SHARP QUANTITATIVE ISOPERIMETRIC INEQUALITIES IN THE L 1 MINKOWSKI PLANE by

Benoît Kloeckner

An isoperimetric inequality bounds from below the perimeter of a domain in terms of its area. A quantitative isoperimetric inequality is a stability result: it bounds from above the distance to an isoperimetric minimizer in terms of the isoperimetric deficit. In other words, it measures how close to a minimizer an almost optimal set must be.

The euclidean quantitative isoperimetric inequality has been thoroughly studied, in particular in [START_REF] Hall | A quantitative isoperimetric inequality in n-dimensional space[END_REF] and [START_REF] Fusco | The sharp quantitative isoperimetric inequality[END_REF], but the L 1 case has drawn much less attention.

In this note we prove two quantitative isoperimetric inequality in the L 1 Minkowski plane with sharp constants and determine the extremal domains for one of them. It is usually (but not here) difficult to determine the extremal domains in a quantitative isoperimetric inequality: the only such kown result is for the Euclidean plane, due to Nitsch [START_REF] Nitsch | The quantitative isoperimetric inequality for planar convex domains[END_REF].

Statement of the results

We consider the plane R 2 endowed with the L 1 metric:

|(x 1 , x 2 ) -(y 1 , y 2 )| = |x 1 -y 1 | + |x 2 -y 2 |.
The notation | • | shall be used to denote the size of an object, whatever its nature. If A is an measurable plane set then |A| is its Lebesgue measure, also called its area ; if v is a vector |v| is its L 1 norm ; if γ is a rectifiable curve, |γ| is its L 1 length. We denote the boundary of a set using ∂.

By a domain of the plane, we mean the closure of the bounded component of a Jordan curve. In particular, domains are compact and connected. All rectangles and squares considered are assumed to have their sides parallel to the coordinate axes. The square centered at 0 with side length 2λ is denoted by B ∞ (λ): it is the λ ball of the L ∞ metric. Squares are known to minimize L 1 perimeter among plane domains of given area.

The measure of the distance between compact plane sets A, B we use in our main result is the L ∞ Haussdorf metric :

d ∞ (A, B) = inf{λ 0 | A ⊂ B + B ∞ (λ) and B ⊂ A + B ∞ (λ)}.
Let us explain why this metric is natural here. One way to prove that almost isoperimetric domains are close to minimizers is to prove that they contain a minimizer of radius r and are included in another of radius R, with small radii difference R -r and same center. In the euclidean space, such inclusions imply that the considered domain is at Haussdorf distance at most (R -r)/2 from some ball. However, balls and minimizers are different in the L 1 plane, so that if A is between concentric squares of radii R and r, one can only say that it is at L 1 Haussdorf distance R -r from some square, while the L ∞ Hausdorf distance bound is the expected (R -r)/2.

It would certainly be possible to use the L 1 Haussdorf metric, and we expect that arguments of the same kind that those we use to prove Theorem 1.1, but more involved, would give a constant better than the 1/16 obtained using the inequality d 1 2d ∞ and Theorem 1.1.

Theorem 1.1. -Let A be a domain of the L 1 Minkowski plane whose boundary is a rectifiable curve, and assume that

(1) |∂A| 2 (16 + ε)|A|.
Then there is a square S such that

(2) d ∞ (A, S) 2 ε|A| 64 .
We shall also see that Theorem 1.1 is sharp and show that up to L 1 isometry and homothety the domains that achieve the bound are the rectangles and the squares with one square deleted at a corner.

A second possible measure of the distance between domains of the same area, which present the advantage to be suitable to higher dimension as well, is simply the gap between their area and that of their intersection. In this respect we prove the following.

Proposition 1.2. -Let A be a domain of the L 1 Minkowski plane whose boundary is a rectifiable curve, and assume that (1) holds with ε sufficiently small. Then there is a square S such that |S| = |A| and:

(3) |S ∩ A| (1 - √ ε 4 + O(ε))|A|.
In terms of Fraenkel asymmetry, this reads:

(4) |S∆A| |A| √ ε 2 + O(ε)
We shall see that the 1/2 constant in (4) is sharp. Surprisingly enough, it seems that these results are new, although similar ones can be deduced from the much more general [START_REF] Figalli | A mass transportation approach to quantitative isoperimetric inequalities[END_REF] (but with a non-optimal constant) and [START_REF] Peri | On Blaschke's extension of Bonnesen's inequality[END_REF] (only when A is convex).

Proof of the inequalities

Assume A satisfies (1) for some ε and let R be the smallest rectangle containing A. This rectangle plays the role of a convex hull. Proof. -Since R is minimal, each of its sides contains a point of the boundary of A. Denote r 1 , r 2 , r 3 , r 4 such points so that r i and r i+1 lie on two adjacent sides of R for all i (modulo 4). It is possible that some r i = r i+1 , but this does not affect what follows.

There are four curves γ i in ∂A that connect r i to r i+1 and meet only at their endpoints (see figure 1). Similarly, the boundary of R is made of four curves η i connecting r i to r i+1 . Since R is a rectangle, the η i are L 1 geodesics. The length of γ i is at least

|r i -r i+1 | = |η i |, so that |∂A| = |γ 1 | + |γ 2 | + |γ 3 | + |γ 4 | |η 1 | + |η 2 | + |η 3 | + |η 4 | = |∂R|.
Let ℓ and α be such that ℓ -2α and ℓ + 2α are the side lengths of R. Next we have

16ℓ 2 (16 + ε)(ℓ 2 -4α 2 ) 0 εℓ 2 -4(16 + ε)α 2 α 2 εℓ 2 4(16 + ε) α 2 ε 64 |A|
and we are done.

Note that this lemma is sufficient to deduce the L 1 isoperimetric inequality and its equality case: if ε = 0, then α = 0 and |A| = ℓ 2 .

2.1. Proof of Theorem 1.1. -We have min S d ∞ (A, S) α (where S runs over all squares, see figure 2) and if there is equality, Lemma 2.2 is sufficient to conclude. We therefore assume δ := min S d ∞ (A, S) > α.

The following is the main step of the proof.

Lemma 2.3. -We have either Proof. -Choose the origin so that R has its bottom side at height 0. Let S η be the square that is at distance δ from each short side of R (so that it has side length ℓ + 2α -2δ) and whose bottom side is at height η. When η ∈ [-δ, 3δ-4α], R and A are contained in the L ∞ neighborhood of size δ around S η , thus there is some point p η ∈ S η that is at L ∞ distance at least δ from A.

|A| ℓ 2 -4α 2 -8δ(δ -α) or |A| ℓ 2 -4α 2 -4δ 2 .
This excludes A from a square centered at p η ; the worst case (with respect to our goal of bounding |A| from above) is when this excluded squares intersect only small parts of R and have maximal overlap. This is achieved when p η is a corner of S η for all η and the short side of R closest to p η is constant.

In this case, for each η, if p η is a lower corner then there is a 2δ ×(δ +η) sub-rectangle of R excluded, else p η is a upper corner and there is a 2δ × (3δ -4α -η) sub-rectangle of R excluded. These values assume that η < δ and 3δ -4α -η < δ respectively, otherwise there is simply an excluded square of area 4δ 2 .

Let x be the supremum of the η such that p η is a lower corner. There is an excluded sub-rectangle of area 2δ×(δ+x) and for all η > x the point p η must be a higher corner, so that there is another excluded sub-rectangle of area 2δ × (3δ -4α -x).

Summing up, either there are excluded sub-rectangles of total area at least 2δ × 4(δ -α), or there is an excluded sub-rectangle of area at least 4δ 2 , and we get the desired bounds on |A|.

We can now conclude the proof of Theorem 1.1. First, if |A| (ℓ 2 -4α 2 ) -8δ(δ -α) then we have

|∂A| 2 (16 + ε)|A| (4ℓ) 2 (16 + ε) ℓ 2 -4α 2 -8δ(δ -α) 0 εℓ 2 -4(16 + ε)(α 2 + 2δ(δ -α)) α 2 + 2δ(δ -α) εℓ 2 4(16 + ε) 2δ 2 -2αδ + α 2 ε 64 |A|
the last inequality coming from (5). Since the function x → 2δ 2 -2xδ + x 2 is minimal when x = δ, we have 2δ

2 -2αδ + α 2 2δ 2 -2δ 2 + δ 2 = δ 2 , so that δ 2 ε 64 |A|.
In the case when |A| ℓ 2 -4α 2 -4δ 2 , we get:

|∂A| 2 (16 + ε)(ℓ 2 -4α 2 -4δ 2 ) 16ℓ 2 16ℓ 2 + εℓ 2 -4(16 + ε)(α 2 + δ 2 ) α 2 + δ 2 εℓ 2 4(16 + ε) δ 2 ε 64 |A| 2.2. Proof of Proposition 1.2. -Let µ = max S |S ∩ A|/|A|
where S runs over the squares having same area than A.

Lemma 2.4. -One of the following holds:

µ 2 - ℓ 2 -4α 2 |A| µ 2 - ℓ + 2α |A| .
Proof. -Define S 0 to be a square that shares a corner of R and intersects its interior, and that have the same area than A (see figure 4). The definition of µ implies that |A ∩ S 0 | µ|A|.

If |A| ℓ -2α we have:

|A| µ|A| + (ℓ + 2α)(ℓ -2α -|A|) + |A|(ℓ + 2α -|A|) µ|A| + ℓ 2 -4α 2 -|A|(ℓ + 2α) + |A|(ℓ + 2α) -|A| ℓ 2 -4α 2 + (µ -1)|A| µ|A| 2|A| -(ℓ 2 -4α 2 ).
Otherwise, we get If the first conclusion holds, using Lemma 2.2 it comes

|A| µ|A| + (ℓ -2α)(ℓ + 2α -|A|) µ|A| + |A|(ℓ + 2α -|A|) µ|A| 2|A| -(ℓ + 2α) |A| R ℓ -2α -|A| |A| ℓ + 2α -|A| S 0
µ 2 - (ℓ 2 -4α 2 )(16 + ε) 16ℓ 2 2 - 16 + ε 16 = 1 - ε 16
If the second conclusion holds, using Lemma 2.2 we get

µ 2 - ℓ |A| - 2α |A| 2 -1 + ε 16 - √ ε 4 1 - √ ε 4 + O(ε)
But for all sufficiently small ε, this second expression is smaller than 1 -ε/16, and Proposition 1.2 is proved.

Sharpness

Two examples showing sharpness of Theorem 1.1 steam out from its proof. The first one is the domain S ′ δ obtained from the unit square by deleting a 2δ × 2δ square at one corner (δ < 1/2)). We have |S ′ δ | = 1 -4δ 2 and |∂S ′ δ | = 4, so that (1) holds with ε = 64δ 2 1 -4δ 2 and inf S d ∞ (S, S ′ δ ) = δ so that equality holds in (2). The second one is the rectangle R α whose side length are 1 -2α and 1 + 2α (where α < 1/2)). We have 2) is an equality once again.

|R α | = 1 -4α 2 , |∂R α | = 4 and inf S d ∞ (S, R α ) = α so that (
Let us show that S ′ δ and R α are the only possible (up to homothety and L 1 isometry) exemple realizing equality in both (1) and (2) for the same ε. In the first case of Lemma 2.3, for 2δ 2 -2αδ + α 2 δ 2 to be an equality it is necessary that α = δ, so that A must be equal to R (otherwise R would have smaller isoperimetric inequality and same distance to squares). In the second case of the lemma, one is lead to α = 0 in the last lines of the proof of Theorem 1.1, so that R is a square and according to the proof of Lemma 2.3, A is contained in a S ′ δ having the same isoperimetric deficit and the same minimal rectangle. They must therefore be equal.

At last, R α shows asymptotic sharpness of Proposition 1.2:

sup |S|=|Rα| |S ∩ R α | = (1 -2α) √ 1 -4α 2 = 1 -2α + o(α) and 1 - 1 4 √ ε = 1 -2α + o(α)
when ε takes the extremal value 64α 2 /(1 -4α 2 ).

  Lemma 2.1. -We have |∂A| |∂R|.

Figure 1 .

 1 Figure1. The L 1 perimeter of A is at least that of R

Figure 2 .

 2 Figure 2. The closest square to R.

Figure 3 .

 3 Figure 3. The domain A avoids one of the grey squares.

Figure 4 .

 4 Figure 4. The domain A is included in R and cannot meet a too large proportion of S 0

Figure 5 .

 5 Figure5. Two domain that are almost isopermetric and as far as possible from squares: a square with a small corner deleted and a rectangle with sides of almost the same lengths.

Benoît Kloeckner


Bonpagna Kann

Sandra Castellanos-Paez

Philippe Lalanda

Evaluation of Regularization-based Continual Learning Approaches: Application to HAR

Keywords: Continual learning, regularization methods, HAR

Pervasive computing allows the provision of services in many important areas, including the relevant and dynamic field of health and well-being. In this domain, Human Activity Recognition (HAR) has gained a lot of attention in recent years. Current solutions rely on Machine Learning (ML) models and achieve impressive results. However, the evolution of these models remains difficult, as long as a complete re-training is not performed. To overcome this problem, the concept of Continual Learning is very promising today and, more particularly, the techniques based on regularization. These techniques are particularly interesting for their simplicity and their low cost. Initial studies have been conducted and have shown promising outcomes. However, they remain very specific and difficult to compare. In this paper, we provide a comprehensive comparison of three regularization-based methods that we adapted to the HAR domain, highlighting their strengths and limitations. Our experiments were conducted on the UCI HAR dataset and the results showed that no single technique outperformed all others in all scenarios considered.

I. INTRODUCTION

Pervasive computing has imposed itself in our lives for several years now and allows the provision of services in many important areas such as building management, industrial production, or driving and transport assistance [START_REF] Becker | Pervasive computing middleware: Current trends and emerging challenges[END_REF]. A particularly relevant and dynamic field is that of health and well-being. There, the use of intelligent sensors worn or inserted in the living environments allows to collect relevant information, to conduct analysis, to give advice, and even to act on some physiological parameters if needed. Human Activity Recognition (HAR) belongs to this latter domain. Its purpose is to automatically determine basic activities of a person (like standing, walking, running, etc.) which, if contextualized, allow to infer more complex activities (like dining, cooking, playing, etc.). The most popular approach today to implement an HAR application, because it is not very intrusive as opposed to cameras, is the use of wearable devices such as smartphones or smart watches. Data from accelerometers and gyroscopes is usually used to feed a (deep) machine learning model (ML). Results obtained today, mainly in controlled conditions, are of high quality and the state of the art is regularly improved.

An important challenge, however, is that models must be adapted to the characteristics of the clients (how they walk, run, stand, etc.). This need for adaptation is not limited to the initial deployment phase of the models, but continues throughout their life cycle. This is due to the physical evolution of the subjects, who for example can be affected by a disease or age [START_REF] Xie | Agerelated imbalance is associated with slower walking speed: An analysis from the National Health and Nutrition Examination Survey[END_REF]. Updating a model with new data is however a difficult exercise, especially when the additional data is non IID (Independent and Identically Distributed).

If it is fine-tuned sequentially over time, a model tends to give excessive weight to new data, and progressively forget older data. This problem of not being able to recognize initially learned labels when new labels are integrated in the training process is known as "Catastrophic Forgetting" [START_REF] French | Catastrophic forgetting in Connectionist Networks[END_REF]. An existing solution to preserve old knowledge is to regularly perform complete retraining with all the data, old and new. This approach is expensive and risky. It requires important computational and storage resources, and is time consuming. It also requires personal, sensitive information to be sent up to a cloud since the devices generally do not have the necessary capabilities. This clearly raises security and privacy concerns.

An alternative method to deals with catastrophic forgetting is called Continual Learning (CL) [START_REF] Wiewel | Entropy-based sample selection for online continual learning[END_REF][START_REF] Jeon | Continual representation learning for images with variational continual auto-encoder[END_REF][START_REF] Fayek | Progressive learning: A deep learning framework for continual learning[END_REF]. This approach seeks to retain the accuracy of a ML model in class-incremental scenarios where new classes are sequentially added for training. Here, we focus on a family of CL algorithms known as regularization-based because they are simple and compatible with limited computational resources. These methods use previous version of a model to maintain accuracy on all classes. Specifically, they add terms to the loss function that penalize the model for forgetting earlier knowledge. Initial studies have been conducted and have shown promising outcomes [START_REF] Usmanova | A distillation-based approach integrating continual learning and federated learning for pervasive services[END_REF][START_REF] Jha | Continual learning in sensor-based human activity recognition: An empirical benchmark analysis[END_REF]. However, they remain specific and difficult to compare.

In this paper, we evaluate three regularization methods to analyze their strengths and weaknesses on the HAR domain. In Section 2 we present the context of HAR and the related work regarding continual learning. In Section 3, we describe the different methods and the methodology used to evaluate them. Section 4 presents our results and, finally, we conclude in Section 5 and sketch our future work II. RELATED WORK Continual Learning (CL), also called lifelong learning or online machine learning, is the ability of a model to learn continually from an infinite stream of data, gradually integrating new acquired knowledge into old knowledge [START_REF] Chen | Lifelong machine learning[END_REF]. Going back to our HAR use case, the idea behind CL is to be able to learn new activities or new ways to perform activities (data drift). Previous studies primarily focused on the adaptation of the model to the continuously changing activities of people. In [START_REF] Gjoreski | Unsupervised online activity discovery using temporal behaviour assumption[END_REF], Gjoreski and Roggen used the agglomerative clustering technique to cluster the real-time sensor data. With the use of temporal constraints, the new incoming data will be clustered as an anomaly if it does not belong to any existing clusters. However, this method can lead to many unnecessary clusters resulting from the constant updates of new activities from users. To address this issue, Cheng et al. introduced in [START_REF] Cheng | Nuactiv[END_REF], the use of the knowledgedriven model, which contains the semantic relationship between user activities and sensor attributes from accelerometer data, to recognize a new activity in the case of small-scale training data. After a new activity is added to the training, the new attributes and their relationship to the existing activities will be recorded by the domain experts manually. However, this method requires a great reengineering effort as the core process is conducted manually by the domain specialists. On top of that, the ML models also need to be rebuilt or re-trained from scratch when extending the new activities.

More generally, continual learning methods have been categorized into three main families [START_REF] Delange | A continual learning survey: Defying forgetting in classification tasks[END_REF]: regularization-based methods, replay methods, and parameter isolation methods. Replay methods store samples from each training task in either raw format or pseudo-samples to be used as the inputs in the next training task, which is essential to avoid forgetting when training on a new task. iCarl [START_REF] Rebuffi | ICaRL: Incremental classifier and representation learning[END_REF] stores exemplars, i.e., a subset of samples with the best approximate means of each class, to represent the information from the previous classes. The exemplars are then used as additional inputs to the new training task to retain the good accuracy on the classes from previous tasks. To overcome the problem of overfitting that iCarl is prone to, GEM [START_REF] Lopez-Paz | Gradient episodic memory for continual learning[END_REF] proposed to constrain the updates of the new task without affecting the previous tasks through the estimated gradient direction projection on the feasible region using first-order Taylor series approximation. However, storing the sample data requires large storage space, and affects user privacy.

Parameter isolation methods create or freeze different parameters for each model training to avoid forgetting. For example, when the new tasks are different from the previous tasks, new neuron branches are created in the network, and the parameters from the previous tasks are frozen, making taskspecific components of the network [START_REF] Rakaraddi | Reinforced continual learning for graphs[END_REF]. However, parameter isolation methods have the problem of scalability, leading the complexity of the architecture to grow larger along with the increasing number of learned tasks.

Regularization-based methods use regularization term in the loss function to avoid forgetting. The aim is to retain the knowledge from the previous training when adding new data without having to modify the network architecture. Li and Hoiem proposed Learning Without Forgetting (LwF) [START_REF] Li | Learning without forgetting[END_REF].

This uses the probabilistic output from the previous model to transfer the knowledge to the new task. This method depends heavily on the relevance of the training tasks as it uses the direct output from the previous task to transfer knowledge. To alleviate this issue, [START_REF] Kirkpatrick | Overcoming catastrophic forgetting in Neural Networks[END_REF] proposed Elastic Weight Consolidation (EWC). EWC estimates the importance of the parameters using the posterior distribution with a Bayesian approach. It penalizes the model when there is a significant change to those important parameters in the new task. Despite this specialization of EWC, this algorithm has shown drawbacks when learning new classes incrementally as shown in [START_REF] Kemker | Measuring catastrophic forgetting in neural networks[END_REF]. As EWC and LwF work in different aspects of the model training, Kim et al. proposed in [START_REF] Kim | Keep and learn: Continual learning by constraining the latent space for knowledge preservation in Neural Networks[END_REF] the combined model of EWC and LwF.

III. EVALUATION

We have implemented the three techniques mentioned here before and applied them to our HAR use case. To assess the results, we have followed the evaluation of class-incremental methodology where we begin by performing two-class classification, and add one different class in each round. In our study, an activity is referred to as a class while a collection of one or more activities added each round is described as a task. We implement 6 rounds of local training for the evaluation of LwF, EWC, and EWCLwF.

A. Regularization Methods

LwF uses knowledge distillation (KD) loss as a regularization term, transferring the knowledge from teacher to student model. Distillation loss is described by [START_REF] Becker | Pervasive computing middleware: Current trends and emerging challenges[END_REF], where is defined as the number of classes, the prediction logits of the teacher model, the prediction logits of the student model for an old class label .

The temperature-scaled logits for both teacher and student model are computed as shown in (2), where T is the temperature scaling parameter.

and (2)

With the use of the softmax cross-entropy for the classification loss from the student model, the final loss in LwF is the combination of the classification loss and the knowledge distillation loss denoted as (3):

(3) where α is a scalar which regularizes the influence of each term.

EWC estimates the importance of the model parameters with the posterior distribution of the parameter given the whole dataset illustrated in [START_REF] Wiewel | Entropy-based sample selection for online continual learning[END_REF], where refers to the conventional loss for task B, the important parameters of task A, and the distribution of the dataset in task B.

However, since the true probability is intractable, it can be estimated with Laplace approximation as a Gaussian distribution with a diagonal precision matrix determined by the Fisher Information Matrix (FIM). The loss function of EWC is written in [START_REF] Jeon | Continual representation learning for images with variational continual auto-encoder[END_REF], where is the loss of the new task B, λ refers to the importance of the old task A to task B, (

) refers to the changes of the important parameters in task A after training on task B, and is the diagonal FIM for each task .

EWCLwF was proposed to combine EWC and LwF. The loss function is denoted in ( 6): [START_REF] Fayek | Progressive learning: A deep learning framework for continual learning[END_REF] where is the total loss of the LwF and is the total loss from the EWC method.

B. Metrics

To evaluate the model specialization, we use the overall average accuracy, the average accuracy at a specific task, and forgetting [START_REF] Masana | Class-incremental learning: Survey and performance evaluation on Image Classification[END_REF].

Overall average accuracy:

The average accuracy of the model on classes which were already learned by the model.

where the accuracy is calculated on classes which have been already learned between round 1 and round .

Average accuracy at task t: The average accuracy of the model on task d after training on task t.

where is calculated by taking all the examples of classes corresponding to an old task d from the test set, then calculating the accuracy of the model on them after learning a new task t.

Forgetting:

The average forgetting of the model on task d after performing the task . The higher is, the more model forgets.

C. Dataset

The UCI HAR dataset contains the data of six activities from 30 volunteers. Each activity contains 128 recordings from the accelerometer and gyroscope for the x, y, z axes. The six activities and their labels in the data are described as follows: Walking (0), Walking Up (1), Walking Down (2), Sitting (3), Standing (4), and Laying [START_REF] Jeon | Continual representation learning for images with variational continual auto-encoder[END_REF]. The dataset contains 10299 records with 561 feature vectors with time and frequency domain variables.

In order to provide a deeper understanding of the dataset, a t-SNE analysis [START_REF] Van Der Maaten | Visualizing data using t-SNE[END_REF] on the last dense layer before the final activation of a pre-trained CNN model using 960 samples was conducted. In Fig. 1, we present a visualization using 160 random samples from each class. It can be seen that "Laying" is clearly separated from other classes while walking-related activities stay in the same cluster, displaying similar characteristics between activities. Additionally, "Sitting" and "Standing" are also located near each other, forming another cluster in the dataset.

D. Evaluation Process of Regularization Methods

Python 3 has been used in this experiment along with TensorFlow 2. We used a CNN model architecture from a previous study [START_REF] Usmanova | A distillation-based approach integrating continual learning and federated learning for pervasive services[END_REF] which includes 196 filters of a 16x1 convolution layer followed by a 1x4 max pooling layer, then 1024 units of a dense layer, and a softmax layer with SGD optimizer. The initial weights for the CNN model were obtained by pretraining the previous study CNN model on a well-balanced dataset of 10 examples per class. We used dropout rate equal of 0.5, and batch size B =32 to both LwF and EWC. The learning rate of [0.001, 0.01], α = 0.1 and T=3 is set for LwF part, while the learning rate of [10 -5 , 5x10 -3 ], λ = 5 is defined for EWC part. We trained the model with the class-incremental method for 6 rounds, starting with two classes by learning the same classes with different data for 2-3 rounds before adding another class for the remaining rounds. The size of the training data for each round is 120 for each class. In order to study how each method performs, the following three scenarios were considered:

• Scenario 0: Learning similar activities before switching to a completely different activity. • Scenario 1: Learning different types of activities continuously. • Scenario 2: Learning similar activities continuously.

More information is given in Table I.

IV. RESULTS AND DISCUSSION

A. Evaluation of Regularization Methods

Scenario 0: The results show that EWC has the highest average accuracy and the lowest forgetting in the final round as the class from the new cluster is added to the model after learning the two similar classes as shown in Table II.

TABLE II. AVERAGE ACCURACY (A), ACCURACY BY TASKS (A1-3), AND

FORGETTING BY TASKS (F2-3) OF SCENARIO 0

In addition, the accuracy by classes also displays how EWC could retain the previous knowledge better in this situation with the higher accuracy after the class from a different cluster is added, which is shown in Fig. 2. From Fig. 3, with more different classes added to the training, the performance of EWC started to decrease, and so does its forgetting. Consequently, this performance showed a similar outcome as the result in [START_REF] Delange | A continual learning survey: Defying forgetting in classification tasks[END_REF], suggesting the weakness of EWC over the long-term as FIM is approximated after optimization of the task, inducing gradients near zero. As a consequence, the regularization strength is initially very high and lowered only to decay stability.

B. Discussion

From our experiments, it appears that there is no clear winner for all scenarios. Indeed, when switching activities during learning (Scenario 0), EWC showed the best performances; when learning different types of activities continuously (Scenario 1), EWCLwF performed best; and, when learning similar types of activities continuously (Scenario 2), LwF outperformed the other methods. To explain the difference in performance, we can look at the problem under the plasticity-stability dilemma 1 prism.

On the plasticity-stability axis, LwF leans on the stability side since it is a technique that directly penalizes changes in the network; EWC is more on the plasticity side since it tries to find parameters that achieve good compromise between different tasks; and, EWCLwF combining both lies somewhere in the middle. Then, Scenario 0 having a drastic change in activities needs a very plastic method; Scenario 2 however, having only very similar activities is served best by more stable methods; and, Scenario 1 having consistently diverse activities needs a trade-off between plasticity and stability. 1 This dilemma relates to the fact that in order to integrate new knowledge a network needs to be plastic, i.e. be able to change its parameters to learn effectively. However, to retain knowledge, a network should be stable, i.e. be able to keep its parameters compatible with already acquired knowledge.

Our results revealed the problem of no universally good regularization method to effectively perform CL in HAR in various scenarios. Although we experimented in a single dataset, UCI is a balanced dataset known in the HAR community as an entry level challenge. We thus believe that the problem will only be exacerbated on more complex datasets.

One research avenue to this problem could be the combination of regularization and replay methods which can be more all-purpose since the latter have access to some store data. To overcome store space limitations, replay methods could be based on learned data generator as in [START_REF] Shin | Continual learning with deep generative replay[END_REF]. Although, learning this generator comes with its own challenges. Another research avenue could be to dynamically switch between several regularization methods. To switch efficiently, one have to automatically identify the scenario in which the learning is performed.

V. CONCLUSIONS AND FUTURE WORK

In this paper, we evaluated three regularization algorithms on their strengths and weaknesses in different scenarios in the HAR domain. Results have shown that while EWC helped the model coping with classes from different clusters occasionally, EWCLwF improved the weakness of EWC in the long-term single method implementation. On the other hand, LwF was more specialized in training similar classes.

In future work, we would like to investigate a dynamic learning technique that can help the model deal with a variety of classes that the model has not seen before by maintaining accuracy while keeping the forgetting low in class-incremental scenarios. To do so, a research avenue is to study and exploit the strengths and weaknesses of each CL method depending on the scenario. Thus, we could first answer to the problem of coarsely clustering the stream of input classes. Secondly, a meta-learner could adapt and choose which strategy in the portfolio of state-of-the-art CL approaches to use depending on the found clustering.

Fig. 1 .

 1 Fig. 1. t-SNE Analysis of UCI HAR Dataset

Fig. 2 .Scenario 1 :

 21 Fig. 2. : The accuracy by class in each round of Case 1 in Scenario 0 Scenario 1: According to Table III, after training with two different classes for 3 rounds, another class from another different cluster is added. The results show that EWCLwF has the best performance with the highest average accuracy and the lowest forgetting.

Fig. 3 .Scenario 2 : 2 Fig. 4 .

 3224 Fig. 3. The accuracy by class in each round of Case 1 in Scenario 1 Scenario 2: From Table IV, LwF has outperformed the other two algorithms when learning similar classes. With the

TABLE I .

 I TRAINING SCENARIOS OF CL ALGORITHMS

TABLE III .

 III AVERAGE ACCURACY (A), ACCURACY BY TASKS (A1-2 ), AND FORGETTING (F2 ) OFSCENARIO 1


Helen E Mundler

Linda Bluestone

Linda Constanzo Cahir

Thomas Hutcheon

Yvonne Leitch

Griggs

George Bluestone

Linda Hutcheon

Thomas Leitch

Yvonne Griggs

Revisiting "Medusa's Ankles" in the light of the 2018 film adaptation

dans l 'oeuvre d'A. S.Byatt (2003), and The Otherworlds of Liz Jensen: A Critical Reading (2016), as well as a range of critical articles several of which tackle the work of

Whatever she does, the bright forms will go on shining in the dark. (Byatt,"The Chinese Lobster," The Matisse Stories 133) Text/Image, Novel/Fragment This article revisits A. S. Byatt's "Medusa's Ankles," one of her short stories which has received a lot of critical attention, in the light of its 2018 film adaptation. The first point which seems striking is that one of the defining characteristics of Byatt's work is its transposition of one system of reference into another, exemplified by its sustained and complex exploitation of intertextual and intratextual techniques. Quotation, allusion, intercalated texts and absent texts ascribed to characters-all these are fertile ground to the Byatt scholar. To this extent, the adaptation of her work into another medium seems particularly appropriate, and the adaptation of this particular piece the more so, because of its interest in the visual and its mixing of media. The very act of making "Medusa's Ankles" into a film seems a particularly Byattian step, a rewriting, a recasting, a recycling, an exterior manifestation of a process inherent in the text.

At the time of publication, The Matisse Stories (1993) marked to some extent a new departure in that the volume of three short stories includes illustrations, which seem to function as integral parts of the stories in that they are carried over from one edition to another. These illustrations, paratexts in Genette's terms (Palimpsestes 12), illuminate the short stories without being indispensable to them: that is to say, the stories could still stand without them. The reader could simply imagine some of the most pertinent features of those of the illustrations which are described in the text, as Matisse's Rosy Nude, in "Medusa's Ankles," and his or her reading of the text would not necessarily be compromised or diminished by not seeing the pen-and-ink illustration of Matisse's La Chevelure with which "Medusa's Ankles" begins.

And yet, Byatt chose to include these illustrations. As Kamilla Elliott observes, "[w]hat we find when we look at the history, theory, and criticism of film and of the novel is a pervasive tendency not only to treat novels as 'words' and films as 'images,' but to treat films as though they had no words and novels as though they had no illustrations" (3). Pointing to the widespread use of ekphrasis in novels, and finding a corollary in the way words in cinema tend to be ignored, she argues that "[n]either film nor novel is 'pure '" (3). This is certainly applicable to Byatt's writing: it mixes media. Still Life is marked by a strong interest in the visual: this second novel of the Frederica Potter tetralogy is dominated and steered by the attempt to reproduce in words the process of applying paint to canvas by writing without metaphor, a project which, while doomed to failure, allows for a very rich and highly original heterodiegetic commentary on the processes of creation, which culminates in an extraordinary meditation on the nature of language and representation, seen through the comparison of painting. This passage interrupts the narrative to take up the title of the novel: You may use the word "bloom" for the haze on this plum, and it will call up in the mind of any competent reader the idea that the plum is glistening, overlaid with a matte softness. You may talk about the firm texture of the flesh, and these words will not be metaphors, bloom and flesh. . . . But you cannot exclude form the busy automatically connecting mind possible metaphors, human flesh for fruit flesh, flower bloom, skin bloom, bloom of ripe youth for this powdery haze . . . (Still Life 164) Having posited, in typically precise and demanding Byattian terms, the nature of metaphor and evoked the impossibility of excluding it, the narrator then goes on to conclude that while paint is clearly not reality and is not taken to be real, the same does not apply to language: It is impossible not to think about the distance between paint and things, between paint and life, between paint and the "real world." . . . It is not at all impossible, it is even common, not to think about the distance between words and things, between words and life, between words and reality. (Still Life 165) These contemplations are only loosely tied to the character of Alexander Wedderburn, who is developing his play The Yellow Chair, itself a response to a painting by Van Gogh. The narratorial voice asserts itself almost essayistically, in a discursive pause, to work through an intellectual problem and formulate a position, a technique appreciated by some (Westlake 37) and condemned by others as intrusive (Buschini 99). Although not the next work to be published after Still Life-the neo-Victorian period intervenes, as well as the publication of a volume of short stories, Sugar-The Matisse Stories can be read as a continuation of this meditation on the nature of painting as against the nature of language. Certain questions which are explicitly posed in the meta-narrative of Still Life continue, in a more submerged way, to be central to The Matisse Stories: can the "language" of painting and colour be separated from the actual, verbal language, or are the two part of a continuum? Does language impede vision? And how may an artwork be represented by a text? Moreover, this meditation on painting/language, image/text can itself be read more generally as part of the preoccupation with metamorphosis which runs through Byatt's work (see, for example, Mundler, Intertextualité 144-90; and Mundler, "Puzzle"): "It is all a perpetual shape-shifting and adjustment," writes Ash to Christabel in Possession (190), and medium-shifting seems a logical next step in this great writerly project and a related step in Byatt studies. Given Byatt's interest in what might be considered mixed-medium narrative, and given this long preoccupation with the comparison between narrative and visual techniques, it seems entirely fitting that "Medusa's Ankles" should have been made into a film. Even if Bonnie Wright's project seems from her own account to have little if anything to do with the questions raised above-she puts the emphasis on the setting of the story, highlighting the ability of film to recreate "sensory interaction" in "the heightened space between the hairdresser and their client" (Pulver)-the film version seems a particularly interesting area of study to further previous readings of this short story. Of course, this is not the first time that Byatt's work has been adapted for other media, and similar arguments might be deployed in favour of analysing the BBC Radio adaptations of the tetralogy (broadcast as The Frederica Quartet). In that case, questions about how words heard as opposed to words read function might be interestingly explored, since the theatrical is foregrounded in both Still Life and The Virgin in the Garden, with the production of Alexander's two plays. Moreover, the BBC itself as well as various broadcasting experiences are mentioned in Byatt's work (Frederica appears in radio and television programmes), which could make for an interesting mise en abyme. Neither is it the first time Byatt's work has been filmed: Possession and Morpho Eugenia (released as Angels and Insects) have both been adapted for cinema, although in terms of being adapted specifically to be shown on a loop in a hairdresser's, "Medusa's Ankles" is certainly a first. However, the brevity and intensity of "Medusa's Ankles," a short story only a few pages long, as well as the fact that it was published as an illustrated text, make the relationships between the original and the adaptation particularly striking.

While analysis of the visual within the literary is an interesting avenue of inquiry, so too is the relation of the short story to the longer form in Byatt's output. The novel-grown increasingly vast and rambling-is Byatt's preferred form, although short stories have always been part of her production. The short stories range from novella-length-"The Djinn in the Nightingale's Eye," "Morpho Eugenia" and "The Conjugial Angel" come into this category-to the fragment. I use the word advisedly, not only for its echo to the explosion of the sarcophagus in "The Glass Coffin," one such short text, but because some of Byatt's short stories are literally fragments of her novels, republished and repackaged. Both "Gode's Story," and "The Glass Coffin" are grafted into Possession and then out of it, appearing in The Djinn in the Nightingale's Eye. By virtue of this displacement and new placement, these two stories provide many interesting avenues for Byatt scholars, who may analyse how each fragment fits polyphonically into the whole of Possession, and how it also enters into dialogue with the story of Gillian Perholt in "The Djinn in the Nightingale's Eye," where the short fragment gains in interpretation, in significance, and gives rise to new analyses of the text into which it is hyperdiegetically inserted (see for example Mundler,. These two pieces, "The Glass Coffin" a rewriting of a tale by the Brothers Grimm, even if significantly different (see Mundler,, "Gode's Story" a pastiche of a Breton folk tale, are of major interest to the study of Byatt's output.

Certain of Byatt's short stories, however, may appear fragmentary in the sense of nugatory or inconsequential, and seem to lack gravitas: some readers have shown surprise that Byatt's refutation of the "unsexed mind" (Introduction, The Shadow of the Sun ix) extends as far as setting a short story in the typically female emporia of the beauty salon ("Medusa's Ankles," in The Matisse Stories) or the shopping mall ("Baglady," in Elementals). Both Daphne in "Baglady" and Susannah in "Medusa's Ankles" are much preoccupied with the processes of transformation and disintegration age brings about. Even if Susannah is vindicated by her intellectual pursuits as a classicist, it might seem surprising that Byatt could be so concerned with the banal and the everyday. "Baglady" might seem even more out of character since Daphne has no academic credentials, and is merely a wife, an appendage to a travelling husband-although on deeper analysis that story proves to throw up many Byattian preoccupations (see Mundler, "Puzzle"). Nonetheless, these stories may initially seem mundane, or disappointing, or "feminine." They may seem, on first reading, to be merely "women's" stories, unredeemed, should such redemption be required, by the gravitas conferred by insertion in Byatt's greater works, and inadequate in relation to the articulate and authoritative analysis of "women's concerns" in her novels. Frederica's cri de coeur, "How to live?" (Still Life 122), Matty Crompton's resounding declaration, "It is what I will do," as she prepares to leave her post as governess to go with William to Rio de Janeiro and "the Amazons" (Morpho Eugenia 157), find no immediate equivalent in "Baglady," in which Daphne is for the most part a passive object, and speaks little-and while in "Medusa's Ankles" Susannah's smashing-up of the hair salon is certainly memorable, what it is not is verbal. It requires the reader to formulate for him/herself what Susannah would say, since she does not actually say it, but rather "talks around" it: "I want my real hair back" (Matisse 24) does not sum up Susannah's dilemma, it displaces it. The reader must deconstruct and reconstruct her story from the ruins of the destruction she wreaks, must discover the previous link in the metonymic chain she references when she talks about her hair. "Medusa's Ankles," then, far from being light or frothy, mobilises interesting and complex narrative strategies. A number of critics have offered readings which take that general line (see Mundler, Intertextualité; Campbell; Wallhead; Fishwick). Secondly, it has become clear that this short story departs from Byatt's usual verbal mode and favours the visual-though I would not go as far as Jane Campbell, who posits that "the visual replaces the literary" in The Matisse Stories (Campbell 170). Building on those arguments, I would like to ask how the film version of "Medusa's Ankles" takes up certain of those narrative strategies using the visual medium of cinema, and how it creates "its own palimpsestic thing," as Hutcheon says an adaptation must (9).

Adaptation/Transposition/Translation

Bonnie Wright's twenty-one-minute film (figure 1) is a very recognisable version of the short story. In Geoffrey Wagner's terms it sits somewhere between "transposition (where the text is given directly on screen with a minimum of interference)" and "commentary (where an original is taken and either purposely or inadvertently altered in some respect)" (222-23). While it might be difficult to imagine an adaptation which is not to some degree a commentary in these terms, Wright's version of "Medusa's Ankles" does not venture into the territory of "analogy (which must represent a fairly considerable departure for the sake of making another work of art)" qtd. in Griggs 2).

Describing Bonnie Wright's film necessitates keeping in mind the various warnings: George Bluestone memorably laid down the principle that comparing a film and a novel as "better" or "worse" than one another is as fruitless as "pronounc[ing] Wright Johnson's Wax Building better or worse than Tchaikowsky's Swan Lake" given that each is characterised by "unique and specific properties" (5-6). Since Bluestone, a vast body of adaptation theory has grown, much of which posits similar arguments. For example, Thomas Leitch warns against seeing an adaptation in terms of "fidelity" (3), and Robert Stam against emitting "moral" judgments about such fidelity or infidelity (qtd. in Griggs 4). Thus, I will seek to consider the nature of the film as object rather than as a compromised text. Of course, it is impossible to avoid comparison between text and film since that is part of the purpose of this article, and, as argued above, techniques of transposition, transplantation and adaptation of all sorts are central to Byatt's work and the analysis of it central to its criticism. To this extent, I will make some comparative analysis, asking, what changes and what stays the same? What stays the same, but is mobilised differently in the film, and what is the effect of this? Inevitably, watching the film when you know the text well incites comparisons. It is impossible not to register in passing any number of minor changes-the female hairdressers in the salon's first manifestation do wear pink overalls, but without the cream binding mentioned in the text-and to be conscious of the effect of, and possible reasons behind, others: the hexagonal black teacups after the makeover of the salon stay the same, but the "elliptical sweets, black and white like Go counters" (Matisse 16) are replaced with French macaroons-a symbol of continental sophistication in recent times. Obviously, a sweet, if sucked, would prevent Susannah (played by Kerry Fox) from speaking clearly, but registering the absence of sweets in the film will inevitably send the Byatt scholar back to "Sugar" and its narrator's account of writing about the visit to a humbug factory as a schoolgirl, during which she witnesses the transformation of "large buckets . . . of gleaming fluid" (Sugar 243) into "gleaming sheets" with "rows of rounded discs" (244). This incident is significant in that it contributes to the narrator's conviction that she must write in order to recreate "the pleasure I usually took only in reading" (245), and thus forms an instance of the typically Byattian metadiscourse on the processes of creativity. In this way a small change sets up a significant association: it reinforces the presence of Byatt, the writer, in the film.

More obvious changes are also made: Lucian (played by Jason Isaacs) does not come across as homosexual as he does in the text, and he has a French accent, whereas he seems to be from the north-west of England in the short story. The dialogue is flattened at key points: Lucian does not talk about "placing" the Rosy Nude-the Pink Nude, as the film version more generically has it-in the salon, or say his wife "didn't think much to" it (Matisse 4), he says, "She didn't think nothing of it," adapting the phrase to his own French-London idiom, but losing the comic, Victoria Wood-like effect of the Yorkshire rhythm of his speech. We can only speculate that it might be confusing in a short film to present the main male character as both effeminate and married to a woman, or that a goodlooking male lead was required to balance out Susannah's more homely looks, or that today's sensibilities might detect stereotyping in the original, or that the hint in the text that Lucian married a woman before understanding his own true nature would compete with Susannah's story and muddy the waters ("'We were married very young . . . before we knew what was what'" [Matisse 14]). Intriguingly, however, the part of Susannah's husband is played by Hugh Sachs, best known for his role as Gavin Ramsbottom, a gay character in the TV series Benidorm. While there is no suggestion either in the text or the film that Susannah's husband is not heterosexual, there is, in both cases, a suggestion that he habitually neglects her and fails to look at her, and the film, in its choice of actor, can be interpreted as hinting at a possible reason for this.

To return to the most striking differences between text and film, Susannah's dialogue is, like Lucian's, flattened, with the line, in italics in the original, "I look like a middle-aged woman with a hair-do" (Matisse 24) neutralised into "I look like a middle-aged woman with a bad hair-do"-the addition of "bad" completely ruining the wit of the original remark. However, Susannah's smiling discomfort is portrayed very well in the film, and the blouse she often wears-both text and film cover several visits to the same salon-is printed with a design in complimentary colours, yellow and blue, which recall both Matisse's work and the design of the cover of the short story collection with its yellow on blue lettering. Susannah's "hair-do" is also of interest. The short story lays the emphasis on the changing texture of her hair and its sparser growth: "her hair began to grow old. The ends split, the weight of it broke, a kind of frizzed fur replaced the gloss" (6-7), and she suspects that she is getting "thinner on top" (15). It does not foreground its present colour, although she regrets the "chestnut" of her young hair (6, 19). However, the Susannah of the film has allowed the natural colour of her youth to grow out and grey to grow in, patchily, without allowing Lucian to make any improvements in this respect. She wants to "render it natural-looking, that was, young" (7), but the film shows the extent of her self-delusion: her hair is unequivocally unbecoming, and Lucian's murmured reassurance when she asks if the grey is now dominant is clearly "tactful" rather than sincere. At the last appointment, where Lucian deserts Susannah and Deirdre takes over, resulting in the cut and blow-dry Susannah deems "hideous" (23), the shorter length harmonises the colour of Susannah's hair. This departure from the text can be understood as a shorthand, a semaphore, a code, in the same way as the Susannah of the film asks for a magazine so that she does not have to chat with Deirdre (played by Chanel Cresswell), and holds it up in such a way that she cannot see herself in the mirror, whereas nothing of this sort happens in the text. It is a way of communicating information so that the viewer will grasp it easily-Susannah does not want to talk to Deirdre-as well as a way of explaining that she does not keep a close eye on what Deirdre is doing to her hair, since she cannot see her own reflection.

To return to Susannah's objections to the "hair-do," in the short story these are complex, there are layers of reference-to the terribly artificial hairdressing of her mother's time, for example (6). While the Medusa reference is retained, if diluted-there are some curls, but it would be stretching it to describe them as the "twining coils" of the text (23)-the change in the colour of Susannah's hair, the visual signalling of the sentiment that she has grey hair and is no longer trying to pretend that her youth has not passed, is very striking. The path to her smashing up the salon is rendered more direct. The short story's reference to Medusa is highly textual rather than visual, and draws on one of Byatt's areas of predilection, the classical myth. Arun Kumar Pokhrel interestingly applies Hélène Cixous's analysis of the Medusa myth to "Medusa's Ankles," arguing that Susannah's act of violence, when she destroys the salon, can be seen as one of empowerment. Indeed, in "The Laugh of Medusa," Hélène Cixous views Medusa as a source of power and creative energy, as well as a source of joy and pleasure. Cixous writes: "You only have to look at the Medusa straight on to see her. And she's not deadly. She's beautiful and she's laughing" (255).

According to this reading, in her act of destruction, Susannah does not humiliate or embarrass herself-rather, she triumphs, and becomes "beautiful" by rebelling against the male gaze. The film seems to arrive at this point too, but by different means. It is perfectly clear that Susannah feels belittled and hurt by Lucian's lack of attention, by his insistence on his need for beauty, by his neglect of a long-term client, and Susannah's rampage is a clear protest against this. The film cannot be a text, it cannot replace the pathway by which the reader may construct for him/herself the resurgence of Medusa in Cixous's terms (besides, the text still stands very well if the reader fails to do this). Nonetheless, and importantly, the film modifies the circuit of signification in the original text; it takes a very complex web and makes of it, or with it, or alongside it, an artefact with fewer and more clearly-connected nodes. The effect is to speed the viewer through a different process of associations, but to revert to the spurned turns of early adaptation theory, this is not necessarily "an infidelity"-rather, it can be interpreted as an echo, or extension, of the narrator's musings in Still Life about the relations between the textual and the visual.

Notwithstanding, the final hairstyle can simultaneously be viewed as a confirmation of Susannah's lament in the story, when she thinks about the importance of appearance versus content in her upcoming television interview, that "words, mere words, go for nothing" ( 20). Yet, paradoxically-to return to my question about what stays the same but is mobilised differently-it also creates a further layer of self-referentiality when the text and the film are studied together, in that the use of colour is very central to The Matisse Stories. In what seems to be a reference to Still Life in the short story "Sugar," the narrator-a narrator whose voice either entirely coincides with, or is extremely close, to Byatt herself-mentions "writing a novel in which the idea of Van Gogh stalked in and out of the text" (Sugar 236). However, in The Matisse Stories, it could be said that the idea of Matisse does not merely "stalk in and out": it is present from beginning to end. Several critics have commented on this over the years: Celia Wallhead argues that Byatt "makes a window through which we are invited to look at pictures" (97), and the framing of the Rosy Nude in the text is certainly important, but I go further, suggesting that Byatt "paints" with words: "Byatt sometimes pauses the onward flow of the narrative with frozen descriptions, often in the present tense, and with evocations of unchanging interiors, to emphasise the idea that words can imitate paint" (Mundler,Intertextualité 241,my translation). Colours are used to set the two disparate scenes, to create the two contrasting backdrops, of the two parts of the text of "Medusa's Ankles": the first backdrop is "rosy"-Lucian chooses the Rosy Nude to go with his "colour scheme" (Matisse 4; figure 2) and that colour-scheme is developed with the references to the pink and cream of the curtains, and the pink cups and biscuits (5). The second backdrop is in the much more threatening colours of maroon and grey ("battleship-grey and maroon. Dried blood and instruments of slaughter, Susannah thought" [15]). In this regard the text relies very much on the visual, and on colour in particular, and so the "shorthand" or "code" employed by the film's use of colour in Susannah's hair is not, finally, out of keeping with the complexity of the text, but rather can be read as a furtherance of it which the transposition into a different medium makes possible.

Tyranny/Liberation

My contention is that paradoxically, the adaptation of this short story into a film serves to heighten how intensely visual the story is, both in the way it is written, regarding the narrative techniques it employs, as demonstrated above, and, importantly, in the critical repertoire it mobilises. The climax of the story, when Susannah manifests her dissatisfaction with her "hair-do" by smashing up the salon, can indeed be read as a return of Cixous's Medusa, a celebration of the female, an act of defiance. Yet, it is clearly also an act born of suffering and sadness, and cannot be read as unalloyed celebration: there is pain in Susannah's lament for her lost sexuality, as she remembers her young self with her student lover (22-23), and in her difficulty in accepting her ageing hair-"I want my real hair back" (24); and there is real, threatening violence in the way she sees the various bottles and pots around the salon as "bombs or grenades" (25). Moreover, to transform this act into film is to invite another interpretation, one which applies Fredric Jameson's ideas of "an aggressive mastering gaze intent on physically possessing its objects," and of a "reduction to the body" (qtd. in . In this reading, the "mastering gaze" against which Susannah reacts is inherent to the medium itself, creating a highly self-referential set of associations which constitute a complication rather than a simplification of the circuits of meaning posited above, in the image of the web with a reduced set of nodal points. In the text, Susannah rebels at once against Lucian's gaze, and against the absence of his gaze, which can be read as part of the same thing. She also rebels against what she herself sees in the mirror. In the film, while those things still hold true, the gaze of the camera is added in as a heightened manifestation of the "mastering gaze." In Linda Constanzo Cahir's terms, the adaptation allows significations from the original "to multiply in its new environment" (14). For her, adaptation of text into film amounts to translation-and a by-product of translation is the new associations its creates and which were not present in the original.

Several such new associations may be suggested here. In refusing the mastering gaze, Susannah is also refusing the reduction to the body. Throughout the text, and film, she suffers Lucian's egotistical anecdotes about himself (they are also badly told-the text notes that he is, as the Djinn remarks of Gillian Perholt, a poor storyteller [Djinn 242, Matisse 13]). When Susannah finally has major news about herself, that she has won a translator's prize and will be on television-the only time she puts herself in the spotlight-Lucian shows little interest, commenting, in both text and film, on how she will have to be made to look for the honour of the salon without congratulating her. Meanwhile the Rosy Nude functions as a silent commentary on this, hanging in the salon in the first part of the film, and often reflected in the salon's mirrors; it is a permanent fixture on the dust jacket of the hardback and the front cover of the paperback editions. The Rosy Nude is, as Lucian puts it, "so calm, so damn sure of herself" (4). Although personified by Lucian, the Rosy Nude is only a body, not a woman, so while by virtue of its peace it highlights those qualities that Susannah lacks-she is not sure of herself at all-it also underscores Susannah's humanity and her subjection to the entropy wrought by timeanother theme much explored in Byatt's short stories, and foregrounded in the portrayal of Gillian in "The Djinn in the Nightingale's Eye" (see Mundler,Intertextualité 206). In Byatt's own critical terms, the Rosy Nude can be read as functioning as a "temporary mirror" (Portraits 5). Byatt writes of the recourse to portraits in fiction: "Novelists have played in different ways with characters who use portraits from other times and places to see themselves with a difference" (Portraits 5). Seen in this way, the Rosy Nude is a reflection not of Susannah, but of Suzie, the young woman with "little round rosy breasts" (Matisse 22) who was sure of her attractiveness (not beauty [19]) and was a highly sexual being-though she is now gone forever.

I would like to suggest, however, that "Medusa's Ankles" allows for an optimistic reading by being considered alongside "The Glass Coffin"-a juxtaposition which seems more than justified by the emphasis on glass in "Medusa's Ankles." The story begins "She had walked in one day because she had seen the Rosy Nude through the plate glass" (Matisse 3); moreover, mirrors are necessarily omnipresent, since the story takes place in a hair salon, and long before Susannah smashes one, they are consistently foregrounded. The mirror is the locus in which Susannah meets Lucian's eyes (4), where she discerns his "excitement" (11), where Susannah's own reflection must be accepted ( 16). The smashing of the salon includes references to damage to glass shelves and "uneven spasmodic falls of glass" (26), and Lucian himself claims to see the salon as resembling "a great glass cage" (28). Given all this, it becomes difficult not to make a comparison with "The Glass Coffin"-not in relation to its different contextualisations in Possession and The Djinn in the Nightingale's Eye, but to the content of the tale itself.

Using "The Glass Coffin" to read "Medusa's Ankles" allows not for threads to be tied up, but for the web of associations-see Susannah's web in the mirror ("one whole mirror became a spider-web of cracks" [Matisse 25])-to grow wider and more complex. Like the lady in "The Glass Coffin," Susannah can be read as having been imprisoned in a period not of Freudian latency (Mundler,"Intratextual Passages" 11), which would be inappropriate to her age, but in a period of inaction, of existing as a function of others, existing as object, in that she is subjected to Lucian's art and to his whims. In Byatt's "The Glass Coffin," the sarcophagus is "whole like a green ice egg," with "no visible cleft or split" (Possession 63), and this can be extended allegorically to Susannah, who has been in a place where she cannot be reached, even by herself: only smashing the glass, it seems, can bring back her own sense of herself. Still, whereas in Byatt's version of the tale the lady requires liberating by a tailor who has a key-the tale in Possession functioning, among other things, as an allegory for Christabel's desire for sexual liberation by Ash-in "Medusa's Ankles," Susannah is the agent of her own liberation. This is an important difference in that it allows for an optimistic and feminist reading of the story, in which even the ageing, unbeautiful woman retains power and the ability to dominate, to "break out," to refuse to lie down and "die" because she has lost her looks, and to know that the glass-in this case the looking glass-does not limit or define her (a knowledge which the beautiful woman must also attain, as Maud reflects in Possession [57]).

The question of "liberation" can also be posed in the context of the release of the film: the "placing" (to use Lucian's term) of the film in a hair salon, where it ran on a loop for three days, in May 2018, while the business of the salon went on as usual (Pulver), seems interesting on this score. Does it point to the exploitation of "consumerist art," in which, novelty value aside, patrons of the salon would be expected to pay as little attention to the film as most of Lucian's customers pay to the Rosy Nude? This does seem one possible way to see it, since critics have noted Lucian's consumerist attitude to Matisse: he buys the reproduction to go with his "colour-scheme" (Matisse 4), although his comments about it, "so damn sure of herself" (4) seems to suggest a deeper connection with it. Pokhrel sees the deployment of the painting in the text as pointing to a critique of consumer culture: "The Rosy Nude, the cynosure of the interior, is an embodiment not only of the young, beautiful, and sensual female body, but also of the naturally stabilized and commodified female body. The painting is an object of spectacle for public consumption" (397). He reads it as symptomatic of Guy Debord's society of the spectacle (397).

However, there is a more interesting way to look at this: taking up Byatt's comment on films as "temporary mirrors" (Portraits 5) for characters, the film adaptation of "Medusa's Ankles" can itself be read as a "temporary mirror" in which the customers in the salon see themselves. Bonnie Wright, echoing Lucian, whether consciously or not, comments on the universality of the film: "We've all sat in a hair salon and been at the mercy of someone else's opinion" (qtd. in Pulver). This suggests that each client will see themselves reflected, that is, mirrored, in the film to some extent. Bonnie Wright makes an interesting reference to degree: "What's so great about this . . . is that we spend so much time experiencing things on the internet and social media that an actual physical film screening is rare" (qtd. in Pulver). This suggests that the images we see in the film-filtered through the camera lens (figure 3) and very often through a mirror, of which the film makes liberal use-have some privileged relation to the real as well as to art. This is a nice mirroring which Byatt scholars will appreciate, her use of mise en abyme, of the mirror in the text, being very central to her body of work as a whole. It allows the film to be received as a step beyond text, as a release of a Byattian narrative technique into the wilds of the "real world." The film itself smashes its way out of the sarcophagus of "mere" adaptation, and becomes its own artefact, its own thing, with a field of reference which goes beyond what could be contained in any frame or screen. To this extent, the film adaptation is indeed, as Linda Hutcheon says of adaptations, "second without being secondary" (9). Bonnie Wright's film, whatever she intended, can be understood as the Djinn bursting out of the bottle, as Byatt's genius let loose, as she approaches her last years, on the world beyond readers of literary fiction.

Figure 1 Kerry

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Figure 2 Anna

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Figure 3

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