Sara Rejeb
email: sara.rejeb@sorbonne-universite.fr

Catherine Duveau

Tabea Rebafka

Self-Organizing Maps for Exploration of Partially Observed Data and Imputation of Missing Values

Keywords: Self-organizing maps, partially observed data, missing data imputation, robustness to missingness mechanism

The self-organizing map is an unsupervised neural network which is widely used for data visualisation and clustering in the field of chemometrics. The classical Kohonen algorithm that computes self-organizing maps is suitable only for complete data without any missing values. However, in many applications, partially observed data are the norm. In this paper, we propose an extension of self-organizing maps to incomplete data via a new criterion that also defines estimators of the missing values. In addition, an adaptation of the Kohonen algorithm, named missSOM, is provided to compute these self-organizing maps and impute missing values. An efficient implementation is provided. Numerical experiments on simulated data and a chemical dataset illustrate the short computing time of missSOM and assess its performance regarding various criteria and in comparison to the state of the art.

Introduction

In chemometrics, data exploration is essential and is generally the first part of the analysis of any dataset. With the explosion of data volume in many fields of application due to big data, the structure of data is often hidden behind the massive data and difficult to detect. This increases the importance of data exploration methods providing meaningful information on the internal structure and correctness of the data, as well as the relationships and redundancies among the variables. The recorded at all for some of the sources.

The impact of missing data on statistical results can be serious, leading to biased estimates, loss of information, decreased statistical power, increased standard errors, and weakened generalizability of findings. However, for a long time, in statistics, missing data have been treated in a very simple and inappropriate way, either by deletion of incomplete measurements or by basic data completion by mean or median values. This has changed during the last decades, by the development of many statistical methods that account for missing data in a meaningful way. There are two general approaches to deal with missing data: either a statistical method is directly adapted to the partially observed data, or first an appropriate imputation method is applied to complete the data such that the statistical method of interest can be used on the completed or augmented data.

In this paper we are interested in self-organizing maps in the presence of missing data. This problem has been considered among others by [START_REF] Cottrell | Missing values : processing with the Kohonen algorithm[END_REF] by simply restricting all vector calculations to the observed entries. As such, all observed data entries are taken into account in the algorithm.

However, the method performs rather poorly when the number of incomplete observations with multiple missing entries is large. Moreover, the imputation of missing data is done afterwards by replacing missing entries by the closest features on the learned map. Related approaches are presented in [START_REF] Folguera | Self-organizing maps for imputation of missing data in incomplete data matrices[END_REF][START_REF] Rustum | Replacing outliers and missing values from activated sludge data using Kohonen Self-Organizing Map[END_REF][START_REF] Nkiaka | Using Self-Organizing Maps to infill missing data in hydrometeorological time series from the Logone catchment, Lake Chad basin[END_REF][START_REF] Adeloye | Neural computing modeling of the reference crop evapotranspiration[END_REF][START_REF] Kalteh | Imputation of missing values in a precipitation-runoff process database[END_REF] and more recently in [START_REF] Junno | Predicting missing seismic velocity values using self-organizing maps to aid the interpretation of seismic reflection data from the Kevitsa Ni-Cu-PGE deposit in northern Finland[END_REF][START_REF] Khotimah | A hybrid self organizing map imputation (SOMI) with naïve bayes for imputation missing data classification[END_REF].

Missing data imputation can be useful to avoid incomplete data, which is crucial in data mining when methods cannot handle any missing entries. We have the ambition to combine the tasks of imputation and learning the map by a principled approach. Our motivation is the fact that any non trivial imputation method is based on some data model, and so it is natural to use the selforganizing map for imputation. Conversely, a better map may be learned when data are complete. Thus, treating both tasks simultaneously may be beneficial for the two of them.

Our approach can be viewed as an extension of the standard Kohonen algorithm for selforganizing maps and the principle of our method is given in Section 2. A mathematical presentation of the method, a new loss function that encodes our double goal of imputation and learning a selforganizing map and two algorithmic solutions are given in Section 3. Moreover, Section 4 provides an extensive numerical study assessing the robust performance of our method in various settings and in comparison to alternative methods from the literature.

The new method in a nutshell

This section presents the principle of our new method for self-organizing maps for partially observed data. To start with, we recall the classical self-organizing map. Let x 1 , . . . , x n be n observations or measurements of dimension p. A self-organizing map represents a nonlinear projection of the high-dimensional data onto a low-dimensional subspace. This subspace is a two-dimensional map represented as a regular grid composed of K fixed neurons. This fixed spatial arrangement of the neurons on the map is the key for the preservation of the topology of the input data when projected onto the map. Every neuron k is associated with a p-dimensional prototype vector w k , also called code vector, that is to be learned. The prototype vectors define a discretization of the data space, and each observation x i is assigned to its closest prototype. Ideally, prototype vectors of neighboring neurons on the grid are close one to another, so that data points x i that are close in the input space are also close on the map.

The Kohonen algorithm computes the self-organizing map for complete data in an iterative fashion. One randomly picked observation x i is treated at each iteration. First, the winning neuron or best matching unit is determined, which is the neuron whose prototype is the closest to measurement x i . Then, all code vectors w k are updated by attracting them towards the measurement x i .

The attraction is the strongest for the winning neuron and very weak for the neurons that are far from the winning neuron. Those updates eventually result in an ordered map, where neighboring neurons have similar prototype vectors. Now, when some of the measurement vectors x i contain missing entries, the Kohonen algorithm is not applicable anymore. We propose to learn the missing entries while learning the map in the following way. Our algorithm is the following: starting from some initial imputed values, like the mean values computed over the observed entries, select every measurement x i once, determine its winning neuron by considering the distance only over the observed entries of x i and then update the code vectors just as in the classical Kohonen algorithm. Then, perform an update of the imputed values using a weighted means of the closest code vectors of the partially observed measurement.

Repeat this procedure until convergence. A full description is provided in Algorithm 3. This algorithm performs both data visualization and imputation of the incomplete data. Interestingly, it is as fast as the standard Kohonen algorithm, since the update of the imputed values is immediate.

Algorithm 1: Standard Kohonen algorithm

Input: Data matrix X, size and topology of the map, neighborhood function V λ , sequence of radii (λ t ) 0≤t≤T and learning steps (ε t ) 0≤t≤T .

Initialize code vectors W (0) ; Initialize the counter of iterations: t = 0 ;

while not converged do

Increment t: Set t = t + 1 ;
Choose an observation i ∈ {1, . . . , n} randomly ;

Assignment: Compute winning neuron = h(x i , W (t-1) ) ;

Update code vectors:

for k = 1, . . . , K do w (t) k = w (t-1) k + ε t V λt (k, ) x i -w (t-1) k . end end Output: Code vectors W (t) .

Self-organizing maps with incomplete data

In this section we first formally state the classical self-organizing map, before introducing the new loss function and two algorithms for the computation of self-organizing maps with partially observed data and missing data imputation.

Classical Kohonen algorithm

The data matrix containing the measurements is denoted by X = [x 1 , . . . , x n ] ∈ R n×p . The arrangement of the neurons on the map is given by some neighborhood function V λ : {1, . . . , K} 2 → R + . The neighborhood radius λ > 0 describes the zone of influence around a neuron. The best prototype vectors of the self-organizing map are defined as the minimum of the loss function F defined by

F (W ) = 1 2n n i=1 K k=1 V λ (k, h(x i , W )) x i -w k 2 2 , (1)
where W = [w 1 , . . . , w K ] ∈ R p×K is the matrix of K prototype vectors and h : R p × R p×K → {1, . . . , K} denotes the allocation function, attributing the closest prototype to a data point x w.r.t. the Euclidean distance, defined as

h(x, W ) = arg min 1≤k≤K x -w k 2 . (2)
The loss F takes into account all distances between every measurement and all code vectors, weighted by the neighborhood function evaluated on the corresponding neurons. As a result, code vectors that minimize the loss are similar if they are close on the map. In the specific case where the neighborhood function satisfies V λ (k, ) = 0 for all k = , the loss F is the criterion minimized by the k-means algorithm. That is, clusters obtained by k-means are independent, while prototypes of a self-organizing map are organized in a topological way.

To compute the minimum of loss F , [START_REF] Ritter | Neural Computation and Self-Organizing Maps: An Introduction[END_REF] showed that in the given framework a gradient descent algorithm can be used, referred to as the Kohonen stochastic algorithm. For a randomly picked observation x i with winning neuron = h(x i , W (t) ), the updates of the code vectors are given by

w (t+1) k = w (t) k + ε t V λ (k, )(x i -w (t) k ), (3)
where (ε t ) t≥0 is a sequence of decreasing learning steps. This update attracts all prototypes towards observation x i . It is also common to shrink the neighborhood by using a decreasing sequence of radii (λ t ) t≥0 in the neighborhood function V λ . The algorithm is summarized in Algorithm 1.

Notation for incomplete data

Now we consider an incomplete n × p data matrix containing missing values. Let the matrix M = (m i,j ) i,j ∈ {0, 1} n×p be the missing-data pattern which indicates where the entries are missing or masked, and that is defined by

m i,j =    1 if x i,j is observed 0 if x i,j is missing
We denote X obs the set of observed data values and X miss the set of non-observed data entries hidden by the missing-data pattern M . The complete data are denoted by X compl = (X obs , X miss ).

Likewise, for the observation vector x i we denote by x obs i and x miss i the observed and unobserved entries, respectively, and, with some abuse of notation,

x compl i = (x obs i , x miss i
) is the complete vector, which also corresponds to the i-th row of X compl .

Our goal is to adapt the model of self-organizing maps to partially observed data, and moreover, learn the values of the missing data. The motivation to treat these tasks simultaneously is that learning missing values requires a data model, and as we are interested in self-organizing maps it is natural to use this model for data imputation. At the same time, learning a map with completed data may give better results compared to using only the observed part X obs of the data.

New loss function

We introduce a new loss function that considers both problems : finding the best self-organizing map and the best values for imputation of the missing data. In other words, by minimizing the new loss function F missom we search for both the best code vectors W ∈ R p×K for the map and the best values for the missing data denoted by X * chosen in the set of all possible values for the missing entries X miss .

To define the new criterion, an adaptation of the definition of the winning neuron is in order.

In the presence of missing values, it is natural to restrict the Euclidean distance in (2) only to the observed entries. More precisely, for any vectors

x obs ∈ R p (p ≤ p), m ∈ {0, 1} p with p j=1 m j = p and code vectors W ∈ R p×K , we set h miss (x obs , m, W ) = arg min 1≤k≤K x obs -w k m 2 ,
where w k m denotes the p -vector made of the elements w k,j of w k such that m j = 1. Now, we define the new loss as

F missom (W, X * ) = 1 2n n i=1 K k=1 V λ k, h miss (x obs i , m i , W ) (x obs i , x * i ) -w k 2 2 ,
where m i ∈ R p is the i-th row of the matrix M and (x obs i , x * i ) denotes the i-th measurement vector completed with x * i . Since

(x obs i , x * i ) -w k 2 2 = x obs i -w k m i 2 2 + x * i -w k (1 p -m i ) 2 2 ,
where 1 p = (1, . . . , 1) T ∈ R p , the criterion F missom can be decomposed into two parts according to the observed and the missing entries as

F missom (W, X * ) = F obs (W ) + F miss (W, X * ),
where F obs (W ) is the part of the loss over the observed entries given by

F obs (W ) = 1 2n n i=1 K k=1 V λ k, h miss (x obs i , m i , W ) x obs i -w k m i 2 2 ,
Algorithm 2: missSOM algorithm Input: Incomplete data X obs , missing-data pattern M , size and topology of the map, neighborhood function V λ , sequence of radii (λ t ) 0≤t≤T and learning steps

(ε t ) 0≤t≤T .
Initialize imputed values X * (0) and code vectors W (0) ; Initialize the counter of iterations: s = 0 ;

while not converged do

Increment s: Set s = s + 1 ;
Update code vectors by Kohonen Algorithm 1 on the augmented data with winning neurons obtained by h miss instead of h:

W (s) ← Kohonen(X aug = (X obs , X * (s-1) )) ;
Update imputed values: for i, j such that m i,j = 0,

x * (s) i,j = K k=1 V λ T (k, h miss (x obs i , m i , W (s) ))w (s) k,j K k=1 V λ T (k, h miss (x obs i , m i , W (s) ))
.

end Output: Code vectors W (s) and imputed data X * (s) . and F miss (W, X * ) is the contribution of the imputed values X * to the loss, defined as

F miss (W, X * ) = 1 2n n i=1 K k=1 V λ k, h miss (x obs i , m i , W ) x * i -w k (1 p -m i ) 2 .
Note that in the complete-data case, where the missing-data pattern is M = 1 n×p , F miss (W, X * ) = 0 for any W and any X * , so that the criterion F missom is equal to the one of the classical selforganizing map, that is, F missom (W, X * ) = F (W ).

Minimization algorithm

For the minimization of (W, X * ) → F miss (W, X * ) on R K×p × X miss we propose to alternate the minimization in W and X * while keeping the other argument fixed.

For fixed X * , the function W → F missom (W, X * ) is similar to the objective function F in (1) in the complete-data case applied to the augmented data X aug = (X obs , X * ). The only difference lies in the definition of the winning neurons by h miss that appear in the neighbourhood function V λ .

Thus, a Kohonen algorithm applied to X aug can be used to find the best code vectors W .

In turn, when W is fixed, the minimization of X * → F missom (W, X * ) boils down to minimize X * → F miss (W, X * ). This problem has a unique explicit solution given for all 1 ≤ i ≤ n and j such that m i,j = 0 by

x * i,j = K k=1 V λ (k, h miss (x obs i , m i , W ))w k,j K k=1 V λ (k, h miss (x obs i , m i , W )) . ( 4
)
That is, the imputed values are a weighted mean of the prototype vectors weighted according to the neighborhood function.

To summarize, the algorithm updates imputed values for the missing data and applies the classical Kohonen algorithm with adjusted winning neuron function h miss to learn the map. This is repeated until convergence or until a maximum number of iterations chosen by the user is attained.

The algorithm is described in Algorithm 2.

As initial values for the imputed values X * , one can simply impute the sample mean or median of the variables obtained over the observed entries.

Accelerated version

Algorithm 2 happens to be expensive in terms of computing time, in particular when the number of iterations is large, since the entire standard Kohonen algorithm is carried out during each While the first version of the missSOM algorithm has some theoretical justification, the accelerated version lacks this foundation. A numerical study given in Appendix 6.1 shows that the Algorithms 2 and 3 provide very similar maps and hence justifies the utilization of the accelerated version, which achieves a significant gain in computing time.

Note that while the selection of the observations x i in Algorithm 3 is deterministic, it is possible to use a random selection scheme.

Numerical experiments

In this section, the performance of the proposed method missSOM is evaluated and compared to alternative methods. A simulation study is conducted to assess the quality of the representation of the data via the map and the accuracy of imputed values under various conditions.

Performance criteria

Let W * be the code vectors of the final self-organizing map, X * the imputed values and xi = (x obs i , x * i ) the completed observation vectors. The quality of the map as a representation of the data can be evaluated by two criteria. First, the quantization error defined as the average of the squared distances between the observations and their nearest prototype vector given by

E = 1 n obs n i=1 x obs i -w h miss (x obs i ,mi,W * ) 2 2 ,
where n obs = i,j m i,j is the number of observed entries in the data, informs on whether the prototype vectors are good representations of the data. Second, the topographic error evaluates the preservation of the topology of the data in the map by the proportion of observations for which the winning neuron and the second closest neuron are not neighbors, i.e. not connected on the grid. It is defined as

T = 1 n n i=1 e(x i ),
where

e(x i ) =    0 if the neurons h miss (x obs i , m i , W * ) and v miss (x obs i , m i , W * ) are adjacent 1 otherwise with v miss (x obs i , m i , W * ) = arg min j =h miss (x obs i ,mi,W * ) x obs i -w * k m i 2
2 the second closest neuron to measurement x obs i . The topographic error is expected to be small when the map is well organized and ordered. As both errors are computed only on the observed entries, the focus is put on the quality of the map with respect to the observed part of the data. The accuracy of the imputed values is assessed separately by the imputation error, which is defined as the root mean square error and quantifies the quality of the imputed values compared to the true missing values.

Datasets

For the numerical experiments, two settings are considered. First, the dataset wines from the UCI machine learning repository [START_REF] Dua | UCI machine learning repository[END_REF] is used, which contains the results of a chemical analysis of 178 wines on 13 quantitative variables. We generate 100 perturbed datasets by adding gaussian noise with mean 0 and standard deviation equal to one tenth of the mean value in each variable.

In the second setting, 100 datasets are simulated from a multivariate gaussian mixture with dimension p = 5, four equal-sized groups and n = 2000 observations. The correlation among all pairs of components is equal to 0.5 and for every dataset, the 4 gaussian means are drawn independently from a centered normal distribution with standard deviation equal to 5.

In both settings, missing values are generated using the ampute function from the R package mice [START_REF] Van Buuren | mice: Multivariate Imputation by Chained Equations in R[END_REF]. Different proportions of missing values, namely 5%, 20% and 40%, and different mechanisms of missingness are considered. The literature on missing data traditionally distinguishes three mechanisms that lead to missingness [START_REF] Rubin | Inference and missing data[END_REF] and it is well known that the performance of imputation methods may be sensitive to the mechansim at work. The mechanism is said to be missing completely at random (MCAR), when the causes of missing values are independent from the data and the probability of being absent is the same for all items. That is, a subset of observations is chosen at random using independent Bernoulli variables with fixed success probability for all entries.

In contrast, when the probability that a value is missing depends on the values of the observed variables, the mechanism is called missing at random (MAR). In our simulations, to obtain MAR, for each variable, missing values are obtained using a logistic regression model depending only on the other variables. Finally, when the probability of being absent also depends on the unobserved value, the mechanism is called missing not at random (MNAR) and we can consider missing data simulated by using a logistic regression model depending on all variables.

Alternative methods

Our study is twofold. On the one hand, we compare missSOM to the state of the art on selforganizing maps. The simplest method (referred to as deletion in the figures) consists in deleting the observations containing missing values and applying the standard Kohonen algorithm to the remaining data. Incomplete observations are classified once the map is built by assigning them to their closest prototypes and missing entries are imputed by the corresponding values of the winning prototypes. Cottrell's approach is a variant of the classical self-organizing maps appropriate to deal with incomplete observations [START_REF] Cottrell | Missing values : processing with the Kohonen algorithm[END_REF]. Cottrell adapts the Kohonen algorithm by simply restricting all vector calculations to the observed entries. The main difference with missSOM is that data imputation is performed only after learning the map by imputation with the values of the closest prototypes.

On the other hand, in our experiments, we compare missSOM to the state of the art on missing data imputation. The literature provides numerous general imputation methods. A basic approach (here referred to as mean) is the imputation by the mean value of the observed variables. A nonparametric approach called missForest predicts missing values using a random forest trained on the observed part of the dataset. Moreover, the method referred to as knn is a k-nearest neighbors approach, which is implemented in the VIM package [START_REF] Kowarik | Imputation with the R package VIM[END_REF]. Missing values are imputed iteratively by a weighted average of the k closest observations. Finally, assuming a gaussian mixture model for the data, the Amelia package [START_REF] Honaker | Amelia II: A program for missing data[END_REF] performs imputation using the expectation-maximization algorithm and a bootstrap approach to iteratively estimate missing values. In our numerical study, the imputation error of missSOM is compared to the one obtained by all these methods. In addition, we also compare the map obtained by missSOM with the maps obtained by the classical Kohonen algorithm applied to the complete dataset, where missing values are imputed by the aforementioned imputation methods.

As a benchmark we consider a self-organizing map trained on the complete data set (corresponding to a missing rate of 0%) and compute the error rates only on the observed values that are provided to the other methods. This method is referred to as complete-data SOM.

Parameters of the methods

The missSOM algorithm is applied with the default parameters in the package missSOM. That is, the neighborhood function is a gaussian and the sequence (λ t ) t≤T decreases from λ 0 = 4.58 to λ T = 0.5 for the wine dataset and from λ 0 = 8.89 to λ T = 0.5 for the gaussian mixture data. The maximum number of epochs is T = 100. Concerning the map, it has a hexagonal topology and the grid is composed of K = 9 × 7 neurons for the wines data and K = 16 × 14 neurons for the gaussian mixture data. The choice of the size of the map depends on the sample size and on the objectif of the analysis. In general, missSOM has the same parameters as classical SOM and they can be chosen in the same way as for SOM.

For the alternative methods, the maximum number of iterations is set to 100 and for the other parameters the default values are used.

Comparison to the state of the art on self-organizing maps

Figure 1 and 2 present the boxplots of the topographic and the quantization error for all SOMtype methods in both settings. They serve to evaluate the quality of the representation of the data using a self-organizing map computed with the different approaches. The corresponding imputation errors in Figure 3 allow to judge their performances as imputation methods.

First, we observe that when the percentage of missing data is low, most methods have very similar performance. With increasing percentage of missingness, the problem becomes harder and differences among the methods appear. But as errors are evaluated only on the observed part of the data, errors are not necessarily increasing. Namely quantization errors for the wines data decrease with increasing missingness. This may be a consequence of the higher variance and less structure of the wines data compared to the Gaussian mixture data. So when deleting entries from the wines data, the data variance decreases and the resulting map is a better representation of the observed part of the data. Concerning the mechanisms of missingness, it appears that the quality of the map does not depend on it for any SOM-type method. However, imputation is impacted by the mechanism. Imputation is the easiest under MCAR and the hardest under MNAR.

Next, we see that the deletion method is very unstable. In some scenarios its error rates are among the worst, and, more importantly, when too many values are missing, the method breaks down and does not produce any result. Indeed, on small datasets as wines with 20% of missingness, the number of complete observations is smaller than the size of the map and thus classical SOM is just not applicable. On the large gaussian mixture dataset, the deletion method produces the best topographic error, but the associated quantization error is disastrous, disqualifying the approach.

The quantization error of the complete-data SOM method is constant when varying the amount of missingness, because the underlying map remains the same. For the topographic error, an increase is observed which is due to the definition of the error. The error determines the closest and second closest prototypes only with respect to the observed entries, while the map was optimized by taking into account the complete data. This explains why missSOM and Cottrell have lower topographic errors, as their maps are learned with a notion of closeness restricted on the observed entries.

Finally, we observe that Cottrell's method always achieves the best quantization errors, directly followed by missSOM. Concerning the topographic error, missSOM is consistently doing better than Cottrell. Thus, in terms of quality of the map and representation of the data, none of the methods outperforms all others, and Cottrell and missSOM have both a good global performance. Now, considering the imputation error, there is a clear winner. When the proportion of missingness is important, missSOM outperforms all other methods in every scenario. This confirms us in the use of missSOM with respect to Cottrell, especially when accurate imputation is desired.

Comparison to the state of the art on missing-data imputation

In the second part of the simulation study, missSOM is compared to general imputation methods.

Figure 4 shows that as an imputation method missForest is unbeaten regardless of the missingness mechanism and the percentage of missing data. On gaussian mixture data, knn also provides accurate imputations. The imputation accuracy achieved by missSOM is high and missSOM is mostly among the best imputation methods.

Finally, Figure 5 and Figure 6 provide insights on the quality of the data representation, when classical SOM is applied to the data with missing values imputed by the different imputation methods. In this respect, missSOM clearly outperforms all the other methods. In particular, missForest provides self-organizing maps of poorer quality, especially in terms of preservation of the topology of the input data. This confirms the initial motivation of missSOM: in the presence of missing values and when a self-organizing map is required, it is better to learn the map simultaneously with the imputed values than to treat the two tasks separately.

A further advantage of missSOM is its computing time, which is about 10 times shorter than the one of missForest.

To summarize, we have seen that missSOM is the method of choice when both tasks data imputation and data representation are desired. However, when the focus is only on data imputation, then better alternatives as missForest exist.

Conclusion

In this paper, we have proposed an extension of the self-organizing map for partially observed data, referred to as missSOM. The proposed method addresses simultaneously the two problems of computing a self-organizing map and imputing missing values. A numerical study assesses the good performance of missSOM regarding various criteria and in comparison to the state of the art.

While this paper focuses on the standard Kohonen algorithm, in future work we may address the transfer of our approach to other existing variants of self-organizing maps on more complex data types such as mixed data to enable them to deal with missing data.

6. Appendix

Validation of accelerated missSOM algorithm

  iteration. A speed up is obtained by interwining updates of the missing data and the iterations of the Kohonen Algorithm 1. More precisely, we propose to update the missing data at every epoch, that is, after every pass through the data. This procedure gives rise to Algorithm 3. As such, the Kohonen algorithm is carried out only once, while in the initial Algorithm 2, the entire Kohonen algorithm is applied repeatedly. Thus the computing time of the accelerated version of missSOM is comparable to the computing time of the standard Kohonen Algorithm 1, since the update of the imputed values is fast.

Figure 1 :

 1 Figure 1: Topographic error of SOM-type methods for various amounts and mechanisms of missingness on the wines (a) and the gaussian mixture data (b).

Figure 2 :

 2 Figure 2: Quantization error of SOM-type methods for various amounts and mechanisms of missingness on the wines (a) and the gaussian mixture data (b).

Figure 3 :Figure 4 :

 34 Figure 3: Imputation error of SOM-type methods for various amounts and mechanisms of missingness on the wines (a) and the gaussian mixture data (b).

Figure 5 :Figure 6 :

 56 Figure 5: Topographic error of missSOM and classical imputation methods for various amounts and mechanisms of missingness on the wines (a) and the gaussian mixture data (b).

Table 1

 1 compares the results of the basic missSOM Algorithm 2 and its accelerated version Algorithm 3 on the simulated gaussian mixture data in various conditions. On the one hand, for all settings the errors are totally equivalent. This indicates that the accelerated version provides the same self-organizing maps and very similar imputations as the basic missSOM algorithm. On the other hand, we see that in terms of computing time we gain two orders of magnitude. Hence, the use of the accelerated algorithm instead of the basic version is completely justified.

			5%			20%			40%
		MCAR	MAR	MNAR	MCAR	MAR	MNAR	MCAR	MAR	MNAR
	Topographic error
	basic missSOM	0.337 (0.047)	0.345 (0.035)	0.336 (0.037)	0.313 (0.032)	0.355 (0.063)	0.339 (0.046)	0.294 (0.058)	0.303 (0.057)	0.306 (0.083)
	accelerated missSOM	0.352 (0.039)	0.345 (0.049)	0.338 (0.030)	0.301 (0.035)	0.299 (0.039)	0.322 (0.042)	0.282 (0.035)	0.280 (0.059)	0.285 (0.040)
	Quantization error
	basic missSOM	0.405 (0.115)	0.394 (0.042)	0.407 (0.145)	0.327 (0.115)	0.326 (0.116)	0.330 (0.112)	0.225 (0.074)	0.233 (0.080)	0.233 (0.077)
	accelerated missSOM	0.413 (0.149)	0.408 (0.139)	0.407 (0.138)	0.351 (0.117)	0.340 (0.116)	0.341 (0.115)	0.261 (0.081)	0.245 (0.071)	0.260 (0.074)
	Imputation error
	basic missSOM	0.601 (0.140)	0.599 (0.154)	0.613 (0.171)	0.659 (0.125)	0.686 (0.141)	0.759 (0.108)	0.758 (0.093)	0.795 (0.095)	0.871 (0.091)
	accelerated missSOM	0.615 (0.097)	0.624 (0.096)	0.654 (0.103)	0.665 (0.092)	0.694 (0.087)	0.750 (0.075)	0.742 (0.069)	0.793 (0.069)	0.848 (0.072)
	ARI
	basic missSOM	0.922 (0.092)	0.929 (0.089)	0.937 (0.071)	0.849 (0.088)	0.846 (0.108)	0.823 (0.089)	0.686 (0.108)	0.640 (0.101)	0.623 (0.086)
	accelerated missSOM	0.923 (0.090)	0.923 (0.086)	0.922 (0.082)	0.841 (0.092)	0.845 (0.081)	0.805 (0.090)	0.647 (0.094)	0.629 (0.109)	0.591 (0.089)
	Computing time
	basic missSOM	33.169 (0.546) 32.940 (0.737) 31.860 (0.396) 33.076 (0.517) 32.988 (0.776) 32.948 (0.530) 33.031 (0.583) 31.920 (0.061) 32.846 (0.0644)
	accelerated missSOM	0.394 (0.010)	0.387 (0.006)	0.376 (0.009)	0.557 (0.008)	0.561 (0.017)	0.556 (0.006)	0.790 (0.016)	0.753 (0.011)	0.784 (0.022)

Table 1 :

 1 Comparison of the basic missSOM Algorithm 2 and its accelerated version Algorithm 3 in terms of different errors and computing time (in seconds) on the gaussian mixture data.

Algorithm 3: Accelerated missSOM algorithm Input: Incomplete data matrix X obs , missing-data pattern M , size and topology of the map, neighborhood function V λ , sequence of radii (λ t ) 0≤t≤T and learning steps

Initialize imputed values X * (0) and code vectors W (0) ; Initialize the number of epochs: t = 0 ; 1) ;

. , n do

Assignment: Compute winning neuron = h miss (x obs i , m i , W (i-1) ) ; Update code vectors:

Update imputed values: for i, j such that m i,j = 0,

.

end Output: Code vectors W (t) and imputed data X * (t) .


Emilie Narni-Mancinelli1

Eric Vivier1

Advancing natural killer therapies against cancer

 show that pro-apoptotic molecules, such as BH3-mimetics, synergize with NK cells to induce mitochondriadriven apoptosis in tumor cells, thereby enhancing the efficacy of NK cell therapies.

Cancer is a major healt11 problem, and cancer t11erapies have been intensively investigated over decades. Patients can benefit from surgery, radiotl1erapy, and systemic chemotherapy depending on cancer indication. Surgery, when possible, remains the most effective treatment of localized primary tumors and associated regional lymphatics. Advanced cancers require systemic cytotoxic chemotherapy. However, the side effects of these treatments limit their application and effi-cacy. Cancer patients can also benefit from targeted therapies, such as mono-clonal antibodies (mAbs) or small mole-cules, which can either deplete tumor cells or target the tumor cell growth signal cascade.

lmmuno-oncology is revolutionizing the treatment of cancers. Recent advances have focused on generating or unleashing tumor antigen-specific cytotoxic T cell re-sponses. However, resistance to treat-ment and toxicity are often reported. Thus, despite major advances in these therapies, several cancer conditions remain true unmet medical needs and prompt the generation of a second wave of immune-oncology treatments [START_REF] Demaria | Harnessing innate immunity in cancer therapy[END_REF]. Along this line, bringing natu-ral killer (NK) cells to the clinic is very promising for a variety of reasons. NK cells can recognize a wide array of tumor cells across all cancer types and contribute to their elimination by direct cytotoxicity and by shaping a multicellular protective immune response via their secretion of cytokines and chernokines [START_REF] Demaria | Harnessing innate immunity in cancer therapy[END_REF][START_REF] Myers | Exploring the NK cell platform for cancer immunotherapy[END_REF]. The presence of functional NK cells correlates with better overall survival in several blood and solid cancers. Therefore, harnessing NK cells to fight cancer is being evaluated in clinic using mAbs or cell therapies. lmportantly, NK-cell treatments appear saler than T cell treat-rnents [START_REF] Daher | Outlook for New CAR-based therapies with a Focus on CAR NK ce lis: what Lies beyond CAR-Engineered T cells in the Race against cancer[END_REF]. ln this issue of Cel/, Pan and colleagues sought to improve antiturnor NK cell activ-ity by making tumor cells more sensitive to NK-cell-induced death [START_REF] Pan | Augmenting NK cell-based immunotherapy by largeting 1 mitochondrial apoptosis[END_REF]. There are several forms of cell death: apoptosis, necrosis, pyroptosis, and ferroptosis. lt has been known for years that NK cells induce the apoptosis of tumor cells. Two pathways can trigger apoptosis: the extrinsic pathway induced by engagement of the death receptors such as TRAIL and Fas, and the intrinsic pathway involving the mitochondria. Pan et al. discovered that the NK-cell-medi-ated killing mechanism requires the mito-chondrial-mediated apoptosis pathway within the tumor cell [START_REF] Pan | Augmenting NK cell-based immunotherapy by largeting 1 mitochondrial apoptosis[END_REF]. This mechanism occurs at a low effector:-target ratio, a condition that is likely found in patients where NK cells are outnum-bered by cancer cells.

Apoptosis via mitochondria is regulated by the pro-and anti-apoptotic proteins of the BCL-2 family. Anti-apoptotic proteins (BCL-2, BCL-w, MCL-1, BFL-1, and BCL-XJ prevent apoptosis by binding and sequestering their pro-apoptotic counterparts (PUMA, BIM, BIO, BAD, BIK, NOXA, and BMF) [START_REF] Singh | Regu-lation of apoptosis in health and disease: the balancing act of BCL-2 family proteins[END_REF]. A subset of BH3 (BCL-2 homology domain 3)-only proteins, including BIM and BIO, activate the effectors of apoptosis (BAX and BAK), which lead to mitochondrial outer membrane permea-bilization (MOMP). MOMP is recognized as the point of no return in programmed cell death and causes the release of cyto-chrome c into the cytosol, resulting in the formation of the apoptosome and, ulti-mately, activating the caspase cascade (caspases 9/7 /3). A cell that expresses just enough pro-survival proteins to contain pro-apoptotic signais is consid-ered "primed" for apoptosis. Consistent with the importance of mitochondrial apoptosis in the regulation of NK cell-mediated killing of tumor cells, Pan and coworkers showed in t11eir model that the silencing of the NK cell cytolytic mole-cule granzyme B impaired NK-cell-medi-ated tumor cell killing. Granzyme B has indeed been reported to induce mito-chondrial apoptosis through the activa-tion of BIO [START_REF] Lord | Granzyme B: a natural born killer[END_REF]. However, the contribution of NK cell cytokines, such as IFN-y, in the induction of mito-chondrial apoptosis in cancer cells re-mains to be analyzed. ln the search of novel antitumor thera-pies, efforts have been made to identify cytotoxic molecules that directly act on BCL-2 protein family, such as the BH3 mimetics [START_REF] Singh | Regu-lation of apoptosis in health and disease: the balancing act of BCL-2 family proteins[END_REF]. These are small molecules that disrupt the interac-tion of the BH3 domain pro-apoptotic molecules with BCL-2 proteins, thus limiting cell survival. The authors showed that NK cell treatment com-bined with low doses of BH3 mimetics synergizes to reach the threshold for apoptosis and cause cell death. The dependence on specific pro-survival BCL-2 family proteins is specific to each type of tumor cell and can be determined via a BH3 profiling assay. Of note, BH3 mimetics sensitize tumor cells for NK-cellmediated killing but do not promote the expression of activating ligands, such as the NKG2O ligand MICNB. The induction of other ligands for other activating receptors remain to be evaluated. lmportantly, NK cell pre-activation with interleukin-2 (IL-2) makes them less sensitive to BH3 mimetics, further supporting their com-bination for effective antitumor treat-ment. ln xenograft mouse models of blood and solid cancers, BH3 mimetics administration in combination with the transfer of activated NK cells signifi-cantly improve tumor cell control and mouse survival.

Several studies have reported that in-creases in the numbers of intratumoral and circulating NK cells are associated with favorable outcomes of treatment with anti-EGFR mAb (cetuximab) and anti-CD20 mAb (obinutuzumab and ritux-imab) [START_REF] Myers | Exploring the NK cell platform for cancer immunotherapy[END_REF]. lndeed, a cell that is dependent on EGFR will upre-gulate pro-apoptotic BIM when EGFR signaling is inhibited, thus lowering the threshold for apoptosis [START_REF] Deng | Proapoptotic BH3-only BCL-2 family pro-tein BIM connects death signaling from epidermal growth factor receptor inhibition to the mitochon-drion[END_REF]. Pan and coworkers showed in their report that rituximab also pushes cells to-ward apoptosis [START_REF] Pan | Augmenting NK cell-based immunotherapy by largeting 1 mitochondrial apoptosis[END_REF]. There-fore, NK-cell-mediated killing of primed tumor cells may be involved in the under-lying mechanism of anti-EGFR-mAb and anti-CD20-mAb therapies.

The findings of Pan et al. could be applied in a combinatorial way to improve NK-cell-based immunotherapies(Figure 1 ), for example, by combining BH3 mimetics with recent therapies that aim al unleash-ing NK cells with immune checkpoint inhib-itors, such as anti-NKG2A (monalizumab), anti-TIGIT (tiragolumab and ociperlimab), and anti-LAG3 (relatlimab) mAbs [START_REF] Demaria | Harnessing innate immunity in cancer therapy[END_REF]. Alternatively, BH3 mimetics could be combined with recent therapeu-tics designed to promote NK cell activity via NK cell engagers, which bind NK cell activating receptors (CD16 and/or natural cytotoxic receptor) and a tumor antigen [START_REF] Demaria | Natural killer cell engagers in cancer immunotherapy: Next generation of immuno-oncology treatments[END_REF]. The benefit of combining these therapies remains to be tested.

Harnessing the potential of apoptosis as a strategic approach to kill cancer cells is not new. Many common types of anti-cancer therapies, such as chemotherapy and radiotherapy, engage the mitochon-drial apoptosis pathway [START_REF] Singh | Regu-lation of apoptosis in health and disease: the balancing act of BCL-2 family proteins[END_REF]. The approval of the BH3 mimetic BCL-2 selective inhibitor venetoclax in several malignant therapies is a milestone in targeting the apoptotic pathway for cancer treatment. Venetoclax provides an additive effect to the anti-CD38 mAb daratumumab for NK-cell-mediated killing of multiple myeloma cell lines (Na-kamura et al., 2021). ln general, cancer cells that are sensitive to these drugs are more prone to apoptosis priming than their non-transformed counterparts, thus creating a therapeutic window for tumor response to pro-apoptotic therapy [START_REF] Singh | Regu-lation of apoptosis in health and disease: the balancing act of BCL-2 family proteins[END_REF]. However, toxicity of these pro-apoptotic agents is a burning malter limiting their efficacy. This issue could be circumvented by reducing the dose of these treatments and combining them to NK cell therapies, sa that the level of apoptotic stress induced within a cancer cell is insufficient to trigger apoptosis but enough ta lower the threshold for apoptosis, thereby sensitizing tumor cells for an efficient NK-cell-mediated killing.

Figure 1. Sensitizing tumor cells to NK-cell-mediated killing

Drugs such as the inducers of the DNA damage response sensitize tumor cells to NK-cell-mecliated killing by increasing the amounts of ligands recognized by NK cell activating receptors [START_REF] Ruscetti | NK cell-mediated cytotoxicity contributes to tumor contrai by a cytostatic drug combination[END_REF]. Pro-apoptotic drugs, such as chemotherapy, radiotherapy, and BH3 mimetics, promote mitochondrial apoptosis. BH3 mimetics diffuse into tumor cells and inhibit specific anti-apoptotic BLC-2 protein family members. Activated NK cells can produce granzyme B that cleaves BIO, which activates the apoptotic ettectors. Bath events synergize and converge toward the MOMP, cytochrome c release, and programmed cell death. Treatments that sensitize tumor cells to NK-cell-mediated killing could thus improve NK-cellbased therapies, including NK cell infusions, or the boost of NK cell functions by immune check point inhibitors (ICls) or NK cell engagers (NKCEs).

ACKNOWLEDGMENTS

The E.V. laboratory at CIML and Assistance-Pub-lique des Hôpitaux de Marseille is supported by funding from the European Research Council (ERG) under the European Union's Horizon 2020 research and innovation program (TILC, grant agreement no. 694502, grant agree-ment no. 875102 -Mlnfla-Tilc), the Agence Natio-nale de la Recherche, including the PIONEER Project (ANR-17-RHUS-0007), MSDAvenir, lnnate Pharma, and institutional grants to the CIML (IN-SERM, CNRS, and Aix-Marseille University) and Marseille lmmunopole.

DECLARATION OF INTERESTS

E. V. is an employee of lnnate Pharma.


Jérôme Denis
email: jerome.denis@minesparis.psl.eu

David Pontille
email: david.pontille@minesparis.psl.eu

Cultivating Attention to Fragility: The Sensible Encounters of Maintenance

Keywords: ecologies, material entanglements, maintenance, sensorial knowledge

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Maintenance practices have been an object of growing interest in recent years, and numerous researchers working in a variety of disciplines have made stimulating forays into a vast continent of jobs and ordinary activities which still appears largely unexplored [START_REF] Jackson | Speed, Time, Infrastructure: Temporalities of Breakdown, Maintenance, and Repair[END_REF][START_REF] Continent | R3pair Volume[END_REF]Strebel, Bovet and Sormani, 2019;Denis and Pontille, 2020b). This body of research works has notably played an active role in acknowledging the importance of material "reproductive work" in the constitution of social life, bringing to light the omnipresence of mundane practices dedicated to making things last. Among these "maintenance and repair studies", some have also initiated a fruitful dialogue with feminist theories of care, especially the work of [START_REF] Tronto | Moral Boundaries: A Political Argument for an Ethic of Care[END_REF], whose definition of care, articulated with Fisher, famously refers to "everything that we do to maintain, continue and repair our 'world' so that we can live in it as well as possible" (Fisher and Tronto, 1990: 40). Extending the empirical and analytical gesture aimed at "surfacing invisible work" [START_REF] Star | The Ethnography of Infrastructure[END_REF], this conversation between maintenance and care has made two important contributions to the ways in which social sciences approach the relationship between humans and artefacts. On one hand, it has participated in decentring the traditional focus on stability and persistence towards material fragility and its various manifestation [START_REF] Denis | Material Ordering and the Care of Things[END_REF]Domínguez Rubio, 2020;[START_REF] Henke | Repairing Infrastructures: The Maintenance of Materiality and Power[END_REF]. On the other hand, it has emphasized, and documented, the ethical and affective dimensions of maintenance [START_REF] Houston | Caring for the 'Next Billion' Mobile Handsets: Opening Proprietary Closures Through the Work of Repair[END_REF][START_REF] De La Bellacasa | Matters of Care: Speculative Ethics in More Than Human Worlds[END_REF].

In this chapter, we propose to further these reflections by exploring a less known, though complementary, aspect of maintenance: its sensorial dimension. If maintenance is a matter of affect, it is also a matter of attention. Material fragility is indeed anything but a transparent feature. In fact, when it appears obvious to anyone, this generally means that maintenance has failed. Fragility of things is thus something that one needs to become sensible to. Making things last entails cultivating a special relationship with them, keeping careful eyes and hands on their condition, and scrutinizing, even sensing, their transformations. Even though some scholars have highlighted the sensory skills involved in specific maintenance activities [START_REF] Dant | The 'Pragmatic' of Material Interaction[END_REF][START_REF] Dant | The Work of Repair: Gesture, Emotion and Sensual Knowledge[END_REF]Cállen and Sánchez Criadio, 2015), little is known yet on how maintainers concretely become attentive to fragility. What is this attention made of? What does it take? How is fragility experienced? How is it dealt with? Answering these questions implies describing maintenance activities as closely as possible, taking maintainers' preoccupations seriously, as mundane as they may seem. This is what we propose to do throughout the next sections, drawing on two recent investigations, the first one on graffiti removal in Paris (Denis and Pontille, 2020a), the second one on water systems management in France [START_REF] Florentin | Gestion Patrimoniale des Réseaux d'Eau et d'Assainissement en France[END_REF]. By looking at contrasting maintenance situations (urban maintenance articulated to city cleanliness on the one hand, infrastructure maintenance in utilities on the other), our aim is not to produce a systematic comparison, obviously. Rather, we take the opportunity of these two ethnographic inquiries to put to the fore various ways of cultivating attention to things, and of enacting fragility during maintenance interventions. As we will show, in both cases, maintenance operations do indeed take the form of situated explorations through which maintenance workers become sensible to the state of things, deal with their tendency to deteriorate and disintegrate, and foresee their capacity to progressively mutate. Such "material politics" [START_REF] Gregson | Performativity, Corporeality and the Politics of Ship Disposal[END_REF] paves the way for a particular "ecological thinking" (Puig de la Bellacasa, 2016), which in the case of urban settings and infrastructures broadens two wellknown approaches: the human ecology of the Chicago School [START_REF] Park | Human ecology[END_REF] and the ecology of infrastructures [START_REF] Star | Steps Toward an Ecology of Infrastructure: Design and Access for Large Information Spaces[END_REF]. Accompanying maintainers at work, and describing the encounters they cultivate within emerging material entanglements, enables us to consider further how humans interact with things and their environment. Such an ecological approach accounts for the daily presence and vitality of heterogeneous biological, animal, mineral entities in mundane urban practices, and the active "capacity of relation-creation, to how different beings affect each other, to what they do to each other, the internal 'poiesis' of a particular configuration" (Puig de la Bellacasa, 2016: 52).

Sensorial explorations

In order to understand how the maintenance workers' explorations are performed, and to familiarize the reader with the domains in which they take place, let's start by following the course of two interventions.

[Intervention 1]. Étienne starts the visit of a site on a mountainside where a reservoir, a valve room and a technical facility are located. For the past two years, he has been travelling the roads in the area to inspect the drinking water system works that are managed by the public utility hiring him. His visits also have an inventory function: the agency has recently incorporated many small municipalities and faces a considerable lack of knowledge about the nature of the facilities and their condition. His intervention starts outside. He goes around the elements that are directly accessible, inspecting the walls, ventilation ducts and roofs when they are visible. He takes pictures with his phone, and uses a voice recognition program that allows him to dictate his remarks: a moved tile, a crack in a wall, a spot of moisture. He enters the valve room and roughly draws the installation on his phone. He then goes down inside the reservoir, which has been emptied in preparation for cleaning by two agents who greeted us on arrival. Once downstairs, he explains: "The first thing to do is to stay put and look down, all the way across. Sometimes there are slopes or trenches where you can fall and get hurt. Then I look at the walls". He sweeps the beam of his flashlight across the floor, then over the walls. He approaches and passes his hand in several areas (figure 1). He then concentrates on the ceiling and finishes by sweeping the raft, which he walks over, still holding the flashlight. Once back up, he adds a few comments to his list and then takes some time to talk to the workers who are former employees of the municipality and who are still operating the facilities on site. He asks them some questions about the history of the site and the interventions that have been carried out in the past. Back at his office, Étienne transfers his photos to his computer and prints out his notes and sketches. He slips everything into a folder on which the name of the site is written, and which already includes a description form that he will fill in later. The file joins the stack of "to be processed" files that Étienne will eventually record in a database that he himself configured and which gathers all the items of each visit. This data informs the other people of the agency of the condition of the structures and makes it possible to prioritize interventions throughout the year. [Intervention 2]. This morning, Tom starts his round at Place de la Nation. After a quick glance at the printed map of the neighbourhood that he has taken out of his pocket, he chooses a street, then he goes into it, phone in hand, staring at the building facades. As a full-time "detector", he has been roaming the streets of Paris daily since 2005, in search of all forms of graffiti. Each of his detections initiates another round, that of an operator in charge of removing the unwanted inscriptions he has noticed. That's why Tom scrutinizes the facades so meticulously: the more accurate the information he provides, the easier it will be for his colleagues to proceed. As soon as he spots graffiti, he opens the dedicated application on his phone: "I create a file for each intervention by filling in the different categories", he explains. "The address and its geolocation, the time of intervention (including opening and closing times for stores), the type of vehicles to be used (for parking), the surface properties, the size of the graffiti calculated by square meters, the recommended intervention technique... Then I take a photograph, and I validate my entry (figure 2). Everything is directly transmitted and connected to the database". Despite the experience accumulated over the rounds, detection is not always so straightforward. Depending on the spread of the graffiti, Tom sometimes takes some time to reflect further before indicating the total surface to be processed. He may also come closer to the facades and frequently touches the graffiti, passing his hand over several times, before filling in his file. Some stone surfaces may be covered with a thin layer of almost invisible paint and require a specific removal technique. Others, made of porous material, may have been degraded over removal interventions and require careful reflection to avoid further deterioration. Apart from their specificities, these two scenes share several important features. They show inspection sessions, common to most maintenance occupations, which consist for the maintainers in making sure that "all is well" and, if not, in identifying what is called in the vocabulary of professional maintenance "disorders". These inspections are linked to a variety of documents (laws, standards, contracts, guides, etc.) that define some of their aspects, and guide part of their accomplishment.

Regarding water networks in France, for instance, the so-called Grenelle I (Article 27) and Grenelle II (Article 161) acts, which were followed by local regulations, have designated water network maintenance as a major national issue, notably by setting objectives in terms of reducing leaks and developing a sound knowledge of the network's components. Together with a profound reorganization of local governance that has redistributed the responsibilities among authorities at different scales, this legislative framework has contributed to transforming maintenance practices by placing them at the forefront of the management of water infrastructures. Etienne's activity is an integral part of this political and managerial agenda, as it draws on numerous forms and texts that identify a series of elements he has to pay attention to, and the main signs of failure he must take into account.

Graffiti removal in Paris is part of a municipal policy that is closely connected to the "broken windows" thesis initially formulated in the United States by [START_REF] Wilson | The Police and Neighborhood Safety: Broken windows[END_REF]. This doctrine instantiates a very particular maintenance epistemology (Denis and Pontille, 2020a) that makes any visual sign of disorder (a broken window, litter on the street, graffiti...) a disturbing element which, albeit minimal, must be taken care of as quickly and systematically as possible, at the risk of their proliferation. This is translated in legal documents and contracts in which it is stipulated that removal must be carried out at a steady pace: every day Tom's inspections are mandatory, and each graffiti he reports has to be erased within ten days.

These political, legal and technical documents, together with the situated inspection activities, emphasize how maintainers' attention combines a slow and routine mode of perception during rounds and a much more pressing and tense organizational rhythm.

These temporalities of maintenance point to distinct, and potentially conflicting, ways of paying attention to fragility. They are an important aspect of the heterogeneous assemblage through which specific features of things come to matter more than others. The expert exploration that maintainers engage in to appraise the condition of things is crucial in this process. Ethnographic observation is a good way to appreciate this expertise. Indeed, the maintainers' ability to perceive certain marks or minute traces contrasts sharply with that of the observer, who is absolutely incapable of distinguishing certain imperfections on the surface of things they struggle to examine scrupulously. During their rounds, maintainers demonstrate "connoisseurship", a form of knowledge, both intimate and skilled, which tends "to an appreciation of details rather than of the [thing] as a whole" (Ginzburg, 1979: 274). They spot sometimes infinitesimal clues, "details usually considered unimportant or even trivial" (p. 280), thanks to which they manage to assess the gaps between what things are and what they should be. This ability to explore the condition of the world is practiced through professional experience, sharpened into a "skilled vision", much like the cattle farmers observed by [START_REF] Grasseni | Skilled Vision. An Apprenticeship in Breeding Aesthetics[END_REF], who learn to estimate herds and cows by breaking away from the habits of ordinary perception.

As the very idea of "rounds" suggests, inspection is also a matter of both proximity and displacement. To take care of things, maintainers get close to them, which tells us that maintenance goes beyond the loose vocabulary of observation. The sensibility to fragility is activated though the encounter between the maintainers' body and that of the things. A matter of co-presence, of "staying with" (Jackson, 2016: 183), such proximity is all the more valuable as it goes with constantly reiterated displacements. The attention cultivated during the rounds is indeed enacted in motion: Etienne walks around the site he is visiting before moving into the tank, and once inside, he never stops walking around; Tom roams streets, his gaze meticulously focused on one building facade after another. These displacements are an attention operator in their own right. Repeated regularly, they fuel the emergence of salient features among things and the identification of material variations, and eventually fragility to manifest itself. In our cases (and in numerous other situations), maintainers' attention relies thus heavily on what [START_REF] Gibson | The Ecological Approach to Visual Perception[END_REF] termed the ambulatory dimension of perception.

Importantly, as researchers who have focused more specifically on diagnostic practices have shown [START_REF] Orr | Talking About Machines: An Ethnography of a Modern Job[END_REF][START_REF] Dant | The Work of Repair: Gesture, Emotion and Sensual Knowledge[END_REF][START_REF] Sanne | Making Matters Speak in Railway Maintenance[END_REF], such expert exploration is largely multisensorial. The use of hands, of course, is crucial for most maintenance workers (one of the origin of the verb to maintain, maintenir in French, is the latin verb "manūtenēre", which means "to hold with the hand"). In our cases, touch allows to appraise substances through manipulation, specify the identified problems, and anticipate further interventions. Though hearing and smelling are more rarely involved, they sometimes allow the detection of signs to which one must be able to pay attention to (an unusual smell that indicates infiltration in a structure for instance), or to supplement the first inspection steps (for example by striking against a wall whose resonance helps to identify its composition). Apparently insignificant and characterized by the monotony of their repetition, these gestures belong to what Dant (2010) calls "sensual knowledge", which both aims to produce knowledge about what happened and to guide the flow of the intervention as the close examination goes on.

The expert exploration at play during inspection sessions is actually twofold. On one hand, it is oriented towards the series of features and signs that are more or less precisely identified in technical documents. Never fully exhaustive, of course, these lists nonetheless constitute important resources for developing a careful scrutinization. On the other hand, maintainers cultivate a more "floating" kind of attention, which is genuinely open to discoveries.

Étienne has just finished his round. He is talking to the two cleaning agents who washed the tank after he made his inspection. One of them is leaning against the outer wall of the valve room. As he lifts his back from the surface of the wall, Stéphane stops in the middle of a sentence and looks at him, puzzled. He exclaims, "Wait, what's that noise there? Do it again…" The agent puts himself back in place, amused, and then pulls himself off the wall again. A loud "ploc" resonates. Étienne steps forward and feels the surface of the wall in several places with his hand, reproducing the suspicious noise (figure 3). "I can't believe it. What the hell is that thing? The paint's completely peeling off". He turns around and explains to the observer: "See, the colours are a bit different, here, it goes up very high. I would never have guessed. Maybe it means that there is an infiltration problem, or simply that the paint has been badly applied. It's outside, so it's not a big deal anyway. I still need to discuss this with my colleagues at the headquarters". He then grabs his phone to take pictures. After several shots, he hesitates, then: "Actually, I'm going to film, it's the only way to make them understand". He records a video of his hand pressing on the wall and reproducing the sound. What is at stake in this form of attention is to let unexpected events happen. Taking into account how material objects "surprise" the people who deal with them, [START_REF] Yaneva | How Buildings 'Surprise': The Renovation of the Alte Aula in Vienna[END_REF] shows, allows to reconsider material agency, beyond the idea of inert artefacts that human can turn into powerful and stable resources to do "politics by other means" [START_REF] Winner | Do Artifacts Have Politics?[END_REF]. Surprise emerges from "what is being added, above and beyond what is done and expected" (Hennion, 2015: 50). By leaving room for surprises, maintainers let walls, paint, doors or pipes act and "speak" [START_REF] Sanne | Making Matters Speak in Railway Maintenance[END_REF]. They acknowledge the capacity of things to demonstrate "a heightened presence" (Hennion, 2015: 39), and disturb the course of human action in manifesting their own signs of existence and transformation. Each maintenance inspection is therefore both a test of conformity and an opportunity for the thing to "makes itself knowable and lets itself being known" (Yaneva, 2008: 17).

Ecologies of fragility

Once inspections are completed, maintenance interventions go through even closer relationships to things, in the course of which maintainers encounter a great variety of materials: plastic, wood, marble, concrete, plaster, bricks, water, sand, cement, stones… as well as soil, plants, various pollutants, in the case of water networks management; and inks and paints (from graffiti itself or already on surfaces) in the case of graffiti removal. Among these heterogeneous materials, some are clearly part of the thing that has to be maintained, while others compose its environment. Some others, in contrast, cannot be easily distributed into this binary categorization beforehand. Much more than identifiable objects with stable properties, maintainers deal with what Bennet (2010) and [START_REF] Ingold | Toward an Ecology of Materials[END_REF] call an "ecology of materials", in which they try to sense the behaviour of material entities.

For instance, before removing a graffiti, the workers we observed spent several minutes touching the graffitied surface so as to estimate the degree of penetration of the graffiti paint into the stone, examine the invasiveness of an acid-based ink on a shop window, or anticipate how well the covering paint will hold in contact with the surface's components depending on the weather conditions (figure 4). [START_REF] Edensor | Entangled Agencies, Material Networks and Repair in a Building Assemblage: The Mutable Stone of St Ann's Church, Manchester[END_REF] terms "entangled agencies". Conversely to a sign of passivity or a lack of agency, material fragility is thus apprehended during these sensible encounters as the result of the active, uncertain, and even surprising relationships between materials. From this point of view, maintainers come close to the figure of the "craftsman" that Deleuze and Guattari (1987: 409) have defined as "one who is determined in such a way as to follow a flow of matter", rather than interacting with previously stabilized entities whose characteristics are clearly circumscribed. They cultivate a careful attention towards material variations, modulations and movements, feeling these singularities in the making, engaging with the flow of matter so they can temporarily reorient its always transitory course.

The metaphor of flow "monitoring" should, however, be used with precaution. It could indeed give the impression of an asymmetrical confrontation between a human being in a situation of detached "perception" on the one hand, and an inert world that would be "apprehended" and transformed by the former on the other. In fact, each intervention involves not only the eyes and sensory organs of the maintainers, but also their flesh, muscles and bones, which are entirely part of the material ecology of the situation. These material relations between workers' bodies and things become particularly salient when tensions arise. For example, removing graffiti involves handling tools (high-pressure water spraying guns, extension arms for paint rollers, spatulas for manually mixing sometimes very thick paints, etc.) which, as for many "physical" jobs, can lead to pain or even musculoskeletal disorders, notably in the back and hands. To these potential sources of pain is added the harmfulness of the solvents the removal workers sometimes use without wearing their protection: without a mask, their prolonged inhalation causes lung problems, and without a cap or gloves, the burn is instantaneous on the skin and hair. Water networks management comes with its own physical stresses, starting with those generated by the very access to the facilities and infrastructures: going down ladders, moving in confined spaces, walking on soils with uncertain surfaces, and so on (figure 5). Just as the bodies of the maintainers, the tools they handle and the products they use obviously participate in the material ecology of maintenance. In the case of graffiti removal, taking into consideration the particular sources of transformation they may generate is a constant concern. For instance, a table included in the particular technical specifications of the call for tenders of the City of Paris (Annex 1, for the 2018 version) lists compatible materials and helps to select the appropriated removal technique by identifying forbidden combinations. Beyond these main principles, the practical implementation of techniques and the use of instruments require adjustments to be made in situation. From one intervention to another, some maintainers pay special attention to the reactions that the tools and products they use are likely to trigger among the heterogeneous elements at play.

Jules walks up to a door full of graffiti, and touches it: "That's baked paint, it's strong, it's factory-made! Nothing has been added to it, I can go straight with bubblegum [one of the four chemical solvents at his disposal]". He pours some product on his scraper and carefully rubs the door, alternating quickly with a cloth pass, without pressing, while specifying: "It is necessary to wipe quickly so as not to let it act too long. Even on this paint it would make a much lighter stain". While he continues (figure 6), he insists on the quality of the rags: "The choice of rags is important. They must not scratch the paint. The recent purchase order they made doesn't work, the new cloths damage the surfaces. It's the same with the scrapers, by the way. I keep some old, tired ones that hardly scratch anymore, for cases like this one. Because on a dark paint, black like here, it scratches very easily. You have to be really careful". Solvents, paints, rollers, rags: maintainers' "intuition in action" (Deleuze and Guattari, 1987: 409) unfolds from the fine-tuning of the gestures that accompany the action of tools and products, to the accuracy of the rhythm of their passage and coating, to the adjustment of their own material qualities. This is what makes Jules prefer here the use of certain worn scrapers whose softened action makes it possible to preserve the paint on the door. The choice of removal instruments and their careful handling are intended to "leave as few traces as possible of my passage", as Jules later explained to us. This meets a professional requirement: the technical specifications of the contract prohibit the removers from producing "cleanness stains" (p. 14, for the 2018 version), under penalty of sanction by the municipality or litigation on the part of the facades' owners. But their level of precision is also a matter of personal meticulousness. On several occasions, Jules showed us traces of intervention by colleagues that he described as less "serious". Whatever the fine-tuning of these gestures and the level of operators' personal commitment, it is in the course of this complex choreography that the "material politics" [START_REF] Gregson | Performativity, Corporeality and the Politics of Ship Disposal[END_REF] of graffiti removal is actualized, in the intertwinement of facades, graffiti, workers' bodies, the instruments they handle and the products they use.

The attention to fragility that unfolds during maintenance interventions is thus a matter of pace, gestural coordination, balanced rotation between the action of tools and products, and the reactions of the maintained thing itself within a material ecology whose boundaries are never fixed once and for all. During these reciprocal transactions where sensibilities are distributed, a lot of materials may interact with each other, those that are maintained as well as those that maintain, those that must be preserved as well as those that must be eliminated. Far from being a stable and intrinsic quality against which maintainers should struggle, material fragility is ecologically enacted through these close encounters during which the decay of some entities (here, Jules' rags) and the destruction of others (the graffiti) are used as means to preserve a thing (a surface in the city), while the degradation of yet others (the very maintainers' body) appears as the unevenly distributed price to pay for this thing to last. In this sense, these ecological encounters are always simultaneously political, affective, sensual and, ultimately ontological.

Thinking with maintenance

By sticking as close as possible to the gestures and words of the maintenance workers, we seek less to produce knowledge about maintenance, though, than to develop a way of thinking through and with it. What does it mean concretely? If "thinking-with" is a way to making worlds happen (Puig de la [START_REF] De La Bellacasa | Nothing Comes Without its World': Thinking with Care[END_REF], what kind of world emerges when thinking with maintenance? Mierle Laderman Ukeles' work offers a fantastic inspiration to answer this question. Author of the pungent Manifesto for Maintenance Art, 1969! Ukeles, a feminist conceptual artist, dedicated her whole career to bring maintenance in the realm of arts, questioning in return what has so far prevented modernity from taking maintenance practices into consideration. In the catalogue of the major retrospective that the Queens Museum dedicated to her work in 2016, Phillips (2016) describes Ukeles' performances as a gesture which incites spectators to pay attention to the maintainers' own attention. Placing maintenance in its very mundaneness operates thus what she calls a "transfusion of attention" (p. 99).

Ukeles' very first performance is particularly telling in this regard. Entitled Transfer: The Maintenance of the Art Object: Mummy Maintenance: With the Maintenance Man: The Maintenance Artist, and the Museum Conservator, it consisted in inviting two museum employees in front of the glass vitrine protecting a 5000-year-old Egyptian mummy at the Wadworth Atheneum in Hartford, Connecticut. The performance was organized in three successive moments. It began with the cleaning of the vitrine surface by the maintenance worker, who did it as he had done every day since the piece was present in the museum. In a second step, the worker's gestures were repeated identically by Ukeles, who had carefully observed them. Ukeles then stamped the vitrine with the words "Maintenance Art Work" written in ink. Such a gesture, directly inspired by Marcel Duchamp's work, turned the pristine vitrine into evidence that this protective furniture of art display had become itself a work of art. In a third step, the conservator was then compelled to authenticate it and write a report. Above all, he became the only person responsible and authorized to carry out future cleaning operations. Through this performance, attention has thus been concretely and even legally transferred: the conservator had now to perform the careful gestures of maintenance the maintenance worker used to do. Such a posture is highly political, since attention engages specific power relationships. Just as the responsibilities between those who "bear the burdens (and joys) of care" and those who "escape them" are distributed according to social and institutional contexts (Tronto, 2013: 32-33), there is a big difference between those who can afford not to pay attention to the fragility of things, and those who have no choice but to be sensible to it. Ukeles' work invites to reduce these asymmetries and redistribute both attentional abilities and responsibilities.

This is also what we try to do, in humble terms, in our own research work. By making ourselves attentive to the attention of maintainers, by conducting inquiries alongside their own inquiries, and then write about it, we strive to produce the conditions for what Citton (2016) calls a "meta-attentional engagement", in which the reader "is found to be plugged into the attentional experience of another more or less strongly subjectivized perception of the world, through which a certain reality is revisited" (p. 139).

In the particular context of this collective volume, we believe that exploring maintainers' attention helps to grasp two important aspects. First, it outlines ways to assume a specific ecological posture. As a situated and political process that implies materials, bodies, instruments and texts, maintenance also draws on a delicate art of provoking and cultivating encounters within emerging material ecologies wherein what acts, and what interacts, is never completely known in advance. This consideration for material ecologies implies to become aware of the constant mutations things are made of, from micro-variations to dramatic transformations, and the flow of materials that animate them. Maintainers, we argue, can help us to cultivate "deep attention to materiality and embodiment in ways that rethink relationality, in ways that suggest a desire for tangible engagements with mundane transformations" (Puig de la Bellacasa, 2017: 112). They are among the best positioned to shed light on the incessant material metamorphosis that characterize every object, which "are continually becoming in the course of their lives" (Gregson et al., 2009: 250).

But cultivating such attention to fragility is also ecological in the sense that it is a matter of relationships and connections. Understanding how maintainers engage with the objects they take care of leads to reconsider how humans interact with things, even the most insignificant ones, way beyond "cultural" or "symbolic" relations. Maintenance both acknowledges and enacts the vital interdependencies and "attachments" which hold that or who is cared for and the ones who care for them together [START_REF] Hennion | Attachments, You Say? … How a Concept Collectively Emerges in one Research Group[END_REF].

Since it invites to be simultaneously attentive to the maintainers' careful physical explorations and to the degradation of their bodies, this ecological approach resonates with Jackson and Houston's call to "think [the] poetics and political economy [of maintenance] together" (Jackson and Houston, 2021: 246). Thinking ecologically helps indeed to identify the intricacies of material fragilities and thus to tackle some of the ambivalences of care (Puig de la Bellacasa, 2017). In our case, this consists in exploring how urban maintenance articulates capitalist and disciplinary considerations with the aesthetic and affective dimensions of maintainers practices. Such posture has theoretical, but also methodological implications. As Domìnguez [START_REF] Rubio | Still Life. Art and the Ecologies of the Modern Imagination[END_REF] has stated in his careful exploration of the worlds of art conservation, because it evolves in the "ecological nexus" (p. 8) through which things are made to last, maintenance (and care) requires imagining ways of conducting ecological inquiries. And what better guides to initiate this kind of inquiries than those who engage in it on a daily basis, those who have become experts in material mutations and entanglements by constantly taking care of things? Finally, the second aspect maintainers attention invites to reflect on concerns the idea of reparation itself. Round after round, intervention after intervention, maintenance performs a kind of reparation that has no end, and cannot be "settled" once and for all. Lying at the core of maintenance, such a form of repair is always in the making [START_REF] Denis | Before Breakdown, After Repair: The Art of Maintenance[END_REF]. Fragility requires a continuous attention: routine and uneventful work, which gains its value from repetition and constancy and acknowledges its never-ending modest, though vital, contribution. What maintenance can teach us is that, just like the artwork [START_REF] Souriau | Les différents modes d'existence. Suivi de Du mode d'existence de l'oeuvre à faire[END_REF][START_REF] Latour | Reflections on Etienne Souriau's Les différents modes d'existence[END_REF] wrote about, things from mundane artefacts to spectacular pieces of urban infrastructure always remain "to-be-done".

Figure 1 .

 1 Figure 1. Inspecting a water reservoir: touching (© J. Denis)

Figure 2 .

 2 Figure 2. Reporting graffiti presence (© J. Denis & D. Pontille)

Figure 3 .

 3 Figure 3. Ploc, ploc, what's that sound? (© J. Denis)

Figure 4 .

 4 Figure 4. Touching the graffitied surface (© J. Denis & D. Pontille)

Figure 5 .

 5 Figure 5. Inspecting a water reservoir: the descent (© J. Denis)

Figure 6 .

 6 Figure 6. Chemical solvents and the dance of rags (© J. Denis & D. Pontille)

Acknowledgements

We warmly thank the people who agreed to answer to our questions and to be shadowed during their working days. We are particularly grateful to Annemarie Mol, Mandy de Wilde, Antoine Hennion, and Alexandre Monnin for their insightful comments and suggestions on an earlier version of the text. We also thank Sara Aguiton, Marc Barbier, and Soraya Boudia for their invitation to the 2019 edition of the Institut Francilien Recherche Innovation Société (IFRIS) summer school during which an early draft has been discussed. Finally, we are extremely thankful to Maddalena Tacchetti, Dimitris Papadopoulos and Maria Puig de la Bellacasa for their careful reading and stimulating comments.