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Developing an ideal emotion model for AEI is a complex problem. Existing models either do not provide enough coverage {{cite:313a51cea15a32262b08a56b60ddd73cfe6e48d4}} or include excessive, overlapping labels to describe the space {{cite:98e677cfaed65973834486a5d3453ecb5532a35b}}, {{cite:cf18193a8e73e17a283e7d80bec6fd536112662a}}, {{cite:04e817e49f4dfa8d3d4fd1c4a8a72ced3728c215}}.
We provide a visual representation of an emotion model's coverage so the power of different models can be compared (Fig. REF ).
These are generated by taking the FastText word vectors {{cite:3c3d1139e3a9117853b558ed11849065b7d40591}} for 1,720 emotion concepts and projecting them down to two dimensions using Uniform Manifold Approximation and Projection (UMAP) {{cite:205d8f9aacd3059aee3596a0fb443b6ad203d7c7}}. This dimensionality reduction technique is similar to t-SNE {{cite:7825361b2257f8e263fe38224daca8d147760c3f}}, but UMAP has also been shown to help preserve global relationships. We then generate the heatmaps using the maximum log cosine similarity between the FastText word vectors for the model and our emotion-concepts list.Since the distribution of the maximum cosine similarity across the entire list is exponential, the log of this distribution is taken to assist in visualization. This step allows us to visualize the coverage of each emotion model in relation to our emotion concepts list. More details will be provided later.
| i | aa2e7ae9ef18863a68317ae4343b72b1 |
{{formula:e9cf07c1-88c6-4c38-9b66-837dcfb4425d}}
classic All samples of target domain (size is 2000) are positive samples and texts generated by DARL with same size are negative samples. The processes of pre-training and training follow the work{{cite:5e5a833ed233e67ac1d0ddeba59c12064bc131e0}} with {{formula:4806a2ef-dff5-40ad-9b46-9622eeb1c5e5}} .
{{formula:9507a1b8-9278-4a0b-9802-4accb2c7d212}}
transition The {{formula:597d51d4-4813-40bd-9006-08a1bf317d85}} samples of target domain are positive samples and texts generated by DARL with {{formula:086e5ad5-7847-4c06-9c41-dc51c27af18d}} size are negative samples. The processes of pre-training and training follow the classic method. And positive samples are kept same in each training.
{{formula:250ec4e7-eb2a-479f-acd9-112e0cee16cf}}
few-shot The processes of pre-training and training follow the work{{cite:10e09a774b6861b2b39ebfe7a6b28693773d8c26}}. In pre-training, CNN is trained with the source domains. At the last iteration of pre-training, it is trained with the {{formula:389b510e-2397-45f3-9b71-d3e44197c31c}} samples of target domain and {{formula:3fca61ae-403d-48a8-83b0-f653e0966d9f}} samples of DARL generated texts.
{{table:50614eb7-be5c-4ad7-b847-9d6d89917e63}}{{table:224cb283-9102-40ce-ba5d-65770946c987}}{{figure:f6a7808f-3b7e-4bb8-919c-4295fdc532ac}} | d | d5aec150121bd353e338f6c0cf388d30 |
In this section, we present the results of utilising a Punzi-net in a search for {{formula:e8bb20f8-5568-4b5b-814f-83e1ae9fedb7}} signals amongst various common backgrounds found in {{formula:141c4972-345d-41e1-b7e5-b3dce80659ee}} collider experiments. At the Belle II experiment, this search was performed with the commissioning data for the specific case of invisible decays of the {{formula:bd1c2e5f-a495-4668-8dfe-ead8fce1fd93}} boson {{cite:933fb2eb2ede6846c87c286c6fcfba3172b905fe}}, a final state in which only the two muons produced by the electron-positron annihilation can be reconstructed. All information about the production and decay of the {{formula:7c67587a-a932-4054-a81e-1415c0c6bdc0}} boson is therefore to be inferred by the two-muon system.
The signal events are generated with MadGraph 5
{{cite:f7c4c9c5fb248ba6ef74b8dedfcd13bf217a6976}}
for a range of candidate {{formula:a7ae5d54-5df9-4ba1-9318-584f3e361d4c}} masses, spanning 0.18.7{{formula:3a90a8b6-21ed-46a5-bfe2-e3a3a5c62eb3}} in steps of 0.1{{formula:398ff93b-d509-4db2-b9fa-3ab36c9c14f8}} with 20000 events produced at each.
Additionally, MC samples for the background process {{formula:5c87599e-db46-46b9-bcbf-04cc4b4bdecc}} , {{formula:826a4732-e996-45b8-8613-e44ea194a109}} and {{formula:79c26eef-bf30-4923-b446-1cb300b41210}} corresponding to 1000{{formula:fb54445d-12f4-441b-a649-8a6a35a986e6}} were used, since these can mimic the signal.
The simulation and reconstruction of the events was done using GEANT4 {{cite:369987ac8b762d7205ee5f2bca10e63b2f38f7ca}} and the Belle II Analysis Software Framework {{cite:d2cf4c5ae3cb0a8fbd3f7ad31ae6999cf1833c87}}.
The analysis is conducted via the search for a peak in the distribution of the squared mass recoiling against the two-muon system. An excess of entries beyond that of the expected background at a given mass would indicate the presence of such a {{formula:148c52ea-4bc3-4ddc-b948-6138125ceded}} particle of that mass. This distribution is divided in (potentially overlapping) bins with bin widths corresponding to {{formula:3beef50e-f46f-45dc-8dd3-8509b0d292ee}} of the fitted {{formula:58fdf6de-f0d0-470b-a36f-69094f9a3b54}} signal distributions.
| r | af8c28e1978162eb2d48a853d217b861 |
For this task, some of the most successul methods are successors of xNetMF: SEGK and RiWalk. Both methods generalize the structural connectivity measure between nodes beyond degree alone, which RiWalk notes can be ambiguous {{cite:f28b25ac9ab40678652b00dab6a3b682a9686990}}. In particular, both methods use the Weisfeiler-Lehman neighborhood aggregation method, a well-known heuristic for graph-level similarity which has its roots in a graph isomorphism test {{cite:6e540f21056d2406d53692cd1ee541925ac486cd}}. The neighborhood aggregation process iteratively relabels each node, capturing degree-based statistics in early iterations but gradually building up higher-order information.
| r | 283b44663dac85fd58132a66892684fc |
Transfer learning and fine-tuning aim at reusing the pre-trained weights of a CNN as an initialization for a new task of interest. The CNN model Inception-v3 {{cite:cbfceea48338288f9d7d5fde51f47e046f356101}} devised a new module named "The inception module" which is a 4 parallel pathway of 1x1, 3x3 and 5x5 convolution filters. The architecture of this module allows the model to recover both local features via smaller convolutions and high abstracted features via large ones. And because of the parallel network implementation in addition to the down sampling layers in each block, the model's execution time beats other state-of-the-art CNNs.
| m | 4a70b38725b8ea972b91013b64543a15 |
Deep neural networks (DNNs) achieve cutting edge performance in many problems and tasks. Yet, it has been shown that small perturbations of the network, which in many cases are indistinguishable to a human observer, may alter completely the network output {{cite:4d4a3b15cf406bb3e1ed9518548b8746569a1030}}, {{cite:8732f4e51adec2aadf0aba8f298a89349bd077c6}}. This phenomenon pose a great risk when using neural networks in sensitive applications and therefore requires a lot of attention.
| i | 27cc407b6a13c9c8e6db9b243e1531e1 |
A major challenge that arises when maximizing the objective in Equation REF , particularly in applications with high-dimensional spaces, is that it becomes trivial for each skill to find a sub-region of the state space where it is easy to be recognised by {{formula:1b41db0c-77e7-4491-af0e-4801153c4aec}} . In preliminary experiments, we observed that the existing methods discovered behaviours that covered small parts of the state space. For the HalfCheetah environment {{cite:15bd224892575bb2bf846515a43d986c8af35370}} this resulted in many skills generating different types of static poses (see Figure REF ) and not many skills exhibiting "interesting" behaviour such as locomotion.
| m | 3e66d9c946c51f7339d7199cd595bc28 |
Prinzipiell basieren alle Analysemethoden für Big-Data-Probleme auf etablierten Datenanalysemethoden, die seit Jahrzehnten erfolgreich eingesetzt werden. Diese Methoden stammen sowohl aus der klassischen Statistik {{cite:d429ddb16de6f9bd94bff970e65a718fce4859fc}}, {{cite:089384a5d862c6b1b10347b91e8ba633c0c14ab2}} als auch aus dem Maschinellen Lernen und der Computational Intelligence {{cite:5bc4979533312e36dc5f5680a087acf4d680754b}}, {{cite:46dc8698bd6b70c4f1dd59745b00f8068373c04b}}. Beispiele für Verfahren des Maschinellen Lernens sind {{formula:6d618e0e-e3a0-45af-aa1f-cc0b390dba83}} -Nearest-Neighbor-Verfahren (Analyse der AusgangsgröÃe ähnlicher Beispiele, {{cite:505729dcd4d079ecf721f9111896c76c88cdcd8c}}) und Support-Vektor-Maschinen (Suche optimaler Trennflächen in höherdimensionalen Räumen mit integrierten nichtlinearen Transformationen {{cite:d5c8e385ca9dad364288479e0cd652eb7b57da37}}). Zur Computational Intelligence gehören u.a. Künstliche Neuronale Netze {{cite:f60cc9694bfe9347467e647a50cee47221d9280b}}, die biologisch inspirierte Verschaltungen einfacher Verarbeitungseinheiten zur Nachbildung nichtlinearer Zusammenhänge nutzen, und Cluster-Verfahren zur automatischen Suche nach ähnlichen Beispielen wie Fuzzy-C-Means {{cite:8273a34d21dc1c011d26173e58cfbd837fc81037}}, {{cite:1fa1fafb772f81ec598f06708ab4537865c4ca87}}.
| m | c0904a00a6272723af303f997db09f11 |
Anomaly detection has always been considered to be a difficult problem due to its subjectivity. {{cite:04c6e0eb97992ea40e111e82f921f84bfa41ca7a}} clearly stated in their classic book on the subject that the major problem in outlier study remains even after surveying the vast literature: “It is a matter of subjective judgement on the part of the observer whether or not he picks out some observation (or set of observations) for scrutiny ...what characterises the `outlier' is its impact on the observer (it appears extreme in some way) ...when all is said and done, the major problem in outlier study remains the one that faced the very earliest workers in the subject—what is an outlier? We have taken the view that the stimulus lies in the subjective concept of surprise engendered by one, or a few, observations in a set of data ...(and that) the concept is a human one". {{cite:04c6e0eb97992ea40e111e82f921f84bfa41ca7a}} foresaw the difficulty of translating the problem into a mechanised form and that “trying to teach the computer what is surprising is difficult".
| i | 9b09001337055f146ace231c9b9ea64e |
Although mathematically simple, the {{formula:94f627f8-4266-4011-8f0e-a4b410e483e5}} CDM model provides an excellent fit to a wide range of cosmological data. However, an exception is emerging in the Hubble constant {{formula:dc7a7733-845a-4147-8398-2c73e8e37a26}} . In 2018, the Planck satellite measured a {{formula:84ca02e7-52e8-44c4-a952-237d743c3f54}} value of {{formula:9a8f0ab8-acb3-4fc9-876c-6d59928ab918}} ({{formula:5de5c283-3e8d-4bac-8cce-13a303002e51}} level) km/s/Mpc from a {{formula:4dfe7ac7-d60a-4eed-ae73-14c16a60d0a8}} CDM fit to the CMB {{cite:a421bf321e35f0ce9b80af6ae986039a697b1d24}}. In 2019, The SH0ES collaboration yielded a latest {{formula:2e55ef52-116c-4df2-b409-baa53954e27f}} value of
{{formula:7fdb4f06-e15f-4ee2-977e-8e30bc6f3bd9}} ({{formula:97afd30e-4e60-4136-851a-60f4e5a91a8a}} level) km/s/Mpc from direct measurements by using so-called standard candles: type Ia supernovae and Cepheid variable stars {{cite:9332d6e2b941e3e87b9ea4deb782bef3e2f2e67e}}. Recently, the H0LiCOW collaboration obtained a {{formula:9daea849-80e0-49d3-b930-d6e13c58c8a8}} value of {{formula:6d10aa37-6dc4-4561-bda6-b728a1da63f8}} ({{formula:2fd55482-a3e3-4140-8698-daa91a8cb8e3}} level) km/s/Mpc by using gravitationally lensed quasar {{cite:ea151584dc09bb2af913e1bcf872b9918ba095c0}}. Combining the SH0ES
and H0LiCOW measurements gives a model independent {{formula:9236517e-d966-4459-9538-7fc0d355032c}} value of {{formula:5c2106bb-c537-40f6-a56b-9268eadb3076}} ({{formula:44e260a6-d67b-4113-81cc-823c11a30b38}} level) km/s/Mpc, which is in 5.3{{formula:1a1d6921-2a47-4c92-955b-7a7f368cbda9}} tension with the {{formula:c44fd929-6428-46a8-acf6-be2c0e043f29}} CDM prediction.
| i | 695131850d87785c8c84d0f92d4bd3e2 |
In this paper,
drawing on the analytical learning theory {{cite:3e16d5d48956f5f81a37e0ee603b1a59ad7d920e}}, we rationalize that
1) disentangling and 2) self-ensembling over the stochastic latent space
will improve the generalization ability of the model.
Based on this rationale,
we investigate using unsupervised disentangled representation learning
as the stochastic input embedding in
self-ensembling.
The presented SSL model
consists of a VAE-based unsupervised embedding of the data,
followed by a semi-supervised
self-ensembling network utilizing
the stochastic embedding as the inherent random augmentation of the inputs.
We evaluate the presented SSL model
on the recently open-sourced Chexpert data set for multi-label classification of thoracic disease using chest X-rays {{cite:b768a8e7b34aa990edc3c1cf3e977d70cabb5f63}}.
To demonstrate the benefits gained by exploiting the stochastic latent space in self-ensembling,
we compare the performance of the presented method with the standard self-ensembling method considering different image-level input augmentation methods {{cite:d2732c62b3ed880f6f33e80653b4114974d6ff0c}}, VAE-based embedding with and without a subsequent deep generative SSL {{cite:af93cf1cef3305cb20327bd671ff1e960d2e1b9f}}, along with a generative adversarial network (GAN)-based SSL {{cite:843fd52a671ee648af8479a675af4d70279b1d0d}}.
We further qualitatively demonstrate the disentangled representation obtained via unsupervised embedding, and discuss its use for data analysis and model interpretability.
| i | 94f4757bb0dcb05af259124cf71aaf1c |
The algorithms in this paper bend themselves to a broad family of problems which have been considered in the quantum computing world. We have illustrated the solution of time-dependent partial differential equations, but the same techniques can be extended to stationary problems. This way, the MPS-simulated quantum register becomes a natural tool to solve the Poisson equation {{cite:48d9f753b481f0a1fbff5c3a4aa3a2a50a91fbe1}}, the wave equation , the fluid equations , or even the Schrödinger equation itself. This implies not just abstract, fundamental studies in Physics, but practical applications in fields such as aerodynamics or finance. We expect new applications of quantum-inspired finance that reach beyond the state-of-the-art , providing new schemes for evaluating financial products {{cite:c77bca3b10107ac9c08fd72319f27c2ba5635563}}, performing risk analysis {{cite:3f3384eab4c4f3f1b10a9cd8f4359b01e9aaf472}} and even more sophisticated time-dependent simulations and tracking.
| d | 278f932aafdd3af91ec3a470ce52b906 |
Lastly, a fertile ground for exploration is extending
BORE with classifier designs suitable for BO more sophisticated paradigms,
such as in the
multi-task {{cite:131a6cb37524b1595f790d8c99b0e97c760e0a0b}},
multi-fidelity {{cite:eb371cdf56a0da15458baf1daf906759e72142dc}}, and
multi-objective settings {{cite:bda81dc0a8f13d8d52e0d3fb34cd091b106b0940}},
in addition to
architectures effective for BO of
sequences {{cite:964f7073c0081357b10f0aa5fa36c6e8d22e33b5}} and by extension,
molecular structures {{cite:70266fc027280d0ea721c05fc0baf22c6f651c29}} and beyond.
| d | c8a15abef7867b09642dd447200288d8 |
may be computed explicitly, to become part of integrable probability. Here {{formula:79c30f4d-ff23-4c96-b2a3-6aafefc36ee3}} stands for the time 0 whole-plane map from {{formula:a66655da-24c3-4815-803b-60e64991c211}} to the slit plane in the corresponding Loewner process. Note that complex values of {{formula:0832193d-06b7-4316-a875-f195e7c06c14}} are considered here in the case of whole-plane SLE with drift. In agreement with the citation by P. Painlevé above, the suggested passage by the complex plane will help us discover the precise form of the associated integral means spectrum in the case of SLE{{formula:69a5c8f3-9008-4c12-8eac-6f6e8f24ddf3}} with drift, via its complex and generalized versions {{cite:225772f035cbd4b671c60a78aab335fd993de60c}}. We shall make use of the so-called Liouville quantum gravity, in the spirit of {{cite:8788c75d02f9b27da6ad9df16a1e3ab7750684f7}}, {{cite:e17aa5467d8589d368dec3067e7ec7e734f10b86}}, {{cite:99487b3d748f3bfa5f4ba4b4539caa4a2e41909f}}, {{cite:b54d62e145b160739c6753661002b905558949fe}}. For the generalized spectrum of LLE processes, we shall concentrate on {{formula:02aaf29c-e444-450b-9c4e-5914f7529fc2}} cases. Precise definitions are given in the next sections.
| i | 36a45b89318abb84fe7cb161a4841378 |
Our techniques have certain advantages and disadvantages compared to simulation with the stabiliser formalism, and its elaborations in Refs. {{cite:05362caed4f0d8a63e1cbfa1fef482ae01d3317a}}, {{cite:0516594436d0970c40ccb0535884cd1d4561d58d}}, {{cite:31b41c315383ca3746e4b0d29b3a1ebfd8a6b845}}.
Those techniques uniformly require {{formula:ab9108df-135a-41e9-80e3-2ba55008dfe1}} time to simulate unitary operations such as {{formula:a63b3477-4bb5-4842-9ffb-256bb2437090}} , {{formula:1b59e6d5-cfd7-4d8d-8c33-34c8fac913d1}} , {{formula:38db2b86-b91c-4fd3-922f-a21254f388c7}} , and {{formula:75f7939f-f22f-4c61-acd6-8f6e5c8d8dc3}} , whereas in the case {{formula:302d3d03-3e77-45f8-a7fb-c0779b42785b}} , our techniques frequently require {{formula:0fcdd9ac-e079-452d-81d1-523c04e0fbc1}} .
Our techniques are no worse than the stabiliser formalism for simulating measurements, as stabiliser-based techniques require time {{formula:36597cea-fa7e-418e-952d-2c2356923ad9}} in general to perform a weak simulation of a single measurement.
More notably, in the case of strong simulation of a few qubits (or of nearly all of the qubits) in the {{formula:3388de6c-b0e0-4e64-9b56-74c51e2ab8a6}} -basis, our techniques provide asymptotic improvements over the stabiliser formalism for {{formula:aaac9164-c964-49e1-899b-35cbc669e45d}} -basis measurements with random outcomes.
Furthermore, whereas Gidney's refinement {{cite:31b41c315383ca3746e4b0d29b3a1ebfd8a6b845}} enables {{formula:b36dfd5e-f6bf-43a5-afef-d553cb6e4ae9}} -time simulation of multi-qubit measurements with deterministic outcomes, our techniques can simulate single-qubit measurements with deterministic outcomes in time {{formula:27333054-c7ed-4d28-b506-33115207e9bd}} .
As we describe in Section REF , our techniques also extends to {{formula:f2c424fc-b84f-4b47-8483-5309e7e2fa00}} -time simulation of fault-tolerant syndrome measurement of `local' Pauli operators {{formula:f3c1e952-58e1-42cd-9c45-8d3a96f6e8e6}} under Pauli noise models including ones which yield non-deterministic measurement outcomes as part of the process.
Thus, our techniques may prove more practical for simulations in which such measurements are frequent.
We speculate that the setting of operations on error-corrected qubits may also have enough additional structure to allow simulation using sparse data structures: this may provide further opportunities to improve the simulation complexity of these circuits, using our techniques.
| d | da8590cfc6d336de2f15d2fa3ff70b18 |
Ablation Study.
We conduct ablation studies on several vital components of the proposed method: dual-path attention network (DPAN), dynamic deep linear kernel (DDLK) and DCLS deconvolution. The baseline model uses DPAN architecture and estimates a single layer kernel (SLK), and adopts the kernel stretching strategy following {{cite:fe77bca8665e80d2452d589d7db075ed84e53179}}, {{cite:c967de1a9f1ee60050ba0e5927dac7daf5c3af71}}. The quantitative results on DIV2KRK are exported in table:ablation. Note that the baseline model with DPAN eliminates artifacts from kernel and thus improves the result. And the DCLS deconvolution can further make use of the estimated kernel and high-level information from deep features to achieve a higher performance (+0.15dB from baseline).
| d | fbc34dd11fd668f63234db5cc9ecef9d |
Based on the above two key observations, we introduce an interactive explanation framework, CX-ToM. Unlike current XAI methods that model the explanation as a single shot response, in CX-ToM, we pose the explanation generation as an iterative process of communication between the human and the machine. Central to our approach is the use of Theory-of-Mind (ToM) {{cite:6d8fab74d5be2873061ca4d9086247cfb52139e2}}, {{cite:ac469db7201a31d6b2ea83832525e75008397bc2}}, {{cite:199d1327143b2906571d4384d7c00fca197ccb22}} in driving the iterative dialog by taking into account three important aspects at each dialog turn: (a) human's intention (or curiosity); (b) human's understanding of the machine; and (c) machine's understanding of the human user. Specifically, in our framework, the machine and the user are positioned to solve a collaborative task, but the machine's mind ({{formula:c97554ca-d57a-4101-99a4-c2584607c095}} ) and the human user's mind ({{formula:5b180da8-3c8c-41fb-9e0e-b3fc3210556a}} ) only have a partial knowledge of the environment (see Figure REF ). Hence, the machine and user need to communicate with each other, using their partial knowledge, otherwise they would not be able to optimally solve the collaborative task. The communication consists of two different types of question-answer (QA) exchanges — namely, a) Factoid question-answers about the environment (W-QA), where the user asks “WH”-questions that begin with what, which, where, and how; and b) Explanation seeking question-answers (E-QA), where the user asks questions that begin with why about the machine's inference.
{{figure:be25f3ec-a2ff-4242-a330-2a9582b39108}} | i | 1b930ea9ddf751f8c11209eaeee4f5dc |
However, other co-citation frequencies do exceed the seemingly modest frequencies noted for delayed co-citations. For example, {{cite:c70f387f2c4f14095f7c84fb3f61fba88160272d}} and {{cite:b3f49258c84b8767192a047ae6ed685f78cb6af9}}, a pair of articles from the field of physical chemistry, have been co-cited over 51,000 times but do not exhibit delayed citation kinetics. It should also be noted that these articles have individually been cited over 70,000 times each. Similarly, 1,357 pairs from the data shown in Fig REF have co-citation frequencies greater than 1,000.
| r | b785fa0deaee1eec6395ddf88e529a7a |
With more accessibility to computational power, several variants of BERT have been proposed for different domains. For example, BERTweet {{cite:b175824ad6c15b6120cc13ea4fbfad38c0efa7e3}} is a bidirectional transformer model trained on twitter data and can be particularly useful for analyzing social media data. CT-BERT {{cite:d06dd06c84bb0aa9ae93030501bdea03595e2de2}} is another domain-specific variant that was specifically pre-trained on Covid-19 tweets. Before transformers changed the landscape of NLP, sequence-to-sequence models such as the Recurrent Neural Network and Long-Short Term Memory (LSTM) {{cite:85531ec1c9cf176e5681b72810116bf75513a0d5}} were the state-of-the-art, beating out nearly all other trivial approaches. For the task of question answering, one of the first sequence models proposed was MatchLSTM {{cite:7d6f1061da4c413c8a04a46acf883b4176998307}}.
| i | 7bd3532daecfbfaf99e3ea07e75bb74a |
In this section, we review the fundamental rsults for abstract linear evolution equations by semigroup theory; see e.g. {{cite:1759d685cd81b039e14a84e5ac642fc857a4c595}}, {{cite:732b5162b824c3a21e73e3e9b6642d5465a1f585}} for more details. We consider the non-autonomous Cauchy problem (NCP) as follows
{{formula:08940bc4-5ed3-4ea8-991c-1bda29155427}}
| m | 8e3edc923df1d6eaa5b4914601c62702 |
In addition, we can explain the latent space of our VAE model by a post-ad-hoc method: mapping an observation to the latent variable and reconstructing the observation by a sample from the neighborhood of the mapped image {{cite:76aa9857f6b1ab61340a07cc6cb053756438b062}}. Since each observation has its own representation on the latent space by the proposed encoder, we can easily compare latent features by investigating a set of observations of interest. By this post-ad-hoc method, we discover that only a part of the latent space determines the characteristics of generated samples. That is, the encoder of our VAE model selectively activates the latent subspace, and the generated samples from our model substantially depend on the values of the subspace. Thus, we call the encoder of our proposed VAE model EXoN (EXplainable encoder Network) by borrowing the term in the field of gene biology. Figure REF shows a series of images obtained by interpolating two points on the latent subspace which are detected by the EXoN.
{{figure:6a0fa77e-2468-45b3-ad84-d04bebd30d71}} | i | 80618ac6ce7a07b29aa3b6c305dff3dc |
Although these two model classes can generate similar sets of neural trajectories, different approaches are typically used for fitting them to neural data: parameters of LDS models are in general inferred by variants of the expectation-maximization algorithm {{cite:deb3702c9d9ce5009eaf80b879813376c6f1cef4}}, {{cite:3d9c813c6a552d2790394d4a1350a381d742cde8}}, {{cite:7e0a87ccbf8c8522b1ec36449f7fa7f5cece45dd}}, {{cite:199ed8701777b3e0b41f2222608f4686d35d022d}}, which include the Kalman smoothing equations {{cite:507703929ec3855366e2647d5f91dcbffbc1e894}}, while RNNs are often fitted with variants of linear regression {{cite:88430843661a874a71839f5f1020bb4be51e9690}}, {{cite:75520a9c8beed43324e5d504421f58585fc0d9e8}}, {{cite:2c512dc67f27a49cc0c8f145e838e55bac030136}}, {{cite:935a07484ae0e553e3ecc4c2f0d629972ce9bc39}} or backpropagation-through-time {{cite:550bf984f99d27e62a959ca669d8310dc209ec14}}. The relationship uncovered here therefore opens the door to comparing different fitting approaches more directly, and in particular to developing probabilistic methods for inferring RNN parameters from data.
| d | 3e589e140f4fd209dd2420f1e52dfab1 |
Deep reinforcement learning (RL) algorithms are in principle capable of learning policies from high-dimensional observations, such as camera images {{cite:04b8311caa7bca7dd0eb4c6564ea1d340771404c}}, {{cite:36846e7958e40d5fcd26b154d740356c04a2aa31}}, {{cite:75b20921939ca2e28091ee00eb307f13ec364f94}}. However, policy learning in practice faces a bottleneck in acquiring useful representations of the observation space {{cite:af513fd187488470cd194177e2b2e6a9ad308971}}.
State representation learning approaches aim to remedy this issue by learning structured and compact representations on which to perform RL.
A useful state representation should be sufficient to learn and represent the optimal policy or the optimal value function, while discarding irrelevant and redundant information.
Understanding whether or not an objective is guaranteed to yield sufficient representations is important, because insufficient representations make it impossible to solve certain problems.
For example, an autonomous vehicle would not be able to navigate safely if its state representation did not contain information about the color of the stoplight in front of it.
With the increasing interest in leveraging offline datasets to learn representations for RL {{cite:6b0f5bb40a6810d4037a49a596a015d982ae8585}}, {{cite:092184dacf3d396a14f5bfae270d6fccf6149d6f}}, {{cite:977b6df95868bbaf3774bd52365d343c421effe4}}, the question of sufficiency becomes even more important to understand if the representation is capable of representing policies and value functions for downstream tasks.
| i | d51723d4202c3410eedcbb7b04c5581d |
Recently, some works {{cite:f231965e3ab1c116cc94ece2b6aa1b1c71b23c26}}, {{cite:4565332c0f3e38b110f7770da2747b3146bd8305}}, {{cite:e9a3cf2bc8067e2918945bf923e5ed3e894f84a8}}, {{cite:9f85512db259efd0791844531797f301c1c7794d}} used Fisher information, natural language, active learning and deep reinforcement learning to develop an intelligent interactive segmentation or annotation tool. In the future, it is of interest to use active learning {{cite:e626033e749c44eac62c61794e4a246b52a82bd4}}and deep reinforcement learning {{cite:4c1395458abc3d5e186139cb01fc965ef3d1dd17}} and uncertainty estimation {{cite:a3a1004f5903edd9bd8d8319281ac982e9fd26bd}} to provide guidance on user interactions for refinement, which has a potential to further improve the efficiency of interactive segmentation.
| d | 0e4589eaa18e4e85c9d0f8aa3f9a34ab |
Another limitation of most DA methods is that they are not guaranteed to satisfy the governing equations and conservation laws. A possible remedy to this is to use physics-informed neural networks (PINN) {{cite:3c9230f9c9d4ecaf4cf4d3538ac2f844c42400d8}}. In recent work, PINN has been used for superresolution and denoising of 4D flow MRI {{cite:0f6ce136e1fe6fa1cfced322de7b19dee40ee852}} and synthetic low-quality hemodynamics data {{cite:b7b8fad4d0ffd106bd761bd63bcfbd69567a4f33}}. We may also think of this approach as a deep learning DA strategy where we use PINN to find spatiotemporal hemodynamics. In such PINN problems, the low-resolution experimental data is the data loss in our neural network and the mathematical equations (physics loss) have replaced our computational or reduced-order DMD model. In general, machine learning and DA are closely related, and they are different approaches to inverse modeling and optimization {{cite:b13d2aef5bda8d45e8a69dfd20fc458d0062a09e}}. For instance, variational DA and neural networks have mathematical similarities {{cite:952ba20f49d7136406570e344d1641092f8acfa7}}. Traditionally, DA could integrate physical laws more readily in its framework (the prediction step); however, with the advent of PINNs, it is now possible to easily incorporate physical laws in machine learning {{cite:3c9230f9c9d4ecaf4cf4d3538ac2f844c42400d8}}.
| d | d39c92f538d4b08ec022d5deba47c6ab |
Baraniuk et al {{cite:c4a1b8da3586da5abdb016f98b6c72172e3ad530}} provides a bound on RICs for a set of random matrices from concentration of measure. For these random measurement matrices, Theorem 5.2 of {{cite:c4a1b8da3586da5abdb016f98b6c72172e3ad530}} shows that
for positive integer {{formula:4938913e-63ba-44a7-b09d-0729a4608e14}} and {{formula:93dfbffa-c58e-46fd-90c3-b12f91654681}} ,
{{formula:891df287-eb5c-4760-8df3-a2f8fb8520ae}}
| d | 2408dd89b56460753648ecd91c82d6d6 |
Attention mechanism {{cite:4e09ebb01dcf59a3fd62fa4b4738c37e5172c46d}} has shown remarkable success in human body {{cite:9fc8feb25bc469d3af568eb1cd0b083c6e0c7e7f}}, {{cite:f42afc3ede109fa06da6a33e1bd8a28eb16dc67c}} and hand pose {{cite:15619b5e6ff31f977223fe0fafbc2fbe2df4a4af}} estimation as it can effectively model long-range correlation and aggregate component features. Hampali et al. {{cite:d383a4516c9cf1b85d14bfa91eb6b771bd57d7f7}} propose to learn attention between a sparse set of sampled hand and object keypoints. In {{cite:f94d4a5c6ce309960282679ecc046df10a48fff8}}, an attention-guided GCN is proposed to effectively aggregate vertex features within either hand or object graphs. The interaction between the hand and the object is explored via the exchange of global features during the iterative process. In contrast, we propose to exploit mutual attention between every hand and object vertex that better learns the interaction dependencies.
| m | f81470bb56ac194f3593b7f22f9cde49 |
Theorem 2.5 ({{cite:2e9c37505b4923ccc96bdda0b75e396053cfd891}})
Let V be a complete metric space and {{formula:86c199e7-6995-4d10-a5a5-61ad1309867e}} be a lower semicontinuous functional on {{formula:bc52c497-bf4c-4e00-ae58-e16801a06c71}} , that is bounded below and not identically equal to {{formula:e5d428c4-f8a6-4b24-bb58-b61ca2d68c45}} . Fix {{formula:e8226c29-a7bf-445b-bd2c-65445d4056c9}} and a point {{formula:ac14da16-54d1-4c87-87ec-6d5adcc02bad}}
such that
{{formula:810c5afd-e381-483e-be75-74f998eff7f2}}
| r | 2b76c2671d64d517bbdecdcbe368d307 |
In conclusion, we have demonstrated a strongly localized and almost radiation-free magnonic defect state introduced by a point defect in a on-chip magnetic array that is coupled in a long range mediated by the surface acoustic waves of the substrate. Such a defect state is demonstrated to be even inertial to the non-Hermitian topology, protected by the long-range nature of the phonon-mediated interaction, although all the other states are skewed to one boundary. We find that the local deformation of the lattice constant induces the interference of the subradiant magnonic states, i.e., those collective magnon modes with longer lifetime than that of the individual one, which is responsible for the localization and a much longer lifetime.
The radiation-free and configuration robust defect state may be desired for many practical applications, such as the high-fidelity information storage and single magnon trapping. Our formalism on the magnonic quantum emitters can be extended into the other quantum dipolar emitters {{cite:284677229eb839c24678de28fb2c4a197bbd17f0}}, {{cite:eef6f18ff03a5a18f6490eebecc120fc7fb8b8a1}}, {{cite:f1b27dcb3c044997254459702b736f2658b764a8}}, {{cite:d177b80802503af59f120434d4a15c5a5c5530b5}}, {{cite:548c1337e39907a41c61315fca99ebb3abee8271}}, {{cite:411c39386925cf64d78c7d45abe15d49a81eac65}}, {{cite:3e68d8205dac2aaae013b5267d5f0505005b6498}}, {{cite:8f4568aef4807042172e72f62494ab3fa8da2d3d}}, and opens new perspective on the realization ultra-long lifetime states for quantum memory.
| d | f0f1dad14366de4ce40c3a469f76c6b9 |
To examine and compare representations from different models, we need to design a fair method to learn these models and collect respective representations. To that end we consider three types of models: discriminative – {{formula:f01a7707-7dfc-4451-bddd-e674227965e0}} , generative based on Autoencoder – {{formula:84762e6a-faff-4519-93ab-d00e12745299}} and generative based on Variational Autonecoder {{cite:f37270fd0eb6d37c8c6f99dcc97a02d507f1e4b8}} – {{formula:64f5285e-191c-40bd-bfb3-ee7b0a7c03df}} . To make a fair comparison, these networks share the same architecture, except for the last layer, defined by the model's objective. In the case of the discriminative task, the last layer has output neurons to discriminate between classes. In the generative task, the final layer outputs vectors of equal size as input data. Depending on the model, we train the networks with different objective functions. {{formula:cd93026c-d4e0-40ac-87da-443305a1c4c8}} model is trained with CrossEntropy, {{formula:49e24410-78a5-4b36-9989-c76965d9cac7}} minimizes the MSE and the {{formula:bbd4a9dc-da45-4afc-a6e2-16c26d484b5d}} uses ELBO. As shown in Fig. ,REF we train all models on the same training sequence {{formula:6f1d6b58-52ec-44bc-8491-ffc7ca5480e0}} but with different objectives. The index of particular task is denoted by {{formula:7de63377-9f7f-4c7b-80ba-04af3733462b}} , where {{formula:1f69e21d-31f8-4257-aed1-fa8032a22900}} . To obtain the representations of data for a particular model, we feed the data to the model and collect the representations from the penultimate layer of the model. We use {{formula:b14fd00c-3460-4243-94b5-2f75ea02416c}} , to denote activations from model {{formula:ad24e7b7-9b99-40d5-ac1d-e46a27ab0ca7}} after finishing task {{formula:cc964218-7b30-485b-82b3-8245b105965a}} for input data {{formula:d76ebf53-bc17-4f52-8005-3deddf2d587f}} from task {{formula:c774dcb6-5bac-4332-9a80-ca0fd77f909a}} .
{{figure:ff092b84-b22b-45de-b844-ba06392b5433}} | m | 76cc1ea908f057c5ffd7c2bbee4006b5 |
We now recall second order estimates for admissible solutions from {{cite:695ed9ce09bcd6055bbe4a4219ef48e5b8bcf5bb}} and {{cite:1dc9ec9ee361977d16dd05528d270c1e188d8f46}}.
| r | ad153c7bbe5c562c27ae000511efeb3c |
Proximal Parameter {{formula:a7836c39-8c9c-4d75-831a-b4faf18f5983}}. To counter client drift in non-IID data distributions, FedADMM and FedProx both use a quadratic proximal term for the local training problem. The proximal coefficient {{formula:eb933d4f-783b-428d-84ce-c76b2e015177}} in FedProx has to be carefully tuned to achieve competitive performance across different settings {{cite:972ff870f0a6cd7bdcc6e04fd041439e68369f07}}, {{cite:dfe72652ed6fee19ca51006e17fc0ca348e524cc}}. Such tuning is dependent on system sizes and data distributions, as we also demonstrate in Table REF . Note that the best {{formula:d8289984-2635-40d2-9f28-830d17dbe2da}} value (0.01) for FedProx in the FMNIST dataset gives the worst performance for MNIST in the 200-client setting. Moreover, the performance of FedProx with respect to {{formula:146c67f2-3f21-4ef6-9523-e6261c8cbecc}} is not monotone, which makes tuning even more challenging. On the other hand, FedADMM dominates all tested instances of FedProx with fixed {{formula:260b4249-cda2-436a-b624-f939695dae95}} . This is also supported by our theoretical analysis (Theorem 1 and Remark 1 supports a constant {{formula:2a671ae0-cd59-4017-ab03-7e4aae1f42d0}} ).
Additional insights can be gained by a simple dynamic adaptation of {{formula:29644e47-5a96-4eb9-9b2f-197a7cefe752}} for FedADMM in Fig. REF . A smaller value (0.01) at initial stages of training allows efficient incorporation of local data when the global model is not informed, while an increase of {{formula:491b7dbc-42e2-41ee-97a6-5580c9f05bf0}} at later stages reduces discrepancies between client models and the global model.
{{table:f7fa137b-a514-4363-8e56-22b21a014b6d}}{{figure:952bac9b-fef3-4bc9-900d-9a4d3df3fb50}} | r | 1d9bed5d73e685f952ac77a200ce480f |
and {{formula:01aa3db9-ac3a-4a69-b99b-118d85a2b58f}} was obtained in Refs. {{cite:e10b1cece7e895f5b34f7a4990e4f1ecd9c725af}}, {{cite:e8095ff050b83975954420bc77f7219d392ffcae}} as
{{formula:a9892728-994c-49cf-9803-844ceb5b2614}}
| d | c395d5d6c46b70c1980d0a094e0561b4 |
We shall impose more assumptions on our domain. For both the “free boundary" and the “direct" results, we will assume that {{formula:7f44bafd-f93f-4166-9c65-91db77815b87}} is a 1-sided Chord Arc Domain (see Definition REF ). For the “direct" result, we will rely on the assumption that {{formula:6d995b86-44f3-4a81-bd52-be022681a27a}} is uniformly rectifiable (see {{cite:b4e820037532b5b23061a6e4ceb17a446e9a5032}}, {{cite:1058d6343cdddfcaa8cb26dec94590f472b00996}} and Section below), and thus ultimately assuming that {{formula:8859ebfc-51d4-4933-9b78-0549cbfd3536}} is a (2-sided) Chord Arc Domain. The optimality of the assumptions on {{formula:51057757-e038-4bae-be6a-37b1e4ae0dbd}} is discussed in more details in the end of this subsection. Since the dimension {{formula:795d93d9-8384-4db1-9a26-a152b6799563}} plays an important role in our paper, and in order to lighten the notion, we shall write {{formula:853a407c-86f5-4fb8-9228-42d9d586ddc8}} for {{formula:f71fbd33-90f2-4fb3-896b-8d07715e79e5}} .
| r | 55ca7165ee1a0c9fa4a95242afafc725 |
Effective model complexity is a relatively new, promising and useful problem in deep learning.
Detecting effective model complexity during training helps to investigate the usefulness of optimization algorithms {{cite:dba7cf5ea21a48bde1510caa659d72ccc4bc89a4}}, the role of regularizations {{cite:63ef9ec4c4713c1dc9a15d67e662f1d40bd34de2}}, {{cite:8bb8ad813b093f00ea11209e21f681f594f37c0a}}, and generalization capability {{cite:d7a12449963acac8757379b2300f7b9de1a16b5e}}, {{cite:aa2622cabb4078baeff42ea4d8aeab8192021218}}.
Furthermore, effective model complexity can be used to describe model compression ratio, since effective model complexity can be considered as a reflection of the information volume in the model {{cite:5c1174f0a9587c05f83a64d1d5e2a9ce438ee90e}}.
Effective complexity can also be used for model selection and design to balance resource utilization and model performance.
| d | afe8f447709ed556f23b3be294a9eb5c |
As shown in Figure REF , Figure REF , Figure REF , Figure REF and Figure REF , we provide qualitative results on all the benchmarks, which includes PASCAL-5{{formula:81e9ac9f-788b-47ea-ac30-b39954c7b027}} {{cite:b7daf20240bd8f388508ad25d9fae9ab94b37fca}}, COCO-20{{formula:c95ec50b-2197-4e6e-86b7-89fe67819597}} {{cite:4dc93ac76caab66e89bad69d35e6f55b496537cc}}, FSS-1000 {{cite:1e5351ea1ade734fc1911db88abecadfebaa16cd}}, PF-PASCAL {{cite:e8481382941710272b474d410f4d4852c2713868}}, PF-WILLOW {{cite:4e8483ef4ace861380bc5c0035bc9cb19b625fa4}} and SPair-71k {{cite:a0b06d5bbbb5deb7cbd38cbd596fe7a034fba0fb}}.
| r | e4f5903c980029445e15f811b2492ca7 |
The proposed adaptive distillation method can be combined with most existing distillation methods. We combine the proposed method with ten state-of-the-art distillation methods to demonstrate the superiority of the proposed method, including FitNets {{cite:ee007c375454b286b48c3850188e725586f79254}}, AT {{cite:8e5a3f69e6ac9db9254a7ba3aa5ccd4039231d0d}}, SP {{cite:dca765911cd8de36434119e4af377456e3cc1db1}}, CC {{cite:be699761cce80c9238cd556804dbdd0e38764bce}}, VID {{cite:0764b33606df9d73dfe66c3e863aaf1bdf590824}}, RKD {{cite:0bb30b76139c113d1584a9c273c17d6b2e51c071}}, PKT {{cite:207879f65b21462fdb36bd6dd427817fe6c19e19}}, FT {{cite:177eec1e3c3c7daefc0608368f76cc21e694bc41}}, NST {{cite:fa4651069ff2a86fa515f0b3912884be55d2ff45}}, CRD {{cite:85a486b51fb2352985ddaec6b00ee889441db35d}}. For all methods, we combine their objectives with the vanilla KD {{cite:b37ebaa87c3e7fbfc8445d1645b4734ef8f48690}} objective, the KL divergence between the softened predictions from teacher and the student models, to boost their performance. Thus all the methods involve at least two distillation spots, turning into a multi-spot version no matter they are originally one-spot or multi-spot distillation methods.
| m | 8e46c57cf7d28dc6d2c05ef522e83537 |
A space {{formula:96f7d349-d5cc-4da6-b17a-48575b2c863c}} is called countably compact if any open cover of {{formula:55e42291-e38d-4a6e-9a9a-6d5d5fb8fe60}} has a numerable subcover. Since each Tychonoff countably compact space is pseudocompact (see {{cite:1fcca68422436ee18bf5d7411a799932ad649027}}), then Corollary REF is a partial answer to the Problem 2.5.2 of {{cite:054074a8994340b8aa5512377d2e6730859ea74b}}.
| r | b704269caf4a206e658b0ce2bfac2035 |
Remark 1.19 We should note the difference between our Generalized Mountain Pass Lemma (GMPL) and the following
theorem of Struwe( {{cite:2c565a4539977685e98ebba5a5962966c8645448}}): Suppose {{formula:a4dc569c-1c4f-468d-9ec3-b4fc60a3e914}} is a closed convex subset of a Banach space
{{formula:16c747cf-141b-4b1d-a66e-2939b9288ecd}} and {{formula:95fbe1ce-f16a-428c-816b-c16f4c5518fa}} satisfies {{formula:2a72ef8f-eca0-4378-9e0c-aaa14841b4b3}} on {{formula:f954d537-ba95-4ba7-8d78-747f39b22e8d}} .
Any sequence {{formula:ad921f9e-5c4c-4114-b677-05d73281c2ca}} such that {{formula:380c0f8c-4b1f-4988-b72e-abe1b838323d}} uniformly, while
{{formula:8ce14bb4-deb7-4379-a7ad-fd6a39ab7db8}}
{{formula:a90b940d-0cf4-405f-9928-1382b41fdba0}} ,
is relatively compact.
Suppose further that {{formula:0bc4d2a2-84ec-4d36-b524-e885518324e3}} admits two distinct relative minima {{formula:869b8250-b4e3-4602-8ff1-805c0e39e4cc}} , {{formula:f3a6a7ee-090d-42be-8c14-5ea03e42e2fd}} in {{formula:0bbb4026-c340-42a0-a650-5c51e044a4bf}} .
Then either {{formula:10272b27-86d7-447f-95e9-b6ee8de86369}} and {{formula:bd8811f7-e7e8-44ac-b99f-d8fb4dc23ec4}} , {{formula:745f2761-633a-4125-81f3-0be37ff67341}}
can be connected in any neighborhood of the set of relative minima {{formula:6d9397e3-6999-4c39-950c-4d199efc3d38}} of
{{formula:371bf1d3-521e-4790-8bdb-f1a3fda560a2}} with {{formula:c34fe4db-0560-48e8-be04-00944e947923}} ,
or there exists a critical point {{formula:c3b9f4c3-caff-4908-b7ec-5c968c1b949b}} of {{formula:b66b8e24-64ce-4191-82d0-fd3380b19902}} in {{formula:a2e2b654-f898-46df-97cf-4484cf2ecad6}} which is not a relative minimizer of {{formula:be8f1a8e-f843-4c64-b93a-7c7b51bd9ce4}} .
| i | 3c33d788854147310d0d9658474518b8 |
These are the solutions studied by Landau and Lifschitz in {{cite:9310cb1f6db2f9df38a047c1ec6ac87fada3e128}}.
| d | 37f205722c5f24d8445642da3dab73c1 |
Our algorithm relies on oblivious sorting, which dominates the asymptotic computational complexity.
For oblivious sorting, we use Batcher’s bitonic sorting networks {{cite:e634f494b03e455ebeecc4a942320da50f61da58}}, which has {{formula:3c24c9da-6b28-4dc3-a7e5-5528ac37abc6}} time complexity.
Although it can be improved to {{formula:81dc6701-631b-4a45-8bbe-f263705fadfa}} by using AKS sorting network {{cite:7ec40b7ed9b368e231f04d482f22a84a53fd6679}} or Zig-zag sort {{cite:7b1a836aa1e89bb878ae21103da3d9247dd3c226}}, they have a huge constant hidden in asymptotic analysis, so we use Batcher's sort and denote computational complexity by it.
Note that the sort can be implemented in a register-level oblivious manner by using oblivious swap (o_swap) to obliviously compare and swap.
| m | a92c41eb5f626bd6722d62a484652adb |
We explored white-box attacks neural networks trained with gradient descent. In our experiments on the large QMNIST dataset (200,000 training examples),
Deep CNNs such as ResNet seem to exhibit both good Utility and Privacy in their “native form”, according to our white-box attacker.
We were pleasantly surprised of our white box attack results, but, in light of the fact that other authors found similar networks vulnerable to attack {{cite:cb061565ef402b0964e3d0dc99cbf2183f5a6426}}, we conducted the following sanity check.
We performed the same supervised learning experiment by modifying {{formula:70a9e6b1-e012-456c-9a57-966dc6f4e699}} of the class labels (to another class label chosen randomly), in both the Defender set and Reserved set. Then we incited the neural network to overfit the Defender set. Although the training accuracy (on Defender data) was still nearly perfect, we obtained a loss of test accuracy (on Reserved data): {{formula:d4b47e0f-e842-4732-9485-468a59ade62e}} . According Theorem REF , this should result in a loss of privacy. This allowed us to verify that our white-box attacker correctly detected a loss of privacy. Indeed, we obtained a privacy of 0.55.
| d | 0282d7cce56dd942e76617117c04fdc6 |
We used nnU-net as a proxy for extensive hyperparameter tuning, as it also integrates hyperparameter selection for medical image segmentation. Whenever possible, DL practitioners should use improved hyperparameter optimization strategies other than grid search. For instance, simple strategies like random search{{cite:1b50b3964e64c064db06cc971bde7aeefef9835a}} to more complex methods such as hypernetworks{{cite:55335868a7cdf40833d535832f676db3b211198c}}, {{cite:8f76c02e2c7171e3088dc43e2bffdbb88df64dc4}} can significantly reduce the computation time needed during model selection.
| d | 9ae3038199a2ee4c43a28a52936fc7bd |
By Proposition REF , it suffices to consider the {{formula:e5b4bc96-bc92-4ec9-8d55-5088128d4b7b}} distribution. Let
{{formula:9f2937d8-205d-4c3e-9c9f-6dd94f61400a}} be its density function, i.e., {{formula:622117ef-703c-42cb-9f85-e7dccf3ae1d7}} as in (REF ).
Fix {{formula:49da6c01-d249-4e17-83e3-f151be03c627}} . To show that {{formula:334651a2-663f-4903-aa4b-f8a4260f787e}} is increasing at {{formula:96971229-886f-44de-8f48-fad4516f9ed1}} , it suffices to
show that the log density ratio of the left and right parts
{{formula:1bb2c5c9-5457-41f8-8e03-1c2a9be2dc12}}
for {{formula:b765aeb1-ff0c-46fc-b530-2afcb5704923}} , has exactly one positive critical point.
Indeed, observe that {{formula:f9d82caf-364c-40b5-9bf4-199aa1798fb2}} as {{formula:6cad8dbe-eb4b-4a2f-bf73-bf657eb0605f}} and that {{formula:fa0e3a65-8f13-4815-8eb5-a2305f28579e}} . Moreover,
(REF ) implies that {{formula:e2fb6a7b-1e41-4cd1-aa2b-229be565477b}} cannot be non-positive for all {{formula:02d21edd-f4d0-44e3-ab4c-b145039601af}} .
Therefore, if {{formula:27885c7d-c61a-492e-b095-4df5fbe796e1}} has a single positive critical point, then it must be a maximum,
and it follows that the conditions of Lemma REF are satisfied and so
{{formula:ed7dcb10-1030-4706-9b21-428289074148}} is increasing at {{formula:c89c88d8-633b-4c0b-8ac4-9fd4ba8aadc4}} .
To identify the positive critical points, observe that
{{formula:2952d5ea-8aa2-457b-adaf-5912adf920a7}}
so the critical points are solutions to the equation
{{formula:f761cff5-6edf-468f-8c7e-b9e3344974c9}}
Further simplification yields a quadratic equation in {{formula:7a531c61-66af-4374-a29d-025e841d7896}} ,
{{formula:bb7e97c8-9e14-4768-909e-d20139f14597}}
which has exactly one root in {{formula:54474339-8404-48c4-9b1f-50a1af4445a3}} if {{formula:2dd2f74e-1ffd-4cc5-ae21-24774a5f39d6}} .
This holds for arbitrary {{formula:a2dc191f-b401-4424-b510-f84c335eeb50}} 1>2/3{{formula:77bb0b61-37d4-4156-abfb-08802d2329ab}} 1>0{{formula:5059c560-978f-42ef-96b3-150a24589a2a}} 0=1/3{{formula:900033a7-ee02-4d4d-b31e-e3fa58f1a32e}} 1=12(1-erfc(12))-2
2.17{{formula:780684e5-fa6b-4f53-b67a-c134948823d8}}
Fix {{formula:ac37bf91-b4b9-43e8-a585-4ca090042124}} and let {{formula:4386f0a5-e34f-4fa0-8abd-9361a74256dd}} be the density function of {{formula:7c1ac1b2-83ff-45dc-bcf9-4c7dee67658b}} , i.e., {{formula:3d910f70-0b7a-4895-83e6-771eca400385}} as in (REF ).
Fix {{formula:792b9c8c-d735-4f52-bc0a-2ac0607471d4}} . There are two cases.
Case 1: {{formula:e6833596-7a9d-4599-9ea7-8dc4a104932a}} . Here, {{formula:b66094d2-eda3-4aa0-86db-6f950cca9c45}} is decreasing on its domain and so true positive skewness follows from
Proposition REF and Theorem REF .
Case 2: {{formula:68457426-407d-41a1-ac83-3b35304aa6ce}} . As in the proof of Theorem REF , to show that {{formula:1280f98e-e2a6-40cf-931e-e74372f9ae22}} is increasing at {{formula:4cc3b221-e9ee-4617-ab6e-c27d1d525d2b}} , it
suffices to show that the ratio of the left and right parts
{{formula:cb29f7cd-0c64-4e25-ab6e-76fb3151c321}}
for {{formula:708ec697-53ce-4937-9bfa-9d7592bb585c}} , has exactly one positive critical point, since {{formula:0a60d78a-b004-4025-9c5f-1c440dc4e361}} as {{formula:e6250ae7-5a10-49f3-b2b5-451be1cf9dd0}} and {{formula:efc494a6-da25-4565-9ae6-609e6030597e}} .
Observe that
{{formula:a20d3a47-82ab-45db-807c-ce4fa1d00498}}
so the critical points are solutions to the equation {{formula:43fc3662-ed0f-4620-81ca-3c2f777476e5}} , i.e.,
{{formula:cc466c25-91c5-4a4f-bc58-37d00dfe475e}}
There is exactly one positive critical point when {{formula:d6cf3820-f7ef-4f63-b15e-19297bf96c94}} . Since {{formula:3ea655ef-5767-4de0-8415-2fe30838e6d5}} has support on the positive half
line, then {{formula:23942906-024b-492a-b75f-338d20be45ad}} is always non-negative by Proposition REF . Thus {{formula:c7a96c66-a092-45d3-b57e-ace4de1712a5}} is increasing at {{formula:cae66dd8-228c-4475-9b9c-a27019cc7323}} if
{{formula:13e5c6fe-7ca2-46d5-9b78-153d8bc55ad7}} . Now by Lemma REF , true positive skewness of {{formula:90066a6b-eb8a-4caf-a7d8-083ddee95f64}} follows if the median satisfies
{{formula:2fa25fc6-9924-4aaa-8b64-87d3747b7e95}} , but this inequality is well-known (see, e.g., Sen {{cite:fb31f29b96a056320611cd5396c92dcfa6dcd992}} and the references therein).
By Proposition REF , it suffices to consider the {{formula:466d6aac-5ce5-433b-b44e-72b85f359a8f}} distribution.
Let {{formula:f0626393-25b3-4dd4-b8b0-2049d27deb9f}} be its density function, i.e., {{formula:43f66c7d-ea37-4e48-8b1e-2278ba80dfcf}} as in (REF ).
It is well-known (see, e.g., {{cite:8fca04b3ff6379a63b6fa9876e7cd7ab5aefddef}}) that the {{formula:649a2e31-2897-4cde-af61-73ea6e966f88}} distribution
has finite moments of all orders and is unimodal with median and mode given by
{{formula:48f89fcd-0ec5-49fd-a075-b6b87db91b4b}}
Notice that {{formula:f084ee17-33aa-48bc-b948-36a132152e7b}} holds if and only if {{formula:ab76ffac-2a3a-4b9f-9cb4-aee0475636c4}} .
Since the Weibull distribution has support on the positive half-line, then the second
part of the theorem follows from Proposition REF . For the first part
of the theorem, the “only if” part follows from (REF ).
It remains to show the “if” part. For {{formula:57b8f0f9-d8f4-4480-bf78-3aa4b1f8ac1a}} , {{formula:206b8541-e182-453e-875d-001746e62323}} is strictly decreasing, so we are
done by Proposition REF .
Suppose {{formula:83592004-769a-400c-81a3-b2f3465ecd94}} and fix {{formula:030bf754-81e5-4d90-9374-7305860732a5}} . As in the proof of Theorem REF , to show that
{{formula:1e90fcdb-32cd-4ce4-b751-bb81f6c437cf}} is increasing at {{formula:55e4ce3e-7d59-46af-81fa-bc4d2686def4}} , it suffices to show that the log density ratio of the
left and right parts
{{formula:983d3de7-f709-454f-934b-7ffb835505bf}}
for {{formula:044f0667-9cb2-40b6-9a51-2a91cd5da515}} , has exactly one positive critical point, since {{formula:24f6afa7-a76e-496d-b144-d35ce1820352}} as
{{formula:bfd9b121-e4b8-4142-94a9-244acdc611ae}} and {{formula:251762c5-0801-40f9-8650-bc3e4107e726}} . Observe that
{{formula:69498b9a-2aac-4029-84e3-1a7c401ec5b2}}
so the critical points are solutions to the equation
{{formula:99f4e3f9-6082-4e28-ad37-114b880e981e}}
Since {{formula:e626e9d2-4571-4a9f-8fe6-5393448ddfb5}} , the binomial series for {{formula:4f0cf5a2-4942-40c4-b7dd-7899b82452d4}} and
{{formula:7574531c-942c-4f92-a864-28a8327372dc}} converge, hence
{{formula:61ec0b4c-5937-48d4-b63d-217b831a3049}}
in the notation of the generalized binomial coefficient. Substituting into (REF )
yields the equation
{{formula:aad1b926-dbc7-4465-80a5-64b5f0fa8e52}}
where
{{formula:d197f033-873a-41b3-9f9e-f330d2b32902}}
We analyze the sign changes of the coefficients in (REF ) to determine the number
of its positive roots by splitting into several cases for the value of {{formula:75f2db90-af4b-48b3-abe7-2996d45a3b84}} .
Case 1: {{formula:85988d99-842d-4dfd-85d7-f8816cf7c07f}} . Then {{formula:144e2413-995f-4929-8b8e-2c8e8f9a26cc}} are positive and {{formula:1fd6916d-dc4b-4f6c-8b7e-66fb0306e962}} are negative.
There are an even number (possibly zero, if {{formula:a106dd5b-fb8d-4baa-b959-a87e9e1ebd2b}} ) of negative factors in {{formula:d5c174b3-8a03-4f2a-8c2c-00c8a466a136}} ,
so it is positive. Also note that {{formula:96a0fa10-6e90-4285-befc-d95702768453}} for all {{formula:16f73b52-d093-4cc8-99ed-f4f04ef34be8}} , hence (REF ) is
negative for all {{formula:6609401a-fd0b-4edb-8dee-514b124773d0}} .
For the series expression {{formula:9560c40c-1fcb-4a59-bf5a-3da1c13011d5}} , we have shown that {{formula:2bf9469a-f40e-41d4-8a04-d9b11dc4ce1f}} holds for
non-zero even {{formula:e0b05235-1b95-4b8a-bc8b-2d9c3994ff4a}} and {{formula:5b80d28c-8738-467a-9610-a9bd3dddbe80}} holds for odd {{formula:94a92b93-4384-4b6d-add7-6bf0f7c2b8f3}} . By Descartes' rule of signs for infinite series,
{{formula:2c4fd6f3-a581-4a0c-85b2-82995b6b7baf}} has no positive real roots if {{formula:351c5d29-41aa-4695-815b-35f85fae609b}} and has at most one positive real root if {{formula:e0ded2db-1b20-435d-afab-df3ddfffe13f}} .
Suppose {{formula:ac9da72c-e2d3-445e-9457-488f14c9d20b}} such that {{formula:7ba3192b-9fd6-4103-b958-773ae31f1619}} has no positive real roots. By extension, {{formula:5251a429-a14c-401a-8b5c-0ed1986a1d43}} has no positive
real roots, so {{formula:8eed81b6-c866-4a20-9fe7-e21c1d24212d}} has no positive extrema and is strictly monotonic on {{formula:56a1e51f-1d2f-40d8-8059-e098dde55795}} . Since
{{formula:35395e69-3bd4-436c-8b8f-73043c701d86}} and {{formula:dbe3978d-66cb-4d41-97c2-8eaecfbfd0e7}} , then {{formula:75ee3cd8-ea65-4b55-90de-e1479230ced2}} is strictly negative on {{formula:e2f19923-5e70-4bf1-addf-e8864c442e8c}} ,
hence {{formula:0d720e25-0ac9-4db1-9ee9-0afd9075a577}} for all {{formula:53d49ddc-bb18-40e7-8a44-fa4fe1ae9c18}} . By monotonicity of the integral, this
contradicts (REF ). Thus {{formula:e87ec5ce-a86f-421b-9fa7-12be71e5f9a9}} holds, and {{formula:58ecaf72-8901-4181-b8f4-4ff43f1e3e88}} has exactly one positive root
in {{formula:9a423a2d-5377-45ba-a955-f90a35f19565}} , which implies that {{formula:491fcee3-475d-4424-8f8c-27260df8e8d2}} is increasing at {{formula:e689fd3f-d78b-414c-8fc2-94db92662e1b}} .
This holds for
arbitrary {{formula:e32039ab-2eb9-4190-bcdb-dc38b3bb9bf5}} , so {{formula:fa9faf15-dd15-4925-8ff7-c1ea5d211fec}} is increasing on {{formula:d431a5ab-3172-41fe-a628-1ca7bf7568f1}} . Moreover, since {{formula:b5bee0a4-2714-457b-8046-9a6a2625443a}} holds, then {{formula:db340c14-0e15-4d9c-997c-55a3b6c953d8}} holds for every
{{formula:99b43c6b-018e-4196-9f48-e4131c9c08a2}} , and we are done.
Case 2: {{formula:ade5bf9c-9ff9-449c-8b9e-ffff72ba3ac5}} . The inequality {{formula:8a39f689-5f46-4cf9-90c1-728e28901c59}} still holds for all {{formula:6477b5a4-c1b1-44db-9f48-8bcd96f4592b}} , but we now have
{{formula:31a9d610-0356-48de-9beb-e827f267215b}} and {{formula:df7923ba-ca0b-470c-a6bd-626c3138dfea}} . If {{formula:6505d5a2-e880-4451-a2ee-a1c35ad96f7d}} , then there are an odd number
of negative factors in {{formula:b9153992-6497-44e8-9ff1-96bf9e565da8}} , so the numerator of (REF ) is positive.
If {{formula:ac205e02-5ec3-4d5b-b49f-3b391c9e93c9}} , then the numerator is simply {{formula:85b52182-c572-4e31-982f-be3889f90c6f}} .
Thus our coefficients in {{formula:cd5da964-0425-488d-a0e5-d8ce7179abf5}} are positive for even {{formula:d1a060fc-d026-4de1-8ed1-afe4bfc4f5c7}} and zero for
odd {{formula:90f85ceb-3a77-434e-b517-60bcd01d3711}} , with {{formula:e29f36d4-fefe-4877-94b3-8976cca85c7b}} . By Descartes' rule of signs, {{formula:fb1051d7-b72c-426d-9006-d026237cdcc9}} has at most two positive real roots if
{{formula:475bc118-4dcb-495b-8886-0e5e621f5e29}} and at most one if {{formula:15bd7fcb-7a30-42ee-a2c6-fcb2a7b89e14}} .
Suppose {{formula:a1dac3fe-6ef6-4226-99b3-01a3e78af15f}} . If {{formula:07a47d5d-0cd8-4799-9093-4ca62f2c9002}} has a single positive root, then {{formula:eca53eb0-4bd4-4e16-950f-0514da9a352a}} . But
by continuity this limit tends to {{formula:483187cd-0e53-4a5f-acd2-10d5c9802ae1}} . If instead {{formula:59a9d2ef-56f5-4200-89ed-b48a1838f342}} has no positive real roots,
then by the same argument above we contradict (REF ). Thus {{formula:fb42f25b-63de-4780-bab5-545a53065d52}} . In this
case, if {{formula:e7a5e57c-89b8-4dca-ba00-2e649c324861}} has zero or two positive roots, then again {{formula:65549a96-9890-47ce-85bc-a771543373af}} and we reach a
contradiction. Hence {{formula:0748f123-c8ca-4f81-aeb6-8f7f22f1198d}} has exactly one positive real root. We may now conclude in the same
fashion as in Case 1.
Case 3: {{formula:0520c626-5e6d-407e-9b94-2ce845a47fea}} . We again look at the numerator of (REF ). If
{{formula:8c1b3f61-b121-4c3e-badc-dff5fafdec9a}} , the numerator is {{formula:1f3abf66-4b95-4321-922f-598ec5be41ee}} . If {{formula:1e9bc7e9-7fd4-429c-8dae-4aae5278cd98}} , the numerator is {{formula:fdfc61e4-7ee0-423c-97c7-4ac75cbf0ec1}} . If {{formula:645fa251-98ef-48dc-ae56-f2b466703933}} , then
{{formula:1d88f784-9280-4055-9250-ae6425af091a}} has positive factors {{formula:d0624625-2fad-4e5b-ac42-6a712d45f266}} , and an odd number of negative factors
{{formula:5ebae578-f288-4f21-a069-9a812e5512d9}} . Additionally, {{formula:d0afadf5-d5f4-43e1-9607-bda2a05914fe}} when {{formula:72876c4d-ca6b-4058-a142-433b1b5cfb1a}} , so the numerator of (REF ) is
positive.
Our coefficients in {{formula:50872f88-34ea-4098-b710-a26e1dfbff1e}} are zero for odd {{formula:ef00ddd4-b46b-44e5-9201-c8fd7bdf3bf8}} , positive for even
{{formula:28f1f308-9abe-4c6e-9b59-efdda163a88f}} , negative for {{formula:352daf49-407c-4fd8-bbf6-43acff248ca1}} , and positive for {{formula:34f87b89-926b-4171-b9de-c900ca26b5d2}} . This yields two sign changes and hence at most
two positive roots of {{formula:82468b5f-9581-4884-a694-a0dd2c5fd76f}} if {{formula:04f4d9f5-45c2-4e5a-bf3a-d727fa4d7e65}} . Recall from Case 1 that {{formula:aa79306e-b2be-471f-b125-799f19c5dc83}} holds if and only if
{{formula:078ae632-3e4b-421d-842d-4bbdea4a12d5}} holds. If the latter holds, then {{formula:2e0dc4fb-61ef-4a91-9f6c-4461c00075c7}} must have exactly one positive root since {{formula:ede93216-0187-4e57-a7c4-506a777a3684}}
is negative at the right limit of its domain, and so {{formula:58a06b2f-4078-48a3-9d9e-42368e6a4f29}} is increasing at {{formula:dc9a2d17-92c6-4148-b0c6-7a0588bacb91}} . Now by
Lemma REF , true positive skewness follows if {{formula:2663a596-3033-4043-8ea2-3a9942e7c301}} , but this holds
immediately from (REF ) and our assumption {{formula:91a172e0-b745-43c8-b52a-c19b1210eeef}} .
Case 4: {{formula:51e45bea-3f04-42d7-86be-5859fad69916}} . We can plug {{formula:c51b0256-2efd-4879-aec0-62ea374dfb10}} directly into (REF ) to obtain the roots
{{formula:ae7b32a5-d8dc-4cce-8db7-35158f026dbd}}
The argument in Case 1 can be adapted to show that {{formula:199d022d-eb96-41aa-9535-b23781abfccb}} holds for every {{formula:03507bdd-3266-4e40-a1eb-9646913870d2}} ,
hence exactly one positive root exists.
Case 5: {{formula:42907b34-8af1-40f3-8f58-c337e3a2024c}} . Similarly, we plug {{formula:d7375850-d52f-4113-95df-258212645e0c}} into (REF ) and obtain the roots
{{formula:dd753b67-1e3e-4045-ba98-0a5d545258c6}}
Again, since {{formula:f9a03069-3f74-4024-97fb-eaddef2677c8}} holds for every {{formula:8e9e2a04-a231-4eae-87b7-94ca845c7e93}} , then exactly one positive root exists.
Recall that {{formula:1139855d-97fe-46b8-8707-6000121e1512}} are the density and distribution functions of the standard Gaussian
distribution. For simplicity, set
{{formula:8fef6950-2023-406d-a9fa-c0fa05a4910e}}
We show true positive skewness for positive shape parameter {{formula:e8e750e4-aa65-4d20-b078-2c7e1c3f7c06}} ; the proof for true
negative skewness for {{formula:bc6ba624-2f28-417c-92e4-269a081370bf}} is analogous, and clearly the skew-normal is simply the normal
distribution, which is symmetric, when {{formula:44fb4bc1-8ecc-42e4-b628-876f5102dce5}} . Thus, fix {{formula:8387e4b2-6888-4b6b-8d28-62008718e20b}} and let {{formula:0fdbfd33-5709-4a2c-af4e-541c344bd167}} be the
density function of {{formula:eadfbd53-f818-4be0-95ed-c4c10961b09e}} , i.e., {{formula:10a451a2-12e5-401a-9dc2-867999bc9e41}} as in
(REF ).
Fix {{formula:daadd09e-9c95-4fe7-a929-18f449b49e82}} . To show that {{formula:ffab1cfe-ef18-4b23-ba09-9cce28a7cef7}} is increasing at {{formula:34b8a749-5cbd-4815-bceb-5d26afc9b324}} , it suffices to show
that the log density ratio of the left and right parts
{{formula:d93fff36-e184-42bf-bb59-610d24e7b1ed}}
has exactly one positive root {{formula:d64f9f99-4ddc-41f7-a5c2-b102d4a0b58e}} , satisfying {{formula:03524982-061c-49fb-97b1-3ba32a551534}} for {{formula:1dfeeab0-1f76-4cd2-8e02-8a6c0f1a8107}} and {{formula:e4a58d39-4974-4600-bf52-717484074494}} for
{{formula:a8d0f1f5-0e39-4e5e-a083-a77948c67c22}} . This condition holds if the following four conditions can be verified:
{{formula:c6b615f9-3f60-45c7-9837-af03180a80e3}} ;
{{formula:1075c360-6852-4de2-88a0-37bf0488a8bd}} for all {{formula:446387c4-a213-4fab-96a0-bbab18b6126b}} .
{{formula:0f5cc5f7-e716-43cf-b672-880c545eaea7}} ; and
{{formula:edb8e832-81b6-43b5-8847-da305aae8ba1}} for some {{formula:df207141-a2e9-4ce2-afba-0113fb0b86ff}} .
Condition (I) holds trivially. To prove condition (II), define
{{formula:bf27c90f-0316-4d54-bb3a-db08205db398}}
hence
{{formula:b7b9389e-9780-40da-8bd0-ec248d12a27e}}
Condition (II) holds if and only if {{formula:cc4082a4-8781-4259-912f-822d25d82f80}} holds
for every {{formula:2ced7f94-ab7d-4814-854c-dd466bff0c7d}} . It suffices then to show that {{formula:f0838742-ff4f-4e3c-8805-458f415cda9d}} is monotonically increasing everywhere,
i.e., that {{formula:910c2471-7bc3-4352-8268-7688626b7dd8}} holds for every {{formula:588431c1-69b4-4cde-8b24-ada06321fc67}} .
The third logarithmic derivative of {{formula:2d804118-429f-4579-872e-8ee1b815e930}} is given by
{{formula:d3ed2dea-4cb8-4df2-a28c-8eb75ce8a6f6}}
One may compute directly
{{formula:862b23e3-d314-4670-980e-35f4df675312}}
so if {{formula:4014d1b8-986f-4717-9ce2-ecf70eea12cd}} has no zeroes, then continuity implies that {{formula:523615cd-c1cc-4970-ab39-2a9b5b10ca88}} is positive everywhere.
Define {{formula:782a6fd0-353f-4111-aec0-3955f04a20fa}} , noticing that {{formula:ef0ff72b-365a-4a27-89f9-067eada02cb5}} is strictly
positive. By (REF ), {{formula:4b8e6c1c-5740-48cf-bc29-dc94632f3bcc}} holds if and only if
{{formula:bc6b85f3-57f8-4f92-b068-463397d88fee}}
which, by the quadratic formula, holds exactly when
{{formula:9c667503-67a1-4436-99cb-89bb04228f25}}
In other words, the zeroes of {{formula:4ea423b9-1249-456c-88b3-45ea9e16a2a5}} are solutions to the equations
{{formula:aa73286c-efd9-4fc2-845c-95fc1a38767a}}
with special care needed for the asymptotic values {{formula:319f1fae-e62c-4a6f-94f5-e5fa805a6827}} for {{formula:129d5bbd-0095-4d7f-8498-ca47a6529d70}} . We handled these
cases in (REF ), so it suffices to consider {{formula:529a0418-f892-4508-bc57-ca91c9e3125d}} .
Observe that {{formula:1321d1ac-a4c1-43bc-a1e0-326ecdd0c8d4}} holds if {{formula:0a07b4dc-4cc5-4cee-bd0c-8d887a68c67e}} , so solutions to (REF ) must fall in
{{formula:fe2bdb15-564e-4ff7-a56c-5af4e7871073}} . If {{formula:603f9a90-0761-4fd8-a074-a3de7f993af5}} , then both {{formula:e73fb866-eb84-48e2-b2a0-0e2c42782deb}} and {{formula:acaa8919-fa30-42a1-82cc-c09958ef3163}} are positive and, one can easily
verify the inequalities {{formula:8d6792a9-bde3-4909-bc30-46679831d6d9}} . On the other hand, if {{formula:83d7f181-1606-40d8-9764-1d1edf5093c3}} , then {{formula:a285a3de-1b22-449b-8ecb-b90346ee3bad}}
whereas {{formula:6cf32480-29f8-449b-ba9f-c75b8ca3eda9}} . Therefore, it suffices to show that {{formula:bbea03c5-1fef-4633-b91c-7db0b7ab6116}} for every {{formula:358df1e6-9ad7-4596-9171-89fd7f02ebfb}} ,
{{formula:7ba104c1-d8cd-42ad-b364-a3ac1dfb4275}} .
The key observation is that {{formula:abb947c1-99e5-46d8-90a5-057814cc098d}} is simply the reflection of the Mills' ratio of the standard
Gaussian, i.e.,
{{formula:8bdfb166-0e00-4e9f-9f99-d02908d9f94d}}
for which very precise bounds in terms of elementary functions exist. In particular, we use
the known bound (see, e.g., {{cite:c3fa0bc01869f192b147d4a6976a995ebc1e5e1d}}, or {{cite:dc9e719fb3259704dadb551ee9b6572d79ea91f0}}, {{cite:5f80b8a01a388e17a2bd73dce4c6be34025f15d8}})
{{formula:1206dbb6-37bf-4792-a1b5-4be2577ee1f3}}
which yields
{{formula:3811ca39-6924-401c-98c5-347112e6ab04}}
We have now shown that (REF ) has no solutions and thus that {{formula:e18a642f-59b8-40a9-a3b3-fa01ff614f41}} has no zeroes,
proving condition (II).
Since {{formula:319474bd-e3e0-4097-9639-ac950e2e6d96}} cannot be non-positive on the entire half-line, as a consequence of
(REF ), then condition (II) implies condition (III). For condition (IV),
observe that {{formula:b7f8754a-aea8-4287-9550-28f748770dcf}} is decreasing and tends to infinity as
{{formula:461084c3-a61c-4789-a966-b15efd303211}} , so (REF ) implies that {{formula:c7a87357-96b6-4e44-92e3-938a80c0035a}} as {{formula:cddbd476-e89a-4965-9917-e243d1004d50}} .
It remains to show that {{formula:557bc0ad-34cc-489c-9b2b-90c68b0896ab}} for every {{formula:ad324055-726a-47da-b095-acdac0709738}} . Notice that
{{formula:aa9afbe7-34a5-484e-ab9a-1045ca82b9ee}} is increasing as a function of {{formula:ff78d74b-703b-4cbb-9e2e-602ebecb9179}} given
the decrease of {{formula:11c5cfc2-eead-41b1-bd80-acb6040bbc54}} . The mode {{formula:543b26af-884c-4527-a71d-f08181571b20}} is the unique value satisfying {{formula:41f804ef-26cd-4855-bebc-d3a3444f2e22}} , which
is equivalent to the equality {{formula:3e7893ef-290c-4eac-a333-f96969d989f1}} . Condition (III)
and the increase of {{formula:278991fa-67bf-4fa8-afc8-2f69aaedeb54}} now yield {{formula:5fb1efb7-b9aa-4b3b-b398-03cbcdf84e44}} for every
{{formula:4d30bdbb-dd9d-4c81-9528-10d04655689b}} .
Proofs of results for Section REF
To prove the first part of the theorem, fix {{formula:ccc593d4-1772-42d6-bc0d-5b39c484d578}} and define
{{formula:c8d2687c-288b-4a8c-8e2a-961f13fef2e3}}
{{formula:91e32dde-5416-4002-b3a7-ed58178c16d5}}
so that {{formula:12eee090-954a-45fa-89e7-e13ff8b5942f}} and {{formula:fd011760-ffe7-4c70-ad81-5e2725c6deed}} are the {{formula:822f916e-2aa8-4676-9609-02d7c215b94f}} -means of {{formula:57af28f5-6416-44da-b8fe-9fcb16ecf288}} and {{formula:2cf7462b-05ef-4d27-bf8b-c601b298bac3}} respectively by Definition
REF .
Let {{formula:a1444363-206a-46df-9220-cd1167824cb0}} be any sequence of numbers converging to {{formula:4a60d149-1980-44a7-940c-e78a5aa56144}} .
Observe that the random variable {{formula:72027e6f-d610-4cb5-a3f6-7be431e758e8}} has distribution function {{formula:0c3d7a1b-e445-4cea-8aff-abf392365e04}} .
Similarly, {{formula:098ab65b-c053-4465-ba79-b3764e0a764a}} has distribution function {{formula:ec1c2c03-3cff-439d-9386-44c46c85a713}} , so the uniform convergence
{{formula:6be773b0-c1d6-4fd1-aecf-fb6412c96aae}} implies the pointwise convergence {{formula:a183e92d-7faa-44dc-8b78-ad9bbdd86ee2}} . Therefore, {{formula:4c8973d7-e787-4e4b-ac84-d14098593db7}}
converges to {{formula:64823538-71d9-4c63-a901-fd7bb96a5ebc}} in distribution.
By Jensen's inequality, we have
{{formula:ab756deb-1eb6-4e9c-bb8f-1f5da9770458}}
where finiteness holds by the convergence of {{formula:2334d46a-7a81-48c4-8db8-76f0144f4cd2}} and by assumption on the {{formula:086c02d3-879a-43d7-a36e-c9fbffb9977c}} 's.
It follows from, e.g., {{cite:cb4c94bf2e0a1f322881e9b7dfe5657853c60faf}}, that
{{formula:3ee59d31-712f-4c52-88bd-7c22dfb4f5b7}}
Suppose that the {{formula:d37a8c4e-5a94-4b4c-8e17-293ee752c482}} 's are contained in a compact interval. Then every subsequence
{{formula:c036f153-53be-4998-8c09-c68c6a1e6d18}} has a limit point {{formula:73098f70-ac43-4f30-b53a-b4e76ec3bea6}} . Since {{formula:8888772c-cc4f-4a54-b5db-74a70221e9c7}} for all {{formula:be345942-dd5f-4513-a201-bec97409904b}} ,
then by taking {{formula:d6d50f0b-f7c4-4874-9137-8d053e13559f}} we obtain {{formula:439ed6e7-2eb5-443b-bb5a-41bac803aab2}} holds for every {{formula:9ae5ae75-c39c-445a-97fb-6ad0e8664642}} via
(REF ). Then {{formula:b2eaa694-3a3a-4233-82ff-5db9484a04f5}} minimizes {{formula:8eaf3521-6a4e-4b88-b1cd-2140d4dc6d12}} , and since the minimizer of {{formula:c001df6f-4109-477b-ad24-8ed3f861f141}} is unique
(see Definition REF ), we obtain {{formula:e2b1c880-d4e5-40e9-b45d-a3d8232f6739}} . Every subsequence of {{formula:b925149a-4b81-4d3c-ba15-a99b684fda98}} converges
to {{formula:8e63c271-ba81-472c-a636-739e1aba8075}} , hence {{formula:01397ad9-48de-4d4d-8ebe-b1f7b0371d46}} .
Consider {{formula:fcd534a1-1c9a-47b7-b7ac-e2e39161183d}} and {{formula:29e9d39b-9bbd-4634-8777-d9d9a63f0589}} as functions of {{formula:3bd4c098-c1f1-451a-8456-ac96a73fc318}} , so {{formula:5c621ab8-6ef7-492f-b08a-bb8aaad412c6}} converges pointwise to {{formula:bf335fc0-ba72-436c-a83d-b1a9ce0c2965}} . True
non-negative skewness of {{formula:e94c3dfb-31e8-4329-891a-23f3da0f6d12}} implies {{formula:c77b7cdd-54bd-4aa2-95fc-a07609e455c0}} is non-decreasing, and the pointwise limit of
monotone functions is monotone, hence {{formula:12b78f5b-d809-4f3b-9c62-0822156be164}} is non-decreasing.
It remains to show that the {{formula:ce2c09fc-d149-47eb-a1cf-d317ae7babcb}} 's are contained in a compact interval. Suppose otherwise,
so we have a subsequence {{formula:71b715ef-8a48-41bb-87d6-c3900a85eb31}} as {{formula:554ad5a6-8b0c-43c6-8398-e954265b6735}} . (The argument is similar if
{{formula:8bc19274-f111-40cd-859a-b0b1f968c3ab}} .) We have from (REF ) the pointwise convergence {{formula:1fe5beac-5525-4dc6-afcf-bdcb4a7a4cc8}} ,
so for fixed {{formula:fc3442b9-0892-4334-a008-84cade2c11ee}} , we may choose large enough {{formula:9bcb0f4f-4028-4647-a45e-a3d3028fe39f}} such that {{formula:afd36fcf-b4b5-47f4-8d14-69e675bd76d8}} and {{formula:40a747a3-d112-412d-a053-d24915394aa3}} for all {{formula:4564a680-6c76-47b6-b4c9-49b778fcbe94}} . Since {{formula:ad4b1d24-c855-4eec-973a-5d5487356fef}} is strictly convex with minimizer
{{formula:3d9afd68-702e-4493-99d3-f99d1c688623}} , then {{formula:d4d18496-1f61-4d9a-b2d4-b0dbd118a33b}} for all {{formula:80a9c291-0a4e-4a58-9a6b-82f4cbfd1fd5}} .
Clearly {{formula:a9c08ae4-f1b6-4c2a-a30c-e944dcbfbda3}} as {{formula:5c94529b-05f0-4c7f-81ab-050a0d0b5c94}} , so choose {{formula:f5067f65-2ce7-4fc8-b7e1-81b2219c7581}} large enough such that
{{formula:5cda9b29-a622-4c72-abbd-7aca1c022b59}} . For any {{formula:578a8e6b-7cea-4256-af60-fb8bf7c2c6ba}} , there exists {{formula:856a29fc-97a1-4da6-be9d-88328480a877}} large enough such that
{{formula:edf209b6-598f-4227-bf95-c1c0d88f7549}} and {{formula:ae837a40-9063-4a4c-a58f-592a8f2f049c}} such that {{formula:a2970361-ed22-45c5-9f94-6383a21ca07e}} as shown above.
This contradicts the pointwise convergence {{formula:55fac69e-896d-4d5d-a4af-60c0c30c4d20}} , and so concludes the proof of
the first part.
The second part of the theorem now easily follows from the fact that {{formula:003544da-aadc-4122-a1f0-78f6ff72d41c}} .
Indeed, for {{formula:e8dc9d9d-199e-4adb-8c35-eff776b2667d}} , we have by assumption that {{formula:82e1fc30-0ed6-4771-8cc3-9fdec4abc8a4}} holds for every
{{formula:1a41a99a-b4f7-4468-a7b1-31eb4cab49dd}} , so taking limits yields {{formula:e4a839cf-a2bf-4563-a0f6-ed0daf468c59}} , hence {{formula:5b1659fc-f2aa-4ae6-9ec0-e6955ee9952a}} is strictly
increasing.
Proofs of results for Section REF
Let {{formula:4f17f7ea-4436-439c-be00-294ea3c11fb0}} be a continuous random variable with density function {{formula:8d4f28d1-6e4c-4577-95b5-32db64a15f41}} decreasing on its support {{formula:8338e36c-9512-4abf-bb25-393966b4140c}} for possibly infinite {{formula:465f2b2f-5576-49ab-aa9f-c6f675ce9cd2}} . Let {{formula:61ad4a5d-5a36-40b8-9d8e-ff461a2b154d}} where {{formula:8415354e-de65-4059-92fe-e710d8e54306}} is convex and strictly increasing on the support of {{formula:f5b36bc2-6156-489d-b528-476beaab7f6f}} . We define {{formula:86a8576b-9e48-48b7-86ed-1805c659795f}} . Note that {{formula:1c62f00d-dda9-41e5-a9e0-f9c267f2f5ec}} . We write {{formula:bc5472e1-0bfa-4d5f-90ba-21596efd8810}} fY{{formula:c4029594-f249-434c-ab5b-ce92f71feef8}} Y{{formula:093d0044-6e21-4459-9f29-55ac7339ae1b}} (0, ){{formula:654dc7a1-1c76-4170-9562-1fe9fca35741}} p. First note that as
{{formula:4f58bd06-252d-473a-b135-61c243d9e2c5}} increases, {{formula:a258d69c-ad2f-405e-be3a-72357d77df04}} approaches 0. Once {{formula:430cc493-4d28-4e75-939b-ae0a33db38dc}} , {{formula:c6f5ce81-b9f8-4fce-8e62-ca541c51f133}} . Thus, on the
interval {{formula:c8184413-7802-416a-a1e6-0c84f21c9ff7}} ,
{{formula:ec38c0a2-2dcf-46cf-89b9-c527dc7fe08e}}
On the interval {{formula:25037ebe-8d82-4138-a95e-0ce87a8508b0}} , we have {{formula:2daa119d-e0b0-4700-b881-19076b0f1eee}} , and {{formula:1207af99-7bb0-43ff-a0f5-c3e088b09b33}} . We expand {{formula:f10a84d4-2483-40f2-bace-43fcfe141c35}} :
{{formula:60fe3538-c754-4fc1-a578-a2c84fd7d740}}
Because {{formula:ebeffde6-df4f-44f1-8e54-e358814ef72c}} is an increasing function and {{formula:98d6ee42-5454-4575-93e5-6a1617708c0b}} is a decreasing function,
{{formula:2bc91763-27cf-48ce-821f-35039a0ccbdb}}
{{formula:cb1d25c5-568f-4118-9240-05bd34c77376}}
The function {{formula:941d3f1a-12ff-4a91-a6b5-218c4a1ca4df}} is convex and increasing, {{formula:19e9fda5-b195-470d-8e1e-6b8de82dfb93}} is concave, and therefore {{formula:8fda6917-749c-4ea7-9eec-9c36c3db0b9a}} is decreasing.
It follows that
{{formula:2a22d4bb-5f8a-48d1-8e41-1bbb53c17589}}
{{formula:c4a5cffd-dd7f-40ea-8372-7f77908a19fd}}
which implies that (REF ) is strictly positive on {{formula:231cdd1e-e6ab-401e-a821-d4cb79f46f3a}} . Since ()
is strictly positive on {{formula:5b3c5013-5448-4b56-b29d-de66d40cd65d}} and strictly negative on {{formula:2df536fe-56c6-4f81-9490-4d5da976d904}} , it changes
sign exactly once on the interval {{formula:1f579800-8f34-47c0-87ad-15675814b24b}} . This satisfies Lemma REF for
all {{formula:2a2c6813-b825-418d-bcc0-464d0b7fdb70}}
Let {{formula:7e6337af-ebaa-41b1-b74e-0a5817d17ff4}} denote the density function of {{formula:34748110-aae7-4ca3-aa20-3f9cf7df3b39}} . For {{formula:959caab4-547e-4a90-87e8-096ac9fde71e}} , define
{{formula:51a3e9af-9aaf-44b5-adb0-f1a8eefa1f95}}
and
{{formula:2956116a-3d6e-4bae-a588-4aae1598321e}}
If {{formula:30cfbd95-7ca0-4a18-88d8-0f330696efdc}} is non-empty, then its infimum
{{formula:2219499a-7f3a-424a-bcb8-67b6432f927f}}
exists and is non-negative. Note that if {{formula:daaf41f8-debb-41ff-9771-47bd565fef91}} is continuous, then so is {{formula:9aa70e3a-ed5f-44c9-9e08-2c5c1e563c47}} . Then {{formula:76c1eba2-e34b-4a8a-9887-0c77c1d846e4}}
is the preimage of an open set under {{formula:59e4a4dd-ce8b-4848-9634-ea2508052697}} , so {{formula:d4a91164-d2ae-4e21-ba79-13b6781a4888}} is also open and {{formula:50b43274-5aed-44fb-bb5a-ea127730b147}} .
Since {{formula:ab19bed5-6411-4078-a86b-3852c9b38855}} , then continuity implies
{{formula:6d9bcd4a-71e9-4da6-b2d5-45c48bd18d2d}}
Similarly, if {{formula:5989d8a0-dcc0-460c-abfc-9f4dc14cb5bc}} is differentiable, then so is {{formula:dff742e5-e233-4c41-9d24-40cc02fee9ae}} . Because {{formula:1a6a86dd-a2fc-49ea-ab4b-864794a877ca}} for all {{formula:89b9ad93-369f-4257-b26c-87d39ca77cb4}} , then
{{formula:0927a676-65b2-47e1-abe6-da731655b2a5}}
To prove the theorem, we will require several lemmas.
Lemma 3.1
The density {{formula:f9e78730-c565-44e6-90e2-64f7e7516997}} is convex on {{formula:7c74de9d-8ac1-4995-9e1e-d8605c842358}} , concave on
{{formula:fb791d62-4a61-42c4-90ca-77527232fbb6}} , strictly increasing on {{formula:90cf6083-827d-4954-bf4d-9e40f6011599}} , and strictly decreasing
on {{formula:3b68018f-98d8-4039-aab3-8937e81a1583}} .
Since {{formula:879e5a57-cb3e-416c-83c2-5673f3a03431}} is positive on its support, then it must be increasing on {{formula:98d406ec-4dbe-4d3a-986c-156d443e7a3e}} . If {{formula:e1149b58-ba44-4e79-855a-9a074eaba86d}}
is concave on {{formula:f3fcbd5c-d113-40f0-af5f-b187ffad5454}} , then it is convex on {{formula:470eb471-53e0-415f-894f-096f8854d5c6}} , contradicting
the fact that {{formula:d288e631-9dbc-4c1a-ba03-a64da2ae60c8}} . The convexity part of the lemma follows, and
since {{formula:cf1a30c8-2b15-4ef7-a645-ee9761444216}} and {{formula:0919f9f4-8758-49d4-9ba8-a4c62e5a436f}} is unimodal, then {{formula:a3bab967-d721-4882-98ec-a2735b0e2c6b}} is decreasing on {{formula:e0ffd3c6-a663-4f6b-8b01-f313045f3bbe}} .
Moreover, since {{formula:1feef65d-142c-4dc5-ba78-632d1f8a7bbe}} near 0 and {{formula:4e1844a8-052a-4164-a3b2-4246ddbfeb4c}} near infinity, then {{formula:62af3ef1-1a0e-4b18-a994-ac2862f5dfa0}} has an odd number of
zeroes. Integrability of {{formula:fd67fb6b-f617-4618-9720-ed31def3c46f}} implies {{formula:00f2f96b-0569-479c-a6a1-a6d68430275b}} , hence {{formula:06b72377-6152-4a2d-a3ed-8e3067685cbd}} is non-zero everywhere on
{{formula:112f783f-f1c4-4d1a-814d-2ed14a15e24f}} except at {{formula:899a053e-f0e6-4731-b4e9-eaca181fdac3}} , otherwise {{formula:c81b2667-774c-42df-a175-1160ab7c22a1}} would have at least three inflection points.
Since {{formula:93f0c110-a759-4625-989a-d132aff92641}} is {{formula:820c0ec5-212f-4ac4-b0ae-bd8c633b1209}} on {{formula:b79abaea-1b4c-4943-8252-12e0825d7b01}} , then (REF ) and (REF ) hold. Moreover,
{{formula:523ae989-b6f2-4850-abc0-5ebd3e7f9415}} is {{formula:5b58099d-c9b5-438d-a915-f895c60d8c81}} . Let
{{formula:b81fbbe9-8864-424a-853f-c804be599897}}
By assumption {{formula:135db01f-e0d0-43c1-ba74-02bb7b9601b8}} . We show that membership in {{formula:b8edc938-fad0-41d9-b9be-1a476d64fabd}} is sufficient to make (REF )
a strict inequality.
Lemma 3.2
{{formula:8ca0bcb8-ebf4-46e2-876e-869b9c472b2b}} for all {{formula:08e6f4bf-378b-4279-8245-96b8a8eb7383}} .
First we locate {{formula:f4d9664d-de30-4205-83b8-92cef8c2ebdf}} and {{formula:01ecbf64-7549-4607-a9d7-4cc195b01e50}} . Note that {{formula:0dc1e9a0-9010-4e7b-8bfe-ec45ee7696e4}} is increasing and {{formula:7b05ae49-fe34-4f23-bd90-a973d0db07f4}}
is decreasing for {{formula:7ee8a078-20ee-4bff-8e80-fc54a5d9f26f}} . The fact that {{formula:d6d17c61-7bee-4074-9edd-be121b1851cb}} is negative near 0 and {{formula:7ee21203-ba40-4952-840e-525c20d4ce56}}
implies {{formula:82998b24-3913-433a-9c3b-b08e9547628d}} . By assumption {{formula:c4c14b60-fdf0-477f-8c68-2ae10f3dff6f}} , so it follows that
{{formula:d652901a-b8dc-4801-b211-96e4a7e79d2e}}
On the other hand, we easily have {{formula:fae15f5c-8909-44d0-bc55-32fce678b44f}} , otherwise {{formula:a68de6e2-139f-4e2e-a96a-7ce0954e6869}} . If
{{formula:d1204ea5-7237-4f2d-9974-4683c5f7b1f0}} , then {{formula:cf695944-fdea-4bcc-879c-3a85e439973b}} by Lemma REF , again contradicting
{{formula:62f2504b-c356-4a2f-ac43-4e95c0786f7b}} . Thus
{{formula:7e100346-e705-47e0-9c0d-5a39019fc963}}
Now suppose for the sake of contradiction that {{formula:516ea0cc-8d73-4091-b2e1-3edac82da890}} . Either {{formula:c35e1f8f-d527-4e9c-8171-cba0b11fe7a6}} or
{{formula:53694a3e-318a-4152-8de8-c0da6a5d32ab}} by (REF ). If the latter holds, then {{formula:e86b5b0e-cf4f-40c2-86dd-67134b8f0a3f}} ,
and (REF ) implies {{formula:c9759eba-544f-4a3c-acc9-b4520d1bd2b7}} . Note that
{{formula:81830f43-021c-4d4d-8359-4388e315eae1}} , hence {{formula:05aaaf6d-ebc5-4048-a991-fdaeeb7a535e}} . By continuity of {{formula:81733838-6e49-431c-ae4c-58f4b22f475f}} and
(REF ), {{formula:b35aac10-b01e-4d3d-8891-e85f67d1c9cd}} is strictly convex and thus positive in a neighborhood of {{formula:084ce190-f8f5-432a-bd7f-f8d4157229c5}} (excluding
{{formula:7682b98c-16d3-4192-82f8-4820b2f5caab}} itself), contradicting the minimality of {{formula:e1efbea3-28e1-4eaf-8faa-166219f3a718}} in {{formula:ebd5a12a-7d1d-4e49-9f9c-a798d5e2f707}} .
Suppose instead that {{formula:4d600a1c-5be1-405c-8190-e8a9e552706b}} . Then from conditions (1) and (2) and equations
(REF ) and (REF ), we have the inequalities {{formula:c0ee9f53-2f5e-4d69-aa25-9823a042fa71}} and {{formula:70a80dee-59ba-4661-8dec-23385afc3eea}} . It follows that {{formula:2d2b8e6d-64b7-4b76-84c7-fa38b6672039}}
and we obtain a contradiction.
Next, we prove some properties of {{formula:12b870a5-3bf2-480a-9c6f-86ac6e7470d1}} .
Lemma 3.3
The map {{formula:412fa43b-407a-4f3b-b617-46b75298c04c}} is continuously differentiable in {{formula:95e5ede5-6991-4e0a-84c0-c2ecfc1c6dac}} .
Define {{formula:c5518d7c-310d-4d25-9970-89587dc924ea}} by {{formula:de1a615c-aed6-4935-8a77-8130c54446d0}} . By Proposition REF , {{formula:3a3d3f7a-a178-4b70-9d24-2afcf17b6e8e}} is
differentiable and has partial derivatives
{{formula:5a7b9dd8-b994-4d17-aa33-4e775a4b7969}}
{{formula:43c1c473-b319-46c0-99bf-3aa17d48677c}}
both of which are jointly continuous in {{formula:1f1ae519-8592-40ae-a5d6-212eea4221af}} and {{formula:cdedcc46-1d4a-46f3-9b5f-3d669d8cb02d}} . Thus {{formula:24e5f083-4062-4c33-8f1b-a0e9971ad9fb}} is continuously differentiable.
By Lemma REF , {{formula:8f8109cd-4b66-4bb3-9c56-6f06377aede9}} for all {{formula:b03d4293-c647-4c34-944e-ed28f4b9140b}} and so the
continuous differentiability of {{formula:32942388-09a9-43b2-8d60-725d3f8ffe86}} follows from (REF ) and the implicit
function theorem.
Lemma 3.4
For any {{formula:4b04f703-ad0a-4728-86c6-5073128d5919}} , {{formula:dd66e7eb-dddf-4e98-bb29-d3a0ab2d3d19}} and {{formula:9df7b2d3-6c17-44c9-acb4-7841c76686f7}} have the same sign, and
{{formula:36db019e-9593-44a1-ad25-e2cc81b49f1b}} if they are non-zero.
By (REF ), we have {{formula:3d8025bf-fb62-4324-9cf3-d737fbcd6556}} . The left side is differentiable in {{formula:fb70f7f4-e703-4b10-9556-c3e1a31fd6a5}}
by Lemma REF , so taking derivatives, we obtain
{{formula:9ff2c95f-e16e-4290-96cb-c5e958ceeea8}}
Rearranging yields
{{formula:f11acc0d-0e34-4fbe-a2b9-e128c511bf46}}
where the fraction is well-defined with positive denominator by Lemma REF . The
numerator is positive by Lemma REF , (REF ), and (REF ), so
{{formula:01285021-741d-415d-9344-71965f9130ad}} has the same sign as {{formula:385b6d07-eb7f-4103-9a0b-6b8edc8dc52e}} . If {{formula:27530310-9eb3-4682-b6a6-6d63c65b7979}} , then one can see immediately from (REF ) that {{formula:c06f0a37-b000-456c-a731-675c88c319d6}} .
The reverse also follows.
Our final lemma concerns a criterion for pointwise increasingness of {{formula:035af941-9b74-4b03-bddf-a323a3de4416}} .
Lemma 3.5
Fix {{formula:a4b21d8a-4b98-43e4-9794-1a46a8938bff}} . If
{{formula:323a2eee-5e72-4742-aa8d-ab5b950095e3}}
then {{formula:de629562-f128-480e-81ec-5213b4682102}} is increasing at {{formula:b85e1e51-3ab2-46fb-a580-85333356eae1}} .
Since {{formula:2498264e-2eda-40ec-a7d2-5e704857e24d}} , we may rearrange to obtain
{{formula:dd602e5f-88f9-4741-a231-35b8bbe396d5}}
Define the line
{{formula:8d19a973-0d1f-4960-9fa7-14434f7cffad}}
and note that
{{formula:3396f9e4-7442-45cc-9ce6-e7c3e9712a15}}
Note that (REF ) implies {{formula:01351044-7212-4064-960e-3d94a4cf108b}} , so by convexity of
{{formula:582ea9c2-81a8-4858-8082-f697687a53f5}} on {{formula:cc5d74db-55fb-4269-ab6b-d32b30f34cef}} , we have {{formula:ae9aa250-4555-40fd-ac5b-a899bc1f2fa0}} on {{formula:77876428-c8ea-4037-b48f-445ce90f0d82}} . Via integration we obtain
{{formula:7b12806e-8c75-464e-97c8-af24a12f0711}}
for {{formula:e5bf059d-6335-4b84-8891-796e54647c49}} . We now split into two cases to show {{formula:a0eec75b-6e1d-420d-b0d3-cecd7eb34248}} for {{formula:6264a8e2-5dd3-45e5-b29b-3d20d29f16fb}} .
Case 1. Suppose {{formula:944f73d0-65a2-47ea-8b40-5edbdff234e7}} . By (REF ) and the convexity of
{{formula:5bbe2ae6-2bb3-4219-906e-2912f70955a4}} on this interval, we have {{formula:8139e2b8-be3b-4100-a57c-3d1b5b907384}} for {{formula:d6305b75-2eb0-4566-86fe-5d78c4fb89cf}} , making use of
the fact that convexity is preserved under reflection and translation. A change of variables
gives the inequality {{formula:00d1a3ce-fd6e-4ac9-a9ea-706a73704919}} for {{formula:e7f4e64d-e3a7-4848-bcbb-46112e8a1f9d}} . Combining with
(REF ) yields {{formula:46770eb7-86b8-44e8-9f91-0235134fdfa0}} for {{formula:887ee497-acb5-4ad2-8bee-a103c0ede28f}} . If {{formula:f6ca17ed-3130-473a-a8c2-4eb494633fc2}} ,
then {{formula:52ee6ac1-57c7-43e5-a9bd-6ef2ca74a929}} and we are done.
Case 2. Suppose instead {{formula:8072e3da-afcf-4594-8589-073436fdae0e}} . If it happens that
{{formula:24a7e626-be8a-4068-8663-a8954b0c6c26}} for all {{formula:2837b44a-7e82-4e43-8a6e-66c0a4f5e3af}} , then the argument in Case 1 applies and we
are done. Otherwise, let
{{formula:707322b3-ec8f-4c91-8825-3d6b8e91c773}}
be the line such that
{{formula:3ca6cf3e-1ab6-402a-bba0-40ed4611ee80}}
and
{{formula:dfcafd6f-5c96-491b-b6ef-c1a9883e29e9}}
Lemma REF and convexity imply {{formula:070af90c-da40-47b3-be65-0aa6a410015d}} and thus
{{formula:5cf42799-97c6-434a-8358-a76976b12339}} for all {{formula:4381e646-8a8e-4a7c-917f-830ea59ee94a}} . By concavity of {{formula:1109b70b-09d1-4d3e-91d3-26cd65bb3176}} near {{formula:55325657-64a6-4e60-910c-8f03a52c6b8b}} , one
can easily see that {{formula:077e23ad-ec8b-4f7c-8794-310bb8c467c9}} for {{formula:81380645-d94d-4990-8a4b-bdea076335d0}} . It
remains to show {{formula:aacc5c02-8c2c-4e4c-ae3f-8dd2be331bb3}} for {{formula:42f2e8e6-dcfc-41bf-bc86-9481cae60e11}} .
We have by assumption that {{formula:b53b4f59-5ea3-4ed5-ae1e-957fe41ca27c}} for {{formula:61bbcb01-666e-4b1e-8888-b63e7de5403f}} in a right neighborhood of {{formula:ff0edafe-416f-4b45-93e7-db5bb236ce90}} .
We also have by (REF ) that {{formula:f6e394a6-414f-422a-bbf6-f563f285c1a8}} . It follows that
{{formula:935177be-d985-413d-b81e-ad4080ea0290}}
Substituting with {{formula:831c0291-0b05-4889-a6e4-bf19f07d0bd0}} with {{formula:e324a268-16f4-4c6e-a5b2-c983c264a8dd}} and rearranging yields
{{formula:cd9d52f8-6758-470e-85aa-87c0c081ef32}}
The left side is precisely the root of {{formula:9ff5e47e-71d0-4f8a-b6eb-fe152688fbb7}} whereas the right side is a root of {{formula:d817a271-ef7f-4a30-9ce1-9016c9d6adf9}} .
We showed previously that {{formula:aa74d631-6c33-4824-a2a0-aadba9d4b19b}} . Convexity of {{formula:45b416f3-8ce8-4b63-81c7-77b897b12e08}} on
{{formula:928cafd5-d7e6-4e8e-8578-f8a437fd4150}} implies {{formula:6582fe43-7bf0-431b-a739-c1840c44d453}} for {{formula:2b4167eb-d076-4baf-bbd6-9364f7af1259}} ,
and we are done.
We have now shown that, in general, {{formula:3dd90a95-7491-4a8f-8323-d1839bfadb29}} for {{formula:c157c8a4-1360-47be-8b05-2fdce7c02fd3}} . We also know by
definition of {{formula:fa3fc17e-7d89-46d4-997a-e0f22122e71b}} that {{formula:2ae83758-cf4e-4823-a160-4bb9456c1c53}} for {{formula:bdc22eff-84fb-4237-afb8-373aa1d4bfb0}} . Thus the conditions of Lemma
REF are satisfied and so {{formula:928e439e-36dd-4fcd-ba0f-1ed10fdeea0b}} is increasing at the point {{formula:4c440c3a-b07a-476a-801f-c299c3e70ee1}} .
To prove the main theorem, it suffices to show (REF ) for all {{formula:5e75240d-6d98-407a-849d-8667acd5e87e}}
and that {{formula:b1caf70c-d606-45d9-8156-f01c8323d4ff}} . If we have conditions (1) and (2) as they are written, then (REF )
holds for all {{formula:82133222-7056-436d-8af1-9839280d7f5f}} immediately by (REF ) and (REF ). Suppose instead we only
have the weaker condition as stated in Remark REF (b):
{{formula:86bff035-d65f-4d87-9b95-460d7de54370}}
Let {{formula:9a5c0310-bda7-400d-a8f8-cd4a02e8f0de}} on {{formula:c3749007-c651-484c-ac1e-9e28ae4ee243}} , so {{formula:0c2db9d5-7758-492e-8e3d-8662ce04f201}} is differentiable on its domain. Also note that {{formula:f4ea32de-5598-410c-b653-a8d704a7a13b}} by
(REF ). Suppose for some {{formula:0d528240-9168-4124-98ae-65e82e014743}} that {{formula:5b91adf4-f2f2-40b4-a351-c6ff111819ae}} . Then
{{formula:d4723fd5-d9ca-452e-b76e-d6c688db8ff7}}
i.e., (REF ) holds for {{formula:20713bdf-bdc7-4b72-a732-13b2437e7375}} and so {{formula:bcae4b0e-2448-49b3-b3e2-dd7fa75c1f26}} is increasing at {{formula:a42a2ab9-b9b0-4c5a-ba29-094db45aee6b}} . Lemma
REF implies {{formula:84041f01-390a-4330-8f32-663587c96f07}} is decreasing at {{formula:ee15582e-984c-4017-9c80-054a08ea7f37}} . It follows that for all {{formula:a92dbd27-78d4-427a-bda3-674c98429975}} , {{formula:3c20b9b0-50fc-4392-b1f3-2746c97e004b}} is
decreasing and so {{formula:a60ab94f-57e5-422d-b855-f220598e1b0b}} . In particular, {{formula:f6e9f1a4-a454-48b3-8195-6ae5bc102af2}} is increasing for all {{formula:9d9accb1-5ebc-41e4-b3dd-725b6c234377}} .
By assumption, {{formula:27e8ebee-1821-495e-a095-0e2667601c04}} and trivially {{formula:d5427f25-56fd-4c1b-98c1-94c9d5effa70}} . Since {{formula:db9a3c09-27b3-40cb-ae1e-9e03071213d8}} , then
{{formula:c9350971-8045-4d3f-9dd3-9fdf9b50546a}} is increasing for all {{formula:785ea413-53f5-4537-ac4d-56f14972bd15}} . Recall that {{formula:33eadc8f-9f2f-41fa-9419-86d15fefbe02}} if {{formula:886cad5e-79be-4b4e-854d-f3b372bf1e10}} .
Since {{formula:b59767f4-d606-472f-a493-1418b9333a0e}} is increasing at {{formula:d42698b8-644a-4b4b-861b-fd36a4122210}} and for all {{formula:d8792808-518a-497f-a3e7-db2c46005ea1}} , then {{formula:78c13306-7ae2-4673-b1b4-053880fd33a7}}
for all {{formula:1d22d45b-e42b-42c4-9fa0-14f53189c457}} , hence {{formula:621c9bbb-a735-41c0-b92c-131fb0c57b50}} . Clearly the argument
for {{formula:322fa77a-29d8-4dee-9428-2fdfca6aac4d}} still applies if we use the alternative definition for {{formula:98b4460f-bf95-4ddc-883d-16e0fc5b6633}} .
The proof directly extends to the case where {{formula:b3fb0a32-659f-4261-9345-a1ec9a106a5b}} only has a single root {{formula:e3939516-cb7e-4b2a-bbd7-f25efa348c0a}} by
setting {{formula:89aa15e4-4a06-4b4c-8282-37b70470c5af}} . In fact, conditions (1) and (2) are not necessary. Indeed, we use them
once in the proof of Lemma REF in the case {{formula:78412473-26e0-4da5-8f14-ebd69297d35c}} , but this
is no longer relevant if {{formula:37cad00c-62cf-4d41-b92a-0bac095a5240}} has only one root. The only other time we use the conditions
is to prove that (REF ) holds for all {{formula:eed94e60-c5f5-4398-8d78-a8524048a8a5}} . However, if {{formula:b264dc76-584b-4bf2-ae94-5b96f94d0a64}} has only
one root then, (REF ) holds automatically. Indeed, note that {{formula:8802cb27-fcef-4736-bc55-2b73bef466eb}} is concave
and increasing near {{formula:0d90515c-61e6-4b63-9614-8173b604463c}} for all {{formula:a45ba3f9-06e2-4c96-9fc7-97922f4e2cb9}} , hence
{{formula:7be77440-374e-45c8-8e64-9a638481e369}}
Since {{formula:188d93e1-8430-4957-9d74-8312e3808610}} , then by substituting {{formula:870759ff-0707-4308-9b7d-50a91207ebe3}} , we have
{{formula:c4c93598-69af-4e3d-9bcf-a3f5bbe23ef7}}
The right side dominates {{formula:0c348c37-fddb-474d-a961-f7d09714e844}} as a consequence of Lemma REF , so we arrive at
(REF ). This proves the corollary.
Discussion
This work attempts to broaden our understanding of true skewness by demonstrating the true skewness of several additional distributions and by establishing simpler criteria for which one can conclude a distribution is truly skewed. Theorems REF –REF , which establish the parameter regions for which the Lévy, chi-squared, Weibull, and skew-normal distributions are truly positively skewed, all rely crucially on showing that the conditions of Lemma REF hold for every relevant value of {{formula:34f45147-0027-4e21-a782-f596c7c46081}} . This is done by analyzing what we have called the “log density ratio of the left and right parts,” which for given {{formula:943fd088-1c5c-4e8b-84c8-21d17b59bec0}} is
{{formula:c6acc048-1da7-4dd4-9273-091f5b9d4c07}}
where {{formula:5f6286a3-ab76-464a-9010-0561ab252848}} is the density function of the distribution in question. The conditions of Lemma REF hold if one can show that {{formula:733cefd8-b765-476e-aa0f-5e7776f863af}} has exactly one positive root {{formula:e7f36ecb-a852-483f-bedd-f576c6d8d5e2}} , satisfying {{formula:ac140be5-3cce-4f93-8264-9421a98c993a}} for {{formula:bf9da058-7ba8-4f42-a999-e32d073f2147}} and {{formula:35b6a0a1-3c49-45d5-aa93-6040edfc0ce2}} for {{formula:72c5507f-b41e-4ec1-96f8-0fa184300cc9}} . In Theorems REF , REF , and REF , this can be shown in quite a straightforward manner simply by finding the critical points of {{formula:1f476ed2-630a-4cf9-80b0-2790283e0ff5}} . Such an approach may likely be applied to other visibly skewed distributions for which a closed-form expression for its density function exists.
However, for the skew-normal distribution, Theorem REF , the absence of a closed-form expression for its density function makes the computation of the critical points of {{formula:fc29bba4-865e-4e93-b0ed-51e357ea12bf}} intractible. Instead, we exploit the fact that the log derivative of a distribution function {{formula:09fb0c2f-0931-4963-8cc0-b635e582e79a}} is simply {{formula:c20b71c1-f1f2-491a-b208-dbd0ecdc6ec1}} , where {{formula:159ab2d9-e840-4784-b6a3-c8ba296bf1b8}} is the corresponding density function. The ratio {{formula:0dfc3334-7123-4171-8a5c-60cbaccc7351}} is the reciprocal of what is commonly referred to as the hazard rate of the distribution {{formula:68ae4607-9ce0-4025-9bdc-386deaf0e4ae}} , which has been studied with some detail for most well-known distributions. In the proof of Theorem REF , we encounter the reciprocal of the hazard rate of the standard Gaussian, which goes by the special name of the Mills' ratio. The Mills' ratio, as one may expect, has been very well-studied, allowing us to employ a closed-form expression, which very tightly bounds the quantities we care about, to prove the required properties of the skew-normal distribution's log density ratio of the left and right parts.
Indeed, the proof of Theorem REF identifies a key yet unsurprising connection between the notion of true skewness, which is fundamentally a comparison of the rate of decay of the two sides of a distribution, and the hazard rate. It suggests that the notion of true skewness is an accurate reflection, and in a much more rigorous fashion than Pearson's coefficients of skewness, of the essence of what we imagine skewness to mean. More practically, the method by which we prove Theorem REF should work for a larger variety of distributions; a reasonable first direction would be skewed versions of other symmetric distributions, as introduced in, for example, Chapter 1 of {{cite:45863800d488a970b8acd4673bd4020c11ae50c5}}.
The remainder of this section discusses sums and products of truly skewed random variables; examines whether true skewness extends to the discrete case; and provides an interpretation of true skewness for multivariate distributions.
Sums and products of truly skewed random variables
The goal of this paper and this discussion is to present tools for establishing a larger class of distributions which are truly skewed. A natural question is whether or not true skewness is preserved under certain “transformations” of truly skewed random variables.
Theorem REF gives an affirmative answer to this question, but with limitations: transforming a random variable with a strictly decaying distribution by an increasing convex function preserves its true positive skewness. One might also expect that sums of truly positively skewed random variables preserve true positive skewness, but the rather strong requirement that the {{formula:0def1d0a-7bae-4299-867d-26feaf0e147e}} -means of a truly positively skewed random variable be strictly increasing allows one to construct rather straightforward counterexamples. Indeed, we may even take the two summands to be identically distributed. For {{formula:97cd4489-57dd-420c-a197-0ec914be3b80}} , let {{formula:876c09cf-e311-49a4-83c4-51c09c9b3af4}} and {{formula:8b78a3b2-40de-458b-b571-f96bff95d7b1}} be i.i.d. with density
{{formula:e4e36630-5dea-4cea-a8e7-e1371c03ae46}}
Since {{formula:07bb49b9-796b-4276-a0c0-5a4e4aa46781}} is decreasing on its support, {{formula:74ade599-8a79-4676-a525-3f27648b6bcb}} and {{formula:b3c645bf-f6ea-4957-9988-8f0359f7ff80}} are truly positively skewed by Proposition REF . One can easily compute the density function {{formula:1adb330d-0466-4a25-9217-625c7d925f7a}} of {{formula:2c88419b-e115-419a-97be-bedd8e30df5f}} , the convolution of {{formula:65c9eb50-d163-416c-abb2-ec10f6a2c9f9}} with itself and see that, depending on the value of {{formula:90e6bc6f-40c0-4661-8505-c1125e162b34}} , the {{formula:0951e440-b62c-4040-ac1c-58d47b3a60d9}} -means of {{formula:12fece59-eafd-430c-aa90-2f204bb1187a}} are not always increasing. For instance, for {{formula:715d27dd-cfb9-4661-960f-99a551e4e336}} , computing numerically the median {{formula:b5799c2c-8162-441c-8f7d-f06bc39b76c7}} and the integrals in (REF ) yields {{formula:23ab4531-6d10-4a5b-a712-40dc863dfe2a}} and
{{formula:68dcdf51-68a6-4e3b-8c77-e5e42dc03318}}
which implies that {{formula:b54e4dd5-c22f-4989-b21b-f3b538fba866}} is decreasing at {{formula:f7fea0bb-94aa-41ec-a9e3-7eff374c5eed}} , and so {{formula:2784924d-b431-4b66-a3d7-6b18f4007504}} is not truly positively skewed. Numerical computations were carried out using Wolfram Mathematica, version 12.3.0.0.
On the other hand, it may be more fruitful to start with random variables with monotone density functions. One class of truly positively skewed densities that is closed under summation is the class of “decreasing linear densities,” of the form
{{formula:3759d5a5-3b1a-4f4c-874d-106c0acd828b}}
for arbitrary {{formula:759387c7-57c0-48f8-976f-9b8fef585521}} .
True skewness of discrete distributions
In the discrete case, even the simplest sums of truly skewed random variables fail to remain truly skewed. Take {{formula:b4766dcc-75f5-459b-806c-62294eae92ee}} and {{formula:e1245393-15ec-478a-b06c-a16f1d11f823}} to be independent {{formula:0ecf86ea-daff-428e-a1d5-0ab686e7de9b}} random variables; one can show that these are truly positively skewed quite easily. Their sum is {{formula:4733e9a7-a600-47cd-8726-f419fe041432}} , which has median 1 and mean 2/3, hence {{formula:d9bc04ad-4793-4af1-b94b-77c5f10279bc}} , and true positive skewness fails to hold. Indeed, the notion of true skewness is finnicky for discrete distributions, at least in the way it has been formulated here and in {{cite:d844093569c5dbd2b703c89a51e0f3ae87da50f5}}: this could be due to the interpretation of discrete distributions as limits of successfully sharper (continuous) bump functions, in which case distributions like the {{formula:8ffb4a2a-8579-4be5-bbfc-542e43487310}} are no longer unimodal but in fact have {{formula:6ae4cb25-3e02-4f97-9738-1cb396d89e09}} modes.
The proof techniques used in this paper often involve taking the log of a ratio of the density function
of a distribution. In particular, the logarithm of a product of two entities is equal to the sum of the individual
logarithms of those entities. For this reason, it seems natural to examine how true skewness behaves under
the products of random variables and whether it is preserved. In particular, it seems that taking the logarithm
of the density of product of two random variables would yield a sum which could then imply true skewness
depending on how one defines the two random variables.
True skewness of continuous multivariate distributions
The definition of Fréchet {{formula:36b4586d-6770-4c02-9110-760ac763993d}} -mean (Def. REF ) extends naturally to multivariate distributions as follows.
Let {{formula:565d5d6a-eb88-427d-be31-f22d3ff2c102}} be a random vector. The {{formula:517b3c18-08b5-4ee9-b54b-d27120c823de}} -mean {{formula:59d909b5-8e93-4630-9470-381570511729}} of {{formula:20038ec5-8390-4378-84a9-d70b0e80fa83}} is defined
{{formula:95a32e54-8c52-4848-921a-9a090d6edec3}}
where {{formula:7a7326e8-e07f-475a-a4d1-850caa376a74}} is the usual Euclidean norm.
In the univariate setting, true skewness corresponds to the sign of {{formula:5762f539-59c8-4624-bb83-f8a7d09450b7}} representing the direction of trajectory {{formula:97ee8ebd-f7af-45e3-863c-d630732595f3}} .
Following {{cite:d844093569c5dbd2b703c89a51e0f3ae87da50f5}}, we adjust true skewness accordingly. We let
{{formula:43a128af-1f85-41a2-b99a-5a1c8692b8f9}}
denote the unit tangent vector for trajectory of {{formula:fce05911-4ee1-4045-8954-f8429946cecc}} in {{formula:e07a9358-5d57-44f5-942f-67ca3b0797a8}} . We will illustrate true skewness in a multivariate setting by means of an example.
It was conjectured in {{cite:d844093569c5dbd2b703c89a51e0f3ae87da50f5}} that the limiting direction vector {{formula:714af971-17e7-4173-b0f8-3852152a52f1}} with {{formula:5258183b-836b-4049-a894-1ea9e2e9e6c5}} increasing to the rightmost bound in its domain {{formula:96845527-7a1e-4fed-ac0e-9f6d2b373026}} may be interpreted as the “direction” of true skewness.
As an example, considered a multivariate skew normal distribution defined in {{cite:4aa5f944ba4e9c4316a97e9303d539c0705b00fc}}, the probability density function of a multivariate skew-normal random vector {{formula:189549c5-b6e2-409e-877e-c786252f9ffa}} is
{{formula:654e4a46-343e-421e-a974-6dcdf10cb93e}}
where {{formula:093f19ad-0fc2-40fe-a922-de581b5041b2}} is the density function of the {{formula:c764792f-47a6-49ff-ae80-33a4535a3ba2}} -variate normal distribution with mean {{formula:b24bc658-a11d-48fb-9dc7-26e050e04828}} and covariance matrix {{formula:a0a608b3-1411-4e3a-8ea4-3f782fa9c6cf}} , and {{formula:e99c25f4-6c05-48e1-a2c2-73134c6ace15}} is the cumulative distribution function of the univariate standard normal distribution. Taking {{formula:264b1e18-2f33-4067-861a-b9f0cab31894}} recovers the standard multivariate normal distribution with density {{formula:473cdbc4-946e-43ff-abec-b3ab8da6201a}} . We refer to {{formula:5bb263eb-9d61-4947-91bb-1b4af5f0cfb9}} as the skewness parameter vector.
{{figure:a7fab1da-4f5d-453b-a446-03f6f99067d9}}The arrows plotted in Figure REF motivate the interpretation of multivariate true skewness via direction vectors {{formula:5cd4c9e1-ab30-4d73-8cbc-d53dfaadefa2}} , which are naturally co-linear with the skewness parameter vector {{formula:8e77e376-fd04-4695-8426-b6cdb48e8cc5}} .
Acknowledgments
The majority of this research was done as a part of a Research Experience for Undergraduates (REU) program at the Oregon State University Department of Mathematics funded by NSF grant DMS-1757995. We would like to thank Javier Rojo of Indiana University for his helpful feedback on the concepts considered in this paper. We would also like to acknowledge professor Holly Swisher of Oregon State University who as the director of the REU program provided invaluable perspective throughout the summer.
| r | 4e61e2521a99c534183512f486a55b04 |
Many complex processes can be viewed as dynamical systems on an underlying network structure. Network with the dynamics on it is a powerful approach for modeling a wide range of phenomena in real-world systems, where the elements are regarded as nodes and the interactions as edges{{cite:88003113759ade809e5dc4f87efd5e416cb056b7}}, {{cite:6239a23e098c6fa97cedc1a6860735d5a33ec644}}, {{cite:1ef3f743b177561ca84681c6470e35b850544023}}. One particular interest in the field of network science is the interplay between the network topology and its dynamics{{cite:fc3955213ae69e891f52b85a15441304f43beb3c}}. Much attention has been paid on how collective dynamics on networks are determined by the topology of graph. However, in real cases, only the performances, i.e., the time series of nodes states are observed, but the network structure and the dynamical rules are not known. Thus, the inverse problems, i.e., inferring network topology and dynamical rules based on the observed dynamics data, is more significant. This may pave a new way to detect the internal structure of a system according to its behaviors. Furthermore, it can help us to build up the dynamical model of a complex system according to the observed performance automatically.
| i | 99018e0e7b2cc71a553098e1c4842721 |
According to Ariel V {{cite:86bfe0414dc1bb31ec3d4c23fbd794e5eb3f9bea}}, CGRO/BATSE {{cite:0b990289e9e6f535d737069fbc1d916a5be5786d}} and Swift/BAT observations, the time separation of {{formula:548cd92d-d912-4ae2-b12a-d1ef78f801cc}} 17 years between the detection of each outburst for 1A 1118–615 suggests that the neutron star does not interact violently with the massive companion during most of the orbit {{cite:e0874905c0ad73a9c3b86289904810c3fa950235}}.
If the outburst is only caused by the approaching of the neutron star to the periastron, the long time separations of the outbursts indicate a very flat elliptical orbit.
| d | afaa685d057fbb6d7070f42318051fd2 |
where {{formula:a609d09f-6baa-466e-81e6-be3527f5c433}} is the quadrature truncation order defined in each subdomain, and {{formula:ff06d91d-2ea0-4a91-9b5c-8abd2351e13b}} denote the values of the integrand at the quadrature collocation points assumed to be the Legendre-Gauss-Lobatto points {{cite:fddb0139a8b014b3e0c2af574ca29d108219b487}}. Also, in evaluating the integrals, we have used the spectral approximations (REF ) - ().
| r | d48a3340978305fd5ca1d83780891ef5 |
Despite the advantages of collaborative learning, there are two major concerns input data privacy and vulnerability of locally trained models to information leakage. For example, model-inversion attacks {{cite:6b594202c77c744a3df9d6e7677c1fb59926ca6b}}, {{cite:265dbda88f245343b35119c22da78a7232b04f50}} are able to restore the original training dataset, hence it is essential to keep the local models private. Apart from the privacy issues, the cloud/aggregation server may also behave maliciously by returning forged aggregated results to the devices for making a wrong impact on model update. Under worse circumstances, a server may also return carefully crafted results to the devices for analyzing statistical characteristics of the shared updates and thereby provoke the devices to unintentionally expose sensitive information {{cite:fd60afc0c406916999d1a070c028ca474f07ba0e}}. Lastly, since the IIoT devices are typically low-powered, they may become inoperable (i.e. dead) due to energy drainage resulting in device dropout at anytime in the network.
| i | cc211d31977c0ccecb9878a48daac93e |
Paired GANs cannot handle such unpaired I2I translation problem (Fig. REF a).
To combat this challenge, researchers have proposed various unpaired GANs, by using the cycle consistency loss {{cite:9d087780906c0a8754025f54d5f5cc039c556aff}}, {{cite:631701e6df286db3dec6bb4d3cc8e8685d57956c}} or learning disentangled representations in latent spaces {{cite:bad909d5da89ac3941d7d921960ef69beabf2231}}, {{cite:4dc17efa3a341bd1fae9ff24bbbde6c801927d99}}. These methods simultaneously learn the mappings {{formula:df25f6e9-992f-4f97-87f9-b19b66896cfe}} and {{formula:e9798af1-79ee-4ef9-8897-c2fcc02a5bce}} (Fig. REF b). Although unpaired GANs perform well in various I2I translation tasks, preliminary experiments show that they fail to generate structure-consistent and stroke-realistic sketches.
| i | 894c3de5055964c2a2861ab6c10d5423 |
Third, in addition to testing, analysis techniques should also developed to understand the root cause of the system violations. To achieve this, more research efforts on the fault localization and repair are necessary.
Threats to Validity. In terms of construct validity, one potential threat is that the evaluation metrics may not fully describe the performance of controllers. To mitigate this threat, we used five evaluation metrics and two falsification tools to comprehensively measure and analyze the performance and reliability of CPS in our benchmark.
In terms of internal validity, one potential threat is that the behavior of a CPS can vary when using different environment parameters. To mitigate this threat, we chose to use the same parameters as described in the documentation of each CPS to keep consistency. Further, we confirmed that our simulation results are consistent with the source descriptions and demos.
In terms of external validity, one potential threat is that our analysis results may not be generalized to other CPS systems. To mitigate this threat, we tried our best to collect a diverse set of CPS with different functionalities, system environments, and control tasks.
Related Work
CPS Benchmarks. As we mentioned in §, collecting benchmarks of CPS is challenging. An annual workshop, namely ARCH,
aims to mitigate this problem by bringing together CPS benchmarks and holding competitionshttps://cps-vo.org/group/ARCH/FriendlyCompetition for different research topics.
The most relevant competitions to this paper are Artificial Intelligence and Neural Network Control Systems {{cite:35503af64364a010aa07f7121987d853a79d3704}} and Falsification {{cite:f457086858c5b658e07b69de9666d2bbc3da6da7}}. However, the benchmark in the second competition only includes traditional CPS rather than AI-enabled CPS. While the benchmark in the first competition includes AI-enabled CPS, their benchmark includes less and simpler CPS such as Cart-Pole, which are not from industrial application domains. Furthermore, their AI controllers are simple feed-forward neural networks (FNN), rather than DRL models.
AI Controllers for CPS.
Duan et al. {{cite:6ce180605178924a065c01effcf37c7dd80c0a14}} proposed a benchmark on continuous control tasks. However, this benchmark involves game scenarios only such as Cart-Pole and Inverted Pendulum, rather than complex real-world environments.
Besides DRL, FNN also has been used in designing a tracking controller for a robot manipulator {{cite:4638c9a8c78a68c1533db3fb2b7d20265e1dc459}}; however, the FNN controller is a subsystem which can only be used to compensate a feedback controller.
CPS Testing and Verification.
Currently, most of the research efforts are devoted to formal verification of such systems, since it can give rigorous proofs on their safety. For example, reachability analysis {{cite:f9503f0b7d826e2e87ce46e9fdbb5cfb5fc9af45}}, {{cite:5f09bf3927194d269e1f5c256a33341a08bfcad6}}, {{cite:9715e869c931e9f2cfd89dcb0046d5b3ae24dfbc}} has been extensively studied and considered as one of the most effective
ways to verify AI controllers.
The other line of formal verification of AI controllers is based on constraint solving, such as DLV {{cite:61ad0db89ef3ea4406e0ac28289b9e10ad7b03da}}, etc.
Although these approaches work effectively and rigorously, due to the intrinsic scalability problem of verification, their evaluations are usually on simple benchmarks.
Falsification is considered as a method that suffers much less from the scalability issue than verification, and this is confirmed by the empirical study {{cite:e0253aea223374f6a8b884692ef4e265c8e8cdff}}, in which they compared the effectiveness of model checking and testing on CPS. However, existing falsification research mostly focuses on CPS with traditional controllers, and does not consider the specific structure of neural networks.
Conclusion
This paper presents a public benchmark of AI-enabled CPS in various domains, which can serve as a fundamental evaluation and testing framework for enhancing the understanding and development of AI-enabled CPS. Based on this benchmark, we collected a series of evaluation metrics and measured the performance and reliability of state-of-the-art deep reinforcement learning (DRL) controllers on various types of CPS. Our
findings reveal some strengths and weaknesses of AI-enabled CPS and highlights an opportunity of strategically combining AI-enabled CPS with traditional CPS.. Furthermore, our analysis of two widely used falsification techniques on AI-enabled CPS motivates further improvement of these techniques to account for the unique characteristics of AI controllers, in order to build safe and reliable CPS in the age of AI.
| d | 46b028ff26dc2bb61577d1922df634be |
According to the long-term photometry, another properties have come out in this case. Although there are some small variations, the main shapes of the light curves are usually the same. There are always two minima, and they are almost constant according to each other. The analyses demonstrated that one of the effects on the light curves is the ellipsoidal effect. The {{formula:0e167bce-6fb3-4a69-9bde-ac485d8243dc}} was found to be -0.0400{{formula:49a377b5-067a-4bf2-b393-949f7b5cf968}} 0.0004 for B, -0.0383{{formula:9a6e994e-254d-4e7e-ac21-3e7eac20d04f}} 0.0004 for V and -0.0358{{formula:3fc944e8-e08d-46a3-ad24-5eb021b47963}} 0.0003 for R-band. These coefficients are larger than all other coefficients in each band. According to {{cite:cc37b29d478452a400fc65095a250aa020127e87}} and {{cite:d8ab8aeb05d22a7a7738d82d6f31d3b80724f7b9}}, in this case, there is an ellipsoidal effect on the light variations. Apart from the {{formula:d2749671-4d43-4261-ac4d-9d4a991588c7}} coefficients, the second dominant coefficient is {{formula:afd55a9f-1e61-4bf2-b51c-55a38a36491e}} . According to {{cite:d8ab8aeb05d22a7a7738d82d6f31d3b80724f7b9}}, if there was only one spotted area on the surface of a star, it would be expected that the {{formula:764360a5-fd8f-4207-9013-735d345cbad5}} term must be dominant. However, {{cite:d8ab8aeb05d22a7a7738d82d6f31d3b80724f7b9}} noticed that the {{formula:fb255f16-7600-48f0-a386-3801974497bc}} term is not be dominated due to only the ellipticity effect, but it can be also dominated because of two spotted areas separated 180{{formula:2edecf5c-9467-4ce4-9396-ffe6de392366}} from each other on the surface of a component. However, there is a way to define the real reason of the {{formula:4128fb3b-0cf0-42f8-a364-98ad1c729724}} term dominated in the Fourier analysis. This is the evolution and movements of the spotted areas on the surface of the star. If the reason of the {{formula:bd0a1226-0535-4b06-a454-408d78562cd9}} term is stellar spots separated 180{{formula:671d81ff-4ac1-4ad4-9f92-f6115ea3b30a}} from each other, the results of the Fourier analyses will change with time. Because, the main shapes of the light curves will change. In this study, we always found the {{formula:381a1541-3fcb-4acf-a9b6-f1495c79f89a}} term to be dominated for each set. This demonstrated that the main effect on the light variation is the ellipsoidal effect. However, one can suspect that V369 Gem may be an eclipsing binary. In this point, the suspect can be tested by the method described by {{cite:cc37b29d478452a400fc65095a250aa020127e87}} and {{cite:569bda3704034fa3d293bf2fe2786633f2855037}}. In the case of V369 Gem, although there is not any available radial velocity, we determined the inclination angle ({{formula:c5e74d9c-2af6-40d0-b8f4-b4b611565188}} ) of the system from the light curve analysis with the PHOEBE V.0.31a software, and it was found to be 44{{formula:b1cb5ab6-da97-419d-aab5-04988200d698}} .25. In this respect, it is not expected that the system exhibits any eclipses.
| r | 6e14e09edd9f2399c966349c00782c50 |
[Proof of coro:1]
When {{formula:1a0ee85e-e274-4f7e-a9fe-d480016d6b13}} in (REF ) is true, {{formula:dbaecffa-d169-40b3-b2ee-79be52cfa472}} is an independent copy of {{formula:2b7d4a74-9345-45bb-b91f-580c577e980c}} for {{formula:651c4b44-cacf-4825-bddf-8fdeb244aa29}} , which, along with the fact that {{formula:73acfdb1-5d3d-484f-ba64-ddd051f9ef4f}} by definition, gives {{formula:939990e9-eef8-4eb5-9f6b-770ec030655f}} for all {{formula:87dc38c3-b0d9-4790-8c64-bd5de1773c2d}} . So, {{formula:8fe8d991-e7ad-4f57-9914-89c672de3e51}} for all {{formula:5abb865a-89de-4bbb-be61-8d58106347b7}} , and consequently, {{formula:0607432c-ee3c-42b9-a465-411fbdc26750}} . Therefore from thm:1, we have {{formula:3bbb41bc-1ca2-4d85-a01f-4a0fbcb8c97f}} as {{formula:c7264092-e9dd-4a49-ab43-1ee5bf30fb8f}} . Hence, from an application of the mapping theorem {{cite:40901d49d85974fc931f5f7bd7f1d02f4a309028}}, we get
{{formula:29ab0367-8b33-416d-9c5d-e0f107849172}} as {{formula:985757c1-4df4-42a0-921c-652c72f3a4d9}} , where {{formula:d026150a-e450-49cf-983e-441340183fc3}} is a random element having distribution {{formula:e86308d6-61d2-404c-a0ca-fccb91cf8f9b}} .
| r | b2eb46d610481f48840751fdc446d17f |
Strongly interacting holographic systems have proved a very useful arena for understanding chaotic dynamics of large-{{formula:e23a89fb-77a7-45b5-ac28-a376480758d1}} theories and phenomena related to thermalization. These have in turn developed our understanding of quantum gravity dual to such strongly coupled systems in the bulk. More specifically the chaotic behaviour of large-{{formula:840c86b2-5ac1-4172-9950-898a66bd9afd}} strongly interacting theories has long since been shown to be mimicked by the scrambling behaviour of black holes in the dual {{formula:a2fb33c2-2230-4a3b-bea1-ec62e8cf6e11}} to in-fallen perturbation {{cite:27c47d317d15e184d2c3cd7de770a87abb9f4334}}{{cite:0dc60934229b3d2f49033be33eabc485fe23c0af}}{{cite:7193ebe5e118bcfd6a5bb8928ed667cdc2482cb5}}. It was famously shown by Shenker & Stanford that the finely tuned mutual information {{formula:1fbe6e5a-ee6e-4c93-bf75-3ef5de3192f6}} contained in the thermo-field double (TFD) state dual to a static black hole in {{formula:13afda1a-6e8e-447a-b3d6-ba08e784eae0}} with temperature {{formula:cbbdac25-fd6c-430f-81d1-f58e5e867e5a}} is perturbed due an in-fallen perturbation of {{formula:ffca81b8-37f9-4678-99b3-0929e2e793f3}} at a rate controlled by
{{formula:28c9fab5-bd7e-40d1-8df1-0cc50e948b99}}
| i | dd0d6cf549bb6f1543378bb03b14ebdd |
Designing separate dnn-aided edge devices to form a deep ensemble is a consideration that has to be accounted for in the training of the models. For once, the individual dnn have to be different from one another in order to benefit from collaboration {{cite:0f99dd943e0d2b52bd801adec4f0950f7cf0305e}}. This can be achieved by either training the dnn jointly while boosting diversity, e.g., by using a regularized objective as in {{cite:3546e772ffaf3d5d4c813789b7f897a03b7cb82b}}, {{cite:77d78d81572ad5dee48cd8098f6c7435b69a0060}} or different randomized initialization {{cite:00275042ec1efa43f5a3335c96d34fb0b3d0f337}}. Alternatively, diverse local models can be trained by modifying distributed learning algorithms to result in different individual networks {{cite:e2376e7cea459a631c1a76e184145ecab9f9f62b}}. In our numerical study reported in Section we use both a centralized training approach where the local models are trained with the same data and diversity is achieved by different initializations, as well as a distributed-oriented training approach in which each model is trained using shared and user-specific data. Both techniques are shown to yield diverse models allowing the users to benefit from forming an edge ensemble.
| d | cb3ab80df92596ca0326e985fd0f3b5a |
In Fig. REF , we compare our method with two recent proposed stroke-based image-to-painting translation methods: 1) “Learning-to-Paint” {{cite:5fbd490b33c1bcec3f0de9233bbad91ff8af126c}}, and 3) “SPIRAL” {{cite:bd16ee7ad6b9988dc0d4a7e5148986a71fa9c171}}, where both of them trains RL agent to paint. We can see our method generates more vivid results with a clear distinction on brush textures, while other methods tend to produce blurred results. We also compare the stylized artworks created by our method with those created manually. Fig. REF shows the comparison, where the second column shows the results create by a famous artist Adam Lister from New York. Those manual artworks are from his official gallery website.
We can see both the manual result and automated results present a low-bit artistic geometry painting effect.
| m | 7596edf7793df0eb439e57389ae4f1c6 |
Due to the non-stationary between individual sessions and subjects of EEG signals {{cite:e6ab9f6ec61fb9685539ece3ae3c61279fa828ee}}, it is still challenging to get a model that is shareable to different subjects and sessions in EEG-based emotion recognition scenarios, which elicits two scenarios: cross-subject and cross-session (i.e., data collected from the same subject at the same session can be very biased, detailed description is given in Section REF ). Besides, the analysis and classification of the collected signals are time-consuming and labor-intensive, so it is important to make use of the existing labeled data to analyze new signals in the EEG-based BCIs. With this purpose, domain adaptation is widely used in research works. As a sub-field of machine learning, DA improves the learning in the unlabeled target domain through the transfer of knowledge from the source domains, which can significantly reduce the number of labeled samples {{cite:3e01954ca22bb6d24aa1c47c86df9113bd541339}}. In practice, we often face the situation that contains multiple source domain data (i.e., data from different subjects or sessions). Due to the shift between domains, adopting DA for EEG data especially when facing multiple sources is difficult. In recent years, the researchers tend to merge all source domains into one single source and then use DA to align the distribution (Source-combine DA in Fig. REF ). This simple approach may improve the performance because it expands the training data for the model, but it ignores the non-stationary of each EEG source domain itself and disrupts it (i.e., EEG data of different people obey different marginal distributions), besides, directly merging into one new source domain cannot determine whether its new marginal distribution still obeys EEG-data distribution, thus brings a larger bias.
{{figure:ecdb3897-91db-4205-919e-c630bb96ab89}} | i | 6fc3154d16564b8f74cf32ebdab8b1c5 |
The first attempt to overcome this dimensionality curse was the {{formula:d8b17633-b146-46c3-94d0-63a3e80d9eb8}} -tree {{cite:5cb9db75ce997152699a1fb714441d8e1face83d}} that subdivides a subset of the reference set {{formula:a27f3561-6aa3-4133-b6fc-1074c73b40af}} at every recursion step into two subsets instead of {{formula:ca051e41-9db3-414e-a05c-82ae9a581ac5}} subsets.
| i | 84a13b88ff99f706177923e1eb751297 |
The loss function, as expressed in Isola et al., 2017 {{cite:f3b0c45ec931d3634d2fd1a16acb1fa1f436e159}}:
{{formula:64ea86cc-8085-4dc1-a504-7a8fc3c96deb}}
{{formula:ca53e3da-4b7b-4f27-8811-33d20603efb4}}
| m | 83a8d861072eef521053a623c6852f87 |
In this section, we set K=3.
To be specific, the lengths are set to 16bits, 32bits and 64bits, respectively.
For fair comparison, the experimental settings of our model and other methods are the same as that of DCMH (please note that the experimental settings of this section are different from Section REF , more details can be referred to {{cite:017242acdf6afb6440027e6d1e188ccf163904b9}}).
To be specific, we adopt CNN-F deep network pretrained on ImageNet to extract the image features, the text samples are described by the BOW features. For MIR Flickr database, 10,000 image-text pairs are arbitrarily
selected for training, and test set is built by randomly selecting 2,000 text-image pairs. For NUS-WIDE, 10,500 samples are selected for training and 2,100 data points are chosen for testing.
We conduct comparison with some recent deep methods on NUS-WIDE and MIR Flickr.
Table REF reports the experimental results.
It can be observed that the proposed MOON achieves good performance on NUS-WIDE and MIR Flickr, the performance is also superior to some comparative deep algorithms.
The above results further demonstrate the efficacy of the developed model.
{{table:6f65514a-cad2-4217-8325-b4aff2f9830a}} | m | b23a652d7fa0a418c81a87c03a9d202c |
Visualization of Feature Distributions. In Figure REF (b), we visualize the distributions of the features using t-SNE {{cite:fc4f1ee19406be73162fc2722ac53354401e6e09}} on MSMT17. We compare the feature distribution with the baseline scheme of All, and observe that the features of different identities are better clearly separated for our scheme MCL, 50%, which demonstrates that the learned ReID representations by MCL are more discriminative.
| r | 59c0985a4f4ac8cfe4d565468f91f8c8 |
Model. We build our models based on ELECTRA {{cite:a52ccfd062ba5931b5ef658d15b31c43253c04c8}}, since it is shown to perform well across a range of NLP tasks recently.
We introduce randomly initialized task-specific parameters designed for each task following prior work on each dataset, and finetune these models on each dataset to report results.
We refer the reader to the appendix for training details and hyperparameter settings.
| r | 44da9825bda491b44ef8cb658c6b335b |
Despite the past three decades of thick-disk studies, there is still no consensus on models for
the formation and evolution of thick disk.
The proposed simulations of thick disk formation can be generally
divided into four groups: (a) accretion from disrupted satellite galaxies {{cite:dd190df89fa8826a5ddda5bc1522c109ec2b14ca}},
(b) heating of the pre-existing thin disk due to minor mergers {{cite:3974ce443787d3ba25a03fed17fd4bc30728e37b}}, (c) in-situ triggered star formation during
and after a gas-rich merger {{cite:9ac35db8acafd38201c2fc1bae5d0e2e3844b6fd}}, {{cite:a9d933e5b0acb01f6396410e9590b8773ea5dc31}}, (d) in-situ formation through radial migration {{cite:33ed9b599ebdca24a0189110c258cbb48530ab07}}, {{cite:980163730bb08b33db3483d22c70db99a1a7a3db}}, {{cite:54c4551cb384a7ab2adc842c0fbb5999d8ec3866}}, {{cite:2250dadd3bb5d40bbf8feeda4f5993d53ca32506}}.
| i | 6d2b4c86b028a3ee3e77370c97df7219 |
We are well into the era of gravitational wave (GW) astronomy with
the rapidly growing catalog of GW events detected by the LIGO-Virgo collaboration {{cite:a16009cc8246474bb73036203f438927eecf2265}}, {{cite:8e897cc197ac5480aaf0db20c1a909fed5bb976d}}.
| i | c9bd7884d0af2ba03397f25e171ef6ca |
Language is not static, it evolves continuously {{cite:b8346ac26e2564b72131194ba70dab2c3d3c1a31}}, {{cite:8f0914bc5994b06a6ed12c44deedaa448d1f9839}}. Social and technological changes are paralleled by changes in the language used to describe them. In the nineteenth century, who or what was performing work was changing dramatically, with different groups of people entering and exiting the (free and unfree) labor pool and becoming key parts of the workforce at different times.
{{table:c8a92dd5-a294-4078-82f1-42d4413c0330}} | d | afdc19a2a009898e21ba300772d8b203 |
Third, pooling of treatment estimates can be done in several other ways than presented.
In general, the pooled treatment effect over clusters is a weighted combination of cluster-specific estimates, where the weights aim to balance aspects that influence estimation and are imbalanced over clusters (e.g. cluster size or variance).
Whereas we applied a cluster size-based approach, several advanced weighing procedures balance unequal variances within clusters via regularization methods {{cite:cc4ec03eb955089f22e76f874fbfcdfa7d4c4dfb}}, {{cite:ac2413980efc97a6a45a5556cb6d8a597f07de47}}, {{cite:ec6e075e939f7228ffca8c3a536c77709f85f8c2}}.
These weighing methods generally produce shrinkage to the mean a) when group level variance is smaller; and/or b) when sample sizes are smaller {{cite:bf61f4416422cd5a91cd7bd33ed8eecf5aa5dc61}}.
Such weighing procedures have interesting balancing properties but are probably less suitable for trials with clusters of single subjects, such as IST-3.
These clusters have no variance, should not be discarded or merged inconsiderately, and call for the exploration of suitable weighing procedures for such data.
| d | 600f951980664fd9ff7e8b5530d9491c |
Next the NN will update its weight values in the policy network based on the gradient descent and back propagation algorithms. We will keep repeating this process time after time for many episodes until we sufficiently minimize the loss {{cite:55f841f286d6d7c20c1b9ce11d32ab8250279597}}.
| m | d0f3b9fca46e1f137e26e323b88f31c7 |
These models {{cite:5cece5899561e931007ea04483404065e8eabbd4}}, {{cite:61111a20831b686fb0ecaa4e6aaf71608b1658cc}}, {{cite:61111a20831b686fb0ecaa4e6aaf71608b1658cc}}, {{cite:13e0da0d60d9ae278ead8d4bddc31abae10b08c2}}, {{cite:5205db0dc479702347d12eeecd362232aa4a31a2}}, {{cite:13e0da0d60d9ae278ead8d4bddc31abae10b08c2}} perform better than reconstruction-based methods but have two main limitations: first, they work with patches of images, which leads to high complexity at the time of inference, second, to measure distances, they usually use flattened descriptors that destroy spatial positional relations of 2D images. In addition, these techniques may require the use of different patch sizes, as using patches that are too large may lead the model to ignore small abnormal areas and vice versa for patches that are too small {{cite:5205db0dc479702347d12eeecd362232aa4a31a2}}.
| m | 00f6947a4e6ceda22136bde7a172f097 |
Network assortativity is computed through the Pearson correlation coefficient {{formula:4d15c5fb-1887-431d-9349-0e77fce5dab4}} between two nodes connected by a link with positive {{formula:d6937da8-7905-4b46-84e9-afbc35d042b7}} with a value in the range {{formula:2ce14eaf-7928-4edc-a0e4-53300289b8b7}} ; while in case of network disassortativity, {{formula:6b5794ed-2e41-4268-b6b2-f3f6e1959e44}} is negative with a value between {{formula:3c654f95-a209-4b62-a6ed-2590a278c50e}} {{cite:627e4a2f349b4fcadaaf7c24bdecec4d543bc01c}}.
| d | 6ba3b03491a49136f1124d056d8448b4 |
The BNNT, filled and annealed samples were all characterized by Raman spectroscopy with excitation by multiple laser sources (see Figure S1, Supporting Information). In most cases, the Raman spectrum was buried under the photoluminescence background originating from color centers present in defective or strained boron nitride structures.{{cite:cdb78feb0d945dfb093cbef7f7242fc8048b1a98}} The 355 nm laser excitation generated the weakest photoluminescence (Figure REF). The BNNT sample shows the well known {{formula:215c71f7-7b0c-471d-ac18-8ad86bbd4147}} mode of BNNTs at 1358 cm{{formula:c3ef4df6-281d-4eac-93c9-c5270d0a016c}} . Apart from this peak the spectrum contains no other features. In the spectrum of the filled sample (TCB@BNNT) the characteristic G-mode of sp{{formula:13861de6-690c-48bf-ba01-9cf750a2bc4c}} hybridized carbon appears at 1598 cm{{formula:0a157898-57da-485d-9212-27d15239ab22}} . The width of this band decreases on annealing, suggesting the formation of more extended structures at elevated temperatures. In the spectrum of the 700 {{formula:1a1814aa-9c39-4aa2-840b-7e7b8bfa61fb}} C sample a new mode at 1223 cm{{formula:fecf0ab8-5e6d-4b2c-9227-9d9f2b91e169}} emerges. A mode in this frequency region has been assigned to a C-H in-plane bending mode in ribbon-like structures {{cite:0a98790e62dd126c92911bf443722f2ee8d6192d}}, {{cite:ee8d556704556526941b59140107e0e68ef7b052}}, {{cite:70cea6036418532af7d142a85a7d2e4a9179cfcd}}, {{cite:73afb7915fb453b8fbe597d235821f85560619c2}}. We regard the appearance of this mode as proof for the presence of nanoribbons, although neither the low-frequency radial breathing-like mode (RBLM, masked by the photoluminescence background) nor the D mode (covered by the BNNT band) can be identified in the spectra. The 1223 cm{{formula:51be0f90-68a5-4134-bac0-c69127c4d629}} band is not observed in the sample prepared at 500 {{formula:255c7cde-df19-4a68-a119-90ea5517740c}} C, contrary to what was found in carbon nanotubes.{{cite:f9835e6ac9e25bff1aedb787158c2d56c0828e52}} We note that polymerization of polycyclic hydrocarbons happens at lower temperature at carbon surfaces{{cite:1d093fe42faba08f690d25f49ac2ecbc9ff1c7a6}} than in the neat materials.{{cite:a1b81c4c3cba5d58cd26b39037844b66609da8a3}} The higher temperature where this spectral feature appears indicates that boron nitride constitutes a neutral surface with no catalytic effect.
{{figure:83233d75-b8e1-41e7-b9ad-6d0df782ec4f}} | r | 145b904853e0bcfe17245193d7ce5d78 |
There are some works employing other distributional approaches to semantic shifts detection. For instance, there is a strong vein of research based on dynamic topic modeling {{cite:61acf271292767940db647dd168ca772d2187278}}, {{cite:bd49392e0cfe498624c7e5c73070e9532dae61ac}}, which learns the evolution of topics over time.
In wijaya2011understanding, it helped solve a typical digital humanities task of finding traces of real-world events in the texts. heyer2016modeling employed topic analysis to trace the so-called `context volatility' of words. In the political science, topic models are also sometimes used as proxies to social trends developing over time: for example, muellerrauh2017 employed LDA to predict timing of civil wars and armed conflicts. frermann:2016 drew on these ideas to trace diachronic word senses development. But most scholars nowadays seem to prefer parametric distributional models, particularly prediction-based embedding algorithms like SGNS, CBOW or GloVe {{cite:aec5ddeadfcb3203722a31acf99d5c4c2299cac1}}. Following their widespread adoption in NLP in general, they have become the dominant representations for the analysis of diachronic semantic shifts as well.
| m | a0dfce504fe156f83a0a9d93e36f6951 |
Beyond the previously mentioned technical limitations and challenges, the methodology does not provide information about a lower limit to the true error, as it would be required for judging the sharpness of the error estimator {{cite:5812f54f220561bda8a07c547529dc7f8434435f}}.
| d | 957470d047f0682ad7008a55ffd9599f |
We have studied the dynamics and propagation of jets launched from a MS star moving through the envelope of a RG star. We followed, using 3D HD simulations, a set of jet models (either self-regulated or constantly powered, and with different kinetic luminosities) in three phases: when the MS is grazing the RG, when the MS star has just started to plunge-in through the CE, and when the MS star is well within the CE and about to end the plunge-in. Our numerical simulations show that jets can be drowned or successful depending on the jet efficiency and on the evolutionary phase. Grazing jets and jets launched at the beginning of the plunge-in stage may be successful. Meanwhile, jets that are launched deep inside the CE are drowned. As shown in Section , it becomes increasingly difficult for the jets to break out through the accreting material as the MS star moves inwards into the CE. Once the jets are launched, independently of whether they are self-regulated or constantly powered, each jet will be successful if its ram pressure ({{formula:f28d048f-3192-4562-8bac-dc8152295287}} ) is larger than the ram pressure of the accreting ambient material ({{formula:e8e97e9d-9e19-41b6-a438-f9126b63ae58}} ). The jet and the accreting material ram pressures are given by (see {{cite:1efa7b37648f0235ef5cd4b9e1d93e4cd32e7ada}} for further details)
{{formula:4ae948d6-39fa-4cd1-a6d4-e8766aaf8ca4}}
| d | 89f79d83e7b5f35901d0c731d9680b05 |
We now turn to the implications of our results on the surface charge. The observed hole depletion and electron accumulation signifies a positively charged surface,
suggesting the presence of empty, positively charged donor states. The change of the surface charge to negative under illumination with blue light
requires the existence of empty surface acceptor states, which are persistently filled by photo-generated electrons.
With red light, no persistent charging occurs, because the photo-generated electrons do not have enough energy to
overcome the surface barrier of about 1.1 eV {{cite:06e1306c8c208a8c94f5ee6ea327cd12138b070b}}. The observed electron accumulation even for the 6n17 sample means that surface
donor states are still not filled with electrons, implying that these donor states must
be located close to the conduction band - otherwise, they would be filled and neutral. The surface acceptor states, filled under blue illumination, must also be similarly high in energy
as the surface donor states - otherwise, they would be filled as well at 6n17 doping, leading to a negative surface charge, and thus changing the band bending to remove
electron accumulation. For a cleaved Ge surface without oxide layer it is well established that Fermi level pinning exists close to the valence band, causing an upward
band bending at the Ge surface with hole accumulation and electron depletion {{cite:6064f4b1e3dd4b13f785cc8db5f76af4ceb1f825}}, {{cite:1b9dc7984fe514cd2e02e1390a59fc7aa05935f1}}, {{cite:7eae616c814929ddf9074e05767eaa39bdbffb3b}}, {{cite:e57e9fd62a344c3b4928aedf1342033ffdea900a}}.
This is different to our commercial Ge wafers with a {{formula:91dd7128-c8ff-4086-bf3f-454671217e2c}} nanometer-thin native oxide layer. The oxide can exist in various oxidation states GeO{{formula:bd8b97dd-c463-4ca6-8978-c258b0d86c7c}} which may strongly
affect the electronic properities/band bending of the Ge/GeO{{formula:dff6d043-731f-49a8-bc2a-bcb0c2183bc8}} interface {{cite:69123a9e5e4409cd1bd9c70a6082b6ff0d9864a0}}, {{cite:e121176a5c5f3e143edc29a31b8dea21ee975b4b}}. The prevailing oxidation state for native
oxide is +4 (GeO{{formula:cf675fb3-750d-41e9-b08c-a3dbc0263687}} ), with the presence of GeO{{formula:a40378ba-6a4a-4f36-beb4-440d5c23bf4d}} with {{formula:1736d244-74bd-41e1-aff6-f9c34dedcb06}} at the Ge/GeO{{formula:bc373ef1-dc13-4056-81df-a573780758b1}} interface {{cite:52ffc666a59d30a9548d432e3c222d473848d04b}}. Assuming that the band structure at the interface
is determined by the band alignment of GeO{{formula:c09f5034-2e40-47ff-a36c-1c2ae5c6f56e}} with a band gap of {{formula:d55211e8-d390-48f9-9f99-69f5d1da6ab0}} eV and a conduction band offset {{formula:b8202d58-80d0-4997-a842-65f5dbe199db}} eV with respect to
Ge {{cite:47cc84c1265a06a4f5787bfb14c41e9a738e222b}}, we speculate that i), the band bending in Ge at the Ge/GeO{{formula:f9129625-9a59-401a-b6ed-8b0d0770e8d8}} interface is opposite to a cleaved Ge surface,
yielding electron accumulation and hole depletion as illustrated for isotype heterojunctions in
Ref. {{cite:b9517ad9699989473d67d318f431d29a6bc98b68}}, and
ii), the surface energy barrier is determined by {{formula:44cf756d-fc99-4f3d-ba23-bcf0764e2b32}} . While the details of the Ge/GeO{{formula:665b0ac7-c1ab-40bf-a764-b482692f052f}} interface are important for device applications, its more detailed
characterization is out of the scope of this study and remains for upcoming work. Here, our intention is to demonstrate the capability of charge carrier profiling in the near-surface region of a
semiconductor which allows getting insights also in the surface characteristics.
| d | bb0e1278bd87d17df338c12c04737051 |
We further find that mergers are important for the formation of the most massive clusters. Mergers are much more frequent in Region 1, enabling massive clusters to form on shorter timescales, compared to Region 2, where they are minimal. We see similar findings in Rieder et al., submitted, where again mergers are more common in massive GMCs found in spiral arms with strongly converging flows, and lead to more massive clusters. A similar picture of massive cluster formation by mergers was also put forward by {{cite:ef009759233e4944aa0e5faae67ff454d9405597}} and {{cite:275e1b058b16f85451940752f9c4b51e089c046d}}, although those simulations started with isolated collapsing GMCs.
| d | 4ab1b389ae3c95d68cbe78ac252727c3 |
To show how different steps affect final performance, we also conduct an ablation study in Tab. REF (step one is required). We find step two is more important than step three in improving final accuracy (90.8% v.s. 89.8%), but both contribute to final performance. To intuitively present adaptation performance and the effects of step two, we utilize t-SNE {{cite:7be495111d0ae2efa8499d00819f20e0d2b53db2}} to visualize the deep features of network activations in 2D space before and after adaptation (domain invariant/specific features) as shown in Fig. REF . Apparently, the distributions of domains A and D become more discriminative after adaptation ({{formula:486fd90a-6d7f-4817-bd21-9552a3886a04}} ), while many categories are mixed in the feature space before adaptation. In addition, the {{formula:dcbe8696-7dcd-4953-88f5-f4f8a88a4220}} can also distinguish the target domain from the source domain. Furthermore, the distributions of {{formula:d5996064-0a52-4da9-a1ba-7d2b59456670}} and {{formula:8e20f2b7-e959-4a16-86b7-9b1f96e3644a}} are different, which implies that contamination between them is minimized. This result indicates that ESD can learn more discriminative representations.
| d | bc92af71b099ce55b284503a01c01c99 |
For each {{formula:7277feaf-54e7-44f4-9dd4-3ac056c50243}} , the {{formula:b9f5ca6b-9b20-4a49-ba92-768d7a9321eb}} is swept from 1025 to generate {{formula:912193a1-334d-4ba6-9e58-62d90678df19}} {{formula:7d383a62-a35c-4e8e-9d7d-fe4de176804e}} values and {{formula:b83435e8-a056-45ef-b1f1-c80df2f73163}} {{formula:2085e40c-04fb-4846-9db0-1939982d1396}} values using Eq: wmmse,Eq:SNRSINRrelation. Input {{formula:f1a6b7d5-455c-4116-ab75-82a74ff02dc1}} , is a parameter measured experimentally in the transmission setup where the technique is being applied. The labelled set of {{formula:204714d3-9c1d-4414-a0ae-1189e1ac46e7}} and {{formula:b5d40be0-c720-4e4e-88a9-b0d9010f5444}} is fed into the ANN shown in fig:generaldiagram(c) as input training features. The ANN receives {{formula:ddc9dd77-f5b1-4bd4-a2f2-567af16d9184}} {{formula:cc3d3edc-a114-4ff8-a3df-921a97101c02}} values and {{formula:4aab34bf-f468-4cb1-a10e-ea689977f762}} {{formula:6ae78711-87fc-4e44-a6b2-126038229335}} values, and provides an estimate of {{formula:12308d63-0cfd-4d59-9371-35c9a17e8bcd}} or {{formula:a055e009-d6f2-4bfc-9e62-fb103fa64846}} . A hidden layer with {{formula:fa375575-47d9-4914-8718-9371746507e0}} neurons, and an output layer with 1 neuron, learn the relation between the input features and the output. The ANN is trained using the Adam optimizer {{cite:5d44773fc9a326bfe999cea965898df97c76da45}} using batches of 5 samples.
| m | fb8abb87e823ad4cc5afe3aeaac8c8e0 |
In conclusion, we list the main results of this review and discuss problems that have not yet been solved. Among these problems, the most important one concerns finding a sufficient condition for collapse. Recall that for the focusing NLSE, such a criterion was first obtained in the two-dimensional case by Vlasov, Petrishchev, and Talanov {{cite:d8a41cce069a1e42b2d50f7183355ea39b4f12bd}}, and then generalized by Zakharov to
three-dimensional NLSE {{cite:d6f3f126432150c6601e454447ad1c858cc36b21}}. The sufficient condition coincides with the negativeness of the
Hamiltonian {{formula:a75c1cbb-6123-4fb6-a950-29391c4d37e9}} for both {{formula:20eafd45-9358-4053-ba25-126b425d8d32}} and {{formula:6bbb553e-ea1f-4a10-8713-39c443622c57}} .
| i | 0edb9323f218647fae5048b2b7cb39b5 |
To closely examine what the weakly and fully supervised ResNet models are learning, we plotted Class Activation Maps {{cite:71cd360c973795a3e658693370793627a01ab73e}} of a normal and PVC beats in Fig. REF . It is evident that to discriminate between PVCs and other beats, our models are primarily paying attention to the QRS complexes in these examples.
Moreover, it also appears from these plots that both models not only perform on par, but they also tend to focus on similar signatures of the ECG signals.
This observation suggests at least some equivalence between the model trained on ground-truth annotation and the one trained on labels inferred from a small number of simple heuristics.
These results reassure us that the more expensive process can be effectively replaced by the proposed framework of weak supervision that uses a few labeling functions based on high-level aspects of domain knowledge derived directly from the time series characteristics.
| r | eb03ddfba2e49c9e8c4d219ad2b34065 |
From Figs REF and REF one can see that for times {{formula:d6740c43-42d6-43c5-989b-cb3586efac24}}
there are such time intervals shorter than {{formula:d2aaca3a-d3eb-4897-a96b-11cdd4f5163e}} , that {{formula:7ae62e53-cd82-4173-9efc-01556d3f2a69}} is positive at some of them and negative for the others. In general {{formula:1c721d59-5509-4b8f-a090-9a40651ee52a}} is oscillatory modulated at this time region
by changing its value smoothly over time from positive to negative and vice versa.
These properties of of {{formula:ad7ab3c0-ea6d-4513-9d88-cc7134dfa070}} are reflected in corresponding, analogous behavior of {{formula:fa9569c8-e37f-4ea7-803c-9a0b18103bd5}} on these time intervals: {{formula:32e5bbbc-e490-4f5a-bb37-892ab3b0405f}} is oscillatory modulated for {{formula:8863c986-0e2d-407d-85dd-92fa0f5375c0}} .
As a result, acceleration {{formula:6cfacaa7-c653-4948-85e3-bab08aef3ffb}} increases or decreases depending on whether time {{formula:da20903c-3e84-47c7-9ee0-89b087a3b9d5}} runs over the interval with a positive {{formula:53ff796a-d53f-45bf-b410-ab04af7b998c}} or a negative {{formula:8ebb73b5-1c0c-41c4-bf54-093dcc5afb10}} and thus the radius {{formula:d6491ccc-68d0-4b10-830c-586eed562629}} of the sphere increases slower or faster: Simply our hypothetical sphere under consideration with the radius {{formula:8876c35f-db27-4c06-99c9-609afd06a5e3}} is vibrating. In other words, within the considered scenario the Universe in the decaying false vacuum state is pulsating for {{formula:7ca0fe1d-2324-438b-b790-d308d29cd1ed}} .
From Eq (REF ), one more conclusion follows: If to consider the time interval {{formula:1bec1645-bded-420a-b5e1-5644c1a6186d}} only and insert the oscillatory modulated {{formula:2c668fd6-ae92-4925-ad13-1cd69653d319}} into this equation than one can conclude that the Hubble parameter {{formula:4db7bf61-b56f-4879-a005-3f60aaeaa47c}} should also be oscillatory modulated at this time region. So, in general, properties of {{formula:da651b63-4cbd-4233-a46b-d95878a59392}} and thus {{formula:d3e4efea-b0f7-4ee3-b2a6-009ad7ae8025}} at times {{formula:391d45d0-fddb-46f3-a727-b4e6bb63aaeb}} generated by quantum mechanism considered in this paper correspond with some properties of Early Dark Energy (EDE) or New Early Dark Energy (NEDE) recently discussed in many papers {{cite:50d4fdfb0536abdebc6dc1db206463f4142d0428}}, {{cite:b5dff9ad28287bf35b94977729d0bf0ed4c84eae}}, {{cite:f6f98d73c976795bff869f16e6709d1c3c50c80c}}, {{cite:3e8efad42f8cac1181cf6b8071bf4089f6739fa9}}, {{cite:589040e4e34f9c7d21221586fff2d35d8a9f2fce}}.
The advantage of the above–described quantum mechanism over the EDE or NEDE theories lies in the fact that this mechanism requires neither additional fields generating EDE nor oscillating potentials (see {{cite:50d4fdfb0536abdebc6dc1db206463f4142d0428}}, {{cite:b5dff9ad28287bf35b94977729d0bf0ed4c84eae}}, {{cite:f6f98d73c976795bff869f16e6709d1c3c50c80c}}, {{cite:3e8efad42f8cac1181cf6b8071bf4089f6739fa9}}, {{cite:589040e4e34f9c7d21221586fff2d35d8a9f2fce}}).
| d | 8eae170dc36edc0b9775d220bddd933f |
VGG-13.
The VGG-13 {{cite:8dac97487066fd90891ec95759ed3acf731170ae}} model, which is trained from scratch, achieves an accuracy of {{formula:961107e0-4e01-463c-9d97-011e0122d05d}} on FER 2013 and an accuracy of {{formula:05897d1a-282c-4cb1-8cd6-591f156f7a93}} on FER+. Since the input of the VGG-13 architecture is {{formula:97c46275-0f79-4797-81f5-df7f7943101d}} pixels in size, it seems to be better suited to the FER 2013 or the FER+ data sets, both containing images of {{formula:d3ec22d3-0ad4-456e-ba6e-a6f078b408c8}} pixels, compared to the VGG-face or the VGG-f architectures, which take as input images of {{formula:f672341c-5f69-4229-8770-93821fdb4562}} pixels. However, its lower performance compared to VGG-face or VGG-f can be explained by the fact that the other CNN models have a better starting point, since they are pre-trained on related computer vision tasks. It is interesting to note that our own implementation of the VGG-13 architecture of Barsoum et al. {{cite:8dac97487066fd90891ec95759ed3acf731170ae}} attains an accuracy of {{formula:56783ed0-8e3d-413e-894a-c2cb980b1d7e}} on FER+, which is {{formula:e61d9ea3-4c86-449e-b76e-6ee85a66cf63}} less than the accuracy reported in {{cite:8dac97487066fd90891ec95759ed3acf731170ae}}. We believe that this difference is a consequence of using the standard softmax loss function instead of probabilistic label drawing.
| r | fe355ae459b941d5e0feff94837182a4 |
ANNs based on photonic technologies developed for telecom applications {{cite:7aab735e3565f2bd292beefc3a98dce1a42a0c48}}, {{cite:dad1df8647e3ea12215e430da3bbedc1fcfd86e2}} can represent a valid alternative to conventional electronic hardware for the achievement of a significant reduction of the operational power and increase of the speed and parallelism {{cite:32d94f945c4e47f929e5d26e243dc954f9f5094c}}. Various photonic ANN models have been reported {{cite:dad1df8647e3ea12215e430da3bbedc1fcfd86e2}}, {{cite:32d94f945c4e47f929e5d26e243dc954f9f5094c}}, {{cite:96e5c4aa8447c41083f4a101d1bfae2eb9643eae}}: as in the case of their electronic counterpart, the architecture of optical ANNs is characterized by a lack of similarity with respect to biological neural systems, where self-organization, redundancy, non-linearity, and non-locality governs both structure and functions {{cite:0d44c76ae12236819df5bb8d330a78f05257871b}}, {{cite:1745473b33468bc06490a125b03b978665b4f446}}, {{cite:7d8e7bfae3ccb21a200fbc1991719359df2b14b0}}. Neurons utilize a wealth of nonlinear mechanisms to transform synaptic input into output firing {{cite:24a28998caaa45acf407986fdd5ac94790396a5c}}, {{cite:ee1f201a84e6068418f9a4e761803aa7ae7c598b}}; the majority of inputs to a neuron is received primarily through synapses made onto elaborate treelike structures called dendrites {{cite:3a538a61ed768cac8f767473657f0ea18860dde9}}, allowing transformations that extend far beyond the simple sum-and-threshold operation {{cite:63599c1d56b1656a67ccca72858f94b631353e4c}}. The morphology and the electrical properties of dendrites define the input-output relationship of neurons and the rules for the induction of synaptic plasticity {{cite:eb4dc3cf1f05417656bf7388f596133183f7e9f5}}.
| i | 312a1ba494f637085979810341b2e922 |
For each theorem, we randomly generated 100000 sample distributions (observational data and experimental data) compatible with the causal diagram (see the appendix for the generating algorithm). Each sample distribution represents a different instantiate of the population-specific characteristics {{formula:61784e08-2435-4e61-a931-a4df090cf988}} in the model. The generating algorithm ensures that the experimental data and observational data satisfy the general relation (i.e., {{formula:fb832b0b-c43d-4e66-afdc-7a13b1505820}} ) {{cite:68043978627e4321e0c6a31a747d69bd4cf473df}}. We set the benefit vector {{formula:d356f6e1-4200-4613-98a7-46a43eb87805}} to be the most common {{formula:c5ef2395-dfdd-49d3-99af-e81b9d7057da}} to encourage compliers while avoiding always-takers, never-takers, and defiers. For the sample distribution {{formula:74be7d9a-6682-4474-acda-8e2ea3943ba7}} , let {{formula:ad1f923a-2b0e-4a5b-900d-14f66f907155}} be the bounds that considered the covariates and the causal diagram from the proposed theorems and {{formula:1e7f1e57-5057-487d-973b-3b46dcdd61f7}} be the bounds that did not consider the covariates and the causal diagram from Li-Pearl's Theorem. We summarized the following criteria for each case:
| r | 5662946f95d0901c8e97ee9e209db4af |
Observations of compact quiescent galaxies, `red-nuggets', at {{formula:ea52687f-850b-481c-a96f-6774574b2240}} have triggered a fruitful debate over the onset of star formation quenching and the factors that may affect it in the early Universe {{cite:638418a1a671c7f7786347e8bf9fa3e84a4ce6e0}}, {{cite:39a15a935b49109f711dc019eb36b69b24548010}}, {{cite:a2867efd3e4d90c48ae143e8dd2e3b7ba3508db7}}. In {{cite:98539be28c40e1fa79b986eeb58c5158c8144dbf}}, we found evidence that massive galaxies after {{formula:d852eeac-6ec9-4ec2-928d-5d42f35f5251}} buildup their stellar mass from ex-situ processes in both MOSEL observations and IllustrisTNG simulations. We also observed a significant increase in the stellar sizes of the massive galaxies at {{formula:5db8a234-da98-4b19-b548-bfbf3152ed94}} . In this paper, we have analysed the TNG100 simulation to gain insights as to why the stellar sizes of some massive galaxies increase dramatically around {{formula:5dd8e74e-cf70-4028-ada5-2cf997f19174}} and how this affects their subsequent evolutionary history.
| d | fcfaf61fd79c46bb2e806788d1a48fad |
Additionally, we have investigated an important limitation of our method: accurate predictions require similar mesh connectivity, i.e. our method is sensitive to remeshing of the input surface. We hypothesise that this limitation can be alleviated by data augmentation. We find that PointNet++ is more robust to remeshing, so it can be an option if heterogeneous mesh size is more important than {{formula:fc6cf01e-ed59-4049-9a19-fa3c1722424f}} symmetry. Furthermore, we see this as an opportunity for discretisation-independent neural networks, e.g. {{cite:da218683d88044c0a830354ffa790ff81fe9300d}}.
| d | 7581f61244abd6eb10ca15c9afb8d589 |
Let us then recapitulate the present work briefly. The study of precision cosmology is not possible in GR frame work until we completely know the physical system, but it provides a way to model the deviations in Einstein-Hilbert GR action. Deviations from Einstein GR theory are indeed predicted mostly in various extra-dimensional theories. Contrasting alternatives to Einstein GR are actually useful to understand precisely which features of the theory have been tested in a particular experiment, and also to suggest new experiments probing different features. Such study can be modelled in the {{formula:cb98ba3b-3056-403a-81f7-78fdcc99b01d}} gravity framework. Therefore, we study the deviations in GR in the form {{formula:53cb182e-bf8a-46c0-8913-f899df779ba2}} with {{formula:c1ade953-1b1e-4e61-ab56-3a22e4f0934f}} being the dimensionless physical observable quantity. We focus on the study of null and non-null geodesics at local scales and explore it under conformal {{formula:4633c587-af98-4d29-b2c8-babf9f9973b9}} theory. Our discussion shows that {{formula:e7b53952-631a-4c8f-926e-b756451ca7f3}} conformal transformation of spacetime metric ({{formula:97e7e245-1fa8-4ac8-96d9-5787821087ff}} ) leaves the null geodesic unchanged (Section III). On the other hand, the effect of such transformation for non-null geodesics produces an additional force on the unit test mass which can be traced under the Newtonian limit (see equation (REF ) or (REF )). We discuss the extent of this additional force in {{formula:76673dd8-009d-4bf7-b985-434071f634bb}} Schwarzschild background and investigate its relativistic effect on the perihelion advance of Mercury orbit by obtaining the expressions for the {{formula:9c747368-21c4-4d75-a05a-e6175d00656f}} conformally related orbital precession of Mercury. We also explore some features of the {{formula:e5fc822e-af9d-4c0f-b80a-be9d9fa3c227}} model for small deviations in Figs. (1), (2) and (3), whereas in Figs. (4), (5) and (8) we investigate the physical observations of the system confronted with different geodesics along with its conformal analogue and explore the precise deviations at different scales.
The main motivation to explore the deviation {{formula:991a3402-a1c2-4bd6-aa17-ba32c7d51899}} in the narrowest value comes from the recent combined investigation of
galactic dynamics for the power-law {{formula:882f8ef0-be2b-4b52-b7c8-c15b749cf684}} model {{cite:6f01ed66cc1744e3ea50babc3781d897a5f2395b}}, {{cite:fba6324f9dcccb0f817aaf2576233d7a0e852e5f}}, {{cite:f3e64a94503ea2f3c7f9186009e65fd356fb43b6}} as well as from {{cite:ba78210d4ea2ff62d6ca035a1213044a58975d1c}}, {{cite:a2b880bae5e10f7ce50970526d90451b91e10fd1}}.
| d | 015c3f247068d1fa5adb370c395fa115 |
Popularity of this GKLS master equation, many times referred just
as Lindblad master equation after one of the inventors, has been
and is remarkably extending in many fields in non-relativistic quantum
physics. It is understood as a Markovian effective equation of
open quantum systems {{cite:5b4760336195cdddb26c882091cb01d9c8caf094}} whose exact dynamics
is non-Markovian. I used to share this view. The only exact Markovian evolutions
are unitary. Exact non-unitary (e.g. dissipative) Markovian evolutions
do not exist. Lorentz invariant Markovian dissipation is impossible.
| i | d5e3f83ce9dbf0cf217831ca94a18782 |
Despite these uncertainties one particular object of interest in the first catalogue of gravitational wave transients is GW170729 {{cite:72a737c7bd929f8fd9f23ad0751f19ca2dad31eb}}, for which the effective spin was reported to be {{formula:76c3afa3-2920-4b5f-a02a-30c36c03560f}} with a mass of the primary BH of {{formula:95a511d8-dbae-47bc-9871-c4802aa1f5b2}} . Although the mass of the primary BH in GW170729 is consistent with the edge of the mass gap as predicted by non-rotating models, most of the {{formula:c9bff945-b4f8-4a02-a836-50b19dc0c8af}} credible interval falls within the gap, which has motivated discussions on GW170729 being a second generation BH merger {{cite:c490e9816d1624b05b5b6e3942bc1ddf9eefff0c}}. Analyzing the posterior distributions provided by the LIGO-Virgo collaboration we find that the individual spin of this black hole is {{formula:05ecfc20-dea2-409e-9914-a051fff2b4fc}} (see appendix ), making it a potential candidate for a BH formed through a rapidly rotating star that underwent PPISN. Even with imperfect measurements, a large number of detections can be used to derive the intrinsic properties of the population {{cite:56bed4fc1ee4676da0184bd295b97a832b729c47}}, which will provide stronger evidence than inferences based on individual objects.
| d | e3f98928d62fa33fda18894133b36e59 |
Matched-filter based analysis defines the False Alarm Rate (FAR) of each detection as the number of false positive detections with an equal or higher ranking statistic, where ranking is assigned to each positive trigger that passes the SNR threshold of the detection pipeline and inter-site travel time requirement {{cite:ff52dade004f986692ae1fc9d1598b0828613411}}. While the FAR threshold used in {{formula:517a6aca-ccab-4a7f-b152-81a17a388e71}} is {{formula:61a2f594-0b77-49ee-8e82-8b6a7c566cac}} , most detections have an FAR that is of the order of years. This is orders of magnitude smaller than the false alarms that are typical to neural networks. In {{cite:57e36c08930528cedb0db03b341553b91326d56f}}, it is noted that the FARs of matched-filter pipelines do not directly translate into the False Alarm Probability (FAP) characteristic of ML based GW search. In our analysis, we define the FAP based on the Peak Amplitude (PA) of a denoised GW signal. We define the FAP of a given detection as the ratio of number of denoised signals with a PA greater than or equal to the PA of a real GW signal. In the case of real events reported in this paper, FAP of an event is calculated using several hours of data before and after the event. This is calculated separately at both detectors, and are presented as `Single detector FAP' in table REF . The single detector FAPs can be further improved by analysing more data around the event and even beyond, hence it is not necessarily a fixed number, nor can it be elevated as a detection standard. Two-detector FAPs are defined as the false alarms from the same data segment at the same time at both the detectors. The two detector FAPs are found to be zero for all real GW events detected in our analysis. This means, no false alarms are produced under the condition that the detection needs to be registered at both the detectors at the same time. We used 10,000 data segments of eight second duration to calculate the FAPs presented in table REF . The PA of the real GW event is typically the highest among the analyzed data, and occasional false positives are counted towards the FAP of the event.
| r | 924520f84583f24d01b4244ea7ac6178 |
The use of handcrafted ORB features, extensively used for SLAM {{cite:ef14116b0652264a435ef5b1fd20d632ae336c2f}}, performs well in between the storage aisles (e.g., view 2 in Fig. REF ) but can also perform poorly at the end of the aisles, where lighting conditions suddenly change, leading to unreliable feature matching. Still, the SLAM back-end is able to recover the track in most cases (see Fig. REF , REF ) but can also fail when, at the same time, the drift in raw odometry becomes too unreliable (see Fig. REF ). This leads to large errors during flight 3 in Table REF .
| r | ae3be212acfbd0861cce35f1ae9678ad |
Beyond our framework design and the compelling motion capture results demonstrated above, there are still something to be discussed or improved.
First, since both video-based Human3.6M and 3DPW do not provide global trajectory for training, our video encoder cannot encode the global motion trajectory information, like other video-based motion capture methods VIBE {{cite:3176643aed0b2bef4de5b34dd46e69446850bf9f}} and MAED {{cite:ab7247fd5c90e736517145c8a654ed68843bd3af}}.
However, our variational motion prior are trained using the motion-only AMASS with global translation. Benefiting from the proposed two-stage prior training, our framework can generate significantly more natural and coherent motions with a certain degree of global trajectory capture ability, see Fig. REF .
Second, although targeted at human body motion, our framework does not rely on any topological constraint and can be easily extended to other articulated objects such as hands.
In future, we expect our method to capture body, hand and face motions in a unified fashion, which can greatly benefit various VR applications.
Besides, our variational motion prior relies heavily on the training data,
therefore, synthesized motion sequences are more related to motion distribution of the training dataset and a more comprehensive MoCap dataset with more complex motions would greatly boost the performance.
Also, our prior can only handle singe-person, extending our motion prior model to multi-person interactions is another promising direction.
Finally, our variational motion prior is trained purely on local motion features, making it less effective in absolute location prediction, such as distinguishing if a person is walking on a treadmill. Combing physical and motion prior constraints is another possibility.
{{figure:e47c7d59-e6d8-4ade-af7c-0601340e76c2}} | d | 67ada5bc3539d444f8bd3ff2a1d05fd0 |
Involving different kinds of noises, the dynamics of the density matrix {{formula:420f40af-1d15-413b-880b-536d8938eb0a}} is govern by the well-known Lindblad equation {{cite:0380a3f64e32d8dbdf338393dddf83730fda123c}}:
{{formula:21658f1a-0d04-486a-b4a7-59f20e9612c5}} . Here, {{formula:3969a2e9-1390-450f-a4da-2b128076b3d0}} is the system Hamiltonian given in Eq. (REF ), the Lindblad operators {{formula:afa34311-20e2-4d86-ae94-6b24495ea235}} ({{formula:d1241ecf-fdb4-4555-90f6-ab1e8cfcdbed}} ) is annihilation (creation) operator for computational qubits and coupler, and {{formula:4d3f8390-5383-452f-93fd-bc0cc6a27518}} represents the energy relaxation rate of computational qubits or coupler (it often relates to the qubit energy relaxation time {{formula:081bab13-9388-4f31-92dd-29e4dbc6ac76}} ). Besides, {{formula:a0989ad8-9e65-4f74-b562-6e10846575e6}} denotes the anti-commutator of two elements A and B.
Using the new representation introduced in this paper, we transform the Lindblad equation to a new form.
To distinguish the new representation from the origin lab frame, we add a symbol “tilde" to every quantity in the Lindblad equation: {{formula:4b219320-ef98-4d4c-874f-856e06d0d838}} . In particular, with the help of SW transformations specified in Sec. , {{formula:9bdef1c2-7c8f-4be2-a551-e2a0edc896db}} is transformed to {{formula:3178a8b2-96dd-4165-94e2-ab5d493f7477}} , and the Lindblad operators are transformed to
{{formula:780656fd-9b5e-4761-ad9b-fd5f9221d67f}} , {{formula:f5fde9ed-2409-40ea-92b9-9059b440a59f}} .
Considering all of these and reducing to computational space,
we ultimately obtain the dynamical equation for the density matrix {{formula:f9136806-66e1-4f68-8630-6657d4b1916b}} in Eq. (REF ). The validation of Eq. (REF ) is verified numerically through comparing with the corresponding results solved from the lab frame Hamiltonian.
{{formula:8ea10069-7590-4f08-adbd-55578602ebd3}}
| r | 302c13c2430f1a6077cb5ebf52cbcb25 |
Figure REF shows the impact of stellar rotation of the Stromlo models on the ionizing photon output. The rotating and non-rotating stars show broad agreement with each other; discrepancies between the (non)rotating models are only significant at the highest energies shortward of 228 Å. The minimal impact of the stellar rotation on the hydrogen ionizing luminosity compared to the helium ionizing luminosity has been demonstrated in prior studies {{cite:58ef818be412760d430a1998adc3fcc9f3865e89}}, {{cite:14030fd2e46a98d51c1669fda75e22e8a0abc6bb}}. The galactic concordance abundance patterns have an equally important role as rotation in determining the ionizing photon output from the stellar properties, especially at lower metallicities; adopting stellar models that are not solar-scaled are critical ingredients in modeled SEDS and the interpretation of observations using these models. Considering non-Solar scaled abundances in stellar models is critical for accurate measurements of the ionizing photon budget from star clusters in local studies as well as measurements for the escape fraction of ionizing photons in high-redshift galaxies during cosmic reionization.
{{figure:96c41b1a-7f8f-4a22-b4e5-3919397898c2}}{{figure:eaa3689e-a45a-4c40-82aa-b001099dceb3}}{{figure:66ad17b3-5569-40af-8598-ae263076214c}} | d | c30906d7a828f926ecfcbf6a5a23c06b |
High dimensional graphical models have become increasingly popular, over the last several decades, for understanding independence and conditional independence relationships
among components of high dimensional random vectors. The challenges posed by the estimation and statistical analysis of a graphical model with many more nodes than the
the number of observations led to renewed interest in these models, and to a very large body of literature, including {{cite:5eef9511039bd85cb078208231259a57921c8a85}}, {{cite:cd7ee55c234c754b3eb4543a2fdbece20d8705dc}}, {{cite:2938c43b9148ad495a8ad56c992daad232479c26}}, {{cite:305d80eb25c9a6936e93f106b47838df4a0987f1}}, {{cite:ccd475434915a43450bbb23fb273f8ebb7c6e4d7}}, {{cite:f85ca78e3c5f68ae3282398532c18fcc7a33423a}}, {{cite:639ab1d1e36050c70373ea1f4ed19b2be6bf98e3}}, {{cite:6038694afbf13572aeb99f2aeff56df93da7c688}}, {{cite:b08c72097038193dc472f218a98f94bddcf9087a}}, {{cite:c5a53efd4d88b70fbb30b64d259ff5fe6c9e6f3d}}, {{cite:fc079b2f5d6888dd5ef6ebba2cd8535f174d8e64}}, {{cite:86ebf9375e58fba9853e504c2f9a37aa8ea390b9}}, {{cite:37658b1b9f69ca2cf477466b047ae9e3cfdd73a4}}, {{cite:2145f227af744d8abe27b301980f0cf6887d1087}}, {{cite:e344383f32b47cc600a4c7700530bc855a6621bc}}, {{cite:5d34d6c468c7fefa3f8aa3aab28f4a175d60c944}}, {{cite:8525121f3484c4a285be5ad461e821557d835d00}}, {{cite:8a1383965f89bdaed4988b216fb96576013c930a}}, to give only an incomplete list. However, in practice, when the dimension is very large and the sample size is small, the dependency among variables may become very weak, if it exists, and difficult to detect without auxiliary information. Moreover, when the number of variables is in the tens of thousands, it is difficult to form an opinion of their overall dependency structure, even at the visual level, from a graph estimated via a graphical model.
| i | 3a4d6211b9543ed13e17133670362e05 |
A first implication of underspecification is that ERM is insufficient to guarantee OOD generalization.
Identified cases of underspecification point at the need for additional task-specific information in the design of reliable learning methods.
If such information cannot be integrated, learned models are at risk of unexpected behaviour when deployed on OOD data, because the depend on stochastic or arbitrary factors (e.g. reliance on texture vs. shape in image classification {{cite:e1dd3b7b370a86804b403b3a7607cd9229ec89c8}}).
| d | 7a1075119396c82777b0fcc24f310321 |
Recently, a multitude of machine learning methods have been proposed to enhance and accelerate physics based numerical solvers in the context of electronic chip simulations. For example, a Deep Neural Networks (DNN) based fast static thermal solver has been proposed in {{cite:163862bad49bacde2fbd910f49c74bb66d7c9087}} to generate a high-resolution Delta T map. In a new branch of study, researchers have employed Physics-Informed Neural Networks (PINNs) {{cite:56dbf1cbc6d747083fa48485f3da74030ce319e5}} to solve chip problems {{cite:4eb39048b9ca56836e663debe81a95fd95ec8dad}}. {{cite:80ec14b7ccd43137a9c79d751b05b6a48cdc45f6}} use PINNs to predict temperature profiles resulting due to variations in the FGPA heat sink geometry for a uniform power map. Similarly, {{cite:a38fe39e9d18e721954e22eab2a313c3ddb6b522}} and {{cite:67c2697d7ab231daf6bace5568389d99e95c3611}} use discretization-based techniques in combination with neural networks to predict temperatures on chips for powermaps sampled from a Gaussian distribution. {{cite:eced2696abc266ca2c8df7b044e2f6706a16bc74}} presented a proof-of-concept approach to provide approximate predictions of steady-state temperatures using convolutional neural networks. {{cite:64c67136262dffae3a07e104ccabafdcc953a9c4}} use a convolutional encoder-decoder network to learn a mapping between powermap and temperature. Most studies in relation to thermal simulation of chips make simplifying assumptions about the power map or other system parameters such as HTC. The high dimensional parameter space involved in on-chip thermal simulations usually makes it challenging for conventional deep learning models to predict accurate temperatures for unseen input parameters.
| i | 9f6ff5f0989ec1bbed2db230ef15c28b |
We test denoising performances on the images corrupted with noise-levels {{formula:1a67ab4d-1a11-453e-b271-c8446014f327}} , on three famous denoising datasets for color images: CBSD68 {{cite:56ba3b5e57e5ab8c957fd2a4855b1c3ca6eb422c}}, Kodak24, and Urban100 {{cite:a3bdb6103e00eb29dfa7c8ae264439a27276fcef}}.
Importantly, considering the practical usage, we mainly aim color image denoising where gray image denoising can also be done with the change of input channels.
| r | 251eda221cd0fc8a642d0b1e3e00570f |
In this section, we evaluate the performance of the proposed algorithm.
There are {{formula:6f8801a5-c056-4a98-b5e9-428d40e63681}} users uniformly distributed in a square area of size 500 m {{formula:8bbcba86-5860-4b29-9ff5-62138fa49da9}} 500 m with the BS located at its center.
The large-scale pathloss model is {{formula:79155871-d563-4cc2-9d2a-9692c51ddfaa}} ({{formula:aacf727d-10e6-4c99-8e71-da3eb4d27350}} is in m), where {{formula:07717d4f-e5e9-4116-9471-5b9763e9e28d}} is the pathloss factor.
The noise power is -114 dBm.
For the channel gain {{formula:a64101b3-c4d2-4180-a3f2-44ecdf5220d1}} , we set {{formula:d226d03e-aa26-467a-9ba5-312606d1a3fd}} , {{formula:2b042363-687f-4a75-bc5c-0479d1e3c36c}} {{cite:85ee69b9c0518b59d52ddcee86b69721d0800495}}, {{cite:e30c44406a170cf99f5f21014f1a0be0764f3e42}}.
Unless specified otherwise, we choose a pathloss factor {{formula:ad7c12e5-aa5b-4447-9a91-1d3428f7d47a}} , a total of {{formula:4b19a727-cd4f-4289-b122-fb2d9821c878}} users, a number of {{formula:c96aa185-4b0f-48a5-b6a7-cb4903b65ef4}} RIS units allocated for each user, a penalty factor {{formula:c7062bf0-1bd9-4ddf-9610-58751ab559ad}} , and an equal SINR requirement {{formula:75239101-23cc-4790-88e8-201e3efd9c6f}} .
Additionally, the effectiveness of the proposed dual method (labeled as “DM”) is verified by comparing with the MRT and ZF methods.
{{figure:8a4b062a-ba08-479d-aa24-5ec50b9b6752}} | r | bb41969e0f61826d70d429789bf3388d |
Due to the large number of possible labels, using standard Transformer models is not feasible. Instead, we cast the type prediction task as an extreme multi-label text classification (XMC) problem: given a question as input text, return the top-{{formula:7c10edf9-35ba-4da1-bd4f-a93af474c3e0}} most relevant types from a large collection of possible types. Vanilla transformer models such as BERT {{cite:16f1200028ee11cd59cf7936dd8aec66a684b487}}, RoBERTa {{cite:2a15a0f0aa3645efdb81b59f966b5210d1e1a654}}, and XLNet {{cite:72e7261fdb99598a16b089f23177a47ee777c8ee}} are ineffective in this scenario due to the memory and computation requirements imposed by the large number of possible labels. This was also confirmed from our experiments that fine-tuning the above mentioned transformer models using the Huggingface framework exhausted all the memory on a 32GB Nvidia Tesla V100 GPU. While this may work on a GPU with larger memory, since we do not have access such a GPU we could not verify and it may still be computationally very expensive to train them. In addition to the computational limitations, as we show in Section , the types are very sparse with most of them having only a few training instances. In order to address these challenges, a model designed for XMC is essential. We use the recent solution to extend the transformer models for XMC coined X-Transformers {{cite:3aa0808b84418b1cbfb22453e2a88aa521f0d094}} for this purpose, which shall be referred to as XBERT in the rest of the paper.
| m | b47117f02c05378299544a453da2a142 |
There are two main unresolved issues with regard to the application of CNNs for medical image segmentation. The first issue has to do with the training procedures and training data. Specifically, the number of manually-labeled images that are available for training is typically very small compared with many non-medical applications. This is because the number of images is small to begin with, and accurate manual annotation is costly because it depends on domain expertise. In recent years, this challenge has led to a surge of interest in such techniques as transfer learning {{cite:bd79fe8d102f08ee6ed443f64aed80d4f3c6d9c5}}, unsupervised learning {{cite:2fad0f10a44ed8b9b0e844a6e741fcabca7b107a}}, and learning from inaccurate and computer-generated annotations {{cite:4af7dd3daf0d88ad9f88f2813ce732a0741901c4}}.
| i | 7b527480b429abf33add07d5223b742d |
using some scalar {{formula:153db040-b326-451b-b3cf-f093679958f9}} . Alternatively, and more accurately, we can use a method akin to automatic differentiation {{cite:604927793700293c746622d7c9ea437fcecd6a72}}
to compute the Hessian-vector product.
| m | 16328d5ff6a98016b7181d79cc70dbb2 |
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