanism, a large amount of gas collapsed rapidly to form galaxies. The speed of collapsed gas determines whether elliptical galaxies, S0 galaxies or spiral galaxies with spheroidal bulge are formed {{cite:741b79c1bc854b5da2ad02950f1522fcd4177206}}, {{cite:443eb1e45d2d3ee150e4f3495b8ea359193fc2af}}, {{cite:c967cf42596e6ae82417b00afb1202fb873a35bb}}, {{cite:c21183395d131b1af510f17095f3bafcb1945bad}}.
S0 galaxies are generally thought to have formed most of their stellar mass 10 Gyr ago, which is why they are quiescent at present.
This view has been reinforced by the paucity of atomic gas {{cite:89418441a75311cbdd9749618ce147654718de76}} and star-forming regions {{cite:3b24fc34a397b3c84f204ab9e623bb1370bcded6}}. {{cite:1e7d2d5886198e6723aa767c456217ff737484f3}} studied the relation between the total mass of atomic and molecular gas and blue luminosity of S0 galaxies and they found that S0 galaxies has lost almost 90% of their gas.
However, the main disadvantage of this theory is that the massive protogalaxies that should be observed have not been observed at any reshift. In addition, S0 galaxies formed in this way should have a bulge component and have little star formation. In our sample, we do not find that these S0 galaxies have a large sérsic index. As {{cite:b4f3085abaa74d1d63dfda71de3bc577c556824f}} pointed out, S0 galaxies with star formation have lower sérsic index than that without star formation.
The deficiencies in the monolithic formation have stimulated investigations of the hierarchical formation idea, where galaxies grew their stellar mass through a series of merger events {{cite:b04660781f6c73caa4a77218dea37347d218d99b}}, {{cite:c6ef1d537d5fa902c4f50245ce9e1c9353fe02aa}}, {{cite:f8665899612be4f03d0124324cd9b98e372032d8}}.
Clumps of different mass and angular momentum interact, which may lead to continuous star formation. At the same time, the evolution of galaxies will continue for a long time, and even at the present epoch we may see the relic of interaction (e.g., the multiple star-forming knots).
| d | 40ec10f88e593fdd27e0b47c8f952108 |
It should be noted that even though our generalized Perfect Hyperfluid construction fits most naturally in a Metric-Affine Gravity approach, this is by no means the only place it can find applications. Indeed, our general construction here can just as well be applied to all Theories that represent special cases of MAG. For instance, Einstein-Cartan {{cite:6d3c5ff8a7a374594f0cfe72dd1d628a9afac42d}} or more generalized torsionful Theories (like Poincare Gravity {{cite:8a9c7c51d9927682402a6b6274b3f1992faacf90}}), non-metric torsionless Theories, and also to all teleparallel Theories such as metric {{cite:7e2abbc4cb9392e60e3343edb1561c72daafe577}}, symmetric {{cite:2decba17303dccf61c666bcf7eb50af0d8e26d42}}, {{cite:591fd763f6049de6dc4c59d66b38c09ca6a55a48}} and generalized teleparallelism {{cite:f7b3ea6ff6a206f5c6ccd71aa90b8d0299b46aed}}. Of course the list could go on and on. In general we expect the Perfect Hyperfluid to describe matter configurations to all Gravity Theories exhibiting a non-trivial connection.
| d | 77b7f9ea21f0b6bd5321605aafc66997 |
Seq2Seq model in this study was trained on KP20k corpus {{cite:2b6668fb406d096aa460cd6ccd901fefb9317e9e}} ({{formula:a0324797-c15b-4aef-9ee4-fb005fae4b19}} documents, providing {{formula:652cf389-4900-4c61-9977-235bfc8729b1}} training sets after preprocessing). For training the model, both sources {{formula:0f50eaa6-7540-4e3a-a50d-cfffd83cb3bb}} and the corresponding keyphrase labels {{formula:5794df1a-a07a-40a8-96b2-510bd29c917e}} are represented as word sequences, such as {{formula:e473b98a-7e1b-4708-b8d8-31d5cf49057d}} and {{formula:9d1e217d-f50b-4d56-81ea-756cab8d63e9}} , where {{formula:8f5dc6e1-507a-4edc-9721-c018b4d88a0f}} and {{formula:6c35866d-c336-4d34-9f11-c29caac6bbb1}} are maximum sequence length of source document and target keyphrase labels respectively. To be noted, each source document corresponds to {{formula:e0badb7d-8bb7-4410-bf08-fa5ff570baac}} multiple keyphrase labels. By splitting the data sample {{formula:7fd1ebbb-f81a-49f8-9c1e-cc89534944bb}} into {{formula:f3a3d39c-f25e-405b-b7b5-e79595d430b7}} pairs {{formula:68e07419-940b-4f2a-98c4-91256f622bf2}} , the training set is presented as text-keyphrase pairs, each contains only one source text sequence and target keyphrase sequence. In inference stage, standard evaluation data sets for keyphrase extraction were used (Inspec {{cite:2f9cee5b5b2ff8b14a4c3f709185100d79466c94}} (201), Krapivin {{cite:ab70f7ff1d5e6cf1485b99606e4a5730a13483bd}} (201), NUS {{cite:bf765c9838f9a5ca5c242118c3c2be852aa414b0}} (156), Semeval-2010 {{cite:8a47700efb2752e33fc08413baf29ea938876c92}} (242).
{{table:3ddd5bfc-a31d-479d-9f2e-1caf9cd996c8}}{{table:0ab0f3e6-db39-4645-a840-e04c08d8c947}}{{table:6fa72ee1-b969-4ea2-8cd8-5098cbd92b2f}}{{table:392aa420-bc59-43b5-82ee-ba5254fe69d8}} | r | 8eff0abb35e5374e5821a9735e3e5b96 |
Iterative search for optimal policy is commonly used in literature {{cite:a7a80b1a65e303cfdb159b757edbafc35f53d79f}}, {{cite:af60152bcd96efa449ba56b6b671c7f07631e381}}, {{cite:70f9d47044e7e67567676c1100962e5fb6ee7146}}.
We formulate the unbiased constrained policy optimization problem over {{formula:a42529de-3ea8-403e-a791-3cc9786a4d32}} and {{formula:26824f49-ff74-445c-a03e-6a4fe61947fa}} as:
{{formula:50a6140d-a12f-4a66-b68b-66ecc021641a}}
| m | 0aa727c106e32e8e2cd5b73331833b83 |
(2) Pseudo-Labeling {{cite:683404915c800d9ab47b3290b1f6ac99f493292f}}:
The approach uses a base model's confident predictions on unlabeled images as labels. Concretely, if the maximum probability of a class is greater than a threshold {{formula:49a59473-4d7e-44ca-8f1f-5c6fee8079b8}} , we then take the class as the target label.
Following the implementation of Oliver et al. {{cite:ffbbb1b17c55ac1f8527852eb8b527aeb79e7578}}, we sample half of the batch from {{formula:1821cdfe-612a-41f5-bed5-6d33be9ae137}} and half from unlabeled data {{formula:e87f658e-ea37-46f6-b5dd-cfa733633238}} during training.
Denote {{formula:42d49cfd-897b-4f42-8127-47165aaadc01}} as a labeled sample, the predictions on unlabeled data {{formula:f47e1ff7-0fdf-4e01-ad8a-789381225746}} of the model {{formula:eacec212-5389-4722-baeb-837d858286de}} as {{formula:55c8a38a-ade1-43b7-a5ad-011a9748c13f}} , pseudo-label as {{formula:b16ddb6f-dc5b-4cbf-a153-db8c2b0f8584}} , and cross-entropy function as {{formula:6192a890-b854-46bd-ba31-545a35fc9d18}} .
Then, the objective for each batch is:
{{formula:48a2c66c-f18d-4b92-b2a7-3d070541ffa2}}
| m | bf19799fb59c0c7e4b08d3a25011e86c |
There have been many proposals of future colliders {{cite:677e082bc8d1acd5954b21a1dd6511d7f5de9428}}, {{cite:cad4eeb60a734123fd19abaf957bbff86bb96e5e}}, {{cite:8b4703729e8b3943c701b999872c4f23fe963ed4}}, {{cite:c07fe9e08869374f93b445bb83c248d819750d74}}, {{cite:11de4430e2afafd33fd0da7efe94aead31232283}}, {{cite:494a86cd5b36adb697adc90f8c31c5658bb0834c}}, {{cite:7fa110db29e95efa98f25b71f1ab475130ff5a76}}, {{cite:599cb5e277d8ded9c2da0ca9481e0b4f3a0a973e}}, {{cite:e4038d85626a2a9e67b1d71c28ae333d8094568a}}. As an input to the Snowmass study, we discuss the connection between luminosity and physics reach at future lepton colliders and hadron colliders as a function of the center-of-mass energy. This is a vast topic. In-depth studies, with extensive simulation taking into account accelerator and detector design details, are needed to generate a full-fledged and precise answer. This is beyond the scope of this short note. Moreover, the detailed designs have not been carried out for some of the more recent proposals included in the Snowmass discussion. At the same time, the current proposals (and the corresponding studies) contain specific run energies and luminosities. It would be helpful to have a sketch of the physics reach beyond those benchmark points, which would help consider variations of the existing proposals and new ones. In this note, we focus on giving such an overview without a specific focus on any proposals.
| i | d6d8ce8643ec4baa3e0891863ff26323 |
The methods in the previous paragraph have various limitations, summarized by Table REF , that affect the design of the experiments.
First, the post-processing methods {{cite:21aed915a28406fd3632c635f2c871d5a05483ab}}, {{cite:1a21ce2f99e3f14f1ea25a4c6504489cea34e0d8}}, {{cite:99d990bd6c59bcce492720d0244803077786f884}} (specifically the WPP variant for {{cite:99d990bd6c59bcce492720d0244803077786f884}}) require knowledge of the protected attribute {{formula:7387ce99-9804-4fc3-bc2e-ed469293197d}} at test time. Accordingly, the experiments presented in this section include {{formula:d91923b7-5a01-436a-97ad-8f44bd8137fa}} in the features {{formula:b939a7f7-c891-4db7-b3f5-6f00367b4f5c}} to make it available to all methods; experiments without {{formula:1228a1a5-516f-4b97-a635-ab8db148b043}} at test time (excluding {{cite:21aed915a28406fd3632c635f2c871d5a05483ab}}, {{cite:1a21ce2f99e3f14f1ea25a4c6504489cea34e0d8}}, {{cite:99d990bd6c59bcce492720d0244803077786f884}}) are presented in Appendix . We also encountered computational problems with {{cite:1d1e42e80fe6fc1dae4acd302f1569a42cbd2f74}}, {{cite:c2feef1ac5fca0bbd63a88dadc9bec09e08bfcfe}} and thus perform separate comparisons with FST on reduced feature sets, also reported in Appendix .
| m | d89960a5d656a9cae1696479dbc412f0 |
We perform our experimental evaluation on CIFAR-100 {{cite:ae8fe7db3df9e4cdbec34371953b882d29c626e7}} and two fine-grained datasets, namely CUB-200 {{cite:b30cf1c3d2eedfa258eab46772574e901b133873}} and Stanford Dogs {{cite:dba73c2688e7d932a39765d1b2270fb423dd31e5}}.
The CIFAR-100 dataset consists of 60000 {{formula:3696ba5f-5846-44ab-971f-d751c9618e45}} images in 100 classes.
The CUB-200 dataset contains 11788 {{formula:7dd3ebc3-4c7b-4d54-9faa-2a586877906c}} images of 200 bird species.
Stanford Dogs includes over 22000 {{formula:b45b5e84-24ad-4d33-901e-ca9f71d7b956}} annotated images of dogs belonging to 120 species.
| r | 0e776dc2a2069e505524c51cc576f706 |
We also make use of the following bound on eigenvalues of normalized covariance matrices given in {{cite:51639a652568ba7bde95313551881390ece2af3e}}:
| r | 60231d91480445f7d6a4d095fb1206dd |
Our method aims to learn a representation of scenes as spatial maps of blobs through the generative process. As shown in Figure REF , a layout network maps from random noise to a set of blob parameters. Then, blobs are differentiably splatted onto a spatial grid – a “blob map” – which a StyleGAN2-like decoder {{cite:fed7fdb6fd9563f87f897b275ec624642f85889a}} converts into an image. Finally, the blob map is used to modulate the decoder We train our model in an adversarial framework with an unmodified discriminator {{cite:df0d400685a6b7fb02883eb483e1de6824cebd76}}. Interestingly, even without explicit labels, our model learns to decompose scenes into entities and their layouts.
| m | 6fdbffca0e5efc52576a1a1a67b90920 |
We assessed the consistency of predictions in BOLD MRI time series using our model, and achieved highly consistent predictions (Dice {{formula:0ea62683-2f21-4d12-b377-6bb5494081cc}} ). For many subjects, we observed modest drops in Dice ({{formula:7f8385b6-d96b-45c7-b58a-a59a33fd0255}} ), which were often due to fetal motion displacing the placenta. However, in a small number of cases, we observed large drops (Dice {{formula:4747a9a0-927b-4ec4-bbe1-feb9198f3377}} ) that we visually verified were caused by segmentation error. Since we apply the model to each volume in the time series independently, imaging artifacts, such as intensity and geometric artifacts, can affect the predicted segmentations. In future work, we will investigate incorporating temporal consistency between consecutive volumes. We will also investigate applying test-time augmentation on image intensity as this has been shown to reduce uncertainty and improve segmentation robustness {{cite:103ee514a2eaef80198ece631f96100e6c288a8e}}.
{{figure:6b307275-d284-464a-8802-c4248b3dee5a}} | d | f3c1019fd2e37234265263d854a03a4f |
Table REF presents the results of our proposed model and the baselines on the MultiSourceFake dataset. Our best result was achieved by using 10 as the number of segments ({{formula:7b58ef83-3998-4ea2-ab81-00ddb6d955ca}} , as found on the validation data). In Figure REF we show the model's performance for segments of different length.In the case of N=1 in Figure REF , we set the maximum segment length to 1500 words instead of 800 to not lose parts of the longer articles. In general, the results show that models that are based on either word ngrams or word embeddings are performing better than other models that use handcrafted features, e.g. {{cite:66bfaacb22d810138bf185b3edb4351a55f169d3}}. Also, despite the huge amount of data used to train the BERT model, the results show that BERT performs worse than FakeFlow and also fails to outperform some of the other models. We speculate that this is due to the fact that the input length in BERT is limited to 512 words, as we mentioned previously, and a large portion of the news articles in the MultiSourceFake dataset has a length greater than 512 words. The results of the Longformer model confirm our claim regarding the documents' length and show a significantly higher F1 score than the BERT model. This emphasizes that despite the strong performance of BERT on multiple NLP benchmarks, it is unable to handle long text documents, in contrast, e.g., to vanilla text categorization {{cite:cb003485d616f8681b1fc5413ef4db093b0a612a}}. In addition, Longformer's results show a higher F1 score than the FakeFlow model, yet, the difference is statically insignificant.
| r | 9bcda200c5624f8296090459c19d7e27 |
For discrete random variables, there are two common strategies for stochastic gradient estimation. The first one involves replacing discrete variables with continuous ones that approximate them as closely as possible {{cite:3e8edb0f1dd3cbc130e21cc6aa0971dba7d9ada4}}, {{cite:39431200398040e19128ab1f8d61bffbde5ea7cf}} and training the resulting relaxed system with the reparameterization trick. However, as after training, the system is evaluated with discrete variables, this approach is not guaranteed to perform well and requires a careful choice of the continuous relaxation. Moreover, evaluating the cost function at the relaxed values instead of the discrete ones is not always desirable or even possible.
The second strategy, involves using the REINFORCE estimator {{cite:6b26ec6fa6ebd26eeb86731fe5f92a663dc4ceaa}}, also known as the score-function {{cite:b655a3745fec1620442d8864b633a7535203b901}} or likelihood-ratio {{cite:3836c65624aff1a31dd72bb290f5a42dc0fefa4d}} estimator, which, having fewer requirements than the reparameterization trick, also works with discrete random variables. As the simplest versions of this estimator tend to exhibit high variance, they are typically combined with variance reduction techniques. Some of the most effective such estimators {{cite:39a0acc629cf882eb1a14ffa867717b498088d00}}, {{cite:27b4f0c279d085f7589a25f9fcb4189378d63cf4}}, incorporate the gradient information provided by the continuous relaxation, while keeping the estimator unbiased w.r.t. the original discrete system.
| i | e9af0f39ac08b11234b8ec71ea5c694b |
We use a dataset containing 46,744 cif files which contains the information describing the unit cells of properly relaxed crystal structures from the Materials Project {{cite:1cbad5efed2ec4b899e02039a86f21297d9d579d}}, {{cite:945e8310d39ac7e2063153f3157fbfc2e4be3fd8}}. We use 80% of the data for testing and the other 20% for training.
Using the Python library pymatgen we preprocess all of the data into the density 3-D matrices {{cite:ef56a0e308177fac244277c017f5c917b9cffbde}} described below.
The boundary box of a crystal structure is controlled by six different degrees of freedom (see Fig. REF ).
Each side can have a different length and the angles between the three sides is also variable.
Additionally, the internal complexity of each crystal can also vary widely– some unit cells in our dataset contain a single atom while other unit cells can have over one hundred different atoms.
This tremendous variability in structure makes a universal representation difficult.
| m | 7b72fa1814f96b59116679a578c10f6b |
Unlike the 0+1d SY model and SYK model which have a large zero-temperature entropy reflecting their “spin glass" like nature {{cite:c8a00b4425168dc59f22fe9239c95b69567706e5}}, {{cite:afef03908f0ee6e185672fafe283442f5afaec1c}}, {{cite:2388cd28946b1d453ac83bea03d234beaef25c06}}, {{cite:56b22b13272dba8c6481e21d07ac4265d722d7c2}}, the 1+1d chiral SY model here and the chiral SYK model in Ref. {{cite:34c5d9ad899b5d0139f456b8843e746cc296e553}} have a vanishing zero-temperature entroy density for {{formula:2965667a-c043-42d5-afb1-e008dbafd62d}} in the quantum chaos regime. This is because as long as the chirality of the model is preserved, any excitations will be energetically lower bounded by the velocity {{formula:75c150c3-d8ba-4e7f-96aa-2cc9c15ac8c6}} times their momenta, which avoids an exponentially large density of states at zero energy.
| d | 0c8d727757c8495d2d066878863ec3a9 |
We skip the proof of the above lemma as it is well-developed in the literature of functional data analysis including {{cite:c1d45c7e341c96fcfe1300c09ab968a93e97fe99}}, {{cite:7d4695023ee116a82c13e33353c46d714d30eafd}}, {{cite:0ebe2c2fa468ef4ced08253056e39ca8586821cf}}.
Next, we show the asymptotic results of the proposed estimation.
| r | 027687c5d1fbed3c189ee44d408a61b3 |
It could be interesting to investigate to what extent the generalisation and overfitting problems could affect predictions of real-life processes, considerably more complex than the artificially created data discussed in this work. Extra overfitting measures may need to be included as well in the future. One option could be found in using a similar resampling method as used here in order to construct the Test log, or a hybrid approach, to create the Validation log. Another course of action would be to alter the loss function used to train the RNN. Important to check further is whether low generalisation scores of merely memorising and overfitting models also lead to less accurate next event predictions. We still assume proper generalisation would be beneficial in predictive task, especially with increasing complexity, and theoretically the power of deep learning models, actually lies in their generalisation capability {{cite:a1319eb16769b98ae2e41d5b66e5aafedb33ad1b}}. Moreover if overfitting would not be an issue, more explainable statistical models could easily be found as well, as opposed to the black-box LSTMS, with comparable accuracy, and preference should be given to these more explainable models.
| d | 8f77295d7f58451caa7970c92718ca1e |
Note that in the smooth case, the higher rank property can be expressed by asking that every complete geodesic admits {{formula:d52875d0-673b-46f8-ab82-1abf54d9e052}} linearly independent parallel Jacobi fields for some {{formula:220a79f9-f8b4-4701-a927-fb5faba191e0}} .
In the presence of a geometric group action these Jacobi fields can then be “integrated” to {{formula:4f7af25f-acae-40d3-be3b-9a44c439b25d}} -flats {{cite:1f626af9a496de14baf93a37e0912fba10c830af}}.
Hence, in the context of metric spaces, the assumption on the existence of {{formula:6d849a60-0113-45d9-8dae-1a41ab8ad52d}} -flats in Theorem REF is naturalSee {{cite:9be4972fd7f0ae08e74bbc2988f9169f1a57fb78}} for a definition of rank for CAT(0) spaces whose isometry groups satisfy the duality condition..
Recall that a flat is periodic, if its stabilizer in the isometry group of the surrounding space contains a subgroup acting geometrically on it.
By {{cite:33ef0e3474095a4bbc594205caf8404bcea90e86}}, the existence of a periodic {{formula:5cc21da2-e499-42a7-8009-25fcc38e710f}} -flat in {{formula:fcf4c506-9a17-456e-9d2f-3fbfe405762b}} is equivalent to the presence of a subgroup {{formula:f4960144-2abd-4612-aeba-698cab4161a3}} .
We point out that Theorem REF does not assume any symmetries besides the existence of a single periodic {{formula:9be1074f-e494-4155-b00b-490a9e3548fb}} -flat.
Also, if {{formula:0ca02323-939d-48d3-a8e1-7982aa9d67e5}} is a rank {{formula:51498323-66f5-453e-86d3-de739fb4716b}} symmetric space or Euclidean building, and {{formula:078729ab-1ec2-4078-9f24-91bf3d6d15e8}} acts cocompactly on {{formula:5f8170f6-3618-42dc-87ca-592be01736eb}} , then {{formula:b7c54d8b-8db2-41ef-955d-f12264594463}}
contains a periodic {{formula:237adcba-79e8-4ecf-8df4-a47a1f953878}} -flat {{cite:2de1da6e0738ff1762adf923873bb61df3d182c9}}, {{cite:f38f54d4766f71ff50a17b6784646dcd8aa94edb}}, {{cite:1fdf7849392a09abc974363fa399f86f63a4d02e}}.
We emphasize that some symmetry is required in order for the conclusion of Higher Rank Rigidity to hold true, since a geodesically complete CAT(0) space whose Tits boundary {{formula:0a482b54-5a12-4dd5-9acd-2494e62fa2ed}} has
diameter {{formula:be65c677-4934-4961-b4b1-01109c463f08}}
does not have to be a product or a Euclidean building {{cite:7981b177b1400a9cb1a2b3faccf68a4f70b51fe8}}.
| r | 210349f594704d3d34d67e9db0eaa303 |
Generalization is a key desideratum for machine learning models to scale to the dynamic nature of the real world.
The standard supervised learning framework assumes that train and test data are from the same distribution (domain).
Domain generalization techniques {{cite:4b1b3a91788f9c36adef632615a0cbd7dae8b151}}, {{cite:9f79788f1aab87ab146929c9a05b230f8db2be5e}}, {{cite:46fa5ce839a0adbe08f6fe42a9e90afc04d00ddc}}, {{cite:91ba5a03becbfc2172cd6653d3b1ac9ba3d42c0a}}, {{cite:d2c16f92d2e4f44a96bb030059232ea28779471f}} demand to train a model in such a way that it can generalize to a novel domain at inference, by gracefully handling domain shift. However, current domain generalization methods assume the same classes to be present in all domains (including unseen test domains), which is a restriction on the application of such methods. Our work attempts to relax this assumption, and allow novel test domains to have new classes that were not present in any training domain. We introduce this harder problem as Zero-Shot Domain Generalization, and to the best of our knowledge, is the first such effort (Fig REF illustrates the setting). We note that the standard zero-shot learning problem {{cite:e489e58ca8abf0d93de7008866911f3a75fe5607}}, {{cite:98b249205d23e01810204fe5d73f8490243f86e3}}, {{cite:c9f1905f794dde90e62c0765c86ecc8cbca97940}}, {{cite:73cd03b0407c1d214bbaa49a9631c6312226f714}}, {{cite:22ff973c61f308aa16d6d8f6ccc661d58e5178b6}} provides a model to generalize to unseen classes, but assumes that datapoints come from a single known domain.
| i | 27e95df43e58f3bfcbf89ec3c3c0b073 |
Recently, there have been a lot of attempts to exploit graph neural networks (GNNs) for graph classification {{cite:dec2d91a5fc39ca616e52d17923b635cea68e4a6}}, {{cite:a8e0a30eb320e36f37c4cac95ff2d78d7c23a374}}, {{cite:cfa766f7abd6d0557863305fa39202800325126c}}, {{cite:e54d0deb2fd715f472a943dad6708ee86744a0bb}}, {{cite:d2b74856f13e504f6a6a38a46b892ef1153f5232}}, and they have shown promising results without expensive computations required by graph kernels.
They aim to learn the effective representations (i.e., low-dimensional latent vectors) of input graphs while being trained to predict the class labels based on the representations.
Most of them basically adopt the message passing architecture such as graph convolutional networks (GCNs) {{cite:a5fa748e53e1497d03b752b2cd08de6db3236286}} that compute each node representation based on its local graph structure.
Through hierarchical graph pooling {{cite:e54d0deb2fd715f472a943dad6708ee86744a0bb}}, {{cite:cfa766f7abd6d0557863305fa39202800325126c}}, {{cite:dec2d91a5fc39ca616e52d17923b635cea68e4a6}}, {{cite:606ce6cea5ef91238c8e68b0a3358dd7062cbccb}} and readout {{cite:db6626e7668b4d01dcad873b8a644c41350a69ba}}, {{cite:d4e87aa3c6926e44a70deeedaf995c0ae09e9ef0}}, {{cite:a8e0a30eb320e36f37c4cac95ff2d78d7c23a374}} which summarize the node representations, they finally obtain a graph-level representation of the entire input graph.
{{figure:d4765f0c-b9e3-4816-adb1-bc665338f4be}} | i | 5e960a6c6ac1c051541af0eb48551c12 |
The comparisons between the trust-region methods (RDA and MHODA) and the conjugate gradient methods (conjRDA and conj-MHODA) shows that the second-order geometry of the trust-region method improves the clustering and classification performance of RDA, although it may be unreliable for MHODA (Table REF). The advantage of RDA may stem from the use of equivalence classes in vertical space, which can effectively isolate the optimal solution in the quotient space {{cite:e6d0ef31b1ba084ef633953f22ab4e8a7f80edbe}}.
| d | 1dce3db848f5ea2b410ee1f9ab313e06 |
Condition REF holds for any compact {{formula:bb57734c-73a5-46ac-a14f-d05d2ffc68c0}} ; indeed, by {{cite:776a2e53010fcefd3a736f5760dfcb424f0a5e9f}}, {{formula:154ae982-ac22-4340-b4d1-7b5db370454b}} where {{formula:3994a96f-47a6-4592-bf7e-795b281feae1}}
is the number of {{formula:000756a9-1757-4ad0-ab47-266660daebd0}} -balls needed to cover {{formula:6373d42c-c97a-40ea-8d7e-e9020911bbd3}} , centered at any points in {{formula:e51fb331-803f-4024-bb04-54ba9aaba7a5}} , and it is straightforward to verify that {{formula:cb1a698a-e935-4de8-b130-d30ced952d7e}} .
| r | 84d50f01c90e57d3ad2f6796e9ffcbd2 |
Recent developments on optical flow estimation {{cite:2afd803b1d200f6c989db6bb560bbcd9439437fd}}, {{cite:b97ead0e3cd3ceb50697f7a00fc24edb84e04fb9}} using supervised learning have yielded state-of-the-art performance. However, such performance benefits have not yet permeated to pose-estimation tasks, where standard multi-view geometry methods still provide the “gold standard”. This work develops a dense indirect framework for monocular VO that takes as input externally computed optical flow from supervised-learning estimators. We have empirically observed that optical flow residuals tend to conform to a log-logistic (i.e. Fisk) distribution model parametrized by the optical flow magnitude. We leverage this insight to propose a probabilistic framework that fuses a dense optical flow sequence and jointly estimates camera motion, pixel depth, and motion-track confidence through a generalized-EM formulation. Our approach is dense in the sense that each pixel corresponds to an instance of our estimated random variables; it is indirect in the sense that we treat individual pixels as viewing rays within minimal feature-based multi-view geometry models (i.e. P3P for camera pose, 3D triangulation for pixel depth) and implicitly optimize for reprojection error. Starting from a deterministic bootstrap of a camera pose and pixel depths attained from optical flow inputs, we iteratively alternate the inference of depth, pose and track confidence over a batch of consecutive images.
| i | 999769753197042efb793c8469811142 |
The effective treatment of the influence of the on-site repulsive Coulomb
interaction on the states of electrons is a central issue of any theoretical
approach, where at the large repulsive {{formula:7b13e159-0923-4d24-b109-b5499b34760e}} , a double occupied state on each
site is strongly suppressed, and the Hilbert space of the electrons is split
into two subspaces: one is composed of the unoccupied and single occupied
states, and another one composed of the double occupied states that are
lifted up high energy levels. In fact, there emerges a single-occupied
constraint condition for electrons on each site produced by the on-site
strong repulsive Coulomb interaction, which is a major difficulty faced by
the present approaches. On the other hand, it is well known that in the both
cases of weak {{formula:b863f18b-408f-4f46-858a-a1cb567f5f38}} and strong {{formula:2661dd1e-e67d-425a-9f4c-ce6b86c2cb44}}
coupling limits, where {{formula:67a807fe-ccd5-4623-899e-9fe0cdab8dec}} is the hopping amplitude of electrons, the
basic property of the ground state of the 2D square lattice Hubbard model is
clear: in the former it is a Fermi liquid{{cite:dd9112f9c9f6b6275b531e7ff65859bc8bf2fbf9}} as apart from the half
filling; and in the latter it is a fully polarized ferromagnetic metallic
phase{{cite:681910dbce14e2abd9c6d123e680a022b7900da8}} away from the half filling, in which there does not appear
any order state.
| i | 27e1b738adc41c925d6d83a0f2da1087 |
We emphasize that CycleNet loses translational symmetry, which is considered a strong feature of CNNs, especially for image classification tasks.
However, recent evidence suggests that this property may not be crucial.
Symmetries do not have to be hard coded in the architecture: they can be learned, if needed, by stochastic gradient descent {{cite:478c5253e25f6e400c31830eed820f618215c3b3}} and data augmentation {{cite:90b591202b7241030b08d67554b7b10952f36f6e}}.
Furthermore, several non-translational symmetric architectures recently achieved near state-of-the-art performance in image classification {{cite:54c10243a33f79ddd4868d6ca2ae3511b1a4f06d}}, {{cite:72f5a82b3dcf24c0b1cdde49780ec99e56e85d27}}.
The good performance of CycleNet on classification adds to this line of research, suggesting that built-in translational symmetry may not be strictly necessary.
| d | fbea6434683ea47b2d5dff8199d82077 |
The realistic modelling of phenomena in nature and science is usually described by nonlinear Partial Different Equations (PDEs). Comparing to their simplified linear counterparts, these nonlinear PDEs in general has no explicit representation of the solutions in terms of initial and/or boundary conditions. Special explicit solutions, if available, are
often associated to certain symmetry groups of the underlying equation {{cite:5c31fdc62a74703169f75eac75977d9714818572}}, {{cite:9a29ee9a5df1b53aa29ddae1905d30ac69d36cb2}},
including the most important ones, the scaling symmetry induced self-similar solutions.
| i | b202e8781882a2565f66862208b5f4cb |
Since the fundamental solution systems of neighborhoods of different regular
singularities are different, and many equivalent fundamental solution systems
can be constructed corresponding to neighborhoods of same regular singularity.
When the Feynman integral is expressed as a linear combination of
hypergeometric functions of corresponding fundamental solution system,
there are obvious differences among the equivalent representations accordingly.
Finding a framework for understanding these rambling results is undoubtedly
important on both theoretical and phenomenological aspects.
The fundamental solution systems of neighborhoods of different regular
singularities can be regarded as analytic continuations of each other.
Thus the analytic expression of Feynman integral can be obtained in the whole parameter space.
Taking the Feynman integral as the linear combinations of hypergeometric functions from
the fundamental solution systems around different regular singularities and embedding
it on the subvariety of Grassmannians, one finds that the dual space of the GKZ-system
satisfied by the Feynman integral in the splitting coordinates determines the integer
lattice matrices, and then write the exponent matrices compatible with the integer
lattice matrices. The geometric representation {{cite:a6d141b6146e68b42dacb0ae5809a4ba2a1f6dbb}} of these exponent
matrices can clearly tell us which hypergeometric functions of the fundamental
solution systems of neighborhoods of different regular singularities are
proportional to each other.
| i | 5953020ad54c9a64268997d64676c37a |
We have demonstrated a reduction in the second-order correlation function of an HSPS of {{formula:f9fbf083-c0a9-443e-9cdf-3d74580fb18b}} by discriminating the photon-number in the herald arm using commercial SNSPDs. On the detection side, we use the information contained in the slope of the rising edge of the electronic traces to extract the photon-number information, obtaining conditional probabilities {{formula:bb30112e-ef49-4dcd-b422-dc59cb66f441}} and {{formula:76976c43-79ee-497f-a561-52c20bb8f60d}} of {{formula:cb9634ea-48d7-43ac-a469-65acab734455}} and {{formula:40784ab0-b70a-456d-b891-1b184492653d}} , respectively. These values reflect that this type of system can be reliably employed for the purpose of discriminating the detection of one photon from the detection of many photons. Note that in this application, the most important feature needed from the detector is to be able to discern the detection of {{formula:1bc8af8e-c122-481d-8cb7-96d264e016e5}} from {{formula:ecd76189-837c-4227-8e49-3c9e78c4bd78}} , while achieving high fidelities in resolving higher order photon-numbers, e.g. two-photon versus three-photon absorptions, is less relevant. The low value for {{formula:ad69ee51-b4df-4309-a414-82eaab35d807}} indicates that the reduction of {{formula:a7b1812c-7e0b-4dba-ac1f-95a2f1d0c8c5}} is primarily limited by the herald efficiency of the source rather than by the PNR capabilities of the detector. We envisage this technique can be straightforwardly applied in other high-efficiency photon sources such as {{cite:41def94488ba797028b0e81c5e8b53b23abc8a35}}, {{cite:c6b847e4c4586488c0f4f0719e6562afaf547e59}}, {{cite:4611c9030aa23729f29b5306d4b4a247f260cc66}}. The procedure shown in this work will either allow for higher detection rates with the same quality in terms of photon statistics or photon statistics with reduced multiphoton noise. Moreover, the substitution of pseudo-PNR schemes by a single SNSPD for multiphoton detection would entail a relevant reduction in hardware requirements.
| d | 366fb505c5db7f2507422fcd2f0aa767 |
We choose 14 baselines to be compared with UGCL on node classification tasks. These baselines consist of MLP and three types of GNNs: supervised-, conventional self-supervised, and GCL approaches. For supervised GNNs, we select three widely-adopted supervised GNNs, which are GCN {{cite:0ba13a52e710a43df677e98048c459702ed444fb}}, GAT {{cite:f7d33a9e9c6b1a2454cb5aaec4efec7a6e5bcbdf}}, SGC {{cite:1b39e57e85ac78f2b853c4e871ad1a84560e0bfd}}.
Four conventional self-supervised methods including DeepWalk {{cite:f3371e7c42fb82a4d69c8319cb5e03479cf8ca9e}}, Node2vec {{cite:741530b5076506e0f344235dae1db2258691074b}}, GAE {{cite:ab155d0b547ad639c2665f09cbeefe3468fc9482}} and VGAE {{cite:ab155d0b547ad639c2665f09cbeefe3468fc9482}}, and six GCL methods including DGI {{cite:d2a55327ae65bcb5018b7cc97938237650685ce0}}, GMI {{cite:6f7317a19e5f803ada0f2392b8a803c86a7b4bd8}}, MVGRL {{cite:7041fd6ec200962940dc5ab045055e00361f237f}}, GRACE {{cite:eeae4b09db79e8450dbce4180342221377c43c3d}}, GCA {{cite:da6d2948416a579402217ffaacc95d70d11d63fa}}, and BGRL {{cite:530a8d3f8327ca6a9cf5f0f66e728382061c87d2}} are chosen to be compared with our model.
| r | ff417e9cc6e282e78fe3ec06f87a47fb |
Structure-based approaches {{cite:150243e5058abb09437f22c43b45e66a0a372326}}, {{cite:e6ca15be74f45a25d18d1f48ce82162d2b8669f6}}, {{cite:ca541e099821742228b9cb9bb2f6d7298a957636}}, {{cite:e8884809517365aa8191ecfde5d6f70b1b86efaf}}, {{cite:5df437b5cf7af34eaba3135cac7f8a97142819b4}}, {{cite:bbf0fb30b8269b106d15ee4bd989d32f7826b14c}}, {{cite:0f9589c06a1670f89e7111197fcff48584ba8820}}, {{cite:5cb84d28ce5107a9521078747241f09882de5933}} adapt the network architecture with a sequence of tasks. PNN {{cite:150243e5058abb09437f22c43b45e66a0a372326}} expands the architecture for new tasks and keeps the function mappings by preserving the previous weights. LWF {{cite:e6ca15be74f45a25d18d1f48ce82162d2b8669f6}} divides the model layers into two parts, i.e., the shared part and the task-specific part, where the former is co-used by tasks and the later grows with further branches for new tasks. DAN{{cite:e8884809517365aa8191ecfde5d6f70b1b86efaf}} extends the architecture per new task, where each layer in the new-task model is a sparse linear-combination of the original filters in the corresponding layer of a base model. These methods can considerably mitigate or avoid catastrophic forgetting via architecture expansion, but the model is monotonically increased, leading to a redundant structure. In contrast to directly expanding model architecture, {{cite:9afafafba3c764ac0adca934288d691782b37aab}} adds additional task-specific parameters for each task and selectively learns the task-shared parameters together. {{cite:0f9589c06a1670f89e7111197fcff48584ba8820}} adapts architecture search to find the optimal structure for each of the sequential tasks.
| m | 1b54496ce73a51e786df67f97111e54d |
Athermal materials represent a class of disordered systems where large scale properties are only weakly affected by ambient thermal fluctuations. Such behaviour emerges in many disordered systems when cooled to low temperatures {{cite:a1082004dfa3ffd510f14e7dd108b085a4aa0866}}. Being governed purely by local constraints of mechanical equilibrium, athermal systems display amorphous disorder that arises from the many possible arrangements of particles in minimum energy configurations. Such amorphous packings are inhomogeneous at the local scale and consequently are not described by the usual elasticity theories in continuum {{cite:fb1869f52bf5ca08a349d6d47c577f5b9c2ef901}}. Since the thermal motion of the constituent particles is irrelevant, athermal materials are not governed by fluctuation dissipation relations {{cite:625261fb1a91adbf9aac19ccccc6b59f8147a7e6}}, and therefore offer interesting arenas to study non-equilibrium behaviour.
Many real-world systems can be classified as athermal, and they arise frequently in physics and biology. Examples include jammed packings of particles {{cite:4d2d1ca1bf508159460bad834fac40f4ad19f282}}, {{cite:cba199f859b584ec36d19044a7d235ee72b59da6}}, {{cite:6167600a30b91eea9d5911b7f1d926a50f88eeba}}, {{cite:e7a9b85a42e0b1fdedd288b6512ab296692d52bd}}, {{cite:fdc76c91c617dbfd77185a07d8a60a8926b8ec84}}, {{cite:85b509862619e45de6d6ca9a3c24203d561063c8}}, {{cite:1933dc3ddfa66ac44919e0f785634c2291096c4e}}, {{cite:0695c73cd08b30466189ee59b32b60d4c96675c8}}, {{cite:53f73a454f69d69f98586b8de0b771c85dfeb20b}}, {{cite:6f7c4695c1b23b1d6f7e4faa24d56f482104b26c}}, low temperature glasses {{cite:67c7c11ffbae600824d215155991cba6c26352dc}}, {{cite:c630d4f07c016663e00c14e00adbda5c1cacb092}}, as well as densely packed tissues {{cite:92d7663166473b7737038d942a1a45923c92de07}}.
| i | 4eb84de0b1521f63d3b199566786369c |
In PFNN-2, the approximate solution of problem (REF ) is found within the hypothesis space {{formula:cc4887ed-a58c-473a-88d0-b0bf7ffc6aa4}} that is formed by neural networks.
Unlike most of the existing methods {{cite:52eae8039aae8344d215fae0396b3d7d6c9f4da5}}, {{cite:2da7a5ec5af1a90a63de4dc085a2f7278efd7d4b}}, {{cite:0d08d74caf50a63c2fca99e7400c4575d4cff229}}, {{cite:05caa56189572796deb5c09653549921031aeed1}}, {{cite:0f35d73b1584f226059bcd8dd2c090975377a9fa}}, {{cite:3efafdb19974d5d71c10875bef78800ade56462f}}, {{cite:d24fc0df57361c31d7fe82b6601a0b2ea990d3b9}} that merely apply a single network to construct {{formula:0c6aa0a7-46f2-42d2-bb77-21091c9136f0}} , PFNN-2 adopts two neural networks, with one network {{formula:7c2ceeaa-91d2-482f-9e4a-9d3689e6b079}} learning the solution with the initial condition and essential boundary condition more accurately and swiftly, and another network {{formula:ce7520cd-523d-4971-a3b3-044809a19eb1}} approximating the solution on the rest part of the domain, where {{formula:b8d6629a-164c-4516-a3de-171038d2bfa5}} , {{formula:d4c49d0c-ac59-47e9-acd1-ed5c9723b12d}} donate the sets of weights and biases forming the two networks, respectively.
To eliminate the influence of {{formula:0542c9da-04e6-4a9f-97ed-4fa466f6eb41}} on the initial condition and essential boundary condition,
a length factor function is introduced to impose appropriate restriction, which satisfies
{{formula:3286b868-264c-44a8-80b8-2d9428ccdf24}}
| m | 13f3d38fe67932da5e6f47446091f596 |
Co-design Model Architecture with SpAtten. Besides the experiments above that leverage existing model architecture, we also explore the potentials of co-designing SpAtten with model architecture by searching a
Hardware-Aware Transformer (HAT) {{cite:0aef3b8c54b085d3965365e2d9b10c36df761cbb}} for SpAtten-e2e. The search space contains [512, 640, 768] for embedding dim, [512, 1024, 2048, 3072] for FFN layer hidden dim, [1, 2, 3, 4, 5, 6] for decoder layer number, and last three layers for arbitrary encoder-decoder attention. Because the FC layers form the bottleneck of the SpAtten performance, we intentionally configure the lower bound of FFN hidden dimension as low as 512 in expectation of reducing the FC ratio. We set different latency constraints and obtain a series of co-designed Transformers as shown in Figure REF . They are compared with layer number scaling and embedding dimension scaling of vanilla Transformer models {{cite:9566843412097c7e1ceafc144acf3b6464750e32}}. The co-designed Transformer-7 can achieve 1.9{{formula:13c0a717-6705-4d16-9615-1e616e960948}} faster speed and 2.8{{formula:601ca21c-29ec-4531-94e5-e08a18ca9227}} smaller size over the vanilla Transformer-Big model. We also show the computation breakdowns of the vanilla Transformer-Base and the co-designed Transformer-3 in Figure REF . The two models have similar accuracy. Since SpAtten-e2e can support attention with better efficiency, the co-designed model has a larger attention FLOPs. By virtue of the increased attention capacity, the FC computation can be largely shrunk without compromising the accuracy.
| r | 82cbff024ede77b593b3c2e7b52922c3 |
Experiments in Earth's ocean demonstrate that hydrate forms rapidly upon contact between liquid CO2 and liquid H2O, with visible masses forming over the course of just a few hours {{cite:8caa9719d6c3fa5e3e513198231659dc4dd21a27}}. This suggests that with surface temperatures below the 282.91 K quadruple point of CO2 hydrates mentioned in the previous paragraph, precipitation of condensed CO2 from the atmosphere into a liquid water-rich ocean would result in immediate formation of solid hydrates that would then sink through the water and settle on the seafloor. This may result in large-scale hydrate build-up on the ocean floor, which would suppress or halt seafloor silicate weathering, similar to high-pressure ice phases on waterworlds with extremely deep oceans {{cite:235316f1369d21421272b92dc2c49ab923b9b490}}, {{cite:6935574e749e38de70fe7f63061583f90ed503d9}}, removing an important CO2 sink and making it even more difficult for a planet to exit a stable CO2 condensing state. Other forms of low-temperature seafloor alteration would also be dramatically altered, with likely major consequences for ocean chemistry {{cite:78f385c2f1d4559f376610032e6e80b22d11d52b}}. Similar layered structures of hydrate and water have also been proposed for icy moons and dwarf planets {{cite:2533f98c445c12909a522229b9aa516f73417993}}, suggesting exploration of such bodies in the solar system may also provide insight into the structure of low-instellation terrestrial planets. Further, assuming a slow rate of subduction, the formation of large seafloor hydrate reservoirs could consume large fractions of the water in planets with Earth-like volatile inventories, since each CO2 molecule in a hydrate is accompanied by 5.75 H2O molecules in the most common hydrate structure {{cite:8caa9719d6c3fa5e3e513198231659dc4dd21a27}}. Thus, under the conditions where CO2 hydrates are stable (at temperatures below 282.91 K), a planet's subduction rate could exert a powerful direct control on ocean depth and salinity, both of which are first-order parameters in determining planetary climate and surface geochemistry {{cite:46ed305f983cd13482327216c166adde806bff81}}. Finally, the coexistence of liquid CO2 and H2O in the air may lead to the formation of aerial CO2 hydrates, altering the atmospheric lapse rate through latent heat release {{cite:5e50beeee8501655fac8517186b7b554b97744ff}}, but it is unclear whether this would be an efficient process given the factor-of-a-thousand difference in vapor pressure between CO2 and H2O at relevant temperatures.
| d | 2672470366bcdf90bb5d65ca31080e02 |
In this study, we have extended the X-ray scaling method to a sample of heavily obscured type 2 AGN with {{formula:73db2e7e-fd99-4638-a443-83c6e3be504a}} already constrained by megamaser measurements.
This dynamical method is rightly considered one of the most reliable; however, the accuracy of the {{formula:025f4d4c-4ba4-4ed3-9393-1b79777cd6f6}} derived with this technique depends on the quality of the radio data, on the assumption that the megamaser emission is produced in an edge-on disk, and that its rotation curve is strictly Keplerian. Additionally, one should bear in mind that this technique measures the mass enclosed within the megamaser emission. As a consequence, the actual {{formula:98c6657c-11da-48b8-bfc0-ece64803e6ab}} may be slightly smaller if the measured enclosed mass encompasses a nuclear cluster or the inner part of a massive disk, or alternatively slightly larger if radiation pressure (not included in the {{formula:36d31257-cebd-4924-bd5e-86c8827ea2cf}} derivation) plays an important role {{cite:9286f9c3e55e1ef9cd8f5b266fc65d91c3d9505f}}.
| d | 9aef3a5b94063c6ef290846dda59f00c |
For non-convex problems, the computation of the Newton step (REF ) may be an ill-posed or not even well-defined problem {{cite:52d25b8eb245f9752313b91a28b831e4e3eb57fa}}. Moreover, the Newton step might not be a descent direction. The Newton step is only a direction of descent if the scalar product with the gradient is negative:
{{formula:ae8aa8e1-3613-41b8-9d79-f18fcdec5c5a}}
| m | cd76675e3c49ae953986ec9ef9960548 |
Each image is resized and cropped to 1024x1024, and a set of masks is generated with thin, medium, and thick brush strokes, using the methodology described in {{cite:838c60006fc15913efbacf7465bfb53ab560f9fc}}.
These different mask-types are evaluated separately to observe the effect the width of the mask has on image inpaint quality. In accordance with recent works {{cite:838c60006fc15913efbacf7465bfb53ab560f9fc}}, {{cite:bbdc326a14b09c356e58cf5058972c8d8f20ada3}}, performance is evaluated using FID scores {{cite:5af14f064fab02191ac6a1dbf4bbed7e9ebe6c0a}} and LPIPS {{cite:80c7a1632998f5d0811cace7a258a8be65362668}}.
| r | fcbc1549f7c2989583cd3ccdf34148a8 |
Despite many efforts to improve its efficiency {{cite:1cc23766a998742c8ea9a2d77f17ddbd8ab81a15}}, {{cite:ad23fb3d04620d669c149139eab02bcf88ee5bef}}, {{cite:95b7255db2bbe4aa21a751f59297e4522a60cbcf}}, {{cite:2c22cc99831f178dddbd078b998dbd3c5e4113f9}}, blockchain still suffers from poor scalability, which introduces high latency to decentralized applications. For instance, the Bitcoin{{cite:57b4ebb6a7b575cdfddc4c9b4f9bd696882dedf7}} network can only process 7 transactions per second on average and the Ethereum{{cite:d31b65ef3a3e259612dd5682f608b14c7ecc1d3e}} network just provides 15 transactions per second. In contrast, Visa{{cite:0d0940f17f95987f6aa021e50797636ac65af8db}} can process 1700 tx/s on average and over 47000 tx/s at the peak. Blockchain is hard to scale since the underlying consensus protocol is not efficient enough. Transactions are passed through the network and need to be processed by all participants in the blockchain to maintain the consistent state change in a decentralized environment.
| i | 5939e91f8024d057f922791fcd415533 |
This result improves the previously best known time {{formula:03ed5f45-7124-418c-bbc5-3d7dc060df0e}} obtained by {{cite:cd25df833f2e4efa9a4cab5e8f499e3d1514b47a}}, {{cite:08fb43701d53c610fd9aa60ab9673105c2514452}}.
| r | b78a2d3a220b75db41277748d930a211 |
Within the enormous theoretical pool of compact objects available to run tests with {{cite:7167fcda31fd8f71f50b77cfaa23bc58b42777de}}, black holes are undoubtedly still the privileged candidate. The uniqueness theorems {{cite:28c1c65ab7e29e97c8a2e4254c29ae23d4f52f14}}, our understanding of gravitational collapse {{cite:5c8e8fe32fd7ee683d4bb767e613f55c288bc785}}, and the electromagnetic phenomenology tested so far {{cite:108f3a5c7b701bcb7a90dabb86abdfee212dffa8}}, singles out the Kerr(-Newman) family of solutions, described by mass, angular momentum and charge (the latter being typically neglected in astrophysical environments, see however {{cite:0b9200cb5724362a2ab3bb1f4dc6bcfb73a87aa3}}) as the embodiment of the black hole paradigm within General Relativity (GR). When looking for theoretical alternatives to it, particularly within gravitational extensions of GR, most attempts typically assume slow-rotation motion {{cite:23ea479e91aa91dae633c10fe73431cfdc7b3dff}}, {{cite:fda065dba40bb853866ea8639e5898012163564e}}, {{cite:99d9e96069b5efb51df254d2aabededfaedaffc4}}, {{cite:fe83f149de00716ed4c0d25d2f5b0abe96a319b0}}, {{cite:e7fd655862b483c307c240a0e71147dc4eb0f88c}}, {{cite:2071537f210ab7d7a0420e249a343537f3ec26c7}}, either in order to decrease the degree of difficulty in solving the field equations of the theory of gravity under consideration or to implement numerical recipes, though fully-rotating solutions are also known {{cite:e3c3dec4db0c3af4444a306ffe75a98131c1f2c2}}, {{cite:f159edae59c6277c6a9a0f4d9024d0971ba69ae7}}, {{cite:6be303f7d02aad65dac8882a30860a309a717167}}, {{cite:b2a042052dfaf68ad793a3c41c9b302ae3802f22}}.
| i | 7e5d416b6e40ed4f350e599670b9c2ab |
the ledger's properties that are at risk, if the
resources' distribution across the relevant parties becomes
centralized.
For example, considering Bitcoin's consensus layer, the resource is hashing power and
the relevant parties are the miners; the properties at risk are
safety, liveness and, to a somewhat lesser degree, stability and privacy.
We employ this framework for each layer, along with relevant
evaluations from the literature regarding deployed systems.The lack
of public data w.r.t. some dimensions makes it impossible to evaluate the same
set of systems throughout our paper, hence, we present findings depending on
the available information. As the most studied systems, Bitcoin and Ethereum
are included in most evaluations, with others (e.g., Solana, Cardano) evaluated
when possible. The main benefit of our systematization effort is thus laying
out the foundations for assessing any given system's decentralization level in
a quantitative manner.
Table REF provides a summary of our systematization, with
resources and relevant parties presented for each identified layer and
sub-category.
A caveat to consider here is that a resource might be modular, with some parts
considered more important than others. For instance, software products are
typically not monolithic, with e.g., documentation being less crucial than a
library or a configuration file. Therefore, the parties that maintain the
former have (arguably) less influence over the resource (i.e., the software
product) than the coders of the latter. To resolve this concern, one could
compute an aggregate level of decentralization, after weighing each
component based on its significance. Notably, such aggregation methodology
could also be applied to compute the decentralization level of the whole
system, assuming weights for each layer (cf. Section ).
Another issue is that one relevant party may encompass multiple
real-world identities. For instance, consider two software products, one
maintained by a single organization with many members, the other maintained by
a handful of independent developers. Although the first may be more
decentralized in terms of people, from a legal perspective the
second may be deemed more decentralized, as the first is authored by a single legal entity.
{{table:5af22a3f-c064-40bd-aee2-daa5db2dea3e}}By projecting the relevant parties of each category to legal persons,
we articulate a test that can be useful in assessing systems w.r.t. their
decentralization in a legal sense (Definition REF ). Here, a
legal person can be
a company (e.g., a foundation that performs the
development of the system), but also an individual (e.g., a so-called “whale”,
an individual who possesses a significant number of tokens). We note that,
although in this work the MDT is considered w.r.t. legal parties, it could be
adapted to treat exogenous centralization aspects as well, e.g., relating to natural disasters.
Definition 1
A blockchain system fails the Minimum Decentralization Test (MDT) if and only if
there exists a layer (cf. Table REF ) for which
there is a single legal person that controls a sufficient number of relevant parties so that it
is able to violate
a property of interest.
Dimensions of Decentralization and Paper Structure
We identify 8 layers in a distributed ledger where decentralization must be examined:
[(1)]
Hardware;
Software;
Network;
Consensus;
Economics (“Tokenomics”);
API;
Governance;
Geography.
Each is divided into sub-categories and analyzed using the framework outlined
above.
Specifically, Section looks into the hardware and diversity in
manufacturers of mining equipment and cloud hosting.
Section focuses on the major types of relevant software,
that realize participation and asset management,
and the diversity of software products and
development teams.
Section considers the network over which the nodes
communicate and evaluates decentralization on both a microscopic level, w.r.t. a
specific node's network access, and a
macroscopic level, w.r.t. the overall network's topology.
Section explores the consensus mechanism and categorizes
systems based on the mode of participation they allow.
Section focuses on the economics of blockchain systems and
evaluates the hazards of a centralized initial token distribution,
subsequent control of the ledger's native tokens by few investors, and
centralization in few secondary markets.
Section considers how developers and users typically
interact with the blockchains system - in essence the
blockchain's API services. In this context we examine
wallets and ledger service nodes, that
enhance efficiency and scalability of a blockchain
by offering its functionality “as
a Service”.
Sections and explore
multi-layer dimensions; Section examines governance, i.e., how
peers propose and decide on system improvements and funding allocation, while
Section delves into the geographic and jurisdictional distribution
of blockchain systems.
Section presents a case study, where all of the above
decentralization dimensions are evaluated for a specific system,
Bitcoin and enable us to apply the MDT.
Finally, Section offers a comparison with related work and
enhances the discussion with various possible research directions.
Hardware
The material base of each computing system is the hardware. The primary interest in blockchains
is machines that host the consensus software. Here, we categorize the different systems based on their Sybil resilience
mechanisms. Many systems, starting with Bitcoin, rely on computational
assumptions in the form of Proof-of-Work (PoW), while the most common alternative is
Proof-of-Stake
(PoS). Less-used options, e.g., Proof-of-Space {{cite:71d7fcc0ba4c7a1e4ee4aa58c2bdceec6213a1ce}} or Proof-of-Space-Time {{cite:ca6204aeb900fd276638d94e9e59f310c2ad286b}},
rely on other physical resources, like storage or
memory. Following, we focus on the two major options, PoW and PoS, though a similar analysis could
identify whether fundamental differences exist for alternative resources.
Proof-of-Work
PoW systems' security is typically guaranteed if the honest parties (i.e., those following the protocol as prescribed) represent an
aggregate majority of computational power {{cite:5e3bbd83d4eec003d786a1dc67dd864ac9619f8f}}. Therefore, the
resource of interest is computational power (e.g., hashes per second) and
the relevant parties are the manufacturers of hardware mining products.
In theory, PoW allows anyone to participate, regardless how small an amount of
power they control. In practice, some machines are more efficient in
solving the PoW challenge than others. For example, Bitcoin mining started from
regular
CPUs, but quickly migrated to GPUs and, eventually, to dedicated devices
(ASICs),
which produced more hashes at lower cost. Such shift to dedicated devices
introduces various centralization tendencies.
First, producing such devices requires a high-level of expertise and is an
investment in the blockchain system itself. In addition, microchip
manufacturing observes economies of scale, since the barrier to enter the
system in terms of designing and packaging the circuits is high. Consequently,
the market tends to concentrate around few companies.
Second, dedicated devices are particularly expensive,As of 2022,
popular Bitcoin ASICs cost (tens of) thousands of USD (e.g., Bitmain).
but significantly more profitable
than generic hardware. ASICs are also intensive in terms of energy consumption,
heat, and noise. Therefore, mining demonstrates economies of scale and
centralization tendencies.
Industrial farms host hundreds of ASICs, with hashing power sometimes offered as a “cloud” service, resulting in a
profitability gap between ASIC and generic
hardware users {{cite:71e85826ddfc6bcef6a8f1e6aa5a67ead31a8c84}}.
This trend has also motivated significant research in “ASIC-resistance”
and the development of PoW algorithms that attempt to facilitate better hardware
diversity {{cite:0457da1eab91d0df4adf4bdf9e115d09515ef2fc}}, {{cite:32a2c9de1d01a971491c2476ad26d8513f5b5156}}, {{cite:7eccb3917471281dcb4fb0ba4f8bdc754f674f91}}.
Concentration around few hardware manufacturers creates various hazards.
Same-vendor products are more susceptible to collective faults, e.g., due to defective parts or
hardware bugs. Such faults could result in sudden drops in the network's
power, lowering the threshold for gaining a computational majority
(safety and liveness hazard) and slowing down block production, at least
until the PoW parameters are recalculated (liveness hazard).
Manufacturers could also introduce backdoors, threatening the
ledger's security and stability, albeit such hazards can possibly be
mitigated via cryptographic techniques {{cite:f6755a4f02271e57eda603f2d2eed65bf35a5c23}}.
Evaluation
Centralization around specialized hardware has been
documented {{cite:396e756c1b1a3992a339c2ed73914535587aeb3f}}, although no academic research could be
found on mining hardware usage in real-world systems. Interestingly, it is
unclear how to even measure the usage of hardware equipment in PoW mining via public data, as
well as how to develop PoW algorithms that promote diversity, thus future
research could aim at answering these questions.
Nonetheless, there exist some reports,
though they often present conflicting assessments. In Bitcoin,
between {{formula:64ff20b4-ba05-46f3-9fcd-a6113c5813f0}} a single mining hardware provider accounted for either
{{formula:b612102e-a978-434e-b43c-0c60ecce2202}} % {{cite:fa49a68a4b39f0ae36c12dac03b33f037e8cbb7d}} or {{formula:3bf07c56-5965-4743-a1c6-f0a03ff29571}} % {{cite:fa8823faa4900be36821162f23fdaa2a204cfdad}} of the
network's hashrate, with 98% of the market controlled by 4 firms.
Proof-of-Stake
All (existing) distributed ledger systems define an internal currency (token).
PoW systems use the token only as a unit of account and a means of value
exchange. However, in PoS systems tokens replace computational power in
terms of Sybil resilience. Here, each user participates proportionally to
their owned tokens (“stake”) and/or the stake “delegated” to
them. Thus, a PoS ledger is safe and live if the honest users
control the staking rights to a majority of tokens.
Decoupling Sybil resilience from physical requirements enables PoS nodes
to run on generic hardware, e.g., even home equipment. However, in practice
portability drives PoS users to employ cloud services, which offer
uptime and connectivity guarantees that a DIY configuration cannot.
This trend is exacerbated when PoS systems apply penalties to absent users and
uptime guarantees become of utmost importance to guarantee profitability.
Therefore, the PoS resource of interest is stake and the relevant
parties are hosting providers.
When nodes that control a large amount of stake are hosted by the same provider,
significant hazards arise for all properties. First, the provider has access to
the users' private keys and is able to create
conflicting blocks (safety hazard) or deanonymize users' privacy.
Second, the provider controls the node's network access, so it could prohibit
communication (liveness hazard). Finally, it could also tamper with the
system's stability, e.g., increasing price volatility via targeted
interference or even stealing user assets.
Evaluation
PoS networks have been scarcely analyzed, compared to PoW.
Therefore, a comprehensive evaluation of the centralization of PoS validators across
multiple systems, in terms of hardware hosting, and how to incentivize
hosting diversity is a promising thread of future research.
Here, we consider two examples of highly-valued PoS
systems.Solana and
Avalanche are #9 and #14 respectively w.r.t. market capitalization. [CoinMarketCap; August
2022]
Solana's validators predominantly operate cloud-based nodes; of the 1873
nodes, more than half are hosted in two services, with more than 50% and
more than 66% of participating stake hosted by 3 and 5 providers
respectively.validators.app.
[August 2022]
Avalanche observes similar concentration issues; 731 out of 1254 validators,
who control {{formula:a7e79dbb-ece3-44f5-9773-051008c06f23}} % of all stake, are hosted by a single
company.Data obtained from
avascan.info. [August 2022]
Software
Diverse software development and usage is a core element of stability and
safety of distributed ledgers, as it increases resilience to
catastrophic bugs in a product's code. A vulnerability in
one implementation may jeopardize a part of the system, but if the system is
sufficiently decentralized, such vulnerabilities would not escalate to systemic threats.
Following, we discuss the development of core blockchain software components,
namely transaction validation and PoW mining (via full nodes) and management of
keys and digital assets (via
wallets).Appendix REF also explores
software testing.
Protocol Participation
The principal type of software in blockchain systems is the full node.
Full nodes implement the ledger protocol by:
[i)]
keeping a local chain;
validating new transactions;
extending the local chain with new blocks;
participating in the consensus mechanism to incorporate new blocks.
For the purposes of analyzing decentralization, we identify two
resources of interest:
[1)]
full nodes (as an absolute number);
participating power (e.g., computational or stake) that is hosted on
full nodes.
The relevant parties are full node software developers.
Relying on a handful of full node implementations introduces
safety, liveness, and stability hazards. A
bug that fails to validate correct transactions would hurt the system's
liveness, whereas accepting incorrect transactions could hurt the system's
safety and stability, e.g., via a network split or token forgery. Such bugs have been observed
in Bitcoin Core and could have resulted in
Denial-of-Service (DoS) attacks {{cite:c956293cdef42ad6714a18e636be3ded5a2ae550}} and
token forgery {{cite:52f23410eda0bf492916cb8fb50b33f737323b0b}}. Implementation bugs could
also threaten privacy, if nodes reveal information
about message origin (e.g., IP addresses).
Similar threats arise when code gets reused across different projects. Often, a
new blockchain is only a “derivative”, that is a project that started by
forking an existing codebase, including copyrighted
information {{cite:34a57b8cb01b042e4f0f6e4ee41ec53690710065}}.
Such projects often remain unpatched, so bugs in the initial implementation
tend to spill over {{cite:b651a7f60a10c377f195755bb190a0ea1b1b18f7}}. Vulnerabilities may also arise when
adapting an existing implementation in a new setting, e.g., copying Bitcoin's
code and replacing PoW with PoS {{cite:1da396ca3f75a67bd9f5eb8294e41b78d93e362e}}.
Thus, widespread usage of
multiple node implementations, developed by different teams, is a
prerequisite for a secure and reliant ecosystem.
Evaluation
The literature is lacking formal analyses on the usage of full node
ledger software, so a rigorous evaluation of the dynamics in software development
and usage could highlight various centralization tendencies.
Various community and commercial projects do keep track of
statistics though. In most systems, a single client software is
predominantly used by the participating nodes in the network. In Bitcoin
99% use Bitcoin Core (aka Satoshi), in Ethereum 78% use geth, in
Litecoin 95% use LitecoinCore, while systems like Zcash are completely
centralized with all nodes using one software (MagicBean); a notable exception
is Bitcoin Cash, where usage is split between BCH Unlimited (33%), Bitcoin
Cash Node (51%), and Bitcoin ABC (12%).Sources:
blockchair,
ethernodes [August 2022]
Some projects are actively managed by a wide network of
developers, e.g., more than 200 contribute to Ethereum {{cite:94f609130cf09354477b0c764e60b2270ae68881}}, while others are particularly
centralized. As of 2018, 7% of all Bitcoin Core files were written by the
same person, while 30% of all files had a single author. In Ethereum,
these figures rise to 20% and 55% respectively {{cite:ec13c4bc75972ce3fd8ac0030a886e01ae8a36b6}}.
Comments observe similar centralization patterns, with 8 ({{formula:f7e5826a-1390-4680-a52d-f8b39a7b7264}} %) and 18
({{formula:4316d9a8-6632-4df8-92ad-980558349b5e}} %) people contributing half of all comments in Bitcoin and Ethereum
respectively {{cite:ec13c4bc75972ce3fd8ac0030a886e01ae8a36b6}}.
Asset Management
The primary use of distributed ledgers is bookkeeping of digital asset
transactions. Thus, securely managing and transferring said assets is a core
necessity. Digital assets are typically managed by private keys and
represented via addresses. The software responsible for managing keys and
addresses is the wallet {{cite:4fee83b3e940e903bb9fda9122f667e363663078}} and its principal
functionalities are:
[i)]
store the user's keys;
prove ownership of the assets (managed by the keys);
issue transactions that transfer assets to other accounts;
retrieve the user's (keys') balance and history information.
Therefore, the resource of interest is the set of all assets managed via
the ledger and the relevant parties are, like before,
software developers.
Naturally, the wallet is a major point of security consideration, so multiple properties rely
on it. A bug which e.g., corrupts the user's keys could not only prohibit a user
from transacting with their assets (liveness hazard), but forever lose access
to them (stability hazard). This was demonstrated in 2017, when
the “Parity” Ethereum wallet saw a vulnerability that allowed a user to take
ownership of multiple assets and then lock them {{cite:f11bfd4e65889a4a7c1059038b74716c87dce8dd}}; as a result,
300m worth of Ethereum tokens were forever lost. In another example, some
Bitcoin wallets possibly displayed incorrect balance, effectively enabling
a double spending attack {{cite:50fe6b4917d5b9e50bb5b915231f0cc3ff7f826d}}. In addition, wallets are
often lightweight, relying on third-party full nodes for ledger interactions,
but also possibly enabling the nodes' operators to link and/or deanonymize
transactions and accounts (privacy hazard). Consequently, if a few
implementations are predominantly used, a vulnerability could result in
assets being unusable or stolen. Such
vulnerability could also turn into a systemic point of failure, with the whole
ledger becoming unusable.A prime such example was “The DAO”, which
in 2016 attracted nearly 14% of all Ethereum tokens and, when hacked,
instigated a change in Ethereum's consensus layer and a hard fork which split
the network {{cite:86065c1027505a39d51a28906a078dbc2d404460}}.
Evaluation
As keys and addresses are wallet agnostic, it is
impossible to identify if two addresses are generated by the same wallet
implementation, unless it purposely reveals such information. Consequently, it
is unclear how to evaluate the wallet market's diversity and how widespread
wallet usage is from public data. To our knowledge, no rigorous investigation has been conducted
on this topic, either analyzing historical data patterns or conducting
usability studies.
Testing
Testing is a core part of software development. In blockchains,
a major means of testing new applications or ledger features is testnets. A
testnet is a separate chain, identical to the main chain in terms of offered
functionalities.
To transact, a user acquires testnet tokens for free, so the
native testnet tokens have no real-world value.
Testnets offer multiple functionalities. Users can test
features without risking losing funds.
Developers test new features and applications in a
scale that closely resembles the main chain.
Adversaries evaluate the efficacy of
attacks {{cite:88eedd0887d3d739d39defbefe410155d0f482df}} or exploit the zero-cost nature of testnet
transactions {{cite:337515ed7eed7669870ad724e3969744b466f960}}. Therefore, testnets indirectly
safeguard all ledger properties.
Fewer testnets increase centralization around
specific full node software products, while
testnets maintained by diverse teams may collect richer
data. Hence, the resource is testnets and the
relevant parties are their operators.
Evaluation
Bitcoin offers a single primary testnet; the same holds for alternative cryptocurrencies like Zcash
and Monero.Sources: Bitcoin
Wiki,
Zcash
Docs, Monero Docs
[August 2022]
In Ethereum, although seemingly multiple testnets exist, most are
deprecated due to the system's transition to PoS, and only one
of the recommended networks is expected to be maintained in the
long term.The deprecation of Ropsten, Rinkeby, Kiln and Kovan
was announced in 2022
[Ethereum blog]. Goerli will be maintained in the long run, while the future
of Sepolia is undecided.
[ethereum.org; August
2022] PoS testnets, e.g., in Cardano and Solana, are also highly
centralized.Cardano and Solana testnets are run by Input Output and
the Solana Foundation resp.
[Cardano Testnets,
Solana Docs; August 2022]
Network
Blockchain nodes communicate over a peer-to-peer (P2P) network.
Systems often employ a message diffusion
mechanism {{cite:1603fe45243496180184c37ca060625302705b40}}, based on a gossip protocol that avoids
full graph connectivity, cf. {{cite:0dbba972013879c1d302a25e09a584a795a81d69}}. Users predominantly rely on the Internet,
on which the P2P network is overlaid, though some attempts try to
use a more autonomous infrastructure {{cite:d6eadbf6116c3374d285ec05bcd28387582f97e4}}. In this section, we explore
different networking aspects which present single points of failure, on the
individual user level and the network as a whole. We note that a relevant
research question that arises organically from our analysis, and touches upon
both following subsections, is how to create a P2P network that is both
permissionless and Byzantine resilient. Some recent works investigating this
direction includes {{cite:909589cc51b324906e4d673c85f416ffee8dd235}}, {{cite:d41343f52afe5627a70af20e322760b07d6dec78}}.
Peer Discovery
Joining a ledger's network and catching up with it
is termed as the “bootstrapping” process. All real-world ledgers rely on an initial
(trusted) setup, the first (“genesis”) block. Thus,
obtaining the correct genesis block is a prerequisite for secure
participation. Following, the node connects to various peers to receive all
available, possibly conflicting, chains and, using the ledger protocol's chain
resolution mechanism, decides which chain to adopt. At that point, the
node is synchronized and ready to participate in
the consensus mechanism.
In these systems, every node maintains a list
of peer connections.
Crucially, message provenance is not typically provided
so no party can know the network origin of an incoming message.
Each node needs to maintain at least one
connection to an honest party to receive all messages and avoid eclipse attacks {{cite:29332520211aae8076044b7538a081d04307ec78}}. Thus, the number and condition of a node's
peers is a significant element of security. Consequently, the
resource of interest here is the typical node's (in and out) connections,
while the relevant parties are the owners of the peers that maintain
said connections. We note that this resource refers to the average node, i.e., the nodes that run the standard ledger implementations (e.g., Bitcoin Core in
Bitcoin), even though alternative, niche node implementations could set
different network parameters.
If a node connects only to few peers, then an adversary could more easily
mount an eclipse attack. However, maintaining many peer connections
comes with a tradeoff, as it jeopardizes message
anonymity {{cite:88f3d3ccfda345db57e6e9be4eb3003ef13bf620}} and increases bandwidth requirements.
Here, we focus on the case of low connectivity, where eclipse attacks threaten
multiple ledger properties.
First, an eclipse attack can result in safety violations. By preventing communication between
a node and the rest of the network, an
attacker reduces the honest computational power and isolates that node. Consequently, it becomes
easier to mount a 51% attack and, particularly, a double spending attack
against the isolated node. For instance, the adversary may supply malicious
chains and stop honest blocks, such that the node holds a skewed view of the
system state.
Second, an adversary can violate liveness by blocking the “eclipsed”
node's transactions from reaching the rest of the network.
Third, an adversary that controls all of a node's connections can link
transactions to the specific node. Therefore, the adversary can both
correlate the user's (otherwise distinct) addresses and associate them with
real-world information, such as an IP address, thus breaking privacy.
Finally, eclipse attacks may break the bootstrapping process.
The genesis block, needed for bootstrapping,
can be either
hardcoded, thus relying on software, or
received from peers.Some proposals rely on
computational assumptions instead of a trusted setup {{cite:f943ac31d88033bc1e7530d62110d40c12f31817}},
but their real-world performance and applicability is still untested. However, catching up with
the other peers using only genesis (“bootstrapping from genesis”)
is not always feasible. Especially in PoS systems,
an attacker can effortlessly assemble an arbitrarily-long, seemingly
correct chain {{cite:cf0eed667813ebf0a7a1f94d570d5eaafa9b21b9}}, {{cite:0ee6b9dfa95a20226c792f0414f6cf64b1b0fc24}}, {{cite:d84caf769d9330c22ebad7f425d9038c0234a579}}. To
counter such attacks, most ledgers employ
checkpoints {{cite:882fda85c30cfdcf2b0ffc0337d072b7fa7b1bc3}}, {{cite:81e84071809b16d842b3cff9c59d6b415f7013fc}}, {{cite:3787b90e9123e732d1a89ad8ea58ab993bbd05b5}}, which
are often issued centrally and, as with the genesis block, are either hardcoded or
received from the peers. Other solutions do exist, e.g., analyzing block density
and relying on key erasures {{cite:593d630d3bb9a092d71fc5f44a3d695886929fb4}} or using VDFs {{cite:dc680328eb15311c2a3e51d5592d45ea37d939eb}},
but enforcing and/or relaxing such assumptions still poses
an interesting research problem.
Evaluation
Bitcoin, as the first blockchain system, has seen multiple eclipse attacks
and defenses {{cite:1b601063eca1b6414c41bb83faaee31aebb340dd}}, {{cite:29332520211aae8076044b7538a081d04307ec78}}, {{cite:8724e5e45b4b1bfa19aa6d9a38a73317ba11b9ea}}. Some works
attempt to increase the number of connections
without reaching prohibitive levels of bandwidth
usage {{cite:3dd9255ad590bed8af95ed6107c7118f8055b202}}. Bitcoin Core defines
8 outgoing connections, selected randomly from known identities, and up to
125 incoming {{cite:7708c11872485ce41c22d644177b029da7559ee2}}. When (re)joining the network, a node
attempts to connect to previously-known identities and, if unsuccessful,
employs a (hardcoded) list of DNS seeds. Other systems, like Ethereum and
Cardano, employ more complex, DHT-based mechanisms {{cite:637be6fd34ad71c4c649f7af70801d7e62b5a203}} that require further
analysis.
Interestingly, Ethereum was found vulnerable to eclipse attacks that do not require
monopolizing a node's connections, but relied on message
propagation {{cite:58ff8ea487ee5e819bb06ef5aec45a14ff6ca676}}. Cardano is also an interesting
implementation, as it assumes two node types:
[(a)]
core nodes that participate in consensus, and
relays that intermediate between core and edge nodes (e.g., wallets);
in the default configuration relays are operated by only a small committee {{cite:7708c11872485ce41c22d644177b029da7559ee2}}.
Topology
Our second networking aspect focuses on the overall network's topology.
Evaluating a real-world network's clustering properties is a well-known
problem. The traditional methodology relies on generating random graphs and comparing
the expected with the observed
values {{cite:214cf6e372a3dd024e203b343a7211b8f79043b5}}, {{cite:27d5d549ccd469a70d8a1366d30f908e64d4cb80}}, {{cite:1f652a77a672bd22cccb0a247acc648c2f580e9e}}.
In our framework, the resource of interest are “bridges” (single nodes or small cluster of nodes) that may exist between
different connected components of the network graph and their owners or operators
are the relevant parties.
Metrics and solutions from traditional distributed networking
could possibly be emigrated to ledger systems.
In blockchains, a distributed network (in the tradition of
Baran {{cite:1f307fdc6314e943aab3809925e29b7cc2b8b275}}) is key in maintaining safety and
liveness. Under the CAP theorem {{cite:8e212cf2c24d8ce21f29c75de1c27203d7d862fe}}, any networked
system can satisfy at most two of the following properties:
[i)]
a consistent data copy;
data availability;
network partition tolerance.
Ledger systems are no exception. If some parties cannot communicate with
the rest of the network, either they halt (violating liveness) or
produce separate ledger versions (violating safety).In longest-chain protocols,
like Bitcoin, miners keep producing blocks in isolation, ending up with
different ledger versions. In BFT-style
protocols, like Algorand, the ledger cannot be updated if a large number of
block producers become unreachable and thus do not adopt new transactions. Also,
a node that acts as a central communication hub, or hosts
a single channel over which two clusters communicate, can possibly obtain
information and even deanonymize some participants (violating privacy).
Evaluation
Bitcoin is notoriously vigilant in hiding its network topology {{cite:9bcdfb77f7412a5a9dc8eee41e6b5dcd4ef2ae9d}}, {{cite:69d2ffe48dbd79761b781afcf35b899a688c6d27}}.
Various works analyze it by inferring
a node's neighborhood {{cite:e3c028d12fffdb523d8e50c47d792c3367a5c354}}, timing
analysis {{cite:03af00c363b8ecf7cd5a1aee655bd768ccd48148}}, or conflicting
transaction propagation {{cite:9bcdfb77f7412a5a9dc8eee41e6b5dcd4ef2ae9d}}. In 2014, it was found that more than half
of Bitcoin nodes resided in 40 autonomous systems (ASs), with 30% in just
10 ASs {{cite:dbb79756c6ab665baf87fe2cbaa735a23445a026}}. In 2017, Bitcoin's and
Ethereum's P2P networks observed similar sizes (3390 nodes for Bitcoin, 4302
for Ethereum). Bitcoin offered lower latency and higher
bandwidth, with nodes being closer geographically and 56% of
them hosted on dedicated hosting services (vs. 28% for
Ethereum) {{cite:6bd9575898239319bcfb43791510421ec16c351c}}. In addition,
68% of the mining power was hosted on 10 transit networks, while 3 transit
networks saw more than 60% of all connections {{cite:e58391fabf6dc6cf7ae9ffd2cb49611440ba2952}}.
In 2019, Ethereum's network presented a large degree of
centralization around clusters, forming a “small world
network” {{cite:bd5ad5f6eec5588d5c6bd3b2128c0be7a6ce14cb}} with 10 cloud hosting
providers accounting for 57% of all nodes and one hosting
almost a quarter {{cite:86fcbc7f64f793b46cd886029d258b6383c8040b}}. This was reaffirmed in 2020, as Ethereum
messages could be sent to most nodes within 6 hops {{cite:5b45d7b3b8ae7e05e32fbff8a0115caa52e3ed06}}.
In 2020, Monero's topology also observed a high level
of centralization, as {{formula:3ccfff46-2ec2-4301-a214-524fa1a8e591}} % of nodes maintained {{formula:7e69cac9-7f73-4ad0-aa2a-54d5bd93fd2d}} % of all
connections {{cite:40a1edde9ce2eb57621cf970760103fba709895a}}. No analysis of PoS systems'
networks could be found; given their non-reliance on specialized hardware and ease of
relocation, a PoS-PoW comparison would be of interest.
Consensus
A key element of any distributed ledger is its consensus protocol.
Protocols in our context are “resource-based”, i.e., they are executed
by parties possessing units of an underlying resource (e.g., hashing power or stake).
As discussed
in Section , the two core consensus properties are
safety and liveness. To guarantee these properties, at least a
majority (or in some cases a supermajority of {{formula:9ca76df6-fe71-41d7-b99b-c7c9ff5a1804}} ) of the ledger's participating resources
should be honestly controlled {{cite:6df5a075fb332771966b3dfc2375994961fe6bda}}. When a handful of actors
control enough resources, a direct point of failure arises.
A useful distinction between consensus protocols for our purposes is in terms
of whether
the protocol works with direct participation from resource holders
or necessitates some form of pooling or delegation
of resources.
Consensus with Direct Participation
This first category includes Bitcoin {{cite:1603fe45243496180184c37ca060625302705b40}},
Algorand {{cite:aa46a8741d8c27d159c8468cfabc4e88acbf9ba0}}, Ouroboros Praos {{cite:1603fe45243496180184c37ca060625302705b40}}, and NXT {{cite:d56ad684dc4e1ad9e9e980adf72304cf003365c6}},
among others.
Such protocols enable resource holders to engage in the protocol directly
with (essentially) whatever amount of resources they have.
An important consideration in this setting is that
block producers can, even though they do not have to, form coalitions called
pools. In PoW, a pool “leader” validates transactions, and organizes
them in a candidate block, while each “member” executes the PoW puzzle for
the leader-made block. If a member is successful, the leader collects the
block's reward and distributes it, proportionately to each member's power. In
PoS, the leader has full control over the block's creation, while the members
only pay fees to delegate their staking rights to the leader and collect
rewards. Pooling
behavior is also driven by temporal discounting {{cite:d313f74db1b614da97005ea2457181bd4e6ed200}}, i.e., the
tendency to disfavor rare or delayed rewards. In essence, a small miner may
prefer small frequent payments, at the cost of some fee, over rare
large payments, when producing a block.
Therefore, there are two resources of interest:
[i)]
owned participating power, e.g., computational or stake;
delegated participating power, including the power to choose
a block's content.
Correspondingly, the relevant parties
are:
[i)]
miners and stakeholders, who own hashing power and stake respectively;
pool leaders and delegates, who control how the resources are used.
In addition to that, one could also consider the actual people that control the nodes, instead of the nodes themselves,
further enhancing the discussion around centralization with impossibility arguments, given the
“permissionless” nature of blockchains and the absence of a central
registry to impose “Sybil”
penalties {{cite:27077a797b29a42ea8184cbe609d5b4cbe1606e7}}, {{cite:71e85826ddfc6bcef6a8f1e6aa5a67ead31a8c84}}.
Gradually, as power is concentrated in fewer pools, the pools' leaders may
become single points of failure {{cite:da5c33220c478aebdb2664a93baa8a49251763dc}}. Liveness can be hurt, if corrupted
leaders censor transactions from the candidate blocks. Also, if pool
members do not perform any validity checks on the candidate blocks they
receive, corrupted leaders may launch a long-range attack to violate
safety. Depending on the pool's internal protocol, the leader
may steal member rewards (stability hazard) and possibly link the
user's resources with information like IP addresses (privacy hazard).
Finally, a threat arises due to the lack of self-healing, i.e., the inability to
recover from a temporary adversarial takeover. In PoW, even if a majority gets
corrupted, honest users can increase their own power and, eventually, overthrow
the adversary and restore the ledger's security {{cite:cb824f212bcea53c0b2db30ff456b3a49e03dddb}}, {{cite:f56db7fc84ba539459accb1bb01b2bb1712f6513}}. In PoS though, power shift
takes place on the ledger, by transferring stake. If an
adversary temporarily obtains a majority, they can prohibit transactions
that shift power away from them, thus retaining control
indefinitely (for example, a large centralized cryptocurrency exchange can make it hard to issue outgoing payments and withdrawals, while enabling payments between different users of the exchange).
Consequently, a diverse stake distribution
(cf. Section ) is paramount to protect against
takeovers.
We note that the above narrative distinguishes between PoW and PoS systems, nevertheless it is also possible to have direct participation systems that combine different resources in the same consensus protocol for various purposes, e.g., see {{cite:2f094166b04f23e9d421bfd2a02f4e7617b2dc2c}}, or {{cite:64e55bb608b9c8ba2b2bda0641c1ae0032ccf52e}}.
Evaluation
In (game) theory, Bitcoin's resistance to centralization has been both
supported {{cite:94b91dc6e78c4db10efc3fc9016181d3bfc56b10}}, {{cite:f23b92fa359cc5d33dab2f84b19eeaf66bdd3658}} and
refuted {{cite:9d4c85b637e19dd21e60263828981d2348589da6}}, {{cite:dbd9212162e6bdd400dc2ba3ea02a94d1f7be959}},
depending on the economic model assumed for the participants' utilities. In practice,
mining pools have been observed as early as
2013 {{cite:6eed1be8f2a1667f37f6cdfb51b32ba28972667e}}. Between {{formula:556dcf18-21f9-41b3-af5a-0938cc526ee4}} ,
pools created {{formula:2c4a0d03-e2bb-4af6-af64-c6c04d6cc6e2}} % of Bitcoin
blocks {{cite:5f2c91dd1d4bd7453b21a3678e4f33eadd051253}}, with 5 pools consistently
contributing between {{formula:c09d6818-22b0-4e7b-ac50-dd7993e6cd9d}} % of the eventual blocks and 25
controlling more than 94% of all hashing
power {{cite:551bf461347aa78fe6aeab386612c517fd8b100d}}, {{cite:99e8dc06d2311b2577a7cc61588e55a7a84e279f}}.
Centralization has also been
observed within mining pools. Between {{formula:ff406a98-f825-42e1-b79b-c18c981ebf66}} , no entity
controlled more than 21% {{cite:6bd9575898239319bcfb43791510421ec16c351c}} of hashing power, but three pools controlled
a majority; within these pools, a few participants
({{formula:36da0aef-6496-4f1f-9227-4dc7fee625ce}} ) received over 50% of
rewards {{cite:1d4d8cd8f7d3f10c34ab89be30b3d5381e102f7c}}. Miners often
participate in multiple pools at the same time, a behavior also observed in
Ethereum {{cite:aa11e06c139f8ba0cb401c67ef74ae4bad57aab0}}. Although
centralization around pools is high in (PoW-based) Ethereum (in 2020, 4
pools controlled a majority of mining
power {{cite:5b0bf0b275b111c99500cd7a3cfe9338d6c5329f}}), power within the pools is
spread across hundreds of
addresses {{cite:aa11e06c139f8ba0cb401c67ef74ae4bad57aab0}}, albeit some possibly
owned by the same parties.
Consensus with Representative Participation
The second category concerns systems that require
parties to delegate
their resources to a representative or “validator” node.
The most relevant resource here, beyond stake and hashing power, is
the validator identities which correspond to the cryptographic keys
that participate in consensus, while the relevant parties are parties
that control the validator nodes. Representative participation is encountered
mostly in PoS systems.
An important consideration in the setting of representative participation is
the “barrier to entry” for becoming an active validator. This is relevant for
understanding the distinction from the previous section. For instance, in the
case of
PoS protocols, if there is no (significant) barrier to entry for running an active validator, then
the protocol is practically a direct participation protocol,
e.g., Cardanohttps://cardano.org or Algorand essentially operate
in this way. Ethereum, post the 2022 transition to PoS, requires 32 ETH in
order to run a validator, nevertheless third-party smart contracts such as
rocketpoolhttps://rocketpool.net can minimize the barrier to
participation.
On the other hand, systems like Cosmoshttps://cosmos.network and
EOShttps://eos.io restrict participation, either directly or
by requiring participants to control a large amount of assets. Any party
without enough stake, i.e., below the system's threshold or less than its
competitors, is required to delegate their staking rights to a
validator. At every “epoch”, a committee of (a fixed number of) parties
is elected to run the protocol based on the resources they have delegated to them. It is worth noting here that
as the committee members are
public, an adversary may choose to target the committee that is elected, thus
taking over the protocol during an epoch if the attack is successful.
This is exacerbated when the
committee is small and, combined with the lack of self-healing properties, such
attacks can be devastating for the systems' safety and liveness. To
make matters worse, the choice of delegates is often based on reputation, so
prominent parties may exert a large influence due to brand awareness.
Depending on the system's internals, delegates may
also control reward allocation and decision-making (cf.
Section ), thus affecting the system's stability.
Evaluation
The following are examples of systems with representative participation. Each
employs their own
consensus protocol and defines a different number of participants per
epoch using an on-chain process:Sources:
hub.cosmos.network,
wiki.polkadot.network,
developers.eos.io,
docs.harmony.one,
near.org
[August 2022]
[i)]
Cosmos: 175;
Polkadot: 297;
EOS: 21;
Harmony: 800;
NEAR: 100.
In all these systems, well-known exchanges are among the top elected
validators.For example, Binance is a validator in all
mentioned systems.
[Cosmos,
Polkadot,
EOS,
Harmony,
NEAR; August
2022]
Interestingly, the stake controlled by the elected validators is mostly
delegated, instead of self-owned. Also, organizations often
control multiple validators, so the number of real actors
is often even smaller than the nominal number of participants
(nevertheless some systems, e.g., Polkadot, go to greater lengths
to ensure the representative participation satisfies desirable properties such
as proportionality, cf. {{cite:ac2dd7f40267f651b846a177513c0b286cb783e0}}).
Consequently, identifying the participation distribution among
real-world users and the refreshment rate of the
elected committee across multiple epochs is an interesting research question.
Similarly for investigating all the desiderata of representative participation
from a social choice perspective.
Cryptocurrency Economics
A core component of ledger systems is their native token.
Tokens compensate system maintenance and accommodate value transfers.
They are treated as currency or
assets by their users, thus forming a market economy.
To record data on the
ledger, e.g., financial payments or interactions with
applications, users obtain tokens to pay
the corresponding fees. System maintainers get compensated in tokens to offset their costs. In this
section, we explore decentralization in blockchain-based economies, in terms of
(initial) token distribution, token ownership, and secondary
markets.
(Initial) Token Distribution
To bootstrap the system, a blockchain protocol defines two parameters:
[i)]
the distribution of tokens at the system's launch, and
how new tokens are generated and distributed as the system evolves.
Thus, the generated tokens form the resource of interest, while the
relevant parties are the token holders.
As with other aspects, Bitcoin led the way and other systems explored alternatives.
In Bitcoin, no coins existed prior to its beginning, i.e., there was
no “pre-mine”. Starting from
genesis, each block creates a predetermined amount of coins, based on a rate that
converges to 21 million tokens in existence.
New coins, along
with transaction fees, are awarded to the miner that
produces each block. Therefore, to acquire new tokens a user
gathers enough computing power to produce a block.
In other blockchain systems, some tokens were sold via traditional
markets before the blockchain was deployed. This approach, termed “Initial Coin
Offering” (ICO), enabled funding the project with the future
proceeds of the token investment. In return, investors acquired a pre-launch amount of
tokens, which was codified in the
chain's first block.
In terms of token generation, most systems employ a variation of Bitcoin's mechanism, e.g., Ethereum blocks yield 2 new tokens, while others, like Cardano,
employ elaborate mechanisms to incentivize pooling around a target
number of pools {{cite:fe7285255e884b6985a315cb38735394998535b2}}.
The initial token distribution is particularly important in PoS systems,
where Sybil resilience relies on it (cf. Section ). If
centralized around a
few parties, e.g., via pre-mining (or “pre-minting”), early investors
have to maintain the system in its early stages, while also
receiving the early blocks' rewards. Fewer consensus
participants during this time lowers the threshold for adversarial
takeover, threatening the system's safety and
liveness, as explained above. In both PoW and
PoS systems, new users are onboarded (and can participate in
consensus, in the case of PoS) if early investors sell tokens
on secondary markets. Consequently, early investors
control the system's expansion and valuation, impacting its
stability.
Evaluation
PoS systems like Cardano, NEO, and Algorand tried to reduce early-stage risks
via a two-phase launch. At first, the ledger was
controlled by either the core development company or
foundation or a committee numbering a small number of
entities. After token ownership was sufficiently distributed,
participation opened widely to all stakeholders.
Beyond the obvious issues in maintaining a
permissioned database, the first phase typically takes years to conclude.
Early users often tend to either not
participate or transfer their tokens to the few exchanges
that support these new tokens {{cite:5be70da376aba6dd59a9ed549e8cf5018173f6f1}}. Therefore, an interesting
question is the relationship of the delay between launch and full
decentralization and the diversity
of early investors.
Token Ownership
Diverse token ownership plays a central role in the usability and security of a
blockchain. Hence, the system's circulating tokens
are the resource of interest, while the relevant
parties are:
[i)]
addresses;
key managers;
(legal) asset owners.
This distinction arises due to the existence of custodians, who control assets
on behalf of other stakeholders, and users controlling multiple
addresses.
If most tokens are owned by a few parties, many hazards arise.
First, PoS systems' security, i.e., safety and liveness, relies
directly on diverse token ownership,
which makes corrupting enough parties to control a majority of tokens more
difficult.
Second, the
token's price may be manipulated, posing a risk on the system's
stability and, indirectly, security, in both PoS and PoW systems. Specifically, participation
cost, e.g., for mining equipment or electricity, is denominated in fiat currency.
However, miner income from block rewards comes
in tokens. Thus, miners need to sell part of the rewards (for fiat) to
pay for their operational costs.
If the market is volatile, profitability is more precarious and miners are
possibly less inclined to participate, which can impact the safety or
liveness of the system by reducing the threshold for conducting a 51%
attack.
There exist various drives towards token ownership centralization. Initial
tokens are often allocated centrally (see above). System incentives, e.g., fixed
token supply, generally favor hoarding tokens instead of spending them.
Finally,
rich participants accumulate capital faster than small
ones, an inevitability in pseudonymous
systems where downwards wealth redistribution is impossible {{cite:71e85826ddfc6bcef6a8f1e6aa5a67ead31a8c84}}.
Evaluation
Bitcoin's wealth ownership and transaction graph has been
analyzed since at least 2012 {{cite:ff8fd6ceac206a2e832f8cba7cd5c0871040ca06}}. Over time, it demonstrated a
three-phase history of distinct (de)centralization patterns, where 100
addresses possess a high centralization degree of assets and wealth flow in the
network {{cite:424235177947bdc5435b4580b49bdbef30268d36}}, {{cite:e663c3e054fbbcf352c1f686ef74525a03588292}}. Similar
analyses exist for Ethereum {{cite:f9d8b0b93469119adc560e43cb3895f0aad6f1ad}},
Zcash {{cite:d7d9e628a99238ea3dd95b85e005846eb20f20e0}}, and other
cryptocurrencies {{cite:4f484778f4ff50a1b4a0df9dce3a0d74e3062bbc}}.
As of 2022, cryptocurrency wealth concentration is particularly
extreme (Table REF ). To establish some context, the income Gini coefficient of the 10
lowest-performing countries ranges between {{formula:380311e6-80da-4298-9699-c00e90424fad}} {{cite:befa44e0826b7e00d40ac012de281edcd2262c2d}}.
Bitcoin has a Gini coefficient of {{formula:b8b866f6-343d-47f6-8e50-a4cff0e74c5c}} ,
considering only the {{formula:b51ee00f-5111-40d0-8ec8-5f657a9140a2}} richest addresses, and a staggering {{formula:a24cc8d1-b6bb-47d3-8f19-8017eb3b6893}} w.r.t. all addresses. In the arguably deeply unequal global
real-world economy, the richest {{formula:519b555d-b040-4bbe-9c3e-5c522b075f49}} % of individuals ({{formula:e97ddf3e-f336-4b0d-b638-7523ec5f44d1}} people) hold
11% of all wealth {{cite:3b6e0fbe75250ba5dad52a0738eedb4e026859cd}}. Bitcoin
manages to beat that figure, with 100 addresses
holding {{formula:16d6e6bd-c378-47cc-9de6-f028a9fd473b}} % of all tokens.
{{table:5c1a988c-ec82-48f6-9b78-2bf5f0c6b28f}}A complexity in measuring wealth decentralization in
cryptocurrencies arises due to their pseudonymous (or even anonymous) nature. Specifically,
the number of addresses often does not correspond to individual people or entities,
cf. {{cite:759df43567a80f7d2c123fc4b35d884f250437dc}}, {{cite:45e131ac6791c2eb9bc36b0aa9f470e04d32d2f8}}.
A user may control multiple addresses, e.g., each with a small
balance. When interpreting the Gini coefficient, this artificially enlarges the population and possibly biases the results
towards decentralization. In addition, an address's assets may be owned by
many users (e.g., exchange addresses), which
biases Gini towards centralization. Thus, developing tools to
compute wealth inequality in blockchain systems, without sacrificing
core features like anonymity and privacy, is a crucial problem for
exploration.
Secondary Markets
Distributing the tokens to a wide population is predominantly made on secondary markets as
[(a)]
the rate of token production is typically slow (depending on block production), and
the new tokens are often distributed to existing users (particularly in PoS systems).
Markets take the form of centralized operations (“exchanges”) and,
at a lesser extent, face-to-face transactions. Therefore, the tokens that are
bought and sold through these markets constitute the resource of
interest, when it comes to measuring the decentralization of secondary markets,
while the relevant parties are
[i)]
the assets for which they are bought and sold (“trading pairs”), and
the exchanges that host these trades.
Many hazards arise when tokens are available on a limited number
of markets.
First, exchanges typically offer little privacy
guarantees, exchange operators have full access on their users' data as
required by KYC regulations.
Second, exchanges are largely
unregulated by financial
authorities and
may engage in market manipulation.
Third, few marketplaces often result in lower liquidity. Thus, the threshold for manipulating
the token's price by some percentage, via selling or buying tokens, also
lowers. All such events threaten the system's stability,
while also, when mining profitability drops due to the token's devaluation,
safety and liveness are indirectly hurt (see above).
Evaluation
Table REF summarizes secondary blockchain market data
across 121 exchanges. Many systems (Bitcoin, Ethereum,
Litecoin, XRP) are traded on all but a few small exchanges. Tether is by far the most
available, in terms of market pairs, and used, in terms of volume.
Interestingly, for all systems, except perhaps Bitcoin, the majority of volume
is not of the highest transparency.For “transparency” see the
methodology and data of Nomics:
https://nomics.com/blog/essays/transparency-ratings. This is consistent
with reports that show market manipulation is endemic in cryptocurrency
markets, with multiple cases of wash trading, fake trading volumes,
and other fraudulent
behavior {{cite:632eda6b63670bbb2388116f1a91264e84633ad8}}, {{cite:35af1449a44cbec589823598d8d1de97ffb7cfbc}}, {{cite:0224814e10e8d6dd69d00ff281badc93372c7bc9}}.
Market transactions are primarily conducted in a handful of
exchanges. By far the most used is Binance (20% of the total daily
volume),CoinMarketCap [August 2022] although
Coinbase is the most recognized in North America {{cite:b32e7c650039015c738f9a2e979315168cb486a4}}.
{{table:2a948926-dc01-4c77-9864-d7ce4278103c}}
API: Blockchain as a Service
To join a blockchain system, full nodes typically need to download and
parse the entire ledger, which often amounts to hundreds of
GBs.Bitcoin: 485 GBs;
Ethereum: 819 GBs. [bitinfocharts,
Etherscan; August
2022] The ledger's state, which is usually stored in memory,
is also largeBitcoin's UTxO set is
{{formula:3de748c8-3079-43f8-848b-242108259701}} GBs.
[Satoshi
info; August 2022] and often poorly
maintained, cf. {{cite:c85256b323b826ea98ae8fb8def1c8fd5c367cb1}}.
Consequently, maintaining a full node requires significant computational and
storage capacity and, eventually, it seems impossible that a node will be
hosted on home equipment.
This concern is well-known and ongoing research tries to resolve it via
ledger compression {{cite:15e9442187fe13335b17a539cd4eb315592ad2d4}}, {{cite:2e656d087e994b35e2885c17932ef4ed58535144}}, {{cite:fa13b682d34ddea4e132a8bc4fbf0099ada59d30}}. In practice though,
users often employ third-party services that offer an interface to
the ledger. Given the
widespread use and variety of applications that rely on such services, “Blockchain as a Service” is a separate layer, with the
resources being
[(i)]
tokens stored in light wallets and
light nodes,
and the relevant parties being the full node operators that service them.
Many properties are at risk by corrupted API services.
By not verifying the entire ledger, a “light” client might be duped in
accepting an invalid transaction.
A possible solution is succinct verification
proofs {{cite:769b30ec55c1b1c34612546aad0560127d17df62}}, {{cite:8220d52722506445d5c4aeb2eb6549af161936dd}}, {{cite:15e9442187fe13335b17a539cd4eb315592ad2d4}}, {{cite:e00e32dd9e810107894081b2c777ebccc01cb3cb}},
although light clients often have to parse the block headers, so
most deployed solutions have linear asymptotic complexity on the
ledger's size. To avoid the overhead, a large portion of users rely on intermediary services,
often without receiving any correctness guarantees. Thus, a double-spending attack
(safety violation), where a user accepts a canonically invalid
transaction, possibly requires corrupting only a few parties that control
such services.
Also, light wallets typically rely on external nodes for transaction processing and
balance computations (cf. Section REF ). Thus, liveness, privacy, and
stability hazards arise if these nodes become corrupted and can block,
de-anonymize or, depending on the implementation, divert a user's funds and
transactions.
Finally, although applications often rely on media, storing even small
images on-chain is prohibitively expensive.Bitcoin and Ethereum
require approx. ${{formula:cdfc966d-1a32-4ea2-be8c-817d0189aa97}} and $23 per byte resp. [August
2022] Blockchain applications often employ a combination of storage types,
with the logic being on-chain and files in servers. For example,
NFTs are digital images whose ownership rights are tracked on a
blockchain. Here, the ledger typically stores only a link to the image, e.g., a
hash or URL, while the file is accessible via a third-party's server.
Naturally, if these off-chain elements become inaccessible, the
stability of the application and, possibly, the underlying ledger are at
risk.
Evaluation
In Bitcoin, most wallets are either SPV or
explorer-based {{cite:4fee83b3e940e903bb9fda9122f667e363663078}}. In the first case, the wallet obtains
the chain's headers and, to verify that a transaction is published,
requests a proof from full nodes. Although SPV does mitigate safety attacks, it
also hurts the user's privacy, as their transaction information is leaked to
the full node operators. Explorer-based wallets instead rely entirely on a
single explorer service and its full nodes, which are trusted completely.
In 2018, {{formula:922b77da-3958-43cc-a169-eb93de0ba782}} % of all Ethereum nodes reportedly relied on a
centralized blockchain API service, Infura {{cite:f8c311adbf2a2e9c5a5842bd218e99e1ef50e4e9}}. This reliance
continued throughout the years. In 2020, a service outage demonstrated in
practice the hazards of such centralization {{cite:db5b4fe65bec7a94bb0a68f99c165d6c7f9c5d91}}. In 2022, a
misconfiguration on Infura's part resulted in wallets (and, thus, user funds)
being inaccessible {{cite:8b76bfe51e7a2ccefbfafb68349063f4755f5869}}. In terms of applications, OpenSea is the
leading hosting service for NFTs. As of 2021, it reportedly handled 98% of
all NFT volume {{cite:0a1ddc59aca07a255ee95aaee30ac24093c392d4}}, charging a {{formula:5778b1e9-e9eb-4854-b282-726f4d8453bc}} % commission on all
sales. As expected, an OpenSea outage in 2022 also resulted in the NFT
market being practically unusable {{cite:1ff4bca65605455601253567752e416464aefe3b}}.
Governance
Governance in blockchain systems is a broad
topic {{cite:abc0536f5db18b6d14dcffa9da5e6672783eaa03}}, {{cite:457cb05247e2032784f9e5038e657f5ec00ee2b4}}, {{cite:70db2fe8099e73d955e97003abc33c795997303b}}.
Here, we focus on two aspects that affect all previous layers:
[i)]
improvements and conflict resolution;
fund allocation for research and development (R&D).
Improvements and Conflict Resolution
Decision-making mainly concerns conflicts and improvement proposals. Proposals
may affect mining, e.g., changing the PoW function {{cite:83fab96a044865355ab25bb7662218d227a32dbb}} or switching to
PoS {{cite:d825a666cb0850bc9511aedd8256be79b0f64f79}}, the consensus protocol, e.g., changing block
structure {{cite:6eba87d8f0d76ccd87a6b10a4c9e887e81e8eacf}}, or token ownership, e.g., blacklisting
accounts {{cite:ed90a0a93ba74f33926de6e76680b03a0cacd4d5}}, just to name a few aspects that are frequently contested.
In theory, anyone can propose changes in blockchain systems and respond in
some way, depending on their role. For example, PoW full nodes pick a
side in a debate by choosing which ledger version to extend.
Token owners influence decision-making directly, by operating PoS full
nodes and/or participating in voting mechanisms,
or indirectly, by choosing which ledger to transact on.
In essence, full nodes assume executive, legislative, and judicial powers
by operating the ledger and choosing its rules,
while other actors voice their opinions by affecting the token's market
price {{cite:a6ae0545980b4d8933a4174cd74f77dc1fcbad95}}. The governance
resource is decision-making power, which
may take various forms, and the relevant parties are all active
entities in the system.
If the other layers are centralized, governance follows suit.
For instance, if mining is concentrated around a few operators, they might
force a choice by mining on one ledger, or if
a handful of investors own most tokens, they could manipulate the market to
force a choice favorable to them.
If disagreement turns into a stalemate, systems may split into distinct
ledgers, that share the same history up to a point but diverge
thereupon {{cite:eff7a49ee3667e8ed773495d5786822bed73ebf4}}, {{cite:ed90a0a93ba74f33926de6e76680b03a0cacd4d5}}. These outcomes harm
the system's stability, and indirectly
threaten its safety and liveness.
An effective governance process should prevent such harmful events.
However, it is not always possible to make
consistent decisions in a decentralized manner, as demonstrated by theoretical
results in social choice such as Arrow's
impossibility theorem {{cite:f7892a2328b277dffcdaed9482ac2537e47cbb8e}}.The
theorem's formal
proof states that no rank-order electoral system can satisfy a set of three
criteria:
i) if every voter prefers X over Y, the group prefers X over Y;
ii) if every voter's preference between X and Y remains unchanged, the group's
preference also remains unchanged (even if voter preferences over other pairs
change);
iii) there is no “dictator”, i.e., no single party can always determine the
group's preference.
This idea dates back to the late 18th century and Condorcet's paradox in
collective decision-making: consider three candidates {{formula:1e32d700-c7e0-4565-b46a-af2f9a8bc358}} and three
voters {{formula:d23d2a95-ce8a-40aa-987d-28aac5c5e364}} , with (ordered) candidate preferences {{formula:d750d6e5-f1f7-4a25-bebc-da8f20c2e1b0}} resp; although each candidate's order of preference is
consistent, the collective preference is cyclical.
Additionally, when agents act in a selfish manner, as is presumed in distributed
ledgers, efficiency can degrade (cf. “Price of
Anarchy” {{cite:1de06f4272caa3ba3b4793d41a2d83dbe562fab7}}).
Therefore, decentralized decision-making processes face a challenge, as they
need to address various social choice theory (e.g., Arrow's theorem) and
game-theoretic (e.g., rational ignorance {{cite:e7acf0ef7f076552b81684380b0de9f697d80d32}}) considerations.
Evaluation
Most systems employ an Improvement Proposal mechanism,
where proposals are posed as issues in Github, a (centralized) system that is extensively used for software development. If a change gathers enough support,
it is incorporated in the codebase. To voice approval for proposals, miners
often include encoded messages in blocks.
From early on, proposals in Bitcoin and Ethereum have been made by a
handful of developers {{cite:6eed1be8f2a1667f37f6cdfb51b32ba28972667e}}, {{cite:ec13c4bc75972ce3fd8ac0030a886e01ae8a36b6}}.
In the discussion phase, many people participate but again only a
few actors contribute most comments, while in cases like
Bitcoin the groups of developers and commenters largely overlap.
Development Funding
Funding for research and development can cover the maintenance of legacy
codebase, research in features like privacy and scalability, market incentives,
e.g., stabilizing the token's price at times of high volatility, and more. Thus,
the resource of interest here is capital and the relevant parties
are the active developers in a ledger's ecosystem.
Ledgers typically make no funding provisions, besides allocating
rewards from coin issuance and transaction fees. R&D is conducted
via corporate vehicles which rely on traditional funding models, such as venture
capital. However, since designing and implementing hardware
and software for distributed ledgers is particularly expensive,
this model can lead to centralization, as
discussed in Sections and . In addition,
lack of funding or concentration around a few teams may delay crucial
updates or new features, thus hindering stability.
A common alternative to traditional financing is ledger “self-funding”.
Here, the system defines a treasury, i.e., a pot which collects
part of each transaction's fee and, over time, accumulates
significant amounts of (token) funds {{cite:99679a80691e8e3726924b181a33938693edfe2c}}, {{cite:86be8850c9b94aa5545468c6a8d1a28120088012}}.
These funds are allocated
via a voting mechanism, open for participation to stakeholders and
other relevant parties. Given the pseudonymous nature of these
systems, voting might be weighed by each participant's wealth, with each token
granted one vote (instead of one person one vote, as in traditional
democratic processes). Presumably, if stakeholders aim at increasing their
assets' value, they support proposals that improve the system. However, if
ownership is concentrated around a few investors (cf. Section ),
the decisions might aim at benefiting these few parties in the short
term, at the expense of the system's long-term benefit.
Evaluation
Most existing blockchain systems follow the first approach, i.e., not making
funding provisions. In many cases,
funding is channeled through a few foundations and
companies.The first usually take the name of the token, e.g., the
{Bitcoin, Ethereum, Cardano} Foundations. Examples of the second are the ASIC
companies discussed in Section REF or software companies
like Blockstream (Bitcoin), Consensys (Ethereum), Input Output (Cardano), etc.
Treasuries are present in some ledgers, like Decred, Cardano, and Dash.
Despite their potential though, widespread funding has yet to be
demonstrated for most systems.Decred's treasury holds
${{formula:fabdc4ce-9e14-4d48-8645-432db95f42c9}} M, and has allocated {{formula:430b4191-630f-430b-a667-de510ab8c628}} K over the past year. Cardano's
treasury holds approx. $500M and has distributed ${{formula:cedd174d-4bcb-448f-aa64-f72639c75479}} M across 939
projects. Dash, one of the first systems to set a treasury, allocated
${{formula:2a06bdfa-3fc9-45b7-9e0a-e49008697b6a}} over 2018, but it appears non-functional as of
2022.
[dcrdata.decred.org,
cardano.ideascale.com,
dashvotetracker; August 2022]
Geography
Geographical decentralization touches upon all dimensions covered in the
previous sections. Accordingly, it involves all resources described so far,
e.g., hashing power or tokens. Nonetheless, it constitutes a
dimension on its own, as parts of a system may be well distributed w.r.t. one
dimension but geographically concentrated or vice
versa.For example, independent
actors may participate in mining within a single
country.
The tendency to centralize in certain areas
arises due to economical, technological, or sociopolitical factors. For
example, miners often set up their operations in countries with
low electricity costs, hardware companies operate in countries with small
production costs, nodes are hosted in areas with high internet speed, and tokens
are accumulated by residents of countries with low taxes and where many
exchanges operate. Geographical centralization poses two main threats to the
properties of a ledger:
[i)]
physical hazards and
legal impediments.
Physical safety
Physical hazards
could threaten a system's infrastructure. For instance, if part of a system is located
in a small area, failures or interruptions in the area's connectivity could destroy or split the ledger system in two. This
concern is particularly relevant in PoW, where equipment is hard to relocate.
All resources examined above
can be impacted (e.g., via drops in hashing power or token loss when mining equipment or cold storage
is damaged), while the
relevant parties are the regions of
resource concentration.
Single points of failure may arise when
geographically-concentrated nodes act as central hubs, harming
either safety or liveness (cf. Section ) and,
indirectly, stability (e.g., due to increased market volatility).
Legal compliance
Failures possibly occur w.r.t. the legal frameworks under which the
system falls. If some layer is concentrated in a specific
jurisdiction, authorities can possibly restrict or
subvert it. Again, this touches upon all resources examined so far, as
all are
influenced by the law, with the relevant parties being legal
jurisdictions. Depending on the occasion, different properties of the system
are impacted. For example, if a country bans Bitcoin mining, the power drop could decrease
the threshold for controlling a majority (safety hazard), while
blocks are produced at a slower pace until the PoW difficulty is
recalculated (liveness hazard). Stability could also be hurt, if
some part of the system, e.g., mining, software access, or asset
ownership is restricted. Additionally,
exchanges often adopt KYC processes to comply with AML
regulations {{cite:d74cc69183a1072bb3199c593a43db347321e91e}}, linking the users' identities to their
activity and thus compromising their privacy. Arguably, a system is
more likely to uphold its properties by falling under many
jurisdictions, such that violating them requires the coordinated
efforts of multiple authorities.
Evaluation
In 2014, 37%
of Bitcoin nodes resided in the US and China {{cite:03f8d332860ce88fdcb474122eed001cf670670a}}.
In 2018, its testnet also showed a concentration in the USA, Central Europe,
and East Asia {{cite:9bcdfb77f7412a5a9dc8eee41e6b5dcd4ef2ae9d}}.
In 2019, Bitcoin mining
hardware was mostly located in China (particularly Sichuan) and the US {{cite:fa49a68a4b39f0ae36c12dac03b33f037e8cbb7d}}, {{cite:5cfd6b3a94f61973b3577557f9f0490656d82773}}.
Notably, the mining pools then-located in China accounted for
68% of all hashrate {{cite:78d389a49c704d9f44edbd174651836e7298b7b1}}.
As of 2022, a large fraction of nodes communicates over
Tor,{{formula:572f0a41-f952-49c0-b3d6-ff1e607e7f53}} % of Bitcoin's nodes operate over Tor.
[bitnodes; August 2022], thus
analyzing the network's topology is often hard.
Nonetheless,
more than {{formula:488eda36-6f37-495c-a1ea-b49787804a1e}} of Bitcoin mining is presumably located in the USA, with Kazakhstan and
Russia following with 18% and 11% respectively {{cite:460faa52a87278d4e80caee17097f682e93270e8}}.
In terms of full nodes, USA and Germany
see roughly equivalent shares, with other countries hosting far fewer;
still, a majority communicates over Tor.
Until 2021 China hosted as high as {{formula:ef90c7a5-45d1-4005-957e-699bc73aca20}} % {{cite:460faa52a87278d4e80caee17097f682e93270e8}} of Bitcoin
mining power; following its ban that year, Bitcoin's hashrate dropped from 197 to
68 Ehash/s in one
month.bitinfocharts.com
Ethereum (pre PoS) observed similar concentration
patterns; by far the most nodes are located in the USA
(37%) and, secondarily, Germany ({{formula:89e9047c-809c-4e7a-8be0-89b12c630399}} %) {{cite:86fcbc7f64f793b46cd886029d258b6383c8040b}}.
Finally, Monero nodes are mostly located in the US and, to a lesser extent,
elsewhere {{cite:40a1edde9ce2eb57621cf970760103fba709895a}}.
Table REF shows various systems' geographical distribution.
In terms of legal jurisdiction, different aspects are
centralized in different countries.
In Bitcoin, the 4 companies that predominantly produce mining hardwareBitmain,
MicroBT, Canaan,
Ebang. are all based in China.
Regarding secondary markets, many exchanges operate in multiple countries (Table REF );
20 of 121 operate in USA, thus falling under US jurisdiction, 17 in
China, and 10 in Japan, with the rest spread across the world.
However, only 8 are based in the US, with most registered in the
Seychelles (13) and other “offshore” locations. Many ICOs
also exclude US investors, following their US classification as
securities {{cite:42085a13c67aa6a71cfe1f7fbbddebac1298fdd7}}.
Finally, an interesting case concerns the Bitcoin Core software, which is not
available via bitcoin.org in the UK, following a
related court ruling {{cite:81dfcb4965875d39c5eae1f7e1b3ec3c8caaa551}}.
{{table:afcf50da-ad42-4987-889f-916e019090d3}}
Case Study: Bitcoin
In this section we apply our methodology to Bitcoin. We review Bitcoin's status
w.r.t. each identified layer. We show that it passes the Minimum Decentralization
Test (MDT) (cf. Definition REF ), though possibly due to the lack of
substantial evidence within some layers.Note that we employ a
“decentralized until proven centralized” approach.
The properties of interest here
are safety and liveness, i.e., the two core security properties
that a ledger should guarantee.
Decentralization Layers
Hardware
No concrete data could be found on the distribution of hashing power across PoW
mining products. Hundreds of efficient products (ASIC) are available, on top of
generic hardware (e.g., GPUs). As of 2022, the market appears centralized
around 4 ASIC manufacturers {{cite:fa8823faa4900be36821162f23fdaa2a204cfdad}}. Interestingly, only some
ASICs are profitable, so, unless the token's price increases without an
increase in PoW difficulty, mining should be expected to concentrate around
these products.A profitability calculator is available at
nicehash.com.
Software
As outlined in Section ,
{{formula:cd7c314f-e4b9-45ac-b8d8-3dab53cc604c}} % of all Bitcoin full nodes run Bitcoin Core.
Therefore, Bitcoin is completely centralized around (different releases of)
this product.
Regarding the distribution of tokens across wallets, no data could be found.
Network
Section offers a brief evaluation of Bitcoin's network.
Regarding peer discovery, Bitcoin Core sets 8 outgoing and 125 incoming
connections, chosen randomly from known and/or hardcoded peers.
Most Bitcoin nodes communicate over Tor, making topology analyses particularly
hard.{{formula:8c298066-2c0f-4d35-9130-87c6d35d476b}} % of Bitcoin's nodes operate over Tor.
[bitnodes; October 2022] Nonetheless,
it is estimated that the network is evenly spread across multiple Autonomous
Systems, thus presenting high levels of decentralization {{cite:e58391fabf6dc6cf7ae9ffd2cb49611440ba2952}}.
Consensus
On the consensus layer, Bitcoin presents mixed results regarding
decentralization (cf. Section ). Hashing power is distributed across
thousands of machines. Although no concrete data could be
found, folklore evidence suggests that these machines are owned by a highly
diverse set of users. However, Bitcoin also observes high levels of
centralization around pools, i.e., w.r.t. block formation and the input to the PoW
module; specifically, at the time of writing,
4 pools control more than 60% of the whole network's
mining power.
Tokenomics
At its onset, no Bitcoin tokens existed. They were generated and allocated as
the system progressed. Early participants were disproportionately favored, with
half of all tokens created within the first two years, when consensus
participation was sparse and mining was conducted by only a few parties. As
more transactions were issued, the tokens were distributed more widely, albeit
wealth is still highly centralized, compared to real-world economies (cf. Section ,
Table REF ). Specifically, approx. 43M addresses own some amount of
tokens, with the top 100 addresses controlling {{formula:a204dd40-df8f-4b34-8bf6-e5acd453da9b}} % of all wealth.
Nonetheless, tokens are traded on more than 100 marketplaces at volumes of
approx. $53B (cf. Table REF .
API
As discussed in Section , most of the available Bitcoin wallet software is either SPV or
explorer-based {{cite:4fee83b3e940e903bb9fda9122f667e363663078}}. In the first case, the wallet downloads
only the block headers, so does not validate each block's transactions, while
in the second case the wallet relies entirely on a server. However, no data
could be found on the ownership of Bitcoin tokens w.r.t. wallet types, therefore
Bitcoin's decentralization w.r.t. the API layer is inconclusive.
Governance
Deciding on improvement proposals and conflict resolution in Bitcoin is
somewhat centralized, but not entirely. Specifically, decisions, which are made
by accepting suggestions via Github, are typically taken by a small set of
developers, who are often the ones to comment during the relevant
discussions {{cite:6eed1be8f2a1667f37f6cdfb51b32ba28972667e}}, {{cite:ec13c4bc75972ce3fd8ac0030a886e01ae8a36b6}} (also see Section ).
In terms of development funding, Bitcoin makes no provisions. Therefore, the
available data are inconclusive on how many sources of funding exist, e.g., companies and foundations, and how much influence each has.
Geography
As mentioned in Section , Tor communication between nodes makes
analyzing the network's topology particularly hard. Nonetheless, Bitcoin
miners, although fairly well distributed with a presence in 95 different
countries, tend to cluster in certain areas.
More than {{formula:536d17ec-7575-42fe-a650-a500880efc75}} of mining is located in the USA, with Kazakhstan and
Russia following with 18% and 11% respectively {{cite:460faa52a87278d4e80caee17097f682e93270e8}}.
In terms of full nodes (which may not participate in mining), USA and Germany
see roughly equivalent shares, with other countries hosting far fewer nodes
(cf. Table REF ), although still a majority communicates
anonymously.
Minimum Decentralization Test (MDT)
As of October 2022, Bitcoin appears to pass the MDT, as no legal
entity, with the power to violate a property of interest in some layer, was
identified. Some layers are highly centralized (e.g., consensus, where only few
pools are responsible for the majority of the produced blocks), but violating
the relevant properties still appears to require the coordination of at least two legal
actors. Particularly regarding software, although almost all nodes run the same
product, Bitcoin Core, this is:
[i)]
developed by over 100 monthly active core developers {{cite:94f609130cf09354477b0c764e60b2270ae68881}};
is provided under an open source license (MIT).
Consequently, we deem that the MDT does not fail because of it, as this
product's development and maintenance is not controlled by a single legal
entity, but rather a team of open source developers. Finally, for some
layers (hardware, network, API, geography) there is inconclusive or no
available data, so further work is needed to offer a definitive and conclusive
answer regarding them. Despite that, we find no reason at present to believe
that a single legal entity is capable of subverting the relevant ledger properties in any of these
layers.
Discussion
Our work explores how to define and measure decentralization in
distributed ledgers. Table REF summarizes our methodology,
i.e., the layers that comprise a ledger and the relevant resources and parties
that guarantee its core properties. Evidently, removing single points of failure via
a diverse distribution of resources among relevant parties is
paramount in guaranteeing security, privacy, and stability
and can also have legal implications (cf. Definition REF ).
Related Work.
To the best of our knowledge, this is the first work that offers a systematic
methodology of defining and evaluating decentralization in distributed ledgers.
Our work is complementary to research of blockchain decentralization from other
perspectives, e.g., economic or
social {{cite:09016babd9efe854dcc18a2566d3329111a5c874}}, {{cite:1c94207026b4924a8260adbd5965364e8424cdcb}}. In computer science, only a couple of works have attempted to tackle the question of blockchain decentralization.
Sai et al. {{cite:78e484f7cc3cd5eab4e814bd77412c1f693f4ee5}} offer a blockchain centralization
taxonomy, based on an algorithmic literature review and expert interviews.
Like our approach, they treat ledgers as multi-layer systems; some
categories overlap, e.g., wealth concentration and geography, while
others differ. Nonetheless, our work goes beyond a standard literature review
and identifies various nuances in how ledgers operate, are organized and how this interacts with decentralization. Also, instead of
employing various non-comparable metrics, our work proposes a unified
evaluation methodology and a directly applicable minimum decentralization
test (Definition REF ).
Zhang et al. {{cite:28847f4311c15b7162013ee0e7d69ff3a0b67508}} propose a taxonomy around five
facets: consensus, network, governance, wealth, transactions. They describe an
“index” based on transactions ceteris paribus, such that “as the number of
transactions increases, the index increases, indicating greater
decentralization.” Nonetheless, it is unclear how this treats transactions
of different utility (e.g., spam) and whether it is meaningful to compare
ledgers with different transaction throughput.
In comparison, we offer a holistic treatment that not only
encompasses these facets, but also offers
a principled way to measure decentralization, independent of
narrow focused performance metrics.
Measuring Decentralization.
Our work offers a framework for analyzing blockchain decentralization, but
not specific metrics to quantitavely measure it. For example, a metric could assign a
single number to reflect how close a system is to a single
point of failure, given a distribution of resources over a set of relevant
parties. Here, we briefly review some metrics,
at a high level, and leave for future work the exploration of alternatives and the computations over
real-world data.
A first option is Shannon entropy {{cite:3ae5ef1744dc2978024da0150f08fcc5e9f9a15c}}. Briefly, a random
variable's entropy measures the uncertainty of its possible
outcomes. In our setting, the more bits of entropy in the resource
distribution, the more diverse it is, thus the more decentralized the measured
component is. Min-entropy, i.e., the smallest of the Rényi family of
entropies {{cite:96025ca4fbb08556a72eaf44f190021c9e8d582a}} can be also used instead since it also offers
a lower bound.
An alternative is the Gini coefficient {{cite:6b8cd90ba55b993311154701d7a602e8889e4322}}. Gini
expresses the percentage of space between the 45o line and the curve
that plots the cumulative wealth {{formula:644023db-4932-4745-a148-c6ec37092a95}} owned by the bottom {{formula:c0c3ad93-c223-4941-8e06-998eaa209498}} of the
population. Intuitively, a Gini value of 0 implies perfect equality, where
each person owns the same amount of resources, while 1 reveals extreme
inequality.Although Gini is used by organizations like the
OECD {{cite:32754eab944761c6834b04938d25907b87c11c97}} and the World Bank {{cite:befa44e0826b7e00d40ac012de281edcd2262c2d}}, it has major
shortcomings. First, what the metric is applied on may skew the
narrative. E.g., the Gini coefficient of real-world economies w.r.t. to
income offers a much more appealing image compared to wealth ownership, as
income differences persist and accumulate over the course of life. Second,
Gini's single value obscures qualitative differences between economies at the
same “level”. For instance, consider the following two wealth
distributions {{cite:63c253929007886985cc69153b7b092025e9174f}}:
i) one person holds 50% of total wealth, while all else share the rest;
ii) half the population shares equally all wealth, while the other half has
none.
Although these economies are vastly different, their Gini coefficient is the
same (approx. {{formula:ec8f6074-c0d2-4e06-b4ad-d7c6e2629dbc}} ).
Alternative metrics could
also help evaluate different aspects of decentralization. Examples from traditional economics
are the Theil {{cite:074e4861d78024bd3cdec407a8f96ed2613c1842}},
Atkinson {{cite:bb80ab0b97af3da80a917fbe90ee5631b8dd79ac}}, and Herfindahl-Hirschman {{cite:f83e6cc50d81c3d6eac76a6e14a5d04e4957cf38}} indices.
Drawing from the blockchain space, an often-used metric is the Nakamoto
coefficient {{cite:fefb865a73211f0202823930ee31156bd00af01f}}, which measures the minimum number of
parties that control a majority of resources. Nonetheless, a systematic
comparison of all alternatives is an interesting question for future research.
Research Directions.
Our work opens various threads for future research, beyond evaluating
metrics of decentralization.
First, off-chain “Layer 2” protocols are an important part of blockchain
ecosystems {{cite:2be607c874d81ae1807d97a1da5eee2e82b87ff5}}. Our work does not treat such protocols as a
distinct stratum. Nonetheless, our methodology can be readily applied to a
combination of Layer 1 (main chain) and Layer 2 protocols. For
example, one could apply our layering methodology on a combination of Bitcoin
and the Lightning Network {{cite:e71bec71d6351a57f12b0aca39d841c2969d6a47}} and assess the decentralization of the combined system as a payment network. Taking it one
step further, one could use our methodology to assess “decentralized
applications” (DApps) that run on blockchains.
This for instance could be used to assess the decentralization of DeFi systems.
In such an investigation,
smart contract development would naturally fall into the software layer. In summary, exploring the
decentralization of Layer 1 and 2 protocol compositions or DApps under our
methodology, and possibly identifying alternative properties and relevant
resources, is a promising thread of future research.
Second, a natural end-goal of all questions posed in this work is to produce a
single, quantitative blockchain decentralization metric or “index.”
Historically, it has been shown that increasing decentralization on one axis
coincides with, or even results in, centralization on
another {{cite:e35460f4bcd805c7f918d0ddcca3760e4856620f}}.
Therefore, future work should further explore the
dynamics between all layers, resolve possibly inescapable trade-offs,
determine the importance (weight) that should be assigned to each
layer, identify appropriate metrics for each layer and
efficiently combine them into an index.
| m | e155909b35f9aff4d1425a9322dd341b |
Recently proposed volumetric neural rendering methods, i.e. NeRF and its variants {{cite:cff19d82b1699101c2663e9da85a93c9105c8f90}}, {{cite:395dd9e36dadfa165f826f5f12ec5081be589ec1}}, {{cite:7e22b2e850fe01fec80454e66dcf0a65049cfb7a}}, have shown great advances in high-quality free-view synthesis for static objects. NeRF models static objects by an implicit radiance field function with multi-layer perceptron (MLP) networks. Inspired by NeRF, recent works {{cite:e8f24c98e913a63a863a84435c623c6ff7d4e8e9}}, {{cite:a593b8d20f075538a032542b75949658cf66cfdf}}, {{cite:038f528e72ff1f66bc4d3f981f841727266adfab}}, {{cite:5553cfc2d787490d219ec96c556a3f28b95e9834}} attempt to model 3D humans by conditioning the radiance field on 3D poses / parametric meshes.
While promising human reconstruction and view synthesis results have been achieved, these methods only focus on the modeling of conditional radiance field itself and require accurate 3D poses or meshes as a prior. This assumption is often too strong to be fulfilled in practical capture setups, especially with monocular video only.
| i | 54776ce6b5bd9ee7fefb01f893682e49 |
Since {{formula:4a3e9b83-724f-4064-bad1-b7756b5b5300}} is a random matrix with each entry i.i.d. {{formula:94f13736-77d5-42c4-8f83-355dc2f485bc}} , this follows by standard arguments (Proposition 2.4 in {{cite:1bfc6e9cc320646dc24150d429f76d2aad89471b}}). Since {{formula:773d472b-4033-4cff-ad8e-5e96cb2b3ccd}} , with probability at least {{formula:72240c45-ecf0-4b10-a32b-960f17099de0}} , {{formula:5153c6a9-13f9-40e5-ad73-94d9c9505d5b}} is at most {{formula:db7219db-d870-4736-a8a9-730847f13d9f}} .
| r | ddd7a06aa4122af02744a64a95c68754 |
Panoramas with depth information are very useful for 3D computer vision tasks such as novel view synthesis {{cite:9cf11f4e8ca6904565e172319f1dd5c727265b0c}}, {{cite:b31c7df0feea61688f6c4b395d1823171b9764c5}}, {{cite:a12ca0a052ba9e1f6bd51439628e261ad6a98ea2}}, 3D scene understanding (room layout estimation {{cite:34467ebf670d9f3ce6cda7a41f98ad93f8e29500}}), omnidirectional SLAM {{cite:354729bd8fb481c554298eb7573872cb979dd965}}, and virtual reality (VR) applications {{cite:3f19d5a1c4b1a46e1cc7b364c8a49e06f7deef8a}}. Traditional monocular depth estimation methods built for perspective images cannot directly work on panoramas, as a panorama cannot be converted to a perspective image since the latter can not have field-of-view (FOV) angles exceeding 180 degrees. Therefore, many depth estimation methods that are specially built for panoramas have been proposed {{cite:6f8a86c8412616e8f9cb87bcacd7bed63432759d}}, {{cite:ce4d02c6464dc322cfd2f53153436994d5435455}}, {{cite:35d5c018c4c5681e44a7fd2a6b9a9b907cc5ff66}}, {{cite:df45cc26c8e60d1fa3bf8146cb862bd11b232af2}}, {{cite:6bb986e4021af013783105cb921786bd614dca82}}, {{cite:51c2312a12b6d9b94618d84ce18a41d9b499137c}}, {{cite:eb6cfc5229ee2b1afc9913ce02def316f6ece74f}}.
| i | d68ebb9e8eb620dc4f16ff6d2fc1a2b9 |
Many cosmologists have considered, wormholes in {{formula:b8bb8bed-3635-412c-a72f-a30c47e621af}} gravity in various perspectives. Now we are going to examine the contribution
of certain cosmologists in {{formula:a9073703-29a0-443e-9cba-4ab79441bcd7}} and {{formula:7c7c98b3-81d9-4348-8a16-5d61f988a673}} gravity theory. Harko {{cite:b1bc61adec95d68c73fb76de882ba175a5744891}} worked on {{formula:5ec3efd9-2537-4431-867f-51ed312099b5}} theory and {{cite:0d557fd21b84cff848244ec0f1e1aa03115265d8}}-{{cite:86aedda952e7aea78303912edffd82c38af9848c}}
by replacing the function {{formula:f371c398-4a16-43f5-bc76-aa0f52992fee}} with an arbitrary function {{formula:ba79a83b-e339-43e2-8a0e-4a3f1f506a1f}} . The anisotropic cosmological models in {{formula:4d461bfa-7762-43d6-8f41-93626f214ea5}} gravity with
{{formula:b2344663-04c6-4390-be64-36d23e502740}} discussed in {{cite:04ea062a6037c5ed0f02e1cb3c0783bf8c958676}}. Moraes et al. {{cite:c6e3b8b3a73874868c2b98dcac5e45cc11788e72}} contemplated the charged wormhole arrangements in {{formula:b46fc5f6-35e6-4fea-b057-7100fba95551}} gravity
and obtained fulfillment of energy conditions. In the same context {{cite:a22aa3e48e2531e089b960d933b61550739b7d23}} utilized the analytic technique to investigate the wormhole
solutions in {{formula:d182b660-a7ad-45af-834c-dafeed4a31a0}} gravity. Bhatti et al. {{cite:97187bbb1d0c4db5b7c83523e72fa3d56f9a3255}} considered exponential {{formula:aff4e8cf-df7a-45b1-970b-890027a133b9}} gravity model and the solutions of wormholes.
Elizalde and Khurshudyan {{cite:d9f66a4845450b342d31569c34d13eabb3c14d09}} explored the transferable wormhole solutions in {{formula:d709cfcb-a143-42cf-84b7-c379a9201372}} gravity by assuming the different sorts
of energy density. Sharif and Nawazish {{cite:8eee90940f51aa2075a69fb49e1385fcf011d04e}} explored wormhole solutions for dust and non dust appropriations by the utilization
of No-ether symmetry technique in {{formula:f30fe956-fd42-4c5a-8dbf-1c6e3144c19d}} gravity.
| i | 4b8f0c913ef62011062da9ad4667fdc0 |
Ever since the introduction of dimensional regularization (DR) {{cite:9ee6a864c28254b077c7cfab3a6a55e9072f8624}}, {{cite:93b4ef502d57cf0f86b38a3b99d78c601a36ce71}}, it has been recognized that special attention is required in the treatment of {{formula:8f8470a5-d13d-48bd-a9b8-31ca01011a15}} , an intrinsically 4-dimensional object.
At the root of the issue is that a fully anticommuting {{formula:32fb5057-f6ba-4a41-af16-ccfbcca589e6}} is algebraically incompatible with the Dirac algebra in D ({{formula:d4ae5bcd-8c13-42b1-97d2-5a2baebce7b4}} ) dimensions, which on the other hand is essential for the concept of chirality of spinors in 4 dimensions and (non-anomalous) chiral symmetries in quantum field theory (QFT).
Furthermore, there is one well-known subtlety related to {{formula:4de5d5e6-6df5-4ce0-9625-ef87e252798c}} and chiral symmetries in QFT, namely, that the flavor-singlet axial-vector currentBelow we denote (flavor-singlet) axial-vector current simply by “(singlet) axial current” for brevity., defined with {{formula:7ce17cd8-3230-43de-ad16-0c9f9511c5f0}} , exhibits a quantum anomaly, the axial or Adler-Bell-Jackiw (ABJ) anomaly {{cite:7f58b410785f19f7399b0bf86e1c1cf75b78c288}}, {{cite:69d7083f8cf2ba1eab1675652a0708f9ac678af0}}, in its divergence.
An anticommuting {{formula:a6d58097-7d57-4cb2-a26c-806b898c630d}} together with the invariance of loop integrals under arbitrary loop-momentum shifts, however, leads to the absence of the axial anomaly, which can be easily checked at one-loop order which subsequently holds to all orders, due to the Adler-Bardeen theorem {{cite:020bfd172a2e4f9eb4e2cddb870580044507316f}}.
| i | 67d7f155a9994dac0525202179055606 |
We consider the large family of multi-view representation learning methods {{cite:1671df6fb5fa69e61db97435fc02d195d2d71fba}}, {{cite:91198b7f71ce6165aa85c0c9596406e915641c89}}, {{cite:a6ce10ce1690ed2d457956047c33603b546ed460}}, {{cite:3fecbe3d7ab162b86b62ee7c572ecbc0fd57e6e2}}, {{cite:9386512a961e342652cc81ee702bc200c5471e0f}}, {{cite:09187cebfa117fd1ab74b2fe9df3ce8a66c2e18c}}, {{cite:51cb672e418e44759d552aec7160871e7b805951}}, {{cite:f4d543408019d24bf59c653aff50d93a9fcd9354}}.
In particular, contrastive methods are a subset of methods that leverage an instance classification problem in order to learn representations. Given a batch of data {{formula:f785780f-5d3c-4b5a-93f0-79c3972e28f6}} these methods learn the representation of an anchor point {{formula:ada28f22-fb11-4edb-b501-dd004fcea712}} sampled from {{formula:8fe06dfc-c9cd-4152-b634-ec3846f3d066}} by comparing it against positives {{formula:34cd98d0-a14f-4a51-b1c7-a05207542e00}} and negatives {{formula:f9ee2c88-b93f-4d23-a5ed-d481ad0de3d8}} .{{formula:cc602c81-7101-4023-8c4d-faaa83c067ea}} and {{formula:a1a33786-050f-485e-b5ff-59eeb57e0a80}} may be sampled from {{formula:a4bdf759-4024-4baf-a7db-a57342f554af}} or more generally from some other data structure which accumulates points or statistics from successive batches such as a queue {{cite:51cb672e418e44759d552aec7160871e7b805951}}, {{cite:a6ce10ce1690ed2d457956047c33603b546ed460}}. The goal of contrastive methods is to classify the positive point as an instance of the anchor point against the negatives.
| m | bbe7576847768281d96c100c0f9f9f76 |
where {{formula:ccc22bac-28ef-47b3-8fc6-2b05ab42a99a}} contains {{formula:9e8d1205-0680-4334-ac67-3404cfc99bcb}} basis vectors for {{formula:beef4ce7-b8f4-473f-9406-6bdb621332c6}} . For 2D images, over-complete wavelet basis ({{formula:ab2e777c-19d0-497b-8692-cf2b92f74be2}} ) are often used in both dimensions. For the EEG data that are channels by time points, wavelet or Fourier basis can be applied to the time dimension. It is of direct interest to seek a sparse low rank representation of the signal {{formula:451cd689-c2a8-4847-9339-6777b979cb23}} in the basis system by solving the regularized regression. In that sense, our proposed regularized matrix regression can be considered as the matrix analog of the classical basis pursuit problem {{cite:60835cdfc77365d39108bc1bf2950558f8f3c559}}.
| d | df67f5de3ff480f40f8ca3fb0a803173 |
In a series of recent studies {{cite:7d3375b9bf727f3e3494c7bc12831bc1aaea51d6}}, {{cite:d516f8c436fd9d4d7d09f9b237eb84294edb0bb3}}, {{cite:6b7c60cdc36015d5af78f12f6df87fcb3081fff2}}, the role of sparsity of the networks for the Bouchaud and BM models has been investigated. For those sparse networks the trap model is formulated as a continuous time Markov chain with a master operator that depends on the connectivity of the network and the landscape. An analytical approach that is suitable for studying such models is the cavity method {{cite:18a51433ea190d3a9de140ecf460c6bae8dcbdd5}}, {{cite:e9ddc030c7cb9c3ff87c2df95acdecdc06506e47}}, {{cite:0edc4331800518adf8112c5088348d88ee72e132}}, which provides access to both spectral and time–domain properties. The overall outcome is that sparsity yields non-trivial features in the dynamics that translate into richer and more realistic physics. In particular, the sparse BM model exhibits a crossover from an initial transient entropic–driven relaxation to a long time activated dynamics as a consequence of the finite connectivity of the network and the eventual lack of “escape directions” towards lower energies {{cite:6b7c60cdc36015d5af78f12f6df87fcb3081fff2}}. This picture is very intuitive if one thinks of the relaxation in a system of particles with short–range interactions from some suitably random initial condition:
rearrangements first take place as a downhill motion in the energy landscape driven by entropic features, while further relaxation implies crossing of energy barriers activated by thermal fluctuations. The crossover between these two types of dynamics may be relevant for describing relaxation in models that present qualitatively similar features, for instance the Random Orthogonal Model {{cite:e826cd17362a2b9258e7597d3a554e70daf6c131}}.
| i | 923097fdc4119a40fb933f8215991e9c |
A permutation {{formula:96c4c0c6-6889-471e-9373-452f32fdc852}} is a bijection from the set {{formula:de484a23-55ef-48a8-a5e5-165428e749b9}} to itself and we will write it in standard representation as {{formula:ac6d58de-842f-4843-9435-79e99680d396}} , or as the product of disjoint cycles. The parity of a permutation {{formula:af1e9acf-6384-4917-8eca-49ee1d48131f}} is defined as
the parity of the number of transpositions (cycles of length two) in any representation of {{formula:4f55472c-f3c1-4760-bc24-944c0284bdf8}} as a product of transpositions. One way of determining the parity of {{formula:19ed3a63-c45d-4d96-9b07-91d35f7400cf}} is by obtaining the sign of {{formula:a55179c1-0ee7-49e5-8151-1caa2e1fefcb}} , where {{formula:81938de5-2980-4c5c-9715-ae624ef772e8}} is the number of cycles in the cycle representation of {{formula:1bdee0eb-62f9-4a19-90f1-eab2dd038c19}} . That is, if the sign of {{formula:9429450b-42a9-4c78-82aa-c4c3ca33c790}} is -1, then {{formula:b5c85d0e-eea2-42b6-a88a-053f8d1780ca}} is called an odd permutation, and an even permutation otherwise.
For example, the permutation {{formula:e1f33a4c-2491-4b7f-9f3d-dff5ee583dd0}} , of length 8, is even since it has sign 1.
All basic definitions and properties not explained here can be found in for example {{cite:784ab8f0c9f8543eb4e0064d909738e8e0addc3b}} and {{cite:1dcde237eb309d1bcc4c67f3abc0384b4d4cfdf0}}.
| i | d4af343ad72a3d5e645692abda0fea6f |
In summary, we find that in all the three tasks, our proposed method out-performs the methods of simply tuning pre-trained language models, as is proposed in {{cite:a11ef9f25f615abd9438da4012731eef547a9be0}}. However, we would like to caution the readers in two aspects when reading the conclusion of this study. First, this study does not argue that our proposed methods are always superior to fine-tuning only methods. For example, all the experiments in our study are based on data sets of relatively large size. In the other spectrum, if one is only given a limited data set, then building complex networks upon pre-trained language models might lead to disastrous over-fitting. If this is the case, then it is possible that deep domain adaptation {{cite:270b67b3cce2f2b27159623705b3ff07e9875a80}} might be a better choice if one desires to stack neural networks on top of pre-trained language models. However, most domain adaptation applications belong to the field of computer vision, therefore, a call for domain adaptations research in the NLP fields.
| d | 64931067c9f1fdcd9667528cd65e9eff |
One can already provide some initial analysis using known results in the literature. For instance, as the protocol of {{cite:ee5ced8143d1ce5da82e0b6db1473bf1aa0c30bd}} requires the client to prepare and communicate single qubit states, one can wonder what additional requirements are needed to ensure the client can detect deviations when the server is not assumed to employ quantum dynamics a priori. In order for it to make good operational sense, we assume the theory used by the server must belong to the framework discussed here. Ref. {{cite:276066480624f90b7ae1f5387bfc7f13eba2ab34}} has shown that quantum theory is the unique generalised probabilistic theory theory where the local systems are qubits, global systems obey tomographic locality, and where there exists at least one continuous interaction between systems. Hence, if one assumes such structure, then the standard correctness of delegated computation discussed above applies. We have replaced the requirement that the server is bound by quantum theory by the assumption of tomographic locality, and the existence of a reversible, continuous interaction between subsystems—as long as local systems are qubits. In fact, one can go further. Recent work in Ref. {{cite:d11cfae3ba2eefad910b822af628cab28f31d331}} has shown that if local systems are {{formula:e957eb6d-73a5-420c-b81e-698d8a7d2418}} -dimensional Bloch balls (a qubit, for instance, lives in a 3 dimensional Bloch ball), tomographic locality and causality are satisfied, and the global transformations form a closed, connected matrix, then the only theory with non-trivial interactions is when {{formula:79ec5b98-448f-420d-97ad-90d039109a3e}} . That is, when local systems are qubits. All other cases only have local transformations, and hence can be simulated on a classical computer.
| d | 2376dba6527c57741daedbb1893a4f30 |
Traditional contrastive learning methods adopt instance discrimination as the pretext task. This kind of refinement of classification treats every sample as a category to conduct discrimination. Thus, it will introduce many false-negative pairs, leading to inefficient supervisory signals. And another problem is that the accurate instance discrimination requires pair-wise comparison on the entire dataset. However, with limited resources, most implementations compromise to adopt a subset of the comparison with a large batch size or memory bank {{cite:20c21737c9211946ba19280b349c06c2821a25e9}}, {{cite:168ee5f39048ae90b0ee17e5f28ec1e6e19ee657}}, which further decreases the efficiency of learning.
| i | cb413f0b656450cac3d9bf8cabfc4c3b |
In fact, during the architecture design process, many slightly different networks are trained for the same task. Apart from their final validation performances that are used to guide exploration, we should also have access to their architectures, weights, training curves etc., which contain abundant knowledge and can be leveraged to accelerate the architecture design process just like human experts {{cite:0ab4ac40e286db548f79fe329759ba553c706da7}}, {{cite:782abf998cf8eab8ca08834088814e729f88f6bc}}. Furthermore, there are typically many well-designed architectures, by human or automatic architecture designing methods, that have achieved good performances at the target task. Under restricted computational resources limits, instead of totally neglecting these existing networks and exploring the architecture space from scratch (which does not guarantee to result in better performance architectures), a more economical and efficient alternative could be exploring the architecture space based on these successful networks and reusing their weights.
| i | 42b9af9b1bdac208069d8ad553cd78e4 |
[leftmargin=*]
FPMC {{cite:532db91dc95f1d3f3da9a18618e6e0afdb49e4da}} is a sequential method based on Markov Chain. In order to adapt it to session-based recommendation, we do not consider the user latent representations when computing recommendation scores.
GRU4REC {{cite:0742b7b74fb84c6e714998d8fb6db44c7a7f3a2a}} utilizes a session-parallel mini-batch training process and adopts ranking-based loss functions to model user sequences.
NARM {{cite:a2e6899f3a80686264e1640236ce0602f2be9182}}: is a RNN-based state-of-the-art model which employs attention mechanism to capture user's main purpose and combines it with the sequential behavior to generate the recommendations.
STAMP {{cite:725fc1a745d44c47608a20b596576b6f5ee0ab42}}: adopts attention layers to replace all RNN encoders in the previous work and employs the self-attention mechanism{{cite:3449f04d1c97dde0f816f7fc1f8dae9bcab25918}} to enhance the session-based recommendation performance.
SR-GNN {{cite:03e2fa49cbc3135d33de21860bd02d185da33650}}:
applies a gated graph convolutional layer to obtain item embeddings and also employs a soft-attention mechanism to compute the session embeddings.
GCE-GNN {{cite:addbc444a903203c23dcd7e96f3d128df8a9214b}}: constructs two types of session-educed graphs to capture local and global information in different levels.
{{formula:002e48e2-7a96-4f1d-b040-e5dd9fbe94b6}} {{cite:c9c98e82c5a2eacb5f1b02e5e2850ad26ae77628}}: constructs two types of hypergraphs to learn inter- and intra-session information and uses self-supervised learning to enhance session-based recommendation.
| m | ef8d24a0df97d133fbcbc36b1c7b1a8e |
This model is a Friedmann–Robertson–Walker (FRW) cosmological model, which is obeying the cosmological principle and Weyl's postulate{{cite:4183c059e4fc92dd915d3874a24ed096565f2ed6}}, {{cite:2314a6439ceb8e616a599e1d42242327e8786066}}. In {{formula:35b941c7-44ae-4561-8cc5-f34d4f549ef2}} universe that space expands at a constant rate, rather than an accelerating rate. Theoretically, there are some controversies with this model, which focus on the zero active mass condition {{formula:4b9375bb-7a63-497e-b90e-402869be5326}}{{formula:83da26bc-dd7f-4ddf-a971-7c2629740ebf}} (for more details see {{cite:97cca43282f0472315daf6110f50b0071e266943}}, {{cite:4fabf7a9ca796ecdcc3b31cab1dee225bec8e3fe}}, {{cite:1c62a6e2045f8fccbcca2bebb4596bf16832791a}}). In terms of the fitting of observational data, the model performs relatively well. The originator of this model himself and his collaborators have done extensive work comparing it with the standard model using many different types of observations, and they have found that the {{formula:1fb6634d-3926-4bc6-8185-05fa61c33e50}} model is better than the Standard Model(e.g.{{cite:f925606e0c50b448221f99939994c7a440655fc2}}, {{cite:88fd1efa41fe18bc76fac5063a853b7ac6175405}}, {{cite:724d628fbcf3f26acf1698108ee2a110d7902eeb}}, {{cite:16854d469b66c66839e3012dbac34d5949c8afdb}}, {{cite:29125ac75248593c4477da3040643a2932ab20ed}}, {{cite:722e68e098dc36a2e3002e79ec7ed738a74f50c1}}). Additional work by others also illustrates this point{{cite:44739c1c36cd06dda4af0d33a96f240e98caef27}}, {{cite:220902f11c4449af72bc093b48491f7865190bbc}}. There is also a lot of work against this model(e.g. {{cite:a5d83e67788ab8ee95c1c5be0342b42ff60135c4}}, {{cite:72ceca008353d9c5a53bdfe9abccbe8a9141138b}}) Thus, despite the theoretical controversy, this model has gained some support for the data, and we can use this model jointly with the standard model to select the data.
| m | 8cca37f5e7087ec5e3c616a62eb6a993 |
When we have the vacuum case outside the star, the OMOTS surface coincides with the null event horizon. Radiation emitted close enough to this surface reaches infinity with an unboundedly large redshift. It is this divergent redshift that is the reason Hawking radiation is emitted just outside the null OMOTS surface and escapes to infinity. The ultimate source of the divergent redshift is the infinite rescaling that takes place between the affine parameter and the group parameter on a bifurcate Killing horizon {{cite:ff8568916a5cb903375bc6ca4976e14f123388be}} (cf. equation (2.16) in {{cite:df3676e3f59b9366e3a00905443aa221bf9094dd}}); hence it is a consequence of the static nature of the exterior vacuum solution, which allows this symmetry group.
When we take into account the infalling CBR radiation and matter flux, Fig.(REF ), the OMOTS surface is spacelike and lies inside the null event horizon {{cite:837c6a7b16b44011218c8d13ebc2de6c00d251d2}}. Outgoing rays reaching infinity from just inside its classical event horizon, or emitted inside that horizon but outside the OMOTS surface, no longer experience this unbounded redshift; and the same applies to the IMOTS surface. That is the reason that no Hawking radiation is emitted in this case.
This is related to the fact that when the flux falls into the black hole, it's mass increases, hence this is no longer a quasi-static spacetime and there is no external Killing vector field with an associated bifurcate Killing horizon. The conclusion is reinforced by the fact that there will be many other kinds of radiation and matter that will also fall into an astrophysical black hole, and increase its mass further. This is a self consistent approximation: as long as there is CBR and matter influx, the OMOTS surface will remain spacelike at all times because there is no incoming negative density Hawking radiation that could make it timelike {{cite:837c6a7b16b44011218c8d13ebc2de6c00d251d2}}.
As depicted in Fig.(REF ), in the particle tunnelling scenario, if a pair of a particle and antiparticle are created near the dynamical horizon, both of them will fall into the singularity and annihilate each other, so no Hawking radiation will be emitted. However a particle created near an isolated horizon or slowly evolving horizon can reach future infinity, and Hawking radiation will occur. On the other hand, since the geometric optics approximation does not generally held near the OMOTS, we cannot apply the particle interpretation for this surface.
At a late enough time in the very far future, the cosmological expansion decreases the CBR and matter density and black hole will devour all the available matter around itself. The black hole horizon then becomes first a slowly evolving horizon {{cite:8b387eac238da7998a0848fcb095409847b207eb}} and then an isolated horizon {{cite:9114f8b51eb3383a53a888b0f264e536dc0a97a5}} Fig.(REF ). Black body radiation could then initiate at that stage, and possibly lead to black hole explosions at a later time.
As long as the matter flux into the black hole is not negligible there is no Hawking radiation, but the black hole radiation scenario is currently applied to every dynamical black hole. However, all black holes in the real universe are surrounded by different types of matter and radiation leading to substantial positive density energy influx for much of their life, and particularly in the very early universe.
Application of this constraint to primordial black hole evaporation modelling may bring in a correction to their abundance in the cosmos. Specifically, primordial black holes are candidate progenitors of unidentified Gamma-Ray Bursts (GRBs) that are supposed to detect by the Fermi Gamma-ray Space Telescope observatory.
Their abundance might be lowered when the above considerations are taken into account.
| d | 9797ced498cc5580cd289ddd39139a63 |
Data. We adopt the datasets used in the vanilla FastGANs {{cite:9420a8de32caf92929b3d0f7cc24239333518c0d}}. There are 5 categories tested at the resolution of {{formula:a2b474b7-8842-4698-872a-6503e3080c0b}} each of which uses around 100 training images (see part one of Table REF ). There are 7 categories tested at the resolution of {{formula:f1d5f444-21cc-4d29-9cd4-d87d75d8152a}} each of which uses around 1000 training images (see part two of Table REF ).
{{table:776e0844-115f-4d9d-ba0d-4998f074bed0}}{{table:89487f58-b37a-41be-b525-dabe9d1f83f6}}{{table:c358964c-46c8-4f0e-a6d9-b89afd38294f}} | r | dfa5418f66c15ffb389b25ff31b33e13 |
MS-CMRSeg Dataset: Table REF shows the quantitative results of different algorithms for the MS-CMRSeg challenge, specifically for the 40 LGE-MRI segmentation test data. We first compare with the model trained with only limited labelled LGE-MRI data {{formula:59e68e22-66cd-41e8-ad57-267baa65d86e}} (referred as Supervised only) and take all other methods (Unsupervised, Inter-Ob) for benchmark study including ADVENT {{cite:2bddd506fe94cbdcce5567e533e603e13e09186f}} and AdaptSeg {{cite:f231d6f39ad8cb9dcd60bc0a5434f7752f89caea}} for comparison. Besides two-stage approaches, we also compare with the end-to-end model and inter-observer study. Meanwhile, we also compare with segmentation methods like Chen et al. {{cite:d1230f00f76262594898cc71d9fa9f4ec72fd802}} (Unsupervised) and Wang et al. {{cite:d5ae9aace7646038bfa898431706104f950ccd82}} (Supervised), that achieved SOTA performance in MS-CMRSeg {{cite:13d3fcf90f3a1ea0b4afdd2f5c21f28c23fc5528}} cardiac challenge segmentation. Supervised methods in MS-CMRSeg challenge had access to only 5 LGE-MRI studies to train their models where 2 or 3 studies are used for training and cross-validation.
| m | 1b11db4e79fdb9b084046059e62ddf28 |
Gravitational Waves (GW), first introduced by Einstein in 1916 {{cite:bdf7873a9e9b447afec51a5e0e5d0788a25f5f36}}, {{cite:7eff67daf0cead082906c55abd3c294e90413548}} as a linear regime of the field equations of General Relativity, have been directly detected for the first time in 2015 by the LIGO/Virgo collaboration {{cite:044b16193147fd8dc5c08d4335cb940613b94558}}. These ground-based detectors use laser interferometry techniques, but there also exists other detection strategies. Einstein's Equivalence Principle implies that all types of energies produce and experience gravity in the same way. The energy of electromagnetic (EM) radiation must therefore source gravity just like compact objects do and, the other way around, gravity (e.g. gravitational waves) can manifest itself in the physical characteristics of electromagnetic radiation. This is the basic principle behind the EM detection (or emission) of GWs.
| i | 920ba3ccc40acb67a231a939320c932c |
The normal viscous force {{formula:1b3d4176-863e-472e-9857-34069c4968b2}} is due to the lubrication effect of liquid bridges between particles.
Its classical expression for two smooth spherical particles is {{cite:67523f4b32cd7d64f868b6e03decd7bbe4a8598f}}, {{cite:3d48e8a6fd2e6a98cdb251eb31dcec9e04ba259c}}:
{{formula:8fdf5510-2626-4041-bd53-a09010488611}}
| m | 330f7e0fea3433daf05d4d2843751664 |
We analyse all protein trajectories in the framework of a non-Markovian, generalised Langevin equation (GLE). The GLE is a low-dimensional representation of some higher dimension system. In the present case, we collapse the all-atom dynamics of the composite water-protein systems onto a one-dimensional reaction coordinate. This process of dimension reduction is known as projection {{cite:907d21442d41444a98e48f654d1582b43cee8b6f}}, {{cite:ba53c1b45a39156583bf5b34902cfd3b02cee8e0}}. We project the all-atom trajectories, as provided by the Shaw group, onto a well-known fraction of native contacts reaction coordinate {{cite:6bc735fdbe90d5b07cda0c78f3a4ac98ecb5d96e}} with soft cut-off {{formula:66e283b2-1ccc-494b-ad29-3b6b955bd575}} {{cite:d2c37e8de5aecb0546be398be692af15bb55f24b}}, and hence extract friction memory kernels from {{formula:b6b0146e-1dc1-458c-92c7-77e42e4da142}} . Free-energy profiles and friction are a property of dimension reduction, and are therefore unique to a given reaction coordinate. The extracted friction kernels encode dissipation across a spread of time scales, representing the finite relaxation rates of solvent processes and internal reconfigurations, as far as they are represented in {{formula:d330bc4a-f990-41bd-a4b6-e42099e54ce0}} . Consequently, the friction kernel contains the information for the full friction acting on the reaction coordinate. From the free energy landscape and friction kernels, we derive predictions for the folding times on different levels of reaction rate theory and compare to the folding times measured in the simulation. In doing so, we show that the best prediction for the measured folding times is given by a multi-time-scale, non-Markovian theory, which has hitherto only been applied to the characterisation of model systems {{cite:9b52e3b1d352a9486b183615ea473eb0dfa3de4c}}, {{cite:b7490371300044bb7fa03ff321c8ce471861028b}}. This validation is only possible owing to the long simulation times and diverse range of proteins that comprise the extensive protein data set. We show that the memory decay-times, i.e., the duration of memory effects, are significantly long for all proteins in this set, in some instances as long as the folding times. This indicates that even for {{formula:d12c0e84-179d-480a-9f3e-3aad4bc71d0b}} , which is typically considered by other measures to be a good reaction coordinate {{cite:d2c37e8de5aecb0546be398be692af15bb55f24b}}, must be considered as a poor reaction coordinate, when judged according to its non-Markovianity. Finally, we also show that, for this particular set of proteins, friction is more important than free-energy barriers in determining folding times, and that this dominance of friction increases for larger proteins. Taken together, our findings suggest that, when represented by a low-dimensional reaction coordinate, the conformational dynamics of proteins are dominated by friction and should be treated as non-Markovian in nature.
| i | 90b1a313bd6d3defe2d9aa5dbfc73a5e |
Results are presented in Table REF .
The main proposed systems are “Harmonic-CNN” (the instant method) and “SpecTNT (24s, CLT)” with CTL loss (the multi-point method). In the ablation study, we compare these to SpecTNT without CTL; a regular Transformer {{cite:d08f2e7327c2a3747126e194d39dcc38d60530ce}} with CTL; and SpecTNT with CTL and a longer chunk size (36s).
The regular Transformer contains only the temporal encoders {{cite:fd22d4abb46590ce5456b00457ba7350a1e18305}}, so it is non-hierarchical and has no spectral encoder.
It is clear that this system performs worse than SpecTNT, most likely because the training data is insufficient.
Comparing SpecTNT with and without CTL loss, we can see that including the loss improves performance slightly on all metrics, demonstrating its usefulness.
We also observe that SpecTNT with longer inputs can improve the function prediction, but the boundary detection performance drops, possibly because the model has over-fit the training data. To prioritize boundary accuracy, we use the shorter window in the next evaluation.
{{table:7a5f723c-f4b2-4f01-98a0-41375291b41d}} | d | 5bcd490318fe7de94f3da9926cc4debf |
One unexplored direction of creating similarity measures is creating a SNN
similarity measure (Type 3) through training {{formula:61364d2c-5bef-4b8b-9877-0b587f2078e7}} as a classifier on
the dataset later being used for measuring similarity. Then using that trained
{{formula:9bf5038e-4095-4512-a8fc-dc387a362567}} to construct a SNN similarity measure. This is in contrast to the
usual way of training SNNs (as seen in e.g
{{cite:53596926d4aa8a3c25b3398216bbfb6b4bbe7859}}, {{cite:695c21f0fbed9b1fbfbe84c924aa6d11da7df9f4}}) where the loss function is a
function of pairs of data points, not single data points. The motivation for
exploring this type of design is that it shows the similarity measuring performance
of using networks pre-trained on classifying data points directly as part of a
SNN similarity measure. This will be detailed in Subsection REF .
| m | 1235c8febd769d3f342730eb531486ec |
Raw data for the results reported in the text are available
from the authors upon request.
Acknowledgments
We thank J.N. Zhang, C.Y. Hsieh, Z.Q. Yin, Z.B. Yang, G.H. Huang, and X. Chen for helpful discussions and comments. We thank the electronics team of Tencent Quantum Lab for preparing the room-temperature electronics.
Author contributions
Z.L.Y., S.M.A., and C.Z.L. developed the theory.
S.M.A., Z.L.Y., and Z.X.Z performed the experiment.
All authors contributed to the data analysis and writing of the manuscript.
Additional information
The authors declare no competing financial interests.
Supplementary information is available for this paper.
Correspondence and requests for materials should be addressed to S.M.A.(shuomingan@tencent.com).
Supplementary Note 1. Exact Input-output Theory
Here we derive the exact input-output formula used to simulate the output signal of the system shown in Fig. REF . Our analysis follows that of Ref. {{cite:c3337a8550f9a6aad492f5a2edc86eda60cdcf12}}, although we additionally account for the distance {{formula:40522bae-25a9-46a8-a965-2118417ef2b3}} from the input capacitor {{formula:964e5d8c-56d9-4a2c-88a8-88d6b0168ba7}} to the filter {{formula:f3e50bdb-5c37-4a84-98d5-7b2d041b7de0}} , which is necessary for the theory to match experimental observations.
For wavenumber {{formula:8e4d11c5-4c7d-4753-8eb8-6a41979a08d0}} , the phase accumulated after passing through this distance is {{formula:2fc220d6-7688-4858-b663-0f3899295a0a}} .
We find that this phase has a profound effect on the final output signal.
{{figure:748f770f-1750-46dc-9cfa-d572450f2228}}Our goal is to determine how the output field {{formula:05c94656-24df-4d94-9d2d-aeec4daee92f}} responds to the input field {{formula:1073e9bb-09ea-4c21-a348-986f33d3f20c}} and its interaction with the system, including filter mode {{formula:e874ff3e-00a4-4644-87de-956ebe3379a3}} , the resonator mode {{formula:0dbdefaf-231b-499a-aaed-4c67b1f58c1e}} and the qubit state. To simplify the calculation, instead of directly including the qubit state, we will account for its effect by modifying other system mode frequencies.
To build the mode network, we start from the most left input port and consider the transition and reflection of {{formula:f25fb49c-8902-4680-bc64-b5bdae25e6be}} as:
{{formula:4ceacc8f-75d0-41ad-8aa1-efdc1644accc}}
where {{formula:c9887ec9-8094-438e-b549-b56c4f0990a2}} is the reflection coefficient, {{formula:a47973d6-7ae9-4f31-8a82-f4b8d17d92bf}} is the impedance of the line and the loaded impedance of {{formula:43a11db3-853e-4aad-8e08-da5a5597c545}} is {{formula:4beb7404-e5c7-4e22-a8ec-aa41d132c826}} .
As a second step, we consider the effect of the microwave length from {{formula:ffee4d2e-9194-49c0-b396-6160264b6103}} to the system as:
{{formula:d083bb82-5a38-4897-af28-22f2cd01241d}}
After this, the microwave reaches the T connection between the filter and the feedline. Its scattering matrix is:
{{formula:ec7674df-f004-4191-aee7-b59a8b131668}}
Here we assume the impedance of the line connecting the capacitor is also {{formula:56393487-ff15-4109-a7af-8fedb94ec90b}} . Part of the wave in the feedline will drive the filter mode {{formula:d54c36dc-0790-4d72-bdf8-f5d91652296f}} , which satisfies the input-output formula:
{{formula:872723d8-abeb-4011-aa6a-cf0a8cb7d6be}}
where {{formula:b1c8b242-3de2-4059-86fe-d7164d3f6da3}} is the leakage rate of the mode {{formula:c380b5ea-ada7-490d-a9c2-abc883724ec9}} .
Assuming no output reflection ({{formula:cf8bc0c4-1558-4ff2-aa3e-4b94b1add32e}} ), the relation between the input field {{formula:f2424046-d9c9-4a59-bfc0-1f3962609683}} , the output field {{formula:95ec2475-f112-4668-8573-87c1c817fdff}} and the filter mode {{formula:be5a9d47-6f63-413f-844c-143baf30c168}} is:
{{formula:2f61dd48-087b-4e52-bb48-9e7b2bbaaabc}}
To determine the relation between the input {{formula:9efcb2a0-da34-4762-b88c-7cefdfd619b7}} and the output {{formula:2407212c-0c57-4e7b-b5ef-557e309ce164}} it suffices to deduce how {{formula:4878cb33-84e9-400e-99b5-616c619659e5}} depends on {{formula:6f6ffb23-5e3e-460a-bb47-1bfb9f4dcc5d}} . Under the rotating wave approximation (RWA), the equations of motion in the drive frequency ({{formula:03606b78-0e96-42cb-9e28-fc7764cd3e18}} ) rotating frame are:
{{formula:2a26f1e6-ddfa-467b-9417-74c7663d05b4}}
where {{formula:3ed96b78-a653-4c11-b408-e21ec028e588}} is the detuning of mode {{formula:b5bf1c08-597e-41ce-9963-3efc3ef06acd}} frequency {{formula:105f75a8-f741-4a48-bc6c-0477f466d26e}} relative to the drive frequency {{formula:a0f83ff8-ff2a-470e-bc7e-75922c758bfb}} , and {{formula:72e381d3-79c7-44f4-b3aa-b4a04eda40db}} is the coupling strength between modes {{formula:3aace1f9-b23a-484e-94c2-6d2fd636a098}} and {{formula:8d57b9e6-603f-4b1b-8fdb-6a3918cec147}} .
It follows from Eq. REF , REF , REF and REF that :
{{formula:d6a5755e-a5d8-4c22-a905-a65e494301c3}}
Combining Eq. REF and REF gives:
{{formula:4cb89cfb-5b50-4c6b-a2ba-d991b8e141ec}}
where the effective detuning, leakage rate, and input field of mode {{formula:72c72dd3-b674-4010-b113-9e58de07565f}} are {{formula:404282e9-2761-428a-b49e-f1575a3d382b}} , and {{formula:01d6df8d-9a99-42e6-a722-7da2b5cc82e5}} , and {{formula:f9f2b457-9665-4144-b957-b0871c19e155}} respectively.
Note that Eq. REF is equivalent, up to redefining various parameters, to Eq. 3 and Eq. 4 of the main text.
According to the designed values of {{formula:6a0e1d07-64f8-4f67-bfb5-036c5af52b98}} and {{formula:fa8774bd-3c8c-4f9a-9645-f6568509fc0a}} , we estimate {{formula:ea1932f2-ffb4-4789-9ce4-281e07c236be}} and {{formula:a32bf0dc-0823-4605-8395-ddb730cbe458}} .
To simulate the dynamics of modes {{formula:5602eee7-5252-4ae1-8524-02bc304b54df}} and {{formula:0fe0c5fb-d4db-42bb-b1b8-1066a47d535c}} , we use a Lindblad master equation with the Hamiltonian:
{{formula:bbd335f3-76c9-4bec-aae9-24c017815b12}}
and the Lindblad operator {{formula:668ff821-3f83-49f1-8620-3210fb175707}} .
The effective driving field {{formula:fb8358a3-1b64-4e8f-bfdd-40d7bc711c80}} follows from Eq. REF and REF as
{{formula:07b572f6-40ff-4d9e-aa8e-9ee03ef5c528}}
which is averaged in the simulation, given the classical (coherent) input field {{formula:ca1d5144-a310-4fbe-a6a4-66bf8c6ddefb}} .
Substituting this into Eq. REF gives our final input-output formula:
{{formula:182c211f-f5be-4598-9cd6-b5b37e6f4827}}
To account for the weak nonlinearity of the resonator and the uncertainty in the estimated design parameters, in the simulation we multiply the {{formula:b2cfea20-1480-44e0-b2d4-4633ebce21ac}} term on the right-hand side of Eq. REF by a complex coefficient, chosen to fit the simulation to experimental data.
Physical interpretation.
Because the filter mode {{formula:53a5c1f9-e1a5-48a0-95b5-b7f9fe86191b}} driven by {{formula:4a733478-008c-4bce-83b2-3e6ae87dcd63}} can be solved using the Lindblad master equation, we can determine the output mode {{formula:d337d85d-fb54-4e0e-9fb3-ceb8e5e902af}} once we know the driving waveform {{formula:130186f6-66a1-4a89-95e5-b60c9c3fc525}} .
The physical meaning of Eq. REF can be interpreted as follows.
The factor 2 before {{formula:e0b1c259-4062-4693-9e2a-9089cc1f4366}} means only half of the input mode {{formula:79bdcee9-2dbd-4c89-93e9-1a903c4b9c84}} is used to drive mode {{formula:094d5981-4dab-4b28-af04-ae713300bc75}} .
The signal that finally reaches the output port is twice the driving.
The {{formula:7607d04d-49ce-45fa-aa7e-977f0c7331ba}} term means half of the leakage of {{formula:a6af5fd3-e874-464a-9aa2-6b03659f987b}} directly goes to the output port, and the other half will go to the input side and be reflected by {{formula:bad9a40a-27d5-4419-8d2a-8ccfb9f38c2a}} .
Finally, these two branches interfere with each other and contribute a complex factor between the input and the system leakage.
Supplementary Note 2. Single-Mode Counterdiabatic Driving: Short Derivation
Here we give a simple, short derivation of the counterdiabatic (CD) driving (Eq. 1 in the main text) for a single driven bosonic mode coupled to a cold bath, based on a mean-field approximation. We leave a rigorous derivation to Supplementary Note (SN) 3.
As we will see in SN 3, the bosonic mode under consideration can be well approximated by a coherent state, and thus we can use a mean-field approximation for the Heisenberg picture bosonic field {{formula:a7b88070-ed1c-476e-b595-63ad5de75c08}} , i.e. {{formula:913f050b-5cc6-491c-9826-2a47ae71695f}} .
Following SN 1, the dynamics are given by the Langevin equation in the drive frame of frequency {{formula:9a2c385f-ff0b-4f3c-b4d0-7f077afd5103}} :
{{formula:ffb9403a-7536-47a4-a45c-ce377deb59c0}}
where {{formula:553e366c-d7b6-4f28-9712-bb9bc6a9b8c2}} is the cavity-drive detuning, {{formula:f97bf716-707d-412b-953e-9db3463d53d0}} is the damping rate due to coupling to the readout line, and {{formula:41b69c84-b178-4ebe-b048-3f1c07a8eb64}} is the effective drive field.
The instantaneous equilibrium state is obtained by setting {{formula:3c9e9c3b-2419-4bac-85f4-7fd255e0ec0a}} .
We denote this instantaneous equilibrium state as:
{{formula:bfdc48d1-7b74-4e33-9773-7f5892c9e0ee}}
If the drive field is varied slowly enough, the adiabatic theorem guarantees that {{formula:c5cd31e2-beae-491e-8ab5-4e232a701d4d}} be the solution of Eq. REF .
Define the instantaneous diabatic excitation {{formula:53976117-c866-4bd6-a935-05f535c31660}} .
It follows from Eq. REF , and Eq. REF and its time derivative, that the dynamics of {{formula:cc03f41d-9eba-435e-ba68-ad7f8a4214a3}} satisfies:
{{formula:53528037-614b-480f-9e22-4c7834a7590f}}
where {{formula:39a0f395-c137-4854-b9da-da46bd3375cd}} is the (new) CD driving. From the boundary conditions {{formula:413cc259-e5f4-49bc-b299-279fdf603f7e}} and {{formula:fee0dc58-6659-4cf9-ae3c-673b8717817d}} , we obtain the desired CD driving as:
{{formula:e5518a0d-9d38-4607-bbc9-8c0232f2f868}}
Then, for an arbitrarily drive {{formula:911fcdd4-f28f-4d14-a650-996703ed0a9c}}
, the instantaneous equilibrium state {{formula:95a47b3c-5075-4abf-b054-9a3576083b78}} is always the exact dynamic solution of Eq. REF .
Supplementary Note 3. Single-Mode Counterdiabatic Driving: Open Quantum Dynamics Approach
In this section, we give rigorous derivations of CD driving for a single driven-dissipative bosonic mode,
based on two approaches: (i) Lindblad dynamics {{cite:a5b05f6b5cc1277d49c028597da5322d28431e58}} and (ii) an adiabatic shortcut of the decoherence free subspace (DFS) {{cite:5b08895ea83d351b9eea85a639e2c92b28979ed7}}. These results justify the mean-field approximation assumed in SN 2, and give additional insight into the adiabatic dynamics of our system.
In what follows, we set {{formula:c2eb4017-5c9d-4ba3-882f-f0fd882640d0}} . After rotating wave approximation, the Hamiltonian in the driving frame is:
{{formula:24a1ad90-6be6-4126-9b31-84d02a593ade}}
where the cavity-drive detuning {{formula:f6b31596-681f-4c86-a74b-ae4649225a14}} depends on the qubit state in the dispersive regime, and {{formula:4158f4f1-707e-4b92-80e4-043394a1e8f7}} is the effective drive amplitude.
The transmission line is viewed as a channel for both driving and dissipation, so the dynamics for the cavity density matrix {{formula:a2ebd112-5abf-40a3-a041-a573c289f1c1}} is described by the master equation:
{{formula:b4f66b33-21cc-4d88-a9f5-b6a8864489c3}}
where the dissipator is {{formula:d54bdce8-83d5-4d4a-9c31-62c373c6b37e}} .
Here, only photon decay is considered, since at the effective temperature {{formula:8ec3470e-e88f-469b-a81a-39edb0224e2c}} , the average photon population is {{formula:074e8516-a3fc-4175-b3eb-9a75a0721382}} at readout frequency {{formula:8a310bad-0127-4a0c-a569-7f56be8f6165}} .
Lindblad dynamics approach.
In the adiabatic approximation for open systems {{cite:d27c2dc6f49961817d4a74a0814ef41ac1d3cfa1}}, in the limit where the Liouvillian {{formula:15e06abb-3639-4338-a282-587d41db8012}} is slowly varying, the density matrix {{formula:6a389600-bd71-4c12-8c8a-08f5e862d523}} evolves independently in each generalized eigenspace of {{formula:0add4e0c-72a0-419d-9fde-14103743a70f}} .
In other words, {{formula:359ec8d7-ae7f-4aa3-934e-a2b58c1d0c77}} can be decomposed into a direct sum of components, one for each independently evolving Jordan block of {{formula:2d9d4f19-784d-46fb-857d-2de919e75e56}} .
The adiabaticity can be made exact by adding a CD Hamiltonian {{formula:396c605c-31cc-4064-a83e-65a14a096da7}} which suppress the inertial part of {{formula:1644037d-b1b4-4727-8d8b-8678d58754e2}} that causes transitions between different Jordan blocks {{cite:a5b05f6b5cc1277d49c028597da5322d28431e58}}.
To determine {{formula:13bd11f0-d181-4ef9-b406-44118cc92ed6}} , we first we find a superoperator
{{formula:27ed364b-b270-4b9c-9161-175682e8b98e}} that transforms {{formula:f9e8abbd-2b03-4492-ac88-d403edf227da}} into Jordan canonical form (JCF). That is, with respect to a certain (not necessarily Hermitian) basis for the density matrix {{formula:3de51566-5da6-474d-9ac7-3b0a5dfebd2e}} , we have
{{formula:a3c1e59c-d764-458e-9240-8673f019e235}}
where {{formula:73b4fb99-8de8-44ee-8ef4-7532535bcf0a}} are the Jordan blocks of size {{formula:a52e12a8-049b-421b-bbb0-2c5ae3f19c28}} .
Second, we transfer to the adiabatic frame defined by {{formula:c5958900-96b7-48f5-8521-90f9736921ce}} and show that the non-JCF part of the new Lindblad superoperator {{formula:f01f4244-f68c-4a5e-9f5e-593ec48989bc}} , i.e. {{formula:c131c6b6-b297-4261-9f51-9b191c30cc24}} , can be exactly cancelled by adding a specific CD driving Hamiltonian {{formula:3c992259-4998-4e22-83e3-a146bcb8f6ae}} to the system.
In the first step, we choose {{formula:4db79280-444e-4be4-9c81-cd7bdaf2781d}} to take the form of a displacement superoperator {{formula:2e99c791-fbac-4f92-8a88-f0ffc2781633}} where {{formula:37b31eb8-d35e-47d6-bd65-58f1d3048ef6}} is the displacement operator {{cite:2d5c0dd4224be9f5190ec553924c0465a5b9079f}}.
Using the fact that {{formula:4dedd976-af82-4a2f-bc49-fd58b6a879fa}} , it is straightforward to show that
{{formula:f1ad5498-76d1-4714-a07a-53bafa5c0b68}}
Choosing {{formula:3bd89485-40bf-496e-8f7e-44408687ee8b}} – i.e. precisely the instantaneous equilibrium state of SN 2, Eq. REF – eliminates the time-dependent driving term in {{formula:8e798c35-cf5e-47bc-9e75-cc5f286e57f9}} . Thus, in the adiabatic frame defined by {{formula:29d7ff3f-4018-4c31-9ad8-5a063ea11e8a}} ,
the Liouvillian {{formula:3fca98e2-639f-4cc6-9a09-3786adaa3211}} is time-independent, so a fixed basis {{formula:1074d0f2-a432-4be1-a296-4b1af73d6144}} can be chosen in which {{formula:13b205ff-1efc-4c29-ad19-503421ce1de6}} is in JCF.
In the second step, in the adiabatic frame {{formula:8101be45-a1d3-4e30-b82c-c47df38b2903}} we have
{{formula:ed03fcbb-9535-4c10-b627-ca32c2045955}}
i.e. the dynamics are exactly in JCF except for an inertial Hamiltonian {{formula:f6ea1638-a947-4564-8431-733a08236f8e}} which mixes the Jordan blocks of {{formula:2d43e854-2517-443f-a17e-db99c8699a25}} . Adding an additional CD term {{formula:b398e2ce-2b5e-41d2-9b56-86e50fe3c7c1}} to {{formula:18b88746-4759-4c8e-a479-90e08de59c96}} exactly cancels {{formula:282f7444-ca17-4d08-973a-049521ba7f89}} if {{formula:0a1a6c2d-74e4-4aff-a546-7fa886949540}} . That is, if
{{formula:9083138d-8a4c-4654-bba0-43cf261f0052}}
consistent with SN 2, Eq. REF .
Decoherence free subspace approach.
The time-dependent decoherence free subspace (DFS) is a subspace of the full system Hilbert space, in which the open system dynamics is unitary and quasi-steady, i.e. its instaneous motion is generated by an effective Hamiltonian {{formula:fdb2b503-a447-4b7a-a690-96db5679cdf9}} defined within the DFS. By identifying the time-dependent DFS of our system, we derive the CD driving (Eq. REF ) and compare it to the Lindblad dynamics approach.
Following the definition in {{cite:5b08895ea83d351b9eea85a639e2c92b28979ed7}}, for system dynamics described by a Lindblad master equation {{formula:0b2c2b16-543e-4a0b-9bfe-a3119b608c96}} where the Lindblad operators {{formula:b77fdba3-08d7-479b-8c60-33edba9933b3}} have possible time dependence, the time-dependent DFS is the space spanned by a set of orthonormal states {{formula:6c44f52e-eb2d-45b3-9c00-e7e3e916a2eb}} , satisfying: (i) the basis states {{formula:076627d5-ed8f-400c-984b-483e3970d781}} are degenerate eigenstates of any Lindblad operator, i.e. {{formula:3c434b34-caf8-4a13-8fab-6c35638149fc}} ; (ii) the DFS is closed under the effective Hamiltonian {{formula:955f4516-9aa6-4e2e-bd8a-4bb27da36144}} , i.e. {{formula:416684a3-300c-4594-a44b-eea1e7fa170b}} acting on a state in the DFS results in a state in the DFS. For a single lossy mode described by Eq. REF , the DFS exists and is spanned by the single state
{{formula:8abf3543-04d3-4118-9312-dd848f44e4c0}} for {{formula:c4cf40e3-e19d-41c5-bd11-2d8c330090b6}} , and {{formula:2a442d7e-874f-4a73-911a-0f72c9b032c2}} takes the form of a displaced oscillator {{formula:5e6c64ef-2d0f-4228-af89-835e17c5f5b7}} .
Suppose evolution of the DFS is given by the unitary transformation {{formula:a50bb386-207c-4f54-a68f-1f00f9a0c5e4}} , i.e. {{formula:5c04ba1f-be65-4f6e-ad21-fb609f3af09c}} . By direct analogy with closed system CD driving, we can transform to the adiabatic frame defined by {{formula:6a1bc17c-1e24-4039-83ef-acded94537b0}} and cancel diabatic excitations out of the DFS by adding a CD Hamiltonian {{formula:873d1514-488e-4d31-9530-17696e76d62e}} . In our example, the natural choice for {{formula:731e8304-d25f-4cd5-9f58-af3dd94552e7}} is the displacement {{formula:eddf7633-f83c-4161-962f-cc85365cb7b6}} , from which we can derive the CD Hamiltonian {{formula:323fae76-c8fb-4226-a8ab-a978b5836f9d}} , equivalent to Eq. REF obtained from the Lindblad dynamics approach.
Comments.
Steady states and adiabatic timescale. In the adiabatic frame, the Liouvillian {{formula:fc8fbbc0-b28d-49a2-8112-5a35da96697b}} in Eq. REF takes the form {{formula:b7807a06-9cd9-455c-97b3-697fd21d5d61}} . Its eigenvalues can be found by observing
{{formula:f446cf27-ed7f-42ff-ac22-4459047a2ff4}}
where {{formula:d56d42b4-9c43-4926-b48e-3a22c6ca84b1}} are the Fock states, indicating {{formula:d0fc876b-17d9-4852-b5cd-4222dfb26ffc}} is upper triangular in the subspace spanned by {{formula:45f90d5b-5dfa-4164-8430-0b12038734bd}} (or their Hermitian conjugates) for natural numbers {{formula:0c8c461e-9e83-4e00-82fe-89e8c63fba59}} .
Hence, {{formula:fed96874-59b6-4cda-855e-4e49ad3d2bd5}} has non-degenerate eigenvalues {{formula:f4be0e12-3e8f-4090-9496-fd5257d336d2}} (or their complex conjugates) in each subspace and can be exactly diagonalized.
We note that the only steady state of {{formula:21319542-5434-4cff-94f6-7f249bf91f08}} ,
i.e. the eigenstate of {{formula:fe8871a1-6974-47c8-ac7f-8cd7d569f181}} with zero eigenvalue, is given by {{formula:5e77766b-13db-4144-abf2-a37e7514efdd}} , which is the vacuum state {{formula:69e29cdb-28c9-4a98-8126-43d9d94d456b}} in the adiabatic frame or the coherent state {{formula:f8bc00a8-1b01-4e07-9a84-81617069ca5d}} in the lab frame.
We also comment on the timescale required for the adiabatic approximation to hold in open systems, following the results of {{cite:d27c2dc6f49961817d4a74a0814ef41ac1d3cfa1}}. Analogously to closed quantum systems, a sufficient condition for adiabatic evolution of an open system is
{{formula:83dfa79c-4ff5-4fce-b6c7-27110f3c2489}}
where {{formula:bfc7e22c-8213-4cbc-bf8e-65585bf97d33}} is the total evolution time, {{formula:f6580bae-fb1e-4648-9655-160519c3ae67}} defines the inner product, {{formula:d0ec4588-cbc1-47e7-8f05-6eec2688e6c0}} are (lab-frame) eigenstates of {{formula:c250654b-dc51-4455-8506-eb82b4d93e38}} with eigenvalues {{formula:ce778b1e-6f9a-4dc4-8058-e4ec47515208}} , and {{formula:7136811e-b565-464a-9553-6c9333b2959b}} are eigenstates of {{formula:9239deac-396b-472a-85a9-57db117df62f}} . Here the adjoint {{formula:5fe757ee-8e30-443f-b087-48c45d774cb2}} is defined as the superoperator that satisfy {{formula:1b0d578f-8e77-4233-ae7d-fde7c09f33e4}} . The LHS is hard to evaluate in practice, and a crude estimate is obtained by setting {{formula:5f7df46f-e98c-42be-802e-18ef7939ddb1}} for normalized time {{formula:a45e44e9-73b7-4d44-a7d1-86de5e91c1e5}} .
The adiabatic condition for the total time {{formula:7ddc3e25-9b73-4b77-86e2-c1558c10fc19}} is then derived as
{{formula:65b9e95e-74c8-4905-bb89-3b550290bd0b}}
For {{formula:aabe2b06-8031-489b-8f3e-49d7b993f5d9}} comparable to {{formula:7c96fd25-790f-4f2b-883b-5d05e5d47557}} , which is a usual experimental scenario, the adiabatic condition is {{formula:5f56786a-a249-4ceb-82af-a2dec4e00639}} , making STA useful for fast protocols operating within unit lifetimes. This adiabatic condition is verified in Fig. REF , where {{formula:20560757-d4d9-4e49-a5e7-8bdaa4c90af8}} -shaped pulses are applied for different durations {{formula:c643d001-dc97-45cd-8e96-cc35b8509d63}} and {{formula:5a125a2e-dd5b-4edc-963d-762108d102fb}} -shaped output signals are observed only for {{formula:79f63604-c807-485b-beee-ec8543676b16}} .
DFS from Lindblad dynamics.
The derivation of CD driving from both Lindblad dynamics and the DFS approach relies on switching to the adiabatic frame defined by {{formula:18977f53-e487-4b37-9ae7-d93b81b2a3d7}} , i.e. {{formula:20854618-04df-496e-8866-451b02a9b3fb}} . We note that the DFS of our system (i.e. the coherent state {{formula:f8304d93-cf57-42e7-bf07-2a20901c1573}} ) is the vacuum state {{formula:0cbba54b-d125-44d1-8d5f-9458a4c6115a}} in the adiabatic frame - the only steady eigenstate (i.e. having an eigenvalue with a non-negative real part) of {{formula:e5e9dc7a-8049-493c-a9c0-ff2b32a41ab4}} . As a result, the steady eigenspace of the Liouvillian {{formula:d27d9550-c880-4fae-955d-4c17d61f853d}} is equivalent to the DFS, whereas this is not true in general since purity of these steady states requires a zero-temperature approximation or negligible thermal photon number {{formula:bb899e88-6763-4440-b15e-986877f7d023}} in the frequency band of interest. For bosonic modes in the high temperature regime ({{formula:7f0d5637-73c9-41f9-8824-33a0b433e438}} or {{formula:ef5f6ae9-7971-4210-a006-c2ddc4c8c73a}} ), CD driving is still possible by the Lindblad dynamics approach even though the pure-state DFS does not exist. In this case, although CD driving does not prevent heating into the steady thermal state in the adiabatic frame, it ensures fast transport of this steady state, which is still of practical interest.
Mean Field Approximation. Here we show that the single driven-dissipative mode remains in a coherent state, which justifies the mean-field approximation used in SN 2. For open quantum systems, the coherent state is known to be the consequence of the zero-temperature approximation of the environment {{cite:3f6a414a1a1d9841c351e842b94ce32d2bb89484}}. Specifically, in the frame defined by a general displacement {{formula:b6c8943d-355c-4175-a436-cdc85364a083}} , the dynamics in Eq. REF can be rewritten as
{{formula:0d6813d6-36ae-4679-b3de-5a07ac260007}}
Choosing {{formula:a268eae2-93c7-43cc-9617-a62f02e783f3}} that satisfies the Langevin dynamics (Eq. REF ) thus eliminates the driving term. Consequently, the system stays in the vacuum state in the displaced frame, corresponding to the coherent state {{formula:e59f6b9f-9faf-4aae-896d-7242dd683273}} in the lab frame.
Supplementary Note 4. Quantum Speed Limit of the CD Driving Protocol
In this section, we discuss the Quantum Speed Limit (QSL) for a driven-dissipative bosonic mode, and show that our CD driving protocol reaches optimal quantum efficiency among all possible experimental controls.
For open quantum systems, the QSL can be formulated as a geometric constraint, i.e. the total length of the system's trajectory is bounded below by the geodesic connecting its initial and final states, where the geometry is defined in terms of the Bures metric {{cite:3f51758b38b34c310b533f3ae73c1daa166f0a2b}} for density matrices.
This metric is interpreted as the statistical distinguishability between neighbouring quantum states, expressed in terms of the quantum generalization of Fisher information, i.e. the Fisher information maximized over all choices of quantum measurements {{cite:6c6c4e25258cff354e36bb0188c3a41a2c2e05ca}}. For our system, the dynamics can be equivalently described by a unitary operator, generated by an effective Hamiltonian {{formula:30d2dc91-97be-42c1-b527-d2ad91448104}} . This reduces the QSL to the Mandelstam-Tamm (MT) bound {{cite:9b774aeccd56d1bd8c9053c4e3042cdcc8f041fb}}:
{{formula:8e20467b-8ef5-490d-ab2a-678804340330}}
where {{formula:21024bbf-6abe-42ac-b86c-6cea807d5e35}} is the initial(final) state and {{formula:0a888fac-cb86-4f4e-8128-f07c4c5d32a9}} . Geometrically, the LHS of Eq. REF is the Bures length {{formula:07e2e12c-e090-45c8-8316-5f81225828fe}} of the geodesic joining the initial and final state and the RHS is the integrated total length of the system trajectory whose velocity is given by {{formula:be98a6b4-44f6-48f8-85d3-0c0a9e0a44b5}} {{cite:d311b34ae24e41da0a27801688f85304432cb296}}. Here, {{formula:d1d2983c-5c98-4056-8fb5-a719a908278e}} is the quantum Fisher information. We define the quantum efficiency of our protocol to be:
{{formula:04bf8da3-4a1d-4e20-b1f6-08e8d8bfdcbe}}
As shown in Eq. REF , for a general driving {{formula:d4f4096a-0e60-4d87-a287-ac5d91d3f147}} and the system intialized in the ground state, the dynamics is described by the displacement operator, i.e. {{formula:3c912a60-eff7-45cc-9dec-7a40fc9ddcd6}} where {{formula:2d6ec533-b102-4ad1-9589-e48dedf1c66f}} is the solution to the Langevin equation {{formula:66ee26f6-4ce7-4c30-a22e-1670c6869c6d}} (SN 3, Eq. REF ). We note that {{formula:3e851cc5-7da9-40a5-9965-6e8b4ccf57d8}} can generate arbitrary dynamics in the space orthogonal to the system state {{formula:054d7728-a97d-45fa-8c8a-caa909a96ee4}} , but the extra freedom can be shown to have no contribution to the uncertainty {{formula:7fdf9027-5d41-452a-9a46-31b79e83121c}} . With this choice of {{formula:6999bfaa-98e3-4c7e-a5f5-6cafd5c2e1db}} it is straightforward to show that {{formula:c92a2a2c-f113-45ad-a84e-e73f6a985b01}} and {{formula:413e716a-283c-4de6-a660-b3dbec675a38}} .
For CD driving, {{formula:9cc7c7ea-f4bc-4cf4-a918-c0c77ed41d3e}} is simply the added drive {{formula:37f4ae05-86f5-4a15-8ecc-6d31130ea6b4}} , which provides the resource for adiabatic speedup in view of the energy-time uncertainty principle.
Identifying {{formula:697af828-ab82-407f-ac0f-fec5a7b72975}} and applying the triangle inequality, we obtain
{{formula:14047351-3f40-4459-81c4-ca52607534bd}}
{{figure:031097d1-fefb-4c77-81f0-efd9e0640514}}with equality achieved by straight-line trajectories – made possible by CD driving – in phase space (see Fig. REF a). The spiral trajectory {{formula:6a8a2d68-93d6-4315-a951-b3236301b09f}} in Fig. REF a is calculated from Eq. REF with parameters {{formula:e57c00c8-4bcb-48b6-9379-fe89828cdadb}} , {{formula:02c4bc2d-9be1-4328-8268-116100860946}} . Fig. REF b shows
{{formula:581eea84-cb3c-4792-a4ea-69eb1400c778}} as a function of target displacement {{formula:007cb415-eb36-47ee-93de-d7603a32e12d}} for direct and CD driving with two resonator-drive detunings {{formula:b8db8218-5836-4dff-80ba-a9f80235c05f}} .
For both detunings, the CD driving reaches the optimal quantum efficiency experimentally (as given by the right hand side of Eq. REF ). In particular, it saturates the MT bound (i.e. {{formula:a149d5d4-404f-4dfc-8d45-281ad68b98d4}} ) in the small driving limit {{formula:77b35d64-531a-4d0e-acac-a73e6ec69bac}} . The inefficiency at large {{formula:72f76505-d8bb-405d-88da-1767dce560f9}} can be explained by the inability to create direct driving to higher-level Fock states, which is a general issue in applying the MT bound for systems with large numbers of energy levels like the bosonic system we consider. Nevertheless, CD driving achieves optimal quantum efficiency within the space of all available pulses, making it favourable for experimental realisation.
Supplementary Note 5. Derivation of the Multi-Mode Optimal Control Protocol
In this section, we derive the multi-mode optimal control (MMOC) protocol used in the main text, which takes the hybrid frequencies of multiple oscillators as input, and generates a single-port waveform that puts these lossy bosonic modes into thermal equilibrium at a desired final time {{formula:89fa336f-9df5-48da-9dc8-9cced180e5bd}} . We first present a general framework which can be applied to multiple port driving, and then analyse the simpler single port case which is analytically and experimentally more tractable, and sufficient for our needs in the main text.
General multiple-port framework. Consider {{formula:fdbce4ac-0256-40ea-8617-55f5be923b3c}} linear bosonic modes {{formula:1cbdf196-590c-4f0b-86dc-4a6d660519ec}} with frequencies {{formula:2d43f584-aa73-4db3-adf2-ce3e6deaca3c}} and linear couplings {{formula:ae1f99d2-1e7a-4117-a433-79173ab0a2bc}} between modes {{formula:e84f3b06-5a1f-4a96-a161-28fbf367eaa4}} and {{formula:1bd5c412-ebef-439c-965d-f153f44f4a56}} . Each mode is coupled to a feedline with strength {{formula:221ee8eb-f4f5-4238-8e9d-ac4bf9160b51}} and driven by an input field {{formula:f583e0a3-6007-4376-990f-d64ededc5794}} at frequency {{formula:4a57b5c4-93db-4336-8326-9aaac345d950}} . In general, the fields {{formula:598a89cd-57aa-480f-b978-2a97b25cd276}} can be linearly dependent if they come from the same feedline. In the rotating frame with frequency {{formula:471522c1-0e52-4ba4-a325-5ce81d8c710d}} and after rotating wave approximation (RWA), the system Hamiltonian has the form
{{formula:a3514ab9-f151-48e7-a846-e3c073aa2fdb}}
where {{formula:450b9c6c-60d7-440a-88ad-bdc2e58cbec8}} is the {{formula:5b2eb487-2cad-4b23-bbed-8d99fd03029f}} th detuning. In the Heisenberg picture, following the input-output formalism {{cite:201c79601d2ea06cd553a2fd99f8f90b8a020786}}, the Langevin dynamics for the {{formula:707d5d38-6448-4d1a-bd88-3e62d4043a3b}} th mode is given by {{formula:2c22c492-74d2-4dc0-bdd7-8488dcd8dfad}} . Adopting the mean-field approximation {{formula:92abdbc9-ec62-45f7-a74e-69cd232ce33b}} for all bosonic modes and defining the effective drive {{formula:2f2c4565-9ffb-4ee4-9638-6a0501b9c0ad}} , we can rewrite the Langevin dynamics in matrix form:
{{formula:defd2bce-0817-4de3-9483-6aa8358ec8e3}}
where {{formula:c19bd326-1e15-4f20-9882-9cbbc70292f4}} is the (complex) frequency matrix, {{formula:b53ec797-2b17-4df1-858e-34b0d13ae307}} is the column vector of the mean fields, and {{formula:d101fa0f-11d5-4116-925e-6eb790f64bd4}} is the column vector of the effective drives.
{{formula:177d55a7-3545-41af-9040-8ec69ca64a0a}} can be diagonalized as {{formula:aed9256e-0092-4823-bfd8-5e238e8a163a}} , where {{formula:2858dff9-0047-4598-a20e-114bfcfe4fb0}} defines the hybrid detunings and linewidths as {{formula:f358a256-f6f7-4dd7-9c91-e0fdbaff98b3}} , {{formula:1e109fa4-e27d-4b40-9d1b-e45c3e37daaa}} .
We use step function driving in our protocol: the time between initial time {{formula:ca841a62-0acf-4b40-ba99-e8205053d74b}} and final time {{formula:db6bb90c-8651-4bce-bea4-c8b9a4ec3786}} is divided into {{formula:c6f9a15c-9483-4059-a296-60cc18e4508a}} equal-length intervals, over each of which the drive strength is constant. i.e. for each drive {{formula:e9d44d2d-44cb-4f9f-b274-ee919cec6348}} :
{{formula:e55323ab-7af2-43d7-928f-017e32bde32f}}
where the {{formula:05665b07-b6b8-467b-8863-c191488703ff}} are constants. Our goal is to put {{formula:fa5e25e6-1c18-4e6f-92cc-a57a258235a8}} into the target equilibrium state {{formula:25345306-a667-4c81-9b7b-7b5cd8f73a6d}} at final time {{formula:82559c61-d369-4a45-a0bf-b0ebc2ef2694}} , starting from initial equilibrium state {{formula:26da01fa-8222-4254-bf33-06ee3139e24d}} . The propagator and general solution of differential equation Eq. REF are
{{formula:1cdb95ad-b6fa-4f95-a2ec-73d774f84ba8}}
where {{formula:8daeba1f-25d1-465d-9bf2-f9cd7f8f86ac}} is the step function. Using {{formula:cf0cdc6c-76b2-44a8-8478-e1574bd551df}} and Eq. , our goal can be achieved by solving the equations
{{formula:4de20fac-1315-451c-8bec-ae94ef62edeb}}
for {{formula:e95df9f8-31b1-4f39-8f25-1e2dc59ff510}} , where {{formula:356b844a-30a3-4fc5-853a-3a887b3065e8}} is the complex hybrid detuning. Treating the piece-wise driving {{formula:b1e6875b-585a-4a15-aed1-e0ee77cb572d}} as a vector {{formula:899a41f4-aedf-464d-a1dc-d93bdfc94f4b}} of dimension {{formula:7537f11e-3e8d-462e-bc6d-ebc116e2b61f}} and defining the {{formula:d04e8654-b793-4c8b-b406-660adc9630db}} matrix {{formula:b810d440-46f9-4067-a31c-90a9bffaebaa}} , Eq. REF reduces to the linear equations
{{formula:560e8804-78d7-47f7-b4c9-94a57136c416}}
If complete information of {{formula:b5daa2d1-e9e8-425e-8309-d16bc092eaf6}} or {{formula:46a75dbc-e436-44ab-bb60-959a0820c252}} are given, the general solution of Eq. REF can be found by performing a singular value decomposition (SVD) of the matrix M. We concentrate instead on the case of single-port driving, which is considerably simpler.
Single port driving. In the special case of single-port driving, all drivings {{formula:dbe72d8f-8d23-4cde-bb29-f567d3a7f0b9}} are linearly dependent, and Eq. REF reduces to
{{formula:bac11683-59d2-412c-92db-b00d58e3c04b}}
for constant coefficients {{formula:acd7f2f4-70af-4525-bc8a-3a97ec35a835}} , a single-port driving vector {{formula:c470540e-409e-4f30-b58c-3bb0b0a312c3}} , and boundary conditions {{formula:583986b7-b2e2-4087-972f-4ad448d4e21c}} , {{formula:14062f46-33ba-48b4-8352-a49d308f3a4c}} . In this case the {{formula:e9173974-38ae-4575-825c-1eff6c9b50db}} terms in Eq. REF cancel, to give
{{formula:28f511b8-fd82-4464-bace-0b35de43a1d3}}
which takes the form of a linear constraint on {{formula:7a4e543b-dceb-45d5-8440-0400972c024b}} .
Eq. REF can be similarly solved via SVD of the {{formula:4ca492cd-af63-41eb-b6cc-62e6b5cc1820}} matrix G, i.e. {{formula:c9c38da0-36a4-4fca-958c-90001d2df456}} for unitary matrices {{formula:8f027d34-161b-4730-982d-37abd94a41b3}} and diagonal matrix {{formula:9f57bfe8-060d-4272-aff6-3c358703db08}} ({{formula:84ff70f5-6a78-4ed1-b003-0b7a17456503}} , 0 is the zero matrix). This gives the general form of {{formula:f086e01d-9d11-47ad-828e-e0d0d8ea5265}} as
{{formula:9368c20b-0629-4cfa-ad00-101d4141da96}}
where {{formula:e9e6dcee-b689-47a0-ba14-24952718c2a9}} are free complex parameters and {{formula:39b2ff76-43d3-42c9-99b7-135d1ed08d4e}} is the {{formula:89f63c62-c2c8-431d-8166-dd99f723c96f}} th column of {{formula:8d03c937-4923-4f70-86d2-d8f7573e16d3}} , which can be chosen to optimize a user-defined objective function such as the maximum power output of the pulse (see SN 6). We note that in the single-port driving case, the only input to the protocol is the complex detuning {{formula:7977eec4-080b-4671-94b2-bab7c68ac866}} , which is simpler to measure experimentally than the multi-port driving case where full information of {{formula:5de76ae0-6c04-431d-a217-6963fa944e50}} is required.
Experimental implementation.
Our single-port driving experiment in the main text corresponds to
{{formula:0711a36b-4667-4294-8375-96c95b8b0f2f}}
where {{formula:c44b6c00-f727-4bd9-87f0-8b2c1cdfe504}} are the Purcell filter and readout resonator field conditioned on qubit state {{formula:8291305c-b5d8-4d73-98a2-009cc6d13e86}} , {{formula:fd10c074-b916-47b9-9172-6ac509f50576}} is the resonator detuning conditioned on qubit state, and {{formula:ad6cb920-5dff-4f39-9729-d863ad24a015}} is the effective driving on the filter port. The drive constants are {{formula:81386ff2-28c3-4a67-89a3-65668b369967}} in Eq. REF , and the solution to Eq. REF determines the two quadratures of the driving function which, after optimization over the parameters {{formula:487f64a2-17aa-44c3-9fa3-79c95bbb38f5}} (discussed in SN 6), yields the waveform used in Fig. 3 of the main text.
Applications.
Two applications of the class of waveforms derived above are fast equilibration of the readout cavity and Purcell filter and the fast reset of them to the vacuum state.
In the first case, we set {{formula:81d178f6-2e05-45ce-867f-90969bd6f752}} and {{formula:9431867e-0ba9-440c-9275-adcd46af9f0e}} in Eq. REF to be the constant drive amplitude after {{formula:251e7a4b-fec6-412c-b978-b56cd1c67c4e}} .
In the second case, reverse {{formula:32d4834d-299a-4eac-9521-42f684107cd4}} and {{formula:cff6ae28-e58e-46e2-9cb5-bee13be8c831}} .
Unlike the continuous driving pulse in the CD case, the MMOC protocol results in many pulse jumps.
In SN 8, we estimate the effect of the distortion induced by the filter in the AWG and confirm we can still use the MMOC pulses safely.
Supplementary Note 6. Numerical Optimization and Speed Limit of the MMOC Protocol
This section covers various numerical aspects of the single-port MMOC protocol of SN 5, including optimization over the maximum power needed, the speed limit of the protocol given limited output power, and the computational complexity of calculating the desired pulse.
Energy consumption.
The total energy consumption (up to an overall constant) of our pulse in Eq. REF is, due to unitarity of {{formula:1a161c38-f9cf-4155-a5a3-33023bcad9e7}} ,
{{formula:7161f01a-006e-4b07-8a61-bc0f06a42170}}
where {{formula:fe59c082-8d31-431c-9601-05fc1c2745a9}} denotes the inner product. From Eq. REF we see that the minimum energy solution {{formula:e6572403-cacb-4c14-be34-4ec0d5f01895}} is obtained by setting {{formula:4fd7c5e0-d24d-4abe-a51c-92083d7f7cf6}} .
{{figure:2a4d4b23-5047-485e-ae25-7317838dd9b8}}Minimizing the maximum power output.
Given the output power limitations of the microwave devices, it is desirable to minimize the maximum output power {{formula:47b0b35c-5537-463f-8887-7ae818081981}} of the pulse. To achieve this, we numerically minimize {{formula:16db1d7d-6731-4828-9a39-12129b47d47d}} (as a function of free parameters {{formula:30e1e40f-11d8-4868-8be0-76223dfab4bd}} from Eq. REF ) using a differential evolution algorithm. The resulting optimized MMOC pulses for both the ring-up and reset stage are those used in the main text.
Fig. REF shows the numerical results of {{formula:3583aea1-bf96-48bc-b950-69d02b19f8ce}} (in dB) in units of the steady power {{formula:a7df7630-c7b2-438e-b6e2-22db603ff2d8}} after {{formula:1fcaa7f6-c65b-4ee7-83e6-30d94570a007}} , plotted for the ring-up stage with different protocol times {{formula:119c3538-a5fe-40d8-8b2d-ba6092cc63ce}} . For comparison, we also plot a lower bound on {{formula:4e8fe080-7ce5-4c1e-bd76-51965edddf75}} , which follows from Eq. REF and the fact that {{formula:024934a9-1bf1-4097-89e4-cb27bb062915}} :
{{formula:4317f069-da77-4490-b62c-76c951f50a5b}}
Given the protocol time {{formula:c1d18a18-3c02-4a2f-85be-9b1543553ae9}} used in the main text, we find {{formula:070129c5-d432-4ff2-a655-51d0d2d2aee8}} after numerical optimization, which is a {{formula:799315fc-4d56-47a8-b9e7-6e3713d0b5fa}} reduction from that of the minimum energy pulse.
For speedup beyond unit resonator lifetime {{formula:b8605b81-f323-4826-8234-041c0daaf07c}} , {{formula:bc651a05-1769-4ef4-bb60-af6150dcc0cc}} grows rapidly and may induce unwanted qubit transitions, which sets a speed limit for the MMOC protocol, as discussed in SN 7.
Computational complexity.
For single-port MMOC, the number of total qubit-state-conditioned bosonic modes {{formula:e95a20ca-8904-4b0b-ba03-8298640b67f2}} (in our experiment {{formula:95798b97-6a1f-4479-8ec5-ba3ca73a73cd}} ) is less than the total number {{formula:86cddc68-672f-439c-a4b9-c0edf4cfcd54}} of pulse sections. In this case, the most time-consuming step in computing Eq. REF is the singular value decomposition of the {{formula:8825e0c2-be21-43c7-b088-1481ad137e4f}} matrix {{formula:633850f3-93be-4c56-b34d-db57772d80da}} , which has time complexity {{formula:ef712ebe-a5fc-4e14-a053-62462d322a21}} . For the general case of {{formula:5d534fe4-e883-4a4b-86e8-8ade1dcb704a}} -port driving, {{formula:f34c9ccf-3856-4664-bc1c-1e88ba8f5c22}} is replaced by the {{formula:7a94d14d-a35d-4336-8834-b6678e3b4b5a}} matrix {{formula:f9956e34-96d5-49d9-baf8-a6f34db9062d}} , with corresponding complexity {{formula:88a04fbd-e16b-467b-bdca-10fa50abed62}} . In either case, the problem admits an efficient polynomial time solution.
Supplementary Note 7. Influences of the Large Drive
We observe that the output signal drifts with a large driving power, which sets a limit on the steady-state driving power of our protocols.
This can be explained by the nonlinearity of the resonator {{cite:53758224cbdcf3c08f2a82098582bdb52af644b1}}.
At the same time, according to the previous study {{cite:8e2a5d6822d796f89a4aa1ef33ed28253f88cb5c}}, higher transmon levels are excited due to the non-RWA part of the qubit-resonator Hamiltonian, which becomes on-resonant as the photon number in the resonator increases through a Raman-like process.
These two observations are shown to be closely related in theoretical simulations {{cite:0ac3ff017d147e3eaa564f42c40d3306078a8307}}.
Here, we conduct two different experiments to confirm this point and find limitations of our protocol when applied to the transmon-resonator cQED system.
In the first experiment (Fig. REF ), we compare the output signal of a small pulse of strength 1 a.u., and another larger pulse of strength {{formula:dbc06693-3192-41a2-8ebc-3ae084491318}} a.u..
Each point is averaged over {{formula:b3d39d22-73a5-4bcc-8274-204d04f526ef}} measurements and moving averaged with a Savitzky-Golay filter (width 21, order 3).
IQ traces of the output signal in Fig. REF (c) show a clear drift even long after {{formula:132dc601-6eb2-4737-8c3e-5603a0f91847}} , which can be qualitatively explained by the nonlinearity of the cavity mode.
In Fig. REF and REF , another experiment is conducted to test the impact on the transition out of the {{formula:b10fff61-24b9-476c-b983-46105a2ae4a0}} state of different pulse amplitudes and durations.
A significant drop in {{formula:04aa1cbe-a1c1-4477-994a-7956025aad2a}} is observed above amplitude {{formula:f41be475-e201-40e2-81b5-e2a453a48669}} .
At this drive amplitude we estimate the steady-state cavity photon number (via qubit spectroscopy) to be roughly the critical photon number {{formula:a0cdffc2-d67a-40a6-b763-024879e9c278}} {{cite:443a9d66fbf7db7a9329e57d896e94fc887e9a67}}.
{{figure:5a6483e2-fb37-443f-af64-17dc3dbfa65c}}{{figure:6141eac5-4578-4758-8823-63d431f7e7e6}}{{figure:9b338ff2-97a4-4ed9-87c7-0c44e3693c64}}
Supplementary Note 8. Propagator Corrections from the
Low-pass Filter in the AWG Driving Line
Here we show that corrections to the MMOC pulses imposed by the low-pass fourth-order Chebyshev filter are negligible. In our experiments, the MMOC pulse (Eq. REF ) from the arbitrary wave generator (AWG) has a carrier driving frequency {{formula:6b3b178c-8aff-4c60-b1f7-d95a84df5ce2}} of 200 to {{formula:307cdc9d-fe82-41a8-bfd1-6e0b2599d1d3}} , which passes through the filter with a cutoff frequency {{formula:f08aebcd-e7bc-47fc-bcd1-9bc636f8981c}} . The piece-wise constant pulse causes the Gibbs phenomenon, a potential source of error.
Here we give a qualitative evaluation of this error. To simplify our calculations, we assume that the passband's transfer function {{formula:4ff39099-72fc-4682-9889-66e377168338}} is 1, and is 0 outside the passband. The waveform after the filter {{formula:3d55cc31-11ec-4280-9b87-bd0f5bd44bbf}} is described by a convolution {{formula:1446b19d-f6e6-4f04-b7d6-fd1208cfb246}} of the pre-filter waveform {{formula:584b1097-8ae3-4839-a863-0f9fb6ce0471}} with the filter function
{{formula:6ea286b6-7534-43d9-b4ec-04ecb150a3e4}}
with frequency cutoffs {{formula:adbc618c-c0b1-472b-9969-e54ede562dd7}} and {{formula:6c066364-295b-46d2-bfec-a97fa424ba25}} . The calculation is done in the rotating frame defined by {{formula:b47c95cb-ac8a-4576-a67c-c96e38443bba}} . Replacing {{formula:5f4a6e47-95b0-4fc6-9b46-6be70b715bc0}} by {{formula:a9b17022-fd4f-4bd0-8099-80b5675a6cc3}} in Eq. REF results in a modified constraint matrix {{formula:bd944075-7e04-48b7-a371-d10d3654ac6f}} , given by {{formula:96989573-142b-40c7-9593-261daf0e1fb3}} , which satisfies
{{formula:db1d2115-efb9-4a66-b5b8-396613c3ca96}}
where {{formula:f5ec2151-53c9-44ea-89e3-f66f0a037754}} is the finite time window in which the corrected pulse {{formula:83c3e4a0-0398-4bd5-800a-a556eaa0d806}} is integrated over, and {{formula:3f87d04b-beb3-44f5-acec-72ee35622e1f}} is the {{formula:3a0c8d22-f11c-4d93-a87a-58fcf663aa80}} th square function which equals 1 for {{formula:5d8c6a14-312c-47df-a220-8567abb963c3}} and 0 elsewhere. The single integral Eq. REF is easy to evaluate numerically and should be compared to the original matrix elements {{formula:e3634e6f-9669-4f1c-966f-b0733a35850d}} . We find that the relative difference between {{formula:c110fdf0-cfd4-4600-a500-1abbb1b586be}} and {{formula:546393de-7334-4446-8ab0-87045f159cc9}} is less than {{formula:3944309c-a1d2-4a7e-ac03-f9a4c7104a4b}} and is generally independent of the choice of window time {{formula:a9183058-a693-45fa-9e91-c599e029a99c}} , {{formula:14517cea-0dc4-40ad-828d-4b9936a13065}} .
Supplementary Note 9. Additional Experimental Data
In this section, we show additional experimental data we have collected.
In Fig. REF , we test the widely used square wave driving with an initial amplitude twice as large as the remaining waveform, and see little decrease in the equilibrium time.
In Fig. REF , we apply the CD pulse designed according to the set of parameters for the {{formula:8ed15c2d-2f9d-4b8f-b014-b9df9bc07d7b}} and {{formula:a4cba766-1c45-41da-b63a-b9902681e24b}} states, and confirm that in the single-port driving situation, CD is only able to accelerate one mode at one time.
The main text shows the 4-mode MMOC protocol controlling all four modes with the same pulse.
In Fig. REF , we design the MMOC for two modes corresponding to only one specific qubit state, then apply it on both qubit states.
In this case, the method only works when the qubit is in the correct state.
In Fig. REF , we compare the output amplitudes of the {{formula:d081396c-858a-4e30-b453-59e9b17aa7e1}} and the corresponding CD ringup drives of different durations.
Fig. REF and Fig. REF show the output signal's IQ trajectories for the {{formula:2616f3f9-6277-4a35-81d3-50863485459c}} and the corresponding CD ringup drives.
In Fig. REF , a large-amplitude and far off-resonant {{formula:a1e3cc01-563b-44d6-957b-bbb56a58667b}} drive , and corresponding CD drive are applied.
The far-detuning guarantees the cQED system will not be excited.
According to the input-output theory, the output signal will be a simple rescaling of the input signal.
The designed input ringup duration {{formula:176dbfce-3d82-4a40-b153-598304be1b31}} is {{formula:5c66aef5-8435-403d-a395-6872ae8a365d}} .
However, we see the output reaches the designed stable value around {{formula:9c3b7cb7-d04e-4348-bc1b-1490fab5e15d}} .
This tail indicates the filtering effect of some low-Q and energy-storing microwave components in the feedline.
This effect is more noticeable when the input signal is larger and can be simulated by a convolution with a low-pass filter transfer function.
{{figure:2a66df68-36c8-405d-a1d6-23c41aa6592b}}{{figure:24e9a643-13a6-4195-acc0-d53672caabd2}}{{figure:83417371-cdf3-4c2f-87f8-10b5a80ed356}}{{figure:78c5a689-29b3-4aaa-9f53-35a2240347ee}}{{figure:d8f091e7-b868-4638-9006-09513cb1cead}}{{figure:48786649-e94f-4b66-899d-9b50b85c8ad5}}{{figure:92360dde-eef4-4a4d-987e-b14f85b3df59}} | m | 9426d94ce730c4931204d536839b2a07 |
To illustrate the effect of the momentum dependence of {{formula:8eab8d7b-5d8d-4507-8e73-cb791808b826}} mixing, we perform two fits simultaneously taking into account the experimental data sets of the pion form
factor {{formula:fcb4c762-587a-4c5b-8c1b-a63db3b578e1}} of the {{formula:4451a750-8e11-4026-98f9-b4e388d1a904}}
process in the energy region of
600{{formula:d6c9737b-6f83-4b6e-bc32-7491cfa82cd8}} 900 MeV measured by the OLYA {{cite:c0da12939616d318484cdcb2d47ac8c8dc5d97d0}}, CMD {{cite:e21d08c7fac2eebc35315122f118173fcc3ab088}}, BaBar {{cite:7c55d64eac087936a3a82ba1411836fc34063de2}}, BESIII {{cite:ca17fcdec2707a24c6115a34f0a23517d1428964}}, KLOE {{cite:ecde937e2405d5ea1f65fb53ccca70a1d4d0d614}}, CLEO {{cite:5586d0b5a7b37604f71f0b91ea5fc82e7a8d71c2}}, and SND {{cite:498a1f73290d968dec936e99c4104276cbe12536}} Collaborations, and the decay width of {{formula:f33343ab-baa5-4306-81dc-a9b7d55913a5}} {{cite:dcad4195101bc79a21d8993e716616d08b68685e}}. To be specific, in Fit Ia we use a momentum-independent {{formula:a636ed96-f82f-46af-9874-4363c2f7e1ed}} . There are five free parameters in Fit Ia: {{formula:fdec0c8c-aa8a-44f9-8398-16ca9da69283}} , {{formula:524e06c1-f184-4dd3-9b13-afa50634fcfb}} , {{formula:c96cedcb-0be3-4d2f-8b76-510498da41da}} , the real and imaginary part of constant {{formula:53175bd9-7a3d-4095-b423-1385db2a1b36}} . While in Fit Ib the momentum-dependent {{formula:f1aa559b-52c6-4d49-9116-1698f852a197}} is considered. There are six free parameters in Fit Ib: {{formula:c029e291-29a8-40a3-a404-65e6bc0ee4c9}} , {{formula:ff817e58-f5db-406f-a100-13922e66aed7}} , {{formula:7ba01687-e91d-40e7-8d1c-ed468e06d22e}} , {{formula:ef84ca1d-126d-460b-bb32-bc714ce7cb45}} , {{formula:4a290a3b-1fbd-497d-884a-e8e19dee9c8f}} , and {{formula:0d6853bc-711b-4b71-af71-2e641c7eb109}} .
The {{formula:fa2b348c-8a61-4b8b-b07b-c4c19dfc1ca7}} and {{formula:42fb49ad-e981-42bd-81d3-ac9e99bd812a}} are the couplings between the {{formula:3106da45-81e9-42bd-b923-002a6c9fc21d}} meson and electromagnetic fields or Goldstone bosons as given in Eq. (REF ).
As defined in the appendix , the parameter {{formula:a3960fe4-c0c7-42d4-ae01-adf8d9a8d7fc}}
corresponds to the coupling constant of the direct {{formula:5667c8a8-1c4c-4f14-9366-3fdf6a77f4dd}} interaction. {{formula:2721f2e9-f4e5-4aa7-9dd8-2e09af209f8d}} , {{formula:a25308de-9959-4a05-bad9-0dfb7abfec49}} , and {{formula:7880f615-6991-4914-865b-edf843a166a6}} are the corresponding parameters for the counterterms. Note that a tension between the two most precise measurements by BaBar and KLOE is observed at and above the {{formula:ef8d8430-afc0-4136-a41e-e54032712cf5}} peak region, while the other measurements are consistent with both within uncertainty. We also perform four further fits. In Fits IIa and IIb we fit all data sets but BaBar with momentum-independent {{formula:ebd8a875-5974-435e-b609-1f808038346b}} and momentum-dependent {{formula:ed8b825c-e774-4657-b797-3b754bafb880}} , respectively.
In Fits IIIa and IIIb we fit all data sets but KLOE with momentum-independent {{formula:f3606c44-8a22-4977-89db-ed5f229ef200}} and momentum-dependent {{formula:7979f874-2b0f-4c01-b8ce-57a8fc0858b5}} , respectively.
{{figure:fbae34ef-3701-430a-898a-763765b6d06c}}{{table:1cc00edf-5139-43ec-ad2c-e86917a220b0}} | d | c7e14ae665685c861a1b62ecad09c3ac |
K-means divides a set of input items on explicitly defined number of clusters. We set this number as the number of all users divided by a coefficient of {{formula:7c95dad0-f577-4593-88d8-249b9478f259}} . Thus we achieved clusters, composed by average of {{formula:2409c201-a423-4e7b-9234-8982b3f4af1f}} users. This coefficient value {{formula:7a158c05-6e3a-42e7-893a-8900262bdc39}} was determined in respect to the results of normalized discounted cumulative gain metric (NDCG) {{cite:a0c554d7034a479fbf39e5640bf809ccae1ba691}}, {{cite:3e5f1eec4dd6c844923cba47f8db5f017e8910b9}} and mean average precision metric (MAP) {{cite:fe534144551e71655160ed6d84adf0887f31a785}} as the point at which these two metrics reached their maximum value (Fig. REF ). To make it clear, on the MovieLens dataset {{formula:e97a9805-ce22-44ff-a393-67bd930ec633}} value 50, and on the Jester dataset {{formula:6be738cb-cea6-4813-8805-2cd839e34938}} value 100 guarantees best performing clusters (from NDCG and MAP metrics respectively).
{{figure:1501eb32-2f93-40cc-af5a-7e7199e4be8e}} | m | 1c6f2ea5ac880a839501f68c67e545e3 |
Figure REF shows the resulting population when running evolution using the RWRL Cartpole environment {{cite:a5b64092ac1bc20807ac976146366e43e977a16b}} and Table REF shows the average fitness scores ({{formula:359a0411-329d-4ded-8b6a-292cfa58dce5}} standard error of the mean) for each algorithm in the Pareto-optimal set.
{{figure:79254fa5-ba34-43d1-b861-c526a6486afd}}{{table:7e0638b3-fe05-4fbe-9eab-effff47658d4}} | r | cb9b31f7ac0dfd0ad9f1a892e50a5363 |
In some extreme cases, there may not exist a pre-trained model. In this situation, we can evaluate weights with models trained from several rounds of FedBN {{cite:9f4cc4f4da50eb15b847101d3bc551d8ca058abd}}.
{{figure:adc22905-357e-41a8-9dee-215eddb6c44a}} | d | 544640de188072332894f4fcf5afde46 |
Let {{formula:4624ad19-64fc-4700-a70e-568440f19d63}} , let {{formula:9d527f18-0677-4279-8c6b-5dd78430e7b9}} be a family of independent exponential distributed random variables with parameter {{formula:ba4b4a72-196c-4005-b57d-0eb39949efc9}} ,
let
{{formula:c59d475b-cb83-40b0-ad62-dc8bf9f700dc}}
and let {{formula:e95d3b1b-1ed5-44ce-bcfb-e2c2ef237424}} be the counting process defined by
{{formula:c7e22c1a-baef-492d-b72a-c65ef01a0133}}
Observe, for any {{formula:72d1be18-1da4-49f0-881e-87500ccbd17d}} , {{formula:25a96898-9538-48b2-86ef-dcd6326c1714}} is a Poisson distributed random variable with parameter {{formula:94dfea71-587e-4a22-aea5-cdf60e4248e2}} .
Let {{formula:610bcc24-9de5-4026-94c3-8c38dca9e628}} be a family of independent, {{formula:b82a2592-5ecd-46f4-84f8-d53906cf95db}} distributed random variables. Then
the Lévy process {{formula:a6509ccd-7cd6-4316-a078-6ffe48d12c96}} given by
{{formula:eff34753-431b-4671-9e66-7e7c9a7334cc}}
can be represented as
{{formula:66a60a79-c2b7-4083-8b2c-eeb52af8f95b}}
where {{formula:07e3aea9-c6eb-4138-9dea-fd1982f552a2}} (see e.g. {{cite:e0293b4fc4d8df809a5b4763515429268015ebee}}).
Now, by a modification of {{cite:526c80284976e6a8c393bbfc376f377d1503aeaf}}
there exists a solution of the deterministic equation
{{formula:9cfc48a9-534f-4c05-bcc5-e18138dc5af4}}
which is in {{formula:a290f15a-c166-442c-900a-c1c8a9cc1993}} ,
with {{formula:fa461eb8-2e85-4595-ab7a-3d29e7403f29}} .
Indeed, setting {{formula:72718600-a835-4e0e-83d5-8188b8878951}} , it is not difficult to check that for any {{formula:48df785a-0ebf-4265-8bed-91c90afe09b9}} we have
{{formula:043b4fa8-dffa-42a1-a284-cfeaa5165ccb}}
and
{{formula:578f6639-000e-46fd-aaad-ac91d78e5db4}}
where the Lipschitz constant is given by
{{formula:73b5ed02-651b-4319-9afc-34bee9894af8}}
Hence, setting {{formula:e52ac36c-e25c-4a66-8c9d-998a928f198b}} and {{formula:1572419b-9f43-4405-8754-49b870d6ac41}} so that {{formula:3ee50260-137d-4799-9ccf-98ab30db5fe5}} is an admissible pair, one may use
as in {{cite:526c80284976e6a8c393bbfc376f377d1503aeaf}} a fixed point in
{{formula:a8cebe34-fe6c-4390-afeb-6a601e8b6f8c}}
equipped with the distance
{{formula:c6d039f5-5c86-438c-8450-520b405e5316}}
and a constant {{formula:223f8958-3c62-4af4-a2a5-327866f3a2aa}} depending on the initial condition (see {{cite:526c80284976e6a8c393bbfc376f377d1503aeaf}}).
For sufficiently small {{formula:37f534c4-8f2d-4c7f-9b87-8b311d157216}} , we obtain the existence of a unique local solution.
By uniform bounds, this local solution can be globalized.
Let us denote the solution by {{formula:ed26e3ef-5c46-45fb-8f9a-a4bb2cd74b75}} . Since at time {{formula:554c0406-656c-4ea7-9963-6ceb7c68e2a2}} a jump with size {{formula:7640e337-58bb-47b8-ae0c-c870f89da38e}} happens,
we put
{{formula:b7ce36d0-4f2c-42c9-8300-921654dc5071}} and consider a second process, starting at time 0 in point {{formula:64990b8f-7b87-4f23-b4f0-73913cfde814}} .
By Hypothesis REF -(i), we know, {{formula:b1a228f0-64ee-400e-8bb5-4da8f835f29b}} .
Hence, again by Theorem {{cite:526c80284976e6a8c393bbfc376f377d1503aeaf}} and the previous arguments,
there exists a unique global solution of the deterministic equation
{{formula:4f2b22b6-c9f7-47bd-b704-4aca1c1cdf23}}
Let us denote the solution on {{formula:8922de78-914e-4b41-a5eb-1101901129b0}} by {{formula:6aa00fef-3210-4043-8233-94a0a9ded7b1}} .
Iterating this step we get a sequence of solutions {{formula:de1e828e-e4e2-4218-a2d2-4d5c9486f8c5}} . To be more precise,
let us assume that we are given a solution {{formula:096ad269-f7be-4825-96cc-b22607015ac0}} on the time interval {{formula:c574ba4a-d459-49f2-97f2-888532649922}} , where
the family of stopping times {{formula:c95088d2-4d2d-4c39-ba17-e7812a7ccb90}} is defined in (REF ).
Let {{formula:06b9fddc-17ca-4ffa-b6d6-7434a3574bee}} .
Then, we denote by {{formula:d5f6cd15-0ce9-40e8-addb-0b19a994a5ac}} the solution of the following (deterministic) problem
{{formula:dda5cbd5-0f18-4dd8-823b-6dc6d2acc19c}}
So, for each {{formula:cb8120f6-68fd-44c4-9163-0a280c40a705}} we can construct a solution {{formula:36dc5dcd-f443-4860-8cea-9e1ee18f1c1c}} on the time interval {{formula:9d6088db-d841-4905-b53f-72ec63cceea2}} . In the next step we glue these solutions together
by putting
for {{formula:c90e7bb4-f9c3-4540-a086-3b8d92ac91f1}} , {{formula:8ef1a99e-338e-4583-9a52-46b163cb146f}}
{{formula:b48d6fe2-cd0b-4484-9c70-428b5edc5d0b}}
Let us observe, that the jumps take place at the end points at each interval and will be taken into account, by taking as initial starting point for the next solution {{formula:4aa8ce23-c1d6-4fb5-b447-3b60772a2e7c}} ,
the solution {{formula:cfc3b78d-7639-4973-a0c1-88567447a8b0}} at the end point {{formula:22d4015c-2f16-4f6b-8634-9e2545579a6f}} plus the jump. In particular, we put {{formula:f8bdde67-83c4-41b0-8b0d-fc3eeed7ea1e}} .
It is straightforward to show, that {{formula:1155e98f-c761-4e83-baf3-474a83d5fb57}} solves (REF ).
Since {{formula:d4903b36-b057-4b00-9a01-863daa318735}} , the solution {{formula:8871b252-d7b6-41d4-93e3-0347eff40ce3}} is a.s. defined on {{formula:3fa6754b-5e45-4cfc-bf6a-ffa246028363}} .
The càdlàg property follows by the fact, that {{formula:dd1019c0-14bc-429e-9768-67f50e1adeec}} and the limit {{formula:2d6b445b-8deb-4e5a-96af-5395d772d9bc}} exists
in {{formula:88fa559e-4407-4baa-ab7d-e71202507096}} .
Summing up, we have shown the existence of a unique solution {{formula:663b4555-c1aa-4f42-87c5-c588abb8a0d3}} belonging {{formula:2bcd5207-7c4d-4b89-8e4c-c9b02d8bf626}} –a.s. to {{formula:a5c815eb-38de-41e2-a1ff-527b9b06fa95}} .
Next, we will show that under the hypothesis of the Lemma, the mass may be estimated, i.e. for {{formula:40ef9d9e-03a0-4dc2-97a9-39553ab1b6a6}} ,
{{formula:1e2c5eb6-f694-4597-9971-f189172286de}}
where {{formula:967c9096-dd74-4c7e-b605-b7bfc6678f4b}} .
In a first step we are aiming to prove
{{formula:8b3a143e-4308-4938-9793-630e0c182026}}
where {{formula:4fe9a2ee-05b4-4608-83d3-cb49ade5c81a}} .
Assume for the time being that (REF ) is true. Then, it follows by the Hölder inequality
{{formula:7cc97827-1b93-42d6-8363-300428cd73ed}}
If {{formula:9f202eb2-040d-4099-93bf-9c9a8df74e81}} is small enough that {{formula:cd92dd98-6180-4675-9c21-73dbb3a03dfc}} , then
{{formula:9ad8c0cb-79da-4983-ba4b-261c9cde9305}}
Iterating this step we get
{{formula:f81d802e-ccda-4e18-9878-cf7c2f71ce53}}
where {{formula:8cff64b3-eb1b-424d-bcaa-ed162429952d}} .
Let us show estimate (REF ). If we denote by {{formula:128a413e-72ba-4d98-ae8a-62b973047971}} the continuous part of {{formula:611a8014-89a4-4abe-8f65-bb3e1de692a6}} and put {{formula:d85c7be3-b656-4185-8d85-e02e5abe2461}} , we get by the Itô formula for a twice Frechet differentiable function {{formula:53124ea8-a9ca-48b7-aeea-4859120db9a3}}
{{formula:2815c1fb-dcd8-49d1-92fe-e8afe47e7312}}
First, note, since on each interval the solution belongs to {{formula:06eced33-5be2-4b76-bb7a-f7882e8f7ad9}} ,
all terms in the above Itô formula are well defined.
Additionally, with
{{formula:742028a9-5a33-4b01-8cc2-83e42297026a}}
one obtains
{{formula:ed2e0bd4-8fd5-4a9f-985e-eba76d1894d4}}
To be more precise, one has by direct calculations
{{formula:c4e1c518-171b-486e-8b6d-38ab9d5330e5}}
An application of the Burkholder inequality and Minkowski inequality yields
{{formula:ac813864-3164-49b5-a169-98dea4cc8787}}
Taking into account Hypothesis REF , we know that there exists a constant {{formula:72c28645-1588-4a19-a6e2-96ea16cfce2b}} such that
{{formula:e9ce0ca6-d67b-4d40-ad13-a35f75c6318f}}
that is (REF ) holds and the estimate on the mass follows as explained above.
If Hypothesis REF is satisfied, one easily deduces from (REF ) that
{{formula:30a2df2b-6773-49d4-9b96-8e4b0b0429d5}} for all {{formula:04be8a6b-de9e-4ca3-8d42-40932fbb5bf9}} .
If only Hypothesis REF is satisfied,
then one easily deduces from (REF ) that
{{formula:d0fbe4fd-88bc-4733-a66c-996de1ef8d6e}} for all {{formula:824b8f78-1580-4600-8ad8-543ea27df0ef}} .
In a second step
we will prove that there exists a constant {{formula:01068033-5416-4ce9-bd12-ee18d9f609f3}} such that
{{formula:ec676a06-1216-46b4-b9d3-6bb62a38544f}}
In order to justify the computation of the Itô formula for the Hamiltonian {{formula:ff636050-70ed-4bfe-a8f9-1aa629663fb7}} , one needs also to regularize the Hamiltonian. In particular, one needs to regularize both terms in the Hamiltonian. One possibility is to define {{formula:5d2f6311-a748-4b81-b533-015f8f317254}} and to consider
{{formula:1f0b3f5e-11a9-4b9f-a53a-4dbc2a52ea6e}}
Note, since {{formula:9919b12f-21fd-4c90-8455-7f2e2183da86}} belongs {{formula:f8c25a66-4684-40d8-b108-1ebb046100c6}} -a.s. to {{formula:23aa6295-f754-4421-bbdb-11fe46698aaa}} for any {{formula:6c730e3b-4cec-4d4b-8315-4be53c78c3eb}} , we know {{formula:f8f51ad7-be73-4df5-ba8e-b579704a994d}} –a.s. {{formula:25137948-8683-4f21-916c-bb14a6ff3245}} for {{formula:5655f068-a693-48ae-becc-55c32d2122b9}} .
Taking into account that {{formula:6b587f96-ac9d-4388-821d-e81a88a6debf}} –a.s. {{formula:0ed77d15-18ff-43e4-a9d4-49bcca0a05a3}} belongs to {{formula:3b48c9a7-6663-466d-a80b-d9894de82165}} , in addition, by Theorem 7.8-(b) {{cite:2edf12cfcb4bd9b056175507b6ddcdb426a30f6b}}, it follows that the process {{formula:2bf36718-d808-44da-b24a-13a2987e31ba}} converges to the process
{{formula:4ca5a5a6-f728-4ebe-9c0f-b14dc6c77d94}} in {{formula:baa942a3-c1a3-4d3f-9179-0db85ce05f57}} .
Let us apply the Itô formula to {{formula:0aa8822e-ee76-4784-b6ee-a1927e27f623}} .
First, note that
{{formula:bac73692-c580-48fb-af08-4664e4a90a36}}
Using
{{formula:1927f543-6fe9-4611-8b69-07edf18f2efb}}
{{formula:9a4f449e-acb4-495c-a77e-6ad17972f682}}
and the Itô formula
{{formula:82ffdc13-54ef-457f-a225-37d0053c0280}}
we get
{{formula:efb1543a-529b-47da-898a-da1ddabace5f}}
In order to analyse the first term, we first use the fact that {{formula:cbb69a68-7bb0-4daa-9454-7cbeb5a86d0b}}
to write
{{formula:5fcd0091-bf00-4e24-8eec-20d829fb37f8}}
Note that all the terms are well defined, since {{formula:552728e2-941d-42d1-b427-702f32f1797b}} implies {{formula:a1f4978c-9bee-4d17-8344-6840aa1ad729}}
and {{formula:f3c468ab-171e-44c7-88c6-b4678f4f128b}} is a.s. in {{formula:8bc6d4ca-4ac9-4623-8bae-9504b0d1770a}} , while {{formula:30b019e6-5074-48a8-82b0-ec634e76e389}} so that
{{formula:28758f38-2735-416e-97e0-03caad7d66c4}} .
Next, we prove that {{formula:901163be-226f-40ae-b147-10676b78a7d5}} tends to zero as {{formula:34267499-9cc0-479b-9e6b-2b78e00ff6ab}} tends to zero, for any
{{formula:b41843e8-084e-4cba-b161-d401f8481fd2}} , a.s. First, note that
{{formula:2562bf27-3d82-42e6-a787-9e30ac8032ed}}
Let us remind that the solution belongs {{formula:ad3dca22-403d-4cc0-ab29-ee807b71d9f7}} -a.s. to {{formula:679b6c15-4140-46b0-93e1-1737d6164903}} .
Then, using Hölder inequalities and Sobolev embeddings, it is easily seen that the above term is bounded independently
of {{formula:1acfde6b-0702-4cab-84d7-cf3e7d7bc0a8}} in {{formula:6ee6d1a7-a696-4c1d-87c3-895df627e609}} ; indeed, one may e.g. bound
{{formula:3d6581f8-fb60-4a4d-8f6f-e4d7c8f6f82b}}
since {{formula:ba2a655a-a53a-46d8-871b-132e8f3df463}} is a bounded operator – with a bound independent of {{formula:6bf34f40-dd04-4c25-8a60-594c6594cc9b}} – in {{formula:99176d8e-631c-490f-bddc-7494bab8bbcb}} . All the other
terms are estimated in the same way. Hence, we know by (REF ) that
{{formula:aed395a4-ac38-4d6d-815f-b47be415a500}}
With the same arguments,
{{formula:35e6ec45-5ba6-4960-be09-515eb3f7b452}}
Let us now decompose
{{formula:3dd2c0f6-5aa2-4a06-b613-a89e77cc5b81}}
and integrate the first term by parts. We then have to consider
{{formula:51503ccb-9f89-43df-9438-669137f12471}}
Using (REF ), {{formula:15a88675-3abe-4f3b-93f1-c01e02f4b5be}} is bounded in {{formula:b301b53e-960d-4381-babc-e45035a6bf4d}} ,
independently of {{formula:5fffcc3b-b1e9-4643-a9cc-ad3b72045f28}} by
{{formula:9ccf2d04-626f-4e4f-ad35-b4f60b771591}}
Hence, {{formula:9e3c7975-d8bd-40ab-bff0-251a53efb366}} converges weakly to 0 in {{formula:2c18ee90-1a69-47d7-86d6-d0b0e2ea89c5}} .
Since {{formula:831d66d9-026e-44df-88f0-a36d663c902d}} , it follows that {{formula:f18ba368-b055-46b5-89c7-68d1ad5878cf}} converges to 0 as {{formula:9922836a-1176-4312-ba1d-3bbca3860972}} .
For the second term, we use the same argument and (REF ) : by (REF ) and the embedding {{formula:3dd14cc0-42e1-48d3-af80-c08a9fcaa3c7}} , the term {{formula:5d7daba0-9952-40a3-9916-4a997aa70fd3}} is bounded uniformly in {{formula:1a969c09-a561-41de-b4fa-065de63d6659}} , in {{formula:8cb810bd-baa2-4097-8953-4cc47d883688}} ,
by {{formula:847a68bd-be7c-424b-af0b-141d17215573}} , and {{formula:d1ff8243-ec71-41d6-8366-82aaffaa1109}} by (REF ), so that again,
{{formula:428abf1e-c36a-4f86-b4fb-7b0fb1588106}}
converges to 0 as {{formula:35de7a8f-2940-48f7-8762-253d79c6201f}} .
Going back to (REF ), the first term which does not vanish for {{formula:3de50253-b53b-4564-b3ca-3058e427fdd7}} is
{{formula:e3aae6d1-8c43-4210-a2fb-672e2665de4a}}
However, taking {{formula:a1a292a4-a595-4bc0-86e8-17830ae3e6de}} we see by similar arguments as before that
{{formula:e6f82a9a-49db-474f-aeca-e1c4b3abf71c}}
Straightforward calculations give
{{formula:fe58183c-538e-4e8c-9ca8-8bb5a1559ef3}}
Applying integration by parts we get for the first summand
{{formula:2d3bb863-5bae-4f74-82ab-0ea642335193}}
Next,
{{formula:c2f919f1-7404-4051-b5a5-31f79452e776}}
In the next lines we calculate the terms arising due to the jumps, that is the terms
{{formula:df2753ce-7504-4590-a4c5-708f3898374b}}
and
{{formula:f42d8d8e-8ab8-4ba7-a63e-46b7149521c9}}
Here, again one has to take the limit. Since, the arguments are similar as before we omit them.
Applying the Burkholder inequality we get
{{formula:9e20fea3-18ae-4dea-b52a-c02552161d4c}}
In order to calculate the inner part of the first term we compare it to (REF ) and get
{{formula:5eff451a-6546-4bcf-a70e-2ee86f1dd860}}
In order to calculate the inner part of the second term we compare it to (REF ) and get
{{formula:1db6c3ea-7ff9-4b47-8fc6-f88d11599e45}}
Now we are going to calculate the term (REF ).
Here, taking into account that
{{formula:628fa8cd-6f4a-4a31-a148-fab90edb23d7}}
for all {{formula:0fb86232-e793-4303-a7a7-4f12b906fc66}} ,
taking the expectation leads on both sides, to
{{formula:1c98b89a-7ea2-4d95-afbc-f10d3d621e15}}
First we will calculate the terms of {{formula:c4ce123c-c76b-473e-987a-9528fd336f01}} involving {{formula:bac648c2-fb01-4742-922f-99f0f887c83c}} .
Here, we will use the identity {{formula:5271fe13-1025-40e0-b3a9-5cb61ccd83b6}} , and taking expectation gives
{{formula:17a5a128-f043-43ee-91ae-f29a86ff9482}}
It remains to calculate the second part of (REF ), i.e.
{{formula:5713e469-2785-4839-a03f-74460dbfb65c}}
The Taylor formula yields
{{formula:62a4f6bb-7791-42ee-b5f8-b9a3fab4d4b6}}
Collecting altogether, taking into account the Hypothesis REF , and rearranging the terms we see that
the bound (REF ) below is satisfied ; indeed, first, observe, since no stochastic integral is involved in the bounds,
we can change from {{formula:11f4c2e2-8bb2-4b8c-8aeb-fddfea8269b4}} to {{formula:da180263-cc3c-479d-b8bb-59071b79664f}} , and we have
{{formula:7f4c863a-45be-4e0a-b8ef-754f8cf7fac8}}
Next,
carefully applying the Hölder inequality and, if necessary, the Young inequality term by term, and using
Hypothesis REF -(i)
one finally arrives at
{{formula:9fdd99a5-16ba-46cb-80c6-6656a489717f}}
where the constant {{formula:db5f080d-44da-404f-986d-ad3d37c8348b}} depends only on {{formula:ed4517ed-a7e6-4f90-bc2f-5bdf63d862a0}} , {{formula:656b1508-87d7-4871-876e-3e1a7b9bb457}} , {{formula:4fca8295-fb96-4895-b766-e22bd4a4f09d}} , and {{formula:b142d611-a0e7-48f4-9ce3-9a7268cac639}} .
We deduce
{{formula:d52d0e15-99dd-43f2-ba54-3b3ddf7be884}}
Now, let {{formula:58eac31d-b63f-423d-8a2f-da986480880d}} be so small that {{formula:28745ea3-ffbf-464e-ac2d-3672c6e54c5a}} . Then, we get
{{formula:aa5bb50b-94c2-4caf-850f-066f6d7edf86}}
and therefore
{{formula:d3dcea9e-88b2-488a-bd72-8fe5c42d1bab}}
Noting that {{formula:c8250393-b233-40c3-b258-009751274c02}} is only depending on {{formula:f3ce6923-d84f-4a81-853c-7d29fcef69e0}} , {{formula:a9b19daf-5c3d-4ebc-acf4-e060d3e4d942}} , {{formula:fd69231b-e3e9-494b-ac20-74eee29399d2}} , and {{formula:20cc63bd-e756-4754-ad1c-6668ea480432}} , one may iterate
the previous step on {{formula:706582b4-6ec7-4899-a8cf-10727ab7fd0c}} , etc, and show that (REF ) holds.
Next, we want to investigate the entity {{formula:4097431f-53a8-41a0-ad6e-4d953e724b9f}} .
First, we will prove the inequality
{{formula:7c23313a-234a-4cf7-bd34-d8b7e5d12fc5}}
where the constant {{formula:f0f7b620-b834-4e02-ad40-df586d5e70de}} depends only on {{formula:d3515486-997d-4361-95f8-e9fd83a64d8c}} , {{formula:02f549ad-a986-49f2-8003-d66a8b86269b}} , {{formula:e3e9bdc8-3871-4cd6-b9ce-c574a7103c4c}} , and {{formula:ed48cd10-58e5-4662-922e-4b9ec6c1930b}} .
Secondly, we will give an estimate of {{formula:41d7c56b-4823-4d81-95d0-849acf829913}} .
In particular, we have by the Itô formula
{{formula:43db6d65-0e6f-470f-b097-791d1d0d9d01}}
Integrating by parts,
{{formula:3eed68e6-6648-4f91-93ae-398e9ad46d45}}
Next,
{{formula:31fcabaf-3b19-4ed8-a502-d0e3deee91f6}}
Since, for any complex valued functions {{formula:1f87a44f-b38b-425a-a2f3-ccc829967f36}} and {{formula:118a42c6-36d9-4351-b308-2d49a5e6cd00}} we have {{formula:e66f8aca-dae2-4b9f-b269-b2b76c3f56e7}} , we have
{{formula:1a96bc5e-7750-4fe8-8d94-ee60daee81c3}}
Collecting altogether, taking expectation we get
{{formula:8d0c408e-2a4a-4c61-853e-7d4ff815bbf9}}
Taking into account Hypothesis REF -(ii)-(b), we get by the Young inequality
{{formula:968c1a81-a8ab-4e14-9bba-64dae3fec974}}
Since the second term is bounded by {{formula:6693a903-5349-4add-b5f9-0b3b212eb7d4}} , the assertion follows by the Grownwall inequality.
Next, observe that
{{formula:9a430ea5-ebf6-48b2-9ee4-ebbd2b0b47fe}}
Hence, we get in addition, by Burkholder inequality
{{formula:d87f45d0-1a37-4dd5-a3c8-22efc38b7abc}}
By similar arguments as before, we can show that
{{formula:2e8f851e-e6c4-47fb-adff-d8bdb27b2c7e}}
| r | 2d45e2db8979cc4748d63b8016fdd36f |
An agent exploring an infinite translation-invariant world would find that the statistics of the local random variables which he can access are constrained by the requirement of infinite TI. However, and despite a long history of research on TI systems, driven by the needs of statistical physics (see, e.g. {{cite:3a03a4ad230185591e71ee046317a151d177ae18}}), it is far from clear what those constraints exactly are.
| i | 1f647fb6de8282ca3fa12282998cd41f |
For anomaly detection, existing Toolboxes are mainly divided into three types: (1) standalone tools implementing single algorithm (like PyNomaly {{cite:464f397748e73bbb73cac844dbdf56ebbd4b0844}} , Jubatus{{cite:87ffad4c92f8dfc3931ff8a65dd741b111377deb}}), (2) part of a general larger framework that doesn't specifically cater to anomaly detection (Novelty and Outlier Detection in Scikit-learn {{cite:9efb592719ed9a26e9457e5b721ec8dafab66de9}}) and (3) toolbox dedicated to Anomaly detection (PyOD {{cite:9f29dc36e2af16ee4e7babd7c807a5d07886860b}}, PySAD {{cite:6bb1c845c76f5e46734a2ee8b6c454f1b071ac2b}} etc). Also some frameworks focused on anomaly detection, such as PyOD and ADTK only target anomaly detection on batch data, whereas some framework ( like PySAD, Jubatus, MOA) targets anomaly detection on streaming data.
| d | d67d4627aa2126d66cdd1ea507c09ab0 |
Which machine annotator should we select?
Ideally, we want the machine annotator to provide precise labels on training images. For this we consider ReLabel generated by a few state-of-the-art classifiers EfficientNet-{B1,B3,B5,B7,B8} {{cite:fd6044fcd0eb00fe185af1c6438efa60a92ec4ad}}, EfficientNet-L2 {{cite:52ed1e3cff690231a696b67afed73721569fb58b}} trained with JFT-300M {{cite:c091a44b94000a97bc2736c4fda6144fe8384016}}, and ResNeXT-101_32x{32d,48d} {{cite:edb5707b86ce50bf12e9606ebfd80713b631ff08}} trained with InstagramNet-1B {{cite:5c8db10eaaba1ef25f12b142e4811f98ff74f521}}.
We train ResNet-50 with the above label maps from diverse classifiers.
Note that ResNet-50 achieves the top-1 validation accuracy of {{formula:24e3b87c-93ca-461c-be2e-2755ca2f029f}} when trained on vanilla single labels.
We show the results in Figure REF .
The performance of the target model overall follows the performance of the machine annotator.
When the machine supervision is not sufficiently strong (, EfficientNet-B1), the trained model shows a severe performance drop (76.1%).
We choose EfficientNet-L2 as the machine annotator that has led to the best performance for ResNet-50 ({{formula:ef1ad6f0-5635-461b-9231-a11ed5d96a7f}} ) in the rest of the experiments.
{{table:e7860b3e-e3c3-465a-b9be-aebab0860362}} | d | b8a1b120a731630dc3ff27056f9a8f7c |
Working with other {{formula:310dddd6-5335-4bce-a327-daee9e7b88ae}} sets.
In certain application domains, it may be difficult to collect a {{formula:a337a124-17b7-4a50-ac52-12c1187efbdd}} set that is at all meaningful/relevant to the ID task (e.g., attempting to use animal images as {{formula:954754c2-03b0-430a-9aea-9112223a7996}} for medical image classification).
In previous experiments, even though we filtered out the holdout-overlapping classes from {{formula:ae53486d-99d7-4c62-96ea-eb5f41f464a9}} , some of the remaining WebVision images still had relevance to the ID tasks (e.g., despite removing dogs, the {{formula:555ca82a-6563-4d5d-b930-1fe3194f9147}} set still contained categories of four-legged animals like cats, deer, and foxes).
To evaluate the effectiveness of MixupOE when {{formula:65a0e25c-a13f-4f8f-9e89-aef364a5631a}} is not at all “close” to {{formula:b0a78446-99cd-4f82-a8d2-ac4f7dbcfe60}} , we now consider using the Materials in Context Database (MINC) {{cite:a13f40d670b1363c1e01d3663612e5c7bf7d69c5}} as {{formula:0b1e5d9d-2b0d-48d6-b673-5f142f653052}} (which contains material images such as metal and leather).
We find that on the [Bird, Dog, Flower, Car, Aircraft] tasks, MixupOE can still improve the holdout OOD AUPR by [{{formula:dd241b05-2c28-4a75-b407-76fcc0d3162d}} , {{formula:703d5638-87e0-4aa1-885d-8cdbbeb3efcd}} , {{formula:d573d06f-8e70-4b3b-ad71-3a8cb790f183}} , {{formula:67cc4799-02e2-43df-a4d7-57c070abae29}} , {{formula:ff26d3af-b32e-4a2f-b1ee-aea4e9ba22fe}} ] over MSP, which is comparable to the results obtained with WebVision in Section REF .
Thus, our key takeaway is that MixupOE can remain effective even if the {{formula:06d45ddc-c44e-4f58-a4fa-9538e60b4dec}} set is not at all relevant to {{formula:7ddbac96-4080-433b-920b-9693b1344bd3}} .
See Appendix for further results and discussion.
| d | b20bcf3a1bb9f33c0d47caa28038756d |
To overcome these problems, network embedding, also known as network representation learning, has gained increasing attention for the past few years as a fundamental tool for analyzing networks {{cite:8af58dfbea0d3052803f3cc00515324c83f44ab6}}, {{cite:131e9adb94d25aa88339753184db4afd450b9f89}}. Network embedding learns a mapping from each node in a graph to a low-dimensional vector in an embedding space while preserving intrinsic network properties, resulting in an efficient representation of the graph in the sense of solving downstream machine learning (ML) problems with little or no modification. Recently, for networks in which node attributes are available, so-called attributed networks, graph neural networks (GNNs) have been widely studied as a powerful means to extract useful features from such networks while performing network embedding or solving other graph mining problems {{cite:f1abe844cc211a149e789c515a5e711f7127593f}}. As a deep learning-based approach for adopting neural networks as a building block, GNNs are known to have high expressive capability via message passing in effectively learning network representations {{cite:4c5d3866e96a2fe429daa933a634150f0b726a46}} and thus to be a successful model for performing various downstream ML tasks.
| i | 04a020cccf9a8faa8594f411a4215ca6 |
We choose three of the most prevalent VAE models for semi-supervised learning as baselines: M2 {{cite:589d94b95d0c3d25a271e4379b33a0124bbf1935}}, Auxiliary Deep Generative Model (ADGM) {{cite:77414f2fd20a780754b0e4b1bcec3be13fe38189}}, and Ladder Variational Autoencoder (LVAE) {{cite:3423ba36b2f14f70e77b969ac2550c28075e47dc}}. We have described M2 in subsection REF . ADGM includes auxiliary variables to improve semi-supervised learning. LVAE enables knowledge sharing between generation and inference model. All three models have distinct mechanisms to improve variational inference and thus help to show the efficacy of our proposed method over a wide variety of variational autoencoders. We change the optimization objective for each of the models to incorporate our proposed modification and refer to the modified models as 'MIM': Mutual Information Maximization. We choose four different benchmark datasets: CIFAR10 {{cite:f44aa970cd3269cd083f409bc3010bc647976e14}}, Fashion-MNIST {{cite:e56135afd89ced031e06c9fe9bc2a2ad7f277945}}, KMNIST {{cite:f7787c8fcef5fa7a5ae558576bff27c918232e09}}, SVHN {{cite:da962d84ea92aafcb4b6fbefe393a65508267062}} for our experiments.
| m | aa6b052b59a0e2bf893d218b322ad203 |
As the fraction of closed RCs, {{formula:58d9b80b-5e74-4b86-9908-95d6f1e9091f}} , increases connected clusters are formed. These clusters are characterised by a typical mass (number of closed RCs) {{formula:8bcf190c-23a3-4bf3-b98f-d264c7c23c02}} and a typical linear size, {{formula:cb4050c8-a7c5-4ced-999f-791690097bd2}} , both monotonously increase with {{formula:5bf5d05f-417a-40b7-97fa-2f06abecef25}} . At a critical value, {{formula:868e6554-f86b-4a2a-9c7e-33af8bf1cc99}} , a giant cluster is formed and its size becomes divergent as: {{formula:70866c37-f692-46fb-97bb-1bc3d3b1b2b6}} , with {{formula:d6d0ec52-98cc-44fa-843b-53892683a8bb}} being the correlation length critical exponent{{cite:598122aa52f259578b4e2d028c725ac3fd688889}}. At the critical point the giant cluster is a fractal, its mass is related to its linear size as {{formula:91d8149d-7314-49a2-ac2f-1ebbbdef25cb}} , and {{formula:d681bac4-0e9b-4471-abc1-a9987a3b50bb}} is the fractal dimension. The fractal structure of the giant cluster is illustrated in Fig.REF .
{{figure:cb4419f6-8c07-4a23-bfce-8db236377e49}} | r | 0be80fbcb899a21810f1014f760bed14 |
This phenomenon of preheating is typically considered after the slow-roll phase (see, e.g., {{cite:2ef94c4ccd2a0e0791bd1ddaae2993afdaea4811}}, {{cite:034c80c3b964a045e546c9b4a1765718d5b0148a}} for a review), but it can also affect the dynamics of the Universe during inflation in the multi-field setup {{cite:b0efe02b3bb30a995e2c62922d21f608d1403992}}, including a premature
end of inflation. There are also other mechanisms which can end inflation prematurely or change its course.
In hybrid inflation {{cite:0bcee8beb97714b6f8e852cedd461cbca50e4c8e}}, {{cite:356cbfeabddd068686294cd996c4c509c3ccb106}}, {{cite:3d4d99569e9ced60ad62e9786b649bb07c3389b0}}, {{cite:ee37e7e0d8a8180b1c70cc9d8556902af6ab5640}},
there is a `spectator' field with an inflaton-dependent mass. This field develops a tachyonic instability which either quickly terminates inflation or the field takes over the role of the original inflaton field
{{cite:a03a63ef41fd6c070b12eb29072db28847bdc600}}, {{cite:3277d05d5c62aacdebb4d63bbbb1abeee63eeee7}}, {{cite:c15e0022f773823555e43b69df29a2f70c9111de}}.
| i | 1e0bf29d6f280e97b7b2aadb60c7dd19 |
Scalar field (SF) configurations in General Relativity and its modifications are interesting for several reasons. Various SF models are extensively used in cosmology {{cite:47e9fb7ccf19d928618c887c5fc1371aa26f1ce4}}, {{cite:b166499dbfca72dd222080022017fcc4e2bf6607}}, {{cite:dc63021dd2f0f64cf2f4db46353a649084dd8713}}, {{cite:93a9c02ba587a841bcae1077909e0357b0114ed7}}. Some propositions to relax the well-known "Hubble tension" involve SF in diverse approaches to dynamical dark energy (see {{cite:e0bea55baf3354719e45077ae389ed3497363fcd}} for a review). It is currently unknown, whether the cosmological fields of the late epoch (if any) are the same as the fields that caused the early inflation, or they are of a completely different nature. In any case, the question arises about possible manifestations of SF in relativistic astrophysical objects. Interest in alternative models of these objects has increased significantly after the image of the accretion disk in the core of M87 had been obtained with the Event Horizon Telescope (EHT) {{cite:f3afb16c18485c2cae8e57b74c505402e488746f}}, demonstrating future prospects to distinguish the black holes from their exotic mimickers.
| i | ba826506ce4db58173d2f595d06b5fa9 |
Price of interpretability.
From the literature that we have reviewed, merely 24% of the studies adopt interpretable models, while the majority of them focus on applying post-hoc explainability.
This may be due to a long-held belief that there exists a trade-off between interpretability and performance, exemplified by the Figure 1 of DARPA's XAI program announcement {{cite:3bce21e4994694dca9313f48a7d2200d0f81ed51}}.
However, some works have provided counter-examples showing that interpretable models can achieve performance similar to black-box models {{cite:20760fadb1dd1ce7a259f658708b40d053b810d7}}, {{cite:5187b5bf8d0f133c2d5095b920d0d7b1e774b7a2}}.
The trade-off between explainability and performance, known as the `price of interpretability' {{cite:00165f8d173d3b2292195e01aa22313762e83881}} is a much-discussed idea in the general ML community, which we believe requires special considerations in cybersecurity. We believe that the presence of an adversary and the prevalence of spurious features will likely make this trade-off less pronounced compared to other fields.
Further research is warranted to identify how big the `price of interpretability' actually is, and under what circumstances it applies.
| d | 1f6632891d557c4989d300ebf303801d |
Visual object detection and tracking become a very challenging problem due to several factors like (i) low-quality camera sensors (including low resolution, low bit depth, low frame rate and color distortion), (ii) challenging factors (like tracking non-rigid object, tracking small object, tracking multiple objects and tracking pose varying object), (iii) requirements for real-time tracking, (iv) multi-view object tracking and (v) variations in object appearance due to several complicated factors (such as illumination variation (Figure REF (a)), background clutter (Figure REF (b)), partial object occlusion (Figure REF (c)), full object occlusion (Figure REF (d)), large variation in object scale and orientation (Figures REF (e) and (f)), partially camouflaged objects (Figure REF (g)), pose variation (Figure REF (h)), shape deformation (Figure REF (i)), rapid camera motion and noise {{cite:9099e29205baa31d4bbb8ba3850e0004389ab02d}}, {{cite:c3b3284d28d00eef3a74f6afc58af5ac906e9f4b}}, {{cite:5549c82e85d13718e985a29cd1ed4f0daf4c87f2}}. Detection and (or) tracking accuracy may be degraded and even failed due to these challenges. Numerous object tracking algorithms have been developed in the literature to handle these challenges. These invented algorithms with different properties and characteristics usually solve different visual object detection and tracking problems.
| i | 889bf94292cd46f8ebb615ccf505862a |
Overview We propose a purely anchor-free architecture named AFSD which is shown in Fig. REF . Concretely, given a video {{formula:5c631665-ab61-4e9a-b742-c6583c02c3ec}} , we first process the video with a backbone network and a feature pyramid network. Take RGB frames as example, for each video {{formula:013b7d02-5797-4f3a-8506-c7f1d5db6858}} , we use a Kinetics pre-trained I3D {{cite:6ffaebef34b5657c65e0368a6ce45af2aba71d50}} model to extract a 3D feature {{formula:30b9d276-a051-473f-89b1-751a8067e246}} , where {{formula:39493751-10c7-4021-8b25-51ae26e1c03e}} denote the time step, channel, height and width individually. This feature is afterwards flattened along the last three dimensions to a 1D feature sequence. Such a sequence can contain the temporal and spatial information of whole video. We then exert a feature pyramid network including several temporal convolutions, of which the detailed architecture is shown in our supplement, to merge the spatial dimension and aggregate the temporal dimension in different levels. The pyramid features are further utilized to generate a coarse proposal sequence {{formula:d8740511-21d5-4238-a10a-346e526487d4}} with a basic anchor-free prediction module (Sec. REF ), which includes a simple regressor and classifier. After that for each proposal, the predicted temporal regions are employed to get the salient boundary features with the boundary pooling (Sec. REF ). The boundary features are exploited together with the feature pyramid to output a fine-grained prediction {{formula:b77acc88-7766-4556-a7d3-d1b8e941ce3a}} for both temporal regression and action classification.
| m | 5e244451157dc363a7bcce0aed0fd8dc |
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