Alexandre Zouaoui

Gedeon Muhawenayo

Behnood Rasti

Fellow, IEEE Jocelyn Chanussot

Julien Mairal

Entropic Descent Archetypal Analysis for Blind Hyperspectral Unmixing

Keywords: Hyperspectral image, remote sensing, blind spectral unmixing, non-negative matrix factorization, archetypal analysis, entropic gradient descent

I. INTRODUCTION

Hyperspectral imaging (HSI) [START_REF] Landgrebe | Hyperspectral image data analysis[END_REF]-[5] consists of measuring the electromagnetic spectrum in a scene by using multiple narrow spectral bands. Thanks to its richer spectral information compared to traditional RGB images, HSI enables more accurate materials identification, leading to a broad range of applications including crop monitoring in agriculture [6], waste sorting [7], food safety inspection [START_REF] Gowen | Hyperspectral imaging-an emerging process analytical tool for food quality and safety control[END_REF], or mineralogy [9].

Remote sensing [START_REF] Clark | Imaging spectroscopy: Earth and planetary remote sensing with the USGS Tetracorder and expert systems[END_REF], [START_REF] Bioucas-Dias | Hyperspectral remote sensing data analysis and future challenges[END_REF], such as airborne or satellite imagery, yields hyperspectral images whose pixels capture several objects or materials. As such, each pixel can include several pure spectral components (called endmembers), mixed in different proportions [12]. Any further analysis hence requires identifying and disentangling endmembers present in a scene before estimating their respective proportions, or fractional abundances, within each pixel of the HSI [13]. Since the endmembers spectrum signatures are not known beforehand and must be estimated from data, this operation is named blind hyperspectral unmixing (HU) [14], [15] owing to its link with blind source separation [START_REF] Adão | Hyperspectral imaging: A review on UAV-based sensors, data processing and applications for agriculture and forestry[END_REF].

Alexandre Zouaoui, Jocelyn Chanussot, and Julien Mairal are with Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France. Gedeon Muhawenayo worked on this project while he was there.

Behnood Rasti is with Helmholtz-Zentrum Dresden-Rossendorf (HZDR) This paper has supplementary downloadable material available at http://ieeexplore.ieee.org. Contact alexandre.zouaoui@inria.fr for further questions about this work.

In this paper, we adopt a linear mixing model since it is often relevant in remote sensing scenes where mixtures occur between macroscopic materials. Therefore, we assume that each observed pixel can be represented as a linear combination of endmembers and some additive noise. In other words, we are interested in tackling unsupervised linear HU [13].

Further assumptions on the nature of endmembers are generally needed to estimate meaningful spectra. For instance, it can be assumed that there exists at least one pure pixel for each material present in the scene. The problem then requires finding these pure pixels within the original image. The pure pixel assumption is at the core of several geometrical endmember extraction methods including pixel purity index (PPI) [17], N-FINDR [START_REF] Winter | N-FINDR: An algorithm for fast autonomous spectral end-member determination in hyperspectral data[END_REF] and vertex component analysis (VCA) [START_REF] Fox | Applications of hyperspectral mineralogy for geoenvironmental characterisation[END_REF]. Once endmembers have been extracted, abundances can be estimated by minimizing the least squares errors between the original input spectra and the linearly reconstructed spectra as long as the abundances fractions satisfy the two physical constraints stating that they should be non-negative and sum to one for each pixel [START_REF] Heinz | Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery[END_REF]. That being said, pure pixels are often missing in real scenarios. In the absence of pure pixels and in the case of linear models, endmembers and abundances can be simultaneously estimated by solving a constrained or penalized non-negative matrix factorization problem (NMF) [START_REF] Lee | Algorithms for non-negative matrix factorization[END_REF]. For example, the authors of [START_REF] Zhuang | Regularization parameter selection in minimum volume hyperspectral unmixing[END_REF] have proposed a formulation that involves a data fidelity term and a minimum volume regularization term on endmembers, whose minimization consists in alternating between solving for endmembers and abundances.

In this work, we do not assume the existence of pure pixels as they are often missing in real data, since, for instance, the spectral signatures of endmembers in HSI can be significantly affected by various changes in atmospheric, illumination, and environmental conditions within the scene [START_REF] Borsoi | Spectral variability in hyperspectral data unmixing: A comprehensive review[END_REF]. There are multiple strategies that can be employed to tackle spectral variability such as (i) augmenting the linear mixing model [24], (ii) using a tensor-based approach, like the Sparsity-Enhanced Convolutional Decomposition (SeCoDe) method [START_REF] Yao | Sparsityenhanced convolutional decomposition: A novel tensor-based paradigm for blind hyperspectral unmixing[END_REF], (iii) performing unmixing in orthogonal subspaces as in [START_REF] Ren | Orthogonal Subspace Unmixing to Address Spectral Variability for Hyperspectral Image[END_REF]. Instead, we mitigate the effect of spectral variability by (i) normalizing each pixel by the ℓ 2 -norm of its spectrum as a pre-processing step and (ii) modeling endmembers as convex combinations of pixels present in the scene. Not only HSI pixels are linear combinations of the estimated endmembers under the linear mixing model, but the estimated endmembers are also convex combinations of pixels. This corresponds to the archetypal analysis (AA) formulation introduced by Cutler and Breiman in [START_REF] Cutler | Archetypal analysis[END_REF]. AA has the advantage to be more interpretable than NMF because the basis elements (i.e. endmembers) are directly constructed from the data points (i.e. pixels). In addition, since the estimated endmembers spectral signatures generally correspond to averaging the contributions of several pixels, the resulting estimation appears to be more robust to noise and spectral variability than pure pixel methods that only select one pixel per endmember. However, AA usually suffers from a high data fitting error because the basis elements are constrained to be contained in the convex cone of the data points [START_REF] Handschutter | Near-convex archetypal analysis[END_REF].

The contributions and innovations of this article are as follows:

1) We propose a new hyperspectral unmixing algorithm relying on entropic gradient descent for archetypal analysis. Our approach (i) provides solutions for hyperspectral unmixing as good as traditional alternating optimization schemes based on projected gradient methods or active set algorithms, and (ii) allows more efficient GPU implementations.

2) The efficiency of our method enables us to make a key practical contribution, consisting of an ensembling mechanism along with an appropriate model selection procedure, which makes our method almost parameterfree and thus easy to use (the only sensitive parameter is the number of endmembers we want to estimate). 3) Our approach, available in an open-source package 1 , outperforms state-of-the-art matrix factorization and deep learning methods on six standard real datasets. The remainder of this paper is organized as follows. Section II presents related works for unsupervised linear hyperspectral unmixing. Section III introduces our method. Section IV presents experimental results highlighting the performance of our proposed approach. Finally, we conclude the article and underline future research directions in Section V.

II. RELATED WORK ON UNSUPERVISED LINEAR HYPERSPECTRAL UNMIXING

In this section, we present the framework of non-negative matrix factorization (NMF) before introducing archetypal analysis (AA). Finally, we mention widely used deep learning architectures that tackle blind hyperspectral unmixing.

a) Non-negative matrix factorization (NMF): NMF [START_REF] Lee | Algorithms for non-negative matrix factorization[END_REF], [START_REF] Paatero | Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values[END_REF] is popular for hyperspectral unmixing. It consists of factorizing a matrix representing the HSI signal into the product of two matrices with non-negative entries, where one factor carries the endmembers whereas the other one represents the abundances for each pixel (typically with the constraint that abundances sum to one). Several variants of NMF have been proposed for this task. For instance, the method of minimum volume constrained non-negative matrix factorization (MVC-NMF) of [30] uses a minimum volume term for endmembers that effectively gets rid of the pure pixel assumption. In minimum dispersion constrained NMF (MiniDisCo) [START_REF] Huck | Minimum dispersion constrained nonnegative matrix factorization to unmix hyperspectral data[END_REF], the regularization function is called dispersion, which encourages endmembers with minimum variance to avoid degenerate 1 Code is available at https://github.com/inria-thoth/EDAA solutions and improve the unmixing robustness to flat spectra. Other approaches design regularization functions for abundances such as [32]- [START_REF] Yao | Nonconvex-Sparsity and Nonlocal-Smoothness-Based Blind Hyperspectral Unmixing[END_REF]. NMF typically uses a least-squares objective function, which may not be suitable for handling noise and outliers in the data. As a result, researchers have proposed alternative approaches, such as general loss-based NMF [35] and self-paced NMF [36]. These methods utilize different optimization criteria that can be more robust to noisy data and outliers.

b) Archetypal analysis (AA): AA was first introduced in [START_REF] Cutler | Archetypal analysis[END_REF] and consists in modeling individual pixels present in the HSI as a linear mixture of archetypes (here the endmembers). Interestingly, archetypes are defined as convex combinations of the individual pixels present in the HSI. Motivated by the good interpretability of AA, Zhao et al.

[37] have proposed a kernelized variant, which enables greater modeling flexibility at the cost of an extra hyper-parameter (the bandwidth of the RBF kernel functions). Notably, they use the relaxation form introduced in [38] to handle the cases where endmembers are located outside of the convex hull of the data.

Inspired by the robust AA formulation introduced in [39] that considers the Huber loss instead of the squared loss to reduce the impact of noise and outliers, Sun et al. [START_REF] Sun | Pure endmember extraction using robust kernel archetypoid analysis for hyperspectral imagery[END_REF] have introduced a robust kernel archetypal analysis (RKADA) method for blind hyperspectral unmixing. Their approach refines the standard AA formulation by adding a binary sparse constraint on the pixels contributions. Consequently, their method ensures that each endmember actually corresponds to real pixels rather than a sparse linear mixture of all pixels. The resulting optimization algorithm is a block-coordinate descent scheme that uses an active-set method [41] to solve the smooth least squares optimization subproblems with a simplicial constraint. As noted by [START_REF] Chen | Fast and robust archetypal analysis for representation learning[END_REF], the active-set algorithm can be seen as an aggressive strategy that can leverage the underlying sparsity of the subproblems solutions.

Recently, Xu et al. [START_REF] Xu | L1 Sparsity-Constrained Archetypal Analysis Algorithm for Hyperspectral Unmixing[END_REF] have proposed an ℓ 1 sparsityconstrained AA algorithm to increase the sparsity of the abundances. Rather than using an active-set algorithm, they adopt a projected gradient method [START_REF] Lin | Projected gradient methods for nonnegative matrix factorization[END_REF] to solve the resulting optimization subproblems inside an alternating scheme. Their optimization approach is similar to the Near-Convex Archetypal Analysis (NCAA) [START_REF] Handschutter | Near-convex archetypal analysis[END_REF] method which aims at combining the advantages of both AA and NMF, by requiring the endmembers to be linear, instead of convex, combinations of the pixels. c) Autoencoders (AE): Interest in deep learning has been rapidly growing in many fields including remote sensing [44] thanks to the increasing amount of available data, rising computational power, and the development of suitable algorithms. Autoencoders for hyperspectral unmixing have been widely used since the pioneering work of Licciardi et al. [45]. In a nutshell, the encoder transforms an input into abundances maps that are then linearly decoded by the decoder. The encoder activations correspond to the fractional abundances while the parameters of the linear decoder layer correspond to the endmembers spectra. While the AE described above corresponds to a basic fully connected architecture, refinements have been introduced such as using a loss functions that involves the spectral angle distance and a dedicated sparsity penalty term as in [46], or convolutions to take advantage of the spatial structure in HSI [START_REF] Palsson | Convolutional autoencoder for spectral-spatial hyperspectral unmixing[END_REF]. See [48] for a comprehensive study on the various existing autoencoder architectures for blind hyperspectral unmixing. In addition to simple autoencoders, researchers have proposed more complicated architectures such as (i) a siamese network architecture, known as EGU-Net [49], which aims to enhance endmember extraction using nearly pure pixels, (ii) cascaded autoencoders as in [START_REF] Gao | CyCU-Net: Cycle-consistency unmixing network by learning cascaded autoencoders[END_REF] whose cycle-consistency property leads to improved unmixing performances. Finally, it is worth mentioning the approach by Rasti et al. [51] to incorporate a geometrically motivated penalty into deep learning-based approaches. Their method, called minimum simplex convolutional network (MiSiCNet), uses both geometrical and spatial information, by means of a convolutional encoder-decoder architecture, to tackle unsupervised hyperspectral unmixing.

Although the surge of deep learning methods has led to improving overall unmixing performances, training a single network per image remains costly due to the required extensive hyperparameters search. Moreover, training such networks requires GPU, and yet it can be considerably slower than traditional methods that run on CPU.

III. METHODOLOGY

In this section, we present our model formulation before describing its optimization. Next, we mention implementation details required to run our approach. Finally, we explain how to leverage our efficient GPU implementation and propose a procedure to make our model robust to hyper-parameter choices and thus easy to use in practice.

A. Model formulation

Let X in R L×N be a hyperspectral image (HSI) where L is the number of channels and N is the number of pixels. We assume a linear mixing model such that

X = EA + N, (1)
where E in R L×p is the mixing matrix composed of the discretized spectra of p endmembers over L channels, and A in R p×N is the abundance matrix that describes, for each pixel, the fraction relative to each endmember. Finally, N represents some classical noise occuring in hyperspectral imaging. We are interested in tackling the blind unmixing setting where both the mixing and abundance matrices are unknown. However, we assume that the number p of endmembers is known. Since E represents the endmembers reflectance over L channels, it follows that its elements should be non-negative, i.e. , E ≥ 0. This is also the case for A, whose columns represent the abundances for each pixel, which in addition, should sum to one. In other words, each column of A = [a 1 , . . . , a N ] belongs to the simplex ∆ p defined as:

∆ p ≜    a ∈ R p s.t. a ≥ 0 and p j=1 a j = 1    .
(2)

The previous model and constraint yields the classical optimization problem

arg min E, A 1 2 ∥X -EA∥ 2 F . s.t. E ≥ 0, a i ∈ ∆ p for 1 ≤ i ≤ N, (3)
which is a variant of non-negative matrix factorization.

The archetypal analysis formulation we consider adds a constraint and forces the endmembers to be convex combinations of the pixels present in X. Formally, it simply means that there exists a matrix B in R N ×p such that E = XB and the columns of B are in the simplex ∆ N (their entries are non-negative and sum to one). This formulation yields the archetypal analysis formulation [START_REF] Cutler | Archetypal analysis[END_REF]:

arg min A, B 1 2 ∥X -XBA∥ 2 F , s.t. a i ∈ ∆ p for 1 ≤ i ≤ N, b j ∈ ∆ N for 1 ≤ j ≤ p (4)
where

A = [a 1 , . . . , a N ] and B = [b 1 , . . . , b p ].

B. Optimization

Minimizing (4) is difficult since the objective function is not jointly convex in (A, B). However, it is convex with respect to (w.r.t.) one of the variables when the other one is fixed [38]. Consequently, it is natural to consider an alternating minimization scheme between A and B, which is guaranteed to asymptotically provide a stationary point of the problem [START_REF] Bertsekas | Nonlinear programming[END_REF]. Yet, because of the non-convexity of the objective, the choice of optimization algorithm is important as different stationary points may not have the same quality in terms of statistical estimation. In other words, different optimization procedures may have a different "implicit bias", a phenomenon that is currently the focus of a lot of attention in machine learning [53], especially for deep learning models, suggesting that it may also be key for blind HU.

In this paper, we adopt an optimization scheme called entropic gradient descent which has shown better theoretical properties in terms of convergence rates than gradient descent when optimizing over the simplex [START_REF] Beck | Mirror descent and nonlinear projected subgradient methods for convex optimization[END_REF]. Our second motivation was the simplicity of the algorithm, which does not require performing orthogonal projections on the simplex, nor dealing with complicated active-set rules as in [START_REF] Chen | Fast and robust archetypal analysis for representation learning[END_REF]. This enables us to take advantage of modern GPUs. We now present mathematical details before discussing implementation.

1) Entropic descent algorithm (EDA): As noted in [START_REF] Beck | Mirror descent and nonlinear projected subgradient methods for convex optimization[END_REF], the entropic descent algorithm is simply a gradient descent method with a particular choice of a Bregman-like distance [START_REF] Bregman | The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming[END_REF] generated by a specific function, here the negative entropy. As explained in [START_REF] Teboulle | Entropic proximal mappings with applications to nonlinear programming[END_REF], the choice of an appropriate distance-like function tailored to the geometry of the constraints, here the simplex, provide theoretical benefits in terms of convergence rates. The minimization over the simplex is precisely the reason why we adopt the negative entropy to derive the updates of the alternating minimization scheme. Formally, the negative entropy function h is defined as follows: for

x in R d , h(x) = d i=1 x i ln(x i ) if x ∈ ∆ d , +∞ otherwise, (5)
with the convention that 0 ln 0 ≡ 0.

h exhibits several desirable properties, including convexity on ∆ d . This enables us to consider D h , the Bregman divergence [START_REF] Bregman | The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming[END_REF] w.r.t. h, defined, for x and

y in R d D h (x, y) = h(x) -h(y) -∇h(y) ⊤ (x -y), (6)
which is also called the Kullback-Leibler divergence. By convexity of h, we naturally have D h (x, y) ≥ 0.

We start by considering the following generic optimization problem: min

z∈∆ d f (z), (7)
where f is a convex Lipschitz continuous function with a gradient at z ∈ ∆ d denoted by ∇f (z) and ∆ d corresponds to the d-dimensional simplex (2). The algorithm to solve (7) we consider performs the following iterates, given

z k ∈ ∆ d , z k+1 ← arg min z∈∆ d ∇f (z k ) ⊤ (z -z k ) + 1 η k D h (z, z k ) .
(8) If D h was simply a squared Euclidean norm, we would recover a projected gradient descent algorithm. By using instead the Bregman distance function D h derived from the negative entropy, we obtain the entropic descent method. Here D h measures the distance between two vectors in ∆ d . As such, the next iterate z k+1 should aim for the optimal balance between taking a gradient step and moving the least from the current iterate z k according to the geometry induced by h, with η k controlling this trade-off.

We will now see how the negative entropy h yields explicit steps that effectively enforce the simplicial constraints. Indeed, after a short classical calculation (see, for instance [START_REF] Beck | Mirror descent and nonlinear projected subgradient methods for convex optimization[END_REF]), it is possible to show that the update ( 8) is equivalent to the following one: for all j in {1, . . . , d},

z k+1 j = z k j e -η k ∇f (z k )j d l=1 z k l e -η k ∇f (z k ) l , (9)
where z k j is the j-th entry of the vector z k and similarly, ∇f (z k ) j is the j-th entry of ∇f (z k ).

We start by writing the Lagrangian corresponding to the constraint d j=1 z j = 1 using (6):

L(z, ν) = ∇f (z k ) ⊤ z + 1 η d j=1 z j ln(z j ) - 1 η d j=1 z j ln(z k j ) + ν   d j=1 z j -1   . (10)
Next, for j ∈ {1, . . . , d}, we consider the derivative of L w.r.t. the j-th component of z ∈ ∆ d :

∇ zj L(z, ν) = ∇f (z k ) j + 1 η ln(z j ) - 1 η ln(z k j ) + ν + 1 η , (11)
We now consider the equation ∇ zj L(z, ν) = 0. Then, we have:

ln(z j ) = ln(z k j ) -η∇f (z k ) j -ην -1, (12)
and

z k+1 j = z k j e -η∇f (z k )j e -ην-1 . ( 13
)
As a result, z k+1 j ≥ 0 and it remains to choose ν such that d j=1 z k+1 j = 1 as in (9). It is thus easy to see that the iterates (z k ) k∈N stay in the simplex ∆ d , and it is possible to show (see [START_REF] Beck | Mirror descent and nonlinear projected subgradient methods for convex optimization[END_REF]) that the sequence (z k ) k∈N converges to the set of solutions of (7) with the appropriate step-sizes η k . Interestingly, the update (9) can be implemented efficiently by using the softmax function, assuming the entries of z k are positive:

z k+1 = softmax log(z k ) -η k ∇f (z k ) . (14)
where log(z k ) is the vector carrying the logarithm of each entry of z k . This update immediately suggests a high compatibility with GPUs.

2) Alternating optimization scheme: We are now in shape to describe an alternating optimization scheme, consisting of performing, alternatively, K 1 updates of EDA for minimizing A when B is fixed, and vice versa using K 2 updates. This strategy is presented in Algorithm 1. Formally, by replacing the generic function f with the functions corresponding to the two optimization subproblems, we obtain the following updates:

A k+1 = softmax log(A k ) + η 1 k B ⊤ X ⊤ (X -XBA k ) , (15)
B k+1 = softmax log(B k ) + η 2 k X ⊤ (X -XB k A)A ⊤ , (16)
where log(A k ) is the matrix carrying the logarithm of each entry of A k while softmax is applied in parallel on the columns of log(A k ) + η 1 k B ⊤ X ⊤ (X -XBA k ). Note that when A and B are initialized with positive values, these iterates keep them positive. In addition, our optimization strategy does not require inverting any matrix contrarily to ADMM-based approaches [START_REF] Bioucas-Dias | Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing[END_REF].

C. Implementation details a) Normalization: The input image X = [x 1 , . . . , x N ] is ℓ 2 -normalized for each pixel: for all i in {1, . . . , N },

x i ← x i ||x i || 2 , ( 17
)
where x i denotes the i-th pixel. This step is important to gain invariance to illumination changes.

Algorithm 1 Entropic Descent Archetypal Analysis (EDAA)

1: Input: ℓ 2 -normalized data X in R L×N ; p (number of endmembers); T (number of outer iterations); K 1 (number of inner iterations for Â); K 2 (number of inner iterations for B). 2: Initialize  ∈ R p×N using [START_REF] Winter | N-FINDR: An algorithm for fast autonomous spectral end-member determination in hyperspectral data[END_REF]. 3: Initialize B ∈ R N ×p using [START_REF] Fox | Applications of hyperspectral mineralogy for geoenvironmental characterisation[END_REF]. 4: Set η 1 according to [START_REF] Heinz | Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery[END_REF]. 5: Set η 2 according to [START_REF] Lee | Algorithms for non-negative matrix factorization[END_REF]. 6: for t = 1, . . . , T do 7:

for k = 1, . . . , K 1 do 8: Â ← softmax log( Â) + η 1 B⊤ X ⊤ (X -X B Â) 9:
▷ log is applied element-wise;

10:

▷ softmax is applied along the first dimension.

11:

end for 12:

for k = 1, . . . , K 2 do 13:

B ← softmax log( B) + η 2 X ⊤ (X -X B Â) Â⊤ 14:
end for 15: end for 16: Ê ← X B 17: Return Ê, Â ▷ Estimated endmembers, abundances.

b) Initialization: We initialize the abundance matrix A uniformly,

A 0 = 1 p 1 p 1 T N , (18)
where 1 d denotes a d-dimensional vector of ones. This corresponds to the maximal entropy configuration for each pixel.

The entropy for each pixel will naturally decrease as a result of the optimization, but the high entropy of the initialization will have a regularization effect. The initialization of the pixel contribution matrix B is then also close to uniform. Nevertheless, we introduce a small random perturbation which is necessary to break the symmetry between the columns of B (otherwise, the updates of A and B will keep them invariant). Concretely, the entries of B are randomly sampled according to the uniform distribution on [0, 1], U [0,1] . Next, they are rescaled by a factor 0.1. Finally, we apply the softmax function on each column so that the columns of B belong to the simplex ∆ N , for j in {1, . . . , p},

b 0 j = softmax(0.1 u), (19)
where u ∼ U [0,1] N . In practice, we observe that such an initialization leads to a matrix B 0 that is very close to a uniform initialization 1

N 1 N 1 T p . c)
Step sizes: We use constant step sizes η 1 and η 2 , for A and B respectively.

η 1 = γ σ 2 max , ( 20
)
where γ is a value in S = {0.125, 0.25, 0.5, 1, 2, 4, 8} and σ max is the largest singular value of the matrix XB 0 . We recover the classical convergence of gradient descent with fixed step size [START_REF] Nesterov | Introductory lectures on convex optimization: A basic course[END_REF] up to the factor γ, since σ 2 max corresponds to the Lipschitz constant of the sub-problem related to (4) when minimizing w.r.t A, B being fixed. Having γ in S allows us to use slightly different step sizes and yields better performance in practice. Note that our model selection procedure, presented later, will automatically choose the right parameter γ, thus removing the burden for the user of having to deal with an extra hyper-parameter. Finally, η 2 is simply a rescaled version of η 1 to account for the matrices being of transposed dimensions:

η 2 = p N η 1 . (21)
d) Hyperparameters: For all experiments, if not stated otherwise, we set T = 100 and K 1 = K 2 = 5 as it provides a good trade-off between convergence speed and unmixing accuracy. Note also that these hyper-parameters are robust to different real datasets as detailed in section IV.

D. Model selection procedure

As stated above, the initialization of the matrix B is random, leading to different solutions for each run of the algorithm since the overall optimization problem is non convex. Besides, we allow for different step-sizes γ, which we draw randomly from the set S in practice, Since the convergence of the algorithm is very fast (see experimental section for concrete benchmarks), we are able to provide a large diversity of solutions given a dataset by running M times our method with different random seeds, while keeping the global computational complexity reasonable. A major question we address next is then how to select optimally the best solution in terms of unmixing accuracy.

For this, we take inspiration from classical model selection and sparse estimation theory [START_REF] Hastie | The elements of statistical learning: data mining, inference, and prediction[END_REF]. First, we measure the fit of each solution in terms of residual error ∥X -XAB∥ 1 , choosing the ℓ 1 -norm which is known to be more robust to outliers than the mean squared error. Second, we select the set of solutions that are in the same ball park as the best solution we have obtained in terms of ℓ 1 fit. This selection process is illustrated by the red dotted line in Figure 1, while the precise criterion is described in Algorithm 2.

From the subset of solutions with good fit, we then choose the one whose endmembers have the best incoherence, a desired property, which is classical in the theory of sparse estimation [START_REF] Elad | A generalized uncertainty principle and sparse representation in pairs of bases[END_REF]-[62]. Indeed, dictionaries (here endmembers) with more incoherence will benefit from better theoretical guarantees in terms of estimation of abundances, making it a natural criterion for model selection in the context of unmixing. Formally, the coherence is simply defined as the maximal pairwise spectral correlation between the estimated endmembers. More precisely, for Ê = [ê 1 , . . . , êp ] the endmembers matrix, the coherence µ is defined as:

µ = max k̸ =k ′ ⟨ê k , êk ′ ⟩, (22)
where ⟨.⟩ denotes the inner product.

To the best of our knowledge, this is the first time the coherence µ is used as a model selection criterion for archetypal analysis. Our experiments, see next section, show that it is highly effective.

In summary, we automatically select the model whose endmembers have the lowest maximal pairwise spectral correlation among the ones that have a good ℓ 1 fit. This strategy is illustrated in figure 1

IV. EXPERIMENTS

We have performed experiments on one simulated dataset with different noise and purity levels as well as six standard real datasets whose descriptions are given below.

A. Hyperspectral data description 1) Simulated dataset: For our study, we chose six endmembers from the USGS library and generated a 1000-pixel data cube using the methodology outlined in [63]. This approach allowed us to vary the purity level of the pixels by adjusting the parameter ρ, which is used in Table I. Specifically, lower values of ρ correspond to pixels that are less pure, while higher values indicate greater purity. By manipulating this parameter, we were able to simulate a range of real-world scenarios and evaluate the robustness of our algorithm under different conditions. 2) Samson: The Samson 2 hyperspectral image is a 95x95 pixels sub-region of a larger image captured using 156 bands spanning from 401 to 889 nm. Three main materials have been identified: Tree, Soil and Water. Note that we use a different ground truth from [51] that we selected for its sharper details on the abundances. 3) Jasper Ridge: The Jasper Ridge 2 hyperspectral image is a 100x100 pixels sub-region of a larger image initially captured using 224 bands spanning from 380 to 2500 nm. In total, 198 bands remain as 26 were removed as a pre-processing step due to dense water vapor and atmospheric effects. Four main materials have been identified: Tree, Dirt, Water and Road. 4) Urban4 and Urban6: The Urban 2 hyperspectral image is a 307x307 pixels image collected by the Hyperspectral Digital Image Collection Experiment (HYDICE) [START_REF] Rickard | HYDICE: An airborne system for hyperspectral imaging[END_REF] 2 downloaded at https://rslab.ut.ac.ir/data sensor using 210 bands spanning from 400 to 2500 nm. In total, 162 bands remain as 48 were removed as a pre-processing step due to dense water vapor and atmospheric effects. There exists three versions of this dataset w.r.t. the number of endmembers. In this study, we focus on the two extremes: Urban4 contains 4 endmembers (Asphalt Road, Grass, Tree and Roof) and Urban6 contains two additional materials: Dirt and Metal, making it more challenging. 5) APEX: The APEX [65] hyperspectral image that we consider in this paper is a 111x122 pixels cropped region 3 of a larger image captured over 285 bands spanning from 413 to 2420nm. Four main materials were identified: Road, Tree, Roof and Water. Note that the ground truth abundance map for water appears to contain shadows due to the sunlight direction (see figure 3). This is a common issue when dealing with real remote sensing data and we cannot expect the semi-automated ground truth abundances map generation to be perfect. 6) Washington DC Mall: The Washington DC Mall (WDC) hyperspectral dataset 3 consists in a 319x292 pixels image captured by the HYDICE [START_REF] Rickard | HYDICE: An airborne system for hyperspectral imaging[END_REF] sensor over 191 bands spanning from 400 to 2400 nm. Six main materials were identified: Grass, Tree, Roof, Road, Water and Trail. According to [66] (Samson, Jasper Ridge and Urban) and [51] (APEX and WDC), the endmembers spectra were manually selected from the images and the ground truth abundances were set by FCLS. Illustrations of the datasets and their ground truth endmembers are available in the supplementary material.

B. Experimental setup

We compare our approach to nine competitive methods from different unmixing categories:

• Geometrical unmixing baseline: FCLS [START_REF] Heinz | Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery[END_REF] using VCA [START_REF] Fox | Applications of hyperspectral mineralogy for geoenvironmental characterisation[END_REF] to extract endmembers. Our implementation of the FCLS algorithm uses the DecompSimplex routine implemented in SPAMS 4 . This method relies on the activeset algorithm [41] that enables significantly faster convergence than generic quadratic programming solvers by leveraging the underlying sparsity of the abundances as noted by [START_REF] Chen | Fast and robust archetypal analysis for representation learning[END_REF]. • Deep learning unmixing: Endnet 5 [46] using VCA [START_REF] Fox | Applications of hyperspectral mineralogy for geoenvironmental characterisation[END_REF] to initialize the endmembers, MiSiCNet 6 [51] and the deep unrolling network ADMMNet 7 [67]. • NMF-based blind unmixing: minimum-volume-constraint non-negative matrix factorization (MVCNMF) 8 [30], non-negative matrix factorization quadratic minimum volume (NMF-QMV) 9 [START_REF] Zhuang | Regularization parameter selection in minimum volume hyperspectral unmixing[END_REF] using the boundary term as the quadratic minimum volume penalty, near-convex archetypal analysis (NCAA) 10 [28] and AA [START_REF] Cutler | Archetypal analysis[END_REF] using the imple- mentation from [START_REF] Chen | Fast and robust archetypal analysis for representation learning[END_REF] developed in SPAMS. The approach denoted as AA involves solving (4) using an activeset method to optimize the convex sub-problems. This method is a conventional alternating approach that uses a fixed initialization (A and B entries are set to 0) and a fixed number of iterations (T = 100). While AA is a well-known method, it has not been thoroughly evaluated on various real-world unmixing datasets. Therefore, we include it as a competing method in our study to compare its performance with other state-of-the-art algorithms. By using AA, we can also assess the improvements in optimization achieved by EDAA since the underlying model ( 4) is the same for both methods. This comparison enables us to gain insights into the benefits of EDAA for hyperspectral unmixing. • Finally, we include a recent technique addressing spectral variability: SeCoDe 11 [START_REF] Yao | Sparsityenhanced convolutional decomposition: A novel tensor-based paradigm for blind hyperspectral unmixing[END_REF].

To quantitatively evaluate the performance of the selected methods, we consider two metrics that are computed both globally and individually for each endmember. On one hand, we measure the quality of the generated abundances by means of the abundances root mean square error (RMSE) in percent between the ground truth and the estimated abundances: 11 implementation at https://github.com/danfenghong/IEEE TGRS SeCoDe RMSE(A, Â) = 100 × 1 pN

p i=1 N j=1 A i,j -Âi,j 2 . ( 23
)
On the other hand, we assess the quality of the estimated endmembers spectra by using the spectral angle distance (SAD) in degrees between the ground truth and the generated endmembers:

SAD(E, Ê) = 180 π × 1 p p i=1 arccos ⟨e i , êi ⟩ ||e i || 2 ||ê i || 2 , (24)
where e i denotes the i-th column of E, i.e. the spectrum of the i-th endmember.

C. Unmixing experiments

Table I presents the results of our unmixing accuracy evaluation on the simulated dataset. Our analysis indicates that the performance of the AA variants (active-set based and EDAA) is lower in scenarios where the pixels are highly mixed (ρ = 0.7) when compared to geometrically motivated methods, such as MiSiCNet, MVCNMF and NMF-QMV, which do not rely directly on the pixel values to estimate the endmembers. However, it is worth noting that the EDAA method performs better than the plain AA method in this setting, thanks to its advanced model selection. In the case where pure pixels are present (ρ = 1.0), the AA model formulation (4) yields the best results, as it estimates the endmembers as convex combinations of the pixels. This approach is more robust to noise than the FCLS method. For the medium case (ρ = 0.85), both AA and EDAA exhibit better performance than MiSiCNet and NMF-QMV for low Signal-to-Noise Ratio (SNR) due to their greater robustness to noise. However, they are outperformed by MiSiCNet and NMF-QMV for high SNR, as there are no pure pixels available for these methods to leverage.

Tables II and III report the unmixing accuracy in terms of abundances RMSE [START_REF] Borsoi | Spectral variability in hyperspectral data unmixing: A comprehensive review[END_REF] and endmembers SAD (24) on six standard real datasets. The datasets are arbitrarily ranked based on their difficulty. For a fair comparison, all methods were evaluated on the ℓ 2 -normalized data (17) which induces slight changes compared to the results reported in [51].

Overall EDAA obtains the best abundances RMSE on five out of six real datasets. The results on Urban6 fall in favor of plain AA but the gap is moderate (12.83 vs 13.63) given that EDAA is more than seven times faster (594 versus 75 seconds) than plain AA on this dataset (see table IV). Note however that plain AA performs poorly on WDC where the ground truth endmembers are highly correlated. We argue that our model selection technique is instrumental in avoiding collapsing runs in which endmembers spectra are highly correlated. This is underlined by the overall competitive SAD results obtained by EDAA across datasets. It should be noted that the SAD metric alone is not sufficient to assess the unmixing performance as a good SAD score does not necessarily lead to better abundance maps. Thus it is not contradictory to have slightly worse SAD scores yet better looking abundance maps.

The FCLS baseline based on VCA obtains rather poor results except for WDC. This is likely due to the pure pixel assumption. Indeed, VCA selects a single pixel to represent the endmembers spectra, which is too stringent in real scenarios where spectral variability is ubiquitous.

Despite its quadratic minimum volume boundary term, NMF-QMV generally obtains worse results than the FCLS baseline. Since it operates the unmixing in a subspace, NMF-QMV cannot prevent the endmembers spectra from having negative values, which breaks the physical interpretability of the estimates and subsequently harms the unmixing performance. This phenomenon can be observed in figures 2, 3 and 4 for several endmembers. The associated abundances show that NMF-QMV produces maps that are too uniform and lack sparsity.

In contrast, MVCNMF is a strong baseline as it does not operate in a subspace.

When it comes to deep learning based methods, Endnet achieves very good results in terms of SAD but tends to create overly sparse abundances which hinders its performance in terms of abundances RMSE. For instance, as can be seen in figure 2, the Road endmember is overlooked by Endnet even though EDAA recovers it neatly. Likewise, in figure 3, the Road endmember spreads too much compared to EDAA which appears closer to the ground truth.

MiSiCNet gives better unmixing results than Endnet in terms of abundances RMSE except for APEX although the SAD results falls in favor of Endnet except for Jasper Ridge. This is likely due to Endnet using the spectral angle distance on the input data in its loss which helps in achieving better SAD accuracy. However a good SAD is not sufficient to obtain good abundance maps, an area where MiSiCNet tends to shine as it incorporates spatial information by using convolutional filters and implicitly applying a regularizer on abundances.

ADMMNet obtains rather poor results on all datasets, which is likely due to using ADMM to blindly solve the linear mixing model rather than leveraging a known library.

SeCoDe works reasonably well on the Samson dataset due to its implementation available online that has been tailored to this dataset. However it does not perform as well on other datasets, which is likely due to requiring precise hyperparameters tuning depending on the data at hand.

NCAA obtains reasonable results due to its fine-tuning procedure but its computational cost is prohibitive as we will be discussing next.

Finally, the plain AA method leveraging an active-set algorithm is a very competitive method for five out of six datasets, yet its performance drops significantly when dealing with the hardest mixing scenario that is WDC. For example, in figure 2 only AA and EDAA are able to uncover the Road endmember in Jasper Ridge whereas all the other techniques fail. Unlike EDAA, AA does not rely on a random initialization of the estimates as they are set to zeros in practice. Instead, EDAA requires starting from feasible points w.r.t. the optimization sub-problems. In addition, the initial estimates in EDAA should not contain any zeros due to the presence of the logarithm in (14). Thanks to the entropic gradient descent speed, it is possible to fit several models that have been randomly initialized as described in III-C and use the model selection procedure presented in Algorithm 2 which selects the model that exhibits the lowest coherence among the pool of candidates that are well fitted. Notably, this approach prevents the estimated endmembers from collapsing into degenerate solutions that would end up being perfectly correlated. This likely explains why the performance of plain AA on WDC drops as the endmembers are highly correlated.

Additional qualitative results for the Samson, Urban4 and Urban6 datasets can be found in the supplementary material. Moreover, we have also included the results on a 250×191 pixels subset of the Cuprite dataset, which is more challenging and does not come with ground truth abundance maps.

D. Computational cost

Table IV reports the processing times for the different unmixing algorithms on the six real datasets. MVCNMF, NMF-QMV, SeCoDe and NCAA were implemented in Matlab (2020b) while FCLS, Endnet, MiSiCNet, ADMMNet and the AA variants were implemented in Python (3.8). NMF-QMV, MVCNMF, SeCoDe, NCAA, FCLS and AA run on CPU whereas Endnet, MiSiCNet, ADMMNet and EDAA run on GPU. The processing times were obtained using a computer with an Intel(R) Xeon(R) Silver 4110 processor (2.10 GHz), 32 cores, 64GB of memory, and a NVIDIA GeForce RTX For example, it takes on average 1.5 seconds for EDAA to perform a single unmixing task on the Urban6 dataset, which is three times faster than FCLS. This demonstrates the efficiency of EDAA which allows us to use an adequate model selection procedure over several runs. Note that the processing time of NCAA is prohibitive due to its fine-tuning component despite using a fast projected gradient method.

E. Ablation study

Finally, we study the sensitivity to hyper-parameters for Algorithm 1 and 2 in figure 5 where the Y-axis corresponds to the overall abundances RMSE. Given a fixed computational budget of 1000 updates, figure 5 (a) shows that the hyperparameters of EDAA are robust provided that the number of runs M in the model selection is large enough (here 100). Only the two extremes configurations (K 1 = K 2 = 1, T = 500 and K 1 = K 2 = 50, T = 10) are slightly worse, especially on Urban6. For the remaining experiments, we use K 1 = K 2 = 5. In figure 5 (b), we see that the number of outer iterations is quite stable except for WDC which requires more updates (1000, i.e. T = 100). Finally, we study the importance of the number of runs M from which to select the best candidate in figure 5 (c). We observe that the model selection procedure requires at least 50 runs to obtain very good performances, hence we use M = 50 in our unmixing experiments. On unknown datasets where real-time unmixing is not required, it is advised to use a large number of runs (at least 100) to ensure that the model selection procedure yields a good candidate. Detailed results for both abundances RMSE and SAD metrics are available in the supplementary material.

V. CONCLUSION

We have proposed a new algorithm based on archetypal analysis for blind hyperspectral unmixing. We have shown how to take advantage of its efficient GPU implementation in order to develop an adequate model selection procedure to obtain state-of-the-art performances. Remarkably, our simple and easy-to-use approach considerably improves the unmixing results on a comprehensive collection of standard real datasets. In addition, we have made our results reproducible by releasing an open source codebase which also includes the plain archetypal analysis variant presented in this study. While this paper was under review, we also investigated in [START_REF] Rasti | SUnAA: Sparse Unmixing using Archetypal Analysis[END_REF] the problem of semi-supervised unmixing by using a variant of archetypal analysis, showing that such a framework may be useful beyond the problem of blind unmixing that we address here. Finally, it is worth noting that our approach does not consider the spatial structure of the data. This feature suggests that a natural extension to our approach would be to incorporate missing spatial information, which could potentially improve the accuracy of our results.

Fig. 1 :

 1 Fig. 1: Illustration of the model selection procedure on six datasets using M = 50 runs. Runs are illustrated by blue dots and the selected one is in black. The selected run is the one with lowest coherence µ under the dashed red line representing the ℓ 1 fit threshold, see Alg. 2.

Fig. 2 :Fig. 5 :

 25 Fig. 2: Estimated endmembers (a) and abundances (b) on the Jasper Ridge dataset. Ground truth abundances are displayed on the right-most column.

  and described in Algorithm 2. In the experiments, the number of runs M was set to 50.Êm , Âm ← EDAA(X, p, T, K 1 , K 2 , s m ) ▷ See (1) : fit min ← min{fit 1 , . . . , fit M } 9: I ← {m | fit m ≤ 1.05 × fit min }▷ Subset of models. 10: best ← arg min{µ i , i ∈ I} 11: Return: Êbest , Âbest .

	Algorithm 2 Model Selection Procedure
	1: Input: M (number of runs); ℓ 2 -normalized data X in
		R L×N ; p (number of endmembers); T (number of outer
		iterations); K 1 (number of inner iterations for Â); K 2 (number of inner iterations for B).
	2: for m = 1, . . . , M do
	3:	Set random seed s m .
	4:
	5: 6:	fit m ← ||X -Êm Âm || 1 Compute coherence µ m on Êm .	▷ See (22)
	7: end for

[START_REF] Gowen | Hyperspectral imaging-an emerging process analytical tool for food quality and safety control[END_REF]

TABLE I :

 I Abundances RMSE and endmembers SAD for the simulated dataset with different noise and purity levels. The best results are shown in bold. The second best results are underlined.

				RMSE			SAD
	Methods	ρ		SNR (dB)			SNR (dB)
			20	30	40	20	30	40
		0.7	16.53 17.15 17.97	7.20	7.18	7.38
	FCLS [19], [20]	0.85	12.96 11.44 10.15	3.86	3.05	2.45
		1.0	8.03	4.43	3.31	1.84	0.58	0.17
		0.7	10.60	8.64	8.31	3.94	1.88	1.58
	MiSiCNet [51] (λ = 0.1)	0.85	13.76	8.62	7.68	4.70	1.20	0.75
		1.0	15.08	6.42	3.73	5.11	1.12	0.39
		0.7	15.49 16.92 16.68	11.44 13.76 13.48
	ADMMNet [67]	0.85	17.18 19.54 19.58	8.52	10.23	9.93
		1.0	21.13 16.57 16.60	9.03	6.26	6.55
		0.7	15.80	9.75	15.53	6.85	2.52	6.85
	MVCNMF [30]	0.85	9.88	9.80	9.54	1.85	1.16	0.76
		1.0	6.84	7.03	3.55	1.12	0.78	0.17
		0.7	10.11	8.32	7.36	4.96	4.76	3.32
	NMFQMV [22]	0.85	10.58	7.11	6.79	2.91	1.18	0.98
		1.0	11.24	6.62	3.80	3.25	1.54	0.41
		0.7	13.49 13.36 13.62	16.06 16.38 16.54
	SeCoDe [25]	0.85	13.56 14.76 14.69	14.62 15.79 15.73
		1.0	7.88	4.74	4.35	7.76	5.07	4.76
		0.7	14.62 13.43 14.64	8.32	8.00	7.44
	NCAA [28]	0.85	12.15	7.66	7.19	5.72	2.56	1.10
		1.0	11.61	3.20	3.25	5.37	1.70	0.71
		0.7	17.67 17.23 17.18	7.61	6.71	6.43
	AA (T = 100)	0.85	9.44	8.23	8.04	2.55	2.09	1.86
		1.0	5.13	2.98	2.75	0.88	0.45	0.34
		0.7	14.35 13.75 14.91	5.50	5.01	6.52
	EDAA (K 1 = K 2 = 30)	0.85	8.84	8.37	8.37	2.63	2.18	1.92
		1.0	5.82	4.26	2.89	0.96	0.51	0.33

TABLE II :

 II Abundances RMSE on six real datasets. The best results are shown in bold. The second best results are underlined.

			FCLS Endnet MiSiCNet ADMMNet MVCNMF NMF-QMV SeCoDe NCAA	AA (T = 100) EDAA
		Soil	11.28	11.61	6.47	18.41	6.62	13.67	7.67	12.37	6.16	5.74
	Samson	Tree Water	9.13 5.05	7.72 6.86	5.38 3.47	14.32 6.67	3.47 4.66	8.40 11.61	6.68 2.57	4.87 7.80	4.00 2.30	3.77 2.59
		Overall	8.88	8.97	5.25	14.00	5.08	11.44	6.05	8.89	4.44	4.24
		Dirt	21.23	18.26	21.68	21.58	20.19	19.97	20.62	18.85	9.29	7.32
		Road	24.72	29.40	24.94	21.37	24.28	26.13	21.42	23.95	8.04	7.61
	Jasper Ridge	Tree	11.20	4.00	3.41	13.19	9.80	14.55	9.57	3.34	7.43	6.63
		Water	13.61	22.38	7.07	13.92	17.83	19.81	9.38	10.36	6.10	5.69
		Overall	18.52	20.70	16.98	17.96	18.79	20.53	16.31	16.18	7.80	6.85
		Road	30.54	12.04	10.30	19.45	12.89	20.25	12.84	11.58	11.13	8.62
		Grass	32.99	17.94	12.35	22.17	14.00	20.22	24.34	7.57	11.47	9.28
	Urban4	Roof	15.40	11.28	8.01	13.31	10.98	13.29	13.20	10.46	7.30	6.37
		Tree	20.02	11.69	8.78	16.45	13.63	21.56	26.36	7.63	8.51	6.27
			25.78	13.52	10.00	18.15	12.93	19.11	20.17	9.48	9.76	7.75
	Urban6

TABLE III :

 III Endmembers SAD on six real datasets. The best results are shown in bold. The second best results are underlined.

			FCLS Endnet MiSiCNet ADMMNet MVCNMF NMF-QMV SeCoDe NCAA	AA (T = 100) EDAA
		Soil	2.76	0.61	1.21	7.48	1.08	4.90	0.79	4.57	0.78	1.64
	Samson	Tree Water	3.05 7.15	1.93 1.48	3.38 5.36	2.48 6.68	2.54 4.82	5.34 11.14	1.96 2.56	2.42 5.15	1.80 1.38	1.98 1.31
		Overall	4.32	1.34	3.32	5.55	2.81	7.13	1.77	4.05	1.32	1.64
		Dirt	13.03	1.63	4.26	21.90	9.41	12.40	9.94	4.08	2.54	2.74
		Road	40.39	32.85	20.04	46.06	36.38	45.66	45.87	41.84	5.26	3.10
	Jasper Ridge	Tree	11.16	1.39	1.27	10.78	8.28	14.46	12.42	2.25	4.68	4.23
		Water	13.24	3.21	4.18	13.14	10.40	14.53	6.44	5.00	3.03	2.80
		Overall	19.46	9.77	7.44	22.97	16.12	21.77	18.67	13.29	3.88	3.22
	Urban4
	(2080 Ti) graphical processing unit. The table shows that
	FCLS is clearly faster than the other unmixing techniques,
	however it is a supervised method that relies on an endmem-
	bers extraction algorithm. In this case, VCA is used which is
	also fast. The deep learning methods are the slowest techniques
	despite running on GPU. Interestingly, EDAA requires a lower
	computational cost than NMF-QMV and AA although our
	approach consists in aggregating 50 runs obtained iteratively.

TABLE IV :

 IV Processing times in seconds on six real datasets. The best results are in bold and the second best underlined. EDAA includes the model selection procedure over M = 50 runs.

		FCLS	Endnet	MiSiCNet ADMMNet MVCNMF NMF-QMV SeCoDe NCAA	AA (T = 100)	EDAA
	Samson	0.3	≈ 560	≈ 80	55.2	3.8	20.1	24.2	113.1	33.0	11.2
	JasperRidge	0.4	≈ 680	≈ 90	54.1	8.0	22.3	28.6	501.4	43.6	9.6
	Urban4	3.6	≈ 720	≈ 411	504.6	25.4	112.5	475.1	3529.9	415.7	66.3
	Urban6	4.4	≈ 1000	≈ 417	507.5	45.0	158.4	707.7	4880.8	593.7	75.2
	APEX	0.6	≈ 720	≈ 92	80.0	11.3	27.2	56.1	441.5	75.4	15.4
	WDC	4.0	≈ 1000	≈ 409	535.8	55.1	174.4	524.7	4179.0	493.1	81.0

Acknowledgments and Funding

This project was supported by the ERC grant number 714381 (SOLARIS project) and by ANR 3IA MIAI@Grenoble Alpes (ANR-19-P3IA-0003).


researcher, ethnologist Christelle Pineau
email: jean.foyer@cnrs.fr

Tasting life and energy with the body: the biodynamic resonance of wine

Keywords: biodynamics, natural wines, vibrations, energy, body, Loire, tasting

Based on two long term research with natural and biodynamic winegrowers in France, this text returns to the principles of biodynamic agriculture in its desire to potentiate the living character of the vine and the soil; living supposed to translate into energy or vibrations in the glass. Why is wine attracted to biodynamics? The hypothesis put forward, which would partly explain the apparently happy marriage between wine and biodynamics, is that the latter helps to give more life and energy to wine and that this energy would be perceived through new way of tasting. Our aim here is not so much to study what this extra life or energy might be as a scientific object as these notions are in any case impossible to stabilise from a scientific point of view. The reason for our use of these unstable categories is that they emerge repeatedly on the subject of biodynamic wines from the discourses of winemakers, oenologists, sommeliers and tasters, whether beginners or advanced. These categories of energy or vibrations, still in gestation, refer to forms of wine tasting that go beyond organoleptic criteria alone to extend to the receptivity of the whole body involved in the act of drinking.

Introduction

The text we offer here is centered on biodynamics, an agricultural technique born 100 years ago from lectures given by the thinker Rudolf Steiner. In addition to rejecting the use of synthetic chemical inputs, which is common to organic farming, biodynamics advocates the use of alchemical principles, combining animal, vegetable and mineral compounds. It also envisages the farm -or the estate -as a whole agricultural organism, or 'social organism' within different human and non-human entities, contributing to the balance of each part [START_REF] Rigolot | Biodynamic Farming as a Resource for Sustainability Transformations: Potential and Challenges[END_REF]. The choice of focusing on biodynamics here was prompted by a recurring question raised by colleagues: why is biodynamics developing so much in wine? This apparently simple question is all the more relevant as the marriage was far from obvious. Because of the predominance of monoculture in wine production and reservations about alcohol consumption beyong biodynamic farmers1 biodynamics has long been reluctant to reach into the world of vines and wines. Similarly, the wine sector, which has largely embraced the principles of agricultural modernisation, has for a long time remained impervious to biodynamics, still controversial today because of its esoteric principles. However, the use of biodynamics in viticulture seems to have developed exponentially over the last two decades2 , especially in France. Although the basic principles are not well known, it has nevertheless gained a form of recognition among consumers and "influencers" (specialist magazines, wine merchants, wine bars, etc.) Why has biodynamics attached itself so well to wine in recent years? Why is wine attracted to biodynamics? In short, what does biodynamics do to wine? The hypothesis put forward, which would partly explain the apparently happy marriage between wine and biodynamics, is that the latter helps to give more life and energy to wine and that this energy would be perceived through new way of tasting, involving the whole body. Our article is conceived as an original and quite experimental contribution to the anthropology of taste and smell, in an equally original context, that of biodynamic production, far from industrial production methods [START_REF] Lahne | Introduction to Accounting for Taste[END_REF]]. 3 Our aim here is not so much to study what this extra life or energy might be as a scientific object and using rigorous metrics, as these notions are in any case impossible to stabilise from a scientific point of view. We also differ here from classic approaches in terms of the sociology of taste, which emphasize the social determinisms and strategies of distinction [START_REF] Bourdieu | La distinction: critique sociale du jugement[END_REF] woven from the expression of a taste. In this vein, the wine object proves particularly relevant for analyzing the making of class boundaries and differences in terms of social, economic and cultural capital [Hartman, 2000;[START_REF] Howland | Distinction by Proxy: The Democratization of Fine Wine[END_REF], de Benedittis, 2019]. As anthropologists, we feel closer to pragmatic approaches inspired by the sociology of science [START_REF] Hennion | Those Things That Hold Us Together: Taste and Sociology[END_REF][START_REF] Shapin | Changing Tastes: How Things Tasted in the Early Modern Period and How They Taste Now[END_REF] and which, in the case of wine [START_REF] Cisterna | Unexpected Moments and the Wine Experience[END_REF][START_REF] Hennion | Le goût du vin : Pour une sociologie de l'attention*[END_REF]Teil, 2021], emphasize the tasting experience itself, the affects and perceptions involved, and the ontologies of the various objects. In keeping with the very principles of biodynamics, which invite a phenomenological approach in which the observed and the observers are intertwined, we propose to take this approach a step further by assuming and even asserting the fact of mixing our own subjectivities and sensitivities with those of the actors and objects, in this case the wine, of course. Then, the reason for our use of these unstable categories of energy and vibrations is that they emerge repeatedly on the subject of biodynamic wines from the discourses of winemakers, oenologists, sommeliers and tasters, whether beginners or advanced. From an "emic" rather than an "etic" perspective (i.e., one that favors internalist and indigenous points of view) [Olivier de [START_REF] Olivier De Sardan | Emique », Alliance, rites et mythes[END_REF]], what interests us here is starting from what these actors tell us about their practice, and taking it seriously in an attempt to account for their rationality as finely as possible. Our focus on actors' discourse is partly explained by the fact that the expressions of the sensitivities we're talking about come either through personal experience or through their verbalization, but can hardly be directly observed or measured. By giving credence to these actors' discourses, we believe that what we lose in terms of "critical distance", we gain in terms of a fine-tuned understanding of their singular experiences. In a certain way, we try to respond to the provocative proposition of Gravina who said : « I think that the so-called maturity of wine criticism today should be put to the test by more radical questions that would allow for the possibility of going back to a critical reexamination of the taste of wine. » [Gravina, 2012 : 150]. Our investigation, which is multi-sited and rooted in the various vineyards of France, is all the more interesting in that it targets a powerful cultural and heritage stronghold. Demossier points the burden of identity of wine drinking in France, and evokes "the construction of the myth of French wine as an element of national heritage and patrimony » [Demossier, 2010 : 13).This French « wine drinking culture » was historically influenced by the Appellations dorigine contrôlée (AOC) system created in the 1930s, the aim of which was to guarantee quality standards, and this influence really took off in the period known as the 'Trente Glorieuses' in France, before spreading internationally [START_REF] Jacquet | Entre discours et réglementations : le développement international des vins français d'AOC durant les Trente Glorieuses[END_REF]. This system of evaluation, which is still in force, is based on tastings carried out by winegrowers, mainly using organoleptic criteria. It should be pointed out that modern tasting standards are based almost exclusively on those organoleptic criteria which, in most cases, can be objectively assessed. Parmi les critères les plus communs, on trouve les différentes acidités, la limpidité, les expressions aromatiques etc. Wine tasting and evaluation sheets are based on international standards (i.e. the world's best sommelier), drawn up by the International Union of Oenologists, and therefore not typically French. It is widely recognized in professional circles as well as among wine lovers. 4Then, the aim of this study is to make visible and comprehensible new ways of tasting wine, based on less commonly expressed and accepted forms of perception, which contrast sharply with the more classic and standardized forms of wine tasting. It also continues our previous analyses on wine concerning the involvement of the body and the sensitivities at work in the vineyard [START_REF] Foyer | Syncrétisme des savoirs dans la viticulture biodynamique. Incorporation dans l'expérience et le sensible et trajectoire initiatique[END_REF][START_REF] Pineau | La corne de vache et le microscope[END_REF]. In the first section, we will see how, beyond the techniques specific to biodynamics, they echo a concern to enliven soils, vines and, with them, wine. Then we will see how the work undertaken with living organisms seems to resonate in a pattern of wine-tasting that engages the whole body and calls on the mobile categories of energy or vibration. This way of understanding wine beyond olfactory and gustatory criteria alone opens up new ways of appreciating and drinking wine.

Materials and Methods

Our discussion is based on ethnographic surveys and interviews carried out in the Anjou, a region where the development of biodynamic viticulture is particularly 'dynamic', and during multi-site fieldwork in French vineyards where 'natural' and/or biodynamic winegrowers are strongly represented (Ardèche, Beaujolais, Jura, Loire, South-West...). It seemed obvious to the authors that this article should be written as a duo. The subject, biodynamic wine, is common to both. Our recent studies intersect and echo each other, even though our paths towards it have different starting points. Christelle Pineau came into contact with biodynamic wine through 'natural' wine, which has been the focus of her research [START_REF] Pineau | La corne de vache et le microscope[END_REF] for more than twelve years, following on from the thinking and the links she has maintained with wine and winetasting in contact with professionals. Jean Foyer came to wine through biodynamics. He led long-time research about biodynamics in Anjou based on 94 qualitative interviews between 2014 and 2020, mostly with winemakers, biodynamic trainers and students in biodynamic training. He also led participatory observation during organic wine salons and biodynamic training. Although a complete beginner in oenological terms at the start of his fieldwork, he began to appreciate the wines of the vintners he interviewed, especially after being struck by the sense of discovering not only the wine in the bottles, but also the people. This nearsymmetry between our pathways has multiplied the possible approaches to the complex and "borderline" object of biodynamics associated with wine. The overlaps or divergences that result from our observations in the field both nourish and enrich our analyses. To support our argument and give credibility to the discursive motives of those involved in the wine industry, we have drawn on the opinion of a recognised expert in the field of wine tasting. Pascaline Lepeltier represented France at the Best World Sommelier's Competition in 2023. She knows all the different styles of tasting, from the most standard to the most alternative.

Results:

1. Life in motion: the principles of biodynamics Biodynamics is anchored in the work done in the vineyard, in the care given to the land and, more precisely, to the balance of the vineyard as a whole. While the same applies to all winemaking methods, with biodynamics, the choice of doing without synthetic chemicals5 and the extra attention [START_REF] Foyer | Quand les actes agricoles sont au care et au compagnonnage : L'exemple de la biodynamie[END_REF] given to the dynamics of the living world refines our understanding of the interactions between plants, animals and humans. According to Aurélie Choné "To be a farmer for Rudolf Steiner is both to be active in nature and to seek more consciousness in the work undertaken, in other words in the social organism." [START_REF] Choné | Changer le monde par l'agriculture ? L'influence des théories et pratiques de l'anthroposophie sur les modèles de pensée écologiques alternatifs[END_REF]. The contributions of different composts and preparations6 mixing animal envelopes and plant containers and stemming from analogical thinking [START_REF] Breda | The Plant in between: Entanglement and Analogism in an Italian Community of Anthroposofists[END_REF]Pineau, 2020] are supposed to contribute to the revitalisation of each particular environment. We should note here that two of the central features of analogism as defined by Philippe Descola are the use of different techniques for establishing correspondences (for example, between microcosm and macrocosm) and the circulation of vital energy [START_REF] Descola | Par-delà nature et culture[END_REF], features that are widely present in our fielwork observations. These fundamental techniques are combined with observations of the position of the moon and the planets (interpreted in a biodynamic calendar). Contemporary biodynamic practice can cover different levels of understanding and appropriation by winemakers, from the highly pragmatic to the highly esoteric energy [START_REF] Grandjean | Biodynamic Wine-crafting in Switzerland: The Translation and Adaptation of Rudolf Steiner's Cosmology into Dark Green Agronomies[END_REF]. Some observe it rather 'distantly', others embrace it in a 'discreet and personal' manner, or approach it in a 'secular and republican' way. 7 Adapting the biodynamic method in the field according to the technical and mental ergonomics of each farmer (or winegrower) is a key point in Steiner's thinking. Another idea, just as central, is based on dynamising, or setting the living in motion, as expressed by the term of biodynamics itself. The question that concerns us here is whether the work thus undertaken with the living can be translated into a chain of traceability from the vine to the glass. In other words, can we find in a glass of wine the "extra vitality" that applying biodynamics in the vineyard is claimed to confer on it, and if so, how can we identify it? This formulation of the living appeared with the rise of the "natural" and biodynamic movements. More often than not, the discourse on wine focuses on the restitution of the terroir in a glass. A lively terroir is supposed to be reflected by a wine which is livelier (which can also be understood as more stimulating), or simply more alive. This discourse contrasting the dead soils on which conventional wines are grown with the living soils of biodynamic wines was developed in particular by the 'Renaissance des Appellations' group in France from 2001, formed around Nicolas Joly and Mark Angeli, winemakers who put forward a radical critique of appellations and a call for their redesign. These appellations, which are supposed to reflect and guarantee the quality of the terroir, become problematic and highly questionable when the land is exposed to the intensive chemical treatments used in conventional viticulture and no longer benefits from the living substratum that gives it its particularity: "The respectability of a brilliant creation is destroyed, or almost, within a few decades. Wine, which has become a fruit of technology rather than a fruit of nature, only has to please the palate of neophytes and the consumer majority". Nicolas Joly continues his argument with the need to reconsider the terroir: "A terroir does not only consist, as is said today, of the geological originality of its soil, but also, and just as importantly, of the living world that accompanies it: in other words, of all these microbes and bacteria that are each adapted to a particular type of soil and climate. By their presence they give the roots 'access' to these original features". Joly, 2007: 29]. Biodynamics, by virtue of its aim of revitalising the agricultural organism, notably through the use of preparations, seems to be particularly suited to the logic of the terroir that is so central to wine production. The "Renaissance des Appellations" winegrowers are therefore not at all opposed to the principle of appellations; rather, they dispute the technicist and technocratic logic at work in the current appellation system, particularly within the local institutions of the INAO (Institut National de l'Origine et de la Qualité), which, they believe, can only lead to the standardisation of wines. While this argument of revitalising wines by revitalising their terroirs and soils has attracted many winegrowers, opinions among biodynamic winegrowers on the promotion of terroirs (and the AOCs attached to them) are variable. Not all biodynamic winegrowers have the same attachment to terroirs and AOCs and some go so far as to break with the system, which they consider is incapable of reflecting and enhancing the living -and therefore potentially unstable -character of the wine they produce. Because of these differences of opinion on the question of the terroir, we prefer to focus on the notion of the living, which is common to all. The terroir, subject as it is to criticism, denial, variability and doubts from both scientific and non-specialist quarters and the object of insoluble laboratory studies and interminable discussions in bars and cafés, is considered to mask the question of the life that resides in wine. From an analytical point of view, we have therefore opted to shift the focus from the terroir to the living, but without abolishing the consubstantial links between the two 8 . Bearing in mind the possibilities of the terroir in the tension between the objective and the subjective, we will focus our interest on the characteristics and effects generated by biodynamic wines to track down the supposed framework of the living which is inherent in the practice, over and above the pedological, climatic and geographical data of the place in question. Before going any further, it should be noted that many of the winegrowers we met practise "natural" vitiviniculture, i.e. without synthetic chemical inputs in either vineyard or cellar, applying a non-interventionist approach in the sense that human action is considered as "indirect" and "negative" with regard to the plant and the animal, to use A.-G. Haudricourt's terminology [1962]. "Natural" and biodynamic practices do not systematically overlap, but neither can they be separated, such is the reciprocal porosity that becomes apparent in the field. While not all 'natural' winegrowers are interested in biodynamics, a kind of benevolence towards it can be observed in the vast majority of cases. Conversely, not all biodynamists make 'natural' wines, but many of them tend towards the same ideal of minimal intervention in the wine cellar. "Natural" and biodynamic methods do not have the same origins, but they link up and intertwine in the flow of daily life and encounters, as evidenced by the many wine salons advertised as "natural" or biodynamic, mingling hybrid and evolving profiles, each choosing to define themselves as one or the other according to backgrounds and convictions.

Tasting with one's body: energy and vibrations

Just as the idea of the terroir has remained controversial and difficult to grasp in its entirety, postulating the existence of a common thread in biodynamic wines is proving to be equally fraught. The burden of proof in an "objective" scientific framework remains the major obstacle. In an attempt to understand the mechanisms governing the expression and definition of taste, researchers in neurophysiology and sensory analysis try to eliminate bias -external or internal influences of a socio-cultural and psychological nature. As Pr Jacques Le Magnen reminds us, some stimuli can be described in more or less the same way by a large number of individuals, as is for sugar, but also "the smell of a pure substance is described in approximately the same way by everyone." But he goes on to say that the same cannot be said of the likes and dislikes, preferences and aversions that subjects, and therefore consumers, express with regard to the foods that produce the sensations." [Le [START_REF] Magnen | Evaluation sensorielle, manuel méthodologique[END_REF][START_REF] Magnen | Evaluation sensorielle, manuel méthodologique[END_REF]: X preface first edition]. Moreover, Jules Chauvet, a wine merchant and winemaker from the Beaujolais region who was also a researcher in microbiology and chemistry 9 and has become an emblematic figure in the "natural" wines network, pointed out that the act of tasting brings into play a whole system that is difficult to define, being at once clear and opaque: "While our apparatus is ultra-sensitive (it can detect certain odours from infinitesimal quantities of matter-substance, of the order of a millionth of a milligram), our analytical faculties have remained very rudimentary. Nature seems to have deployed every obstacle to put us off the scent: in olfaction, there are no elementary units as there are in vision, for example, where colours can be expressed in wavelengths. There are no elementary odours: all odours and scents are complex and it is not possible to measure their intensities in practice." 10 Being aware of these difficulties -which we fully acknowledge -that arise either from idiosyncrasy or from the inability to analyse and then to state the complexity [Teil, 2019], we could simply state that our attempt to identify a common thread in biodynamic wines is futile. Nevertheless, on the basis of fieldwork, repeated as a finished and defined object-thing. However, she specifies: "The impossibility of predetermining a terroir does not make it a pure invention, but a relatively unpredictable production of which the winegrower is not the only master". [Teil, 2011 : 18]. 9 This was to better understand fermentation, but also to break away from synthetic chemistry, which he did from 1951 10 Jacques Néauport, Jules Chauvet ou le talent du vin, published by Jean-Paul Rocher, Paris, 1997, p.57. interviews, informal observations, personal experience and critical tastings over the long term, it seemed to us that the hypothesis could not be dismissed so easily. A dominant feature emerged in the data collected, contrasting with the usual expressions used for smells and tastes: the bodily involvement associated with tasting biodynamic wines. This involvement of the body is directly associated with energies and vibrations, the whole body being seemingly more affected by the circulation of the wine. Clearly, any type of tasting involves the body, but what we are trying to capture here is the involvement of the body as a whole, as an organ of perception in the manner of an antenna. A fragmented tasting experience could be contrasted with a synaesthetic tasting experience, in which the body is no longer fragmented into different organs of perception and senses. Pascaline Lepeltier, who was elected best sommelier of France 2018 and best worker of France in 2018 in the sommelier category, knows all the registers of wines, from the most conventional taught during her training and encountered during official competitions to the most "alternative", such as "natural" and biodynamic wines. Today, thanks to her wide-ranging expertise, she is able to give more precise descriptions of these "differences" and to discuss them, even though the divide between conventional and biodynamic methods occurred at the beginning of her career: "There were expressions, tasting sensations, mouth-feelings and lingering finishes that were appealing even for a novice like me at the time. And especially, this was at a time when, even during the competition, we were blind-tasting a Sauvignon blanc, a Sancerre probably and an Anjou, a village Brissac, I still remember thinking that these were very hard angular wines, and saying to myself that I preferred the wines I had tasted the day before at the Renaissance des Appellations fair 11 . That weekend was quite an experience for me, and after that it continued, of course." 12 After explaining that she was introducing contrasts between organic wines ("to me, organic can only be a palliative"), biodynamic wines with too much sulphur ("very safe in terms of winemaking, from 30 mg/l 13 onwards they get a bit hard"), and biodynamic wines which she prefers (except for "very large companies, and mainstream biodynamics"), she expands on her description of tasting sensations:

"There is an undeniable salinity on the palate and harmonies between the different elements, with the alcohol responding to the acids, to the sugars, to the bitterness, to the phenols, coherently and without asperity. You'll get wines with crazy acidity but you won't feel the acidity, it won't dominate, it won't be aggressive, or you'll get wines with a lot of alcohol, but you won't get the alcoholic sensation, the alcohol burn, because it's all coherent."

The question of mineral characteristics in the description of wines was only introduced some 20 years ago and there are still problems with multiple meanings [START_REF] Deneulin | Minéralité du vin : représentations mentales de consommateurs suisses et français[END_REF]Deneulin and Bavaud, 2015]. Some associate it directly with the terroir, others with the presence of chemical additives in the cellar (such as PVPP 14 , used in conventional methods to "glue" and filter the wines or many other additives). But it would seem that the mineral characteristics called for in biodynamic winegrowing do not only correspond to descriptions of rocks such as the classic "gunflint, pebble, flint or limestone". They appear to open up a new field of perception that combines with the register of energy and vibration: 11 The Renaissance des Appellations is a biodynamic wines fair held in Angers" => Renaissance des Appellations is a "group" as it calls itself and organises a well-known wines fair in Angers every year. https://renaissance-des-appellations.com/fr/ 12 Interview, 7 April 2020. Pascaline Lepeltier has been working as a sommelière in New York for over ten years. She represented France at the World's Best Sommelier competition in 2023. 13 Added SO2 in the wine. The Demeter biodynamic label allows added sulphur up to 90mg/l depending on the colour of the wine. Up to 150 mg/l can be added to organic wines, and 200mg/l to conventional wines. The recent charter published by the nature'L or "vin méthode nature" winemakers' union, the S.A.I.N.S. wine association (no inputs or sulphite) and the older AVN (association des vins naturels) do not allow any added sulphur. There is a tolerance of up to 30mg/l of total sulphur (generated by the wine and added) within the union, provided this is mentioned on a specific logo. 14 Polyvynilpolypyrrolidone is a synthetic glue. "I'm very sensitive to the energies, the vibrations in the wines, the fullness in the mouth, the length. I can feel the energy of a wine when I taste it. [...] I see energy as a vibration. [...] And this vibration of a finished product is capable of being in harmony with a living body like mine. Afterwards, this vibration can linger more or less strongly. [...] It's as if my body recognises the living nature of the wine. [...] Once you start tasting these wines, your body will remember them, there's a reptilian memory."

The mention of energies and vibrations is one of those most frequently made by tasters and winemakers, both advanced and neophyte. But what kind of energy is it? Here again, we are faced with a "versatile" object, as Sébastien Carcelle describes it in his study on energy in rural homeopathy in Brazil [START_REF] Carcelle | Les secrets de Fatima. Expérimentation, similitude et énergie dans l'homéopathie rurale au Brésil[END_REF], a category that can take on dimensions that are at once physical (thermal or electro-magnetic energy), metaphysical (energy as a vital flow) or moral (good or bad energy) [START_REF] Bordes | ¿De qué hablamos cuando hablamos de energía? Una aproximación a las nuevas culturas terapéuticas desde la sociología del conocimiento[END_REF]. Jo Landron, a winegrower from the Nantes region, refers more to vibrations: "Biodynamics is a way of accompanying the vine, in particular to modify the vibratory level of the wine. The vibratory dimension was the trigger that made me switch to biodynamics. When the wine speaks to us on another level, we don't really know why. That intense happiness that gives you a thrill. Trying to feel within yourself what makes you feel good. I wonder if in ancient times, wine might also have been like that: there were people with powers who I'm sure could make wines that were divine, with that kind of awareness." (Jo Landron, 27/07/18) Others attempt to measure this energetic-vibratory quality with a pendulum, or try to detect the tuning of their vats with a tuning fork. Others again have music playing in their cellars. Experiments on the borderline between science and art are conducted to record the musical scores of fermentation and to understand the living dimension of the microbiota in the musts [Blanc et al., 2021]. The words living/energy/vibration that alternate throughout the discourse seem to evoke a single category that encompasses even the registers of health and well-being. Some winemakers are looking for the "drinkability" or "digestibility" of the wine, categories that are still vague, but which refer to the "health-giving" character or beneficial effects of their products for the body. Some even go so far as to evoke the ideal of wines that heal, not so much by the chemical compounds they contain as by their capacity to transmit their vital and energetic dimension. Pierre Overnoy, a Jura winegrower and pioneer of "natural" wines (Jules Chauvet school), opened out to biodynamics through his "spiritual son" and partner, Emmanuel Houillon. He frequently receives Japanese tasters who express the benefits of this type of wine for the body, its "nourishing" quality. Listening to these words, what emerges from all the variations is above all the circular dimension, from vine to wine and from wine to taster, which is inherent to the holistic vision of biodynamics. What is said is that the information and the intention of early care (given to the vineyard) must be transmissible.

Renegotiating wine-tasting: beyond organoleptics

Validating the transmission of energies, as a combination of spiritual -even esoteric -and pragmatic influences, comes up against the academic scientific framework, but it also comes up against the very foundations of the standard organoleptic tasting exercise. When Pascaline Lepeltier evokes these resonances between bitterness, sweetness and acidity, or when Jacques Néauport (a winemaker of natural wines and friend and pupil of Jules Chauvet) specifies that "a wine that is balanced when analysed will not necessarily produce a harmonious wine in the glass", we realise that analytical grids are not enough to give full expression about a wine. Nathalie Dallemagne, a wine-making and wine-tasting engineer specialising in organic and biodynamic viticulture, goes further: "There is something intimate about wine. We're working with the completely unknown, because even if oenology understands things and can intervene, we have to admit that most of the time we don't understand. When you really want to accompany a wine, there's just you and your own sensitivity." 15Tasting here seems to be renegotiated and presented as an intimate experience. This intimacy of tasting -shared or not -can only be experienced through a framework for interpretation which is open to sensitivity. It is an intimacy that "counteracts the stabilisation of binary categories of inside and outside, of space and time. It disturbs what is binary. It is effectively a democratic experience of sharing what can be sensed, in which we find ourselves on an equal footing because there is no longer any power relationship. " Laplantine, 2018: 82]. On this point of the democratic character of these new ways of appreciating wine -we are aware that the efforts made here to account for these new tasting criteria may seem to be the sophisticated or elitist considerations of insiders -it appears that wine tasting in a more sensitive mode has, on the contrary, been able to open up the world of wine to a new and non-specialist audience. Over and above the effect of fashion, which also plays a fundamental role, and the stories being told about biodynamic wines, it may be that it is precisely these bodily resonances that make these wines accessible to a non-specialist audience which reacts intuitively and sensitively to these wines, without worrying about the intimidating codes of organoleptic tasting. Because, looking into the wineglass, all the stories about the vineyard, the winemaker, the wine and the self intertwine, giving the tasting a unique, unrepeatable and virtually incomparable character. The taster's self is embedded in other selves (animals, plants, humans, spiritual entities, things...), in an extended "ecology of selves" [Kohn, 2013] that tells each sequence of a story in which each has deployed "energy" to transmit "information". This is perhaps the direction we need to take to understand the energetic and vibratory supposed powers expressed by winemakers and tasters, in the perception of the "chain of energy" contained in the living, which is neither stabilised or oriented by oenological products and techniques:

"After a while, this living thing also becomes very sensitive to the winemaker's touch. There's such transparency that you can really feel the agony or the delight of the person making the wine. [...] When there's this much depth to the winemaker's work, you sense it all the more in the bottles. [...] You take a sip and it makes you reflect, open up fields of thought that you didn't necessarily have. What I expect from a real bottle of wine is organoleptic, hedonistic, satisfaction, but I also expect intellectual satisfaction in the sense that the wine offers ways of thinking that I didn't have before...like a piece of music." (Pascaline Lepeltier) At this point, we understand at the very least that expressions around the tasting of biodynamic wines shifts the framework of organoleptic tasting into a different psychological, ontological and artistic terrain 16 . Some professional sommeliers are quite prepared to consider a rethink of tasting codes that would bring in other dimensions, such as energy, aesthetics and poetry... In a way, this would mean moving away from the ethnocentric in the tasting exercise, reconfiguring a model constructed by modern Western oenology. It would mean tasting "in three dimensions" as the Angevin winemaker Didier Chaffardon puts it, giving precedence to the "sculpture and shape of a wine", to understand the "mobility" of a sulphite-free (and biodynamic) wine that "multiplies individual feelings" like Calder's mobiles, writes Alsatian winemaker Jean-Pierre Frick, to analyse the musicality of wines, as proposed by winemaker Gilles Azzoni, a natural winemaker in the Ardèche, who is not a biodynamist but is open to the practice: "Biodynamic wines are tranquil, more gentle... to me it's almost like listening to chamber music... they interest me, they challenge me, everything is about sensing... there's this idea of seeking a lasting balance, harmony. I don't mind imbalance, although I will try to restore a balance. It's not a matter of taste, but of sensations when you drink. In biodynamic wines, there's nothing that jars. Non-biodynamic "natural" wines don't always have that gentleness, they vibrate but more like rock and roll, or reggae. Biodynamic "natural" wines aren't rock and roll, they're baroque, or maybe free jazz." 17To get an idea of the changes underway in the development of evaluation criteria, it's worth mentioning the recent work carried out with the Haute Ecole de Changins in Switzerland, which specialises in viticulture and oenology. Following discussions between the winegrowers of the Syndicat Suisse des Vins Naturels,created in 2021, most of whom practise biodynamic viticulture, and various partners including Pascale Deneulin, Professor of Sensory Analysis, new criteria have been added to the conventional evaluation. The new tabs invite us to take into account "the individual hedonic appreciation, the overall sensations perceived, the taste aspect of the wine", and "the emotion provided by the wine. This judgement is personal, although regularly shared. It does not represent the quality of a wine, as perfect things are not necessarily harmonious or a source of emotion." 18The understanding or perception of what can emanate from a biodynamic wine seems to come from a combination of different tasting models that recognise the importance of organoleptic tasting while linking it with other categories of expression, including synaesthetic tasting19 and the use of metaphor. These categories, which resonate with the underlying analogical structure of biodynamic wine, seem to be better suited to descriptions of the evanescent and subtle otherness of biodynamic wines, but they are also able to broaden the lexical and perceptual field of tasting so that it is accessible to all, from the highly professional to the complete novice.

Conclusion

In this article, we have tried to give a relatively integrated interpretation of the different ways in which winegrowers or professional tasters talk about their biodynamic or "natural" wine practices, in an attempt to grasp what the common thread of these wines might be. We have seen that the work done by the winegrower to stimulate life in the vineyard is said to be reflected in wines that give back this "extra life" in the form of energy and vibrations that are more or less consciously perceptible to the winemakers as well as to those who taste and drink their wine. These ways of understanding wine not only from an organoleptic point of view but also from the point of view of energy, broaden out the horizon to encompass new ways of tasting, drinking and appreciating wine, and with them, new forms of sociability around the act of drinking. In these forms of sociability, a kind of distinction exists, no longer based solely on class boundaries, but on different ways of engaging the body in the act of tasting. This distinction is based on another, perhaps more fundamental, disctinction between conventional and biodynamic production methods. It is the adherence to the other modes of production and consumption that creates a community of knowledge and experience. Alain Corbin's work has enabled us to understand the break with the sensitive apprehension of the world that took place during the period of modernity [START_REF] Corbin | Le miasme et la jonquille. L'odorat et l'imaginaire social. XVIIIe-XIXe siècles[END_REF], a loss of reference to the senses that led to a loss of meaning by cutting itself off from sensitive and intuitive evaluation to the benefit of hygienism and standardisation, a dynamic that applies perfectly to the industrialisation of the wine industry to the detriment of traditional methods.

Like the objects analysed, this article is experimental in the sense that it gives an account of ongoing experiments captured by a process of semantic trial and error that calls on unstable categories such as energy and vibration. By intertwining our object and our model of interpretation, we highlight the separation of appreciation criteria in the 'modern western' organoleptic model, such as the split between body and mind or subject and environment, which all derive from a naturalist matrix. The warp and weft of biodynamic wines could then be found in a certain ontological model of tasting. This would no longer be just a matter of drinking a glass of wine (which is already highly charged with history and symbolism), but also of taking an extract from the world into oneself, an image of other plant, mineral and animal worlds -in other words, of making room for otherness within one's intimate self, as Joël Candau explains "As the same time as odours, other elements of the world are mobilised, thought about and described" [Candau and Jeanjean, 2006 : 9] This way of tasting offers a kind of simplicity but there is also a more ambitious dimension, that of amplifying the resonance of the living world and, in taking a sip of wine, promoting its diversity.

Steiner himself says nothing specific about the question of wine.

Demeter, the main biodynamic certification organisation, had certified 468 winegrowers in France by 2021, and had certified more than 1000 vineyards worldwide. To these figures, we should add the 150 or so members of the Biodyvin Union. While a few estates are certified by Demeter and are also Biodyvin members, many winegrowers practice biodynamics more or less without seeking to advertise the fact. A total of 600 winegrowers practising biodynamics in France today seems to be a conservative but realistic estimate.

See the International Organisation of Vine and Wine website: https://www.oiv.int/public/medias/7943/oivpatronage-concours-norme-ed-2021.pd

Doing without chemical inputs increases risks compared to the conventional method, which can (partly) guarantee the harvest, or correct it using synthetic products, including pesticides.

Of which the 500 and 501 preparations are the most emblematic and widely used. The 500 is a preparation made from cow dung inserted into cow horns that have been left to overwinter, dug up and then dynamised and sprayed in infinitesimal quantities on the vines. This is supposed to nourish and restore the vigour of the soil. The 501 preparation contains quartz silica and is based on the same process, except that it is put out to dry. This is designed to address the foliage of the vine.

Quoted from interviews conducted between 2012 and 2017 with winemakers in the Anjou and Minervois regions.

Among the studies describing the pitfalls of attempting to reify the notion of the terroir[Van Leuwen and Seguin, 2006], we note those of M.-F. Garcia-Parpet for the way they highlight the changes in the socially established hierarchies between terroirs and appellations[START_REF] Garcia-Parpet | Le terroir, le cépage et la marque : stratégies de valorisation des vins dans un contexte de mondialisation[END_REF], 2014], and Geneviève Teil's proposal for a specific ontology of the terroir. Teil analyses the terroir as an object-product 'in the process of being made', not

Also expressed by JulesChauvet (1907Chauvet ( -1989)), a winegrower who became a chemistry and microbiology researcher in later life. Voir Jules Chauvet, Le vin en question. Wine in question. Entretien avec Hans Ulrich Kesselring, Jean-Paul Rocher éditeur, 1998.

Although a study of ten conventional and biodynamic wines did find differences in favour of biodynamic wines from an organoleptic point of view[START_REF] Reilly | Moonshine, Part 2. A blind sampling of 20 wines shows that biodynamics works. But how ?[END_REF]. Study not published in a peer-reviewed journal. Another study of organic and biodynamic wines found less clear-cut differences across vintages[START_REF]Difference Testing of Merlot Produced from Biodynamically and Organically Grown Wine Grapes[END_REF]]. Yet another study found that biodynamically produced grapes had higher levels of sugars, phenols and anthocyanins (the latter contributing to aroma, taste and mouthfeel),[Reeve et al. 2005].

Interview, 28 April 2020.

© Changins -ChG et PaD. See the website : https://www.vin-nature.ch/copie-de-deutsch

The ability to bring multiple senses into convergence. See Victor Segalen, Les Synesthésies et l'école symbolique, Fata Morgana, 1981. Or, on a more poetic note, Baudelaire's well-known poem Correspondances.

"En même temps que des odeurs, d'autres éléments du monde sont mobilisés, pensés et décrits" (author's traduction)