Equal contribution
October 16, 2020
Abstract
For composite nonsmooth optimization problems, Forward-Backward algorithm achieves model **identification** (e.g., support **identification** for the Lasso) after a finite number of iterations, provided the objective func- tion is regular enough. Results concerning coordinate descent are scarcer and model **identification** has only been shown for specific estimators, the support-vector machine for instance. In this work, we show that cyclic coordinate descent achieves model **identification** in finite time for a wide class of functions. In addition, we prove explicit **local** **linear** **convergence** rates for coordinate descent. Extensive experiments on various estimators and on real datasets demonstrate that these rates match well empirical results.

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(PS) penalties and general C 2 loss functions with Lipschitz gra-
dient. Under nonlinear complementarity requirements analogous to the non-degeneracy assumption “(x ∗ ) > 0” of Theorem 1(b), and rank constraints analogous to the requirement that the Jacobian matrix T 0 (x ∗ ) have spectral radius less than 1 (in Theorem 1(c2)), the authors of [22, 23] prove finite-time activity **identification** and **local** Q-**linear** **convergence** at a rate given in terms of Friedrichs angles, via direct application of [20, Theorem 3.10]. The authors show that their arguments are valid for a broad variety of problems, for example the anisotropic TV penalty. Still in the framework of partly-smooth penalties, [25] showed **local** Q-**linear** **convergence** of the Douglas-Rachford algorithm on the Basis Pursuit problem. Comparison with [22, 23]. The works which are most compara- ble to ours are [22] and [23], already presented above. Let us point out some similarities and differences between these papers and ours. First, though our constructions are entirely different from the tech- niques developed in [22, 23], one notes that both approaches are ultimately rooted in the same idea, namely the work of B. Holmes [16] on the smoothness of the euclidean projection onto convex sets, and other related functionals (Minkowski gauges, etc.). Indeed, The- orem 1 builds directly upon [16], whilst, [23] and [22] are linked to [16] via [26], which builds on [27], and the latter builds on [16].

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Abstract – We consider the class of inertial Forward–Backward (iFB) proximal splitting algorithms, to minimize the sum of two proper lower semi-continuous convex functions, one of which having a Lipschitz continuous gradient and the other being partly smooth relative to an active manifold M. Special cases of this class include the FB and, for an appropriate choice of the inertial parameter, FISTA-like schemes. We propose a unified analysis, under which we show that iFB-type splitting, (i) correctly identifies the active manifold M in a finite number of iterations, and then (ii) enters a **local** (**linear**) **convergence** regime, which is characterised precisely. This gives a grounded justification to the typical behaviour that has been observed numerically for many problems encompassed in our framework, including the Lasso, the group Lasso, total variation minimization and the nuclear norm regularization to name a few. These results may have numerous applications including in signal/image processing processing and machine learning.

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predictions. The beginning of the dashed lines are the points when x k
identifies the manifold M x ⋆ . As one can observe, FISTA has the
fastest manifold **identification**, however, locally it is the slowest for all tested examples. Indeed, when the manifold is affine, it can be shown from Theorem III.1 that ρ k ∈]η k , √η k ] for a k > η k , i.e. FISTA is locally slower than FB.

13. Liang, J., Fadili, M.J., Peyré, G., Luke, R.: Activity **identification** and **local** **linear** **convergence** of Douglas–Rachford/ADMM under partial smoothness. In: Scale Space and Variational Methods in Computer Vision, pp. 642–653. Springer (2015)
14. Borwein, J.M., Sims, B.: The Douglas–Rachford algorithm in the absence of convexity. In: H.H. Bauschke, R.S. Burachik, P.L. Combettes, V. Elser, D.R. Luke, H. Wolkowicz (eds.) Fixed-Point Al- gorithms for Inverse Problems in Science and Engineering, Springer Optimization and Its Applications, vol. 49, pp. 93–109. Springer New York (2011)

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Abstract: We consider the problem of finding a singularity of a vector field X on a complete Riemannian manifold. In this regard we prove a unified result for **local** **convergence** of Newton’s method. Inspired by previous work of Zabrejko and Nguen on Kantorovich’s majorant method, our approach relies on the introduction of an abstract one-dimensional Newton’s method obtained using an adequate Lipschitz-type radial function of the covariant derivative of X. The main theorem gives in particular a synthetic view of several famous results, namely the Kantorovich, Smale and Nesterov-Nemirovskii theorems. Concerning real-analytic vector fields an application of the central result leads to improvements of some recent developments in this area.

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For both dynamics (2) and (3) it is dicult for a general potential V to obtain sharp theoretical bounds on the rates of **convergence** (see [13] for consideration on this matter in the metastable case, namely the regime " ! 0 with the potential V " = 1 " V where V has several **local**
minima). A particular simple situations is the case where V is quadratic or, in other words, is a Gaussian measure. Of course MCMC algorithms are not really relevant in practice regarding sampling according to Gaussian measures, but then the exact rates of **convergence** for (2) and (3) are trackable (see [2, 12] and below).

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Formed as a historian and canonist, the young lay theologian began taking interest in dogmatics: an interest awakened in part by his study of Ibas of Edessa and in part by his contact with Father Bulgakov. 38 It seems that his studies in philosophy, history, canon law, New Testament, dogmatics, etc. transformed him from a canonist into an ecclesiologist and his lessons in canon law—that dry subject that he was asked to teach—into lessons of ecclesiology. This transformation is visible in the two articles he wrote between 1931 and 1932: “The Two conceptions of the universal Church,” 39 “The Canons and Canonical Conscience.” 40 The publication of the first of these articles marked a landmark in the theological career of Afanasiev. In this article, originally prepared as an exposition for Father Bulgakov’s ‘Seminar,’ 41 we find the first sketches of his Eucharistic Ecclesiology. Distinguishing between the two conceptions of the universal Church in vigour—that of Rome (a juridical universality around the successor of Peter) and that of Constantinople (ecumenical universality)—the author states that in both these concepts the catholicity of the Church is understood in quantitative terms, whereas in reality it is a qualitative reality which has its foundation in the Eucharist. As he saw it, the unity of the Church is the unity of the **local** Churches, which are united to one another through their communion to the one and unique Table of the Lord. Contrary to the expectations of the author, the article was accorded only a cold reception by his mentor, S. Bulgakov, because the latter did not find any relationship between the Eucharistic Ecclesiology and the doctrine of Sobornost’. Rather discouraged, Afanasiev decided to return to his earlier preoccupations, viz. canonical and historical questions. Thus, during the years preceding the World War II, he launched an ambitious project of writing a work on Ecclesial Councils and Their Origin. But the outbreak of the war interrupted the work.

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able, whereas, the second approach suggests to find a set of output subsequences that have time-invariant behaviors. This makes it possible to derive an **identification** algorithm for these subsequences, which is close to the classical time- invariant algorithm. Among the few attempts on the sub- ject, one can cite Liu [1997] where a discrete time domain state space model, transition matrices and pseudo-modal parameters are used to describe and identify recursively periodic systems, and Verhaegen and Yu [1995] in which subspace model **identification** algorithms that allow the **identification** of an LPTV state space model from a set of input-output measurements are presented.

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i
t 2 , ∀ t 2 > t 1.
Assumption 1 is satisfied by most residential tariffs and feed-in-tariffs (FIT, the price at which prosumers can inject their surplus energy into the grid), and it guarantees that storing energy in the battery for selling it later is never optimal. Assumption 2. Since players have inflexible usage of their appliances, their utility v i is set to be 0 when consuming their desired load profile , and it is set to −∞ if they do not. Fur- thermore, users have quasi-**linear** utilities: u i (z i ) = v i (z i )−p. Because the load will always be satisfied, their utility can be summarized by the negative amount of money that they need to pay for electricity.

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3-LQG.
Such a **convergence** has been established in the case of “pure quantum gravity”, corresponding to uniform planar maps and γ = p 8/3, in the impressive series of papers by Miller and Sheffield [57, 58, 59]. Obtaining such a result for a model of decorated maps outside the pure-gravity class seems out of reach for the moment. However – building on the so-called mating-of-trees approach initiated by Sheffield [61] and which has allowed to obtain various **local** **convergence** results for models of decorated maps (see e.g.[27, 12, 44, 46, 45]) – Gwynne, Holden and Sun [42] managed to prove that for some models of decorated maps, including the spanning-tree decorated maps, bipolar oriented maps and Schnyder wood decorated maps, the volume growth of balls in their **local** limit is given by the “fractal dimension” d γ , for the

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a fraction of the nonzero entries over the total length of the vector φ(θ 3 ).
4.2 **Identification** of the PVs
The second objective is to test the statistical robustness of the **identification** algorithm. For this purpose, we use 100 different independent realizations of the input, the discrete state and the output noise (SNR=30 dB) to generate 100 data sequences of length N = 600 each. The **identification** algorithm (Algorithm 2 indeed) is then run on each of these different 100 data sequences. The user- specified parameters of Algorithm 1 and Algorithm 2 are set to ε = 0.1, Tol = 0.001 and Thres = 0.05. At each run, the first 300 points are used to identify a model and the whole sequence of length 600 is used to validate the estimated model, i.e., to verify its ability to reconstruct the system output from the true input and an estimated discrete state. This is evaluated with the criterion [20]

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Remark 4.1. Obviously, the scheme being nonlinear (which is nearly a require-
ment for monotone methods; see the introduction), the computation of its solution is more costly than for **linear** methods. We will nevertheless see that, in many tests, the number of iterations in the algorithm stays relatively low; it should also be noticed that the **linear** system solved at each iteration comes from an M -matrix that is roughly as well conditioned as the matrices of usual **linear** schemes and is also well adapted to multigrid methods. Finally, we notice that realistic models (see section 4.2.3) are non- **linear** and that, in practice, the “additional” nonlinearity introduced by our nonlinear method increases very little the cost of solving the scheme.

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L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

(4) confidence sets for a = θ − β and any **linear** transformation w ′ a may then be derived by projec-
tion; these confidence sets have level 1 − α ;
(5) confidence sets for σ Vu and w ′ σ Vu can finally be built using the relationship σ Vu = Σ V a . For inference on a, we develop a finite-sample approach which remains valid irrespective of as- sumptions on the distribution of V. In addition, we observe that the test statistics used for inference on β [the AR-type statistic] and θ enjoy invariance properties which allow the application of Monte Carlo test methods: as long as the distribution of the errors u is specified up to an unknown scale parameter, exact tests can be performed on β and θ through a small number of Monte Carlo simula- tions [see Dufour (2006)]. For inference on both regression and covariance endogeneity parameters

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Most of the existing contributions on the subject deal with the class of Piecewise ARX models. These models are defined on a polyhedral partition of the regression space, each submodel being associated to one polyhedron. Hence, the main challenge in the identifica- tion of this class of systems is to determine the right partition of the regressors. Once this task is complete, the estimation of the different submodels follows by means of standard **linear** regression techniques [4]. To find the regions, it is proposed in [1] to group the regressors associated with each mode by performing the K-means clustering algorithm in a special space and then, to compute the parameters of each submodel. The solution in addi- tion to requiring a knowledge of the system order is suboptimal. In a similar framework, [5] uses a statistical clustering approach instead and provides a method to derive the number of submodels from batch data, the orders being assumed to be available a priori. Another category of methods alternates between assigning the data to submodels and estimating simultaneously their parameters by using a weights learning technique [6], solving a Mini- mum Partition into Feasible Subsystems (MIN PFS) problem [7], or resorting to bayesian learning [8]. In [9], the hybrid system **identification** problem has been transformed into a **linear** or quadratic mixed integer programming problem for which there exist efficient tools for solving it in an optimal way. Conversely, this algorithm suffers from a high computa- tional complexity. Another optimal, but deterministic algorithm is the algebraic geometric approach developed in [3]. Under the assumption that the data are perfect (in the sense that they are not corrupted by noise), the authors recast the problem into one of computing and deriving a homogeneous polynomial from which the submodels are deduced without any iteration [10]. For a comprehensive review of hybrid systems **identification** methods, we refer the interested reader to the survey paper [11].

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Figure 2: Example of inverse **identification** with 4 constitutive models
Discussion
The central tendon membrane of porcine clearly showed a non-**linear** behavior with a characteristic “toe-region”, similar to those obtained for many other soft tissue behaviors.

The Hypothesis 1.1 is needed in order to cover the case of large initial conditions, this is compatible with the global ellipticity condition we assumed in [8] and with the ones given in [13, 17]. As already said, this hypothesis is not necessary in the case of small data.
We discuss briefly the strategy of our proof. We begin by performing a para-linearization of the equation `a la Bony [3] with respect to the variables (u, u). Then, in the same spirit of [8], we construct the solutions of our problem by means of a quasi-**linear** iterative scheme `a la Kato [11]. More precisely, starting from the para-linearized system, we build a sequence of **linear** problems which converges to a solution of the para-linearized system and hence to a solution of the original equation (1.1). At each step of the iteration one needs to solve a **linear** para-differential system, in the variable (u, u), with non constant coefficients (see for instance (4.93)). We prove the existence of the solutions of such a problem by providing a priori energy estimates (see Theorem 4.1). In order to do this, we diagonalize the system in order to decouple the dependence on (u, u) T up to order zero. This is done by applying changes of coordinates generated by para-differential operators. Once achieved such a diagonalization we are able to prove energy estimates in an energy-norm, which is equivalent to the Sobolev norm.

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légales de calcul et des statistiques.
Ce n’est que lorsque cette différence est mesurée jusqu’au 31 décembre que l’impact fiscal pourrait être mesuré.
Dans le cas d’une création d’entreprise et d’emplois, par exemple via la création d’une zone d’activités économiques, l’évaluation de l’impact économique et fiscal attendu sera plus complexe et peut être insuffisamment scientifiquement maîtrisable : à l’estimation de la différence (ici, positive et sur une durée annuelle) de salaire, il faudra ajouter un facteur peu évident à cerner : celui de la part des postes qui pourraient être occupés par des personnes habitant le territoire communal ou **local**. Les facteurs qui influencent chez l’employeur l’engagement de personnes domiciliées sur la commune dépendent de facteurs plus ou moins concrets (les personnes en recherche d’emploi ont-elles les capacités requises ?) mais aussi souvent de facteurs moins perceptibles ou plus subjectifs (la stratégie RH de l’entreprise, qui peut délibérément et stratégiquement choisir de recruter ailleurs, la capacité de négociation de l’élu **local**…).

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an operator means identifying one of its characteristic quantities. As the operator to be identiﬁed is **linear**, a convenient and rather general approach consists in working in the frequency domain where any causal operator can be well-deﬁned by its symbol H(iω), that is the Fourier transform of its impulse response. Then, the problem of identifying H(iω) can be classically solved from physical measurements by means of Fourier techniques. Note how- ever that purely frequency identiﬁcation presents some well-known shortcomings. In particular, the so-identiﬁed symbol H(iω) is in general ill adapted to the construction of eﬃcient time-realizations. This is partly due to exces- sive numerical cost of quadrature approximations resulting from the intrinsic convolution nature of the associated operator, sometimes with long memory (Rumeau et al. [2006]) or even delay-like behaviors (Montseny [2007]). Another shortcoming is that frequency methods are incom- patible with real-time identiﬁcation (and so with pursuit when the symbol has the ability to evolve slowly). But above all, the number of unknown parameters is excessive, what makes the problem excessively sensitive to measure- ment noises.

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