context stringlengths 80 2.25k | correct_answer stringlengths 80 2.91k | distractor1 stringlengths 82 3.71k | distractor2 stringlengths 80 5.71k | distractor3 stringlengths 80 6.62k | __index_level_0__ int64 0 363 |
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Further evidence shows that even for k=7,9,11𝑘7911k=7,9,11italic_k = 7 , 9 , 11 the ratio qnsubscript𝑞𝑛q_{n}italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT converges to ρ𝜌\rhoitalic_ρ if we increase d𝑑ditalic_d. | This is confirmed in Figure 2. We should point out that we also experimented with | We should point out that in the case when ξ𝜉\xiitalic_ξ is real valued (which is not the case here), | qnsubscript𝑞𝑛q_{n}italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT exhibits an oscillatory behavior and does not appear to converge to ρ=0.8𝜌0.8\rho=0.8italic_ρ = 0.8. This is confirmed by Figure 1. | For d=128𝑑128d=128italic_d = 128 we solved the system (19) using Orthomin(k)𝑘(k)( italic_k ) with k=1,2,3,4,10𝑘123410k=1,2,3,4,10italic_k = 1 , 2 , 3 , 4 , 10. In Figure 4 | 0 |
ℳ,s⊧νpsuperscriptmodels𝜈ℳ𝑠𝑝\displaystyle\mathcal{M},s\models^{\nu}pcaligraphic_M , italic_s ⊧ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT italic_p | iff s∈ν(p)iff s∈ν(p)\displaystyle{\rm iff}\,\,\,\text{ $s\in\nu(p)$}roman_iff italic_s ∈ italic_ν ( italic_p ) | iff ℳ,s⊧νΦ1iff ℳ,s⊧νΦ1\displaystyle{\rm iff}\,\,\,\text{ $\mathcal{M},s\models^{\nu}\Phi_{1}$}roman_iff caligraphic_M , italic_s ⊧ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | iff ℳ,s⊧̸νΦiff ℳ,s⊧̸νΦ\displaystyle{\rm iff}\,\,\,\text{ $\mathcal{M},s\not\models^{\nu}\Phi$}roman_iff caligraphic_M , italic_s ⊧̸ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT roman_Φ | iffℳ,π[1]⊧νΦiffsuperscriptmodels𝜈ℳ𝜋delimited-[]1Φ\displaystyle{\rm iff}\,\,\,\text{$\mathcal{M},\pi[1]\models^{\nu}\Phi$}roman_iff caligraphic_M , italic_π [ 1 ] ⊧ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT roman_Φ | 1 |
We adopt a Bayesian perspective in which, in the absence of data, u𝑢uitalic_u is distributed according to a prior measure μ0subscript𝜇0\mu_{0}italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT on ℝKsuperscriptℝ𝐾\mathbb{R}^{K}blackboard_R start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT, with μ0(X)=1subscript𝜇0𝑋1\... | Assumption A. We have ‖f‖Lμ02(X)≤CA‖f‖L2(X)subscriptnorm𝑓subscriptsuperscript𝐿2subscript𝜇0𝑋subscript𝐶Asubscriptnorm𝑓superscript𝐿2𝑋\|f\|_{L^{2}_{\mu_{0}}(X)}\leq C_{\mathrm{A}}\|f\|_{L^{2}(X)}∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ sta... | ‖f−mNf‖L2(X)2superscriptsubscriptnorm𝑓superscriptsubscript𝑚𝑁𝑓superscript𝐿2𝑋2\|f-m_{N}^{f}\|_{L^{2}(X)}^{2}∥ italic_f - italic_m start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_f end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_X )... | \mu_{0}}(X;\mathbb{R}^{J})},≤ italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∥ caligraphic_G - italic_m start_POSTSUPERSCRIPT caligraphic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ start_P... | ^{2},≤ italic_C ∥ roman_Φ ( italic_u ) - italic_m start_POSTSUPERSCRIPT roman_Φ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_u ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIP... | 2 |
{n/2+j}\right\}\right)>V\left({\cal F}_{W};U\cup W\right)italic_V ( caligraphic_F start_POSTSUBSCRIPT italic_W ∪ { italic_σ start_POSTSUBSCRIPT italic_n / 2 + italic_j end_POSTSUBSCRIPT } end_POSTSUBSCRIPT ; italic_U ∪ italic_W ∪ { italic_σ start_POSTSUBSCRIPT italic_n / 2 + italic_j end_POSTSUBSCRIPT } ) > italic_V ( ... | # does item σn/2+jsubscript𝜎𝑛2𝑗\sigma_{n/2+j}italic_σ start_POSTSUBSCRIPT italic_n / 2 + italic_j end_POSTSUBSCRIPT contribute to | we consider item σn/2+jsubscript𝜎𝑛2𝑗\sigma_{n/2+j}italic_σ start_POSTSUBSCRIPT italic_n / 2 + italic_j end_POSTSUBSCRIPT, i.e. Wj≜W∩{σn/2+1,…,σn/2+j−1}≜subscript𝑊𝑗𝑊subscript𝜎𝑛21…subscript𝜎𝑛2𝑗1W_{j}\triangleq W\cap\left\{\sigma_{n/2+1},\dots,\sigma_{n/2+j-1}\right\}italic_W start_POSTSUBSCRIPT italic_j end_PO... | solution increases when we add σn/2+jsubscript𝜎𝑛2𝑗\sigma_{n/2+j}italic_σ start_POSTSUBSCRIPT italic_n / 2 + italic_j end_POSTSUBSCRIPT. | item, i.e. Uj≜{σj+1,…,σn/2}≜subscript𝑈𝑗subscript𝜎𝑗1…subscript𝜎𝑛2U_{j}\triangleq\left\{\sigma_{j+1},\dots,\sigma_{n/2}\right\}italic_U start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≜ { italic_σ start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT , … , italic_σ start_POSTSUBSCRIPT italic_n / 2 end_POSTSUBSCRIPT }. | 3 |
3}{n-1}.italic_μ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_SRW end_POSTSUPERSCRIPT ( | italic_x | > italic_a ) ≥ italic_μ start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_SRW end_POSTSUPERSCRIPT ( | italic_x | > itali... | μGSRW(|x|>a)superscriptsubscript𝜇𝐺SRW𝑥𝑎\displaystyle\mu_{G}^{\mathrm{SRW}}(|x|>a)italic_μ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_SRW end_POSTSUPERSCRIPT ( | italic_x | > italic_a ) | 3}{n-1}.italic_μ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_SRW end_POSTSUPERSCRIPT ( | italic_x | > italic_a ) ≥ italic_μ start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_SRW end_POSTSUPERSCRIPT ( | italic_x | > itali... | If μLsubscript𝜇𝐿\mu_{L}italic_μ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT is the empirical measure for the eigenvalues of L𝐿Litalic_L, then μGSRW(|x|>a)=μL(|x|>a)subscriptsuperscript𝜇SRW𝐺𝑥𝑎subscript𝜇𝐿𝑥𝑎\mu^{\mathrm{SRW}}_{G}(|x|>a)=\mu_{L}(|x|>a)italic_μ start_POSTSUPERSCRIPT roman_SRW end_POSTSUPERSC... | Similarly, μGnSRW→μGSRW→subscriptsuperscript𝜇SRWsubscript𝐺𝑛subscriptsuperscript𝜇SRW𝐺\mu^{\mathrm{SRW}}_{G_{n}}\to\mu^{\mathrm{SRW}}_{G}italic_μ start_POSTSUPERSCRIPT roman_SRW end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT → italic_μ start_POSTSUPE... | 4 |
We further design a message passing based method to discover general abnormal random variables, even if the abnormal random variables have the same mean as the regular random variables. Our message passing based algorithm uses bipartite graph to perform statistical inference. We remark that message passing algorithms h... | Let us denote the j𝑗jitalic_j-th linear mixed function by 𝒂jT𝑿superscriptsuperscript𝒂𝑗𝑇𝑿{{\boldsymbol{a}}^{j}}^{T}{\boldsymbol{X}}bold_italic_a start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_italic_X, where 𝑿𝑿{\boldsymbol{X}}bold_italic_X is a random... | In our message passing based algorithm, messages are exchanged between variable nodes and check nodes. The messages are the probabilities that a random variable Xisubscript𝑋𝑖X_{i}italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is abnormal. More precisely, the message sent from a variable node Xisubscript𝑋𝑖X... | Figure 3: Illustration of a factor graph (a) from a matrix (b). A random variable Xisubscript𝑋𝑖X_{i}italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and Yjsuperscript𝑌𝑗Y^{j}italic_Y start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT are considered as a variable node and a check node in the graph respectively... | i}^{j}X_{i}^{j}=\langle{\boldsymbol{a}}^{j},{\boldsymbol{X}}^{j}\rangle,italic_Y start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_g start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , italic_X start_POSTSUB... | 5 |
In Japan, a fatal accident of long distance bus occurred in which 15 people were dead in January 2016 [14]. Thus, needs for safety management with wearable sensors for drivers are increasing because dangerous driving posture like picking up things or fatigue accumulation of bus or taxi (hereafter, vehicle) drivers may ... | Regarding to sensors for acquiring vital data, wearable terminals have been spread. There are various terminals such as watch type, list band type, eyeglass type, T-shirt type and so on. Apple Watch[16] is a watch type computer, contains heartbeat sensor, acceleration sensor and can collect vital data continuously. Son... | For method 2 verification, hitoe wearer got on a regular bus and changed posture while seating like picking up things to confirm feasibility of posture detection analysis from hitoe acceleration data. Figure 3 shows acceleration data on a regular bus. From Fig.3, acceleration change from bus is mainly slight changes of... | One of IoT application areas (e.g., [1][2]), there are solutions of safety management which acquire users’ vital data by wearable sensor and analyze health and work status of them. Sensors acquire vital data such as acceleration and heart rate, and the data is managed and analyzed using cloud technologies [3]-[7]. A cl... | In Japan, a fatal accident of long distance bus occurred in which 15 people were dead in January 2016 [14]. Thus, needs for safety management with wearable sensors for drivers are increasing because dangerous driving posture like picking up things or fatigue accumulation of bus or taxi (hereafter, vehicle) drivers may ... | 6 |
Step 6: A shoplifting prevention application checks product item stock data in item DB because image analysis accuracy is not sufficient. If there is a shoplifting, there is inconsistency between stock in item DB and actual stock in a shelf. Actual stock in a shelf also can be detected by security camera image. | Step 7: A shoplifting prevention application notifies an alert with suspected customer image to smart phones of shop staffs when item DB check leads a high possibility of shoplifting. | Based on these backgrounds, this paper targets a low cost shoplifting prevention SaaS service for small shops using cloud technology and data analysis technology. In our proposal, machine learning framework Jubatus[6] on a small computer deployed in a shop analyzes security cameras movie, detects anomaly behavior and n... | We proposed a low cost shoplifting prevention service for small retail shops. In our proposal, Jubatus on small computers deployed in shop sites analyzed security camera movie, detected anomaly behaviors of customers and notified to a cloud, and a shoplifting prevention application on a cloud checked product item DB in... | Step 5: When Jubatus detects a shoplifting suspicion such as anomaly score is high, image data and related data are sent to a shoplifting prevention application on a cloud. | 7 |
In utilizing POVM, first an ancilla (i.e., auxillary quantum system) is prepared in a known state, say ρancsubscript𝜌𝑎𝑛𝑐\rho_{anc}italic_ρ start_POSTSUBSCRIPT italic_a italic_n italic_c end_POSTSUBSCRIPT. Combining this ancilla with the original quantum state gives an uncorrelated state. Now, the combined Hilbert... | To do that, we need to set up a close indistinguishability such that the coefficient of |00⟩J1,J3subscriptket00subscript𝐽1subscript𝐽3\left|00\right\rangle_{J_{1},J_{3}}| 00 ⟩ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is a0... | _{3}}+a_{0}\left|00\right\rangle_{J_{1}J_{3}}± start_POSTSUPERSCRIPT ( italic_ς ) end_POSTSUPERSCRIPT ∓ start_POSTSUPERSCRIPT ( italic_σ ) end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | 11 ⟩ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 3 end_P... | _{J_{1}J_{3}}+a_{0}\tau_{0}\left|00\right\rangle_{J_{1}J_{3}}.± start_POSTSUPERSCRIPT ( italic_ς ) end_POSTSUPERSCRIPT ∓ start_POSTSUPERSCRIPT ( italic_σ ) end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | 11 ⟩ start_POSTSUBSCRIPT italic_J start_POST... | \tau_{1}\left|11\right\rangle_{J_{1}J_{3}})caligraphic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 2 start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | 00 ⟩ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRI... | 8 |
In device layer, it is necessary to switch a device that satisfies the needs of user at first. In the case of tracking cameras, it means to select the camera in which the image of the child appears based on location of the child. At this time, if we analyze images with all cameras connected to the network and use only ... | Furthermore, in device layer, where to allocate analyze function on a gateway from gateways which accommodate devices affects operation costs much. In the case of tracking cameras, an image analysis function such as OpenCV is used. We are considering that image analysis functions are arranged in the gateways of the are... | As a processing of tracking cameras, Tacit Computing discovers the camera in which the child appears, and delivers movies of the camera to the parents’ mobile terminals when the parents request movies. And for watching, image analyzing functions such as OpenCV library are arranged on gateways or network edge SSE (Subsc... | In cloud layer, where to process in the cloud greatly affects cost and performance. Firstly, we deploy the processing function to the cloud of the DC (Data Center) that has a small delay from the network edge that accommodates devices frequently used. Furthermore, since the size of the cloud resource also affects the o... | For example, we suppose a situation that a user’s friend participates a marathon contest and a user wishes to see movies of cameras that the friend appears. In this case, bib number of the friend is requested for searching key to Tacit Computing. Tacit Computing distributes image analyzing functions such as OpenCV to g... | 9 |
Figure 2: The Babai partition (rectangular partition with solid lines) and the Voronoi partition (hexagonal partition with dotted lines) for a lattice in ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. The first stage of the algorithm determines the cell of the Babai partition whi... | We note that NLP is equivalent to finding an optimum partition (Voronoi partition) of ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, where each cell of the partition is associated with a unique lattice point and is the set of vectors which are closest to that lattice poin... | Figure 2: The Babai partition (rectangular partition with solid lines) and the Voronoi partition (hexagonal partition with dotted lines) for a lattice in ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. The first stage of the algorithm determines the cell of the Babai partition whi... | The solution to the nearest lattice point problem partitions ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT into Voronoi cells, as described above, and the resulting partition of ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ... | The set of 𝒙∈ℝn𝒙superscriptℝ𝑛{\bm{x}}\in\mathbb{R}^{n}bold_italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT mapped to 𝒚np∈Λsubscript𝒚𝑛𝑝Λ{\bm{y}}_{np}\in\Lambdabold_italic_y start_POSTSUBSCRIPT italic_n italic_p end_POSTSUBSCRIPT ∈ roman_Λ by (1) and (2) is called the Babai cell associa... | 10 |
The methods explain the experimental gait data recordings, the model description and its validation. | To validate the model, the gait data from 12 healthy subjects (7 males and 5 females) were recorded. All the subjects have given written informed consent to the experiment that was approved by the corresponding Institutional Review Board (IRB-2016-09-015). | The results are validated by comparing the mediolateral amplitude of the CoM and the vertical amplitudes generated from the proposed model with the CoM trajectories recorded during our experiment. The regressions were computed with a 95%. | C.T. formulated the model, conceived the experiment, conducted the data collection, performed the data analysis and wrote the paper, M.J.F. helped in the data collection, K.C.V. worked in the model formulation, experimental design and writing, A.F.C. helped in the writing of the manuscript, data analysis, and data pres... | This work presents a new model to estimate and predict the trajectory of the CoM during gait. It includes an accurate model of heel-strike and toe-off to remove the tracking error of the CoM trajectories. Afterwards, the trajectories generated from the model simulation were compared with experimental data of healthy su... | 11 |
i.e. the operations which replace two series arcs f,g∈E𝑓𝑔𝐸f,g\in Eitalic_f , italic_g ∈ italic_E by the single arc w𝑤witalic_w with the cost cw𝝃=cf𝝃+cg𝝃subscriptsuperscript𝑐𝝃𝑤subscriptsuperscript𝑐𝝃𝑓subscriptsuperscript𝑐𝝃𝑔c^{\boldsymbol{\xi}}_{w}=c^{\boldsymbol{\xi}}_{f}+c^{\boldsymbol{\xi}}_{g}italic_c ... | 𝝃∈𝒰𝝃𝒰\boldsymbol{\xi}\in\mathcal{U}bold_italic_ξ ∈ caligraphic_U and xw∗=xf∗subscriptsuperscript𝑥𝑤subscriptsuperscript𝑥𝑓x^{*}_{w}=x^{*}_{f}italic_x start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = italic_x start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIP... | {\xi}_{K}))bold_italic_F ( italic_X , caligraphic_U ) = ( italic_F ( italic_X , bold_italic_ξ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_F ( italic_X , bold_italic_ξ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) ) we use the maximum criterion, that is | _{i}}c_{e}^{\boldsymbol{\xi}}x^{\prime}_{e}italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = roman_max start_POSTSUBSCRIPT bold_italic_ξ ∈ caligraphic_U end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ] end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_e ∈ ... | \in E_{i}}c_{e}^{\boldsymbol{\xi}}x^{\prime}_{e}=C^{\prime}= roman_max start_POSTSUBSCRIPT bold_italic_ξ ∈ caligraphic_U end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_L start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ] end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_i... | 12 |
Biclustering is an unsupervised machine learning technique which attempts to detect meaningful data patterns that are distributed across different columns and rows of the input dataset. This allows biclustering to capture heterogeneous patterns that manifest only in subsets of genes and subsets of samples. Biclustering... | With exponentially increasing sizes of the input datasets, there is an emerging need for effective and efficient methods that would scale well with growing amounts of data. Although there was discussion on possibility of applying biclustering to larger datasets Kasim et al. (2016); Padilha and Campello (2017), hardly a... | One of the recent advancements in biclustering area was introduction of EBIC - a parallel biclustering method, which takes advantage of multiple evolutionary computation strategies Orzechowski et al. (2018). This representative of hybrid biclustering algorithms Orzechowski and Boryczko (2016a, c, b) has been shown to o... | Biclustering is an unsupervised machine learning technique which attempts to detect meaningful data patterns that are distributed across different columns and rows of the input dataset. This allows biclustering to capture heterogeneous patterns that manifest only in subsets of genes and subsets of samples. Biclustering... | In this paper we present the recent advancements in one of the leading biclustering methods. The algorithm was wrapped into a framework, which is conveniently integrated with R and allows multiple input file formats. In Supplementary Material we also demonstrate that even for such a large genomic dataset, the results p... | 13 |
Suppose that the aggregate outcome of a balanced configuration is not a Nash equilibrium. | If a configuration (μ,s)𝜇𝑠(\mu,s)( italic_μ , italic_s ) is stable in (G,Γ0(μ))𝐺subscriptΓ0𝜇(G,\Gamma_{0}(\mu))( italic_G , roman_Γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_μ ) ), then the aggregate outcome of (μ,s)𝜇𝑠(\mu,s)( italic_μ , italic_s ) is a Nash equilibrium of G𝐺Gitalic_G. | In (G,Γ0(μ))𝐺subscriptΓ0𝜇(G,\Gamma_{0}(\mu))( italic_G , roman_Γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_μ ) ), a configuration (μ,s)𝜇𝑠(\mu,s)( italic_μ , italic_s ) is stable if it is balanced, | Let (μ,s)𝜇𝑠(\mu,s)( italic_μ , italic_s ) be a balanced configuration in (G,Γ0(μ))𝐺subscriptΓ0𝜇(G,\Gamma_{0}(\mu))( italic_G , roman_Γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_μ ) ) with the aggregate outcome x𝑥xitalic_x. Then | then it is stable in (G,Γ0(μ))𝐺subscriptΓ0𝜇(G,\Gamma_{0}(\mu))( italic_G , roman_Γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_μ ) ) for some μ∈ℳ(Θn)𝜇ℳsuperscriptΘ𝑛\mu\in\mathcal{M}(\Theta^{n})italic_μ ∈ caligraphic_M ( roman_Θ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ). | 14 |
of the cut-free proof in H𝐻Hitalic_H. The only complicated cases are the modal rules in Table 4. We only check the case where the last rule is (𝖢?α)𝖢superscript?𝛼(\mathsf{C}?^{\alpha})( sansserif_C ? start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ): | \AxiomCΓ,?αφ,?αφΓsuperscript?𝛼𝜑superscript?𝛼𝜑\Gamma,?^{\alpha}\varphi,?^{\alpha}\varphiroman_Γ , ? start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_φ , ? start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_φ | \UnaryInfCΓ,?αφΓsuperscript?𝛼𝜑\Gamma,?^{\alpha}\varphiroman_Γ , ? start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_φ | \UnaryInfC!αΓ~α⇒!αp!^{\alpha}\tilde{\Gamma}_{\alpha}\Rightarrow!^{\alpha}p! start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT over~ start_ARG roman_Γ end_ARG start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⇒ ! start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_p | !^{\alpha}p! start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT over~ start_ARG roman_Γ end_ARG start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , ! start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT over~ start_ARG roman_Σ end_ARG start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⇒ ! start_POSTSUPERSCRIPT italic_α end_POST... | 15 |
lim supn→∞𝔼[Xn]n=limn→∞n−12nlimn→∞n(πη2nϕ+∑l=0f(n)−1sl64π(2l+1)nϕ+sf(n))subscriptlimit-supremum→𝑛𝔼delimited-[]subscript𝑋𝑛𝑛subscript→𝑛𝑛12𝑛subscript→𝑛𝑛𝜋superscript𝜂2𝑛italic-ϕsuperscriptsubscript𝑙0𝑓𝑛1superscript𝑠𝑙64𝜋2𝑙1𝑛italic-ϕsuperscript𝑠𝑓𝑛\displaystyle\limsup_{n\to\infty}\frac{\mat... | \sum_{l=0}^{f(n)-1}s^{l}64\pi\frac{(2l+1)}{n}\phi+s^{f(n)}\right)lim sup start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT divide start_ARG blackboard_E [ italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] end_ARG start_ARG italic_n end_ARG = roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT divide s... | 1}s^{l}64\pi\frac{(2l+1)}{n}\phi+s^{f(n)}\right).≤ ( FRACOP start_ARG italic_n end_ARG start_ARG 2 end_ARG ) ( italic_π divide start_ARG italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_n end_ARG italic_ϕ + ∑ start_POSTSUBSCRIPT italic_l = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_f ... | )-1}s^{l}64\pi(2l+1)\phi+\lim_{n\to\infty}s^{f(n)}n\right)divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_π italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ϕ + roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_l = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ital... | \infty}\frac{(n-1)\Pr[X_{n}^{pq}=1]}{2}<\infty.lim sup start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT divide start_ARG blackboard_E [ italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] end_ARG start_ARG italic_n end_ARG = lim sup start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT divide start_ARG ( italic_n ... | 16 |
𝒒Dsubscript𝒒𝐷\bm{q}_{D}bold_italic_q start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT need not even be continuous. This necessitates a | ∫Ωϕ𝒒D𝑑V=∫𝒯ϕ𝒒D𝑑V=∑𝒯′∈ℳD∫𝒯∩𝒯′ϕ𝒒D|𝒯′dV.subscriptΩitalic-ϕsubscript𝒒𝐷differential-d𝑉subscript𝒯italic-ϕsubscript𝒒𝐷differential-d𝑉evaluated-atsubscriptsuperscript𝒯′subscriptℳ𝐷subscript𝒯superscript𝒯′italic-ϕsubscript𝒒𝐷superscript𝒯′𝑑𝑉\int_{\Omega}\phi\,\bm{q}_{D}\,dV=\int_{\mathcal{T}}\phi\,\bm... | ∥𝒒T−𝒒D∥2=min𝒒∈𝒱T∥𝒒−𝒒D∥2\left\lVert\bm{q}_{T}-\bm{q}_{D}\right\rVert_{2}=\min_{\bm{q}\in\mathcal{V}_{T% | ∫Ω𝒒D𝑑V=∫Ω𝒒T𝑑V.subscriptΩsubscript𝒒𝐷differential-d𝑉subscriptΩsubscript𝒒𝑇differential-d𝑉\int_{\Omega}\bm{q}_{D}\,dV=\int_{\Omega}\bm{q}_{T}\,dV.∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT bold_italic_q start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT italic_d italic_V = ∫ start_POSTSUBSCRIPT roman_Ω end_POS... | ∫𝒯0∩𝒯1F(x,y)𝑑V=∫𝒫F(x,y)𝑑V+∫𝒫′F(x,y)𝑑V+⋯.subscriptsubscript𝒯0subscript𝒯1𝐹𝑥𝑦differential-d𝑉subscript𝒫𝐹𝑥𝑦differential-d𝑉subscriptsuperscript𝒫′𝐹𝑥𝑦differential-d𝑉⋯\int_{\mathcal{T}_{0}\cap\mathcal{T}_{1}}F(x,y)\,dV=\int_{\mathcal{P}}F(x,y)\,% | 17 |
2) We exploit the stacked encoder-decoder structure by connecting multiple contextual hourglass modules from end to end. This architecture can effectively extract features from various scales and increase feedback loops for better learning contextual semantics through intermediate supervision. | Due to the connection to the original hourglass network, we call our network structure, Contextual Hourglass Network (CxtHGNet). To validate the performance of our CxtHGNet, we test it on both Potsdam and Vaihingen open datasets acquired for the purposes of urban classification and semantic labeling. The test results s... | We develop a novel Contextual Hourglass Network (CxtHGNet) for semantic segmentation of high-resolution aerial imagery. Our CxtHGNet can extract rich multi-scale features of the image and learn the contextual semantics in scenes, due to the incorporation of bottom-up, top-down inference across various scales, attention... | In this section, we will provide the details of the contextual hourglass module and our Contextual Hourglass Network (CxtHGNet). | We also show some visual examples of images with a size of 256×256256256256\times 256256 × 256 in Fig. 2. In the 1st and 2nd images, CxtHGNet captures the small red and white regions well, whereas FCN and SegNet are almost not capable of extracting these small-sized regions. In the 3rd image, we can see that compared t... | 18 |
Given history hlsubscriptℎ𝑙h_{l}italic_h start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT, define | μα(θ^αl)=∑j=1Tπ0α(Sπ0(l−1))XαjTπ0α(Sπ0(l−1)).subscript𝜇𝛼superscriptsubscript^𝜃𝛼𝑙superscriptsubscript𝑗1superscriptsubscript𝑇superscript𝜋0𝛼subscript𝑆superscript𝜋0𝑙1superscriptsubscript𝑋𝛼𝑗superscriptsubscript𝑇superscript𝜋0𝛼subscript𝑆superscript𝜋0𝑙1\mu_{\alpha}(\hat{\theta}_{\alpha}^{l})=\frac{\su... | μα(θ¯^αl)=∑j=1Tπ0α(Sπ0(l−1))XαjTπ0α(Sπ0(l−1)) and σα2(θ¯^αl)=∑j=1Tπ0α(Sπ0(l−1))(Xαj−μα(θ¯^αl))2Tπ0α(Sπ0(l−1)).subscript𝜇𝛼superscriptsubscript^¯𝜃𝛼𝑙superscriptsubscript𝑗1superscriptsubscript𝑇superscript𝜋0𝛼subscript𝑆superscript𝜋0𝑙1superscriptsubscript𝑋𝛼𝑗superscriptsubscript𝑇superscript𝜋0𝛼sub... | μα(θ¯αKα)=μα(θ¯^αl)+σα(θ¯^αl)(Sπ0(l−1)2Tπ0α(Sπ0(l−1))−2−1)1/2.subscript𝜇𝛼superscriptsubscript¯𝜃𝛼subscript𝐾𝛼subscript𝜇𝛼superscriptsubscript^¯𝜃𝛼𝑙subscript𝜎𝛼superscriptsubscript^¯𝜃𝛼𝑙superscriptsubscript𝑆superscript𝜋0superscript𝑙12superscriptsubscript𝑇superscript𝜋0𝛼subscript𝑆superscript𝜋0𝑙12... | μα(θ¯αKα)=μα(θ¯^αl)+σα(θ¯^αl)(Sπ0(l−1)2Tπ0α(Sπ0(l−1))−2−1)1/2,subscript𝜇𝛼superscriptsubscript¯𝜃𝛼subscript𝐾𝛼subscript𝜇𝛼superscriptsubscript^¯𝜃𝛼𝑙subscript𝜎𝛼superscriptsubscript^¯𝜃𝛼𝑙superscriptsubscript𝑆superscript𝜋0superscript𝑙12superscriptsubscript𝑇superscript𝜋0𝛼subscript𝑆superscript𝜋0𝑙12... | 19 |
Because TIMEg𝑇𝐼𝑀subscript𝐸𝑔TIME_{g}italic_T italic_I italic_M italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT is asymptotically bounded by the number of the traversed edges before the simulation terminates, we have | ∑g∈𝒢Pr[g]⋅TIMEg≤∑g∈𝒢Pr[g]⋅m=O(m).subscript𝑔𝒢⋅Pr𝑔𝑇𝐼𝑀subscript𝐸𝑔subscript𝑔𝒢⋅Pr𝑔𝑚𝑂𝑚\sum_{g\in\operatorname{\mathcal{G}}}\Pr[g]\cdot TIME_{g}\leq\sum_{g\in% | ∑g∈𝒢Pr[g]⋅x(𝒫g,S)subscript𝑔𝒢⋅Pr𝑔𝑥subscript𝒫𝑔𝑆\displaystyle\sum_{g\in\operatorname{\mathcal{G}}}\Pr[g]\cdot x(\operatorname{% | T=∑g∈𝒢Pr[g]⋅Tg=∑g∈𝒢Pr[g]⋅(TIMEg+∑v∈VrgTIMEgv).𝑇subscript𝑔𝒢⋅Pr𝑔subscript𝑇𝑔subscript𝑔𝒢⋅Pr𝑔𝑇𝐼𝑀subscript𝐸𝑔subscript𝑣superscriptsubscript𝑉𝑟𝑔𝑇𝐼𝑀superscriptsubscript𝐸𝑔𝑣T=\sum_{g\in\operatorname{\mathcal{G}}}\Pr[g]\cdot T_{g}=\sum_{g\in% | T=O(m)+∑(u,v)∈E∑g∈𝒢Pr[g]⋅∑v∈Vrgy(Pvg,{u}).𝑇𝑂𝑚subscript𝑢𝑣𝐸subscript𝑔𝒢⋅Pr𝑔subscript𝑣superscriptsubscript𝑉𝑟𝑔𝑦superscriptsubscript𝑃𝑣𝑔𝑢T=O(m)+\sum_{(u,v)\in E}\sum_{g\in\operatorname{\mathcal{G}}}\Pr[g]\cdot\sum_{% | 20 |
In Figures 3(a) and 3(b), we present the variations in processing speed (FPS) and accuracy (MOTA) under varying confidence thresholds. As the confidence threshold is elevated, the incremental speed advantage is reduced, and correspondingly, the detriment to accuracy becomes less pronounced. This observation suggests a ... | Figure 3(c) illustrates the direct relationship between speed (FPS) and accuracy (MOTA). It highlights that CTD consistently achieves superior accuracy levels at the same speed compared to the fixed frame skipping strategy (0% confidence threshold), underscoring the efficacy of CTD. Furthermore, it offers valuable guid... | Figure 2: illustrates the trade-off between speed (FPS) and accuracy (MOTA) using different confidence thresholds.100% Threshold: Require detection for each frame, considered as baseline. 0% Threshold: Same as fixed frame skipping, never triggers detection based on confidence score but triggers when reaching the maximu... | In Figures 3(a) and 3(b), we present the variations in processing speed (FPS) and accuracy (MOTA) under varying confidence thresholds. As the confidence threshold is elevated, the incremental speed advantage is reduced, and correspondingly, the detriment to accuracy becomes less pronounced. This observation suggests a ... | The analysis demonstrates superior accuracy levels compared to the fixed frame skipping strategy. Furthermore, it yields valuable insights into optimizing CTD’s parameters for different tracking scenarios, enhancing its adaptability and effectiveness in real-world applications. | 21 |
If the end-effector orientation is expressed via a triple of Euler angles, ZYX𝑍𝑌𝑋ZYXitalic_Z italic_Y italic_X, ΦesubscriptΦ𝑒\Phi_{e}roman_Φ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT, the differential kinematics (12) can be rewritten in terms of the vector χ˙e=[p˙eT,Φ˙eT]Tsubscript˙𝜒𝑒superscriptsubscriptsu... | χ˙esubscript˙𝜒𝑒\displaystyle\dot{\chi}_{e}over˙ start_ARG italic_χ end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT | {\chi}_{e},\quad\ddot{e}_{e}=\ddot{\chi}_{e,r}-\ddot{\chi}_{e},italic_e start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = italic_χ start_POSTSUBSCRIPT italic_e , italic_r end_POSTSUBSCRIPT - italic_χ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT , over˙ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_e end_POSTSU... | \dot{\chi}_{e,r}-\dot{\chi}_{e})-\hat{F}_{e},over¨ start_ARG italic_χ end_ARG start_POSTSUPERSCRIPT italic_d italic_e italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = over¨ start_ARG italic_χ end_ARG start_POSTSUBSCRIPT italic_e , italic_r end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT it... | where ev,e=χ˙edes−χ˙esubscript𝑒𝑣𝑒superscriptsubscript˙𝜒𝑒𝑑𝑒𝑠subscript˙𝜒𝑒e_{v,e}=\dot{\chi}_{e}^{des}-\dot{\chi}_{e}italic_e start_POSTSUBSCRIPT italic_v , italic_e end_POSTSUBSCRIPT = over˙ start_ARG italic_χ end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d italic_e itali... | 22 |
This separation allows us to find the noise floor and the maximum significant frequency via a cutoff. This process for finding the cutoff and associated maximum frequency is illustrated in Fig. 8. The following paragraphs give an overview of the modified z𝑧zitalic_z-score and cutoff analysis. | The modified z𝑧zitalic_z-score zmsubscript𝑧𝑚z_{m}italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT is essential to understanding the techniques used for isolating noise from a signal [35]. The standard score, commonly known as the z𝑧zitalic_z-score, uses the mean and the standard deviation of a data set to fi... | where x𝑥xitalic_x is a data point, μ𝜇\muitalic_μ is the mean, and σ𝜎\sigmaitalic_σ is the standard deviation of the data set, respectively. The z𝑧zitalic_z-score value is commonly used to identify outliers in the data set by rejecting points that are above a set threshold, which is set in terms of how many standard... | where 𝐱𝐱\mathbf{x}bold_x is an array of data points and x~~𝑥\tilde{x}over~ start_ARG italic_x end_ARG is its median. The MAD is substituted for the standard deviation in Eq. (7). To improve the modified z𝑧zitalic_z-score, Iglewicz and Hoaglin [38] suggested to additionally substitute the mean with the median. The r... | We can now use the modified z𝑧zitalic_z-score zmsubscript𝑧𝑚z_{m}italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT for evaluating the “significance" of each point in the sublevel set persistence diagram of the Fourier spectrum. A threshold for separating noise in the persistence domain is discussed in the follo... | 23 |
=dji+2m+ς−ℓki−dr(αr,i−2m+ςηi)+2m+ςtiabsent𝑑subscript𝑗𝑖superscript2𝑚𝜍ℓsubscript𝑘𝑖𝑑𝑟subscript𝛼𝑟𝑖superscript2𝑚𝜍subscript𝜂𝑖superscript2𝑚𝜍subscript𝑡𝑖\displaystyle=dj_{i}+2^{m+\varsigma-\ell}k_{i}-\frac{d}{r}\left(\alpha_{r,i}-2% | ^{m+\varsigma}\eta_{i}\right)+2^{m+\varsigma}t_{i}= italic_d italic_j start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 2 start_POSTSUPERSCRIPT italic_m + italic_ς - roman_ℓ end_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - divide start_ARG italic_d end_ARG start_ARG italic_r end_ARG ( italic... | {r,i}-2^{m+\varsigma}\eta_{i}\right)\right\}_{2^{m+\varsigma}}= { italic_d italic_j start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 2 start_POSTSUPERSCRIPT italic_m + italic_ς - roman_ℓ end_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - divide start_ARG italic_d end_ARG start_ARG italic_r en... | \alpha_{d,i}=\{dj_{i}+2^{m+\varsigma-\ell}k_{i}\}_{2^{m+\varsigma}}.italic_α start_POSTSUBSCRIPT italic_r , italic_i end_POSTSUBSCRIPT = { italic_r italic_j start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_m + italic_ς end_POSTSUPERSCRIPT end_POSTSUBSCRIPT and italic_α... | (\alpha_{r,i}-2^{m+\varsigma}\eta_{i}\right)\right\}_{2^{m+\varsigma}}.= divide start_ARG 2 italic_π end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_m + italic_ς end_POSTSUPERSCRIPT end_ARG { italic_α start_POSTSUBSCRIPT italic_d , italic_i end_POSTSUBSCRIPT - divide start_ARG italic_d end_ARG start_ARG italic_r end_A... | 24 |
Let Wb(Mt−1b,t)superscript𝑊𝑏superscriptsubscript𝑀𝑡1𝑏𝑡W^{b}\left(M_{t-1}^{b},t\right)italic_W start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( italic_M start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT , italic_t ) denote the value function for | a buyer who starts period t𝑡titalic_t holding a portfolio Mt−1b∈ℝ+Nsuperscriptsubscript𝑀𝑡1𝑏superscriptsubscriptℝ𝑁M_{t-1}^{b}\in\mathbb{R}_{+}^{N}italic_M start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT star... | a seller who enters period t𝑡titalic_t holding a portfolio Mt−1s∈ℝ+Nsuperscriptsubscript𝑀𝑡1𝑠superscriptsubscriptℝ𝑁M_{t-1}^{s}\in\mathbb{R}_{+}^{N}italic_M start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT sta... | The value for a buyer holding a portfolio Mtbsuperscriptsubscript𝑀𝑡𝑏M_{t}^{b}italic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT in the DM is | }^{b},t\right)\right]italic_W start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( italic_M start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT , italic_t ) = start_UNDERACCENT ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_... | 25 |
To simplify notations, we write this pair again as ((uvnwzω)n∈ℕ,(uvω)n∈ℕ)subscript𝑢superscript𝑣𝑛𝑤superscript𝑧𝜔𝑛ℕsubscript𝑢superscript𝑣𝜔𝑛ℕ\left(\left(uv^{n}wz^{\omega}\right)_{n\in\mathbb{N}},\left(uv^{\omega}\right)% | _{n\in\mathbb{N}}\right)( ( italic_u italic_v start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_w italic_z start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT , ( italic_u italic_v start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ) start_POSTSUBSC... | )_{n\in\mathbb{N}}\right)( ( italic_u italic_v start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT , ( italic_u italic_v start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_w italic_z italic_t start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ) start... | _{n\in\mathbb{N}}\right)( ( italic_u italic_v start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_w italic_z start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT , ( italic_u italic_v start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ) start_POSTSUBSC... | \right)_{n\in\mathbb{N}}\right)( ( italic_u italic_v start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_w italic_z ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT , ( italic_u italic_v start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_z... | 26 |
The general idea is to move from a combinatorial problem, namely computing a set c⊆Q𝑐𝑄c\subseteq Qitalic_c ⊆ italic_Q, to a continuous problem, namely computing a vector μ→∈ℝQ→𝜇superscriptℝ𝑄\vec{\mu}\in\mathbb{R}^{Q}over→ start_ARG italic_μ end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT. | To illustrate this, note that since we can choose as μ→→𝜇\vec{\mu}over→ start_ARG italic_μ end_ARG the characteristic vector of an arbitrary cut, we may also choose a convex combination of such vectors, leading to a normaliser μ→→𝜇\vec{\mu}over→ start_ARG italic_μ end_ARG with entries other than 00 or 1111. | Indeed, we choose as normaliser μ→→𝜇\vec{\mu}over→ start_ARG italic_μ end_ARG a vector that is orthogonal to ℱℱ\mathcal{F}caligraphic_F. | Any cut vector μ→→𝜇\vec{\mu}over→ start_ARG italic_μ end_ARG is a normaliser, i.e., μ→⊤z→D=1superscript→𝜇topsubscript→𝑧𝐷1\vec{\mu}^{\top}\vec{z}_{D}=1over→ start_ARG italic_μ end_ARG start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over→ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = 1. | We define pseudo-cuts over s𝑠sitalic_s to be fibres μ→→𝜇\vec{\mu}over→ start_ARG italic_μ end_ARG over s𝑠sitalic_s such that μ→⊤Δ(w)z→=μ→⊤z→superscript→𝜇topΔ𝑤→𝑧superscript→𝜇top→𝑧\vec{\mu}^{\top}\Delta(w)\vec{z}=\vec{\mu}^{\top}\vec{z}over→ start_ARG italic_μ end_ARG start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRI... | 27 |
Table 7: Archival URLs for sources of data reported in Table 6. Note that URLs are not necessarily those of citations in Table 1. | URLs are relative to https://web.archive.org/web/. For the original URL, delete the numerical prefix and first slash. | 20220428151919/https://fc-solve.shlomifish.org/mail-lists/fc-solve-discuss/archive/0974.html | Table 7: Archival URLs for sources of data reported in Table 6. Note that URLs are not necessarily those of citations in Table 1. | Development history of Solvitaire is also available on Github at URL https://github.com/thecharlesblake/Solvitaire. | 28 |
hyppo’s Fast Dcorr, which uses a fast statistic [33] and p-value approximation [35] is the fastest, even though both energy and kernlab both use highly optimized C++ versions. | Next, we verify that hyppo’s test statistics are equivalent to existing R implementations of the tests. | We have included a host of notable and novel hypothesis tests that we determined to be useful for the end user. | shows the computational efficiency of hyppo’s implementations against existing implementations in commonly used R packages—specifically energy [51], kernlab [52], and HHG [53]. | Figure 1: Benchmarks of hyppo implementations against corresponding R implementations for tests in the independence testing module. Average wall times (over 3 repetitions) (left) are shown for Dcorr in energy, Mmd in kernlab, and Hhg in Hhg as compared against hyppo implementations of Mgc, Hhg, Dcorr, Mmd, and Fast Dco... | 29 |
DRL methods have been applied to a variety of tasks involving sequential decisions and uncertainty. These tasks span the domains of healthcare (Liu et al., 2017), image recognition (Choi et al., 2018), and autonomous driving (Sallab et al., 2017), to name a few. In the operational realm, members of our team have applie... | To test our hypothesis, we consider how to gamify the vehicle routing problem with stochastic requests (VRPSR). The VRPSR is an important problem in modern logistics. It is the problem of dynamically routing a vehicle to service customer requests that occur at random times across an operating horizon and in random plac... | The vehicle routing problem with stochastic requests (VRPSR) dispatches a single vehicle to meet customer requests arriving at random times across a given operating horizon and at random locations across a known service area. The objective is to design a dynamic routing policy, beginning and ending at a depot, that max... | The expected values of agents’ decisions and of each benchmark policy are estimated via simulation. We do this by randomly generating request realizations across the operating horizon and service area, executing the policy, recording the number of serviced requests, then repeating a total of 250 times. The average numb... | We benchmark agents’ performance against two reoptimization policies and the expected value with perfect information (EVPI), which serves as a dual bound on the value of an optimal policy (Brown et al., 2010). At a given state, both policies select an action by solving a mixed-integer linear program (MILP). The MILP se... | 30 |
\right),over→ start_ARG italic_β end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2 ( blackboard_E [ ( roman_ℓ start_POSTSUBSCRIPT italic_j ( 1 ) end_POSTSUBSCRIPT , … , roman_ℓ start_POSTSUBSCRIPT italic_j ( italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) ∣ italic_ω = 1 ] × Cov [ ... | where Cov[ℓj(1),…,ℓj(di)∣ω=1]−1Covsuperscriptdelimited-[]subscriptℓ𝑗1…conditionalsubscriptℓ𝑗subscript𝑑𝑖𝜔11\textsc{\emph{Cov}}[\ell_{j(1)},...,\ell_{j(d_{i})}\mid\omega=1]^{-1}Cov [ roman_ℓ start_POSTSUBSCRIPT italic_j ( 1 ) end_POSTSUBSCRIPT , … , roman_ℓ start_POSTSUBSCRIPT italic_j ( italic_d start_POSTSUBSCR... | β→i=2(𝔼[(ℓj(1),…,ℓj(di))∣ω=1]×Cov[ℓj(1),…,ℓj(di)∣ω=1]−1),subscript→𝛽𝑖2𝔼delimited-[]conditionalsubscriptℓ𝑗1…subscriptℓ𝑗subscript𝑑𝑖𝜔1Covsuperscriptdelimited-[]subscriptℓ𝑗1…conditionalsubscriptℓ𝑗subscript𝑑𝑖𝜔11\vec{\beta}_{i}=2\left(\mathbb{E}[(\ell_{j(1)},...,\ell_{j(d_{i})})\mid\omega=% | \right),over→ start_ARG italic_β end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2 ( blackboard_E [ ( roman_ℓ start_POSTSUBSCRIPT italic_j ( 1 ) end_POSTSUBSCRIPT , … , roman_ℓ start_POSTSUBSCRIPT italic_j ( italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) ∣ italic_ω = 1 ] × Cov [ ... | the log versions. Write ℒi∗(ℓj(1),…,ℓj(di),λi)superscriptsubscriptℒ𝑖subscriptℓ𝑗1…subscriptℓ𝑗subscript𝑑𝑖subscript𝜆𝑖\mathcal{L}_{i}^{*}(\ell_{j(1)},...,\ell_{j(d_{i})},\lambda_{i})caligraphic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_ℓ start_POSTSUBSCRI... | 31 |
Further, compared to the previous AD experiment, this multi-class classification is particularly difficult especially with the severe class imbalance (64.4%percent64.464.4\%64.4 % random test set prediction accuracy). | Thus, it is crucial that a method benefits all four evaluations in this analysis. | The classification results of all baselines and MENET across four evaluation measures (accuracy, precision, recall, and F1-score) averaged across the folds are shown in Table 3. | We expect that this constraint will set many elements in the first layer of 𝐖hsuperscript𝐖ℎ\mathbf{W}^{h}bold_W start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT to zeros and identify the edges that | by focusing at a specific scale s𝑠sitalic_s. It is given as a matrix operation as | 32 |
A constant region is a region of the genome that is exactly the same in both old and new reference genomes (colored in blue). The start and end positions of a constant region are not necessarily the same in the old and new reference genomes. | An updated region is a region in the old reference genome that maps to at least one region in the new reference genome within a reasonable error rate, i.e., differences from the old reference (colored in orange with some differences marked with black bars). | A new region is a region in the new reference genome that does not map to any region in the old reference genome (colored in green). | A retired region is a region in the old reference genome that does not map to any region in the new reference genome (colored in pink). | updated region of the old reference genome with an e=5%𝑒percent5e=5\%italic_e = 5 % error rate (one | 33 |
Figure 1: (Top) Adversarial exemplars X~~𝑋\tilde{X}over~ start_ARG italic_X end_ARG were created with the addition of noise found commonly in the MR setting. (Middle) Didactic examples were created by placing an image (a panda) in a location that varied according to the class label. The image was added to the top rig... | We used axial slices derived from T1-weighted structural MRI data. We randomly selected a subset of subjects from the ADHD200 and UCLA TRECC data sets. In total, there were 1000 scans included. Of these, 500 subjects were diagnosed with ADHD, while the remaining 500 subjects were typically developing (TD) adolescents. ... | Figure 1: (Top) Adversarial exemplars X~~𝑋\tilde{X}over~ start_ARG italic_X end_ARG were created with the addition of noise found commonly in the MR setting. (Middle) Didactic examples were created by placing an image (a panda) in a location that varied according to the class label. The image was added to the top rig... | The addition of noise (up to λ𝜆\lambdaitalic_λ = 0.2) to the entire set of data exemplars appeared to have little effect on the classifier performance (Figure 1). This prompted us to vary the percentage of images that were corrupted, as would be expected if data acquisition took place at different instrumentation site... | Within the field of neuroimaging, however, the goal is often not only to predict accurately but also to interpret which aspects of the image gave rise to a prediction (Kriegeskorte and Douglas 2019). For example, visualization of hidden layer representations has provided insight into the functional response patterns in... | 34 |
iteration of the nodes in C𝐶Citalic_C is finished, swap the meaning of C𝐶Citalic_C and C′superscript𝐶′C^{\prime}italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and | repeat this process iteratively with h+1ℎ1h+1italic_h + 1 as the new height hℎhitalic_h until, | have to compute the classification numbers for the subtrees of height h+1ℎ1h+1italic_h + 1. | After computing the classification number for each node of height h+1ℎ1h+1italic_h + 1 in | We iterate over every subtree with root u𝑢uitalic_u and height h+1ℎ1h+1italic_h + 1 by using the | 35 |
Here we prove that (16) is the unique time-invariant solution by studying all cases. | Case 1: R(ψ^ij)𝒗j−𝒗i=0𝑅subscript^𝜓𝑖𝑗subscript𝒗𝑗subscript𝒗𝑖0R(\hat{\psi}_{ij})\boldsymbol{v}_{j}-\boldsymbol{v}_{i}=0italic_R ( over^ start_ARG italic_ψ end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) bold_italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - bold_italic_v start_POSTSU... | }}_{ij})+R(\hat{\psi}_{ij})\boldsymbol{v}_{j}.bold_italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( over^ start_ARG bold_italic_p end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT - over¯ start_ARG bold_italic_p end_ARG start_POSTSUBSCRIPT ... | \boldsymbol{p}}_{ij}-\hat{\boldsymbol{p}}_{ij})\neq 0.italic_R ( italic_ψ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) bold_italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - bold_italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_k start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( over¯... | 𝒑˙ij=0=R(ψij)𝒗j−𝒗i.subscript˙𝒑𝑖𝑗0𝑅subscript𝜓𝑖𝑗subscript𝒗𝑗subscript𝒗𝑖\dot{\boldsymbol{p}}_{ij}=0=R(\psi_{ij})\boldsymbol{v}_{j}-\boldsymbol{v}_{i}.over˙ start_ARG bold_italic_p end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = 0 = italic_R ( italic_ψ start_POSTSUBSCRIPT italic_i italic_... | 36 |
- an interactive and customizable dashboard with model explanations and EDA visualizations. Here, we present a screenshot of its exemplary layout for the black-box model predicting a player’s value on the FIFA-20 data, see https://iema.drwhy.ai. | This solution puts a vast emphasis on implementing the grammar introduced in Section 3, performing IEMA like in Section 4, and overcoming the challenges discussed in Section 6. From our experience and the users’ feedback, working with the produced dashboard is engaging and effective. modelStudio lowers the entry thresh... | Automation and customizability make the framework approachable for diverse stakeholders apparent in the XIML domain. Interactivity allows for a continuous model analysis process. Standard and well-established libraries for model interpretability and explainability documented by Adadi and Berrada (2018) are not entirely... | Driverless AI (Hall et al., 2019) is a comprehensive state-of-the-art commercial machine learning platform. It automates variable engineering, model building, visualization, and explainability. The last module supports some of the instance and model explanations and, most importantly, does not require the user to know ... | Research in cognitive sciences shows that there is a lot to be gained from the interdisciplinary look at XIML. Miller et al. (2017) and Miller (2019) continuously highlight that there is room for improvement in existing solutions, as most of them rarely take into account the human side of the black-box problem. While d... | 37 |
Figure 1: Screenshot of a Discord Info room, where it is explained how the affiliation system works. | To communicate and organize the pump, the groups typically use Discord servers and Telegram channels. Telegram is an instant messaging service, and a Telegram channel is a special kind of a chat in which only the owner of the channel can broadcast public messages to all the members. Discord is a VoIP and text chat serv... | Usually, the messages written in the news and in the pump-signal rooms are broadcasted to the Telegram channel as well. | With more than 104,000104000104,000104 , 000 members on Discord and more than 72,0007200072,00072 , 000222Data retrieved on January 2019 members on Telegram, Big Pump Signal (BPS) started in December 2017 over Telegram and is arguably the largest pump and dump public community in the Internet. Reading the pump announce... | Pump and dumps are performed by self-organized groups of people over the Internet. These groups arrange the frauds out in the open on the Telegram [6] instant messaging platform or Discord server [7], thus everyone can join the groups without prior authorization. | 38 |
Our main results in this section are the construction of a cohomological form of | integrated information, and using this to show that there is a way to keep track | using a slightly different form of integrated information. We show here that indeed, | divergence, integrated information, in the way that the information cohomology does | Thus, this is a way to account in a consistent way for a setting where the actual | 39 |
What can be noticed in the previous section is a third reason, or case as we can call it, which is the case where some physicians and surgeons decides to either not provide authorization to the neurotechnological intervention altogether or to provide authorization to the neurotechnological intervention but not to under... | Figure 2: Diagram displaying the types of groups that would exist in the case where undergo types of augmentations for physicians and surgeons are optional, and a possible distribution of each group. To avoid providing false information, population percentages were omitted. | What can be noticed in the previous section is a third reason, or case as we can call it, which is the case where some physicians and surgeons decides to either not provide authorization to the neurotechnological intervention altogether or to provide authorization to the neurotechnological intervention but not to under... | A limitation with our work is that there were no medical professionals that directly contributed to the paper. As such, there are some notable ethical issues that were not included in this paper. Additionally, as our work did not include any discussions on health policy and cyber security, we were not able to discuss s... | For our case, we will only mainly focus on (1) and (4) as (2) and (3) are more distant to the scope of the paper. However, those are areas that can and should be further investigated. With there being four different groups, there is the change that those that physicians and surgeons that are fully augmented (i.e. have ... | 40 |
0),over¯ start_ARG italic_π end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT := ( 0 , … , 0 , - over¯ start_ARG italic_h end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , 0 , … , 0 , over¯ start_ARG italic_h end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , 0 , … , 0 ) , over¯ start_ARG ita... | we define the following subspace of SyzB(h¯1,…,h¯m)d=KerφdsubscriptSyz𝐵subscriptsubscript¯ℎ1…subscript¯ℎ𝑚𝑑Kersubscript𝜑𝑑{\rm Syz}_{B}(\overline{h}_{1},\dots,\overline{h}_{m})_{d}={\rm Ker}\,\varphi_% | ρ¯∈SyzB(h¯1,…,h¯m)d∖TSyzB(h¯1,…,h¯m)d,¯𝜌subscriptSyz𝐵subscriptsubscript¯ℎ1…subscript¯ℎ𝑚𝑑subscriptTSyz𝐵subscriptsubscript¯ℎ1…subscript¯ℎ𝑚𝑑\overline{\rho}\in{\rm Syz}_{B}(\overline{h}_{1},\dots,\overline{h}_{m})_{d}% | In particular, ρ¯∈SyzB(h1¯,…,hm¯)d¯𝜌subscriptSyz𝐵subscript¯subscriptℎ1…¯subscriptℎ𝑚𝑑\overline{\rho}\in{\rm Syz}_{B}(\overline{h_{1}},\dots,\overline{h_{m}})_{d}over¯ start_ARG italic_ρ end_ARG ∈ roman_Syz start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( over¯ start_ARG italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCR... | TSyzB(h¯1,…,h¯m)d:=⟨b¯ijπ¯ij,b¯iτ¯i∣b¯ij∈Bd−2d0,b¯i∈Bd−d0q⟩𝔽q.assignsubscriptTSyz𝐵subscriptsubscript¯ℎ1…subscript¯ℎ𝑚𝑑subscriptinner-productsubscript¯𝑏𝑖𝑗subscript¯𝜋𝑖𝑗subscript¯𝑏𝑖subscript¯𝜏𝑖formulae-sequencesubscript¯𝑏𝑖𝑗subscript𝐵𝑑2subscript𝑑0subscript¯𝑏𝑖subscript𝐵𝑑subscript𝑑0𝑞subscript... | 41 |
The filter size in each convolutional layer is 4×\times×4 as opposed to the traditional 3×\times×3, and the activation function is Leaky-ReLU [55] with a slope value of 0.2. The final one-dimensional output is obtained by applying a convolution operation after the last layer, predicting the input’s nature with a certai... | To train the network, the authors employed ADAM optimization [39] with weight initialization using the guidelines from [26]. The system’s performance is assessed using CIFAR10 [40] and Places356 datasets [104]. Overall, the visual performance of the ICGAN [58] is favorable compared to traditional CNNs. | During the convolution operations, the stride is set to 1111 to keep the spatial information the same across all layers. The performance of the proposed method was assessed using the Turing test methodology. The authors provided questionnaire surveys to 80 subjects, asking them 20 questions regarding the results produc... | The low-level features subnet consists of six convolution layers; the mid-level features subnet consists of two convolutional layers; and the global features network consists of four convolutional layers and three fully connected layers. The colorization network comprises a fusion layer, four convolutional, and two ups... | On the other hand, the generator of the PCN network has two subnetworks: the colorization network based on U-Net [69] to colorize the images and the conditioning network to apply the palette colors to the generated image. The PCN discriminator is based on the DCGAN architecture [67]. In the PCN’s discriminator, first, ... | 42 |
In contrast to constrained delegation, the optimal signaling scheme for constrained persuasion does not necessarily have the “credibility” property, i.e., it might not represent a best response against the induced behavior of the receiver in the game without commitment power. As such, it is a natural question to ask fo... | Inspecting the reduction of Theorem 5.5, we observe that in each of these instances the optimal signaling scheme has this property, i.e., it is a best response against the induced action scheme of the receiver. As such, constrained persuasion with equilibrium schemes is also NP-hard to approximate within a factor of n1... | For any constant ε>0𝜀0\varepsilon>0italic_ε > 0, constrained persuasion with equilibrium schemes is NP-hard to approximate within a factor of n1−εsuperscript𝑛1𝜀n^{1-\varepsilon}italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT, even for instances with degree-2 states and degree-1 rejects. | For any constant ε>0𝜀0\varepsilon>0italic_ε > 0, constrained persuasion is NP-hard to approximate within a factor of n1−εsuperscript𝑛1𝜀n^{1-\varepsilon}italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT, even for instances with degree-2 states and degree-1 rejects. | For any constant ε>0𝜀0\varepsilon>0italic_ε > 0, constrained persuasion is NP-hard to approximate within a factor of n1−εsuperscript𝑛1𝜀n^{1-\varepsilon}italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT, even for instances with degree-2 states and degree-1 accepts. | 43 |
Notice that y(1)(t)=ρ(t)superscript𝑦1𝑡𝜌𝑡y^{(1)}(t)=\rho(t)italic_y start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) = italic_ρ ( italic_t ) and in this case there exists no β>0𝛽0\beta>0italic_β > 0 making (13) true, and no rate estimate can be deduced from it. | With T(≫1)annotated𝑇much-greater-thanabsent1T~{}(\gg 1)italic_T ( ≫ 1 ) we can define ρ~(t)∈C(1)(ℝ)~𝜌𝑡superscript𝐶1ℝ\tilde{\rho}(t)\in C^{(1)}(\mathbb{R})over~ start_ARG italic_ρ end_ARG ( italic_t ) ∈ italic_C start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( blackboard_R ) such that ρ~(t)≡ρ(t)~𝜌𝑡𝜌𝑡\tilde{... | H~t(𝒙)≔∫0txt−sρ~(s)𝑑s,≔subscript~𝐻𝑡𝒙superscriptsubscript0𝑡subscript𝑥𝑡𝑠~𝜌𝑠differential-d𝑠\tilde{H}_{t}(\bm{x})\coloneqq\int_{0}^{t}x_{t-s}\tilde{\rho}(s)ds,over~ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_italic_x ) ≔ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_P... | Assume that m>m∗𝑚superscript𝑚m>m^{*}italic_m > italic_m start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, and the initialization is bounded and satisfies ρ^(t;θ0)≈ρ¯(t)^𝜌𝑡subscript𝜃0¯𝜌𝑡\hat{\rho}(t;\theta_{0})\approx\bar{\rho}(t)over^ start_ARG italic_ρ end_ARG ( italic_t ; italic_θ start_POSTSUBSCRIPT 0 end_POSTSU... | (T^{-\omega}+\frac{\omega}{m}T^{1-\omega}\right).roman_sup start_POSTSUBSCRIPT italic_t ∈ blackboard_R end_POSTSUBSCRIPT ∥ italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - over^ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ ≤ roman_sup start_POSTSUBSCRIPT italic_t ∈ blackboard_R e... | 44 |
fi+Nλ2λ1∑j=1nM^i,jfj= 0.subscript𝑓𝑖𝑁subscript𝜆2subscript𝜆1superscriptsubscript𝑗1𝑛subscript^𝑀𝑖𝑗subscript𝑓𝑗 0f_{i}+\frac{{{N}}\lambda_{2}}{\lambda_{1}}\sum_{j=1}^{n}\hat{M}_{i,j}f_{j}\;=% | \;0\;.italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG italic_N italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ov... | N}}}f_{i}+\lambda_{2}\sum_{j=1}^{n}(\hat{M}_{i,j}+\hat{M}_{j{{,}}i})f_{j}\;.0 = - divide start_ARG ∂ end_ARG start_ARG ∂ italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG italic_L ( bold_f ) = divide start_ARG 2 italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_N end_ARG italic_f s... | \left(\frac{\lambda_{1}}{N}I+\lambda_{2}\hat{M}\right)}}\mathbf{f}\;.italic_L ( bold_f ) = italic_c - divide start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_N end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f star... | gi:=λ∑j=1iaM^i,j−M^i,N−j+1.assignsubscript𝑔𝑖𝜆superscriptsubscript𝑗1subscript𝑖𝑎subscript^𝑀𝑖𝑗subscript^𝑀𝑖𝑁𝑗1g_{i}\;:=\;\lambda\sum_{j=1}^{i_{a}}\hat{M}_{i,j}-\hat{M}_{i,N-j+1}\;.italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := italic_λ ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POS... | 45 |
One can think of T⊗T′tensor-product𝑇superscript𝑇′T\otimes T^{\prime}italic_T ⊗ italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT as multiplying T𝑇Titalic_T and T′superscript𝑇′T^{\prime}italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT as polynomials, but then ‘merging’ together pairs of x𝑥xitalic_x-variables... | For a positive integer n𝑛nitalic_n, we write T⊗n:=T⊗T⊗T⊗⋯⊗Tassignsuperscript𝑇tensor-productabsent𝑛tensor-product𝑇𝑇𝑇⋯𝑇T^{\otimes n}:=T\otimes T\otimes T\otimes\cdots\otimes Titalic_T start_POSTSUPERSCRIPT ⊗ italic_n end_POSTSUPERSCRIPT := italic_T ⊗ italic_T ⊗ italic_T ⊗ ⋯ ⊗ italic_T (n𝑛nitalic_n times) for the ... | \times X^{n},\mathcal{Z}:=Z^{n}\times X^{n}\times Y^{n}caligraphic_X := italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × italic_Y start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × italic_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , caligraphic_Y := italic_Y start_POSTSUPERSCRIPT italic_n end_P... | \alpha_{X_{i}}^{-\alpha_{X_{i}}}\right)^{n-o(n)}.( FRACOP start_ARG italic_n end_ARG start_ARG [ italic_α start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋅ italic_n ] start_POSTSUBSCRIPT italic_i ∈ [ italic_k start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ] end_POSTSUBSCRIP... | }_{k^{\prime}}).italic_T ⊗ italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT | italic_X | end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT | italic_Y | end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT ... | 46 |
\mathbb{N}}( start_ARG start_FLOATSUPERSCRIPT italic_m end_FLOATSUPERSCRIPT end_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT satisfies non-negativity as well as superadditivity in n𝑛nitalic_n, and there exists x∗m∈ℝc... | \hskip 2.5pt\cdot n^{-1}=x_{*}^{m}roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT start_ARG start_FLOATSUPERSCRIPT italic_m end_FLOATSUPERSCRIPT end_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_x start_... | \hskip 2.5pt=x_{*}^{m}roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT start_ARG start_FLOATSUPERSCRIPT italic_m end_FLOATSUPERSCRIPT end_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_... | }{}^{\hskip 1.0pt\prime\prime}}\overline{r}_{n,m}=x_{*}^{m}roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT start_ARG start_FLOATSUPERSCRIPT italic_m end_FLOATSUPERSCRIPT end_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT - 1 end_POSTSUPE... | \phantom{r}{}^{\hskip 1.0pt\prime\prime}}\overline{r}_{n,m}\big{|}<2^{-M}| italic_x start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT - start_ARG start_FLOATSUPERSCRIPT italic_m end_FLOATSUPERSCRIPT end_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT italic_n end_POSTSU... | 47 |
is the conditional probability mass function that satisfies (6) and (7). We can construct the random codebook consisting of 2n(I(X;Y)+δ)superscript2𝑛𝐼𝑋𝑌𝛿2^{n(I(X;Y)+\delta)}2 start_POSTSUPERSCRIPT italic_n ( italic_I ( italic_X ; italic_Y ) + italic_δ ) end_POSTSUPERSCRIPT sequences yn(v)superscript𝑦𝑛𝑣y^{n}(... | \hat{C}_{k}(y(n,v)))]italic_θ start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ( italic_v ) = over^ start_ARG italic_C end_ARG start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_y start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_v ) ) ≜ [ over... | }_{k}}(c_{k}|y,\hat{c}_{k})}\right)= ∑ start_POSTSUBSCRIPT italic_y , over^ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_Y , over^ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y , o... | (c_{k}(i),\hat{c}_{k}(i)),~{}~{}\forall k=1,...,K.italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , over^ start_ARG italic_c end_ARG start_POSTSUBSC... | \hat{C}_{k})\right]+H(C_{k}|Y,\hat{C}_{k}),= blackboard_E start_POSTSUBSCRIPT italic_Y , over^ start_ARG italic_C end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_D start_POSTSUBSCRIPT italic_K italic_L end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic... | 48 |
omitted rules are as abovemissing-subexpressionomitted rules aremissing-subexpression as above\displaystyle\begin{aligned} &\text{omitted rules are}\\ | &\text{ as above}\end{aligned}start_ROW start_CELL end_CELL start_CELL omitted rules are end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL as above end_CELL end_ROW | &\text{ are as above}\end{aligned}\end{aligned}start_ROW start_CELL sansserif_emulate start_POSTSUPERSCRIPT sansserif_fE end_POSTSUPERSCRIPT start_POSTSUBSCRIPT sansserif_n end_POSTSUBSCRIPT ( italic_Γ ⊢ italic_t : italic_τ italic_τ ≗ italic_σ italic_Γ ⊢ italic_t : italic_σ ) start_RELOP overdef start_ARG = end_ARG end... | \rrbracket_{\operatorname{\triangledown{}}}}}\end{cases}\end{aligned}\right\}start_ROW start_CELL either end_CELL start_CELL ⋅ sansserif_v = sansserif_inr sansserif_unit and sansserif_p = sansserif_imprecise end_CELL end_ROW start_ROW start_CELL or end_CELL start_CELL ⋅ { start_ROW start_CELL ⋅ end_CELL start_CELL omit... | creftypecap~refnum}{sec:rel-bt})}\end{aligned}start_ROW start_CELL typewriter_repEmul start_POSTSUPERSCRIPT typewriter_fE end_POSTSUPERSCRIPT ( ⋅ ) : end_CELL start_CELL over^ start_ARG italic_τ end_ARG → italic_τ end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL typewriter_repEmul start_POSTSUPERSCRIPT typew... | 49 |
A machine learning model in the wild (e.g., a self-driving car) must be prepared to make sense of its surroundings in rare conditions | that may not have been well-represented in its training set. This could range from conditions such as mild glitches in the camera to strange weather conditions. | A machine learning model in the wild (e.g., a self-driving car) must be prepared to make sense of its surroundings in rare conditions | Algorithms for OoD generalization. Due to the empirical shortcomings of ERM, a wide range of sophisticated algorithms have been developed for domain generalization. | We also show that in any easy-to-learn task that does not have these geometric or statistical skews, these | 50 |
An m𝑚mitalic_m-regular set of portals on a d𝑑ditalic_d-dimensional hypercube CC\mathrm{C}roman_C is an | orthogonal lattice grid(C,m)gridC𝑚\mathrm{grid}(\mathrm{C},m)roman_grid ( roman_C , italic_m ) of m𝑚mitalic_m points in the cube. If the cube | We will normally have m𝑚mitalic_m be chosen as kdsuperscript𝑘𝑑k^{d}italic_k start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT for some integer k⩾2𝑘2k\geqslant 2italic_k ⩾ 2, and as a consequence, grid(C,m)gridC𝑚\mathrm{grid}(\mathrm{C},m)roman_grid ( roman_C , italic_m ) will always contain the corners of CC\mat... | now the grid grid(F,g)grid𝐹𝑔\mathrm{grid}(F,g)roman_grid ( italic_F , italic_g ) in F𝐹Fitalic_F. Let Tgrid(F,g)subscript𝑇grid𝐹𝑔T_{\mathrm{grid}(F,g)}italic_T start_POSTSUBSCRIPT roman_grid ( italic_F , italic_g ) end_POSTSUBSCRIPT be the subtree of T𝑇Titalic_T | points in grid(F,r2/|G|)grid𝐹superscript𝑟2𝐺\mathrm{grid}(F,r^{2}/|G|)roman_grid ( italic_F , italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / | italic_G | ). The leftmost point and the points with | 51 |
where Aπθ(s,a)}A^{\pi_{\theta}}(s,a)\}italic_A start_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_s , italic_a ) } is the advantage function [28]. The agent uses an empirically computed advantage as an unbiased estimate of Aπθ(sc,t,ac,t)}A^{\pi_{\theta}}(s_{c,t},a... | θ←θ+α∑c=1C∇θlogπθ(sc,t,ac,t)Aπθ(sc,t,ac,t)←𝜃𝜃𝛼superscriptsubscript𝑐1𝐶subscript∇𝜃subscript𝜋𝜃subscript𝑠𝑐𝑡subscript𝑎𝑐𝑡superscript𝐴subscript𝜋𝜃subscript𝑠𝑐𝑡subscript𝑎𝑐𝑡\theta\leftarrow\theta+\alpha\sum_{c=1}^{C}\nabla_{\theta}\log\pi_{\theta}(s_{% | where α′superscript𝛼′\alpha^{\prime}italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is the learning rate of the critic, and Vπθsuperscript𝑉subscript𝜋𝜃V^{\pi_{\theta}}italic_V start_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is the estimate of 𝒱πθsuperscript𝒱su... | where Aπθ(s,a)}A^{\pi_{\theta}}(s,a)\}italic_A start_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_s , italic_a ) } is the advantage function [28]. The agent uses an empirically computed advantage as an unbiased estimate of Aπθ(sc,t,ac,t)}A^{\pi_{\theta}}(s_{c,t},a... | To train our RL based system, we use the Asynchronous Advantage Actor-Critic (A3C) framework [28] which is the state-of-art actor-critic algorithm. A3C involves training two neural networks. The agent selects an action based on a policy defined as a probability over actions π:π(sc,j,ac,j)←[0,1]:𝜋←𝜋subscript𝑠𝑐𝑗sub... | 52 |
Since the interest mainly centres on the prior structure given to the score parameter(s) a𝑎aitalic_a (ai′s)superscriptsubscript𝑎𝑖′𝑠(a_{i}^{\prime}s)( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_s ) we consider the same vague gamma prior on the concentr... | To delve deeper into these particular cases, one can see that the marginal distribution of each aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is given by | By doing so, the set of jumps pisubscript𝑝𝑖p_{i}italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT can be thought as the probability that an observation regardless of the class has the corresponding n𝑛nitalic_n-gram, and for each of the d𝑑ditalic_d correlated groups these weights are perturbed by the scores mj... | 5.2.1 Full conditional distribution of aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | The full conditional distribution for the individual score parameters aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT follows the same reasoning and steps as for the global parameter a𝑎aitalic_a by noting that in this case the full conditional density of each aisubscript𝑎𝑖a_{i}italic_a st... | 53 |
To retrieve reliable and versatile risk clusters, we design a loss function that considers the clustering performance in terms of internal quality bSI𝑏𝑆𝐼bSIitalic_b italic_S italic_I, inter-cluster variation σ(Sk)𝜎subscript𝑆𝑘\sigma(S_{k})italic_σ ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ), and... | loss(x)=(S∗−bSI)+λ⋅σ(Sk)+φ⋅cV,𝑙𝑜𝑠𝑠𝑥superscript𝑆𝑏𝑆𝐼⋅𝜆𝜎subscript𝑆𝑘⋅𝜑subscript𝑐𝑉loss(x)=\left(S^{*}-bSI\right)+\lambda\cdot\sigma\left(S_{k}\right)+\varphi% | σ(Sk)=(1|k|Σk∈[1,n](Sk−bSI)2)1/2.𝜎subscript𝑆𝑘superscript1𝑘subscriptΣ𝑘1𝑛superscriptsubscript𝑆𝑘𝑏𝑆𝐼212\sigma(S_{k})=\left(\frac{1}{|k|}\Sigma_{k\in[1,n]}(S_{k}-bSI)^{2}\right)^{1/2}.italic_σ ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = ( divide start_ARG 1 end_ARG start_ARG | italic_k | en... | yt(a,h)={s(i),bSI,σ(Sk),…}|t(a,h),superscriptsubscript𝑦𝑡𝑎ℎevaluated-at𝑠𝑖𝑏𝑆𝐼𝜎subscript𝑆𝑘…𝑡𝑎ℎy_{t}^{(a,h)}=\left\{s(i),~{}bSI,~{}\sigma(S_{k}),\dots\right\}|_{t}^{(a,h)},italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a , italic_h ) end_POSTSUPERSCRIPT = { italic_s ... | σ(Sk)𝜎subscript𝑆𝑘\sigma(S_{k})italic_σ ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | 54 |
First, as a preparation, I proposed an automatic offload method for loop statements for a many core CPU as one of various offloading destination environments, with reference to the evolutionary computation method for a GPU. Next, I studied the order of offload trials for each offloading device and the speedup method wh... | I implemented the proposed method, evaluated its automatic offloading of several applications to mixed offloading destination environments, and confirmed its effectiveness. | Since the automatic offload of loop statements to GPU and FPGA are evaluated by [36][40], and the automatic offload of function blocks to GPU and FPGA is evaluated by [41], this paper confirms that applications can be offloaded to appropriate devices in mixed offloading destination environment. | I proposed an automatic offloading method for mixed offloading destination environments with various devices of GPU, FPGA and many core CPU as a new element of my environment-adaptive software. | The purpose of this paper is to automatically offload applications with high performances in mixed offloading destination environments in which various devices of GPU, FPGA and many core CPU exist. First, I propose a method that automatically offloads to a single device of GPU, FPGA or many core CPU. Next, I propose a ... | 55 |
One way of modeling the blast pressure is with a piece-wise linear function, as is the approach of Krauthammer and Altenberg, (2000), also recommended in USACE, (2008). In this case, the positive phase is approximated as a linearly decaying triangular pulse (see Figure 1(b)) | Pr(t)=Pro(1−tt¯o)(1−ℋ[t−to]),subscript𝑃𝑟𝑡subscript𝑃𝑟𝑜1𝑡subscript¯𝑡𝑜1ℋdelimited-[]𝑡subscript𝑡𝑜P_{r}(t)=P_{ro}\left(1-\dfrac{t}{\bar{t}_{o}}\right)\Big{(}1-\mathcal{H}[t-t_{% | Pr(t)=Pro(1−tto)(1−ℋ[t−to])exp(−dtto),subscript𝑃𝑟𝑡subscript𝑃𝑟𝑜1𝑡subscript𝑡𝑜1ℋdelimited-[]𝑡subscript𝑡𝑜𝑑𝑡subscript𝑡𝑜P_{r}(t)=P_{ro}\left(1-\dfrac{t}{t_{o}}\right)\Big{(}1-\mathcal{H}[t-t_{o}]% | iro=∫0toPr𝑑t=[e−d+d−1]Protod2.subscript𝑖𝑟𝑜superscriptsubscript0subscript𝑡𝑜subscript𝑃𝑟differential-d𝑡delimited-[]superscript𝑒𝑑𝑑1subscript𝑃𝑟𝑜subscript𝑡𝑜superscript𝑑2i_{ro}=\int_{0}^{t_{o}}P_{r}\,dt=\left[e^{-d}+d-1\right]\dfrac{P_{ro}\,t_{o}}{% | 𝒥oθ¨+mgrsin[αsgn(θ)−θ]=SrPr(t)cos[αsgn(θ)−θ],subscript𝒥𝑜¨𝜃𝑚𝑔𝑟𝛼sgn𝜃𝜃𝑆𝑟subscript𝑃𝑟𝑡𝛼sgn𝜃𝜃\mathcal{J}_{o}\ddot{\theta}+mgr\sin\left[\alpha\;\text{sgn}\left(\theta\right% | 56 |
Precision=#(correctly_predicted_ tags)#(predicted tags)×100%Precision#(correctly_predicted_ tags)#(predicted tags)percent100{\text{Precision}}=\frac{\text{\#(correctly\_predicted\_ tags)}}{\text{\#(% | Precision = divide start_ARG #(correctly_predicted_ tags) end_ARG start_ARG #(predicted tags) end_ARG × 100 % | Precision=#(correctly_predicted_ tags)#(predicted tags)×100%Precision#(correctly_predicted_ tags)#(predicted tags)percent100{\text{Precision}}=\frac{\text{\#(correctly\_predicted\_ tags)}}{\text{\#(% | italic_F 1 -score = divide start_ARG 2 × Precision × Recall end_ARG start_ARG Precision + Recall end_ARG × 100 % | Recall = divide start_ARG #(correctly_predicted_tags) end_ARG start_ARG #(true_tags) end_ARG × 100 % | 57 |
g(P)=∑i=1nx(‖Xi‖22−(Xi⊤e)2)≥0.𝑔𝑃superscriptsubscript𝑖1subscript𝑛𝑥superscriptsubscriptnormsubscript𝑋𝑖22superscriptsuperscriptsubscript𝑋𝑖top𝑒20g(P)=\sum_{i=1}^{n_{x}}(\|X_{i}\|_{2}^{2}-(X_{i}^{\top}e)^{2})\geq 0.italic_g ( italic_P ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT... | If P𝑃Pitalic_P is rank-1, then the dimension r𝑟ritalic_r of the vectors e𝑒eitalic_e and Xisubscript𝑋𝑖X_{i}italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is equal to 1111. That is, e,Xi∈ℝ𝑒subscript𝑋𝑖ℝe,X_{i}\in\mathbb{R}italic_e , italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R. H... | g(P)=∑i=1nx(‖Xi‖22−(Xi⊤e)2)≥0.𝑔𝑃superscriptsubscript𝑖1subscript𝑛𝑥superscriptsubscriptnormsubscript𝑋𝑖22superscriptsuperscriptsubscript𝑋𝑖top𝑒20g(P)=\sum_{i=1}^{n_{x}}(\|X_{i}\|_{2}^{2}-(X_{i}^{\top}e)^{2})\geq 0.italic_g ( italic_P ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT... | g(P)≔∑i=1nx(‖Xi‖22−(Xi⊤e)2)=0,≔𝑔𝑃superscriptsubscript𝑖1subscript𝑛𝑥superscriptsubscriptnormsubscript𝑋𝑖22superscriptsuperscriptsubscript𝑋𝑖top𝑒20g(P)\coloneqq\sum_{i=1}^{n_{x}}(\|X_{i}\|_{2}^{2}-(X_{i}^{\top}e)^{2})=0,italic_g ( italic_P ) ≔ ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERS... | By the Cauchy-Schwarz inequality, we have that |Xi⊤e|≤‖Xi‖2‖e‖2superscriptsubscript𝑋𝑖top𝑒subscriptnormsubscript𝑋𝑖2subscriptnorm𝑒2|X_{i}^{\top}e|\leq\|X_{i}\|_{2}\|e\|_{2}| italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT italic_e | ≤ ∥ italic_X start_POSTSUBSCR... | 58 |
Alice sends the pre-trusted BCADDB𝐵𝐶𝐴𝐷subscript𝐷𝐵BCADD_{B}italic_B italic_C italic_A italic_D italic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, its address ADDA𝐴𝐷subscript𝐷𝐴ADD_{A}italic_A italic_D italic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, its public key PKA𝑃subscript𝐾𝐴PK_{A}it... | Upon reception of the Authentication Request from Alice in 1), Bob first checks if the claimed PKA𝑃subscript𝐾𝐴PK_{A}italic_P italic_K start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT can derive BCADDA𝐵𝐶𝐴𝐷subscript𝐷𝐴BCADD_{A}italic_B italic_C italic_A italic_D italic_D start_POSTSUBSCRIPT italic_A end_POSTSU... | Alice sends the pre-trusted BCADDB𝐵𝐶𝐴𝐷subscript𝐷𝐵BCADD_{B}italic_B italic_C italic_A italic_D italic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, its address ADDA𝐴𝐷subscript𝐷𝐴ADD_{A}italic_A italic_D italic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, its public key PKA𝑃subscript𝐾𝐴PK_{A}it... | After receiving the Authentication Response from Bob, detailed in 2), Alice checks if this message is from Bob by validating PKB𝑃subscript𝐾𝐵PK_{B}italic_P italic_K start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT and BCADDB𝐵𝐶𝐴𝐷subscript𝐷𝐵BCADD_{B}italic_B italic_C italic_A italic_D italic_D start_POSTSUBSCR... | 2Thash+2Tverify+2Thmac2subscript𝑇ℎ𝑎𝑠ℎ2subscript𝑇𝑣𝑒𝑟𝑖𝑓𝑦2subscript𝑇ℎ𝑚𝑎𝑐2T_{hash}+2T_{verify}+2T_{hmac}2 italic_T start_POSTSUBSCRIPT italic_h italic_a italic_s italic_h end_POSTSUBSCRIPT + 2 italic_T start_POSTSUBSCRIPT italic_v italic_e italic_r italic_i italic_f italic_y end_POSTSUBSCRIPT + ... | 59 |
under_anchor←0←𝑢𝑛𝑑𝑒𝑟_𝑎𝑛𝑐ℎ𝑜𝑟0under\_anchor\leftarrow 0italic_u italic_n italic_d italic_e italic_r _ italic_a italic_n italic_c italic_h italic_o italic_r ← 0 | for i=2𝑖2i=2italic_i = 2 to |Cv|subscript𝐶𝑣|C_{v}|| italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT | step 2222 do | 16 for i=𝑖absenti=italic_i = |Cv|subscript𝐶𝑣|C_{v}|| italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT | downto 1111 do | 10 for i𝑖iitalic_i from 1111 to |Cu|subscript𝐶𝑢|C_{u}|| italic_C start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT | do | vi,ni←Cv[i]←subscript𝑣𝑖subscript𝑛𝑖subscript𝐶𝑣delimited-[]𝑖v_{i},n_{i}\leftarrow C_{v}[i]italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ← italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT [ italic_i ] | 60 |
\Pi_{B(0,R)}(w_{\mathcal{A}}(X)),caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_X ) = italic_w start_POSTSUBSCRIPT caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_X ) := roman_Π start_POSTSUBSCRIPT italic_B ( 0 , italic_R ) end_POSTSUBSCRIPT ( italic_w start_POST... | where ΠB(0,R)(z):=argminw∈B(0,R)‖z−w‖2=zmax{1,‖z‖R}.assignsubscriptΠ𝐵0𝑅𝑧subscript𝑤𝐵0𝑅subscriptnorm𝑧𝑤2𝑧1norm𝑧𝑅\Pi_{B(0,R)}(z):=\arg\min_{w\in B(0,R)}\|z-w\|_{2}=\frac{z}{\max\left\{1,\frac% | w~(X)≈argminw∈ℝdF(w,X).~𝑤𝑋subscript𝑤superscriptℝ𝑑𝐹𝑤𝑋\tilde{w}(X)\approx\arg\min_{w\in\mathbb{R}^{d}}F(w,X).over~ start_ARG italic_w end_ARG ( italic_X ) ≈ roman_arg roman_min start_POSTSUBSCRIPT italic_w ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F ( italic_w ,... | where wλ∗(X)=argminw∈B(0,R)Fλ(w,X)subscriptsuperscript𝑤𝜆𝑋subscript𝑤𝐵0𝑅subscript𝐹𝜆𝑤𝑋w^{*}_{\lambda}(X)=\arg\min_{w\in B(0,R)}F_{\lambda}(w,X)italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_X ) = roman_arg roman_min start_POSTSUBSCRIPT italic_w ∈... | Assume for now that F(⋅,X)𝐹⋅𝑋F(\cdot,X)italic_F ( ⋅ , italic_X ) in Equation 2 is strongly convex, so that there is a unique minimizer w∗(X)=argminw∈ℝdF(w,X)superscript𝑤𝑋subscript𝑤superscriptℝ𝑑𝐹𝑤𝑋w^{*}(X)=\arg\min_{w\in\mathbb{R}^{d}}F(w,X)italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X ) ... | 61 |
ynsubscript𝑦𝑛y_{n}italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is y𝑦yitalic_y coordinate of the nth vertex and ycsubscript𝑦𝑐y_{c}italic_y start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is the y𝑦yitalic_y coordinates of the center | We argue that our method is more accurate than (Xie et al., 2020), which approximates an arbitrary polygon to a spherical shape to get its area, but our method doesn’t make this aggressive approximation. | First, we follow the loss provided by (Xie et al., 2020) where the the vertices are converted from Cartesian to polar. The approach relies on approximating the area that is produced by the predicted polygon and the ground truth polygon to spherical shape, which enables us to convert the exact equation | The second method is more complex but efficient enough to guarantee the most accurate representation for the object mask. | FourierNet (Benbarka et al., 2020) uses polygon representation to represent each mask. It is a fully convolutional method with no anchor boxes, a shape vector is predicted and then converted into contour points using Fourier transform. The predicted boundaries are smoother than PolarMask. FourierNet achieves 24.3% mAP ... | 62 |
Besides the useful insight about the feasible limit of power, the oracle also provides insights on the optimal weights under each alternative. For example, in simulations we find high colinearity between the approximate oracle weight vector 𝝁~ℒsubscript~𝝁ℒ\tilde{\bm{\mu}}_{\mathcal{L}}over~ start_ARG bold_italic_μ en... | Although the optimal weight 𝝁ℒsubscript𝝁ℒ\bm{\mu}_{\mathcal{L}}bold_italic_μ start_POSTSUBSCRIPT caligraphic_L end_POSTSUBSCRIPT or 𝝁~ℒsubscript~𝝁ℒ\tilde{\bm{\mu}}_{\mathcal{L}}over~ start_ARG bold_italic_μ end_ARG start_POSTSUBSCRIPT caligraphic_L end_POSTSUBSCRIPT is unknown in practice, an unbiased and asymptoti... | \mathcal{L}}/\|\tilde{\bm{\mu}}_{\mathcal{L}}\|.over~ start_ARG italic_B end_ARG start_POSTSUBSCRIPT oracle end_POSTSUBSCRIPT = over~ start_ARG bold_italic_μ end_ARG start_POSTSUBSCRIPT caligraphic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over¯ start_ARG bold_italic_S end_ARG start_POSTSUB... | \mu}_{\mathcal{L}}\|_{2}italic_B start_POSTSUBSCRIPT oracle end_POSTSUBSCRIPT = bold_italic_μ start_POSTSUBSCRIPT caligraphic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over¯ start_ARG bold_italic_S end_ARG start_POSTSUBSCRIPT caligraphic_L end_POSTSUBSCRIPT / ∥ bold_italic_μ start_POSTSUBSC... | _{\bm{U}}.bold_italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , bold_italic_V start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ∼ end_ARG start_ARG italic_i . italic_i . italic_d . end_ARG end_RELOP bold_P start_POSTSUBSCRIPT bold_italic_U end_POSTSUBSCRIPT . From the binary expan... | 63 |
Below, we summarize the quantitative results of the segmentation methods discussed earlier on some typical public datasets and analyze these results qualitatively. | We report the results of RGB-D-based semantic segmentation methods on SUN-RGB-D [158] and NYUDv2 [156] datasets using mean accuracy (mA), overall accuracy (OA), and mean intersection over union (mIoU) as the evaluation metrics. These results of various methods are taken from the original articles and shown in Table 7. | We report the results of projected images, voxel, point clouds, and other representation semantic segmentation methods on S3DIS [2] (both Area 5 and 6-fold cross-validation), ScanNet [21] (test sets), Semantic3D [46] (reduced-8 subsets), and SemanticKITTI [5] (only xyz without RGB). We use mA, OA, and mIoU as the evalu... | Table 7: Evaluation performance regarding for RGB-D semantic segmentation methods on the SUN-RGB-D and NYUDv2. Note that the ’%’ after the value is omitted and the symbol ’–’ means the results are unavailable. | We report the results of 3D part segmentation methods on ShapeNet [217] datasets and use Ins. mIoU as the evaluation metric. These results of various techniques are taken from the original papers and shown in Table 10. We can find that the part segmentation performance of all methods is quite similar. One underlying as... | 64 |
\mathcal{A}^{U}=1\right].( 1 - italic_ε ) roman_Pr start_POSTSUBSCRIPT italic_U ∼ italic_μ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ caligraphic_A start_POSTSUPERSCRIPT italic_U end_POSTSUPERSCRIPT = 1 ] ≤ roman_Pr start_POSTSUBSCRIPT italic_U ∼ italic_S end_POSTSUBSCRIPT [ caligraphic_A start_... | Let ⟨ω⟩delimited-⟨⟩𝜔\langle\omega\rangle⟨ italic_ω ⟩ be the group of (t+1)𝑡1(t+1)( italic_t + 1 )th roots of unity. We first claim that given t𝑡titalic_t copies of the Choi state of U𝑈Uitalic_U (i.e. |ϕU⟩⊗tsuperscriptketsubscriptitalic-ϕ𝑈tensor-productabsent𝑡|\phi_{U}\rangle^{\otimes t}| italic_ϕ start_POSTSUBSCR... | which is t𝑡titalic_t copies of the Choi state of controlled-φU𝜑𝑈\varphi Uitalic_φ italic_U, averaged over all phases φ𝜑\varphiitalic_φ that are (t+1)𝑡1(t+1)( italic_t + 1 )th roots of unity. Assuming this claim holds, then | Above, we are using the fact that the first t𝑡titalic_t moments of ⟨ω⟩delimited-⟨⟩𝜔\langle\omega\rangle⟨ italic_ω ⟩ are the same as the first t𝑡titalic_t moments of the full group of complex units. | Let ℬ(|ϕU⟩⊗t)ℬsuperscriptketsubscriptitalic-ϕ𝑈tensor-productabsent𝑡\mathcal{B}\left(|\phi_{U}\rangle^{\otimes t}\right)caligraphic_B ( | italic_ϕ start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ⟩ start_POSTSUPERSCRIPT ⊗ italic_t end_POSTSUPERSCRIPT ) be the algorithm from Lemma 23. Then: | 65 |
=\mathrm{sign}(\mbox{\boldmath$s$})roman_sign ( bold_italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = roman_sign ( bold_italic_x ) = roman_sign ( bold_italic_s ), we have | xj(yj′−xj)≤0⇒sj(yj′−xj)≤0,j=1,2,…,n,formulae-sequencesubscript𝑥𝑗subscriptsuperscript𝑦′𝑗subscript𝑥𝑗0⇒subscript𝑠𝑗subscriptsuperscript𝑦′𝑗subscript𝑥𝑗0𝑗12…𝑛x_{j}(y^{\prime}_{j}-x_{j})\leq 0\Rightarrow s_{j}(y^{\prime}_{j}-x_{j})\leq 0% | (Ri𝒙)j={xj,j≠i,−xj,j=i.subscriptsubscript𝑅𝑖𝒙𝑗casessubscript𝑥𝑗𝑗𝑖subscript𝑥𝑗𝑗𝑖(R_{i}\mbox{\boldmath$x$})_{j}=\begin{cases}x_{j},&j\neq i,\\ | ui=∑j:{j,i}∈E2wijzij,zij=−xj/‖𝒙‖∞∈Sgn(xi−xj),formulae-sequencesubscript𝑢𝑖subscript:𝑗𝑗𝑖𝐸2subscript𝑤𝑖𝑗subscript𝑧𝑖𝑗subscript𝑧𝑖𝑗subscript𝑥𝑗subscriptnorm𝒙Sgnsubscript𝑥𝑖subscript𝑥𝑗u_{i}=\sum\limits_{j:\{j,i\}\in E}2w_{ij}z_{ij},\,\,\,z_{ij}=-x_{j}/\|\mbox{% | =∑j:{i,j}∈Ewijsign(xi−xj)−qi,qi=∑j:{i,j}∈E and xi=xjwij,formulae-sequenceabsentsubscript:𝑗𝑖𝑗𝐸subscript𝑤𝑖𝑗signsubscript𝑥𝑖subscript𝑥𝑗subscript𝑞𝑖subscript𝑞𝑖subscript:𝑗𝑖𝑗𝐸 and subscript𝑥𝑖subscript𝑥𝑗subscript𝑤𝑖𝑗\displaystyle=\sum_{j:\{i,j\}\in E}w_{ij}\mathrm{sign}(x_{i}-x_{j})-q_{i},% | 66 |
To obtain the bound for w𝖾𝖼=w′+w0subscript𝑤𝖾𝖼superscript𝑤′subscript𝑤0w_{\mathsf{ec}}=w^{\prime}+w_{0}italic_w start_POSTSUBSCRIPT sansserif_ec end_POSTSUBSCRIPT = italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and conclude the proof, | supσsupt∑i=0t𝔼μσ(w𝖾𝖼(vi))≤0+Bsubscriptsupremum𝜎subscriptsupremum𝑡superscriptsubscript𝑖0𝑡superscriptsubscript𝔼𝜇𝜎subscript𝑤𝖾𝖼subscript𝑣𝑖0𝐵\sup_{\sigma}\sup_{t}\,\sum_{i=0}^{t}{\mathbb{E}}_{\mu}^{\sigma}(w_{\mathsf{ec% | supσsupt∑i=0t𝔼μσ(w𝖾𝖼(vi))≤12⋅n8⋅W⋅(1α)n3+n,subscriptsupremum𝜎subscriptsupremum𝑡superscriptsubscript𝑖0𝑡superscriptsubscript𝔼𝜇𝜎subscript𝑤𝖾𝖼subscript𝑣𝑖⋅12superscript𝑛8𝑊superscript1𝛼superscript𝑛3𝑛\sup_{\sigma}\sup_{t}\,\sum_{i=0}^{t}{\mathbb{E}}_{\mu}^{\sigma}(w_{\mathsf{ec% | supσsupt∑i=0t𝔼μσ(w(vi))subscriptsupremum𝜎subscriptsupremum𝑡superscriptsubscript𝑖0𝑡superscriptsubscript𝔼𝜇𝜎𝑤subscript𝑣𝑖\displaystyle\sup_{\sigma}\sup_{t}\,\sum_{i=0}^{t}{\mathbb{E}}_{\mu}^{\sigma}(% | supσsupt∑i=0t𝔼μσ(w′(vi))≤0.subscriptsupremum𝜎subscriptsupremum𝑡superscriptsubscript𝑖0𝑡superscriptsubscript𝔼𝜇𝜎superscript𝑤′subscript𝑣𝑖0\sup_{\sigma}\sup_{t}\,\sum_{i=0}^{t}{\mathbb{E}}_{\mu}^{\sigma}(w^{\prime}(v_% | 67 |
(c) the stacking of multiple KANS modules, where the Iterative Supervision (IS) can leverage the supervision signals to accurately refine the SR image reconstruction. | The proposed KASR framework is divided into three parts: the Kernel Adversarial Noise Simulation (KANS) to implicitly and dynamically simulate the image degradation process for robust SR model training; the High-frequency Selective Objective (HFSO) to constrain the model to focus on high-frequency regions of the images... | (c) the stacking of multiple KANS modules, where the Iterative Supervision (IS) can leverage the supervision signals to accurately refine the SR image reconstruction. | (b) the High-frequency Selective Objective to force the model to focus on high-frequency regions within images due to the higher importance placed in the image regions containing high-frequency for the SR task; and | Figure 2: The overall framework of the proposed KASR framework. The framework consists of three parts: (a) the Kernel Adversarial Noise Simulation (KANS) to adaptively simulate the image degradation process; | 68 |
11: for l=1,…,α𝑙1…𝛼l=1,\dots,\alphaitalic_l = 1 , … , italic_α do | 12: Send the most updated model 𝒘^k,l−1(d)superscriptsubscript^𝒘𝑘𝑙1𝑑\hat{\bm{w}}_{k,l-1}^{\left(d\right)}over^ start_ARG bold_italic_w end_ARG start_POSTSUBSCRIPT italic_k , italic_l - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT to its one-hop neighbors 𝒩dsubscrip... | \hat{\bm{w}}_{k,l-1}^{(j)},l=1,2,\dots,\alpha,over^ start_ARG bold_italic_w end_ARG start_POSTSUBSCRIPT italic_k , italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT ← ∑ start_POSTSUBSCRIPT italic_j ∈ caligraphic_N start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ∪ { italic_d } end_POSTSUBS... | 10: Set 𝒘^k,0(d)superscriptsubscript^𝒘𝑘0𝑑\hat{\bm{w}}_{k,0}^{\left(d\right)}over^ start_ARG bold_italic_w end_ARG start_POSTSUBSCRIPT italic_k , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT as 𝒘~k(d)superscriptsubscript~𝒘𝑘𝑑\tilde{\bm{w}}_{k}^{(d)}over~ start_ARG bol... | 15: Update 𝒘~k(d)subscriptsuperscript~𝒘𝑑𝑘\tilde{\bm{w}}^{\left(d\right)}_{k}over~ start_ARG bold_italic_w end_ARG start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT as 𝒘^k,α(d)superscriptsubscript^𝒘𝑘𝛼𝑑\hat{\bm{w}}_{k,\alpha}^{(d)}over^ start_AR... | 69 |
modalities. Formally, we obtain the language Lscsubscript𝐿scL_{\textsf{sc}}italic_L start_POSTSUBSCRIPT sc end_POSTSUBSCRIPT | (where p∈Prop𝑝Propp\in{\textsf{Prop}}italic_p ∈ Prop and G⊆Agt𝐺AgtG\subseteq{\textsf{Agt}}italic_G ⊆ Agt) with the following | where for every w∈W𝑤𝑊w\in Witalic_w ∈ italic_W and G⊆Agt𝐺AgtG\subseteq{\textsf{Agt}}italic_G ⊆ Agt, | V(w)⊆Prop𝑉𝑤PropV(w)\subseteq{\textsf{Prop}}italic_V ( italic_w ) ⊆ Prop. In addition, it satisfies the following | M,t⊧timepsubscriptmodelstime𝑀𝑡𝑝M,t\models_{\textsf{time}}pitalic_M , italic_t ⊧ start_POSTSUBSCRIPT time end_POSTSUBSCRIPT italic_p iff p∈g(t)𝑝𝑔𝑡p\in g(t)italic_p ∈ italic_g ( italic_t ), when p∈Prop𝑝Propp\in{\textsf{Prop}}italic_p ∈ Prop | 70 |
Usually, the messages written in the news and in the pump-signal rooms are also broadcasted to the Telegram channel. | The levels of activity of the many pump and dump groups on the Internet differ considerably. The most active ones perform roughly one pump and dump operation a day. Less active groups perform one operation a week. Other groups perform operations only when they believe the market conditions are good. The steps during th... | During our study, we found a large number of signal groups. These groups are more significant than the pump and dump groups and arrange operations more frequently. As future work, it would be interesting to study the impact of these groups and their activity on the market. | Although we retrieved 2 weeks of data for each pump and dump scheme, initially, we use only 3 days—the day of the fraud, the day before, and the day after. We can reasonably assume that no other scams are present for the same coin in this time frame. Indeed, among the market manipulations we collected, different groups... | As highlighted by Kamps et al. (Kamps and Kleinberg, 2018), it does not exist a dataset in the literature of the confirmed pump and dumps. Thus, we need to build one for this work. From the 20 groups we joined, we selected only the pump and dump schemes carried out on Binance. We made this choice for two main reasons. ... | 71 |
re←ru(v)⊕rv(u)←subscript𝑟𝑒direct-sumsubscript𝑟𝑢𝑣subscript𝑟𝑣𝑢r_{e}\leftarrow r_{u}(v)\oplus r_{v}(u)italic_r start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ← italic_r start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_v ) ⊕ italic_r start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_u ) | and update 𝚜𝚝𝚎𝚙esubscript𝚜𝚝𝚎𝚙𝑒\mathtt{step}_{e}typewriter_step start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT accordingly. | state of 𝚜𝚝𝚎𝚙esubscript𝚜𝚝𝚎𝚙𝑒\mathtt{step}_{e}typewriter_step start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT. | 𝚜𝚝𝚎𝚙esubscript𝚜𝚝𝚎𝚙𝑒\mathtt{step}_{e}typewriter_step start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT in conjunction. | nodes u𝑢uitalic_u and v𝑣vitalic_v update 𝚜𝚝𝚎𝚙esubscript𝚜𝚝𝚎𝚙𝑒\mathtt{step}_{e}typewriter_step start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT in accordance with the policy of 𝙿𝙿𝚂𝙿𝙿𝚂\mathtt{PPS}typewriter_PPS | 72 |
Before we go ahead and look at spectral efficiency, note that if we do not consider the MMSE solution as a candidate, then RI-MIMO will have an error floor as well. However, due to a much higher probability of success than the conventional approach, the error floor associated with RI-MIMO will be much lower and reduced... | Figure 11: Spectral Efficiency: Comparison between throughput of MMSE, TRIM, RI-MIMO (using adaptive MCS for all) in large/massive MIMO scenarios: (left) 16 antennas at the base station and we use Na=64subscript𝑁𝑎64N_{a}=64italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = 64, (right) 32 antennas at the base s... | Figure 12: Spectral Efficiency: Comparison between throughput of MMSE and RI-MIMO for massive MIMO scenarios with large number of antennas at the base station. | Figure 9: BER comparison of MMSE and RI-MIMO-512 for massive MIMO systems with very large number of antennas at the base station. | Figure 8: Bit Error Rate Curves for various Massive MIMO system with BPSK modulation and 16 antennas at the base station, illustrating the performance of RI-MIMO. | 73 |
J(w)=1N∑n=1N(acknlogpw(ackn|sinrn,mcsn)+(1−ackn)log(1−pw(ackn|sinrn,mcsn)))→maxw𝐽𝑤1𝑁superscriptsubscript𝑛1𝑁𝑎𝑐subscript𝑘𝑛subscript𝑝𝑤|𝑎𝑐subscript𝑘𝑛𝑠𝑖𝑛subscript𝑟𝑛𝑚𝑐subscript𝑠𝑛1𝑎𝑐subscript𝑘𝑛1subscript𝑝𝑤|𝑎𝑐subscript𝑘𝑛𝑠𝑖𝑛subscript𝑟𝑛𝑚𝑐subscript𝑠𝑛→subscript𝑤J(w)=\frac{1}{N}\sum_{n=1}... | \limits_{w}start_ROW start_CELL italic_J ( italic_w ) = divide start_ARG 1 end_ARG start_ARG italic_N end_ARG ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ( italic_a italic_c italic_k start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT roman_log italic_p start_POSTS... | ,mcs_{n})\big{)}^{2}\rightarrow\min\limits_{w}italic_F ( italic_w ) = divide start_ARG 1 end_ARG start_ARG italic_N end_ARG ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ( italic_q start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_a italic_c italic_k start... | Here ackn∈{0,1}𝑎𝑐subscript𝑘𝑛01ack_{n}\in\{0,1\}italic_a italic_c italic_k start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ { 0 , 1 } is the "true" acknowledgement, which we get to know after the action is completed, and pw(ackn|sinrn,mcsn)subscript𝑝𝑤conditional𝑎𝑐subscript𝑘𝑛𝑠𝑖𝑛subscript𝑟𝑛𝑚𝑐sub... | sinr)\big{\}}over~ start_ARG italic_m italic_c italic_s end_ARG start_POSTSUBSCRIPT italic_S italic_E end_POSTSUBSCRIPT ( italic_s italic_i italic_n italic_r ) = roman_arg roman_max start_POSTSUBSCRIPT italic_m italic_c italic_s end_POSTSUBSCRIPT { italic_q start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT start_POSTSUPER... | 74 |
Let us stress that the partial differential equation (3.14) here is not a consequence of the Feynman-Kac formula, but holds true, simply based on the definition (3.4) without imposing a variety of additional sufficient conditions. | It may still not be of use on its own in the absence of concrete information about the boundary ∂D𝐷\partial D∂ italic_D, while we still present it here because it plays a crucial role not only in deriving the hard bounding functions (3.15), but also in proving the upcoming Theorem 3.8, as well as reappears in Theorem ... | This result is not only possibly effective from a numerical point of view (for the obvious reason of the absence of cumbersome time-state dependent jumps), but also insightful from a broader perspective as the approximate solution can then be put in contrast with the Feynman-Kac representation of a suitable initial bou... | In the proposed framework, a sequence (Theorem 3.2), in fact, two sequences (Theorem 3.4) of approximate solutions are set up with a jump time of the underlying dynamics as an information relay point in passing the past information on to a previous iteration step to fill in the missing information on the unobserved tra... | In addition, the modes of convergence of the sequences of approximate solutions and associated hard bounding functions can be strengthened (Theorem 3.8). | 75 |
Then there is a block of G−X𝐺𝑋G-Xitalic_G - italic_X that intersects all the branch sets of M𝑀Mitalic_M avoided by X𝑋Xitalic_X, and this block is unique. | This block is called the main block of G−X𝐺𝑋G-Xitalic_G - italic_X (w.r.t. the model M𝑀Mitalic_M). | Then there is a block of G−X𝐺𝑋G-Xitalic_G - italic_X that intersects all the branch sets of M𝑀Mitalic_M avoided by X𝑋Xitalic_X, and this block is unique. | Thus we may assume that Theorem 13 gives a vertex subset X𝑋Xitalic_X of G𝐺Gitalic_G with |X|<8ℓ=24k𝑋8ℓ24𝑘|X|<8\ell=24k| italic_X | < 8 roman_ℓ = 24 italic_k such that the main block U𝑈Uitalic_U of G−X𝐺𝑋G-Xitalic_G - italic_X is bipartite. | For two edges e,e′𝑒superscript𝑒′e,e^{\prime}italic_e , italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT in G−X𝐺𝑋G-Xitalic_G - italic_X, we write e≡e′𝑒superscript𝑒′e\equiv e^{\prime}italic_e ≡ italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT if e=e′𝑒superscript𝑒′e=e^{\prime}italic_e = italic_e start_POST... | 76 |
4σ2mkλk+1,4superscript𝜎2subscript𝑚𝑘subscript𝜆𝑘1\displaystyle\frac{4\sigma^{2}}{m_{k}\lambda_{k+1}},divide start_ARG 4 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_ARG ... | 𝔼𝝃k[‖∇~f(xk+1,𝝃k)−∇f(xk+1)‖22]subscript𝔼superscript𝝃𝑘delimited-[]superscriptsubscriptnorm~∇𝑓superscript𝑥𝑘1superscript𝝃𝑘∇𝑓superscript𝑥𝑘122\displaystyle\mathbb{E}_{\boldsymbol{\xi}^{k}}\left[\left\|\widetilde{\nabla}f% | 𝔼𝝃k[‖∇~f(xk+1,𝝃k)−𝔼𝝃k[∇~f(xk+1,𝝃k)]‖22]subscript𝔼superscript𝝃𝑘delimited-[]superscriptsubscriptnorm~∇𝑓superscript𝑥𝑘1superscript𝝃𝑘subscript𝔼superscript𝝃𝑘delimited-[]~∇𝑓superscript𝑥𝑘1superscript𝝃𝑘22\displaystyle\mathbb{E}_{\boldsymbol{\xi}^{k}}\left[\left\|\widetilde{\nabla}f% | ‖𝔼𝝃k[∇~f(xk+1,𝝃k)]−∇f(xk+1)‖2subscriptnormsubscript𝔼superscript𝝃𝑘delimited-[]~∇𝑓superscript𝑥𝑘1superscript𝝃𝑘∇𝑓superscript𝑥𝑘12\displaystyle\left\|\mathbb{E}_{\boldsymbol{\xi}^{k}}\left[\widetilde{\nabla}f% | ‖∇~f(xk+1,𝝃k)−𝔼𝝃k[∇~f(xk+1,𝝃k)]‖2≤2λk+1.subscriptnorm~∇𝑓superscript𝑥𝑘1superscript𝝃𝑘subscript𝔼superscript𝝃𝑘delimited-[]~∇𝑓superscript𝑥𝑘1superscript𝝃𝑘22subscript𝜆𝑘1\left\|\widetilde{\nabla}f(x^{k+1},\boldsymbol{\xi}^{k})-\mathbb{E}_{% | 77 |
Suppose φksubscript𝜑𝑘\varphi_{k}italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is atomic, for every k∈[m]𝑘delimited-[]𝑚k\in[m]italic_k ∈ [ italic_m ]. Then k∼ℓsimilar-to𝑘normal-ℓk\sim\ellitalic_k ∼ roman_ℓ, i.e., k𝑘kitalic_k and ℓnormal-ℓ\ellroman_ℓ are lopsidedly dependent, iff φk∨φℓsubscript𝜑𝑘subscri... | Suppose that Φnormal-Φ\Phiroman_Φ is an satisfiable quasi-extremal instance with lopsided dependency graph ([m],∼)delimited-[]𝑚similar-to([m],\sim)( [ italic_m ] , ∼ ). Suppose also that there exists a sequence of reals (xk∈(0,1):k∈[m])normal-:subscript𝑥𝑘01𝑘delimited-[]𝑚(x_{k}\in(0,1):k\in[m])( italic_x start_POST... | For k∈[m]𝑘delimited-[]𝑚k\in[m]italic_k ∈ [ italic_m ], let pk=Pr𝒟(¬φk)subscript𝑝𝑘subscriptPr𝒟subscript𝜑𝑘p_{k}=\mathop{\mathrm{Pr}}\nolimits_{\mathcal{D}}(\neg\varphi_{k})italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = roman_Pr start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT ( ¬ italic_φ start_POS... | Given a satisfiable instance Φ=φ1∧⋯∧φmnormal-Φsubscript𝜑1normal-⋯subscript𝜑𝑚\Phi=\varphi_{1}\wedge\cdots\wedge\varphi_{m}roman_Φ = italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ⋯ ∧ italic_φ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, let k,ℓ∈[m]𝑘normal-ℓdelimited-[]𝑚k,\ell\in[m]italic_k , roman_ℓ ∈ [ ital... | \in[m]}\varphi_{k}\bigg{)}= roman_Pr start_POSTSUBSCRIPT caligraphic_D end_POSTSUBSCRIPT ( ⋀ start_POSTSUBSCRIPT italic_k ∈ [ italic_m ] end_POSTSUBSCRIPT italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | 78 |
SFS_lsvc, which is the forward SFS algorithm with a linear support vector classifier, | SKB_clf, which is the SKB algorithm ranking features with ANOVA (analysis of variance) F-statistic, and | Figure 8: Classification accuracy corresponding to the features selected by the SKB-based algorithm. SKB_clf refers to the SKB algorithm ranking features with ANOVA F-statistic. SKB_mic refers to the SKB algorithm ranking features with mutual information. | SKB_mir, which is the SKB algorithm ranking features with mutual information for regression, | SSC_h, which is the fast SSC-based feature selection algorithm accelerated with hℎhitalic_h-correlation, | 79 |
Figure 2: The overall architectural diagram of CustomGNN. Here, ⊗tensor-product\otimes⊗ denotes matrix multiplication operation. There are two paths in CustomGNN. The first path highlighted in blue uses random walk to generate sub-paths (a subgraph), then, the path re-weighting module is utilized in these sub-paths for... | \mathbf{F}}\}{ over¯ start_ARG bold_F end_ARG ∣ over¯ start_ARG bold_F end_ARG = over^ start_ARG bold_F end_ARG ⊕ over˙ start_ARG bold_F end_ARG } into classifier to obtain the predictions. | 𝐅=𝐅^⊕𝐅˙𝐅direct-sum^𝐅˙𝐅\mathbf{F}=\hat{\mathbf{F}}\oplus\dot{\mathbf{F}}bold_F = over^ start_ARG bold_F end_ARG ⊕ over˙ start_ARG bold_F end_ARG is the final concatenated embeddings. | }},over˙ start_ARG bold_F end_ARG = divide start_ARG 1 end_ARG start_ARG italic_H + 1 end_ARG ∑ start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_h = 0 end_POSTSUBSCRIPT bold_A start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT over~ start_ARG bold_F end_ARG , | 𝐅¯(s)=𝐅˙⊕𝐅^(s)superscript¯𝐅𝑠direct-sum˙𝐅superscript^𝐅𝑠\overline{\mathbf{F}}^{(s)}=\dot{\mathbf{F}}\oplus\hat{\mathbf{F}}^{(s)}over¯ start_ARG bold_F end_ARG start_POSTSUPERSCRIPT ( italic_s ) end_POSTSUPERSCRIPT = over˙ start_ARG bold_F end_ARG ⊕ over^ start_ARG bold_F end_ARG start_POSTSUPERSCRIPT ( italic_s )... | 80 |
to 𝐱isubscript𝐱𝑖{\mathbf{x}}_{i}bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, i.e., yi=ksubscript𝑦𝑖𝑘y_{i}=kitalic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_k if 𝐱i=𝐱j(k)subscript𝐱𝑖superscriptsubscript𝐱𝑗𝑘{\mathbf{x}}_{i}={\mathbf{x}}_{j}^{(k)}bold_x start_POSTSUBSCRIPT italic_i end_... | j𝑗jitalic_j. Let 𝐦ksubscript𝐦𝑘\mathbf{m}_{k}bold_m start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, for k=1,…,l𝑘1…𝑙k=1,\dots,litalic_k = 1 , … , italic_l, denote the sample | _{k}{\mathbf{m}}_{k}bold_m ≔ italic_n start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_n start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT... | _{i}})^{\textsc{T}}bold_S start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT ≔ ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_m start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_m ) ( bold_m start_POSTS... | }}_{k}^{\textsc{T}}+(\lambda_{k}+\lambda){\bm{\mathrm{I}}}_{d}bold_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( roman_Λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT bold_I ) bold_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT T en... | 81 |
ℒfoc=1K∑s=1S∑h=1H∑w=1W𝟙{𝐘h,w=s}FL(𝐎s,h,w),subscriptℒ𝑓𝑜𝑐1𝐾superscriptsubscript𝑠1𝑆superscriptsubscriptℎ1𝐻superscriptsubscript𝑤1𝑊1subscript𝐘ℎ𝑤𝑠𝐹𝐿subscript𝐎𝑠ℎ𝑤{\mathcal{L}}_{foc}=\frac{1}{K}\sum_{s=1}^{S}\sum_{h=1}^{H}\sum_{w=1}^{W}% | \mathbbm{1}\{{\bf Y}_{h,w}=s\}FL({\bf O}_{s,h,w}),caligraphic_L start_POSTSUBSCRIPT italic_f italic_o italic_c end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_s = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_h ... | =1}^{W}\alpha_{s}\mathbbm{1}\{{\bf Y}_{h,w}=s\}FL(\widetilde{{\bf O}}_{s,h,w}),over~ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT italic_f italic_o italic_c end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_s = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_... | w}D_{KL}({\bf O}_{:,h,w}||\widetilde{{\bf O}}_{:,h,w}).caligraphic_L start_POSTSUBSCRIPT italic_p italic_e italic_r end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_Q end_ARG ∑ start_POSTSUBSCRIPT italic_h = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT ital... | :,h,w}),over~ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT italic_p italic_e italic_r end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_Q end_ARG ∑ start_POSTSUBSCRIPT italic_h = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_w = 1 end_POSTSUBSCR... | 82 |
We use a standard strategy in graph algorithms and parameterized complexity (see, e.g., the book by Cygan, Fomin, Kowalik, Lokshtanov, Marx, Pilipczuk, Pilipczuk, and Saurabh [14, Chapter 7]): we show by dynamic programming that the problem can be solved efficiently for graphs of bounded branchwidth, and then, using an... | We can solve the embeddability problem of graphs into two-dimensional simplicial complexes in time (c+w)O(c+w)⋅n⋅superscript𝑐𝑤𝑂𝑐𝑤𝑛(c+w)^{O(c+w)}\cdot n( italic_c + italic_w ) start_POSTSUPERSCRIPT italic_O ( italic_c + italic_w ) end_POSTSUPERSCRIPT ⋅ italic_n time, where c𝑐citalic_c is the number of simplices ... | One can solve the embeddability problem of graphs into 2-dimensional simplicial complexes in 2poly(c)⋅n2⋅superscript2poly𝑐superscript𝑛22^{\text{poly}(c)}\cdot n^{2}2 start_POSTSUPERSCRIPT poly ( italic_c ) end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where c𝑐citalic_c is the number of... | In this paper, we describe an algorithm for deciding the embeddability of graphs into topological spaces that are, in a sense, as general as possible: two-dimensional simplicial complexes (or 2-complexes for brevity), which are made from vertices, edges, and triangles glued together. (We remark that every graph is embe... | Given a 2-complex 𝒞𝒞\mathscr{C}script_C with c𝑐citalic_c simplices, and a graph G𝐺Gitalic_G with n𝑛nitalic_n vertices and edges in total, we can, in 2poly(c)⋅n2⋅superscript2poly𝑐superscript𝑛22^{\text{poly}(c)}\cdot n^{2}2 start_POSTSUPERSCRIPT poly ( italic_c ) end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRI... | 83 |
Let 𝗍×f(x)=supj∈ℤbjxjsubscript𝗍𝑓𝑥subscriptsupremum𝑗ℤsubscript𝑏𝑗superscript𝑥𝑗\operatorname{\mathsf{t\!}_{\times}\!}{f}(x)=\sup_{j\in\mathbb{Z}}b_{j}x^{j}start_OPFUNCTION sansserif_t start_POSTSUBSCRIPT × end_POSTSUBSCRIPT end_OPFUNCTION italic_f ( italic_x ) = roman_sup start_POSTSUBSCRIPT italic_j ∈ blackbo... | Laurent series. We define the Newton polygon 𝒩𝗍×fsubscript𝒩subscript𝗍𝑓\mathcal{N}_{\operatorname{\mathsf{t\!}_{\times}\!}{f}}caligraphic_N start_POSTSUBSCRIPT start_OPFUNCTION sansserif_t start_POSTSUBSCRIPT × end_POSTSUBSCRIPT end_OPFUNCTION italic_f end_POSTSUBSCRIPT | 𝒩𝗍×fsubscript𝒩subscript𝗍𝑓\mathcal{N}_{\operatorname{\mathsf{t\!}_{\times}\!}f}caligraphic_N start_POSTSUBSCRIPT start_OPFUNCTION sansserif_t start_POSTSUBSCRIPT × end_POSTSUBSCRIPT end_OPFUNCTION italic_f end_POSTSUBSCRIPT | Let 𝗍×f(x)subscript𝗍𝑓𝑥\operatorname{\mathsf{t\!}_{\times}\!}f(x)start_OPFUNCTION sansserif_t start_POSTSUBSCRIPT × end_POSTSUBSCRIPT end_OPFUNCTION italic_f ( italic_x ) be a max-times tropical Laurent series, and 𝒩𝗍×fsubscript𝒩subscript𝗍𝑓\mathcal{N}_{\operatorname{\mathsf{t\!}_{\times}\!}f}caligraphic_N st... | 𝒩𝗍×fsubscript𝒩subscript𝗍𝑓\mathcal{N}_{\operatorname{\mathsf{t\!}_{\times}\!}f}caligraphic_N start_POSTSUBSCRIPT start_OPFUNCTION sansserif_t start_POSTSUBSCRIPT × end_POSTSUBSCRIPT end_OPFUNCTION italic_f end_POSTSUBSCRIPT has a rightmost segment of infinite length and | 84 |
and call an element of I𝐼Iitalic_I a player. (Intuitively, the role of ι𝜄ιitalic_ι is to specify which player controls the action at each decision node.) It will be assumed that ι𝜄ιitalic_ι is measurable (footnote 32) in the sense that it is measurable as a function from π1Esubscript𝜋1𝐸π_{1}Eitalic_π start_POSTSU... | the σ𝜎σitalic_σ-algebra whose elements are arbitrary unions of cells from the information-set partition ℋℋ{\mathcal{H}}caligraphic_H) into the player set I𝐼Iitalic_I (endowed with the discrete σ𝜎σitalic_σ-algebra). This is equivalent to | Thus each F(w)𝐹𝑤F(w)italic_F ( italic_w ) is the set of actions that label the edges leaving w𝑤witalic_w. Call F(w)𝐹𝑤F(w)italic_F ( italic_w ) the set of actions that are feasible at w𝑤witalic_w. Then let ⁅F(w)⁆w∈π1E˙:˙π1E˙→˙𝒫(A):⁅𝐹𝑤subscript⁆𝑤∈subscript𝜋1𝐸˙˙subscript𝜋1𝐸˙→˙𝒫𝐴⁅F(w)⁆_{w∈π_... | . The next component of a traditional game is a partition ℋℋ{\mathcal{H}}caligraphic_H of the decision-node set π1Esubscript𝜋1𝐸π_{1}Eitalic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_E. The members (i.e. cells) of this partition are called information sets. (Intuitively, a player will be informed that they are... | where in (32a), for each decision node w⋅∈⋅π1E𝑤⋅∈⋅subscript𝜋1𝐸w⋅∈⋅π_{1}Eitalic_w ⋅ ∈ ⋅ italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_E, Hwsubscript𝐻𝑤H_{w}italic_H start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT is the cell of the information-set partition ℋℋ{\mathcal{H}}caligraphic_H that contains w�... | 85 |
remaining eight vertices are v∅−y0;v{x,y}−z0;v{y,z}−y1;v{x,z}−z1subscript𝑣subscript𝑦0subscript𝑣𝑥𝑦subscript𝑧0subscript𝑣𝑦𝑧subscript𝑦1subscript𝑣𝑥𝑧subscript𝑧1v_{\emptyset}-y_{0};v_{\{x,y\}}-z_{0};v_{\{y,z\}}-y_{1};v_{\{x,z\}}-z_{1}italic_v start_POSTSUBSCRIPT ∅ end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT... | (y0z0y1z1)subscript𝑦0subscript𝑧0subscript𝑦1subscript𝑧1(y_{0}z_{0}y_{1}z_{1})( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). We can then define an involution on ... | {y0,y1}subscript𝑦0subscript𝑦1\{y_{0},y_{1}\}{ italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT } and {z0,z1}subscript𝑧0subscript𝑧1\{z_{0},z_{1}\}{ italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT } setwise. It is e... | remaining eight vertices are v∅−y0;v{x,y}−z0;v{y,z}−y1;v{x,z}−z1subscript𝑣subscript𝑦0subscript𝑣𝑥𝑦subscript𝑧0subscript𝑣𝑦𝑧subscript𝑦1subscript𝑣𝑥𝑧subscript𝑧1v_{\emptyset}-y_{0};v_{\{x,y\}}-z_{0};v_{\{y,z\}}-y_{1};v_{\{x,z\}}-z_{1}italic_v start_POSTSUBSCRIPT ∅ end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT... | leaves a six-cycle v{x,y}−x1−v{x,z}−z1−v{y,z}−y1subscript𝑣𝑥𝑦subscript𝑥1subscript𝑣𝑥𝑧subscript𝑧1subscript𝑣𝑦𝑧subscript𝑦1v_{\{x,y\}}-x_{1}-v_{\{x,z\}}-z_{1}-v_{\{y,z\}}-y_{1}italic_v start_POSTSUBSCRIPT { italic_x , italic_y } end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_v start... | 86 |
Figure 2: Overview of commonsense reasoning benchmarks categorized by the types of commonsense knowledge they evaluate. | The majority of benchmarks are structured as multiple-choice questions, where the evaluated model is presented with several options and must select the correct one or more. In one common scenario, referred to as question answering, the benchmark poses a specific question, and the model is required to identify the corre... | Some benchmarks are designed as binary-choice questions, where the model must select the correct answer from only two options rather than multiple options, as is the case with multiple-choice questions. Consequently, these tasks are inherently easier due to the reduced number of choices. Similarly, a common example of ... | In addition to the question-answering scenario, some benchmarks employ a completion task, where the model is required either to select the option that best continues the given context or to choose the option that best fits into a blank space. Prominent benchmarks in this category include: | In another scenario, the evaluated model performs a natural language inference (NLI) task, where it must select the option that is entailed by the given text. Some benchmarks in this category include: | 87 |
We want to find n𝑛nitalic_n such that for a given δ𝛿\deltaitalic_δ the following inequality holds: | P(missing the caps after n tries)=(1−Isin2ϕ(d~−12,12))n≤δ,𝑃missing the caps after n triessuperscript1subscript𝐼superscript2italic-ϕ~𝑑1212𝑛𝛿\displaystyle P(\text{missing the caps after $n$ tries})=\left(1-I_{\sin^{2}% | \phi}\left(\dfrac{\tilde{d}-1}{2},\dfrac{1}{2}\right)\right)^{n}\leq\delta,italic_P ( missing the caps after italic_n tries ) = ( 1 - italic_I start_POSTSUBSCRIPT roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ϕ end_POSTSUBSCRIPT ( divide start_ARG over~ start_ARG italic_d end_ARG - 1 end_ARG start_ARG 2 ... | 1−Isin2ϕ(d~−12,12).1subscript𝐼superscript2italic-ϕ~𝑑1212\displaystyle 1-I_{\sin^{2}\phi}\left(\dfrac{\tilde{d}-1}{2},\dfrac{1}{2}% | Isin2(αn)(d−12,12)=1B(d−12,12)∫0sin2(αn)ud−32(1−u)−12𝑑u,subscript𝐼superscript2subscript𝛼𝑛𝑑12121𝐵𝑑1212superscriptsubscript0superscript2subscript𝛼𝑛superscript𝑢𝑑32superscript1𝑢12differential-d𝑢\displaystyle I_{\sin^{2}(\alpha_{n})}\left(\frac{d-1}{2},\frac{1}{2}\right)=% | 88 |
Vhc={Φh∈C(Ω¯)∩Vc|Φh∈Pn(𝒦),∀𝒦∈𝒯h}.superscriptsubscript𝑉ℎ𝑐conditional-setsubscriptΦℎ𝐶¯Ωsuperscript𝑉𝑐formulae-sequencesubscriptΦℎsubscript𝑃𝑛𝒦for-all𝒦superscript𝒯ℎV_{h}^{c}=\{\Phi_{h}\in C(\bar{\Omega})\cap V^{c}~{}|~{}\Phi_{h}\in P_{n}(% | \mathcal{K}),\forall\mathcal{K}\in\mathcal{T}^{h}\}.italic_V start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = { roman_Φ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∈ italic_C ( over¯ start_ARG roman_Ω end_ARG ) ∩ italic_V start_POSTSUPERSCRIPT italic_c end_POSTSUPER... | \mathcal{K}}\in P_{n}(\mathcal{K}),\forall\mathcal{K}\in\mathcal{T}^{h}\right\},= { roman_Φ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∈ italic_C ( over¯ start_ARG roman_Ω end_ARG ) ∩ italic_V | roman_Φ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT | start_POSTSUBSCRIPT caligraphic_K end_POSTSUBSCRIPT ∈ italic_P s... | \right\},= { roman_Φ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∈ italic_C ( over¯ start_ARG roman_Ω end_ARG ) ∩ italic_V start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT | roman_Φ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT | start_POSTSUBSCRIPT caligraphic_K end_POSTSUBSCRIPT ∈ italic_P start_POSTSUBSCRIPT itali... | \mathcal{T}^{h}\right\},= { italic_λ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∈ italic_C ( roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) | italic_λ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT | start_POSTSUBSCRIPT caligraphic_K ∩ roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∈ italic_P ... | 89 |
We naturally generalize this notion to RAM machine. We denote by \delay[M][x]\delaydelimited-[]𝑀delimited-[]𝑥\delay[M][x][ italic_M ] [ italic_x ], \amdelay[M][x]\amdelaydelimited-[]𝑀delimited-[]𝑥\amdelay[M][x][ italic_M ] [ italic_x ] and \avdelay[M][x]\avdelaydelimited-[]𝑀delimited-[]𝑥\avdelay[M][x][ ital... | These notions of delay induce natural complexity classes. Some of them have been originally introduced by Johnson, Yanakakis and Papadimitriou in [27]. We are mostly interested in two classes. | On particular consequence of Theorem 13 is that one cannot use regularization schemes to prove that classes DelayP\polysuperscriptDelayP\poly\mathrm{DelayP}^{\poly}roman_DelayP start_POSTSUPERSCRIPT end_POSTSUPERSCRIPT and AmDelayP\polysuperscriptAmDelayP\poly\mathrm{AmDelayP}^{\poly}roman_AmDelayP start_POSTSUPERSCRIP... | An enumeration problem is the task of listing a set of elements without redundancies. It is an important and old class of problems: the Baguenaudier game [32] from the 19191919th century can be seen as the problem of enumerating integers in Gray code order. Ruskey even reports [38] on thousand-year-old methods to list ... | In Section 4.4, we generalize geometric regularization to amortized incremental delay algorithms. Section 5 proves lower bounds establishing the optimality of some of our results in the blackbox oracle model. In particular, we show that a regularization scheme without knowledge of the amortized delay d𝑑ditalic_d of an... | 90 |
\underbrace{0\cdots 0}_{i+1},\underbrace{0,\dots,0}_{k-1},\varepsilon,\dots,\varepsilon)( italic_ξ start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT , … , italic_ξ start_POSTSUBSCRIPT italic_p + 1 end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_b start_POSTSUBSCRIPT italic_t - 1 end_POSTS... | where the string of i+1𝑖1i+1italic_i + 1 zeroes is at index 2j−12𝑗12j-12 italic_j - 1. | where the string of i+1𝑖1i+1italic_i + 1 zeroes is at index 2j−12𝑗12j-12 italic_j - 1. | where the string of i+1𝑖1i+1italic_i + 1 zeroes is at index 2j−12𝑗12j-12 italic_j - 1. | where the string of i+1𝑖1i+1italic_i + 1 zeroes is at index 2j−12𝑗12j-12 italic_j - 1. | 91 |
_{1}\leq d(L|\tilde{B})_{\Lambda}\leq\mathcal{O}(\varepsilon).∥ italic_ρ start_POSTSUBSCRIPT italic_L italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_Y italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_M end_POSTSUBSCRIPT - italic_U start_POSTSUBSCRIPT italic_k / 4 end_POSTSUBSCRIPT ⊗ italic_ρ sta... | Since 𝗇𝗆𝖤𝗑𝗍𝗇𝗆𝖤𝗑𝗍\mathsf{nmExt}sansserif_nmExt is comprised of a sequence of applications of seeded extractors, we need to argue that sufficient min-entropy is retained throughout Protocol 1 in the sources on which seeded extractors are applied. | This is necessary since the procedure of alternating extraction is based on repeated application of seeded extractors, which need some sufficient entropy in the source. | Most of our arguments here are similar to the case of seeded extractor; so we note the modifications that we need to take care of in case of 2nmExt. | To show the security of 𝗇𝗆𝖤𝗑𝗍𝗇𝗆𝖤𝗑𝗍\mathsf{nmExt}sansserif_nmExt, we first explain the correspondence between Algorithm 1 and Protocol 1. Note that 𝗇𝗆𝖤𝗑𝗍𝗇𝗆𝖤𝗑𝗍\mathsf{nmExt}sansserif_nmExt as defined in Algorithm 1 is a generation of sequence of random variables until we finally output L=𝗇𝗆𝖤𝗑𝗍(X... | 92 |
there exist constants c>0𝑐0c>0italic_c > 0 and cα>0subscript𝑐𝛼0c_{\alpha}>0italic_c start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT > 0 such that | g(z)≥cand|Dαgij(z)|≤cα for every multiindex αformulae-sequence𝑔𝑧𝑐andsuperscript𝐷𝛼subscript𝑔𝑖𝑗𝑧subscript𝑐𝛼 for every multiindex 𝛼\sqrt{g(z)}\geq c\qquad\text{and}\qquad|D^{\alpha}g_{ij}(z)|\leq c_{\alpha}\ % | \text{ for every multiindex }\alphasquare-root start_ARG italic_g ( italic_z ) end_ARG ≥ italic_c and | italic_D start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_z ) | ≤ italic_c start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT for every multiind... | for every 0<ϱ<inj(M)0italic-ϱinj𝑀0<\varrho<\operatorname{inj}(M)0 < italic_ϱ < roman_inj ( italic_M ) and every multiindex α∈ℕ0d𝛼superscriptsubscriptℕ0𝑑\alpha\in\mathbb{N}_{0}^{d}italic_α ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | where α∈ℕ0d𝛼superscriptsubscriptℕ0𝑑\alpha\in\mathbb{N}_{0}^{d}italic_α ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is a multiindex, |α|=α1+⋯+αd𝛼subscript𝛼1⋯subscript𝛼𝑑|\alpha|=\alpha_{1}+\dots+\alpha_{d}| italic_α | = italic_α start_POSTSUBSCRIPT 1 end... | 93 |
Given 𝒟sourcelabeled={(xi,yi)}superscriptsubscript𝒟sourcelabeledsubscript𝑥𝑖subscript𝑦𝑖\mathcal{D}_{\text{source}}^{\text{labeled}}=\{(x_{i},y_{i})\}caligraphic_D start_POSTSUBSCRIPT source end_POSTSUBSCRIPT start_POSTSUPERSCRIPT labeled end_POSTSUPERSCRIPT = { ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSC... | p(y|hi)=softmax(𝐖hi+𝐛)∈ℝN,𝑝conditional𝑦subscriptℎ𝑖softmax𝐖subscriptℎ𝑖𝐛superscriptℝ𝑁\begin{split}p(y|h_{i})=\text{softmax}\left(\mathbf{W}h_{i}+\mathbf{b}\right)% | where hi∈ℝdsubscriptℎ𝑖superscriptℝ𝑑h_{i}\in\mathbb{R}^{d}italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is the feature representation of xisubscript𝑥𝑖x_{i}italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT given by the [CLS]delimited-... | θ∗=argminθℒce(𝒟sourcelabeled;θ).superscript𝜃subscriptargmin𝜃subscriptℒ𝑐𝑒superscriptsubscript𝒟sourcelabeled𝜃\begin{split}\theta^{*}=\operatorname*{arg\,min}_{\theta}\mathcal{L}_{ce}\left% | \in\mathbb{R}^{N},\end{split}start_ROW start_CELL italic_p ( italic_y | italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = softmax ( bold_W italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_b ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT , end_CELL end_ROW | 94 |
Many cryptocurrencies that offer complete anonymity or transaction anonymity have not been widely accepted due to their primary use-cases being money laundering and other illicit activities. Cryptocurrencies like Monero (XMR) are not listed on crypto-exchanges due to their hard-to-track privacy features. Often, fewer p... | In this paper, we distinctly identified the tiers of privacy and compared prominently known Cryptocurrencies based on their privacy offerings. We also studied various privacy-preserving measures and algorithms proposed to satisfy the two primary properties of untraceability and unlinkability. Several privacy problems a... | Bitcoin falls in the \sayPseudonymity tier from the four distinctly defined tiers of privacy. Many privacy attacks have been attempted on the Bitcoin blockchain by finding loopholes or exploiting the evident facts/limitations of the protocol. The attacks listed in this section are not limited to the Bitcoin blockchain ... | Bitcoin does provide some in-place countermeasures that can tackle the problem of privacy. Some of these features have been incorporated within the blockchain through soft forks and some of them are still on the soft-fork wishlist. Not all of the measures presented in this section have been incorporated or actively use... | Since its inception in 2009, Bitcoin has been heavily studied by researchers to look for flaws and improvements. Undoubtedly, Bitcoin has been one of the most successful Cryptocurrencies. Consequently, it has been targeted by adversaries and is under constant surveillance by government entities. Many attacks have been ... | 95 |
1 choose parameter s𝑠sitalic_s and base sizes d1,d2,d3subscript𝑑1subscript𝑑2subscript𝑑3d_{1},d_{2},d_{3}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_d start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | 3 set mi=2sdisubscript𝑚𝑖superscript2𝑠subscript𝑑𝑖m_{i}=2^{s}d_{i}italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2 start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and draw random functions Fi:[mi]→[ni]:subscript𝐹𝑖→delimited-[]subscript𝑚�... | _{i}\vee z_{i}-6/7)italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 4 / 7 , italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 4 / 7 , italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 4 / 7 , italic_x start_POSTSUBSCRIPT italic_i end_P... | In our algorithms, the hash functions Fisubscript𝐹𝑖F_{i}italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are chosen uniformly at random. The number of preimages is thus a Binomial random variable with mean 2sdinisuperscript2𝑠subscript𝑑𝑖subscript𝑛𝑖\frac{2^{s}d_{i}}{n_{i}}divide start_ARG 2 start_POSTSUPER... | \frac{\sigma_{0}}{n_{1}n_{2}n_{3}}C_{i,j}blackboard_E [ italic_C start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ] = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT ( italic_x , italic_y , italic_z ) ∈ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_POSTSUBSCRIPT divide start_A... | 96 |
where 𝚎=x^−x𝚎^𝑥𝑥\mathtt{e}=\hat{x}-xtypewriter_e = over^ start_ARG italic_x end_ARG - italic_x, d¯a=ϵℓ0+d¯e0subscript¯𝑑𝑎subscriptitalic-ϵℓ0subscript¯𝑑𝑒0\bar{d}_{a}=\epsilon_{\ell 0}+\bar{d}_{e0}over¯ start_ARG italic_d end_ARG start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_ϵ start_POSTSUBSCRIPT roman... | {2})({2}{\alpha_{G}}^{-1}+1)+\epsilon_{\ell 0}^{2}\alpha_{d}^{-1}italic_C start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT = ( over¯ start_ARG italic_g end_ARG start_POSTSUBSCRIPT italic_e 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + over¯ start_ARG italic_ρ end_ARG start_POSTSUPERSCRIPT 2 end_POSTSU... | where Vsℓ=∫xx^δq⊤Wδqsubscript𝑉𝑠ℓsubscriptsuperscript^𝑥𝑥𝛿superscript𝑞top𝑊𝛿𝑞V_{s\ell}=\int^{\hat{x}}_{x}\delta q^{\top}W\delta qitalic_V start_POSTSUBSCRIPT italic_s roman_ℓ end_POSTSUBSCRIPT = ∫ start_POSTSUPERSCRIPT over^ start_ARG italic_x end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POST... | \rho}^{2}\bar{c}^{2}\bar{g}_{e1}^{2}(\alpha_{G}+{1}/{2})/2italic_L start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT over¯ start_ARG italic_g end_ARG start_POSTSUBSCRIPT italic_e 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_α start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT + 1 / 2 ) / 2 + italic... | \overline{m}^{2}over¯ start_ARG italic_g end_ARG start_POSTSUBSCRIPT italic_e 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + over¯ start_ARG italic_ρ end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT over¯ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT over¯ start_ARG italic... | 97 |
𝑳Y𝑽Ysubscript𝑳𝑌subscript𝑽𝑌{\bm{L}}_{Y}{\bm{V}}_{Y}bold_italic_L start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT bold_italic_V start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT, respectively, i.e., | 𝒄¯X,j:=maxi=1,…,n[𝑳X𝑽X]i,j,\displaystyle\overline{{\bm{c}}}_{X,j}\mathrel{\mathrel{\mathop{:}}=}\max_{i=1% | 𝒄¯Y,j:=maxi=1,…,n[𝑳Y𝑽Y]i,j,\displaystyle\overline{{\bm{c}}}_{Y,j}\mathrel{\mathrel{\mathop{:}}=}\max_{i=1% | 𝒄¯X,j:=mini=1,…,n[𝑳X𝑽X]i,j,j=1,…,mX,\displaystyle\underline{{\bm{c}}}_{X,j}\mathrel{\mathrel{\mathop{:}}=}\min_{i=% | 𝒄¯Y,j:=mini=1,…,n[𝑳Y𝑽Y]i,j,j=1,…,mY.\displaystyle\underline{{\bm{c}}}_{Y,j}\mathrel{\mathrel{\mathop{:}}=}\min_{i=% | 98 |
+ℙ(a,c,d⊆S and b⊈S)+ℙ(b,c,d⊆S and a⊈S)+ℙ(a,b,c,d⊆S)ℙa,c,d⊆S and b⊈Sℙb,c,d⊆S and a⊈Sℙ𝑎𝑏𝑐𝑑𝑆\displaystyle+\mathbb{P}(\text{$a,c,d\subseteq S$ and $b\not\subseteq S$})+% | $a,b,c,d\subseteq S$})+ blackboard_P ( italic_a , italic_c , italic_d ⊆ italic_S and italic_b ⊈ italic_S ) + blackboard_P ( italic_b , italic_c , italic_d ⊆ italic_S and italic_a ⊈ italic_S ) + blackboard_P ( italic_a , italic_b , italic_c , italic_d ⊆ italic_S ) | $a,b,d\subseteq S$ and $c\not\subseteq S$})+ blackboard_P ( italic_b , italic_d ⊆ italic_S and italic_a , italic_c ⊈ italic_S ) + blackboard_P ( italic_a , italic_b , italic_c ⊆ italic_S and italic_d ⊈ italic_S ) + blackboard_P ( italic_a , italic_b , italic_d ⊆ italic_S and italic_c ⊈ italic_S ) | $b,c\subseteq S$ and $a,d\not\subseteq S$})blackboard_P ( italic_a , italic_c ⊆ italic_S and italic_b , italic_d ⊈ italic_S ) + blackboard_P ( italic_a , italic_d ⊆ italic_S and italic_b , italic_c ⊈ italic_S ) + blackboard_P ( italic_b , italic_c ⊆ italic_S and italic_a , italic_d ⊈ italic_S ) | \text{ and }b\subseteq S)\Big{)}( blackboard_P ( italic_a ⊆ italic_S and italic_b ⊈ italic_S ) + blackboard_P ( italic_a ⊈ italic_S and italic_b ⊆ italic_S ) + blackboard_P ( italic_a ⊆ italic_S and italic_b ⊆ italic_S ) ) | 99 |
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