Patrick J C Tardivel
email: tardivel@math.uni.wroc.pl

Ma Lgorzata Bogdan

On the sign recovery by LASSO, thresholded LASSO and thresholded Basis Pursuit Denoising

Keywords: Multiple regression, Basis Pursuit, LASSO, Sparsity, Active set estimation, Sign estimation, Identifiability condition, Irrepresentability condition

Basis Pursuit (BP), Basis Pursuit DeNoising (BPDN), and LASSO are popular methods for identifying important predictors in the high-dimensional linear regression model Y = Xβ + ε. By definition, when ε = 0, BP uniquely recovers β when Xβ = Xb and β = b implies b 1 > β 1 (identifiability condition).

Furthermore, LASSO can recover the sign of β only under a much stronger irrepresentability condition.

Meanwhile, it is known that the model selection properties of LASSO can be improved by hard-thresholding its estimates. This article supports these findings by proving that thresholded LASSO, thresholded BPDN and thresholded BP recover the sign of β in both the noisy and noiseless cases if and only if β is identifiable and large enough. In particular, if X has iid Gaussian entries and the number of predictors grows linearly with the sample size, then these thresholded estimators can recover the sign of β when the signal sparsity is asymptotically below the Donoho-Tanner transition curve. This is in contrast to the regular LASSO, which asymptotically, recovers the sign of β only when the signal sparsity tends to 0. Numerical experiments show that the identifiability condition, unlike the irrepresentability condition, does not seem to be affected by the structure of the correlations in the X matrix.

Introduction

Let us consider the high dimensional linear model

Y = Xβ + ε, (1)
where X = (X 1 | . . . |X p ) is a n × p design matrix with n ≤ p, ε is a random vector in R n , and β ∈ R p is an unknown vector of regression coefficients. The sign vector of β is S(β) = (S(β 1 ), . . . S(β p )) ∈ {-1, 0, 1} p , where for x ∈ R, S(x) = 1 x>0 -1 x<0 . Our main goal is to recover S(β). This goal is somewhat more general than the goal of recovering the active set, supp(β) := {i ∈ {1, . . . , p} | β i = 0}. Indeed, given S(β) one may recover supp(β) since supp(β) := {i ∈ {1, . . . , p} | S(β i ) = 0} but conversely, given supp(β), one cannot recover S(β).

Moreover, some theoretical works recommend sign recovery instead of active set recovery [START_REF] Gelman | Type s error rates for classical and bayesian single and multiple comparison procedures[END_REF][START_REF] Stephens | False discovery rates: a new deal[END_REF].

BP, BPDN and LASSO

First introduced in signal processing [START_REF] Chen | Basis pursuit[END_REF], BP estimator is a solution of the following optimization problem βBP := argmin b 1 subject to Y = Xb, which can be solved using linear programming. In the noiseless case, when ε = 0 and Y = Xβ, we say that β is identifiable with respect to X and the 1 norm if βBP uniquely recovers β i.e. if b = β, Xb = Xβ implies b 1 > β 1 . Note that a geometric characterization of the identifiability condition, depending on X and the sign of β, is given in [START_REF] Schneider | The geometry of uniqueness and model selection of penalized estimators including slope, lasso, and basis pursuit[END_REF]. This condition implies that β is sparse. Indeed, Lemma 3 in Tardivel et al. [START_REF] Patrick | Sparsest representations and approximations of an underdetermined linear system[END_REF] shows that card{i ∈ {1, . . . , p} | βBP i = 0} ≤ n i.e. βBP has at least p -n zeros. Consequently having k ≤ n,

where k is the number of non-zero elements in β, is necessary for the identifiability condition. On the other hand, some assumptions on the sparsity of β guarantee that β is identifiable with respect to the 1 norm. For example, the identifiability condition holds if the number of non-zero elements of β satisfies the mutual coherence condition [START_REF] David | Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization[END_REF][START_REF] Foucart | A mathematical introduction to compressive sensing[END_REF][START_REF] Gribonval | Sparse representations in unions of bases[END_REF]. In the special case where the elements of X are iid normal variables, the transition curve ρ : (0, 1) → (0, 1) [START_REF] Donoho | Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing[END_REF] characterizes the identifiability condition with respect to the signal sparsity ξ = k/n. In particular, it is known that the identifiability condition is satisfied with probability converging to 1 if ξ < ρ(δ)

or converging to 0 if ξ > ρ(δ).

In the case where ε has a continuous distribution, almost surely BP has n non-zero components and thus BP cannot recover the sign of β once k = n. To account for noise, when ε has iid N (0, σ 2 ) entries for some σ > 0, Chen and Donoho [START_REF] Chen | Basis pursuit[END_REF] extended BP by proposing the following estimation algorithm:

β L := argmin b∈R p 1 2 Y -Xb 2 2 + λ b 1 with λ = σ 2 log p. (2)
In the seminal work of Tibshirani [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF], the algorithm was extended for fitting the general multiple regression models where different values of λ are often more appropriate. The method gained great popularity in the statistical community under the name LASSO (Least Absolute Shrinkage and Selection Operator), while in the signal processing community it is often called Basis Pursuit Denoising (BPDN). In this paper, we will use the term BPDN for a slightly different form of this estimator, where association with BP is even clearer

β BPDN := argmin b∈R p b 1 subject to Y -Xb 2 2 ≤ R, where R ≥ 0. (3)
Note that X is fixed but Y is random, and that for a given λ > 0 for LASSO we cannot choose a fixed R > 0 for BPDN under which the two estimators are almost surely equal. To illustrate this fact, we note that β BPDN = 0 if and only if Y 2 2 ≤ R, while for LASSO β L = 0 if and only if X Y ∞ ≤ λ.

Sign recovery by LASSO

Properties of the LASSO sign estimator S( β L (λ)) := S( β L 1 (λ)), . . . , S( β L p (λ)) (or properties of the active set estimator supp( β L (λ)) := {i ∈ {1, . . . , p} | β L i (λ) = 0}) have been intensively studied [START_REF] Dossal | Sharp support recovery from noisy random measurements by 1 -minimization[END_REF][START_REF] Lounici | Sup-norm convergence rate and sign concentration property of lasso and dantzig estimators[END_REF][START_REF] Meinshausen | High-dimensional graphs and variable selection with the lasso[END_REF][START_REF] Martin | Sharp thresholds for high-dimensional and noisy sparsity recovery using constrained quadratic programming (lasso)[END_REF][START_REF] Zhao | On model selection consistency of lasso[END_REF][START_REF] Zou | The adaptive lasso and its oracle properties[END_REF].

According to Theorem 2 of Wainwright [START_REF] Martin | Sharp thresholds for high-dimensional and noisy sparsity recovery using constrained quadratic programming (lasso)[END_REF], the irrepresentability condition is necessary for LASSO to recover S(β) with high probability. Indeed, if ker(X I ) = {0}, X I X I (X I X I ) -1 S(β I ) ∞ > 1 and both ε and -ε have the same distribution, then for any choice of the tuning parameter λ > 0, P(S( β L (λ)) = S(β)) ≤ 1/2.

This result also holds for the noiseless case, in which LASSO does not recover S(β). Moreover, Bühlmann and van de Geer [START_REF] Bühlmann | Statistics for High-Dimensional Data: Methods, Theory and Applications[END_REF] (pp. [192][193][194] have shown that if ε = 0 and the irrepresentability holds strictly (i.e. if X I X I (X I X I ) -1 S(β I ) ∞ < 1) then the nonrandom set supp( β L (λ)) is equal to supp(β) as long as the non-zero components of β are sufficiently large. The proof given in [START_REF] Bühlmann | Statistics for High-Dimensional Data: Methods, Theory and Applications[END_REF] can be easily adapted for sign recovery.

Proposition 1, proved in the Appendix, shows that the identifiability condition is weaker than the irrepresentability condition.

Proposition 1 Let X be an n × p matrix with p ≥ n columns in general position. Moreover, let β ∈ R p , I := supp(β) and suppose that ker(X I ) = {0}. If the irrepresentability condition holds, then the parameter β is identifiable with respect to the 1 norm.

For the case where the elements of X are iid random variables from the normal distribution, the irrepresentability condition is satisfied with high probability if and only if k < n 2 log p (see [START_REF] Martin | Sharp thresholds for high-dimensional and noisy sparsity recovery using constrained quadratic programming (lasso)[END_REF]). This implies that LASSO cannot recover S(β) when k/n → ξ > 0, even if ξ < ρ(δ).

Sign recovery by thresholded LASSO

It is well known that LASSO can consistently estimate β under much weaker assumptions than the irrepresentability condition (see, e.g., [START_REF] Meinshausen | Lasso-type recovery of sparse representations for high-dimensional data[END_REF] or [START_REF] Van De | On the conditions used to prove oracle results for the lasso[END_REF]). This suggests that an appropriately thresholded version of LASSO can recover S(β) under weaker assumptions than the irrepresentability condition [START_REF] Pokarowski | Improving lasso for model selection and prediction[END_REF]. Concerning sign recovery by thresholded BP, first theoretical results were given by Saligrama and Zhao [START_REF] Saligrama | Thresholded basis pursuit: Lp algorithm for order-wise optimal support recovery for sparse and approximately sparse signals from noisy random measurements[END_REF]. More recently, Descloux and Sardy [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF] show that the stable null space property is a sufficient condition to recover the sign of β by thresholded BP.

To our knowledge, the stable null space property has so far been the weakest sufficient condition known to guarantee the recovery of S(β). In Theorem 1, we show that "identifiability", a weaker condition than the stable null space property, is sufficient to recover S(β) by thresholded BP, thresholded BPDN or thresholded LASSO when the non-zero elements of β are large enough. We also show that "identifiability" is a necessary condition for the recovery of S(β) by these methods, regardless of the signal magnitude.

Theorem 1 implies that in the linear sparsity regime of [START_REF] Donoho | Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing[END_REF] for Gaussian matrices, thresholded BP, thresholded BPDN and thresholded LASSO recover S(β) if, asymptotically, k < nρ(n/p) holds and the signal magnitude tends towards infinity. Table 1 summarizes known conditions for sign or support recovery by LASSO and related methods, when ε = 0 and the signal is large enough.

Method

Graphical illustration of the main result

By definition, the irrepresentability condition depends on β by S(β). Moreover, as asserted in Proposition 2, the identifiability condition similarly depends only on S(β). Thus, the comparison of these two conditions can be performed by considering parameter vectors such that β = S(β). In Figure 1, we plot the irrepresentability and identifiability curves representing the proportion of sign vectors for which the identifiability condition or the irrepresentability condition is satisfied among k-sparse sign vectors. For each value of k, this proportion was estimated by generating 1000 sign vectors from a uniform distribution on the set {u ∈ {-1, 0, 1} p | card(supp(u)) = k}.

Figure 1 shows a large potential for improvement in sign detection by thresholded LASSO compared to the base LASSO. If the columns of the design matrix are independent Gaussian vectors, then the irrepresentability condition is satisfied with high probability only if k ≤ 10, which is also a limiting sparsity for the recovery of S(β) by LASSO. Instead, the identifiability condition, which provides the limiting sparsity for the recovery of S(β) by the thresholded LASSO, holds as long as k ≤ 30. The simulations are in strong agreement with the asymptotic results in [START_REF] Martin | Sharp thresholds for high-dimensional and noisy sparsity recovery using constrained quadratic programming (lasso)[END_REF] and [START_REF] David | Precise undersampling theorems[END_REF], which predict the upper bounds for k such that the irrepresentability and identifiability conditions hold for Gaussian design matrices with independent entries.

The difference between identifiability and irrepresentability curves becomes even more apparent when there are strong correlations between different columns in the design matrix. As expected, the irrepresentability curve shrinks towards zero. Instead, and rather unexpectedly, the simulated identifiability curve remains intact. Figure 1: Simulated identifiability and irrepresentability curves for two specific design matrices of dimensions 100 × 300. In both cases the rows of X were generated as iid random vectors from the multivariate normal distribution N (0, Σ), with Σ = I in the left panel and Σ in the second panel having a compound symmetry form with Σ ii = 1 and Σ ij = 0.9 for i = j. The x-axis represents the sparsity k and the y-axis the fraction of sign vectors that satisfy the identifiability condition (respectively, irrepresentability condition). In the left panel, the vertical lines represent the values k = 100 2 log(300) and k = 100 × ρ(100/300). These values correspond to the asymptotic upper bounds for k so the irrepresentability and identifiability conditions hold for Gaussian design matrices with independent entries.

Organization of the article

In Section 2, Theorem 1 shows that identifiability is a necessary and sufficient condition for LASSO to separate the non-zero components of β from the noise and to recover asymptotically S(β) with thresholded LASSO and thresholded BPDN. Corollary 1 shows that under the asymptotic linear sparsity regime for Gaussian matrices thresholded LASSO and thresholded BPDN can recover the sign of β when asymptotically k < nρ(n/p).

In Section 3, Proposition 2 shows that the identifiability condition depends only on S(β) and not on the size of the non-zero components of β. Here we also introduce the irrepresentability and identifiability curves, which respectively give the fraction of k-sparse sign vectors satisfying the irrepresentability condition and the identifiability condition. Section 4 is devoted to numerical experiments showing that sign estimators derived from the thresholded LASSO and the thresholded BPDN are better than sign estimators derived from LASSO and the adaptive LASSO, and that the knockoff method allows appropriate threshold selection for both methods.

The proofs are in the Appendix. In this section, we also formulate Proposition 3 providing a tight upper bound on the probability of recovering S(β) through LASSO.

Notation and assumptions

In this article we always assume that the design matrix X has columns in general position1 (see, e.g., [START_REF] Ryan | The lasso problem and uniqueness[END_REF] or the supplementary material for this manuscript). This assumption guarantees that the minimizer of (2) (resp. minimizer of (3)) is unique and that therefore the LASSO estimator (resp. BPDN estimator) is well-defined.

This assumption is very weak and generically holds. Indeed, when X is a random matrix such that the entries (X 11 , X 12 , . . . , X np ) have a density on R np then, almost surely, X is in general position [START_REF] Ryan | The lasso problem and uniqueness[END_REF].

Notation used in the following sections is as follows:

• Let I be a subset of {1, . . . , p}. We denote by I the complement of I, namely I := {1, . . . , p} \ I.

• The notation X I stands for a matrix whose columns are indexed by the elements of I: (X i ) i∈I . • LASSO and BPDN estimators depend on X, β, ε and the tuning parameter λ > 0 or the regularization parameter R ≥ 0. When appropriate, we use the parentheses to indicate these dependencies. The estimator β indiscriminately represents the LASSO estimator or the BPDN estimator.

To formulate our asymptotic results, we will often consider a sequence of regression parameters β (r) ∈ R p , r ∈ N, for which non-zero components tend to infinity in the following way.

Assumption 1

1) The sign of β (r) is invariant, namely there exists a sign vector s 0 ∈ {-1, 0, 1} p such that for any r ∈ N, S(β (r) ) = s 0 .

2) The following limit holds lim r→+∞ min{|β (r) i |, i ∈ supp(s 0 )} = +∞.

3) There exists q > 0 such that

∀r ∈ N, min{|β (r) i |, i ∈ supp(s 0 )} β (r) ∞ ≥ q.
2 Identifiability is a necessary and sufficient condition for the sign recovery If β does not satisfy the irrepresentability condition, then the LASSO sign estimator S( β L (λ)) is very unlikely to recover S(β). However, one may relax this condition. In fact, Theorem 1 shows that an appropriately thresholded version of LASSO (or thresholded version of BPDN) recovers S(β) if only the non-zero elements of β are sufficiently large and the identifiability condition is satisfied.

Theorem 1 Let X be a n × p matrix in general position and such that rank(X) = n and let β be the LASSO or BPDN estimator with any fixed value of the tuning parameter λ > 0 or with any fixed regularization parameter Necessary condition for sign recovery: If S(β) is unidentifiable with respect to the 1 norm, then the sign estimator derived from the thresholded LASSO or thresholded BPDN cannot recover S(β). Indeed, for each fixed ε ∈ R n , the sign of at least one non-zero component of β is not correctly estimated by β(ε):

∃i ∈ supp(β) such that β i (ε)β i ≤ 0.
Sufficient condition for sign recovery: Let β (r) be a sequence in R p satisfying Assumption 1. If s 0 is identifiable with respect to the 1 norm, then for any fixed ε ∈ R n and sufficiently large r > r 0 (ε) the estimator

β(ε, r) separates negative components of β (r) (i.e i ∈ supp -(β (r) )), zero components of β (r) (i.e i / ∈ supp(β (r) ))
and positive components of

β (r) (i.e i ∈ supp + (β (r) )): i) ∀i ∈ supp(β (r) ), β i (ε, r)β (r) i > 0.
ii)

max i∈supp -(β (r) ) β i (ε, r) < min i / ∈supp(β (r) ) β i (ε, r) ≤ max i / ∈supp(β (r) ) β i (ε, r) < min i∈supp + (β (r) ) β i (ε, r) .
Let us note that the assumptions about X are very weak and hold in general if n ≤ p. The assumption that rank(X) = n ensures that the BPDN estimator is well-defined for any R ≥ 0. The general position condition ensures uniqueness of both the LASSO and the BPDN estimator (see, e.g., Proposition 1 in the Supplementary Material).

Theorem 1 emphasises that one cannot recover S(β) with a sign estimator derived from LASSO or BPDN if β is not identifiable with respect to the 1 norm. If β is identifiable with respect to the 1 norm, Theorem 1 implies that S(β) can be recovered by deriving sign estimators from the thresholded LASSO or thresholded BPDN. In Section 4, we show how the appropriate thresholds can be obtained using control variables constructed by the knockoff method (see, e.g., [START_REF] Barber | Controlling the false discovery rate via knockoffs[END_REF][START_REF] Candes | Panning for gold: Model-x knockoffs for high-dimensional controlled variable selection[END_REF]).

In the asymptotic linear sparsity regime for Gaussian matrices, the transition curve ρ(•) described in [START_REF] Donoho | Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing[END_REF] allows us to characterize the identifiability condition, and so Theorem 1 yields the Corollary 1.

Corollary 1 Let X be a n×p n standard Gaussian matrix and let k n denote the number of non-zero components

of β (n) ∈ R pn .
Necessary condition for sign recovery: If n/p n → δ ∈ (0, 1), k n /n → ξ ∈ (0, 1) and ξ > ρ(δ) then asymptotically the sign of at least one non-zero component of β (n) is incorrectly estimated by β:

lim n→+∞ P X,ε ∃i ∈ supp(β (n) ) such that β i β (n) i ≤ 0 = 1.
Sufficient condition for sign recovery: Given n, let (β (n,r) ) r∈N be a sequence satisfying the Assumption 1 (where q does not depend on n). If n/p n → δ ∈ (0, 1), k n /n → ξ ∈ (0, 1) and ξ < ρ(δ) then the estimator β asymptotically separates the negative components of β (n,r) , zero components of β (n,r) and positive components of β (n,r) : i)

lim n→+∞ lim r→+∞ P X,ε ∀i ∈ supp(β (n,r) ), β i β (n,r) i > 0 = 1.
ii)

lim n→+∞ lim r→+∞ P X,ε max i∈supp -(β (n,r) ) β i < min i / ∈supp(β (n,r) ) β i ≤ max i / ∈supp(β (n,r) ) β i < min i∈supp + (β (n,r) ) β i = 1.

Identifiability and irrepresentability curves

Given a certain design matrix X, we now define the irrepresentability indicator of the sign vector s.

Sign irrepresentability indicator: Given a certain design matrix X, the sign identifiability indicator is defined as follows.

Φ X IC : s ∈ {-1, 0, 1} p →                1 if s = (0, . . . , 0) 1 if ker(X I ) = {0} and X I X I (X I X I ) -1 s I ∞ ≤
Sign identifiability indicator:

Φ X Idtf : s ∈ {-1, 0, 1} p →        0 if s = argmin b∈R p b 1 subject to Xb = Xs 1 otherwise
.

This identifiability indicator shows whether the sign estimators obtained by thresholded LASSO and thresholded BPDN can recover S(β). Namely, if φ X Idtf (S(β)) = 0 then the thresholded LASSO (respectively, thresholded BPDN) sign estimator fails to recover S(β) even if the non-zero components of β are extremely large.

According to Proposition 2 in the Supplementary Material, β does not satisfy the identifiability condition if the columns

(X i ) i∈supp(β) are not linearly independent. Consequently, if card(supp(β)) > n then φ X IC (S(β)) = φ X Idtf (S(β)) = 0.
Let us give some basic properties and comments about the two indicator functions.

1. Both φ X IC and φ X Idtf are even.

Given Proposition 1, for each

s ∈ {-1, 0, 1} p , Φ X IC (s) ≤ Φ X Idtf (s).
3. The computation of Φ X IC requires only a simple matrix calculus; the computation of Φ X Idtf requires only solving the basis pursuit problem.

Graphs of identifiability and irrepresentability curves

The number of sign vectors is very large (3 p ) and therefore we cannot explicitly specify Φ X Idtf and Φ X IC for each sign vector. Instead, we define the identifiability and irrepresentability curves as the following functions of the sparsity k of the vector β, k = card(supp(β)) ∈ {1, . . . , n}:

• identifiability curve is defined as p X Idtf (k) := E U (Φ X Idtf (U )),
• irrepresentability curve is defined as

p X IC (k) := E U (Φ X IC (U )),
where U is uniformly distributed on {u ∈ {-1, 0, 1} p | card(supp(u)) = k}. for two specific 100×300 matrices (one generated with iid N (0, 1) entries and the other generated with positively correlated N (0, 1) entries). In addition, for the case where the design matrix X has positively correlated entries, we also consider a situation where U is uniformly distributed on {u ∈ {0, 1} p | card(supp(u)) = k}. Specifically, we consider the following setting:

Positively correlated entries and positive components: X is a fixed design matrix with n = 100, p = 300, whose rows are generated by independent draws from the multivariate normal distribution N (0, Γ),

with Γ ii = 1 for i ∈ {1, . . . , p} and Γ ij = 0.9 when i = j (the same matrix as the one used for the right panel in Figure 1). The distribution of the sign vectors is uniform on {u ∈ {0, 1} p | card(supp(u)) = k}.

Figure 2 shows an interesting behavior of the irrepresentability and identifiability curves (k → p X IC+ (k) and k → p X Idtf+ (k)) in the above setting. Here we observe that the irrepresentability condition becomes even more stringent than in the case when the distribution of the elements of the sign vector is symmetric. At the same time, the identifiability condition now becomes much weaker and is satisfied under a much wider range of sparsity levels compared to the identifiability curve in the right panel of Figure 1.

Numerical comparisons of sign estimators

Theorem 1 states that the sign estimators provided by thresholded LASSO or thresholded BPDN can recover S(β) as long as the identifiability condition is satisfied. Another way to recover S(β) is to use a sign estimator provided by the adaptive LASSO proposed in [START_REF] Zou | The adaptive lasso and its oracle properties[END_REF]. Indeed, as claimed in [START_REF] Zou | The adaptive lasso and its oracle properties[END_REF] or [START_REF] Huang | Adaptive lasso for sparse high-dimensional regression models[END_REF], if the weights for adaptive LASSO are based on an accurate estimator for β one obtains a sign estimator that is consistent for S(β) under much weaker assumptions than the irrepresentability condition. In this section, we numerically compare the sign estimators obtained by LASSO, thresholded LASSO, thresholded BP and adaptive LASSO. We also include a comparison with LASSO-zero [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF], which is a modification of the thresholded BP aimed at controlling the number of false discoveries.

Selection of the tuning parameter

As explained in [START_REF] Weinstein | A power analysis for knockoffs with the lasso coefficient-difference statistic[END_REF][START_REF] Wang | Which bridge estimator is the best for variable selection ? arxiv[END_REF], a value of the optimal tuning parameter for sign recovery by thresholded LASSO is much smaller than the optimal value of the tuning parameter for LASSO sign estimator. Specifically:

• For the LASSO sign estimator, the tuning parameter must be large enough to prevent the inclusion of false detections.

• For the thresholded LASSO sign estimator, the tuning parameter should be chosen to achieve good separation between zero and non-zero elements of β. This requires good estimation rather than good selection properties. Here, the tuning parameter does not need to be large, as the threshold allows false detections to be eliminated.

The tuning parameter for the LASSO sign estimator

If S(β) satisfies the irrepresentability condition, then Proposition 3 in the Appendix implies that one can choose a tuning parameter λ L such that, for sufficiently large β, P(S (β(λ L )) = S(β)) is arbitrarily close to a given value from (0, 1) (e.g. 0.95). According to the irrepresentability curve in Figure 1, the irrepresentability condition is satisfied with probability close to 1 if the elements of X are iid N (0, 1) random variables and β contains k = 5 non-zero elements. Therefore, in this setting, we choose λ L such that the probability of sign recovery converges to 0.95 and the probability of at least one false discovery (Family Wise Error Rate, FWER) converges to 0.05 when the signal magnitude tends to infinity. Since the irrepresentability condition is not satisfied for the remaining scenarios in our simulation study and thus the FWER cannot be controlled at a low level, the performances of LASSO are not reported.

The tuning parameter for the thresholded LASSO sign estimator

The tuning parameter can be chosen using the asymptotic theory of the Approximate Message Passing (AMP) algorithm for LASSO provided, for example, in [START_REF] Bayati | The LASSO risk for Gaussian matrices[END_REF][START_REF] Weijie | False discoveries occur early on the lasso path[END_REF][START_REF] Weinstein | A power analysis for knockoffs with the lasso coefficient-difference statistic[END_REF]. In this theory, the elements of the design matrix are iid Gaussian N (0, 1) variables and the components of β are iid random variables from the mixture distribution Π = (1 -γ)δ 0 + γΠ , where δ 0 is a point mass distribution concentrated at 0 and Π is an arbitrary fixed distribution. The asymptotic mean squared error of LASSO is derived under the assumption that the number of observations n and the number of explanatory variables p tend to infinity and that n/p tends to δ > 0. Then the tuning parameter λ AM P is chosen such that the asymptotic mean squared error is minimal (see, e.g., the prescription in [START_REF] Weinstein | A power analysis for knockoffs with the lasso coefficient-difference statistic[END_REF][START_REF] Wang | Which bridge estimator is the best for variable selection ? arxiv[END_REF]). As discussed in [START_REF] Weinstein | A power analysis for knockoffs with the lasso coefficient-difference statistic[END_REF][START_REF] Wang | Which bridge estimator is the best for variable selection ? arxiv[END_REF], such a tuning parameter for any fixed value of Type I error allows maximizing the asymptotic power of the LASSO. In our simulation study, we used this asymptotically optimal λ AM P (k, t) for the parameter values: δ = n/p = 100/300, γ = k/p = k/300 and Π = 1/2δ t + 1/2δ -t , where δ t is a point mass distribution at t. The values λ AM P (k, t) were then used for the simulations in the independent case. In the "correlated" case, these values proved to be sub-optimal, so in this case we additionally obtained results for tuning parameters equal to 0.5λ AM P (k, t), which provide much better empirical performance.

Selection of the threshold

We define the thresholded LASSO estimator (or the thresholded BP estimator) to be ∀i ∈ {1, . . . , p},

β τ i := β i 1 {| βi|>τ } . ( 4
)
Given a threshold τ > 0, we now define FWER as

FWER(τ ) := P ∃i / ∈ supp(β), β τ i = 0 .
By taking τ α as the 1 -α quantile of the distribution of max | β i |, i / ∈ supp(β) we would control FWER exactly at the α level. However, τ α cannot be readily determined since β is not known.

To obtain a threshold greater than τ α (and thus control for FWER at the α level), it seems appealing to consider the distribution of the supremum norm of the LASSO estimator (or BP estimator) in the complete null model when β = 0 [START_REF] Giacobino | Quantile universal threshold[END_REF]. For the BP estimator, Descloux and Sardy [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF] propose the threshold τ fn α defined as the 1 -α quantile of max β fn 1 , . . . , β fn p where β fn is the following estimator

β fn := argmin β 1 subject to Xβ = ε, where ε ∼ N n (0, σ 2 I).
Unfortunately, if the vector β contains some non-zero elements, this intuitive method yields a threshold τ fn α smaller than τ α and thus FWER(τ fn α ) > α (see also Su et al. [START_REF] Weijie | False discoveries occur early on the lasso path[END_REF] for further explanation).

The recently developed knockoff methodology [START_REF] Barber | Controlling the false discovery rate via knockoffs[END_REF][START_REF] Candes | Panning for gold: Model-x knockoffs for high-dimensional controlled variable selection[END_REF] provides control of False Discovery Rate (FDR). This control is achieved by adding additional control variables to the design matrix. Originally designed to control the FDR, the control variables also allow us to approximate the distribution of estimators corresponding to the zero components of β. In this numerical study, we informally use the model-free knockoffs proposed in [START_REF] Candes | Panning for gold: Model-x knockoffs for high-dimensional controlled variable selection[END_REF] to approximate a threshold that controls the FWER at a certain level. The approach developed below is suitable for the situation when X is a random matrix whose distribution is invariant to the permutation of the columns.

In this setting, we can generate the knockoff variables one by one instead of generating the full n × p knockoff matrix (see Weinstein et al. [START_REF] Weinstein | A power and prediction analysis for knockoffs with lasso statistics[END_REF] for a similar approach). Since adding the controlled variables may change some relevant properties (e.g. the identifiability condition for β), we should ideally add only one knockoff variable at a time when computing LASSO estimates. However, this would lead to a heavy computational burden on the procedure for estimating the relevant threshold. Therefore, in our simulation study, we use model-free knockoffs [START_REF] Candes | Panning for gold: Model-x knockoffs for high-dimensional controlled variable selection[END_REF][START_REF] Weinstein | A power and prediction analysis for knockoffs with lasso statistics[END_REF] to generate 30 = p/10 controlled variables. Then Lasso or BP is run on the matrix augmented with these additional columns and the maximum of the absolute values of the regression coefficients over 30 controlled variables is stored. This step is repeated 10 times and the total maximum of p = 300 absolute values of regression coefficients over controlled variables is calculated. The whole procedure is repeated many times (1000 in this case) and the 0.95-quantile of the obtained maxima is used as a threshold to identify zero components of β. To be consistent with the setup of the simulations used to derive the irrepresentability and identifiability curves, we used the same two fixed design matrices X as in Figure 1, while the positions of the k sparse signals and the error terms were randomly generated for each of the 1000 replicates.

LASSO and Adaptive LASSO

In our numerical experiments, we chose the following values of tuning parameters for LASSO and adaptive LASSO:

• For LASSO, we chose λ L = 81.18 to control for FWER at the 0.05 level when k = 5 and the covariates are independent.

• For adaptive LASSO, the weights are derived using initial estimates β L (λ AM P ), where the tuning parameter is chosen according to the AMP theory, described above. For i ∈ {1, . . . , p}, the weights w i are defined as

w i := 1/(| β i L (λ AM P )| + 10 -7
). With these weights and the tuning parameter λ L described above, the adaptive LASSO has the following expression

β adapt := argmin β∈R p 1 2 Y -Xβ 2 2 + λ L p i=1 w i |β i |. (5)
In all our simulations, LASSO is computed using glmnet R package.

LASSO-zero from [9]

Given iid n × q standard Gaussian matrices G (1) , . . . , G (M ) , we consider the following basis pursuit minimizers:

∀k ∈ {1, . . . , M }, ( β(k) , γ(k) ) := argmin b∈R p ,c∈R q b 1 + c 1 subject to y = Xb + G (k) c.
Now, given a threshold τ > 0, we can define the LASSO-zero estimator βlass0(q,M) τ as follows:

∀i ∈ {1, . . . , p}, βmed i = median{ β(1) i , . . . , β(M) i } and βlass0(q,M)

τ,i = βmed i 1(| βmed i | > τ ).
One can use the knowledge that σ = 1, to compute the QUT threshold τ as the 1 -α quantile of βmed ∞ when β = 0. Otherwise, one can also pivotize the βmed ∞ statistic to compute this threshold (see [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF] for details).

Numerical comparisons

All numerical experiments are performed with the two 100 × 300 design matrices X already used in Figure 1 and in Section 3.1. We set β ∈ R p such that k := card(supp(β)) with k ∈ {5, 20} and the k elements of supp(β) are drawn without replacement from {1, . . . , p}. The non-zero components of β are sampled from the distribution P (β i = t) = P (β i = -t) = 0.5 with t > 0. Additionally, when the columns of X are correlated, we consider the constellation in which all non-zero coefficients are equal to t. In all simulations, the error term is generated as

ε ∼ N (0, Id n ).
Figures 34show the comparison between the following sign estimators.

• The sign estimator L from LASSO with λ = λ L .

• The sign estimator aL from the adaptive LASSO, described in (5).

• The sign estimator BPk from the thresholded BP, where the threshold is given by the "knockoff" method described above.

• The sign estimator Lk from the thresholded LASSO with λ = λ AM P and with a threshold given by the "knockoff " method described above.

• The sign estimator Lks from the thresholded LASSO with λ = 0.5λ AM P and with a threshold given by the "knockoff " method described above.

• The sign estimator Lz of LASSO-zero, where M = 30, q = n. The QUT threshold is not data driven and is computed as the 0.95 quantile of βmed ∞ when β = 0.

• The sign estimator Lzp of LASSO-zero when M = 30, q = n. The QUT threshold is data-driven and is computed by pivotizing the statistic βmed ∞ , as explained in [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF].

We give the curves representing the following statistical properties as a function of t > 0:

• Probability is the proportion of 1000 replicates for which the sign is recovered.

• FWER is the proportion of 1000 replicates for which at least one zero component of β is not estimated to be zero.

FWER is directly related to the probability of sign recovery, namely F W ER ≥ α implies that the probability of sign recovery is less than 1 -α. The sign estimators L, BPk, Lk and Lks were specifically designed to control for FWER at the 0.05 level.

If k = 5 and the elements of X are iid normal variables, then the irrepresentability condition holds and LASSO can recover the true model. Figure 3 shows that in this case the upper bound on the probability of recovering the sign given in Proposition 3 is achieved by LASSO and the FWER is controlled. However, for moderately large signals, the probability of recovering S(β) by LASSO is much smaller than by the thresholded versions of BP and Lasso, as well as by Lasso-zero. If k = 20, the irrepresentability condition is not satisfied, but the identifiability condition holds, thus the thresholded LASSO and the thresholded BP identify the sign of β when non-zero elements are large and the threshold is properly calibrated.

Figure 3 shows that thresholds for BP or Lz, which are not data-driven and are computed in the complete null model (i.e. when β = 0) do not control the FWER when β has many large components (intuitively, when β is far from 0). Consequently, BP or Lz cannot recover S(β) with a large probability. Instead, our heuristic application of the knockoff method, as well as LASSO-zero when the threshold is data-driven, allow us to almost perfectly control FWER at 0.05. Consequently, when the non-zero components of β are large enough, the sign Figure 4 shows the results of the simulation study for the design matrix X with very highly correlated columns. In this setting, the probability of recovering S(β) by LASSO-zero was very close to zero, and for clarity of presentation, LASSO-zero is not included in Figure 4.

Figure 4 confirms the numerical experiments given in Figure 3. In particular, thresholded BP and thresholded LASSO can recover S(β) when the threshold is well calibrated and the non-zero components of β are large enough. If the tuning parameter is chosen correctly, thresholded LASSO recovers S(β) with a higher probability than thresholded BP. Similar to the case of independent entries, our heuristic knockoff approach allows to control FWER at the 0.05 level.

Numerical experiments with the design matrix from the riboflavin dataset

In this section, we consider the same simulation setting as in [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF], which deals with the riboflavin dataset [START_REF] Bühlmann | High-dimensional statistics with a view toward applications in biology[END_REF]. The design matrix X contains expression levels of p = 4088 genes for n = 71 Bacillus subtilis bacteria. The matrix is mean-centered (for each j ∈ {1, . . . , p}, n i=1 X ij = 0) and standardized (for each j ∈ {1, . . . , p}, The identifiability and irrepresentability curves shown in Figure 5 confirm the numerical experiments reported in Descloux and Sardy [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF]. In particular, the probability of sign recovery by LASSO reported in panel (c) of Figure 6 in [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF] is below the irrepresentability curve from Figure 5 and the probability of sign recovery by LASSO-zero is below the identifiability curve.

1 n-1 n i=1 X 2 ij = 1).
We also ran the simulations comparing LASSO-zero and thresholded LASSO on the riboflavin dataset. In this study, we set k := card(supp(β)) ∈ {1, 2, 3} (so that the identifiability condition holds), and the non-zero coefficients are set to 2 with random signs. In addition to the probability of sign recovery and the FWER, we also estimate the False Discovery Rate (FDR) and the True Positive Rate (TPR).

• FDR is the mean of 1000 replicates of the False Discovery Proportion (FDP), where the FDP is defined as follows:

F DP = card({i / ∈ supp(β)| βi = 0}) max{card(supp( β)), 1} .
• TPR is the mean of 1000 replicates of the True Positive Proportion (TPP), where the TPP is defined as follows:

T P P = card(supp(β) ∩ supp( β)) card(supp(β)) .

For the thresholded LASSO, we selected the tuning parameter by cross-validation. To estimate the threshold, we used the heuristic knockoff procedure described above, where for each run of LASSO we added q = p/8 = 511 columns to the design matrix X, drawn from the multivariate normal distribution N (0, Γ), where Γ ii = 1 and Figure 6 shows that LASSO-zero and thresholded LASSO have FWER above the assumed value of 0.05. This is not surprising since the gene expressions are not exactly normally distributed. However, the FWER of both methods is still quite low (below 0.3) and the FDR is kept close to 0.1. Since the correlations between columns in the design matrix are not too large, the thresholded LASSO and LASSO-zero show similar performance.

Γ ij = 1 p(p-1)(n-1) i =j X i X j = 0.0292.
However, the probability of sign recovery and TPR is systematically larger for thresholded LASSO, which also has smaller FWER and FDR.

The focus of this article is on the theoretical properties of sign estimators derived from LASSO, thresholded LASSO, and thresholded BPDN . When S(β) is identifiable with respect to the 1 norm and when non-zero components of β tends to infinity, we have shown that sign estimators derived from thresholded LASSO and thresholded BPDN recover S(β). On the other hand, if S(β) is not identifiable with respect to the 1 norm, sign estimators derived from thresholded LASSO and thresholded BPDN cannot recover S(β).

We introduced irrepresentability and identifiability curves that provide information about the probability of sign recovery by LASSO or thresholded LASSO and thresholded BPDN as a function of the number of non-zero elements in the vector of regression coefficients.

The performances of the sign estimators derived from LASSO, thresholded LASSO and thresholded BPDN obviously depend on the tuning parameters and the threshold. We have illustrated that AMP theory and the knockoff method are useful to select these parameters. Our simulations show that thresholded LASSO and thresholded BPDN sign estimators outperform the adaptive LASSO and LASSO sign estimators.

Proposition 3 Assume that the assumptions of Lemma 1 hold. Moreover, assume that the sequence (β (r) ) in R p satisfies Assumption 1. If s 0 is identifiable with respect to the 1 -norm, then the following asymptotic results hold.

Sharpness of the upper bound: Asymptotically, the upper bound is obtained.

lim sup r→+∞ P S( β L (λ, r)) = s 0 ≤ P ζ X,λ,s 0 ∞ ≤ 1 , lim inf r→+∞ P S( β L (λ, r)) = s 0 ≥ P ζ X,λ,s 0 ∞ < 1 . Asymptotic control of FWER: Let us define P ζ X,λ,s 0 ∞ < 1 = γ and P ζ X,λ,s 0 ∞ ≤ 1 = γ.
The sign of the non-zero elements of β (r) is correctly identified with probability converging to 1 and the FWER is controlled at the 1 -γ level.

lim r→+∞ P ∀i ∈ I, S( β L i (λ, r)) = s 0 i = 1, lim sup r→+∞ P ∃i / ∈ I, β L i (λ, r) = 0 ≤ 1 -γ, lim inf r→+∞ P ∃i / ∈ I, β L i (λ, r) = 0 ≥ 1 -γ.
Remark 1 The results in Proposition 3 are quite straightforward when X is orthogonal (i.e. when X X = I).

Indeed, in this case the upper bound is simply the probability that zero components of β are simultaneously estimated to 0, namely P(∀i / ∈ supp(β), βL i (λ) = 0)

If ε has a covariance matrix σ 2 Id n , one can obtain asymptotic results by fixing β and letting σ tend to 0.

However, unlike in our asymptotic setting described in Assumption 1, such asymptotic results are very poor and give no information about the FWER. Indeed, when σ tends to 0, the upper bound tends to 0 or 1, depending on whether the irrepresentable condition for β holds or not.

Let us remind that the FWER is equal to P(∃i / ∈ supp(β), β L i = 0). According to Proposition 3, when non-null components of β are infinitely large, the FWER is controlled at level 1 -P ζ X,λ,sign(β) ∞ < 1 . To our knowledge, it is first theoretical result providing a formula for the FWER. Hereafter, let us provide some comments about the FWER control.

• To provide a specific value for λ allowing to control the FWER one needs to know the distribution of ε.

For example, when the distribution of ε is known and β = 0 one controls the FWER at level α by taking λ as the 1 -α quantile of X ε ∞ . Let us point out that when ε 1 , . . . , ε p are iid and the variance σ 2 of these components is unknown then σ 2 can be consistently estimate as explained in [START_REF] Lee | Variance estimation in high-dimensional linear models[END_REF].

• It is easier to control the FWER when X is a random matrix whose distribution is symmetric and invariant by columns permutation than when X is a fixed design matrix. Indeed, when X is random, the distribution of ζ X,λ,S(β) just depends from the sparsity of β and not on S(β).

• Let k ∈ {1, . . . , p} and let U k be the set {u ∈ {-1, 0, 1} p | card(supp(u)) = k} and let us assume that β ∈ U k . When X is a random matrix whose distribution is symmetric and invariant by columns permutation by taking λ α such that P( ζ X,λα,u0 < 1) = 1 -α one controls asymptotically the FWER at level α (where u 0 := (1, . . . , 1, 0, . . . , 0) ∈ U k ). Such a tuning parameter λ α is easy to infer by Monte Carlo simulations.

When X is fixed, by taking λ α as follows

1 card(U k ) u∈U k P( ζ X,λα,u < 1) = 1 -α, (6)
then the average value of the FWER with respect to β ∈ U k is 1 -α (where non-null components of β are infinitely large). Again λ α is easy to infer. In the numerical experiment, the tuning parameter was selected by solving equation ( 6).

Proof of the Proposition 3

First, let us provide lemmas which are useful to prove both Proposition 3 and Theorem 1. Lemma 3 partially proves Proposition 3. Indeed, according to this Lemma, when (β (r) ) r∈N is a sequence of R p satisfying assumptions 1 then the following asymptotic result holds

lim r→+∞ P ∀i ∈ supp(s 0 ), S( β L i (λ, r)) = s 0 i = 1.
Lemma 2 Let (β (r) ) r∈N be a sequence of R p satisfying the conditions 1) and 2) of Assumption 1, let us assume that s 0 is identifiable with respect to the 1 norm and let us set u r = β (r) 1 then

lim r→+∞ β L (ε, r) -β (r) u r = 0.
Proof: Because β L (ε, r) is the LASSO estimator as defined in (2) then the following inequality occurs

1 2 Y -X β L (ε, r) 2 2 + λ β L (ε, r) 1 ≤ 1 2 Y -Xβ (r) 2 2 + λ β (r) 1 .
Since Y -Xβ (r) = ε one may deduce the following inequalities

λ β L (ε, r) 1 ≤ 1 2 ε 2 2 + λ β (r) 1 , ⇒ β L (ε, r)/u r 1 ≤ ε 2 2 2λu r + 1. (7)
Proof: Let ε be a fixed vector in R p . According to the third condition of Assumption 1 we have min{|β

(r) i |, i ∈ supp(s 0 )}/ β (r)
∞ ≥ q > 0, consequently the following inequalities occur ∀i ∈ supp(s 0 ),

s 0 i β L i (ε, λ, r) -β (r) i β (r) ∞ = s 0 i β L i (ε, λ, r) β (r) ∞ - |β (r) i | β (r) ∞ ≤ s 0 i β L i (ε, λ, r) β (r) ∞ -q.
According to Lemma 2, the following inequality occurs

0 = lim inf r→+∞ s 0 i β L i (ε, λ, r) -β (r) i β (r) ∞ ≤ lim inf r→+∞ s 0 i β L i (ε, λ, r) β (r) ∞ -q.
Which implies that for r large enough s 0 i β L i (ε, λ, r) > 0 and thus S( β L i (ε, λ, r)) = s 0 i . When ε is no longer fixed then, for i ∈ supp(s 0 ), almost surely S( β L i (r)) converges to s 0 i and consequently

lim r→+∞ P ∀i ∈ supp(s 0 ), S( β L i (λ, r)) = s 0 i = 1.

Proof of Proposition 3:

Let us remind that the vector β L (λ) is the LASSO estimator if and only if the following two inequalities occur simultaneously.

X A (Y -X β L (λ)) = λS( β L A (λ)), where A = supp( β L (λ)), (9)
X A (Y -X β L (λ)) ∞ ≤ λ. ( 10
)
Sharpness of the upper bound) Since the upper bound depends only on s 0 and not on how large the non-null components β (r) are then

lim sup r→+∞ P S( β L (λ, r)) = s 0 ≤ P ζ X,λ,s 0 ∞ ≤ 1 .
Finally, it must be proven that lim inf r→+∞ P S( β L (λ, r)) = s 0 ≥ P ζ X,λ,s 0 ∞ < 1 . Let us remind that I = supp(s 0 ) and let us assume that the following events hold simultaneously

X I (Y -X β L (λ)) = λs 0 I and X I X I (X I X I ) -1 λs 0 I + X I Id -X I (X I X I ) -1 X I ε ∞ < λ = ζ X,λ,s 0 ∞ <1 . ( 11
)
We aim to show that the inequalities given above imply that β L I (λ) = 0. For convenience, let us set H be the projection matrix H := X I (X I X I ) -1 X I . When [START_REF] Donoho | Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing[END_REF] occurs then the following inequalities holds

X I H(Y -X β L (λ)) + X I (Id -H) ε ∞ < λ, X I H(Y -X β L (λ)) + (Id -H)ε ∞ < λ, X I Y -X β L (λ) + X I β L I (λ) -HX I β L I (λ) ∞ < λ. (12)
Inequality ( 12) comes from the following two identities

HY = H(Xβ (r) ) + Hε = H(X I β (r) I ) + Hε = X I β (r)
I + Hε = X(β (r) ) + Hε and,

HX β L (λ) = HX I β L I (λ) + HX I β L I (λ) = X I β L I (λ) + HX I β L I (λ) = X β L (λ) -X I β L I (λ) + HX I β L I (λ).
Let v be the vector v := X I Y -X β L (λ) + X I β L I (λ) -HX I β L I (λ) . We are going to see that inequality (12) implies that β L I (λ) = 0. Let us assume that β L I (λ) = 0 then, on the one hand, the following inequality occurs

β L I (λ) v ≤ β L I (λ) 1 v ∞ < λ β L I (λ) 1 . (13)
According to [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF] the identity

β L i (λ)X i (Y -X β L (λ)) = λ| β L i (λ)| occurs.
Consequently, on the other hand, the following inequalities hold

β L I (λ) v = β L I (λ) X I Y -X β L (λ) + X I β L I (λ) -HX I β L I (λ) , = λ β L I (λ) 1 + β L I (λ) X I (Id -H)X I β L I (λ), ≥ λ β L I (λ) 1 . (14)
The last inequality occurs because the projection matrix Id -H is positive semi-definite. Inequalities ( 13) and ( 14) provide a contradiction which implies that β L I (λ) = 0. According to [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF], the following implication holds

S( β L I (λ, r)) = s 0 I ⇒ X I (Y -X β L (λ, r)) = λs 0 I .
Because s 0 is identifiable with respect to the 1 norm then, according to Lemma 3, the following convergence in probability occurs

lim r→+∞ P(S( β L I (λ, r)) = s 0 I ) = lim r→+∞ P(X I (Y -X β L (λ, r)) = λs 0 I ) = 1. ( 15
)
Using this asymptotic result and since when [START_REF] Donoho | Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing[END_REF] Asymptotic full power and asymptotic control of the FWER) According to [START_REF] Ewald | On the distribution, model selection properties and uniqueness of the lasso estimator in low and high dimensions[END_REF], asymptotically the power is equal to 1, namely lim r→+∞ P(∀i ∈ I, S( β L i (λ, r)) = s 0 i ) = 1. Now let us prove that the FWER is controlled asymptotically. Let us remind that P ζ X,λ,s 0 ∞ < 1 = γ and P ζ X,λ,s 0 ∞ ≤ 1 = γ. Using asymptotic results given above one may deduce the following inequalities.

γ ≥ lim sup r→+∞ P(S( β L (λ, r)) = s 0 ), ≥ lim sup r→+∞ P ∀i ∈ I, S( β L i (λ, r)) = s 0 i and ∀i / ∈ I, β L i (λ, r) = 0 , ≥ lim sup r→+∞ P(∀i / ∈ I, β L i (λ, r) = 0). ( 16
)
The last inequality comes from [START_REF] Ewald | On the distribution, model selection properties and uniqueness of the lasso estimator in low and high dimensions[END_REF]. Similarly, we have

γ ≤ lim inf r→+∞ P(∀i / ∈ I, β L i (λ, r) = 0). (17)
Consequently, by taking the complement to 1 of the inequalities given in ( 16) and ( 17), one may deduce that

lim inf r→+∞ P(∃i / ∈ I, β L i (λ, r) = 0) ≥ 1 -γ and lim sup r→+∞ P(∃i / ∈ I, β L i (λ, r) = 0) ≤ 1 -γ.

Proof of Theorem 1

Lemma 4 provides the same result for BPDN as does Lemma 2 for LASSO. These both lemmas are the keystones to prove Theorem 1.

Lemma 4 Let (β (r) ) r∈N be a sequence of R p satisfying conditions 1) and 2) of Assumption 1, let us assume that s 0 is identifiable with respect to the 1 norm and let set u r = β (r) 1 then

lim r→+∞ β BPDN (ε, r) -β (r) u r = 0.
and the 1 norm.

Proof: Let us remind that when X is in general position then the minimizer β is unique. Let us assume that β is not identifiable with respect to X and the 1 norm, then there exists b ∈ R p such that Xb = X β and b 1 ≤ β 1 . Consequently, for LASSO, one may deduce that

Y -Xb 2 + λ b 1 ≤ Y -X β L 2 + λ β L 1 .
This inequality contradicts β L as the unique minimizer of (2). Similarly, when β BPDN is not identifiable with respect to the 1 norm then β BPDN is not the unique minimizer of ( 3), which provides a contradiction.

For the proofs of Theorem 1 and the proof of Proposition 2 we need to introduce the following inequality which characterizes the identifiability condition [START_REF] Daubechies | Iteratively reweighted least squares minimization for sparse recovery[END_REF]. A vector b ∈ R p is identifiable with respect to X and the 1 norm if and only if the following inequality holds

∀h ∈ ker(X) \ {0}, i∈supp (b)

S(b)h

i < i / ∈supp(b) |h i |. (20)
Proof of Theorem 1:

Necessary condition: Let us assume that S(β) is not identifiable with respect to the 1 norm. Let us show that when the following events hold

supp -(β) ⊂ supp -( β(ε)) and supp + (β) ⊂ supp + ( β(ε)), (21)
then inequality [START_REF] Huang | Adaptive lasso for sparse high-dimensional regression models[END_REF] occurs which contradicts that S(β) is not identifiable with respect to the 1 norm. Let h ∈ ker(X) \ {0}. On the one hand, when (21) occurs, we have

i∈supp(β) S(β i )h i = - supp -(β) h i + supp + (β) h i , = - i∈supp -( β(ε)) h i + i∈supp -( β(ε))\supp -(β) h i + i∈supp + ( β(ε)) h i - i∈supp + ( β(ε))\supp + (β) h i , ≤ - i∈supp -( β(ε)) h i + i∈supp + ( β(ε)) h i + i∈supp( β(ε))\supp(β) |h i |.
On the other hand, according to Lemma 5, β(ε) is identifiable with respect to the 1 norm then (20) occurs implying the following inequality

- i∈supp -( β(ε)) h i + i∈supp + ( β(ε)) h i + i∈supp( β(ε))\supp(β) |h i | < i / ∈supp( β(ε)) |h i | + i∈supp( β(ε))\supp(β) |h i | = i / ∈supp(β) |h i |.
Consequently the following inequality holds

∀h ∈ ker(X) \ {0}, i∈supp(β) S(β i )h i < i / ∈supp(β) |h i |,
which, according to [START_REF] Huang | Adaptive lasso for sparse high-dimensional regression models[END_REF], contradicts that S(β) is not identifiable with respect to the 1 norm.

Sufficient condition: Let us remind that according to condition 3) of Assumption 1 the following inequality holds ∀r ∈ N, min{|β

i |, i ∈ supp(s 0 )} β (r) ∞ ≥ q > 0. (r)
According to Lemmas 2 and 4, when s 0 is identifiable with respect to the 1 norm then

lim r→+∞ β(ε, r) -β (r) β (r) ∞ = 0.
Therefore, there exists r 0 (ε) ≥ 0 such that

∀r ≥ r 0 (ε), β(ε, r) -β (r) β (r) ∞ ∞ < q/2 ⇔ ∀i ∈ {1, . . . , p}, ∀r ≥ r 0 (ε), β i (ε, r) -β (r) i β (r) ∞ < q/2.
Consequently, when r ≥ r 0 (ε), whatever i / ∈ supp(s 0 ) (thus when β

(r) i = 0) the following inequalities hold ∀i / ∈ supp(s 0 ), β i (ε, r) β (r) ∞ < q/2, ⇒ -β (r) ∞ q/2 < min i / ∈supp(s 0 ) β i (ε, r) ≤ max i / ∈supp(s 0 ) β i (ε, r) < β (r) ∞ q/2. Whatever i ∈ supp + (s 0 ) (thus when β (r) i > 0) the following inequalities hold ∀i ∈ supp + (s 0 ), β i (ε, r) β (r) ∞ ≥ - β i (ε, r) -β (r) i β (r) ∞ + β (r) i β (r) ∞ , ⇒ min i∈supp + (s 0 ) β i (ε, r) β (r) ∞ > -q/2 + q = q/2, ⇒ min i∈supp + (s 0 ) β i (ε, r) > β (r) ∞ q/2. Whatever i ∈ supp -(s 0 ) (thus when β (r) i < 0) the following inequalities hold ∀i ∈ supp + (s 0 ), β i (ε, r) β (r) ∞ ≤ β i (ε, r) -β (r) i β (r) ∞ + β (r) i β (r) ∞ , ⇒ max i∈supp -(s 0 ) β i (ε, r) β (r) ∞ < q/2 -q = -q/2, ⇒ max i∈supp -(s 0 ) β i (ε, r) < -β (r) ∞ q/2.
Finally, when r ≥ r 0 (ε) we have i) β i (ε, r) .

supp -(s 0 ) ⊂ supp -( β i (ε, r)) and supp + (s 0 ) ⊂ supp + ( β i (ε, r)).
These achieve the proof of the sufficient condition.

Proof of propositions

The proof of Proposition 1, provided below, is the one reported in the PhD manuscript of Tardivel [START_REF] Tardivel | Représentation parcimonieuse et procédures de tests multiples: application à la métabolomique[END_REF].

Proof of Proposition 1: From Daubechies et al. [START_REF] Daubechies | Iteratively reweighted least squares minimization for sparse recovery[END_REF], β is a parameter having a minimal 1 norm, namely

Xβ = Xγ ⇒ γ 1 ≥
We are going to show that when the irrepresentable condition holds for β then the inequality (20) holds. Let us notice that when the inequality in the irrepresentable condition is strict, Theorem 1 remains true without assuming that X is in general position.

Proof of

|h i |.
Consequently, according to [START_REF] Huang | Adaptive lasso for sparse high-dimensional regression models[END_REF], parameter b is identifiable with respect to the 1 norm.

Comparisons of conditions for sign recovery and support recovery

Let X = (X 1 | . . . |X p ) ∈ R n×p and β ∈ R p .

• When X 1 2 = • • • = X p 2 = 1, we say that β satisfies the mutual coherence condition once the following inequalities occurs card(supp(β)) < 1 + 1/M (X) 2

, where M (X) = max i =j |X i X j |.

• We say that β satisfies the irrepresentability condition once X I X I (X I X I ) -1 sign(β I ) ∞ < 1, where I = supp(β).

Moreover, we say that the irrepresentability condition uniformly hold on the support I of β once ∀θ ∈ {-1, 1} |I| , X I X I (X I X I ) -1 θ ∞ < 1.

• We say that β satisfies the stable nullspace property once ∀h ∈ ker(X) \ {0}, i∈supp(β)

|h i | < i / ∈supp(β) |h i |.
• We remind that β is identifiable with respect to X and the

=⇒ Stable nullspace Stable nullspace =⇒ Identifiability

Table 2: This figure provides implication between above conditions. The implication "stable nullspace ⇒ identifiability" is straightforward and the implication "Irrepresentability ⇒ Identifiability" is given in Proposition 1. Other implications are proved in [START_REF] Descloux | Model selection with lasso-zero: adding straw to the haystack to better find needles[END_REF][START_REF] Gribonval | Sparse representations in unions of bases[END_REF][START_REF] Sara | High-dimensional generalized linear models and the lasso[END_REF]. Note that an exhaustive scheme is given in [START_REF] Bühlmann | Statistics for High-Dimensional Data: Methods, Theory and Applications[END_REF][START_REF] Sara | High-dimensional generalized linear models and the lasso[END_REF].

One may observe that the mutual coherence is a very strong condition once two columns of X are almost equal (namely when M (X) is close to 1). As an example, M (X) = 0.99 when X is the matrix from the riboflavin dataset and thus the mutual coherence condition is extremely strong. Nevertheless, when β has only one non null component, the mutual coherence condition holds and thus β satisfies both the irrepresentability and identifiability conditions as illustrated on Figure 5.

Finally when X is a n × p standard Gaussian matrix and n < p are both very large then β is identifiable with respect to X and the 1 norm with a very large probability (resp. very small probability) once card(supp(β)) < nρ(n/p) (resp. once card(supp(β)) > nρ(n/p)), where ρ(•) is the transition curve of Donoho and Tanner [START_REF] David | Precise undersampling theorems[END_REF],

presented in Figure 7. already stressed that when β is identifiable with respect to the 1 norm then β is sparse. We show that when the identifiability holds for β then the family (X i ) i∈supp(β) is linearly independent and thus the number of components of β equal to 0 is larger than p -n. Finally, a proof that the stable nullspace property implies the identifiability condition is given

•

  For b ∈ R p , b I denotes the subvector containing elements of b with indices in I. • The symbols supp(b), supp + (b) and supp -(b) denote, respectively, the sets {i ∈ {1, . . . , p} | b i = 0}, {i ∈ {1, . . . , p} | b i > 0} and {i ∈ {1, . . . , p} | b i < 0}.

.Proposition 2

 2 1 where I := supp(s) 0 otherwise This irrepresentability indicator shows whether the LASSO sign estimator can recover S(β). Namely, if φ X IC (S(β)) = 0 then S(β) cannot be recovered using the LASSO sign estimator, even if the non-zero components of β are extremely large. Proposition 2 shows that the identifiability condition also depends only on S(β) and not on the sizes of the non-zero components of β. Consider two vectors b ∈ R p and b ∈ R p such that S(b) = S( b). Then b is identifiable with respect to the matrix X and the 1 norm if and only if b is identifiable with respect to the matrix X and the 1 norm.

Figure 1

 1 in the Introduction provides identifiability curves k ∈ {1, . . . , 60} → p X Idtf (k) and irrepresentability curves k ∈ {1, . . . , 60} → p X IC (k)

Figure 2 :

 2 Figure 2: Graphs of functions k → p X Idtf+ (k) and k → p X IC+ (k) where X is described in the setting above. The vertical lines represent the values k = 100 2 log(300) and k = 100 × ρ(100/300).

Figure 3 :

 3 Figure 3: This figure shows the FWER and the probability of recovering S(β) for each sign estimator and when X contains iid N (0, 1) entries. The graphs on the left show the probability of recovering S(β) (on the y-axis) as a function of t (on the x-axis), where t indicates how large the non-zero components of β are. The graphs on the right show the FWER (on the y-axis) as a function of t (on the x-axis).

Figure 4 :

 4 Figure 4: This figure shows the FWER and the probability of recovering S(β) for each sign estimator when X has strongly correlated columns. The plots on the left show the probability of recovery of S(β) (on the y-axis) as a function of t (on the x-axis), where t measures how large the non-zero components of β are. The graphs on the right show the FWER (on the y-axis) as a function of t.

Figure 5

 5 shows irrepresentability and identifiability curves of this design matrix.

Figure 5 :

 5 Figure 5: Graphs of the functions k → p X Idtf (k) and k → p X IC (k) where k ∈ {1, . . . , 12} and X is the matrix from the riboflavin dataset. The equation of the vertical line is k = nρ(n/p).

Figure 6 :

 6 Figure 6: This figure provides the probability of sign recovery, FWER, FDR and TPR of LASSO-zero (Lzp) and thresholded LASSO (Lk) as a function of the sparsity k ∈ {1, 2, 3}.

  occurs then β L I (λ, r) = 0, one may deduce the following inequalities lim inf r→+∞ P S( β L (λ, r)) = s 0 = lim inf r→+∞ P S( β L I (λ, r)) = s 0 I and β L I (λ, r) = 0 , = lim inf r→+∞ P( β L I (λ, r) = 0), ≥ lim inf r→+∞ P X I (Y -X β L (λ, r)) = λs 0 I and ζ X,λ,s 0 ∞ < 1 , ≥ lim inf r→+∞ P ζ X,λ,s 0 ∞ < 1 .

  s 0 ) β i (ε, r) < min i / ∈supp(s 0 ) β i (ε, r) ≤ max i / ∈supp(s 0 ) β i (ε, r) < min i∈supp + (s 0 )

  Let h ∈ ker(X) and let us remind that h I and h I denote respectively vectors (h i ) i∈I and (h i ) i / ∈I . Then the following equality holds i∈IS(βi )h i = h I S(β I ) = h I X I X I (X I X I ) -1 S(β I ).Because 0 = Xh = X I h I + X I h I , one may deduce the following inequalities|h I S(β I )| = h I X I X I (X I X I ) -1 S(β I ) , ≤ h I 1 X I X I (X I X I ) -1 S(β I ) ∞ .(23)Consequently, when the irrepresentable condition holds for β, namely when X I X I (XI X I ) -1 S(b * I ) ∞ ≤ 1,then the inequality (23) gives |h I S(β I )| ≤ h I 1 . Thus, by the equivalence given in (22), β is a solution of the following basis pursuit problem minimize γ 1 subject to Xγ = Xβ Because X is in general position the previous optimisation problem has a unique solution (see, e.g., Proposition 1 in Appendix) thus Xβ = Xγ and γ = β implies that γ 1 > β 1 , namely β is identifiable with respect to the 1 norm.

Proposition 2 :

 2 Because b is identifiable with respect to the 1 norm and because S( b) = S(b) implies supp( b) = supp(b), then the following inequality holds ∀h ∈ ker(X) \ {0}, i∈supp( b) S( bi )h i < i / ∈supp( b)

Figure 7 :

 7 Figure 7: This figure illustrates the transition curve of Donoho and Tanner [13].

Table 1 :

 1 Some theoretical conditions under which LASSO and related methods can recover the sign or the support of β when ε = 0 and non-zero elements of β are large enough. Additional comments related to these conditions are given in Appendix.

		Conditions	Type	Related articles
	Regular	Mutual coherence	sufficient	[6, 21]
	LASSO	Irrepresentability necessary and sufficient [14, 22, 35, 39, 40]
	Thresholded Stable null space	sufficient	[9]
	LASSO, BP	Identifiability	necessary and sufficient	current article

  β 1 holds if and only if the following inequality occurs ∀h ∈ ker(X),

i∈I

S(β

i )h i ≤ i / ∈I |h i |.

Actually, general position is just a sufficient condition under which, for a fixed X ∈ R n×p , independently of Y ∈ R n the LASSO minimizer is unique. Recently, this condition was relaxed by Ewald and Schneider[START_REF] Ewald | On the distribution, model selection properties and uniqueness of the lasso estimator in low and high dimensions[END_REF] to a geometric criterion which is both sufficient and necessary.

Acknowledgments

We would like to thank Emmanuel J. Candès and Wojciech Rejchel for helpful comments. Ma lgorzata Bogdan's research was supported by NCN grant 2016/23/B/ST1/00454. We gratefully acknowledge the grant from the Wroc law Center of Networking and Supercomputing (WCSS), where most of the computations were performed.

Appendix

Sign recovery with LASSO sign estimator

The upper bound for sign recovery given in Proposition 3 appears in Lemma 3 of Wainwright [START_REF] Martin | Sharp thresholds for high-dimensional and noisy sparsity recovery using constrained quadratic programming (lasso)[END_REF] as a technical result establishing the irrepresentability condition (Theorem 2 in Wainwright [START_REF] Martin | Sharp thresholds for high-dimensional and noisy sparsity recovery using constrained quadratic programming (lasso)[END_REF]). According to Proposition 3, if β is identifiable with respect to the 1 -norm, this upper bound is asymptotically reached as soon as min{|β i |, i ∈ supp(β)} tends to +∞.

Lemma 1 (Lemma 3 from [START_REF] Martin | Sharp thresholds for high-dimensional and noisy sparsity recovery using constrained quadratic programming (lasso)[END_REF] ) Let I := supp(β) and let X I , X I be matrices whose columns are (X i ) i∈I and (X i ) i / ∈I , respectively. Let us assume that ker(X I ) = {0} and let ζ X,λ,S(β) := X I X I (X

The following upper bound for the sign recovery holds.

In addition, Cauchy-Schwarz inequality gives the following implications

Because u r tends to +∞ then, according to [START_REF] Chen | Basis pursuit[END_REF], the sequence (( β L (ε, r) -β (r) )/u r ) r∈N * is bounded since the following superior limit is finite

Consequently, to prove that lim r→+∞ ( β L (ε, r) -β (r) )/u r = 0 it is sufficient to show that 0 is the unique limit point of this sequence. Let (( β L (ε, φ(r)) -β (φ(r)) )/u φ(r) ) r∈N * be a converging subsequence to l (with φ : N * → N * strictly increasing) and without loss of generality, let us assume lim r→+∞ β L (ε, φ(r))/u φ(r) = v and lim r→+∞ β (φ(r)) /u φ(r) = v so that l = v -v . By [START_REF] Chen | Basis pursuit[END_REF] and ( 8) one may deduce that

Since, whatever r ≥ 0, we have S(β (φ(r)) /u φ(r) ) = s 0 where s 0 is identifiable with respect to the 1 norm then, according to Proposition 2, one may deduce that β (φ(r)) /u φ(r) is an unitary vector satisfying the identifiability condition. Consequently, v 1 = 1 and v is identifiable with respect to the 1 norm. Consequently, v = v and thus l = 0 is the unique limit point, which implies that

For the proof of Lemma 2, we have not used the third condition of Assumption 1. This condition, under which the smallest non-null component of β (r) is not asymptotically infinitely smaller than β (r) ∞ , is useful to prove Lemma 3.

Lemma 3 Let (β (r) ) r∈N be a sequence of R p satisfying Assumption 1, then

Proof: Let us define u(ε) ∈ R p as follows

Because X(u(ε)) = ε, we have Y (ε) = X(β (r) + u(ε)) and because β BPDN (ε, r) is an admissible point of (3), one deduces the following inequality

Because β (r) + u(ε) is an admissible point of problem (3) and because β BPDN (ε, r) is the minimizer of (3), one may deduce that the following inequalities hold

Because u r tends to +∞ then, according to [START_REF] Gribonval | Sparse representations in unions of bases[END_REF], the sequence (( β L (ε, r) -β (r) )/u r ) r∈N * is bounded since the following superior limit is finite

Consequently, to prove that lim r→+∞ ( β BPDN (ε, r) -β (r) )/u r = 0 it is sufficient to show that 0 is the unique limit point of this sequence. Let (( β L (ε, φ(r)) -β (φ(r)) )/u φ(r) ) r∈N * be a converging subsequence to l (with φ : N * → N * strictly increasing) and without loss of generality, let us assume lim r→+∞ β BPDN (ε, φ(r))/u φ(r) = v and lim r→+∞ b (φ(r)) /u φ(r) = v so that l = v -v . By [START_REF] Giacobino | Quantile universal threshold[END_REF] and ( 19) one may deduce that Xv = Xv and v 1 ≤ 1.

Since, whatever r ≥ 0, we have S(β (φ(r)) /u φ(r) ) = s 0 where s 0 is identifiable with respect to the 1 norm then, according to Proposition 2, one may deduce that β (φ(r)) /u φ(r) is an unitary vector satisfying the identifiability condition. Consequently, v 1 = 1 and v is identifiable with respect to the 1 norm. Consequently, v = v and thus l = 0 is the unique limit point, which implies that

Lemma 5 is useful to prove in Theorem 1 that when s 0 is not identifiable then sign estimator derived from thresholded LASSO cannot recover s 0 .

Supplementary material

We have already said that when X is in general position the minimizer of problem (2) (resp. problem (3)) is unique. Concerning LASSO, a sketch of proof given in Tibshirani [START_REF] Ryan | The lasso problem and uniqueness[END_REF] shows the uniqueness of the LASSO estimator when X is in general position. In order to provide a self-contained article, we show that when X is in general position, the minimizer of problem (3) is unique when R = 0 as well as when R > 0. We have


Etsuko Mizutani
email: mizutani02@tobunken.go.jp

Daisuke Ogura

Masaru Abuku

Hannelore Derluyn

Preliminary investigation of change of pore structure due to salt precipitation during evaporation in brick with X-ray computed tomography

Keywords: synchrotron X-ray CT, water evaporation, sodium chloride, salt precipitation

Pore clogging by salt crystallization changes mass tranport phenomena. This paper reports on our preliminary attempt to quantify the spatial distribution of salt crystals through time in fired clay brick during drying by using synchrotron radiation X-ray computed tomography (CT; beamline BL20B2 at SPring-8, Japan) that allows fast 3D imaging with high spatial resolution. Specimens saturated with NaCl solution are dried at ambient conditions and scanned at certain moments in time. The mass loss of the specimens during drying is also measured gravimetrically to investigate the macroscopic effect of salt precipitation on water evaporation, and is used for the evaluation of the validity of the image analysis. Since the amount of salt precipitation identified from CT images and the amount of salt precipitation calculated from gravimetric measurements is almost the same, the usefulness of synchrotron X-ray CT imaging for verifying the temporal evolution of crystal formation in our type of porous bricks is confirmed. Image analysis suggests that salt precipitation on the surface of the material (i.e. efflorescence) mainly occurs during the first 2 hours of drying, and inside the material (i.e. subefflorescence) in the latter stages of drying.

Introduction

Salt weathering is a phenomenon in which materials are physically destroyed due to the build-up of crystallization-induced stress, and it is one of the degradation factors of brick or stone buildings and sculptures. This deterioration phenomenon has been confirmed in many historical sites which have been constructed in brick like Mohenjohdaro [START_REF] Fodde | The influence of soluble salts on the decay of Moenjodaro, Pakistan[END_REF]) and at the Hagia Sophia in Istanbul [START_REF] Mizutani | The influence of infiltrated rain water on degradation of the wall paintings in Hagia Sophia[END_REF].

A prediction model for salt weathering is very useful to develop an appropriate conservation and restoration plan in different climate conditions. The frame work for numerical analysis including salt precipitation and deformation has been constructed recently, and several numerical models have begun to be proposed [START_REF] Koniorczyk | Modelling of salt crystallization in building materials with microstructure -Poromechanical approach[END_REF][START_REF] Derluyn | Deformation and damage due to drying-induced salt crystallization in porous limestone[END_REF]. However, there are many points where the physical phenomena related to salt precipitation in the porous material itself are still unknown.

One of the issues for improving the numerical analysis of salt weathering is how to express the change in mass transport properties due to the precipitation of salt. The influence of salts dissolved in solution on the mass transport properties such as the sorption isotherm and the moisture diffusivity have been well studied. It should be noted that there are exceptions [START_REF] Pel | Saline absorption in calcium-silicate brick observed by NMR scanning[END_REF], but in many cases the explanation is given by considering the increase of viscosity and density with salt concentration, and the decrease of surface tension as well as of water vapor pressure. On the other hand, although the significance of the effect of pore clogging due to salt precipitation on mass transport is obvious from experimental research in the literature [START_REF] Espinosa-Marzal | Impact of in-pore salt crysatallization on transport properties[END_REF], the quantitative relationship between the crystal formation and the change in the mass transport properties is not known because of the difficulty in quantifying the salt crystals in the porous material. In order to relate salt precipitation and change of transport properties, it is needed to determine where and how much salts precipitate and how the salt crystals change the pore network structure. The pore network structure is mainly characterized by the connectivity, the pore size distribution and the tortuosity, which are directly affecting gas and liquid transfer in porous media.

In this paper, a series of drying experiments using fired clay brick saturated with NaCl solution is performed to quantify the salt precipitation on the surface and inside the porous media, as well as its influence on mass transport. Synchrotron radiation X-ray computed tomography, which offers 3D images with high spatial resolution, is used to quantify the spatial distribution of salt crystals through time in fired clay brick during drying. Gravimetric measurements of the mass loss during drying are also performed to investigate the macroscopic effect of precipitated salt on the drying rates. In addition, the total amount of salt crystals is calculated from the mass loss, and is compared with the results obtained from CT image analysis.

2

Materials and methods

Materials

Japanese fired clay brick (red brick produced by Okamoto co.) was used for the experiments. The capillary moisture content of this brick is 0.17 [m 3 /m 3 ], and the total open porosity is 0.24 [m 3 /m 3 ]. Figure 1 shows the result of mercury intrusion porosimetry (Autopore Ⅳ9500) of a dry brick. Most of the pores are capillary pores (pore diameter between 1μm and 1mm [START_REF] Klopfer | Experimental study of the effect of sodium chloride on drying of porous media: The crusty-patchy eflorscence transition[END_REF]), implying that the moisture uptake transfer in this brick is primarily caused by capillary forces. Brick specimens of 5mm x 5mm x 8mm were prepared, and an epoxy resin was applied on the lateral sides, making these surfaces water and vapor tight. This ensures a one-dimensional mass transfer during the drying experiments. Drying was monitored after three or four saturating specimens under each of four conditions: (1) with pure water, (2) with an unsaturated NaCl salt solution of 8g NaCl/100g water, (3) with a saturated NaCl salt solution and (4) with a 2nd saturation with saturated NaCl salt solution after experiment (2).

The monitoring was conducted both gravimetrically and through X-ray CT scanning at beamline BL20B2. The gravimetric measurements were performed outside of the Xray CT hutch, at 25°C and 35.8 % RH, whereas the climate inside the hutch was slightly warmer, being 26.5°C and 33% RH on average. The specimen were placed in a holder as shown in Figure 2. The specimen was hold by an upper polyproylene cup in which the pure water or salt solution was absorbed. Then the upper cup was mounted in a lower cup with pure water (condition (1)) or NaCl saturated solution (conditions ( 2) -( 4)) in order to suppress evaporation from the bottom surface of the sample.

Gravimetric experiments

The specimens were saturated with water or salt solution, by injecting the liquid into the upper cup with a syringe (Figure 2). The liquid was absorbed for 30 minutes to ensure capillary saturation. Thereafter, the solution in the upper cup was wiped with a tissue and the brick was dried via the upper surface of the test sample.

The time curve of the accumulated amount of evaporation was measured with a balance with a 0.0001g accuracy (Sartorius AG). Only the mass of the upper part, consisting of the upper polypropylene cup and the specimen was measured so as not to be affected by the mass loss of the solution in the lower polypropylene cup. The mass was measured at intervals of about 15 minutes during the initial stage of drying, and thereafter at 30 minutes intervals.

X-ray CT experiments and image processing

Synchrotron radiation X-ray CT imaging was performed at beamline BL20B2 (SPring-8, Hyogo prefecture, Japan) using a deflection electromagnet as a light source. The critical energy at BL20B2 is 28.9 keV, and the beam size is about 75 mm (horizontal) × 5 mm (vertical). A Hamamatsu Photonics' ORCAFlash4.0 CCDcamera was used as the image detector. Scans were acquired with a spatial resolution of 2.74 μm/pixel, the duration for one scan acquisition was about 5 minutes. The resulting 3D reconstructed volume of each scan consists of 2048 × 2048 × 1700 voxels (± 6.5 x 6.5 x 4.7 mm in width x depth x height). In order to cover the full height of the specimen, three scans were taken at different heights and stitched together to obtain a reconstruction of the complete brick specimen.

First, a three-dimensional CT imaging of a dry specimen was performed. After that, the specimen was wetted in the same way as for the gravimetric experiments, and the wet sample was imaged. Subsequently, several X-ray CTscans were performed during the drying process, i.e. after 30 minutets, 60 minutets, 120 minutets, 180 minutets and 480 minutets of dyring. The specimen was kept on the CT-stage until the end of the scanning series, and extreme care was taken to ensure that the specimen position did not change, particularly during solution injection and wiping.

The image processing starts with reducing the influence of noise, for which a median filter was used on the 3D reconstructed volumes of the brick specimen. The gray values of the CT images reflect the density and atomic number of the components present in the specimen. The specimen consists of solid brick and its pore space, which can be partly or fully filled with fluid (water or salt solution) and salt crystals.

Each of these components have a different gray value distribution, and by thresholding gray value ranges, the different components can be extracted, i.e. segmented, from the images. However, in our case, the overlap of the gray values of the substances is large, as shown in Figure 3 containing four phases determined from different CT images of condition (3). Therefore, the solid phase was segmented from the CT images of the dry brick, and subtracted from the images acquired during drying. The salt crystal phase is then segmented based on grayvalue thresholding of these differential images.

Results and Discussion

Gravimetric experiments

Figure 4 shows the accumulated amount of evaporation from the brick specimen. The evaporation rate becomes slower as the salt content increases. After 300 minutes from the start of drying, 89% of the initial absorbed water was dried on average in the three cases of pure water, while 77% was dried for the unsaturated sodium chloride solution (8g/100g water), 53% for the saturated solution and 33% for the saturated solution during the second cycle.

Though it is reported that salt precipitation may increase or decrease the evaporation rate depending on precipitation patterns affecting the pore size distribution and pore shape of porous media (Eloukabi vapor diffusion is dominant. The average evaporation rates of the capillary regime and the diffusive regime are calculated and shown in table 1 for the brick specimen. The average evaporation rate for salt containing brick specimens decreases in both the capillary and diffusive regime when compared to a brick containing pure water, indicating that salt inside and outside the material affects the movement of both the liquid and gas phase. Especially in cases ( 3) and ( 4) for bricks containing saturated NaCl salt solution the evaporation rate in the diffusive regime becomes slow.

Figure 4: Accumulated amount of evaporation over time, obtained from gravimetric measurements.

X-ray CT experiments

In this section, we focus on the image analysis to quantify the spatial distribution of salt crystals for the experiment with NaCl saturated salt solution (condition ( 3 Table 1: The average evaporation flux per unit surface area amount of precipitated salt obtained from gravimetry and from CT image analysis. The amount obtained from gravimetry is calculated from the amount of water evaporation measured, assuming that salt crystals precipitate as soon as the salt concentration exceeds the saturated concentration. The amount of salt identified by image analysis is 1.68 times larger at 60 minutes, 0.91 times at 120 minutes, 0.87 times at 180 minutes and 0.92 at 480 minutes. Although there would be some differences between specimens in Figure 5, the overall trend of salt precipitation can be rather sufficiently identified, which would confirm the suitability of the use of synchrotron X-ray CT imaging for verifying the temporal evolution of crystal formation in porous materials and its influence on the liquid transport phenomena. The data points of estimation from gravimetry give the values averaged for three specimens as well as the minimam and maximam values.

Figure 6 shows the time evolution of the spatial distribution of salt crystals identified by the image analysis, focused at the evaporation surface and the inside of the brick. The salt precipitation occurs on the surface of the material (i.e. efflorescence) during the initial 120 minutes, and thereafter, proceeds inside (i.e. subefflorescence). This result indicates that salt solution could be transported to the surface by capillary force up to 120 minutes, whereafter the drying front recedes, causing salt precipitation inside. Figure 7 shows segmentated images of the horizontal cross sections taken at a depth of 0mm in Figure 6 at 60 minutes and 480 minutes after the start of drying. Salt precipitation only occured in large pores at 60 minutes, but was found in smaller pores at 480 minutes. This result seems to be qualitatively correct as it is consistent with the theory of evaporation, saying that evaporation is likely to empty first large pores. Later on in the evaporation, salt precipitation increases in smaller pores, which becomes more difficult to identify due to the limit in spatial resolution. This may be the reason why the identified salt amount values by image analysis become smaller than the values from gravimetry after 120 minutes. This difference should be taken into account when the influence of salt precipitation on the mass transport phenomenon is discussed further using the whole data set obtained from the drying experiments with X-ray CT in the future.

Conclusion

A series of drying experiments using fired clay brick filled with NaCl salt solution is performed to quantify the salt precipitation on the surface and inside the porous media and its influence on mass transport. The gravimetric tests have shown that the salt reduces the evaporation rate both during the initial capillary drying regime and the subsequent diffusive drying regime.

Since the amount of salt precipitation identified from CT images and the amount of salt precipitation calculated from gravimetric measurements were almost the same, the suitability of the use of synchrotron X-ray CT imaging for verifying the temporal evolution of crystal formation in porous materials was confirmed.

In the future, the segmented images consisting of the void, solid and salt crystal phases shown in this paper will be used for the study of influences of the pore size distribution, pore connectivity and tortuosity on the gas diffusivity and moisture permeability. Although the CT images used in the study of this paper were limited, our future study will include more data from the drying experiments presented in this paper.

Figure 1 :Figure 2 :

 12 Figure 1: Pore size distribution of the brick sample at the beginning of the experiment measured by mercury intrusion porosimetry.

Figure 3 :

 3 Figure 3: Gray value of each substance determined from CT images of condition (3).

  Figure4shows the accumulated amount of evaporation from the brick specimen. The evaporation rate becomes slower as the salt content increases. After 300 minutes from the start of drying, 89% of the initial absorbed water was dried on average in the three cases of pure water, while 77% was dried for the unsaturated sodium chloride solution (8g/100g water), 53% for the saturated solution and 33% for the saturated solution during the second cycle. Though it is reported that salt precipitation may increase or decrease the evaporation rate depending on precipitation patterns affecting the pore size distribution and pore shape of porous media(Eloukabi et al. 2013), in our case, the salt in the specimen causes a clear decrease. The influence of salt was also quantified on two different regimes during drying[START_REF] Desarnaud | Effect of primary and secondary nucleation[END_REF][START_REF] Scherer | Theory of Drying[END_REF][START_REF] Shokri | What determines drying rates at the onset of diffusion controlled stage-2 evaporation from porous media ?[END_REF]: capillary regime chatacterised by a fast constant evaporation at the initial stage of drying where capillary force is active and diffusive regime, characterized by a small evaporative flux, where

  )). The analysis of the other experiments is ongoing. The total open porosity of the whole sample identified by image analysis is 0.195 [m3 / m3] which corresponds to about 81% of the total open porosity of 0.240 [m 3 / m 3 ]. This difference seems to be reasonable considering the ratio of voids below the spatial resolution (2.74 μm/pixel), as shown in Figure 1, that cannot be identified. Figure 5 shows the comparison of the evolution as a function of time of the total NaCl( 8g/100gwater) ▲;(3)NaCl (saturated solution)

Figure 5 :

 5 Figure 5: Total amount of precipitated salt as a function of time for samples of condition (3). The data points of estimation from gravimetry give the values averaged for three specimens as well as the minimam and maximam values.

Figure 6 :

 6 Figure 6: The temporal evaluation of spatial distribution of salt crystals identified by X-ray CT image analysis focused on (a) the evaporation surface and (b) inside the specimen.

Figure 7 :

 7 Figure 7: Segmented images (400 x 400 pixcels) at (a) 60 min and (b) 480 min after the start of drying (black: void/liquid, orange: solid, green: salt crystals)

Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers 16H06363, 18H01596, 19K23565, and 19H05511. H. Derluyn gratefully acknowledges the financial support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 850853). The synchrotron radiation experiments were performed at the BL20B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2018A1714 and 2018A1756).