Amitoz Azad

Julien Rabin

Abderrahim Elmoataz

Variational models for signal processing with Graph Neural Networks

Keywords: Graph Processing, Neural Network, Total Variation, Variational Methods, Message Passing Network, Unsupervised learning

This paper is devoted to signal processing on point-clouds by means of neural networks. Nowadays, state-of-the-art in image processing and computer vision is mostly based on training deep convolutional neural networks on large datasets. While it is also the case for the processing of point-clouds with Graph Neural Networks (GNN), the focus has been largely given to high-level tasks such as classification and segmentation using supervised learning on labeled datasets such as ShapeNet. Yet, such datasets are scarce and time-consuming to build depending on the target application. In this work, we investigate the use of variational models for such GNN to process signals on graphs for unsupervised learning. Our contributions are two-fold. We first show that some existing variational -based algorithms for signals on graphs can be formulated as Message Passing Networks (MPN), a particular instance of GNN, making them computationally efficient in practice when compared to standard gradient-based machine learning algorithms. Secondly, we investigate the unsupervised learning of feed-forward GNN, either by direct optimization of an inverse problem or by model distillation from variational-based MPN.

Introduction

Variational methods have been a popular framework for signal processing since the last few decades. Although, with the advent of deep Convolutional Neural Networks (CNN), most computer vision tasks and image processing problems are nowadays addressed using state-of-the-art machine learning techniques. Recently, there has been a growing interest in hybrid approaches combining machine learning techniques with variational modeling have seen a (see e.g. [START_REF] Meinhardt | Learning proximal operators: Using denoising networks for regularizing inverse imaging problems[END_REF][START_REF] Kobler | Variational networks: connecting variational methods and deep learning[END_REF][START_REF] Bertocchi | Deep unfolding of a proximal interior point method for image restoration[END_REF][START_REF] Combettes | Deep neural network structures solving variational inequalities[END_REF][START_REF] Hasannasab | Parseval proximal neural networks[END_REF]).

Motivations. In this paper, we focus on such hybrid approaches for processing of point-clouds and signals on graphs. Graph representation provides a unified framework to work with regular and irregular data, such as images, text, sound, manifold, social network and arbitrary high dimensional data. Recently, many machine learning approaches have also been proposed to deal with high-level computer vision tasks, such as classification and segmentation [START_REF] Qi | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF][START_REF] Qi | Pointnet++: Deep hierarchical feature learning on point sets in a metric space[END_REF][START_REF] Wang | Dynamic graph cnn for learning on point clouds[END_REF], pointcloud generation [START_REF] Yang | Pointflow: 3d point cloud generation with continuous normalizing flows[END_REF], surface reconstruction [START_REF] Williams | Deep geometric prior for surface reconstruction[END_REF], point cloud registration [START_REF] Wang | Deep closest point: Learning representations for point cloud registration[END_REF] and up-sampling [START_REF] Li | Pu-gan: a point cloud upsampling adversarial network[END_REF] to name a few. These approaches are built upon Graph Neural Networks (GNN) which are inspired by powerful techniques employed in CNN, such as Multi-Linear Perceptron (MLP), pooling, convolution, etc. The main caveat is that transposing CNN architectures for graph-like inputs is not a simple task. Compared to images which are defined on a standard, regular grid that can be easily sampled, data with graph structures can have a large structural variability, which makes the design of a universal architecture difficult. This is likely the main reason why GNN are still hardly being used for signal processing on graph for which variational methods are still very popular. In this work, we investigate the use of data-driven machine learning techniques combined with variational models to solve inverse problems on point clouds.

Related work

Graph Neural Networks GNN are types of Neural Networks which directly operate on the graph structure. GNN are emerging as a strong tool to do representation learning on various types of data which, similarly to CNN, can hierarchically learn complex features.

PointNet [START_REF] Qi | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF] was one of the first few successful attempts to build GNN. Rather than preprocessing the input point-cloud by casting it into a generic structure, Qi et al. proposed to tackle the problem of structure irregularity by only considering permutation invariant operators that are applied to each vertex independently. Further works such as [START_REF] Qi | Pointnet++: Deep hierarchical feature learning on point sets in a metric space[END_REF] have built upon this framework to replicate other successful NN techniques, such as pooling and interpolation in auto-encoder, which enables to process neighborhood of points.

Most successful attempts to build Graph Convolutional Networks (GCN) were based upon graph spectral theory (see for instance [START_REF] Bruna | Spectral networks and locally connected networks on graphs[END_REF][START_REF] Defferrard | Convolutional neural networks on graphs with fast localized spectral filtering[END_REF][START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF]). In particular, Kipf and Welling [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] proposed a simple, linear propagation rule which can be seen as a first-order approximation of local spectral filters on graphs. More recently, Wang et al. introduced an edge convolution technique [START_REF] Wang | Dynamic graph cnn for learning on point clouds[END_REF] which maintains the permutation invariance property. Interestingly, [START_REF] Wu | Simplifying graph convolutional networks[END_REF] showed that GCN may act as low pass filters on internal features.

Message Passing Networks At the core of the most the aforementioned GNNs, is Message Passing (MP). The term was first coined in [START_REF] Gilmer | Neural message passing for quantum chemistry[END_REF], in which authors proposed a common framework that reformulated many existing GNNs, such as [START_REF] Bruna | Spectral networks and locally connected networks on graphs[END_REF][START_REF] Qi | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF][START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF]. In the simplest terms, MP is a generalized convolution operator on irregular domains which typically expresses some neighborhood aggregation mechanism consisting of a message function, a permutation invariant function (e.g. max) and an update function.

Inverse problems on Graphs. PDEs and variational methods are two major frameworks to address inverse problems on a regular grid structure like images. These methods have also been extended to weighted graphs in non-local form by defining p-Laplace operators [START_REF] Elmoataz | On the p-laplacian and ∞-laplacian on graphs with applications in image and data processing[END_REF]. Typical but not limited applications of these operators are filtering, classification, segmentation, or inpainting [START_REF] Peyré | Non-local regularization of inverse problems[END_REF]. Specifically, we are interested in this work in non-local Total-Variation on a graph, which has been thoroughly studied, e.g. for non-local signal processing [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF][START_REF] Gilboa | Nonlocal operators with applications to image processing[END_REF][START_REF] Tabti | Symmetric upwind scheme for discrete weighted total variation[END_REF], multi-scale decomposition of signals [START_REF] Hidane | Nonlinear multilayered representation of graph-signals[END_REF], point-cloud segmentation [START_REF] Lozes | Nonlocal segmentation of point clouds with graphs[END_REF] and sparsification [START_REF] Tenbrinck | Variational graph methods for efficient point cloud sparsification[END_REF], using various optimization schemes (such as finite difference schemes [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF], primal-dual [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF], forward-backward [START_REF] Raguet | Preconditioning of a generalized forward-backward splitting and application to optimization on graphs[END_REF], or cut pursuit [START_REF] Raguet | Cut-pursuit algorithm for regularizing nonsmooth functionals with graph total variation[END_REF]).

Contributions and Outline

The paper is organised as follows. Section 2 focus on variational methods to solve inverse problems for graph signal processing. After a short overview of graph representation and non-local regularization, we demonstrate that two popular optimization algorithms can be interpreted as a specific instance of Message Passing Networks (MPNs). This allows using efficient GPU-based machine learning libraries to solve the inverse problem on graphs. Experiments on point cloud compression and color processing show that such MPN optimization is more efficient than standard machine learning optimization techniques. Section 3 is devoted to the use of a trainable GNN to perform signal processing on graphs in a feed-forward fashion. We investigate two techniques for unsupervised training, i.e. in absence of ground-truth data, using variational models as prior knowledge to drive the optimization. We show that either model distillation from a variational MPN or optimizing the inverse problem directly with GNN gives satisfactory approximate solutions and allows for faster processing.

Solving Inverse Problem with MPN optimization

Notations and definitions

A graph G = (V, E, ω) is composed of an ensemble of nodes (vertices) V, an ensemble of edges E = V × V and weights ω : E → R + , where R + indicates the set of non-negative values [0, ∞). Let f i ∈ R d represents the feature vector (signal) on the node i of G, w i,j the scalar weight on the edge from the node i to j, and j ∈ N (i) indicates a node j in the neighborhood of node i, such that ω i,j = 0. For sake of simplicity, we only consider non-directed graph, i.e. defined with symmetric weights ω i,j = ω j,i .

We refer to the weighted difference operator as

∇ ω : (∇ ω f ) i,j = √ ω i,j (f j -f i ) ∈ R d , ∀ (i, j) ∈ E. Assuming symmetry of ω, the adjoint of this linear operator is (∇ * ω g) i = j∈V √ ω i,j (g j,i -g i,j ) ∈ R d , ∀ i ∈ V.
We denote as . p the p norm and . 1,p the composed norm for parameter p ∈ R + ∪ {∞}:

∀ g ∈ R |V|×|V|×d , g 1,p = i∈V g i,. p = i∈V j∈V,1≤k≤d |g i,j,k | p 1/p .

Non-Local Regularization on Graph

In this work, we focus on inverse problems for signal processing on graph using the non-local regularization framework [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF]. The optimization problem is formulated generically using 2 fidelity term and a non-local regularization defined from the p norm, parametrized by weights ω and power coefficient q ∈ R + , and penalized by

λ ∈ R + inf f J(f ) := 1 2 f -f 0 2 + λ q R q p (f ) where R q p (f ) = ∇ ω f q 1,p . (1)
Here f 0 is the given signal which requires processing. In experiments, we will mainly focus on two cases: when p = q = 2, the smooth regularization term relates to the Tikhonov regularization and boils down to Laplacian diffusion on a graph [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF]. For q = 1, the regularization term is the Non-Local Total Variation (NL-TV), referred to as isotropic when p = 2 and anisotropic when p = 1. Other choice of norm might be useful: using p ≤ 1, as studied for instance for p = 0 in [START_REF] Tenbrinck | Variational graph methods for efficient point cloud sparsification[END_REF], results in sparsification of signals; using p = ∞ [START_REF] Tabti | Symmetric upwind scheme for discrete weighted total variation[END_REF] is also useful when considering unbiased symmetric schemes.

As the functional J is not smooth, an ε-approximation J ε is often considered to circumvent numerical problems (see e.g. [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF]); this is for instance achieved by substituting the term R q p with (respectively for the case p = 2 and p = 1)

R q 2,ε (f ) = i∈V ∇ ω f i,. 2 p + ε 2 q 2 and R q 1,ε (f ) = i,j∈V ω q 2 i,j (|f i -f j | + ε) q . (2)

Variational Optimization

We consider now two popular algorithms to solve inverse problem (1), described in Alg. 1 and 2 with update rules on f (t) , where (t) is the iteration number.

Algorithm 1: Gauss-Jacobi [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF] Initialization:

f (0) = f0, set ε > 0 compute γ (t) i,j using Eq.(3) f (t+1) i = (f0)i + λ j∈N i γ (t) i,j f (t) j g (t+1) i = 1 + λ j∈N i γ (t) i,j f (t+1) i = f (t+1) i /g (t+1) i
Algorithm 2: Primal-Dual [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] Parameters: τ, β > 0 and θ ∈ [0, 1]

Initialization: f (0) = f (0) = f0 g (t+1) = Prox βA * (g (t) + βK f (t) ) f (t+1) = ProxτB(f (t) -τ K * g (t+1) ) f (t+1) = f (t+1) + θ(f (t+1) -f (t) )
For any p, q > 0, one can turn to the Gauss-Jacobi (GS) iterated filter [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF] described in Alg. 1. It relies on ε approximation to derive the objective function, giving for instance the following update rule when using Eq. ( 2)

γ (t) i,j = w i,j ( (∇ w f ) i,. 2 + ε 2 ) q-2 2 + ( (∇ w f ) j,. 2 + ε 2 ) q-2 2 if p = 2 2w i,j q 2 (|f i -f j | + ε) q-2 if p = 1 . (3)
When considering the specific case of NL-TV (q = 1), one can turn to the primal-dual algorithm [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] described in Alg. 2, as done in [START_REF] Lozes | Nonlocal segmentation of point clouds with graphs[END_REF][START_REF] Hidane | Nonlinear multilayered representation of graph-signals[END_REF]. To do so, we recast the problem (1) as min f max g Kf, g -A * (g) + B(f ) where

K ≡ ∇ ω , A ≡ . 1,p , B ≡ 1 2λ . -f 0 2 2 .
Parameters must be chosen such that τ β||K|| 2 < 1 to ensure convergence. For numerical experiments, we have used θ = 1 and τ = β = (4 max i∈V j∈V ω i,j ) -1 . Note that other methods might be used, such as pre-conditioning [START_REF] Raguet | Preconditioning of a generalized forward-backward splitting and application to optimization on graphs[END_REF], and cut pursuit [START_REF] Raguet | Cut-pursuit algorithm for regularizing nonsmooth functionals with graph total variation[END_REF]. Proximal operators corresponds to

Prox τ B (f ) = λf +τ f0 λ+τ , and Prox βA * (g) = Proj B ∞,p (g) (4)
where the dual norm parameter p > 0 verifies 1/p + 1/p = 1. The projection on the unit ball B ∞,p for p = 2 and p = 1 is given respectively by,

∀ g ∈ R |V|×|V|×d Proj B∞,2 (g) i,j,k = g i,j,k max{1, g i,. 2 } , Proj B∞,∞ (g) i,j,k = g i,j,k max{1, |g i,j,k |} . (5)

Message Passing Network Optimization

Message Passing Networks [START_REF] Gilmer | Neural message passing for quantum chemistry[END_REF] can be formulated as follows

f (n+1) i = ψ (n) f (n) i , j∈N (i) φ (n) f (n) i , f (n) j , ω i,j ∀ i ∈ V, (6)
where (n) indicates the depth in the network. As already shown in [START_REF] Gilmer | Neural message passing for quantum chemistry[END_REF], many existing GNN can be recast in this generic framework. Typically, is the summation operator but can be any differentiable permutation invariant functions (such as max, mean); ψ (n) (update rule) and φ (n) (message rule) are the differentiable operators, such as MLPs (i.e. affine functions combined with simple non-linear functions such as RELU). Note that they are independent of the input vertex index to preserve permutation invariance. We now show that Alg. 1 and 2 can be also formulated within this framework.

G-J algorithm. This is quite straightforward for the anisotropic case (p = 1) using two MPNs to compute the update of f in Alg. 1. A first MPN is used for the numerator f , identifying ψ (n) (f

(n) i , F (n)
i ) = (f 0 ) i + λF (n) i , = and φ (n) (f (n) i , f (n) j , ω ij ) = γ (n) i,j f (n) j
where γ (n) i,j , as defined in Eq. ( 3), is a function of the triplet (f

(n) i , f (n) j , ω i,j ). A second MPN is used to compute the denominator g using ψ (n) (f (n) i , F (n) i ) = 1 + λF (n) i and φ (n) (f (n) i , f (n) j , ω ij ) = γ (n)
i,j . For the isotropic case p = 2, an additional MPN is ultimately required to compute the norm of the gradient (Kf (n) ) i at each vertex i, as detailed in the next paragraph. P-D algorithm. Two nested MPNs are now required to compute the update of the primal variable f and the dual variable g in Alg. 2. For f , the MPN reads as the affine operator ψ (n) (f

(n) i , F (n) i ) = τ τ +λ (f 0 ) i + τ τ +λ (f (n) i -τ F (n) i ).
Similarly, for the update of g one computes the projection:

ψ (n) (g (n) i , G (n) i ) = Proj B ∞,p (g (n) i + βG (n) i ).
The gradient operator K and its adjoint K * (see definition in §. 2.1) can be applied using the processing functions φ and φ j , respectively

G (n) i = (K f (n) ) i = φ (n) ( f (n) i , f (n) j , ω i,j ) = ω i,j ( f (n) j - f (n) i ) F (n) i = (K * g (n+1) )) i = φ (n) j (g (n+1) i , g (n+1) j
, ω i,j ) = j∈Ni ω i,j (g

(n+1) j,i -g (n+1) i,j )
Observe that φ j is a slightly modified message passing function using index j to incorporate the edge features from not only source i to target j (i.e. g i,j and ω i,j ) but also from target j to source i (g j,i and ω j,i ). These MPN formulations of Alg. 1 and Alg. 2 allows to use machine learning libraries to solve the inverse problem Eq. 1 and to take full advantage of fast GPU optimization. In the following experiments, we illustrate the advantage of such MPN optimization in comparison with standard gradient-based machine learning optimization techniques relying on auto-differentiation.

Experiments on point-clouds and comparison with auto-diff

Experimental setting. Edges on point-clouds are defined with weights ω i,j {0, 1} using the indicator function of a k-nearest neighbor search (k = 4) on points coordinates using 2 norm, then imposing symmetry by setting ω i,j ← ω i,j ∨ ω j,i . Two types of features f are tested for q = 1, as illustrated in Fig. 1: 3D point coordinates x for point-cloud simplification (Fig. 2), and colors c ∈ [0, 1] |V|×3 for denoising (Fig. 3). We also consider the non-convex case where q = 0.1 for point cloud sparsification in Fig. 2.

Auto-differentiation.

As a baseline, we compare these MPN-based algorithms with algorithmic auto-differentiation. In this setting, the update of features at iteration (t) is defined by gradient descent on the loss function J(f (t) ) defined in Eq. ( 1), e.g. f (t+1) = f (t) -ρ (t) D (t) . Since the NL-TV regularization term is not smooth, we use the ε approximation of Eq. ( 2) to compute the gradient D (t) . We consider here various standard gradient descent techniques used for NN training: Gradient Descent (gd), adam [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] and lbfgs [START_REF] Liu | On the limited memory bfgs method for large scale optimization[END_REF].

Implementation details. All algorithms are implemented using the Pytorch Geometric library [START_REF] Fey | Fast graph representation learning with pytorch geometric[END_REF] and tested with a Nvidia GPU GTX1080Ti with 11 GB of memory. Note that reported computation time is only indicative, as implementation greatly influence the efficiency of memory and GPU cores allocation. Different learning rates are used for each method in the experiments. The learning rate of adam algorithm is always set to lr = 0.001. gd is also used with lr = 0.001 for point-cloud simplification but with lr = 0.1 for color processing. lbfgs is set with lr = 0.01 for q = 1, and with lr = 0.1 for q = 0.1.

Results.

Observe that the noise on the raw color point cloud in Fig. 1 results from the registration of multiple scans with different lighting conditions, for which we do not have any ground-truth. In such a case, variational models like in (1) are useful to remove color artifacts without more specific prior knowledge.

Figures 2 and3 show the visual results of the aforementioned algorithms after 20k iterations. The objective function J(f (t) ) is plotted for each one in Fig. 1. ) depending on iteration (t), in logarithmic scale, for various setting of (p, q) and for different algorithms.

Since the computation time per iteration is roughly same for all methods (about 0.5 ms for mpn-pd and 1 ms for others on 'elephant' and 'Napoléon' pointclouds with |V| = 3k and 11k respectively), curves are here simply displayed versus the number of iterations. As one can observe from these experiments, MPN based algorithms perform better than gradient based algorithm when it comes to precision. Indeed, for q = 1, Alg. 2 is the only algorithm that can be used without approximation (i.e. ε = 0, which is numerically intractable for other methods) and that gives ultimately an optimal solution. As for q = 0.1, Alg. 1 reaches an (approximated) solution much faster than gradient based methods. Among these, it is interesting to notice that adam is quite consistent and performs quite well. For this reason, we will consider this method for the next part devoted to unsupervised training of GNN.

We have shown that the two algorithms can be implemented efficiently as MPN, a special instance of GNN. In the next section, we demonstrate the interest of training such MPNs to design feed-forward GNNs allowing fast processing.

Variational training of GNN

One of the most attractive aspect of machine learning is the ability to learn the model itself from the data. As already mentioned in the introduction, most approaches rely on supervised training techniques which require datasets with ground-truth, potentially obtained by means of data augmentation (i.e. artificially degrading the data), which might be a simple task for some applications such as Gaussian denoising on images. However, it is challenging and time consuming to achieve this for signal processing tasks on point-clouds. Besides, most of the available datasets are only devoted to high-level tasks like segmentation and classification on shapes. mpn-pd mpn-gj gd adam lbfgs Inspired from the previous results, we investigate the unsupervised training of a GNN to reproduce the behavior of variational models on point-clouds while reducing the computation time. To achieve such a goal, two different settings are considered in this section: Unsupervised training using the variational energy as a loss function ( § 3.1), and model distillation using exemplars from the variational model defined as an MPN ( § 3.2).

q = 1 p = 2 q = 1 p = 1 q = 0.1 p = 2

Unsupervised training

We denote by G θ : R |V|×d × R E + → R |V|×d a trainable GNN parametrized by θ, processing a graph defined by the vertex features f and the edge weights ω. The idea here is simply to train a GNN in such a way that it minimizes in expectation the objective function (1) of the variational model for graphs from a dataset inf θ E X∼P data J (G θ (X, ω(X))) .

(

) 7
where X is a random point-cloud (and its associated weights ω(X)) from the training dataset with probability distribution P data . Once again, in practice, we need to consider the ε-approximation J ε (2) to overcome numerical issues.

Model distillation with MPN

As seen above, the main caveat in variational training is that it requires smoothing if the objective function is not differentiable. to avoid this limitation, we consider another setting which consists of model distillation, where the goal is to train a gnn to reproduce an exact model defined by a mpn. Let F : R |V|×d × R E + → R |V|×d denotes a MPN designed to solve a variational model, such as mpn-pd in the previous section for the NL-TV model. Note that in practice, we have to restrict this network to n updates. Using for instance 2 norm, the distillation training boils down to the following optimization problem

inf θ E X∼P data G θ (X, ω(X)) -F (X, ω(X)) 2 (8)
Other choices of distance could be as well considered, such as 1 norm or optimal transport cost for instance [START_REF] Williams | Deep geometric prior for surface reconstruction[END_REF].

Experiments

Dataset setting. We use the same experimental setting as in the previous section, except that k nearest-neighbor parameters ω i,j ∈ {0, 1} are now weighted using features proximity: ω i,j ← ω i,j e -κ (f0)i-(f0)j 2 where κ = 10 4 . For point-cloud processing, we use the ShapeNet part dataset [START_REF] Yi | A scalable active framework for region annotation in 3d shape collections[END_REF]. Point-clouds are pre-processed by normalizing their features to 0 and

i , f j , √ w i,j (f j -f i )).
Note that, this is quite similar to edge-convolution operation introduced in [START_REF] Wang | Dynamic graph cnn for learning on point clouds[END_REF], in which the MLP layers act on features (f i , f j -f i , w i,j ). The operator is and there is no update rule (ψ).

The MLP layers have respectively 4.8k, 20.6k, and 20.6k parameters. These layers operate on 64 dimensional features, which are then fed to a linear layer to return 3-dimensional features (color or coordinates).

Training settings. Using the dataset and the architecture described above, two GNNs are trained using each of the techniques described in § 3.1 and § 3.2, with k = 4, p = 2, q = 1, λ = 0.05 and ε = 10 -8 . Training is performed using Adam algorithm with 60 epochs and a fixed learning rate of 0.01. Regarding the model distillation, rather than re-evaluating the MPN F in (8) for every sample X at each epoch, we precompute F (X, ω(X)) to save the computation time.

Results. Fig. 4 gives a visual comparison of results of trained GNNs with the exact mpn-pd optimization on test data, showing that both training methods give satisfactory results, close to mpn-pd model. Fig. 5 displays the objective loss function in [START_REF] Elmoataz | On the p-laplacian and ∞-laplacian on graphs with applications in image and data processing[END_REF] during training, on the training and the validation set. The average error relative to mpn-pd on the test set is similar for both methods (2.2% for distillation and 1.2% for unsupervised training). One can observe that model distillation is more stable on the validation set, but need extra computation time to pre-process the mpn-pd on the dataset. Last, but not least, the average computation time for mpn-pd on the test set is compared in Fig. 5 to the feedforward GNNs, showing several order of magnitude speed-up for the latter, which makes such approaches very interesting for practical use.

Fig. 1 .

 1 Fig. 1. (left) Raw data f0 used for experiments in Fig. 2 and Fig. 3. (right) Comparison of the evolution of the objective function J(f (t)) depending on iteration (t), in logarithmic scale, for various setting of (p, q) and for different algorithms.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Comparison of numerical results of MPN implementations of algorithms 1 (mpngj) and 2 (mpn-pd), with gd, adam and lbfgs after 20k iterations. Point-cloud simplification is defined by problem (1) with λ = 0.2 and various settings of p and q. For all methods except mpn-pd, the approximated formulation (2) with = 10 -8 is used. mpn-pd mpn-gj gd adam lbfgs

Fig. 4 .

 4 Fig. 4. Comparison of NL-TV regularization (1) of point-clouds X (for p = 2, q = 1, λ = 0.05) with the MPN algorithm mpn-pd and trained GNN, using model distillation (8) or the unsupervised setting with variational optimization (7). Input here is either unseen point-clouds during training, or from the test category, hold out during training.

Fig. 5 .

 5 Fig. 5. (left) Objective loss function (7) during GNN training versus epochs for the training and the validation sets. (right) Comparison of the average objective function J in logarithmic scale versus average computation time for GNN, adam and mpn-pd.

  1 using minmax normalization; 15 classes out of 16 are used for training and a hold out class for testing. For each training class, 20% of shapes are also hold-out for validation. In total, 16.5k point clouds are used during training, composed of an average of 2.5k nodes. GNN architecture. The GNN is based on MPNs. The architecture is shallow and composed of 3 MPN layers followed by a linear layer. Inspired from MPNs formulated in Section 2.4 from Alg .1 and Alg .2, we define the message function (φ) as a MLP acting on the tuple (f

Acknowledgments This work has been carried out with financial support from the French Research Agency through the SUMUM project (ANR-17-CE38-0004)

Features f 0 q = 1, p = 2 q = 1, p = 1 q = 0.1, p = 2


Hannafi Cyrine

Le Gall

Expert workshop 'non-take-up and coverage' of social benefits 11

I. Introduction

The non-take-up of social benefits has become an important issue that most public policy makers are trying to address. Since people who are entitled to receive social benefits does not automatically take them, the non-take-up questions the efficiency of a social policy and the access to social rights.

Ultimately, it interrogates the determinants of the demand of public policy that emerged to address poverty and exclusion.

France is particularly concerned by this issue taking into account the complexity of its social protection system and the multiplicity of the social benefits [START_REF] Palier | Ambiguous agreement, cumulative change: French social policy in the 1990s[END_REF]. This complexity is stemmed from the successive devices that were created in order to develop the social protection, which is the creation of multiple covers against the risk to loose employment, which appear in the guise of the different branch of Sécurité Sociale. At the end of the cycle of durable growing of wealth from as of the 1970's, the social policy targets the "new risks of poverty" that is the long-term unemployment and the reconfiguration of the solidarity, as an illustration the increasing part of single parent families in the total population. However, the complexity is also close to the development of new measures directed towards fragile population that were not the main targets such as the dependant people or the people under inabilities. The tangle of different social rights and their different eligibility thresholds are therefore the first obstacles to applying for aid, when a person in a fragile situation no longer knows who to turn to for care.

Therefore, measuring the non-take-up phenomenon and its causes is more than essential in order to improve the effectiveness and the relevance of poverty reduction policies with facilitating the access to the social benefits to all persons that are entitled to them. In particular, the "Revenu de solidarité active" (RSA) and the "Prime d'activité" (PA) are concerned by this phenomenon whereas they are targeted to reduce poverty.

The Revenu de solidarité active (RSA) is a differential allowance designed to supplement a family's initial resources in order to reach a guaranteed income threshold (or lump-sum "amount"). At the time of its application in June 2009, the RSA merged two devices, the single parent allowance (API) and the minimum guaranteed income (RMI), which make up the "base" for beneficiaries. The RSA innovates in two ways: on the one hand, by developing an "activity" component intended to provide an income supplement for the working poor; on the other hand, by strengthening the methods of social and professional support for people experiencing poverty. In 2010, the RSA was extended to young people from 18 to 25, with a condition of working hours.

In the architecture of social benefits, the RSA remains the "last safety net", after all other redistribution mechanisms have been exhausted, and in particular, after unemployment benefits have been withdrawn.

Its scale depends on the resources of the adults in a family: therefore, it is not an individual benefit. The RSA is differential because its amount depends on the difference between the "lump sum" and the sum of the household's resources (work, assets, and other social benefits).

The activity dimension of the RSA is fully in line with the notion of "activation" specific to schemes aiming to "make work pay", following the recommendations of the Lisbon Strategy of 2000 which aims at rebuilding the European social model [START_REF] Barbier | Tracing the fate of EU "social policy": Changes in political discourse from the "Lisbon Strategy" to "Europe 2020[END_REF][START_REF] Eydoux | Du RMI (et de l'API) au RSA, les droits sociaux des femmes à l'épreuve des politiques d'activation des allocataires de minima sociaux[END_REF]. Two mechanisms indicate this: the contractualisation between the beneficiary and a referent social worker, initially created within the framework of the RMI, by listing the objectives that must be achieved by the recipient in terms of social and professional inclusion (training, active job search, etc.); the temporality of reevaluating the resources of the recipients, defined on a quarterly basis, by forcing the recipients to keep the accounts of their family.

The RSA is a tax-financed device. The funds are allocated by the State to the departments. By agreement, the family branch of social security is responsible for paying the benefit. The support procedures are shared between the placement operator of the public employment service, the social services of the departments and the family allowance funds. In 2018, approximately 2 million families will be receiving this benefit, and the number of families covered by this benefit has been steadily increasing since its creation (see figure 1 in the appendix).

The prime d'activité (PA) is a work income supplement that became effective in January 2016. It replaces an old tax credit scheme, the employment bonus (PPE) and the RSA activité in order to address the shortcomings of these two schemes [START_REF] Sirugue | Réforme des dispositifs de soutien aux revenus d'activité modestes[END_REF]: time lag between individual situation and payment of the benefit as well as the effect of poverty traps for the one; high non-take-up and low amount for the other. These observations led to the creation of an income support benefit for the working poor that takes into account family composition. As the RSA, benefit entitlements are assessed quarterly.

The amount of the premium is based on a lump sum from which 61% of the household's earned income, other household resources and an individual bonus are subtracted. All persons who have reached the age of majority and are in employment are eligible. The lump sum amount of the PA is equal to 551.51 euros for a single person without children. This amount varies according to family composition. It may be temporarily increased in the case of a single parent with one or more children. The State finances the benefit through the general social contribution (CSG) and the family allowance funds assess the eligibility conditions for the payment of the benefit. At the end of 2018, approximately 4 million households were receiving the activity bonus.

In the social benefit scheme, when a household applies for RSA, it is equivalent to an application for PA. However, the reverse is not automatic. This symbiosis is an important argument for studying the determinants of non-take-up of these two income support schemes.

The non-take-up issue has aroused the interest of economists in France since the early nineteen-twenties but there was a lack of reliable quantitative studies and relevant data on this phenomenon, which provided mixed results, and a wide range of estimations [START_REF] Math | Le non-recours en France: un vrai problème, un intérêt limité[END_REF][START_REF] Van Oorschot | Les causes du non-recours [Des responsabilités largement partagées[END_REF][START_REF] Van Oorschot | La question du non-recours aux prestations sociales[END_REF][START_REF] Simon | Le non-recours aux aides personnelles au logement: l'exemple de la CAF du Havre[END_REF][START_REF] Reinstadler | Les causes du non-recours et du quasi-non-recours à l'allocation parentale d'éducation [Quelles solutions pour limiter l'ampleur de ces phénomènes?[END_REF][START_REF] Terracol | Coûts de perception et taux de non-recours aux prestations sous conditions de ressources[END_REF].

In particular, for RSA, only one survey addressing the non-take-up issue was conducted in 2010-2011 by the Direction de l'animation de la recherche, des études et des statistiques (DARES) of the Minister of Labor with the aim to study this phenomenon and its impact on the labour market. The first « reliable » estimated non-take-up rate for the "base" RSA provided by this survey is about 36% and that for the "activity" RSA is about 65%. The main type of non-take-up is rather the lack of awareness [START_REF] Domingo | Le non-recours au revenu de solidarité active[END_REF][START_REF] Domingo | Impact du non-recours sur l'efficacité du RSA activité seul[END_REF].

A big issue arising from this survey is the fact that it is limited only to people living in ordinary households, which do not include mobile home or collective house. Moreover, its implementation is costly because of the difficulty of targeting and sampling and the fragility of the eligibility test since the calculation of the entitlement to the RSA requires a knowledge of infra-annual income and an information on the resources of the « social household » and its family configuration. Consequently, this survey was not reconducted during the past ten years and the studies dealing with the non-up are scarce.

Although these limits, there were some researches that exploited this survey in order to investigate the determinants of the non-take-up. For instance, [START_REF] Chareyron | Pauvreté et non-recours aux dispositifs sociaux: l'étude du RSA «socle seul[END_REF] found that the non-take-up to the RSA concerns mainly those with low incomes and high RSA entitlements. Furthermore, [START_REF] Chareyron | Take-Up of Social Assistance Benefits: The Case of the French Homeless[END_REF] use a survey among homeless people and show that the non-take-up to the RSA is lower for this population than that computed among ordinary households. [START_REF] Anne | Une analyse spatiale du non-recours aux dispositifs sociaux[END_REF] showed a positive significant impact of the proximity to public transport on the knowledge of the benefit. Following those early researches and because the data usually used for the measurement became dated, there was a public need to develop a method that allow to estimate in a regular basis the non-take-up of social benefits in France. Our main aim is to answer the question that is in which way is it possible to quantify the non-take-up of the RSA and the PA on a regularly and contemporary basis? Once one allows quantifying the non-take-up, in which way is it possible to analyse its determinants as well as assess the effect size of a public policy on its magnitude?

We try to address those questions in a research project funded by the French inter-ministerial delegation of preventing and fighting poverty, conducted in the French Ministry of Social Solidarity and Health, and co-managed by the ERUDITE laboratory (UPEC) and the national family allowances fund (CNAF).

We structure our technical note in the following manner. First, we describe the technical strategy in which we present the main tools used in order to measure the non-take-up namely the data (the tax and social security income survey (ERFS)) as well as the microsimulation model INES, which is the public tool to evaluate the socio-fiscal reform in France, based on the ERFS. Moreover, we present the limits and the issues from which suffer these two tools. Finally, we expose the forthcoming works in order to overcome these limits and to provide a reliable measure of non-take-up.

II. Technical strategy 1. The tax and social security income survey (ERFS)

The Tax and Social Income Survey (ERFS) is the main source of data for producing statistics and conducting studies on the income, standard of living and poverty of households residing in ordinary housing in metropolitan France.

It is not strictly speaking a survey, since part of the data relating to income and social benefits is not collected by survey but by using administrative sources for a representative sample of the entire population. Since the creation of the ERFS in 1956, INSEE has successively modified its mode of production. Now, the latest production methodology follows four stages.

 Methodology

The first step consists of a statistical matching, for the same year N, between a tax data component and an individual information base, in particular to take account of family composition and activity status in determining the standard of living.

The first data set contains, on the one hand, the income tax file which lists all income declared by persons liable to tax whether they are employed, self-employed, retired or inactive and, on the other hand, the files relating to the housing tax, which is a local tax depending on the characteristics of the main or secondary dwelling, to which owners, tenants or free of charge occupants are subject.

The second data component corresponds to the Labor Force Survey (LFS) for the fourth quarter of year N. The LFS is a sample survey, in other words, a representative sample of the population is interviewed six times every quarter. Although the survey is primarily used to quantify the number of persons in employment as defined by the ILO, its iteration in the fourth quarter of year N falls within the scope of the ERFS because it is the sample that offers the best match rate with tax files. As an illustration, the original sample for the EFRS 2017 is composed of almost 116,900 individuals for about 53,000 households.

The principle of matching between these two datasets is to try to retrieve the tax returns of the individuals constituting the households surveyed in the LFS. Matching with the personal income tax file is called statistical matching because it is performed on anonymous data at the individual level (i.e., without the use of surnames). Matching is based on criteria common to both data components: first name; date of birth; sex; address (municipality and department of residence). Information on the marital status of the respondent's spouse may also be used. Then, for the individuals found, the data on income are extracted from the file containing all the taxation elements of the tax household.

These data are then combined with the benefits paid to each recipient in year N by the National Family Allowances Fund (Cnaf) and the National Fund of the Agricultural Social Mutual Insurance Fund (Ccmsa), and the pensions established and paid in December N by the National Old Age Insurance Fund (Cnav). The family allowance funds manage the payment of nearly 93% of the benefits taken into account in the ERFS. The data of individuals that have been matched with the tax files are completed with the information available for each taxpayer and his/her spouse: usual name and/or name of birth (encrypted information), first name, date of birth and tax address for taxation purposes. In the first quarter of year N+2, the CNAF retransmits to INSEE all the benefits paid to recipients on this list during year N. Some households not found in the tax sources are found in the CNAF file. In this database, they are integrated into the ERFS sample and considered as respondents to the ERFS. They will then be part of the non-response treatments in the fourth stage of production. Thus, for each individual, we observe a benefit amount paid at least once in the year for all known social benefits: family benefit; housing allowance; disability and handicap support; RSA; PA; minimum retirement pension.

In a third step, information on income and benefits not provided by the tax and social security source is filled in by imputations made by INSEE.

Finally, the ERFS is subject to corrections for non-response and an adjustment in order to be representative at the level of metropolitan France. Indeed, for some matched individuals, the declarations are not systematically found. As soon as a tax return is found for a LFS household (this is the case for approximately 97% of EEC households), the household is considered as a "respondent" to the ERFS. Individuals in the household may not be located. These represent what is known as partial non-response.

For these households, missing reports are reconstructed.

 Scope of the dataset

The scope of the survey is therefore that of individuals from so-called "ordinary" households residing in metropolitan France. Persons living in collective housing (retirement homes, religious communities, university halls of residence, prisons, etc.) as well as persons living in precarious housing (workers' hostels, hotels, etc.) and homeless persons are therefore excluded.

The ERFS is representative of metropolitan households (approximately 97% of the total metropolitan population) and the household is its statistical unit (and not the individual or tax household). More accurately, the sample for the Tax and Social Income survey stricto sensu is made up of households responding to the Employment survey and for whom at least one tax return or social file (file in one of the social organizations) was found.

 Advantages and limits

The way the ERFS is produced helps to make this data source a first working basis for calculating the non-take-up of a given social benefit because the matching with social data sources makes it possible to establish the number of beneficiaries.

In addition, traditional criteria for eligibility for a social benefit, such as the level of resources and family composition, are also available in the ERFS.

Finally, the questions asked by the LFS make it possible to recompose the annual activity calendar of the respondents, which makes it possible to assess each individual's position with regard to employment in each month of the year.

On the other hand, the strength of the ERFS is also its weakness. Typically, two years are required to complete all matches from the different data sources, which implies producing a measure of non-use with a lag of two years.

In addition, the scope of the sample is restricted to persons residing in ordinary housing, which de facto reduces the share of persons potentially eligible and potentially resorting to social minima.

Let us add, despite the richness of the activity calendar contained in the ERFS, resources are counted at the annual level. As a result, the data source does not allow for all the infra-annual variability of resources that would make it possible to refine the eligibility criteria for benefits whose entitlements are assessed several times during the year, which is the case in particular for the RSA and the PA.

 Elements of comparison to attest to the reliability of the ERFS

In order to weigh up the strengths and weaknesses of the ERFS in measuring non-take-up, we carried out an initial comparison between the number of RSA and PA beneficiaries recorded by the ERFS and the number actually observed by the Cnaf, the organization that pays these benefits. By limiting ourselves to common fields of comparison (definition of the boundaries of families in the sense of the benefit, ordinary housing), we diagnosed that the number of beneficiaries was under-represented by the ERFS, on the range of 400,000 social households for RSA and 350,000 social households for PA. In addition, single individuals without children receiving RSA and PA are under-represented while couples with and without children are over-represented. However, the labor income and replacement income distributions of PA recipients in the ERFS and administrative data are confounded. This is also the case when considering these same distributions for RSA, except in the case of zero income: the RSA beneficiary share of this bracket is under-represented in the ERFS.

These numerous biases in the measurement of take-up, whether upward or downward, do not allow providing a reliable measure of non-take-up. Among the different alternative ways of measuring nontake-up, we resort to a microsimulation model. The INES model, developed by INSEE, DREES and CNAF, is designed for this purpose, and its supply is already based on the ERFS. Its main advantage resides in the fact that it simulates the eligibility of most French social benefits.

Microsimulation model INES

Because the measurement of non-up of social benefits from the ERFS presents some limits, the preferred method consists in estimating the take-up rate of a given benefit by relating, in the same field, the beneficiary population, as identified by the paying agencies, to the eligible population, simulated using the INES microsimulation model. This model, managed jointly by the "Institut national de la statistique et des études économiques" (Insee), Drees and Cnaf, simulates, based on the socio-fiscal legislation, most of the social benefits and direct taxation. It is based on the tax and social security income survey (ERFS).

 Methodology

According to the availability of the data, the INES model is based on an ERFS with a time difference of two years. Thus the legislation of a year N is applied to the ERFS of a year N-2, « aged » by two years and adjusted, based on aggregated information from other sources (census, finance survey, …), to reflect the structure and income of the population in the year N. The INES model has several advantages over an estimate based on surveys that were positive arguments used in other research programme (e.g. [START_REF] Bargain | No claim, no pain. Measuring the non-take-up of social assistance using register data[END_REF]): it is cheaper, it reduces the processing time and it increases the durability of the monitoring. Moreover, Ines could also simulate the impact of fiscal and social reforms on wealth indicators and ultimately on the non-take-up behaviour.

Therefore, many steps are followed in INES in order to simulate most taxes and social benefits:

 A first step is the "data aging", which considers the changes in population characteristics by modifying the weightings using "margin calibration". Moreover, differentiated rates of change are applied to each type of income according to its nature (earned income, wealth, replacement income), using all available economic information. For example, wages are adjusted using the "Labour force activity and employment conditions" (Acemo) survey, which takes into account the socio-professional category and sector of activity of each employee.

 The second step consists on simulating social and fiscal levies: social security contributions, income taxes, family and other social benefits, according to family configuration (identification of social households adapted to each benefit and construction of resources data that will be used to simulate eligibility), and N-2 income (in general).

 The simulation of the RSA and the PA comes at the end of the process because the resources taken into account for the calculation of entitlements include a good part of the benefits simulated beforehand; it requires the following steps: o Construction of "social household": allocation of a PA or RSA "social household" identifier.

o Quarterly earnings simulation thanks to the activity calendar of the LFS. This step is essential to simulate the eligibility for RSA or PA since the benefits are reassessed every three months.

o Construction of resource data at "social household" level.

o Simulation of legislation and determination of eligibility.

o Drawing beneficiaries from among the eligible, based on quarterly targets provided by the Cnaf. Precisely, the draw is made only for PA considering the fact that "social households" eligible for the PA are automatically considered as beneficiaries if they have already been drawn in the previous quarter (regardless of their status with respect to the RSA). Moreover, the probability of take-up for an eligible social household increases with the simulated amount of entitlement. For the RSA, it is assumed that those who do not use PA do not use RSA. For the other eligible to the RSA, they are automatically assumed to be beneficiary.  Nevertheless, these assumptions are under revision by the Ines team and would be updated soon.

 Elements of comparison to attest the power of simulation

As we made for ERFS, we also compared the data from INES to that from CNAF in order to investigate the gaps between the two data sources in terms of headcount and financial mass of the beneficiaries of RSA and PA following many configurations (family configuration, income level, annual amount of benefit, age and employment status). This comparison allows determining the discrepancies that potentially are due either from the representativeness of the ERFS (on which Ines is based), the consistency of the fields, the eligibility simulation process in Ines or the quality of the probabilistic draw of take-up among the eligible persons.

 For PA, we find that Ines underestimates single-parent families and overestimates couples without children and with one child. For RSA, it underestimates single individuals and singleparent families, and overestimates couples with and without children.

 Compared to the exhaustive administrative data of the family branch, Ines underestimates the proportion of PA recipients who receive less than 600 euros per year, and overestimates the proportion of recipients with higher amounts.

 For the RSA, Ines overestimates the number of beneficiaries receiving RSA amounts up to 3,000 euros per year and underestimates the number of beneficiaries receiving more than 3,000 euros. This is also the case in particular for the bracket between 5,400 and 6,000 euros per year, which corresponds to the maximum amount that a single person without resources and receiving housing benefit can receive over the whole year 2017.

 On the other hand, the income distribution of PA beneficiaries in INES seems to be poorly simulated compared to those in CNAF. This may refer to a problem in simulating eligible households or to the take-up assumptions applied in Ines (or both).

As a whole, we detect that the microsimulation model needs to be more calibrated in order to reproduce the corresponding characteristics of the beneficiaries as well as the eligible population. The main explanation of this discrepancy is the fact that Ines does not provide a take-up simulation for the RSA.

Nevertheless, this work is under progress.

Non take-up in ERFS

In our project, we would like to realise a diagnosis that consists on computing a non-take-up rate of RSA and PA from the ERFS itself using the program in INES that allows to simulate the quarterly earnings and that used to simulate the eligibility to RSA and PA in order to make up for the missing information on the infra-annual income. This work could be done since the last available version of 2017 until that of 2011 in order to compare the non-take-up rate with that computed in 2011 from the Dares survey [START_REF] Domingo | Impact du non-recours sur l'efficacité du RSA activité seul[END_REF] and also to have an overview of its evolution over time.

households that respond over two consecutive years : in this case households that respond twice in Q4 of the year N-1 and N.

The "aging" programs would be applied to the N-1 year data to age the information over a single year and then compare the "simulated" information with the "actual" information for the year N.

Improvement of microsimulation

Once the last diagnosis cited are made, we are potentially able to determine which programs are suffering from imprecision and focus on them in order to bring some improvements.

In any case, the non-take-up draw programme is under consideration by the team working on the microsimulation model INES at the DREES with developing a non-take-up drawdown programme for the RSA. Further improvements would be to modify the quarterly targets provided by the Cnaf with refining them according to the family structure.

IV. Conclusion

This technical note allows us to conclude about the possibility to develop a regular and contemporary non-take-up measure of French social benefits from the available tools: administrative data (ERFS) and micro-simulation model Ines. However, the discrepancies found with the exhaustive data of the benefits' recipients (CNAF data) show that there are still some issues to address in order to obtain a reliable measure of non-take-up. Nevertheless, even if we succeed to produce a reliable indicator of non-takeup of social benefits, it would still be affected by inevitable statistical issues related to the ERFS survey process and some assumptions used in Ines.

One possible recommendation for official statistics would be to expertise in new data such as the monthly resource data (MRB) in order to control for the infra-annual variation in the ERFS and then to incorporate changes from this new kind of data in the ERFS. This new administrative source is the result of an improved gathering process of the income declarations on a monthly basis in the public and private sectors since 2017 in France.

The advantage of this computation is to use only some programs from INES mainly that for the eligibility computation and the estimation of quarterly earnings since the benefits are reassessed every three months and the beneficiary population is already known in the ERFS. Moreover, we would compute the non-take-up rate at the same year of the ERFS, so that we would not resort to "aging" the data. The aim from this diagnosis is to limit the points of uncertainty identified in INES. Besides, the estimate could be biased by the lack of infra-annual resources.

 Advantage: The beneficiary population is already known in the ERFS (we would not resort to estimate it). Only the eligibility is simulated using INES (quarterly earnings program and legislation simulation for eligibility).

 Disadvantage: The aim of this project is to develop a contemporary measure of non-take-up (the ERFS is available at N-2 of each year N). Moreover, we would like also to simulate the contemporary legislation in order to evaluate public policies and this is possible only using the micro-simulation model INES.

III. Work in progress 1. Alternative diagnosis

Once these diagnosis are made, further analyses are necessary in order to determine the sources of discrepancies obtained from the comparison between the different data sources mainly in INES model.

Since we cannot modify directly the production of the ERFS, we could only suggest some possible improvements in the weighting procedure in order to have a better representativeness of social benefits' beneficiaries and also people living in ordinary households.

Initially, the idea was to apply only the quarterly earnings simulation programme in the ERFS 2017 to determine eligibility for RSA and PA in the ERFS 2017 and to compute a non-take-up rate of RSA and PA from the ERFS itself. However, a further idea is to apply the non-take-up draw module in order to obtain the "simulated" take-up in the ERFS 2017. This work would help us to compare several elements:  Eligible ERFS 2017 VS eligible INES 2017, which allows to question about the "aging" and quarterly earnings simulation procedures.

 "Simulated" ERFS 2017 vs. "real" ERFS 2017 recipients: This comparison would make it possible to question the non-take-up drawdown programme and the simulation of the quarterly earnings.

In parallel with these activities, the idea is to dig into the "aging" income programs in INES, because they raise several hypotheses in the simulation. To move in this direction, the idea is to exploit the panel dimension of the ERFS (that came directly from the trimestral dimension of the LFS) by considering

Appendix


Muhammad Salman
email: muhammad.salman@iub.edu.pk

Usman Ali
email: uali@bzu.edu.pk

On maximal det-independent (res-independent) sets in graphs

Keywords: determining set, resolving set, independent set, hereditary system 2010 AMS Subject Classification Numbers: 05C12, 05B35

In this writing, we point out some errors made in [D. L. Boutin, Determining sets, resolving sets and the exchange property, Graphs and Combin., 25(2009), 789-806], where the author claims that a maximal independent set in a hereditary system is a minimal determining (resolving) set. Further more, the author claims that if the exchange property holds at the level of minimal resolving sets, then, the corresponding hereditary system is a matroid. We give counter examples to disprove both of her claims. Besides, we prove that there exist graphs having such maximal independent sets which are not necessarily determining (resolving) sets. Also, we give necessary and sufficient conditions for a class of graphs to have a maximal independent set which is not minimal determining (resolving).

Introduction

A set U of vertices of a connected graph G is called a determining set for G if every automorphism of G is uniquely determined by its action on the vertices of U . If the vertices in U are unable to determine all the automorphisms of G, then we call U a non-determining set for G. The minimum cardinality of a minimal determining set for G is called the determining number of G, denoted by D(G) [START_REF] Boutin | Identifying graph automorphism using determining sets[END_REF]. A set W of vertices of G is called a resolving set for G if every vertex in G is uniquely determined by its distances to the vertices of W . If the vertices in W are unable to determine all the vertices of G, then we call W a non-resolving set for G. The minimum cardinality of a minimal resolving set for G is called the metric dimension of G, denoted by β(G) [START_REF] Ćaceres | On the metric dimension of Cartesian products of graphs[END_REF][START_REF] Chartrand | Resolvability in graphs and the metric dimension of a graph[END_REF].

Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolving) sets for G and r ∈ R, then there exists s ∈ S so that S -{s} ∪ {r} is a minimal determining (resolving) set [START_REF] Boutin | Determining sets, resolving sets and the exchange property[END_REF]. It is noteworthy that if determining (resolving) sets have the exchange property in a given graph, then every minimal determining (resolving) set for that graph has the same cardinality.

A set system is a finite set H together with a family F of subsets of H and is denoted by the pair (H, F). A set system (H, P) is said to be a hereditary system (an independence system) if for every subset X of H possessing the property P, each subset of X also possesses the property P. That is, for each X ⊆ H such that X ∈ P, Y ∈ P for all Y ⊆ X. In fact, in a hereditary system (H, P), P is identified by the family of subsets of H possessing the property P. A subset X of H which possesses the property P is said to be an independent set, and a dependent set otherwise.

A subset H of the vertex set V of a graph G is a det-independent (res-independent) set if no proper subset of H is a determining (resolving) set for G. This concept was first introduced by Boutin in [START_REF] Boutin | Determining sets, resolving sets and the exchange property[END_REF]. If we denote the property of being det-independent (res-independent) by det (res), then the hereditary system in G corresponds to this property is as follows:

(V, det (res)) = {H ⊂ V | H possesses the property det (res)},
where V is the vertex set of G. It is also worth mentioning here that, by definition, a nondetermining (non-resolving) set is always a det-independent (res-independent) set.

Counter examples

Formally, determining and resolving sets are defined as follows: Let Aut(G) be the automorphism group of G and U be a set of vertices of G. For every two automorphisms ϕ, ψ ∈ Aut(G), if ϕ(u) = ψ(u) for all u ∈ U implies ϕ = ψ, then U is called a determining set for G [START_REF] Boutin | Identifying graph automorphism using determining sets[END_REF]. Let d(x, y) be the number of edges in a geodesic (shortest path) between the vertices x and y, and is called the distance between x and y in G. A set W of vertices of G is a resolving set for G if for every two vertices v and z of G, there is a vertex w in W such that d(v, w) ̸ = d(z, w) [START_REF] Ćaceres | On the metric dimension of Cartesian products of graphs[END_REF][START_REF] Chartrand | Resolvability in graphs and the metric dimension of a graph[END_REF].

Boutin formally defines det-independence (res-independence), so that a set S of vertices in a graph G is a det-independent (res-independent), if for every s ∈ S, S -{s} is not a determining (resolving) set for G [START_REF] Boutin | Determining sets, resolving sets and the exchange property[END_REF]. With this definition, she claims that a maximal det-independent (resindependent) set is a minimal determining (resolving) set. (line -6 at page 791 of [START_REF] Boutin | Determining sets, resolving sets and the exchange property[END_REF]). But, her claim is unfortunately wrong, as we observe in the following two examples.

Example 1.

Let us consider the graph G of Figure 1. Minimal determining sets for G are: {v 4 } and {v 5 }. Thus, D(G) = 1. According to the property of det-independence, we have the following hereditary system for G:

(V, des) = {{v i } ; 1 ≤ i ≤ 5} ∪ {{v 1 , v 2 }, {v 1 , v 3 }, {v 2 , v 3 }} ∪ {{v 1 , v 2 , v 3 }}.
(1)

v 4 v 5 v 3 v 2 v 1
{v 1 , v 4 }, {v 2 , v 4 }, {v 1 , v 5 }, {v 2 , v 5 } and {v 4 , v 5 }. Thus, β(G) = 2.
According to the property of res-independence, we have the following hereditary system for G:

(V, res) = {{v i } ; 1 ≤ i ≤ 5} (2) ∪ {{v 1 , v 2 }, {v 1 , v 3 }, {v 1 , v 4 }, {v 1 , v 5 }, {v 2 , v 3 }, {v 2 , v 4 }, {v 2 , v 5 }{v 3 , v 4 }, {v 3 , v 5 }, {v 4 , v 5 }} ∪ {{v 1 , v 2 , v 3 }}.
Here the set {v 1 , v 2 , v 3 } is a maximal res-independent, but not a minimal resolving set.

In [START_REF] Boutin | Determining sets, resolving sets and the exchange property[END_REF], the author also claims that if the exchange property holds at the level of minimal determining (resolving) sets, then the property holds for maximal independent sets in the corresponding hereditary system (lines 1-5 at page 792 [START_REF] Boutin | Determining sets, resolving sets and the exchange property[END_REF]). Consequently, the hereditary system is a matroid [START_REF] West | Introduction to graph theory[END_REF]. But, it is not true regrettably, because Examples 1 and 2 provide that a maximal det-independent (res-independent) set need not to be a minimal determining (resolving) set, and so the corresponding hereditary systems (1) and ( 2) are not matroids. Since the graph considered in Examples 1 and 2 is a tree graph, so the hereditary system (produced due to Theorem 3 at page 793 of [START_REF] Boutin | Determining sets, resolving sets and the exchange property[END_REF]) for tree graphs is not a matroid.

Necessary and sufficient conditions

Let v be a vertex of a graph G having the vertex set

V . Then, the open neighborhood of v is N (v) = {u ∈ V : u is adjacent with v in G} and the closed neighborhood of v is N [v] = N (v) ∪ {v}.
Two vertices of G are said to be twin vertices (simply called twins) if they have the same (open or closed) neighborhoods. The relation of twins between the vertices of G is an equivalence relation, which produces classes of twin vertices in G, called the twin classes. That is, a twin class in G is a set of vertices of G such that its every two elements are twins [START_REF] Hernando | Extremal graph theory for metric dimension and diameter[END_REF]. A non-singleton twin contains two or more elements. The next lemma is due to the definition of twins. Lemma 1. [START_REF] Hernando | Extremal graph theory for metric dimension and diameter[END_REF] If u and v are twins in a graph G, then d(u, x) = d(v, x) for each x ∈ V -{u, v}.

Due to Lemma 1, the following remark is directly followed.

Figure 1 :Example 2 .

 12 Figure 1: A graph G

Remark 2. If u and v are twins in a graph G and U (or W ) is a determining (resolving) set for G, then either u ∈ U (W ) or v ∈ U (W ). In other words, if T is a twin class of order t ≥ 2 in G, then every determining (resolving) set for G must contain at least t -1 elements of T .

The removal of only two twins from the vertex set of a graph G makes it a maximal detindependent (res-independent) set for G. Lemma 3. Let T be a twin class of order t ≥ 2 in a graph G with vertex set V . Then, for any two elements u, v ∈ T , the set V -{u, v} is a maximal det-independent (res-independent) set for G.

Proof. Since d(u, x) = d(v, x) for all x ∈ V -{u, v}, by Lemma 1, so no subset of V -{u, v} is a determining (resolving) set for G. It follows the result.

In next result, we observe the condition when a hereditary system for a graph corresponds to the det-independence (res-independence) property is not a matroid. Theorem 4. If there is a non-singleton twin class in non-complete graph (not a complete graph), then there is a maximal det-independent (res-independent) set which is not a minimal determining (resolving) set. In fact, the corresponding hereditary system for the graph is not a matroid.

Proof. Let G be a non-complete graph with vertex set V and let T be a twin class in G of cardinality at least two. Then, for any u, v ∈ T , the set X = V -{u, v} is a maximal detindependent (res-independent) set for G, by Lemma, 3. But, X is not a determining (resolving) set for G, because d(u, x) = d(v, x) for all x ∈ X due to Lemma 1 and Remark 2. Thus, the hereditary system for G corresponds to det-independence (res-independence) property contains a det-independent (res-independent) set which is not a minimal determining (resolving) set, and hence it is not a matroid.

The next result provides necessary and sufficient conditions for a class of graphs to have a maximal independent set which is not minimal determining (resolving).

Theorem 5. The following assertions are equivalent for a non-complete graph G in which determining (resolving) sets have the exchange property.

G has a non-determining (non-resolving) set of cardinality greater than D(G) (or β(G)).

G has a maximal det-independent (res-independent) set which is not minimal determining (resolving).

Proof. Suppose that G has a non-determining (non-resolving

Since the exchange property holds in G for determining (resolving) sets, so the cardinality of all the minimal determining (resolving) sets is the same, and is D(G) (or β(G)). Thus, any nondetermining (non-resolving) set Y with |Y | > D(G) (or β(G)) is a a maximal det-independent (res-independent) set which is not minimal determining (resolving).

Conversely, suppose that G has a maximal det-independent (res-independent) set X which is not minimal determining (resolving). Then, of course, X is a non-determining (non-resolving) set with |X| > D(G) (or β(G)) due to holding of the exchange property.


U Elu

T Steinle

D Sánchez

L Maidment

K Zawilski

P Schunemann

U D Zeitner

C Simon-Boisson

J Biegert

Table-top high-energy 7 μm OPCPA and 260 mJ Ho:YLF pump laser

We present a state-of-the-art compact high-energy midinfrared (mid-IR) laser system for TW-level eight-cycle pulses at 7 μm. This system consists of an Er:Tm:Ho:fiber MOPA which serves as the seeder for a ZGP-based optical parametric chirped pulse amplification (OPCPA) chain, in addition to a Ho:YLF amplifier which is Tm:fiber pumped. Featuring all-optical synchronization, the system delivers 260 mJ pump energy at 2052 nm and 16 ps duration at 100 Hz with a stability of 0.8% rms over 20 min. We show that chirp inversion in the OPCPA chain leads to excellent energy extraction and aids in compression of the 7 μm pulses to eight optical cycles (188 fs) in bulk BaF 2 with 93.5% efficiency. Using 21.7 mJ of the available pump energy, we generate 0.75 mJ energy pulses at 7 μm due to increased efficiency with a chirp inversion scheme. The pulse quality of the system's output is shown by generating high harmonics in ZnSe which span up to harmonic order 13 with excellent contrast. The combination of the passive carrierenvelope phase stable mid-IR seed pulses and the high-energy 2052 nm picosecond pulses makes this compact system a key enabling tool for the next generation of studies on extreme photonics, strong field physics, and table-top coherent X-ray science.

Ultrafast and intense > 10 14 W∕cm 2 pulses with carrierto-envelope-phase (CEP) stability give rise to extreme photonic applications such as strong field ionization [1,2], or the generation of isolated attosecond pulses in the extreme ultraviolet range [3,4]. High-energy mid-infrared (mid-IR) pulses through ponderomotive energy scaling provide unique capabilities for the generation of soft x-ray harmonics up to the kilo electron volt level [5][6][7][8][9][10]. Intense mid-IR pulses can be generated from compact table-top systems, providing an exciting approach towards a new generation of miniature particle accelerators though wakefield acceleration [11]. Low atmospheric losses at wavelengths between 8 and 14 μm provide an opportunity to exploit self-guided pulses via filamentation over kilometer distances without losing intensity through absorption or diffraction [1]. Thus, there is an important demand on high-energy mid-IR pulses which provide CEP-controlled electric field waveforms.

The challenge of generating broadband mid-IR pulses has been approached by direct laser generation, as well as difference frequency generation (DFG) or optical parametric amplification. Solid-state lasers based on Cr:ZnSe and Cr:ZnS have been demonstrated producing nanojoule-level short pulses in the 2-3 μm spectral range [12,13], but amplification to the millijoue level has not been realized, and no realistic materials exist to directly operate in a mode-locked mode at mid-IR wavelengths. DFG has been studied exploiting different nonlinear crystals such as CSP, GaSe, and OPGaP [14][15][16]. All of these crystals show low-absorption coefficients in the mid-IR, as well as in the near-infrared (NIR) regime, making them suitable to seed the DFG process with commercial NIR fiber lasers for the generation of mid-IR pulses. Unfortunately, the generated mid-IR pulse energy is typically limited to a few pico-joules, mainly due to the low damage threshold of these mid-IR crystals and the high multi-photon absorption coefficients [17,18]. Recent intrapulse DFG experiments [19] have generated broad mid-IR spectra with nanojoule-level pulse energy [20,21] which are very promising for spectroscopy applications [22,23], but do not reach ultra-intense levels.

Therefore, efficient energy scaling systems with suitable mid-IR nonlinear crystals are required to increase the energy to the millijoule level. Optical parametric chirped pulse amplification (OPCPA) can address this problem [24,25], though it has been recognized that in order to amplify wavelengths beyond 5 μm a new pump platform had to be developed to optimize the quantum efficiency [26,27] and mitigate crystal defect-related damages [28]. Previously, we have demonstrated the first compact 7 μm millijoule level platform based on a Ho:YLF based chirped pulse amplifier (CPA) at 2 μm as the pump [26] and a ZGP-based OPCPA for the energy scaling of the mid-IR pulses [24] with all-optical synchronization [29]. However, using a diffraction grating-based pulse compressor in this spectral range at the end of an OPCPA restricts the pulse energy of the compressed output pulses due to the low efficiency of commercially available gratings.

Here we present a new scheme which incorporates a chirp inversion inside the OPCPA chain, thus permitting us to mitigate adverse chirp-related amplification effects and to use bulk material for stretching and compression. While the relative energy loss due to the insertion of the chirp inverter remains equal, placing it earlier in the amplification chain reduces the absolute energy spent for dispersion tailoring and thereby increases the overall system conversion efficiency. Thus, the overall efficiency of the OPCPA after compression could be doubled with the chirp inversion scheme. We have also upgraded the pump laser system from 40 to 260 mJ at a 100 Hz repetition rate, while maintaining the high stability and reliability of the CPA system. The high-energy 2052 nm pump laser is used for amplification of the optically synchronized passively CEP stable 7 μm seed pulses in an OPCPA chain separated by the chirp inverter, analog to the previous demonstration with our 3 μm OPCPA system [30]. We start by discussing the 2052 nm CPA pump laser system and then report on our results in energy scaling and characterization of millijoule-level 7 μm femtosecond pulses.

The conceptual layout of the high-energy Ho:YLF CPA is shown in Fig. 1. Narrowband nanojoule pulses from the Er:fiber laser are temporally stretched to 340 ps by a chirped volume Bragg grating (CVBG), and the repetition rate is reduced to 100 Hz with a Pockels cell (PC) pulse picker. The 100 Hz pulses are then used to seed a regenerative amplifier based on a water-cooled Ho:YLF crystal, which is pumped by 24 W from a CW Tm:fiber laser (IPG Photonics) centered at 1940 nm.

The regenerative amplifier achieves amplification from 2 nJ to 4 mJ pulse energy. Its cavity is operated in saturation ensuring highly stable performance and excellent beam quality, both being crucial for optimum performance of the entire system. Subsequently, the 4 mJ pulses are introduced in a booster multi-pass amplifier for energy scaling purposes. The booster amplifier is based on a loose focusing triple-pass geometry using a 50 mm long Ho:YLF active material which is cryogenically cooled to 95 K. The booster is pumped in a single-pass loose focusing geometry using a 120-W CW thulium fiber laser (IPG Photonics) centered at 1940 nm. The booster amplifier increases pulse energy from 4 to 42 mJ in the first pass, to 173 mJ in the second pass, and to 260 mJ in the third pass.

The pulse-to-pulse stability of 0.8% rms over 20 min at 260 mJ (26 W) is shown in Fig. 2(a). Together with excellent beam quality, this makes the system ideal for pumping the mid-IR OPCPA chain.

To temporally compress the high-energy 2052 nm pulses, the beam size is increased to 6 mm (FWHM) to avoid damage to optics. The grating compressor consists of two dielectriccoated gratings (Fraunhofer IOF), with 900 lines/mm and has a footprint of 140 cm by 25 cm. We measure an overall efficiency of 85% corresponding to a final compressed pulse energy of 220 mJ.

Figure 2(b) shows the results from the characterization of the high-energy picosecond pulses via a non-collinear intensity autocorrelation using a 200 μm thick BBO crystal for secondharmonic generation (HHG). The pulse duration, assuming a Gaussian deconvolution is 16 ps; the wings results from uncompensated for higher-order phase. While the 2052 nm spectrum allows pulse compression down to 5 ps, we set the pulse duration to 16 ps in order to maintain a reasonable peak intensity within the limited aperture on the ZGP nonlinear crystals for the OPCPA stages.

Figure 3 shows a schematic of the 7 μm OPCPA system. The two femtosecond outputs from the Er:Tm:Ho:fiber front-end are combined in collinear geometry in a 1 mm thick GaSe nonlinear crystal for type I DFG. Mixing the 1.5 μm, 70 fs, and 2 μm, 100 fs pulses in GaSe, we are able to generate 7 pJ of mid-IR pulses centered at 7 μm.

The 7 μm DFG pulses are stretched in a 16 mm long BaF 2 rod to a pulse duration of 4 ps. All the OPCPA stages are based on non-collinear geometry with an external angle between the pump and seed of 6.5 deg and type I phase-matching configuration in ZGP nonlinear crystals (BAE Systems). The OPCPA consists of two sections, a pre-amplification and a booster amplification section with an intermediate chirp inversion stage. The pre-amplification section is based on two 3 mm thick uncoated ZGP-based OPCPA stages pumped by 200 μJ and 1.5 mJ 2052 nm pulses and achieving 7 μm amplification to 0.1 and 20 μJ, respectively.

The chirp inversion stage between the amplification sections is based on a Martinez-type stretcher which provides control on the stretching factor of the 7 μm pulses, optimizing the Fig. 1. Layout of the 2052 nm Ho:YLF solid-state laser system. The narrowband 2052 nm pulses from the three-color fiber front-end are stretched in a CVBG, and the repetition rate is reduced in a PC before amplification in a water-cooled regenerative amplifier and cryogenic cooled booster amplifier consecutively. Finally, the stretched pulses are back compressed using a pair of dielectric-coated gratings. efficiency of the booster OPCPA stages and allowing the final pulse compression in a bulk dielectric medium. The chirp inverter is aligned to stretch the 7 μm pulses from a negatively chirped 4 ps duration (-1.77 × 10 5 fs 2 ) to positively chirped 11 ps pulse duration (5.3 × 10 5 fs 2 ) in order to optimize the temporal overlap between mid-IR seed and 2052 nm pump pulses. Moreover, the chirp inverter not only gives us the opportunity to tailor the seed pulse duration, but also ensures an efficient compression scheme in dielectric bulk at the end of the OPCPA. This is important as in the current implementation of the chirp inversion stage exhibits an overall efficiency of 10% due to the limitations imposed by commercially available mid-IR gratings. The chirp inverter was placed in the preamplification stages of the OPCPA after reaching an acceptable power level at 7 μm for the alignment of the corresponding optics.

The booster amplifier is based on three consecutive OPCPA stages, with the first stage based on 5 mm thick uncoated ZGP largely to compensate for the losses imparted by the chirp inverter, to restore the pulse energy to 50 μJ. The next and the last stage are composed of two identical 2 mm thick anti-reflection (AR)-coated ZGP crystals and pumped by 3.5 and 15 mJ. This configuration results in amplified pulse energies of 0.25 and 0.75 mJ, respectively. The pump peak intensity in the OPCPA stages was set to be between 2 and 4 GW∕cm 2 to avoid damage on the ZGP crystals. The pump energy used in the OPCPA is presently limited to 21.7 mJ due to the aperture size of 8 mm × 8 mm of the ZGP crystals. Due to the challenges associated with proper characterization and propagation of the mid-IR pulses, all the linear and nonlinear processes were numerically pre-simulated using the Simulation System for Optical Science (Sisyfos) nonlinear wave propagation code [31].

The final compression of the mid-IR pulses occurs in a 48 cm long BaF 2 rod that compensates for the dispersion acquired in the chirp inversion stage and shows an overall transmission efficiency of 93.5%, corresponding to a final compressed pulse energy of 0.7 mJ. Characterization of these pulses is done using a home-built all-reflective optics-based third-harmonic generation frequency-resolved optical gating (THG FROG) device using a thin ZnSe crystal for the generation of the third-harmonic. This characterization technique allows us to measure the signal of the FROG using a thermo-electric cooler-based NIR spectrometer without the need for using mid-IR nitrogen cooled detectors or less precise mid-IR photodiodes. Furthermore, the THG FROG has allowed a fast reconstruction of full pulse amplitude and phase of our high-energy mid-IR pulses showing a FWHM pulse duration of the 7 μm pulses of 188 fs, see Fig. 4.

To test the fidelity of the compression we employed the high-energy broadband mid-IR pulses for HHG in a ZnSe sample. Owing to the materials' polycrystalline nature, we observe both even and odd harmonics up to the 13th order (Fig. 5). The high-energy 7 μm pulses are focused in the sample to a spot size of 136 μm at FWHM, which is shown in the inset of Fig. 5, using a 75 mm off-axis parabolic mirror, and the visible and NIR harmonics are then collimated with a f 75 mm fused silica lens. Using these focusing conditions and employing 50 μJ of 7 μm energy (limited to avoid damage to the ZnSe), we estimate the peak intensity to be 0.45 TW∕cm 2 . By splitting the NIR and visible spectral parts using a silicon thin plate, we measure all the harmonics below the bandgap simultaneously with integration times lower than 500 ms. The HHG emission is fiber-coupled into the spectrometers using a NIR256 spectrometer (Ocean Optics) for the spectral range of 900-2550 nm and a HR4000 visible spectrometer (Ocean Optics) for the range of 200-1100 nm. Fig. 3. Layout of the 7 μm OPCPA. The mid-IR seed is generated using the two broadband femtosecond outputs from a three-color fiber frontend via DFG. Afterward, the mid-IR pulses are stretched in a dielectric bulk and consecutively amplified in a pre-amplifier and a booster amplifier separated with a chirp inversion stage. Maximum efficiency of the OPCPA is achieved by tailoring the seed-to-pump pulse durations in the preamplifier and booster amplifier. The broadband high-energy mid-IR pulses are recompressed using a dielectric bulk rod of BaF 2 . In conclusion, we have demonstrated an all-optically synchronized CPA and OPCPA system generating 260 mJ pulses centered at 2052 nm and 0.75 mJ mid-IR pulses at 7 μm at a 100 Hz repetition rate. We demonstrated that the chirp inversion scheme improved the total OPCPA efficiency and allowed reaching the millijoule-level with a demonstrated pulse energy of 0.7 mJ at 188 fs after compression (eight optical cycles). The quality of the compressed high-energy 7 μm pulses was demonstrated through HHG in ZnSe by achieving a spectrum expanding over all the visible and NIR regime up to order 13. This mid-IR OPCPA platform features the highest demonstrated energies for both 2052 nm and 7 μm pulses and represents a significant step towards mid-IR high-field applications such as coherent kilo electron volt x-rays or self-similar atmospheric propagation and laser-plasma acceleration.

Fig. 2 .

 2 Fig. 2. (a) Measured stability at full power from the Ho:YLF pump system. The inset shows the beam profile measured at full power. (b) Measured intensity autocorrelation (blue) of the 2052 nm pump pulses after compression in the dielectric-coated gratings, together with a Gaussian fit (dashed black). The FWHM duration of the autocorrelation signal was measured to be 19.5 ps corresponding to a pulse duration of 16 ps at FWHM.

Fig. 4 .

 4 Fig. 4. Pulse characterization of the 7 μm compressed pulses. (a) Measured third-harmonic FROG, (b) the retrieved pulse envelope with 188 fs FWHM duration, and (c) measured (filled gray) and retrieved spectrum (red line) and phase (green line).

Fig. 5 .

 5 Fig. 5. Measured high-harmonic spectrum in bulk ZnSe crystal (red curve) and the harmonic orders from the third to 13th harmonic, both even and odd, shown in dashed blue lines. The black thick dashed line refers to the bandgap of the crystal. The inset shows the measured beam profile of the 7 μm pulses at the generation plane of the harmonics.

0146-9592/19/133194-04 Journal © 2019 Optical Society of America

Vol. 44, No. 13 / 1 July 2019 / Optics Letters

Vol. 44, No. 13 / 1 July 2019 / Optics Letters

Funding. Ministerio de Economía y Competitividad (MINECO) (FIS2017-89536-P); "Severo Ochoa" Programme for Centres of Excellence in R&D (SEV-2015-0522); FET-OPEN "PETACom" (829153); Agència de Gestió d'Ajuts Universitaris i de Recerca (AGAUR); Fundación Cellex (SGR 1639); Army Research Laboratory (ARL) (W911NF-17-1-0565); H2020 European Research Council (ERC) (788218, 840010); Horizon 2020 Framework Programme (H2020) (654148).