Datasets:

saier

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unarXive_citrec

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f0fac466-5ba7-4854-9774-102235d25de5

After the deviation of the appropriate GS-TWDP statistical expressions, it is also necessary to verify the applicability of proposed distribution for modeling propagation in mmWave bands. In that sense, GS-TWDP model is fitted to a measurement results performed at 28 GHz and published in [1]}. Obtained results are then expressed in terms of a fitting error calculated using modified version of the Kolmogorov-Smirnov (KS) [2]}, in order to outweigh the fit in amplitude values closer to zero, for which the fading is more severe [2]}. Accordingly, goodness of fit between empirical and theoretical CDFs, denoted by \(\hat{F}_{r_c}(x)\) and \(F_{r_c}(x)\) respectively, is expressed as [2]}: \(\epsilon = \max \left|\log _{10}\hat{F}_r(x)-\log _{10}F_{r_c}(x)\right|\)

[2]

https://openalex.org/W3098595577

e20733bf-804a-41be-8126-65c9b0a600f9

Heart Rate Variability (HRV) [1]}, [2]} is a reliable measure used in physiological and psychological research, since HRV reflects cardiac autonomic nervous system (ANS) regulation variation (i.e., the changing balance between the sympathetic and parasympathetic nervous systems). Beyond a purely cardiovascular measure, HRV has been used as a tool to detect acute illness (e.g., early detection of COVID-19 [3]}), common cold [4]}, and inflammatory response from infection [5]}). HRV also aids diagnostics [1]}, and is predictive of various cardiovascular-related diseases [7]}. Numerous studies have investigated the connection between HRV and a range of conditions, including cardiovascular [8]}, [9]}, [10]}, blood pressure [11]}, [12]}, and myocardial infarction [13]}, [14]}, [15]}.

[1]

https://openalex.org/W2026699230

4d5b2a49-a1c3-4ab2-af91-2288e1dee2ec

Note that joint estimation of the coupling and map, as equation (REF ) does, is shown to be effective in a few OT studies [1]}, [2]}, but this is the first time optimal transport map estimation is used in a multi-task regression.

[1]

https://openalex.org/W2963237040

86b00c84-b217-4bd2-a134-b84ea79194aa

When fixing \(\mathbf {W}\) , the second term of equation (REF ) is related to a category of OT formulation that can be solved by alternatively minimizing [1]} over coupling \(\pi \) and transformation \(\mathbf {F}\) . Specifically, when solving for \(\pi \) with \(\mathbf {W}\) and \(\mathbf {F}\) fixed, the objective of equation (REF ) becomes an \(O(n^3 \log n)\) linear program. Here, adding an entropy regularization will allow us to use the celebrated Sinkhorn algorithm Ag.(REF ) [2]} that just requires \(O(n^2)\) operations [3]}, as: \(\pi ^* = \arg \min _{\pi \in \Pi } \sum _{i,j=1}^T \pi _{i,j} {\mathbf {C}}_{i,j} + \gamma H(\pi ),\)

[1]

https://openalex.org/W2963237040

c3cac511-7df7-429e-b03a-bee14aaa56f5

To tackle domain adaptation limitations, we incorporate optimal transport map estimation theory with multi-task learning. Optimal transport (OT) [1]} has attracted considerable interest in the machine learning community. OT provides a meaningful distance between probability measures that do not share the same support; OT also formulates a transport map [2]}, [3]}, [4]} that pushes forward one distribution onto another. Also, the addition of entropic regularization [5]} yields fast computation, thus enabling scalability of OT to larger problems.

[4]

https://openalex.org/W2963237040

cc10035e-587e-4381-8fd4-4415a0fa9efd

Remote sensing (RS) ship detection has attracted extensive attention in recent years due to its large potential in both civilian and military applications (e.g., port management, target surveillance). As a key factor of ship detection, high-resolution (HR) images can provide abundant appearance information and thus introduce improvement to the detection accuracy [1]}. However, obtaining an HR image posts a high requirement on the satellite sensors and generally results in an expensive cost. Consequently, using image super-resolution (SR) techniques to recover the missing details in RS images has become a popular research topic and has been widely investigated in recent years.

[1]

https://openalex.org/W2963606888

84c69712-176d-45a4-922d-67df64963b37

Aiming at the aforementioned issue, in this paper, we propose an end-to-end network named ShipSRDet to fully use the super-resolved feature representation for RS ship detection. Different from existing SR-based detection methods, we not only feed the super-resolved images to the detector, but also integrate the intermediate features of the SR network with those of the detection network. In this way, more informative cues can be transferred from the SR network to the detection network to enhance the detection performance. Experiments on the HRSC dataset [1]} demonstrate the effectiveness of our method. As shown in Fig. REF , our ShipSRDet achieves notable improvements on detection performance, and recovers the missing details in the input image. <FIGURE>

[1]

https://openalex.org/W2594177559

9f848584-569e-468f-b530-ff6f5c10a6c6

As shown in Fig. REF (a), our SR module takes a medium-low resolution image \(\mathcal {I}_{LR}\in \mathbb {R}^{H\times W\times 3}\) as its input to produce an SR image \(\mathcal {I}_{SR}\in \mathbb {R}^{\alpha H\times \alpha W\times 3}\) and two intermediate features \(\mathcal {F}^{LR}_{out}\in \mathbb {R}^{H\times W\times 64}\) and \(\mathcal {F}^{HR}_{out}\in \mathbb {R}^{\alpha H\times \alpha W\times 64}\) , where \(H\) and \(W\) represent the height and width of the input image, and \(\alpha \) denotes the upscaling factor. Here, we use the residual dense block (RDB) [1]} as the basic block in our SR module since it can fully use features from all preceding layers to generate hierarchical representations, which is demonstrated beneficial to SR reconstruction [2]}.

[1]

https://openalex.org/W2964101377

b692f859-80b4-4398-b582-35e96f5cbba5

As shown in Fig. REF (a), our SR module takes a medium-low resolution image \(\mathcal {I}_{LR}\in \mathbb {R}^{H\times W\times 3}\) as its input to produce an SR image \(\mathcal {I}_{SR}\in \mathbb {R}^{\alpha H\times \alpha W\times 3}\) and two intermediate features \(\mathcal {F}^{LR}_{out}\in \mathbb {R}^{H\times W\times 64}\) and \(\mathcal {F}^{HR}_{out}\in \mathbb {R}^{\alpha H\times \alpha W\times 64}\) , where \(H\) and \(W\) represent the height and width of the input image, and \(\alpha \) denotes the upscaling factor. Here, we use the residual dense block (RDB) [1]} as the basic block in our SR module since it can fully use features from all preceding layers to generate hierarchical representations, which is demonstrated beneficial to SR reconstruction [2]}.

[2]

https://openalex.org/W3175560234

2e816274-aee1-4030-a298-d676cc3ccb73

Specifically, the input image \(\mathcal {I}_{LR}\) is first fed to a \(3\times 3\) convolution to generate initial feature \(\mathcal {F}_0 \in \mathbb {R}^{H\times W\times 64}\) . Then, \(\mathcal {F}_0\) is fed to 8 cascaded RDBs for deep feature extraction. Within each RDB, we use 5 convolutions with a growth rate of 32. As shown in Fig. REF (b), features from all the layers in an RDB are concatenated and fed to a \(1\times 1\) convolution for local fusion. Similarly, features from all the RDBs in our SR module are concatenated for global fusion. Afterwards, the fused feature \(\mathcal {F}^{LR}_{out}\) is added with the initial feature \(\mathcal {F}_0\) and fed to a sub-pixel layer [1]} to generate the upsampled feature \(\mathcal {F}^{HR}_{out}\) . Finally, \(\mathcal {F}^{HR}_{out}\) is fed to a \(3\times 3\) convolution to produce the residual prediction which is further added with the bicubicly upsampled input image to generate the final SR image \(\mathcal {I}_{SR}\) .

[1]

https://openalex.org/W2476548250

78d34211-ad27-4864-aa59-857bffb2de07

Single shot multi-box detector (SSD) [1]} is used as the detection module in our ShipSRDet. In our detection module, VGG-16 network [2]} is used to extract multi-scale features from the input SR image \(\mathcal {I}_{SR}\) . Simultaneously, feature adaption is performed to integrate features from the SR module with those extracted by the VGG network. Specifically, two transition convolutions are performed on \(\mathcal {F}^{LR}_{out}\) and \(\mathcal {F}^{HR}_{out}\) to adjust their feature depth to 256 and 64 to produce feature \(\mathcal {F}^{LR}_{SSD}\) and \(\mathcal {F}^{HR}_{SSD}\) , respectively. Then, \(\mathcal {F}^{HR}_{SSD}\) and \(\mathcal {F}^{LR}_{SSD}\) are added to the features in the 2nd and 7th layer of the VGG network to achieve feature integration. After initial feature extraction with the VGG network, an encoder is further employed for high-level feature extraction. Finally, seven feature maps of different resolutions are generated and fed into the detector\(\&\) classifier to predict the location and category candidates. The final prediction is produced by performing fast non-maximum suppression (fast NMS) on the candidates.

[1]

https://openalex.org/W3106250896

1c122b70-874c-4a7f-8db3-045ffd43c376

Single shot multi-box detector (SSD) [1]} is used as the detection module in our ShipSRDet. In our detection module, VGG-16 network [2]} is used to extract multi-scale features from the input SR image \(\mathcal {I}_{SR}\) . Simultaneously, feature adaption is performed to integrate features from the SR module with those extracted by the VGG network. Specifically, two transition convolutions are performed on \(\mathcal {F}^{LR}_{out}\) and \(\mathcal {F}^{HR}_{out}\) to adjust their feature depth to 256 and 64 to produce feature \(\mathcal {F}^{LR}_{SSD}\) and \(\mathcal {F}^{HR}_{SSD}\) , respectively. Then, \(\mathcal {F}^{HR}_{SSD}\) and \(\mathcal {F}^{LR}_{SSD}\) are added to the features in the 2nd and 7th layer of the VGG network to achieve feature integration. After initial feature extraction with the VGG network, an encoder is further employed for high-level feature extraction. Finally, seven feature maps of different resolutions are generated and fed into the detector\(\&\) classifier to predict the location and category candidates. The final prediction is produced by performing fast non-maximum suppression (fast NMS) on the candidates.

[2]

https://openalex.org/W1686810756

13d5b6d0-eed6-4da9-a8ca-7233500e5b05

We used the HRSC dataset [1]} for both training and test. We followed the original dataset split and performed \(2\times \) and \(8\times \) bicubic downsampling to generate training and test image pairs. Consequently, each HR image has a resolution of \(512\times 512\) and its corresponding medium-low resolution conterpart has a resolution of \(128\times 128\) . Random horizontal flipping, random vertical flipping and random rotation were performed for data augmentation.

[1]

https://openalex.org/W2594177559

df6b961f-0994-448b-9ef9-6a7c5bd6f53b

Our ShipSRDet was implemented in PyTorch on a PC with an RTX 2080Ti GPU, and trained in a two-stage pipeline. In the first stage, we followed [1]}, [2]} to train our SR module using the generated image pairs with an \(L_1\) loss. Adam method [3]} is used for optimization. The batch size was set to 4 and the learning rate was was initially set to \(1\times 10^{-4}\) and halved for every 200 epochs. The training was stopped after 450 epochs. In the second stage, we concatenated the SR module with the pre-trained detection moduleWe used the publicly available SSD network which was pre-trained on the COCO [4]} dataset., and performed end-to-end finetuning for global optimization. In the finetuning stage, the learning rate was initially set to \(1\times 10^{-4}\) and decreased by a factor of 0.1 for every 10 epochs. The finetuning process was performed for 24 epochs.

[1]

https://openalex.org/W3110151525

f4006391-4fa6-49c9-9051-846aae464bb7

Our ShipSRDet was implemented in PyTorch on a PC with an RTX 2080Ti GPU, and trained in a two-stage pipeline. In the first stage, we followed [1]}, [2]} to train our SR module using the generated image pairs with an \(L_1\) loss. Adam method [3]} is used for optimization. The batch size was set to 4 and the learning rate was was initially set to \(1\times 10^{-4}\) and halved for every 200 epochs. The training was stopped after 450 epochs. In the second stage, we concatenated the SR module with the pre-trained detection moduleWe used the publicly available SSD network which was pre-trained on the COCO [4]} dataset., and performed end-to-end finetuning for global optimization. In the finetuning stage, the learning rate was initially set to \(1\times 10^{-4}\) and decreased by a factor of 0.1 for every 10 epochs. The finetuning process was performed for 24 epochs.

[2]

https://openalex.org/W3111051040

009cc96c-ac28-42f5-8382-4c3885a04194

Our ShipSRDet was implemented in PyTorch on a PC with an RTX 2080Ti GPU, and trained in a two-stage pipeline. In the first stage, we followed [1]}, [2]} to train our SR module using the generated image pairs with an \(L_1\) loss. Adam method [3]} is used for optimization. The batch size was set to 4 and the learning rate was was initially set to \(1\times 10^{-4}\) and halved for every 200 epochs. The training was stopped after 450 epochs. In the second stage, we concatenated the SR module with the pre-trained detection moduleWe used the publicly available SSD network which was pre-trained on the COCO [4]} dataset., and performed end-to-end finetuning for global optimization. In the finetuning stage, the learning rate was initially set to \(1\times 10^{-4}\) and decreased by a factor of 0.1 for every 10 epochs. The finetuning process was performed for 24 epochs.

[3]

https://openalex.org/W2964121744

b4f1e715-40e3-4a17-8251-919eb1eca7a5

Our ShipSRDet was implemented in PyTorch on a PC with an RTX 2080Ti GPU, and trained in a two-stage pipeline. In the first stage, we followed [1]}, [2]} to train our SR module using the generated image pairs with an \(L_1\) loss. Adam method [3]} is used for optimization. The batch size was set to 4 and the learning rate was was initially set to \(1\times 10^{-4}\) and halved for every 200 epochs. The training was stopped after 450 epochs. In the second stage, we concatenated the SR module with the pre-trained detection moduleWe used the publicly available SSD network which was pre-trained on the COCO [4]} dataset., and performed end-to-end finetuning for global optimization. In the finetuning stage, the learning rate was initially set to \(1\times 10^{-4}\) and decreased by a factor of 0.1 for every 10 epochs. The finetuning process was performed for 24 epochs.

[4]

https://openalex.org/W1861492603

5dbf03f1-8c74-4586-9f7f-4eba6e8c4a56

Discontinuous Galerkin (dG) finite element methods were introduced in the early 1970s for the numerical solution of first-order hyperbolic problems [1]} and the weak imposition of inhomogeneous boundary conditions for elliptic problems [2]}. In the past several decades, dG methods have enjoyed considerable success as a standard variational framework for the numerical solution of many classes of problems involving partial differential equations (PDEs); see, e.g., monographs [3]}, [4]}, [5]} for reviews of some of the main developments of dG methods. The interest in dG methods can be attributed to a number of factors, including the great flexibility in dealing with \(hp\) -adaptivity and general shaped elements [6]}, [7]}, [8]}, as well as in solving convection-dominated PDEs; see, e.g., early works [9]}, [10]} concerning hyperbolic conservation laws and convection-diffusion problems.

[1]

https://openalex.org/W49087508

c9f5de7e-3bd7-4a4e-8a28-cbab896746de

Discontinuous Galerkin (dG) finite element methods were introduced in the early 1970s for the numerical solution of first-order hyperbolic problems [1]} and the weak imposition of inhomogeneous boundary conditions for elliptic problems [2]}. In the past several decades, dG methods have enjoyed considerable success as a standard variational framework for the numerical solution of many classes of problems involving partial differential equations (PDEs); see, e.g., monographs [3]}, [4]}, [5]} for reviews of some of the main developments of dG methods. The interest in dG methods can be attributed to a number of factors, including the great flexibility in dealing with \(hp\) -adaptivity and general shaped elements [6]}, [7]}, [8]}, as well as in solving convection-dominated PDEs; see, e.g., early works [9]}, [10]} concerning hyperbolic conservation laws and convection-diffusion problems.

[2]

https://openalex.org/W2079423956

7f766b4a-6a4c-45ca-af74-13a8ce83f059

Discontinuous Galerkin (dG) finite element methods were introduced in the early 1970s for the numerical solution of first-order hyperbolic problems [1]} and the weak imposition of inhomogeneous boundary conditions for elliptic problems [2]}. In the past several decades, dG methods have enjoyed considerable success as a standard variational framework for the numerical solution of many classes of problems involving partial differential equations (PDEs); see, e.g., monographs [3]}, [4]}, [5]} for reviews of some of the main developments of dG methods. The interest in dG methods can be attributed to a number of factors, including the great flexibility in dealing with \(hp\) -adaptivity and general shaped elements [6]}, [7]}, [8]}, as well as in solving convection-dominated PDEs; see, e.g., early works [9]}, [10]} concerning hyperbolic conservation laws and convection-diffusion problems.

[3]

https://openalex.org/W1600997507

a7d17117-3f2d-4066-801a-28149e845fc0

Discontinuous Galerkin (dG) finite element methods were introduced in the early 1970s for the numerical solution of first-order hyperbolic problems [1]} and the weak imposition of inhomogeneous boundary conditions for elliptic problems [2]}. In the past several decades, dG methods have enjoyed considerable success as a standard variational framework for the numerical solution of many classes of problems involving partial differential equations (PDEs); see, e.g., monographs [3]}, [4]}, [5]} for reviews of some of the main developments of dG methods. The interest in dG methods can be attributed to a number of factors, including the great flexibility in dealing with \(hp\) -adaptivity and general shaped elements [6]}, [7]}, [8]}, as well as in solving convection-dominated PDEs; see, e.g., early works [9]}, [10]} concerning hyperbolic conservation laws and convection-diffusion problems.

[4]

https://openalex.org/W628726389

bfcabf28-8f8a-4268-807d-412c038aa7b6

Discontinuous Galerkin (dG) finite element methods were introduced in the early 1970s for the numerical solution of first-order hyperbolic problems [1]} and the weak imposition of inhomogeneous boundary conditions for elliptic problems [2]}. In the past several decades, dG methods have enjoyed considerable success as a standard variational framework for the numerical solution of many classes of problems involving partial differential equations (PDEs); see, e.g., monographs [3]}, [4]}, [5]} for reviews of some of the main developments of dG methods. The interest in dG methods can be attributed to a number of factors, including the great flexibility in dealing with \(hp\) -adaptivity and general shaped elements [6]}, [7]}, [8]}, as well as in solving convection-dominated PDEs; see, e.g., early works [9]}, [10]} concerning hyperbolic conservation laws and convection-diffusion problems.

[6]

https://openalex.org/W2116404756

d5540343-5dd2-4bb3-a1e6-8fc1c081c4d4

Discontinuous Galerkin (dG) finite element methods were introduced in the early 1970s for the numerical solution of first-order hyperbolic problems [1]} and the weak imposition of inhomogeneous boundary conditions for elliptic problems [2]}. In the past several decades, dG methods have enjoyed considerable success as a standard variational framework for the numerical solution of many classes of problems involving partial differential equations (PDEs); see, e.g., monographs [3]}, [4]}, [5]} for reviews of some of the main developments of dG methods. The interest in dG methods can be attributed to a number of factors, including the great flexibility in dealing with \(hp\) -adaptivity and general shaped elements [6]}, [7]}, [8]}, as well as in solving convection-dominated PDEs; see, e.g., early works [9]}, [10]} concerning hyperbolic conservation laws and convection-diffusion problems.

[8]

https://openalex.org/W2116404756

0080b7dd-e3b1-4448-ab08-dfb6b33f8596

Discontinuous Galerkin (dG) finite element methods were introduced in the early 1970s for the numerical solution of first-order hyperbolic problems [1]} and the weak imposition of inhomogeneous boundary conditions for elliptic problems [2]}. In the past several decades, dG methods have enjoyed considerable success as a standard variational framework for the numerical solution of many classes of problems involving partial differential equations (PDEs); see, e.g., monographs [3]}, [4]}, [5]} for reviews of some of the main developments of dG methods. The interest in dG methods can be attributed to a number of factors, including the great flexibility in dealing with \(hp\) -adaptivity and general shaped elements [6]}, [7]}, [8]}, as well as in solving convection-dominated PDEs; see, e.g., early works [9]}, [10]} concerning hyperbolic conservation laws and convection-diffusion problems.

[10]

https://openalex.org/W2029017626

cf380ab6-7244-478c-8fd6-3aa779f92df3

Due to missing tools in the analysis, the convergence rate always contains suboptimality in terms of the polynomial degree \(p\) . In [1]}, the first optimal convergence rate of \(hp\) -dG methods is derived for linear convection problems by using the SUPG stabilisation. However, the authors provide numerical evidence that the \(hp\) -optimal convergence rate is achieved even without such stabilisation. In seminal work [2]}, based on (back then) novel optimal approximation results for the \(L^2\) -orthogonal projection, the \(hp\) -optimal convergence rate is derived for dG methods applied to hyperbolic problems, under the technical assumption that the convection field is piecewise linear. Moreover, whenever the above assumption is violated, the theoretical analysis in [2]} leads to error bounds that are suboptimal in terms of \(p\) by \(3/2\) order. Such suboptimality is yet not observed in the numerical experiments. Over the last two decades, the above mentioned technical assumption became standard in \(hp\) -dG methods for convection-diffusion-reaction and hyperbolic problems; see, e.g., [4]}, [5]}, [6]}, [7]}. It is still an open question, whether the \(p\) -suboptimality for dG methods by \(3/2\) order is true or not in general.

[6]

https://openalex.org/W2116404756

aa26ed01-0eda-416e-9e96-bb3eb4d3248e

Due to missing tools in the analysis, the convergence rate always contains suboptimality in terms of the polynomial degree \(p\) . In [1]}, the first optimal convergence rate of \(hp\) -dG methods is derived for linear convection problems by using the SUPG stabilisation. However, the authors provide numerical evidence that the \(hp\) -optimal convergence rate is achieved even without such stabilisation. In seminal work [2]}, based on (back then) novel optimal approximation results for the \(L^2\) -orthogonal projection, the \(hp\) -optimal convergence rate is derived for dG methods applied to hyperbolic problems, under the technical assumption that the convection field is piecewise linear. Moreover, whenever the above assumption is violated, the theoretical analysis in [2]} leads to error bounds that are suboptimal in terms of \(p\) by \(3/2\) order. Such suboptimality is yet not observed in the numerical experiments. Over the last two decades, the above mentioned technical assumption became standard in \(hp\) -dG methods for convection-diffusion-reaction and hyperbolic problems; see, e.g., [4]}, [5]}, [6]}, [7]}. It is still an open question, whether the \(p\) -suboptimality for dG methods by \(3/2\) order is true or not in general.

[7]

https://openalex.org/W2948386797

05f7ef6f-1751-418a-a811-2583aca3b99d

The well-posedness of method (REF ) can be found, e.g., in [1]}.

[1]

https://openalex.org/W628726389

03c27614-2a23-4287-8959-21db6783fe1f

Thanks to assumption (REF ) and an \(\) -polynomial inverse estimate involving bubbles, see, e.g., [1]}, we deduce \(\left\Vert \frac{b_j - \mathcal {I}_j b_j}{\sqrt{w_j}} \right\Vert _{\infty ,K} \lesssim K\vert b_j \vert _{W^{2,\infty }(K)}, \quad \quad \Vert \sqrt{w_j} \partial _j \xi \Vert _{0,K} \lesssim \sqrt{p(p+1)} \Vert \xi \Vert _{0,K}.\)

[1]

https://openalex.org/W253824085

fd39448c-619d-4b61-88cc-28d0f3b91908

Integrable turbulence is a fascinating topic of nonlinear physics. Integrable turbulence is theoretically and numerically described in the framework of integrable equations amongst which the KdV equation [1]}, [2]}, the Gardner equation [3]}, and the 1D-nonlinear Schrödinger equation [4]}, [5]}. In these systems an infinite number of degrees of freedom can be excited randomly. As such there is no energy transfer between these modes and the word turbulence does not refer to the usual energy cascade between scales. Nevertheless, these systems can exhibit complex random behaviors that require a statistical description. Integrable turbulence applies to many fields of physics: hydrodynamics, optics, and plasmas. [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [1]}, [13]}, [14]}, [15]}, [16]}.

[2]

https://openalex.org/W2963740033

6e15c130-6949-48c6-a908-b3493a4ba259

Integrable turbulence is a fascinating topic of nonlinear physics. Integrable turbulence is theoretically and numerically described in the framework of integrable equations amongst which the KdV equation [1]}, [2]}, the Gardner equation [3]}, and the 1D-nonlinear Schrödinger equation [4]}, [5]}. In these systems an infinite number of degrees of freedom can be excited randomly. As such there is no energy transfer between these modes and the word turbulence does not refer to the usual energy cascade between scales. Nevertheless, these systems can exhibit complex random behaviors that require a statistical description. Integrable turbulence applies to many fields of physics: hydrodynamics, optics, and plasmas. [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [1]}, [13]}, [14]}, [15]}, [16]}.

[4]

https://openalex.org/W2080649718

ebe2ea2d-f925-4311-8198-e2b06200ddd5

Integrable turbulence is a fascinating topic of nonlinear physics. Integrable turbulence is theoretically and numerically described in the framework of integrable equations amongst which the KdV equation [1]}, [2]}, the Gardner equation [3]}, and the 1D-nonlinear Schrödinger equation [4]}, [5]}. In these systems an infinite number of degrees of freedom can be excited randomly. As such there is no energy transfer between these modes and the word turbulence does not refer to the usual energy cascade between scales. Nevertheless, these systems can exhibit complex random behaviors that require a statistical description. Integrable turbulence applies to many fields of physics: hydrodynamics, optics, and plasmas. [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [1]}, [13]}, [14]}, [15]}, [16]}.

[7]

https://openalex.org/W2303595150

f83996a6-6079-4d3c-bf4e-4d8efb6186cc

Integrable turbulence is a fascinating topic of nonlinear physics. Integrable turbulence is theoretically and numerically described in the framework of integrable equations amongst which the KdV equation [1]}, [2]}, the Gardner equation [3]}, and the 1D-nonlinear Schrödinger equation [4]}, [5]}. In these systems an infinite number of degrees of freedom can be excited randomly. As such there is no energy transfer between these modes and the word turbulence does not refer to the usual energy cascade between scales. Nevertheless, these systems can exhibit complex random behaviors that require a statistical description. Integrable turbulence applies to many fields of physics: hydrodynamics, optics, and plasmas. [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [1]}, [13]}, [14]}, [15]}, [16]}.

[8]

https://openalex.org/W2340178336

0fa1a5d3-2230-4a73-a88a-e24334ece9d5

Integrable turbulence is a fascinating topic of nonlinear physics. Integrable turbulence is theoretically and numerically described in the framework of integrable equations amongst which the KdV equation [1]}, [2]}, the Gardner equation [3]}, and the 1D-nonlinear Schrödinger equation [4]}, [5]}. In these systems an infinite number of degrees of freedom can be excited randomly. As such there is no energy transfer between these modes and the word turbulence does not refer to the usual energy cascade between scales. Nevertheless, these systems can exhibit complex random behaviors that require a statistical description. Integrable turbulence applies to many fields of physics: hydrodynamics, optics, and plasmas. [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [1]}, [13]}, [14]}, [15]}, [16]}.

[13]

https://openalex.org/W3104785954

cf1b01ca-41ff-46ce-a1f7-4bd1d436f0bb

The theory of integrable turbulence in water wave problems is found to be analytically tractable for two “asymptotic” situations. On the one hand when the waves are of small amplitude the expansion in powers of non linearity yields kinetic equations that model wave resonant interactions. In 2D situations such as for Kadomtsev-Petviashvili type equations [1]} resonant interactions are three-wave resonances. It is known that in the case of the KdV equation the first nontrivial resonances are five-wave interactions but with zero amplitudes [2]}. This result tends to indicate that integrable weak wave turbulence in 1D shallow water cases such as for the KdV equations is precluded.

[1]

https://openalex.org/W2080649718

d3ed370b-92d3-4ff7-98d8-638f3887a711

On the other hand when the turbulence can be considered as a collection of solitons with random amplitudes and phases, kinetic theories of rarefied soliton gas [1]} or dense ones [2]}, [3]}, [4]}, [5]} can be derived . A soliton gas is thus a random state in which solitons behave as quasi-particles due to the fact that their collisions are elastic, only altering relative phases, and thus changing the mean phase speed [1]}. Solitons in the shallow water framework are localized waves which propagate without change of shape due to a balance between linear dispersive effects that tend to flatten out any surface perturbations and nonlinear effects that steepen the fronts. Solitons are at the core of integrable dynamics of the KdV equation.

[1]

https://openalex.org/W3146553400

34a77ff5-816c-4d43-b8d2-dd086485c972

On the other hand when the turbulence can be considered as a collection of solitons with random amplitudes and phases, kinetic theories of rarefied soliton gas [1]} or dense ones [2]}, [3]}, [4]}, [5]} can be derived . A soliton gas is thus a random state in which solitons behave as quasi-particles due to the fact that their collisions are elastic, only altering relative phases, and thus changing the mean phase speed [1]}. Solitons in the shallow water framework are localized waves which propagate without change of shape due to a balance between linear dispersive effects that tend to flatten out any surface perturbations and nonlinear effects that steepen the fronts. Solitons are at the core of integrable dynamics of the KdV equation.

[2]

https://openalex.org/W2158432659

e1d61888-8d11-47be-8409-dc62d9e1baf5

On the other hand when the turbulence can be considered as a collection of solitons with random amplitudes and phases, kinetic theories of rarefied soliton gas [1]} or dense ones [2]}, [3]}, [4]}, [5]} can be derived . A soliton gas is thus a random state in which solitons behave as quasi-particles due to the fact that their collisions are elastic, only altering relative phases, and thus changing the mean phase speed [1]}. Solitons in the shallow water framework are localized waves which propagate without change of shape due to a balance between linear dispersive effects that tend to flatten out any surface perturbations and nonlinear effects that steepen the fronts. Solitons are at the core of integrable dynamics of the KdV equation.

[3]

https://openalex.org/W2098476508

590da5b5-962b-4f8b-bbf1-b4c8a6ef6d89

On the other hand when the turbulence can be considered as a collection of solitons with random amplitudes and phases, kinetic theories of rarefied soliton gas [1]} or dense ones [2]}, [3]}, [4]}, [5]} can be derived . A soliton gas is thus a random state in which solitons behave as quasi-particles due to the fact that their collisions are elastic, only altering relative phases, and thus changing the mean phase speed [1]}. Solitons in the shallow water framework are localized waves which propagate without change of shape due to a balance between linear dispersive effects that tend to flatten out any surface perturbations and nonlinear effects that steepen the fronts. Solitons are at the core of integrable dynamics of the KdV equation.

[4]

https://openalex.org/W3188857046

7aaaddca-11a2-4423-81c8-5308fd38f937

On the other hand when the turbulence can be considered as a collection of solitons with random amplitudes and phases, kinetic theories of rarefied soliton gas [1]} or dense ones [2]}, [3]}, [4]}, [5]} can be derived . A soliton gas is thus a random state in which solitons behave as quasi-particles due to the fact that their collisions are elastic, only altering relative phases, and thus changing the mean phase speed [1]}. Solitons in the shallow water framework are localized waves which propagate without change of shape due to a balance between linear dispersive effects that tend to flatten out any surface perturbations and nonlinear effects that steepen the fronts. Solitons are at the core of integrable dynamics of the KdV equation.

[5]

https://openalex.org/W3150535889

e452cbef-b1c3-4e66-847e-d001d918ae23

Empirical confirmation that soliton gases can be generated were given in optics [1]}, for deep water waves [2]} and for shallow water wave motion [3]}, [4]}. In the experiments energy dissipation cannot be avoided and this seems at first glance strongly incompatible with the concept of integrability. However, [3]} observed that due to a large scale separation between the nonlinear scale related to the (short) duration of soliton collisions and the (long) dissipative time scale, the dynamics is overall ruled by integrability. A stationary random soliton gas in a long wave flume in shallow water conditions can thus be sustained with continuous energy input by the wavemaker. Even though not labeled as soliton gases some 1D flume experiments of [6]}, in which the wavemaker has a sinusoidal displacement, lead to random wave motions. Therefore, the role of the wavemaker in the outcome of these random stationnary wave states needs to be explored.

[2]

https://openalex.org/W3039663288

61857c2d-a49c-4f6e-9e3b-6b79f025e50b

Empirical confirmation that soliton gases can be generated were given in optics [1]}, for deep water waves [2]} and for shallow water wave motion [3]}, [4]}. In the experiments energy dissipation cannot be avoided and this seems at first glance strongly incompatible with the concept of integrability. However, [3]} observed that due to a large scale separation between the nonlinear scale related to the (short) duration of soliton collisions and the (long) dissipative time scale, the dynamics is overall ruled by integrability. A stationary random soliton gas in a long wave flume in shallow water conditions can thus be sustained with continuous energy input by the wavemaker. Even though not labeled as soliton gases some 1D flume experiments of [6]}, in which the wavemaker has a sinusoidal displacement, lead to random wave motions. Therefore, the role of the wavemaker in the outcome of these random stationnary wave states needs to be explored.

[3]

https://openalex.org/W2942459581

4054b07c-1333-4e6a-90a3-dffab0b0547c

\(50\,\) years have elapsed between the first theoretical description of soliton gases by [1]} and the first hydrodynamic experiments. A possible reason is the requirement of highly resolved instruments to capture the space-time evolution of a random state. Another more fundamental issue relates to the difference between an infinite or periodic domain setting, usually used in theoretical approaches, and finite length experimental set-ups. This difference also combines with how initial conditions are easily prescribed in theory and numerics while boundary conditions are most of the time the only options at least in hydrodynamic experiments. A recent notable exception are the deep water soliton gas experiments by Suret et al. [2]} in which the Inverse Scattering Transform for the 1D Schrödinger equation is used to compute boundary conditions in a very long flume. In these experiments an ensemble of random spectral values associated to solitons are prescribed which then evolve towards a soliton gas.

[1]

https://openalex.org/W3146553400

5665867a-b6cf-42e2-9de7-baf8b55d2bb9

\(50\,\) years have elapsed between the first theoretical description of soliton gases by [1]} and the first hydrodynamic experiments. A possible reason is the requirement of highly resolved instruments to capture the space-time evolution of a random state. Another more fundamental issue relates to the difference between an infinite or periodic domain setting, usually used in theoretical approaches, and finite length experimental set-ups. This difference also combines with how initial conditions are easily prescribed in theory and numerics while boundary conditions are most of the time the only options at least in hydrodynamic experiments. A recent notable exception are the deep water soliton gas experiments by Suret et al. [2]} in which the Inverse Scattering Transform for the 1D Schrödinger equation is used to compute boundary conditions in a very long flume. In these experiments an ensemble of random spectral values associated to solitons are prescribed which then evolve towards a soliton gas.

[2]

https://openalex.org/W3039663288

fd5dcfd8-e130-4e30-a87f-20dca440ad79

The constraints on the experiments mentioned above, require to find other routes to the generation of integrable turbulence and soliton gases, an aspect investigated in the present work. In this context the question of the statistical properties of the stationary state of integrable turbulence needs to be addressed since it remains largely open and was mostly investigated by numerical studies [1]}, [2]}, [3]}, [4]}.

[2]

https://openalex.org/W2963740033

03774173-49de-4477-8de2-474000f218cc

The details of the experimental setup and data analysis tools can be found in [1]} and some aspects are discussed in [2]}, [3]} as well. We only recall here the main features of the experimental set-up.

[2]

https://openalex.org/W2942459581

08e0f71a-9c7e-48b6-a79e-872701bf55e0

Our goal is to obtain integrable turbulence steady states such as soliton gases. Thus, one would like to generate as many solitons as possible with the wavemaker. In order to ultimately sustain a large number of solitons in the flume we take advantage i) of the well-known fission phenomenon in shallow water of a sine wave train into solitons as evidenced numerically by Zabusky & Kruskal [1]} and observed experimentally by Zabusky & Galvin [2]} and in a more comprehensive way by Trillo et. al [3]} and ii) of the amplitude amplification of the non-linear modes by their interactions with the moving wavemaker. Mention should be made of the experiments by [4]} who, by slightly detuning the wavemaker motion with respect to the periodic longitudinal seiching mode of the channel, were able to find a route towards the generation of integrable turbulence.

[1]

https://openalex.org/W2050639649

2cfbe990-b2a0-4149-b846-4bf32fa9030d

ridge a: signature of weak dispersive shallow water waves following the Airy dispersion relation \(\omega ^2 = g \, k \, \tanh (k \, h).\) These dispersive waves originate from the bound waves of the wavemaker monochromatic sinusoidal wave forcing and from the weak non integrable effects during soliton collisions [1]}, [2]}; ridge b: signature of shallow water non-linear waves which Fourier modes all travel at a velocity close to \(c_0 = \sqrt{g \, h}\) . ridges c: signature of transverse waves. The uni-nodal transverse wave would be at a frequency of \(0.9\,\) Hz. The energy of the uni-nodal transverse waves is three orders of magnitude smaller than that of longitudinal waves.

[2]

https://openalex.org/W2942459581

bcc85d17-b805-4693-a0c2-c982066fd1c2

The case described in Fig. REF and REF and analyzed in [1]} is typically an example of integrable turbulence containing a significant number of solitons. Solitons are responsible for “ridge b” of Fig. REF and they also clearly leave straight line signatures in the \((x,t)\) plane of Fig. REF . This was discussed in detail in [2]}.

[1]

https://openalex.org/W2942459581

2030a331-09fc-4d62-8cde-a9bbe8ee5dc1

Gardner et al. [1]} made a significant leap forward with the Direct Scattering Transform (DST) of the KdV equation that extracts the spectrum of the associated Schrödinger equation which potential is the nonlinear wave signal to be analyzed. This method decomposes a time series into nonlinear solitonic modes that evolve independently in time and space without change of shape along with radiating shallow water weakly nonlinear waves. Once the time independent spectrum of nonlinear modes is determined, the signal can be reconstructed at any time by the Inverse Scattering Transform (IST). DST theory for the case of a localized initial condition [1]} differs from the much more complex case of the spatially periodic initial condition known as the finite band theory [3]}, [4]}, [5]}, [6]}. In our experimental case we are more concerned with the periodic case due to our configuration in which the waves propagate back and forth in the flume. Although the waves in the present experiments are not in most cases periodic, the wave motion is however confined in space and does not decay at infinity. This approach has been used by [7]} to analyze oceanic field data.

[1]

https://openalex.org/W2030669316

106232d0-808c-408d-bee3-46dc11e7b8a8

Gardner et al. [1]} made a significant leap forward with the Direct Scattering Transform (DST) of the KdV equation that extracts the spectrum of the associated Schrödinger equation which potential is the nonlinear wave signal to be analyzed. This method decomposes a time series into nonlinear solitonic modes that evolve independently in time and space without change of shape along with radiating shallow water weakly nonlinear waves. Once the time independent spectrum of nonlinear modes is determined, the signal can be reconstructed at any time by the Inverse Scattering Transform (IST). DST theory for the case of a localized initial condition [1]} differs from the much more complex case of the spatially periodic initial condition known as the finite band theory [3]}, [4]}, [5]}, [6]}. In our experimental case we are more concerned with the periodic case due to our configuration in which the waves propagate back and forth in the flume. Although the waves in the present experiments are not in most cases periodic, the wave motion is however confined in space and does not decay at infinity. This approach has been used by [7]} to analyze oceanic field data.

[3]

https://openalex.org/W1973107894

34421b73-5ceb-4610-9098-1baa7cfae64e

Gardner et al. [1]} made a significant leap forward with the Direct Scattering Transform (DST) of the KdV equation that extracts the spectrum of the associated Schrödinger equation which potential is the nonlinear wave signal to be analyzed. This method decomposes a time series into nonlinear solitonic modes that evolve independently in time and space without change of shape along with radiating shallow water weakly nonlinear waves. Once the time independent spectrum of nonlinear modes is determined, the signal can be reconstructed at any time by the Inverse Scattering Transform (IST). DST theory for the case of a localized initial condition [1]} differs from the much more complex case of the spatially periodic initial condition known as the finite band theory [3]}, [4]}, [5]}, [6]}. In our experimental case we are more concerned with the periodic case due to our configuration in which the waves propagate back and forth in the flume. Although the waves in the present experiments are not in most cases periodic, the wave motion is however confined in space and does not decay at infinity. This approach has been used by [7]} to analyze oceanic field data.

[4]

https://openalex.org/W2118575947

8f74a0fb-a471-4bdf-9804-1dbfca1d0935

Gardner et al. [1]} made a significant leap forward with the Direct Scattering Transform (DST) of the KdV equation that extracts the spectrum of the associated Schrödinger equation which potential is the nonlinear wave signal to be analyzed. This method decomposes a time series into nonlinear solitonic modes that evolve independently in time and space without change of shape along with radiating shallow water weakly nonlinear waves. Once the time independent spectrum of nonlinear modes is determined, the signal can be reconstructed at any time by the Inverse Scattering Transform (IST). DST theory for the case of a localized initial condition [1]} differs from the much more complex case of the spatially periodic initial condition known as the finite band theory [3]}, [4]}, [5]}, [6]}. In our experimental case we are more concerned with the periodic case due to our configuration in which the waves propagate back and forth in the flume. Although the waves in the present experiments are not in most cases periodic, the wave motion is however confined in space and does not decay at infinity. This approach has been used by [7]} to analyze oceanic field data.

[5]

https://openalex.org/W1661143721

d1990f89-ad4c-490c-af55-d7d6f20f4336

Gardner et al. [1]} made a significant leap forward with the Direct Scattering Transform (DST) of the KdV equation that extracts the spectrum of the associated Schrödinger equation which potential is the nonlinear wave signal to be analyzed. This method decomposes a time series into nonlinear solitonic modes that evolve independently in time and space without change of shape along with radiating shallow water weakly nonlinear waves. Once the time independent spectrum of nonlinear modes is determined, the signal can be reconstructed at any time by the Inverse Scattering Transform (IST). DST theory for the case of a localized initial condition [1]} differs from the much more complex case of the spatially periodic initial condition known as the finite band theory [3]}, [4]}, [5]}, [6]}. In our experimental case we are more concerned with the periodic case due to our configuration in which the waves propagate back and forth in the flume. Although the waves in the present experiments are not in most cases periodic, the wave motion is however confined in space and does not decay at infinity. This approach has been used by [7]} to analyze oceanic field data.

[7]

https://openalex.org/W1984085074

e204fb89-a3c4-470a-afce-57fe8f5cdc11

We implemented the Periodic Scattering Transform algorithm (PST) for the KdV equation developed by Osborne [1]}, [2]}, [3]} in a program to compute the nonlinear spectrum of the experimental free surface records. In the periodic case, the non linear modes are described by hyper-elliptic functions. These waves similar to cnoidal waves are characterized by the so-called spectral modulus \(m\) which quantifies the level of nonlinearity and which is an output of the PST [2]}. For a vanishing small modulus the modes are close to sine waves. For \(m\) very close to 1, the modes appear as localized pulses in the periodic box that resemble solitons. The delineation between solitonic modes and radiation modes thus breaks down to the choice of the threshold modulus. This threshold modulus also defines the reference depth \(h_{ref}\) [1]}, [2]}, [3]} on which these solitonic modes propagate. Osborne [1]} suggests that modes with \(m>0.99\) can be considered as solitons a definition also used by [9]}. We will not discuss here the details of our implementation of the PST which is described in [10]} with different validation cases and an analysis of various limitations since this method has been validated various times [11]}, [9]}. The main output of the PST is the spectrum that lists the nonlinear modes and their moduli.

[1]

https://openalex.org/W366502457

cff6340d-026f-4599-a0b8-7b601a118cd4

One of the relevant non-dimensional number in the shallow water context is the Ursell number [1]}. It is even the only dimensionless number for KdV unidirectional wave motion dynamics. The Ursell number is the ratio in order of magnitude of the nonlinear to dispersive terms of the KdV equation. The dimensional version of the KdV equation is, \(\partial _t \eta + c_0 \, \partial _x \eta + \frac{3 \, c_0}{2 \, h} \, \eta \, \partial _x \eta + \frac{h^2 \, c_0}{6} \, \partial _{xxx} \eta = 0. \)

[1]

https://openalex.org/W2030188624

ad6415bc-34cb-44b5-bb59-58cb824dc748

Equation (REF ) shows that for very small \(U\) the equation becomes linear and dispersive, the so-called Airy equation. Under this condition the forcing wave remains linear but disperses (different frequency components propagate at different speeds) as it propagates and no soliton emerges. For large values of \(U\) equation (REF ) becomes non-linear of the Burgers type. A sinusoidal forcing wave that fulfills such condition will undergo nonlinear steepening up to the gradient catastrophe producing a steep front face (shock wave). At that point the wave front face characteristic length is small and dispersion comes into play. Dispersion forces the shock wave to fission into a train of solitons [1]}, [2]}, [3]}. The Ursell number of the forcing wave therefore indicates how many solitons are expected [3]} . Since our set-up does not allow for initial condition recurrence to take place because of the end wall reflection, the Ursell number also measures how disorganized the regime is. Indeed, solitons will reflect back on the end wall and the wavemaker, interacting with others, generating phase shifts and therefore possibly disorganizing the initial periodicity. <FIGURE>

[1]

https://openalex.org/W2050639649

d438e96c-172f-4aad-a87a-3ba40cfc26bf

However, we observed that the sole value of the forcing wave Ursell number \(U\) is insufficient to discriminate between the periodic and random states. Fig. REF shows the frequency spectra of 9 experiments for the left-running component of the wave motion at \(x=40\,\) m which is the best compromise between soliton separation after fission and dissipation. All these experiments have very close Ursell number values of \(U = 0.53\) , but distinct values of the frequency \(f\) and amplitude \(a\) . Signatures of different stationary states are observed from very periodic ones at low forcing frequency and low forcing amplitude to random ones at high forcing frequency and high forcing amplitude and going through continuously varying power spectra shape in between. This observation shows that another dimensionless parameter must be taken into account to sort the different states out. Fig. REF shows the corresponding distribution of non-linear mode amplitudes given by the PST. The periodic cases (3 top subplots of Fig. REF and correspondingly in Fig. REF ) exhibit sets of non-linear modes of nearly constant amplitudes indicating that these cases remain strongly organized. For instance the case \(f=0.18\,\) Hz for a 60 s wave motion recording corresponds to roughly 10 periods of the forcing wave for which the PST gives 10 soliton modes of \(0.15\,\) cm amplitudes and hardly any other modes. A small soliton of \(0.15\,\) cm amplitude is locked to each wave period. By contrast the soliton modes in the random cases are much more numerous and their amplitudes distributed over a large range indicating indeed that these cases are random. These states (3 bottom subplots of Fig. REF and correspondingly in Figure REF ) are considered to be what is called integrable turbulence [1]}. Discussion of the amplitude distribution is post-poned to section . <FIGURE>

[1]

https://openalex.org/W2942459581

086fee98-0e9b-45cb-95ad-29acd8071272

We conclude from these observations that another way to synthetically discriminate the various regimes, within our experimental framework, is to compute the frequency spectrum as given in Fig. REF . In the periodic case, the spectrum is mostly made of narrow peaks with the fundamental peak at the forcing frequency and the other peaks corresponding to higher harmonics. In the random case the spectrum is quite distinct and it is seen to be continuous with a flat plateau [1]}, [2]} that extends from the forcing frequency down to the smallest resolved frequency. At frequencies higher than the forcing frequency the spectrum decays exponentially. At this point, a question arises on how to quantify the state of the wave field. At this end, we define a dimensionless randomness index \(I_m\) based on the shape of the power spectrum: \(I_m=\log _{10}\left(\frac{E_{forcing}}{E_{av}}\right) \)

[2]

https://openalex.org/W2942459581

ed66017e-8e15-4a5a-9ecf-dce2ac0b5326

In this section statistical distributions and moments of the experimental integrable turbulence are presented. Statistical information complements PST analysis. PST in our present study is a key tool to assess the existence of a soliton gas and characterize the modal content of the gas. Nonetheless integrable turbulence encompasses also shallow water situations where solitonic modes coexist with radiation modes. As discussed in the introduction the literature on the statistical description of integrable turbulence is sparse while such situation can be present in field measurements [1]}. A noticeable exception is the numerical study by [2]} on KdV random wave fields, a reference study on these aspects.

[1]

https://openalex.org/W1984085074

1c887344-279b-4c32-b11e-98c8f4790da1

In a periodic box and with an initial condition made of a sine wave, Zabuski & Kruskal [1]} numerically predict a recurrence, i.e. the fact that the wave system retrieves in a finite time a state very close to the initial condition [1]}. In our experimental setup, it is not possible to start from an arbitrary initial condition which is not rest. What can only be achieved in a controlled way, is to start wave forcing at one end with a wavemaker and a body of water at rest in the entire flume.

[1]

https://openalex.org/W2050639649

19758ae5-9c66-4b83-a41d-ea1755f0f377

Our experimental set-up differs from the integrable framework of Zabusky & Kruskal [1]} on various points. The most obvious is dissipation that imposes some continuous energy flux input for the wave motion to possibly reach a statistically stationary wave regime. However, the time scales of dissipation, well represented by the Keulegan law, are much larger than those involving soliton interactions which suggest that integrability still holds locally. Indeed, we show that once a soliton gas is formed dissipation slowly alters the amplitude distribution but not to the point where it would be obliterated in a flume length propagation time.

[1]

https://openalex.org/W2050639649

aca6b0a2-0ae3-48c4-8e94-2f3fcbe40f60

where \(\nu _0=310\)  MHz. This is substantially larger than what can be explained by the observed counts of radio-emitting galaxies [1]}. This excess is consistent across several independent measurements, and its origin remains unknown [2]}, [3]}, [4]}. In what follows we assume that this is indeed a cosmic radio background that is of extragalactic origin, consistent with the non-detection of bright radio halos around galaxies similar to ours [5]}. Its existence at high redshift would have important implications for searches for redshifted 21cm emission from the epoch of the first stars [6]}. Assuming that this radio background is truly cosmological and isotropic, this same background should be present in galaxy clusters.

[4]

https://openalex.org/W3111523546

0f5270f0-9a3a-4d82-a7ee-2897bf77509c

The predicted radio SZ has a steep scaling towards low frequencies (see Fig. REF ). As such, it could become an important foreground for standard 21cm fluctuation measurements. The signal is expected to spatially correlate with the standard Compton-\(y\) map and therefore can be modelled by combining SZ and 21cm measurements. This is of particular interest to radio experiments such as LOFAR [1]} and the Square Kilometer Array [2]}, and CMB experiments like the Simons Observatory [3]} and CMB-S4 [4]}, which all promise high resolution wide-area maps of the sky.

[3]

https://openalex.org/W2888259247

e1bebd61-f036-43cd-89ba-e5430d278e1e

The predicted radio SZ has a steep scaling towards low frequencies (see Fig. REF ). As such, it could become an important foreground for standard 21cm fluctuation measurements. The signal is expected to spatially correlate with the standard Compton-\(y\) map and therefore can be modelled by combining SZ and 21cm measurements. This is of particular interest to radio experiments such as LOFAR [1]} and the Square Kilometer Array [2]}, and CMB experiments like the Simons Observatory [3]} and CMB-S4 [4]}, which all promise high resolution wide-area maps of the sky.

[4]

https://openalex.org/W2958360979

bc06101c-2bd6-49c5-9623-3d88d01e479c

Researchers have investigated the hierarchical RL problem under various settings. Existing theoretical analyses [1]}, [2]}, [3]}, [4]} typically assume that the options are given. As a result, only the high-level policy needs to be learned. Recent advances in deep hierarchical RL (e.g., [5]}) focus on concurrently learning the full options framework, but still the initialization of the options is critical. A promising practical approach is to learn an initial hierarchical policy from expert demonstrations. Then, deep hierarchical RL algorithms can be applied for policy improvement. The former step is named as Hierarchical Imitation Learning (HIL).

[1]

https://openalex.org/W2133458291

568e6b45-a590-4ebe-a9e0-5d8f49a852fd

Researchers have investigated the hierarchical RL problem under various settings. Existing theoretical analyses [1]}, [2]}, [3]}, [4]} typically assume that the options are given. As a result, only the high-level policy needs to be learned. Recent advances in deep hierarchical RL (e.g., [5]}) focus on concurrently learning the full options framework, but still the initialization of the options is critical. A promising practical approach is to learn an initial hierarchical policy from expert demonstrations. Then, deep hierarchical RL algorithms can be applied for policy improvement. The former step is named as Hierarchical Imitation Learning (HIL).

[5]

https://openalex.org/W2964227312

12576e89-c067-4f1d-8010-74e940003f3b

In this paper, we investigate HIL from a theoretical perspective. Our problem formulation is concise while retaining the essential difficulty of HIL: we need to learn a complete hierarchical policy from an unsegmented sequence of state-action pairs. Under this setting, HIL becomes an inference problem in a latent variable model. Such a transformation was first proposed by [1]}, where the Expectation-Maximization (EM) algorithm [2]} was applied for policy learning. Empirical results for this algorithm and its gradient variants [3]}, [4]} demonstrate good performance, but the theoretical analysis remains open. By bridging this gap, we aim to solidify the foundation of HIL and provide some high level guidance for its practice.

[1]

https://openalex.org/W2498991332

acea0ad9-2e8f-42cf-8179-92601bcdd74a

In this paper, we investigate HIL from a theoretical perspective. Our problem formulation is concise while retaining the essential difficulty of HIL: we need to learn a complete hierarchical policy from an unsegmented sequence of state-action pairs. Under this setting, HIL becomes an inference problem in a latent variable model. Such a transformation was first proposed by [1]}, where the Expectation-Maximization (EM) algorithm [2]} was applied for policy learning. Empirical results for this algorithm and its gradient variants [3]}, [4]} demonstrate good performance, but the theoretical analysis remains open. By bridging this gap, we aim to solidify the foundation of HIL and provide some high level guidance for its practice.

[3]

https://openalex.org/W2605016475

ad2951cc-3942-41c1-839c-ab2e96bfe99f

Due to its intrinsic difficulty, existing works on HIL typically consider its easier variants for practicality. If the expert options are observed, standard imitation learning algorithms can be applied to learn the high and low level policies separately [1]}. If those are not available, a popular idea [2]}, [3]}, [4]}, [5]} is to first divide the expert demonstration into segments using domain knowledge or heuristics, learn the individual option corresponding to each segment, and finally learn the high level policy. With additional supervision, these steps can be unified [6]}. In this regard, the EM approach [7]}, [8]}, [9]} is this particular idea pushed to an extreme: without any other forms of supervision, we simultaneously segment the demonstration and learn from it, by exploiting the latent variable structure.

[7]

https://openalex.org/W2498991332

ce670411-341d-4cff-b992-657e6efbe938

Due to its intrinsic difficulty, existing works on HIL typically consider its easier variants for practicality. If the expert options are observed, standard imitation learning algorithms can be applied to learn the high and low level policies separately [1]}. If those are not available, a popular idea [2]}, [3]}, [4]}, [5]} is to first divide the expert demonstration into segments using domain knowledge or heuristics, learn the individual option corresponding to each segment, and finally learn the high level policy. With additional supervision, these steps can be unified [6]}. In this regard, the EM approach [7]}, [8]}, [9]} is this particular idea pushed to an extreme: without any other forms of supervision, we simultaneously segment the demonstration and learn from it, by exploiting the latent variable structure.

[8]

https://openalex.org/W2605016475

b67af927-f3dc-4673-bd19-de48b385aaac

Recent ideas on EM algorithms [1]}, [2]}, [3]}, [4]} focus on the convergence to the true parameter directly, relying on an instrumental object named as the population EM algorithm. It has the same two-stage iterative procedure as the standard EM algorithm, but its \(Q\) -function, the maximization objective in the M-step, is defined as the infinite sample limit of the finite sample \(Q\) -function. Under regularity conditions, the population EM algorithm converges to the true parameter. The standard EM algorithm is then analyzed as its perturbed version, converging with high probability to a norm ball around the true parameter. The main advantage of this approach is that the true parameter usually has a large basin of attraction in the population EM algorithm. Therefore, the requirement on initialization is less stringent. See [3]} for an illustration.

[1]

https://openalex.org/W2962737134

41fccfa2-0804-4bb4-b4f6-81c61f988db3

The \(Q\) -function adopted in the population EM algorithm is named as the population \(Q\) -function. To properly define such a quantity, the stochastic convergence of the finite sample \(Q\) -function needs to be constructed. When the samples are i.i.d., such as in Gaussian Mixture Models (GMMs) [1]}, [2]}, [3]}, the required convergence follows directly from the law of large numbers. However, this argument is less straightforward in time-series models such as Hidden Markov Models (HMMs) and the model considered in HIL. For HMMs, [4]} showed that the expectation of the \(Q\) -function converges, but both the stochastic convergence analysis and the analytical expression of the population \(Q\) -function are not provided. The missing techniques could be borrowed from a body of work [5]}, [6]}, [7]}, [8]} analyzing the asymptotic behavior of HMMs. Most notably, [7]} provided a rigorous treatment of the population EM algorithm via sufficient statistics, assuming the HMM is parameterized by an exponential family.

[1]

https://openalex.org/W2962737134

95943076-33c3-4a11-8c72-58f096324f9d

The \(Q\) -function adopted in the population EM algorithm is named as the population \(Q\) -function. To properly define such a quantity, the stochastic convergence of the finite sample \(Q\) -function needs to be constructed. When the samples are i.i.d., such as in Gaussian Mixture Models (GMMs) [1]}, [2]}, [3]}, the required convergence follows directly from the law of large numbers. However, this argument is less straightforward in time-series models such as Hidden Markov Models (HMMs) and the model considered in HIL. For HMMs, [4]} showed that the expectation of the \(Q\) -function converges, but both the stochastic convergence analysis and the analytical expression of the population \(Q\) -function are not provided. The missing techniques could be borrowed from a body of work [5]}, [6]}, [7]}, [8]} analyzing the asymptotic behavior of HMMs. Most notably, [7]} provided a rigorous treatment of the population EM algorithm via sufficient statistics, assuming the HMM is parameterized by an exponential family.

[2]

https://openalex.org/W2963626260

d0cd4757-56a8-426c-a733-1e0db2d7e288

The \(Q\) -function adopted in the population EM algorithm is named as the population \(Q\) -function. To properly define such a quantity, the stochastic convergence of the finite sample \(Q\) -function needs to be constructed. When the samples are i.i.d., such as in Gaussian Mixture Models (GMMs) [1]}, [2]}, [3]}, the required convergence follows directly from the law of large numbers. However, this argument is less straightforward in time-series models such as Hidden Markov Models (HMMs) and the model considered in HIL. For HMMs, [4]} showed that the expectation of the \(Q\) -function converges, but both the stochastic convergence analysis and the analytical expression of the population \(Q\) -function are not provided. The missing techniques could be borrowed from a body of work [5]}, [6]}, [7]}, [8]} analyzing the asymptotic behavior of HMMs. Most notably, [7]} provided a rigorous treatment of the population EM algorithm via sufficient statistics, assuming the HMM is parameterized by an exponential family.

[5]

https://openalex.org/W4250389103

c2d7905c-2371-4ee1-8cd6-1ebf8f9ca1ee

The \(Q\) -function adopted in the population EM algorithm is named as the population \(Q\) -function. To properly define such a quantity, the stochastic convergence of the finite sample \(Q\) -function needs to be constructed. When the samples are i.i.d., such as in Gaussian Mixture Models (GMMs) [1]}, [2]}, [3]}, the required convergence follows directly from the law of large numbers. However, this argument is less straightforward in time-series models such as Hidden Markov Models (HMMs) and the model considered in HIL. For HMMs, [4]} showed that the expectation of the \(Q\) -function converges, but both the stochastic convergence analysis and the analytical expression of the population \(Q\) -function are not provided. The missing techniques could be borrowed from a body of work [5]}, [6]}, [7]}, [8]} analyzing the asymptotic behavior of HMMs. Most notably, [7]} provided a rigorous treatment of the population EM algorithm via sufficient statistics, assuming the HMM is parameterized by an exponential family.

[6]

https://openalex.org/W2964126880

a0d1acd7-4f7f-477c-9a61-d395d5f5d69f

Finally, apart from the EM algorithm, a separate line of research [1]}, [2]} applies spectral methods for tractable inference in latent variable models. However, such methods are mainly complementary to the EM algorithm since better performance can usually be obtained by initializing the EM algorithm with the solution of the spectral methods [3]}.

[1]

https://openalex.org/W2105724942

bff40df6-c965-498a-bec7-76d16b9a3a90

In this paper, we establish the first known performance guarantee for a HIL algorithm that only observes primitive state-action pairs. Specifically, we first fix and reformulate the original EM approach by [1]} in a rigorous manner. The lack of mixing is identified as a technical difficulty in learning the standard options framework, and a novel options with failure framework is proposed to circumvent this issue.

[1]

https://openalex.org/W2498991332

766c3f1b-35a0-4a09-9f3d-bccebc340fa7

Inspired by [1]} and [2]}, the population version of our algorithm is analyzed as an intermediate step. We prove that if the expert policy can be parameterized by the options with failure framework, then, under regularity conditions, the population version algorithm converges to the true parameter, and the finite sample version converges with high probability to a norm ball around the true parameter. Our analysis directly constructs the stochastic convergence of the finite sample \(Q\) -function, and an analytical expression of the resulting population \(Q\) -function is provided. Finally, we qualitatively validate our theoretical results using a numerical example.

[1]

https://openalex.org/W2962737134

d28e065f-d8f2-4090-a425-0aa04e1d51b5

The options with failure framework is adopted to simplify the construction of the mixing condition (Lemma REF ). It is possible that our analysis could be extended to learn the standard options framework. In that case, instead of constructing the usual one step mixing condition, one could target the multi-step mixing condition similar to [1]}.

[1]

https://openalex.org/W4250389103

c6efe7a9-b412-49de-8928-732778119dd0

Two comments need to made here. First, it is common in practice to observe not one, but a set of independent observation sequences. In that case, the problem essentially becomes easier. Second, the cardinality of the option space and the parameterization of the expert policy are usually unknown. A popular solution is to assume an expressive parameterization (e.g., a neural network) in the algorithm and select \(\textrm {card}()\) through cross-validation. Theoretical analysis of EM under this setting is challenging, even when samples are i.i.d. [1]}, [2]}. Therefore, we only consider the domain of correct-specification.

[1]

https://openalex.org/W3113313164

7268d357-99f7-4cc9-a0bb-82a49d832592

Adopting the EM approach, we present Algorithm  for the estimation of \(\theta ^*\) . It reformulates the algorithm by [1]} in a rigorous manner, and an error in the latter is fixed: when defining the posterior distribution of latent variables, at any time \(t<T\) , the original algorithm neglects the dependency of future states \(S_{t+1:T}\) on the current option \(O_t\) . A detailed discussion is provided in Appendix REF .

[1]

https://openalex.org/W2498991332

31d90fbb-99e6-4f06-a02a-f20bc8fcd81b

Although we require finite state and action space for our theoretical analysis, Algorithm  can be readily generalized to continuous \(\) and \(\) : we only need to replace \(\pi _{lo}\) by a density function. However, generalization to continuous option space requires a substantially different algorithm. The forward-backward smoothing procedure in Theorem REF involves integrals rather than sums, and Sequential Monte Carlo (SMC) techniques need to be applied. Fortunately, it is widely accepted that a finite option space is reasonable in the options framework, since the options need to be distinct and separate [1]}.

[1]

https://openalex.org/W2111967991

c41f11b1-2e34-4c9f-9053-a765c7713194

Our analysis of Algorithm  has the following structure. We first prove the stochastic convergence of the \(Q\) -function \(Q_{\mu ,T}(\theta ^{\prime }|\theta )\) to a population \(Q\) -function \(\bar{Q}(\theta ^{\prime }|\theta )\) , leading to a well-posed definition of the population version algorithm. This step is our major theoretical contribution. With additional assumptions, the first-order stability condition is constructed, and techniques in [1]} can be applied to show the convergence of the population version algorithm. The remaining step is to analyze Algorithm  as a perturbed form of its population version, which requires a high probability bound on the distance between their parameter updates. We can establish the strong consistency of the parameter update of Algorithm  as an estimator of the parameter update of the population version algorithm. Therefore, the existence of such a high probability bound can be proved for large enough \(T\) . However, the analytical expression of this bound requires knowledge of the specific parameterization of \(\lbrace \bar{\pi }_{hi},\pi _{lo}, \pi _b\rbrace \) , which is not available in this general context of discussion.

[1]

https://openalex.org/W2962737134

d0bce762-5b2a-4745-aeac-de75d7eb2cf0

[Proof Sketch] The main difficulty of the proof is that, \(Q_{\mu ,T}(\theta ^{\prime }|\theta )\) defined in (REF ) is (roughly) the average of \(T\) terms, with each term dependent on the entire observation sequence; as \(T\rightarrow \infty \) , all the terms keep changing such that the law of large numbers cannot be applied directly. As a solution, we approximate \(\gamma ^{\theta }_{\mu ,t|T}\) and \(\tilde{\gamma }^{\theta }_{\mu ,t|T}\) with smoothing distributions in an infinitely extended graphical model independent of \(T\) , resulting in an approximated \(Q\) -function (still depends on \(T\) ). The techniques adopted in this step are analogous to Markovian decomposition and uniform forgetting in the HMM literature [1]}, [2]}. The limiting behavior of the approximated \(Q\) -function is the same as that of \(Q_{\mu ,T}(\theta ^{\prime }|\theta )\) , since their difference vanishes as \(T\rightarrow \infty \) . For the approximated \(Q\) -function, we can apply the ergodic theorem since the smoothing distributions no longer depend on \(T\) .

[1]

https://openalex.org/W4250389103

5efa87c9-1ba8-410c-969d-fe3260382ba7

In Assumption , we require the strong concavity of \(\bar{Q}(\cdot |\theta ^*)\) over the entire parameter space since the maximization step in our algorithm is global. Such a requirement could be avoided: if the maximization step is replaced by a gradient update (Gradient EM), then \(\bar{Q}(\cdot |\theta ^*)\) only needs to be strongly concave in a small region around \(\theta ^*\) . The price to pay is to assume knowledge on structural constants of \(\bar{Q}(\cdot |\theta ^*)\) (Lipschitz constant and strong concavity constant). See [1]} for an analysis of the gradient EM algorithm.

[1]

https://openalex.org/W2962737134

0ed42d85-3e80-4280-bd2d-dc7e3aeb5662

The proof is given in Appendix REF , where we also show an upper bound on \(\gamma \) . The idea mirrors that of [1]} with problem-specific modifications. Algorithm  can be regarded as a perturbed form of this population version algorithm, with convergence characterized in the following theorem.

[1]

https://openalex.org/W2962737134

34d4b257-3c00-4449-843d-583404f1a87e

The proof is provided in Appendix REF . Essentially, we use Theorem  to show the uniform (in \(\theta \) and \(\mu \) ) strong consistency of \(M_{\mu ,T}(\theta ;\omega )\) as an estimator of \(\bar{M}(\theta )\) , following the standard analysis of M-estimators. A direct corollary of this argument is the high probability bound on the difference between \(M_{\mu ,T}(\theta ;\omega )\) and \(\bar{M}(\theta )\) , as shown in the first part of the theorem. Combining this high probability bound with Theorem REF and [1]} yields the final performance guarantee.

[1]

https://openalex.org/W2962737134

0419cf57-e1d1-4860-8111-19a2195c2fea

Appendix  presents discussions that motivate Assumption REF . In particular, we show that Assumption REF approximately holds in a particular class of environment. Appendix  provides details on Algorithm , including the comparison with the existing algorithm from [1]}, the forward-backward implementation and the derivation of the \(Q\) -function from (REF ). In Appendix , we prove our theoretical results from Section . Technical lemmas involved in the proofs are deferred to Appendix . Finally, Appendix  presents details of our numerical example omitted from Section .

[1]

https://openalex.org/W2498991332

cfa1a178-4b46-4915-8604-f928e5d9603c

A Markov chain convergence result is restated in the following lemma, tailored to our need. [[1]}, Theorem 4.3.16 restated] With the Doeblin-type condition in (REF ), the Markov chain \(\lbrace X_t;\theta \rbrace _{t=1}^\infty \) with any \(\theta \in \) has a unique stationary distribution \(\nu _\theta \) . Moreover, for all \(\theta \in \) , \(x\in \mathcal {X}\) and \(t\in _+\) , \({Q^t_\theta (x,\cdot )-\nu _\theta }_\le (1-\delta )^{{t/2}}.\)

[1]

https://openalex.org/W4250389103

b2cb1693-c508-462b-ba4c-333270d53411

First, we point out a technicality when comparing Algorithm  to the algorithm from [1]}. The algorithm from [1]} learns a hierarchical policy following the standard options framework, not the options with failure framework considered in Algorithm . To draw direct comparison, we need to let \(\zeta =0\) in Algorithm . However, an error in the existing algorithm can be demonstrated without referring to \(\zeta \) .

[1]

https://openalex.org/W2498991332

053074b3-012a-4fec-a76e-2cad0da795e6

For simplicity, consider \(O_0\) fixed to \(o_0\in \) ; let \(2\le t\le T-1\) . Then, according to the definitions in [1]}, the (unnormalized) forward message is defined as \(\alpha ^\theta _{t}(o_t,b_t)=¶_{\theta ,o_0,s_1}(A_{1:t}=a_{1:t},O_t=o_t,B_t=b_t|S_{2:t}=s_{2:t}).\)

[1]

https://openalex.org/W2498991332

e4a192c1-4adb-4cd9-99d1-172a3bd9d003

We use the proportional symbol \(\propto \) to represent normalizing constants independent of \(o_t\) and \(b_t\) . [1]} claims that, for any \(o_t\) and \(b_t\) , \(\gamma ^{\theta }_{t|T}(o_t,b_t)\propto \alpha ^\theta _{t}(o_t,b_t)\beta ^\theta _{t|T}(o_t,b_t).\)

[1]

https://openalex.org/W2498991332

7c555f24-b294-49a8-8c91-e462500f4984

For the claim in [1]} to be true, \(\mathbb {P}_{\theta ,o_0,s_1}(S_{t+1:T}=s_{t+1:T}|S_{t}=s_{t},A_{t}=a_{t},O_t=o_t,B_t=b_t)\) should not depend on \(o_t\) and \(b_t\) . Clearly this requirement does not hold in most cases, since the likelihood of the future observation sequence should depend on the currently applied option.

[1]

https://openalex.org/W2498991332

739a4549-e9eb-4bf8-97ef-b82f6a645adb

In our algorithm, as motivated by Section , we effectively consider the following joint distribution on the graphical model shown in Figure REF : the prior distribution of \((O_0,S_1)\) is \(\hat{\nu }\) , and the distribution of the rest of the graphical model is determined by an options with failure policy with parameters \(\zeta \) and \(\theta \) . From the EM literature [1]}, [2]}, the complete likelihood function is \(L(s_{1:T},a_{1:T},o_{0:T},b_{1:T};\theta )=\hat{\nu }(o_0,s_1)\mathbb {P}_{\theta ,o_0,s_1}(S_{2:T}=s_{2:T},A_{1:T}=a_{1:T},O_{1:T}=o_{1:T},B_{1:T}=b_{1:T}).\)

[1]

https://openalex.org/W2962737134

8bfd937c-7571-4f26-81f4-556256598729

where \(const\) is a constant independent of \(T\) and \(\omega \) . The proof is provided in Appendix REF . Now we are ready to present the proof of Theorem REF step-by-step. The structure of this proof is similar to the standard analysis of HMM maximum likelihood estimators [1]}.

[1]

https://openalex.org/W4250389103

0b741f40-eaef-463f-bd67-44db8aee486e

2. We need to prove the uniform (in \(\theta ,\theta ^{\prime }\in \) and \(\mu \in \mathcal {M}\) ) almost sure convergence of the \(Q\) -function \(Q_{\mu ,T}(\theta ^{\prime }|\theta ;\omega )\) to the population \(Q\) -function \(\bar{Q}(\theta ^{\prime }|\theta )\) . The proof is separated into three steps. First, we show the almost sure convergence of \(Q^s_{\infty ,T}(\theta ^{\prime }|\theta ;\omega )\) to \(\bar{Q}(\theta ^{\prime }|\theta )\) for all \(\theta ,\theta ^{\prime }\in \) using the ergodic theorem. Second, we extend this pointwise convergence to uniform (in \(\theta ,\theta ^{\prime }\) ) convergence using a version of the Arzelà-Ascoli theorem [1]}. Finally, from Lemma REF , the difference between \(Q_{\mu ,T}(\theta ^{\prime }|\theta ;\omega )\) and \(Q^s_{\infty ,T}(\theta ^{\prime }|\theta ;\omega )\) vanishes uniformly in \(\mu \) as \(T\rightarrow \infty \) .

[1]

https://openalex.org/W3144951960

b6c60e82-beea-43a5-8f99-87359946f1a6

To extend the pointwise convergence in (REF ) to uniform (in \(\theta ,\theta ^{\prime }\) ) convergence, the following concept is required. The sequence \(\lbrace Q^s_{\infty ,T}(\theta ^{\prime }|\theta )\rbrace \) indexed by \(T\) as functions of \(\theta \) and \(\theta ^{\prime }\) is strongly stochastically equicontinuous [1]} if for any \(>0\) there exists \(\delta >0\) such that \(\limsup _{T\rightarrow \infty }\sup _{\theta _1,\theta ^{\prime }_1,\theta _2,\theta ^{\prime }_2\in ;{\theta _1-\theta _2}_2+{\theta ^{\prime }_1-\theta ^{\prime }_2}_2\le \delta }\left|Q^s_{\infty ,T}(\theta ^{\prime }_1|\theta _1;\omega )-Q^s_{\infty ,T}(\theta ^{\prime }_2|\theta _2;\omega )\right|<,~P_{\theta ^*,\nu ^*}\text{-a.s.}\)

[1]

https://openalex.org/W3144951960

31051bfa-05c3-49aa-a3d8-b4c0d4607677

[[1]}, Theorem 21.8 restated] Given (REF ) and (REF ), as \(T\rightarrow \infty \) we have \(\sup _{\theta ,\theta ^{\prime }\in }\left|Q^s_{\infty ,T}(\theta ^{\prime }|\theta ;\omega )-\bar{Q}(\theta ^{\prime }|\theta )\right|\rightarrow 0,~P_{\theta ^*,\nu ^*}\text{-a.s.}\)

[1]

https://openalex.org/W3144951960

acabf6b8-eb6b-44db-8d50-fd68ac791845

2. The proof of the second part mirrors the proof of [1]}. The main difference is the construction of the following self-consistency (a.k.a. fixed-point) condition. [Self-consistency] With all the assumptions, \(\theta ^*=\bar{M}(\theta ^*)\) .

[1]

https://openalex.org/W2962737134

7bec1fd7-b743-440b-b293-2830c99b0527

The proof of this lemma is presented in Appendix REF . Such a condition is used without proof in [1]} since it only considers i.i.d. samples, and the self-consistency condition for EM with i.i.d. samples is a well-known result. However, for the case of dependent samples like our graphical model, such a condition results from the stochastic convergence of the \(Q\) -function which is not immediate.

[1]

https://openalex.org/W2962737134