Anne-Marie Kermarrec

Afshin Moin

Data Visualization Via Collaborative Filtering *

Collaborative Filtering (CF) is the most successful approach to Recommender Systems (RS). In this paper, we suggest methods for global and personalized visualization of CF data. Users and items are first embedded into a high-dimensional latent feature space according to a predictor function particularly designated to conform with visualization requirements. The data is then projected into 2-dimensional space by Principal Component Analysis (PCA) and Curvilinear Component Analysis (CCA). Each projection technique targets a different application, and has its own advantages. PCA places all items on a Global Item Map (GIM) such that the correlation between their latent features is revealed optimally. CCA draws personalized Item Maps (PIMs) representing a small subset of items to a specific user. Unlike in GIM, a user is present in PIM and items are placed closer or further to her based on their predicted ratings. The intra-item semantic correlations are inherited from the high-dimensional space as much as possible. The algorithms are tested on three versions of the MovieLens dataset and the Netflix dataset to show they combine good accuracy with satisfactory visual properties. We rely on a few examples to argue our methods can reveal links which are hard to be extracted, even if explicit item features are available. * This work is supported by the ERC Starting Grant GOSSPLE number 204742.

1

. The prize was to improve precision of the Netflix recommender system called CineMatch by 10%. In 2009, a union of three teams won the prize. Their approach consisted of a mixture of different CF techniques.

]. In the same way, one user account may delegate several persons, each have different taste and needs. Though, ranked lists do not unveil the correlation between items unless the user knows them a priori. This issue limits the benefit of recommender systems as it prevents users from making a correct selection based on their current needs.

. Nevertheless, there are two barriers against their application to draw 2-dimensional maps. First of all, the existing predictor functions are not apt for visualization purposes. Namely, users link intuitively the relevance between users and items to their Euclidean distance on the map. However, the existing predictors, generally variants of the inner product of vectors, need other information (e.x. the angle between user and item coordinates vector). Secondly, MF approaches need more than 2 dimensions to reach their optimal precision. Consequently, proper projection methods must be picked up preserving as much information as possible.

Introduction

Collaborative Filtering (CF) is the prevalent approach to Recommender Systems (RS). The goal of RSs is to guide users towards their favorite items through the big mass of alternatives available on the Web. They have been for a few years at the center of researchers' attention due to the importance of Our approach is composed of two phases: the embedding phase and the projection phase. The first phase embeds users and items into a highdimensional space based on a predictor aimed at meeting visualization requirements. This predictor is a decreasing function of the Euclidean distance between the corresponding user and item. In the second phase, data is projected from the high-dimensional space into a 2-dimensional map. Two types of maps are presented, each aimed at a different goal. A global Item Map (GIM) is laid out using Principal Component Analysis (PCA). GIM shows the whole set of items in the 2-dimensional space from the most informative point of view. It is advantageous to view how items are grouped according to the overall opinion of users. Independently of the underlying application, it is a general solution to the visualization of CF data. In order to represent a a limited number of items to a specific user, we suggest personalized Item Maps (PIMs). They are interesting in applications like visual recommending. PIMs place items around the active user such that their relevance decreases with their distance from her. They prefer to preserve the local structure of the high-dimensional space. In other words, their priority is to preserve short distances rather than long ones. PIMs are generated using Curvilinear Component Analysis (CCA). The original CCA approach is altered in order to give more importance to preservation of distances w.r.t the active user. All PIMs are presented with the same visual parameters such that comparison between PIMs of different users is meaningful.

The precision of our approach is validated running experiments on the three versions of the MovieLens dataset and the Netflix dataset. The concentration in this work is on recommending movies. Despite, the same approach can be used for any other type of data like musics, restaurants, books, etc. CF does not require content information of the items. Despite, we show with a few examples that our maps are very good in unveiling semantic correlations between items.

Related Works

The literature related to this work is essentially in two subjects of CF and information visualization. CF is the most popular strategy for recommender systems thanks to advantages like good precision and no need to content information of items. Two common approaches to CF are Neighborhood-based methods [START_REF] Linden | com recommendations: item-to-item collaborative filtering[END_REF][START_REF] Sarwar | Item-based collaborative filtering recommendation algorithms[END_REF][START_REF] Herlocker | An algorithmic framework for performing collaborative filtering[END_REF][START_REF] Bell | Scalable collaborative filtering with jointly derived neighborhood interpolation weights[END_REF][START_REF] Deshpande | Item based top-n recommendation algorithms[END_REF] and Matrix Factorization (MF) [START_REF] Hofmann | Latent semantic models for collaborative filtering[END_REF][START_REF] Koren | Collaborative filtering with temporal dynamics[END_REF][START_REF] Paterek | Improving regularized singular value decomposition for collaborative filtering[END_REF][START_REF] Salakhutdinov | Restricted boltzmann machines for collaborative filtering[END_REF][START_REF] Khoshneshin | Collaborative filtering via euclidean embedding[END_REF]. Both techniques have been widely studied and a rich literature on them is available. Neighborhood-based methods rely on the k Nearest Neighbors (kNN) of an item (item-based approach [START_REF] Linden | com recommendations: item-to-item collaborative filtering[END_REF][START_REF] Bell | Scalable collaborative filtering with jointly derived neighborhood interpolation weights[END_REF][START_REF] Deshpande | Item based top-n recommendation algorithms[END_REF]), or equivalently the k nearest neighbors of a user (user-based approach [START_REF] Herlocker | An algorithmic framework for performing collaborative filtering[END_REF]), to predict missing ratings. The neighborhood is chosen based on a similarity measure like Cosine or Pearson Correlation. Missing ratings are then estimated through a weighted function of neighbors' ratings. Matrix Factorization-based RSs leverage the whole ratings at the same time. They embed users and items in the same latent feature space minimizing a global cost error function. Missing ratings are predicted as the inner product of user and item coordinates vectors.

Visualization of CF has been previously discussed in a few works. In [START_REF] Chen | Visualizing collaborative filtering in digital collections[END_REF] for example the NEAR panel is suggested to manage digital collections. A graphical interface is provided to show the latent similarities between items to the end user in a visual manner. This interface is a hierarchical browser, but provides no map. In [START_REF] Kuhn | Social audio features for advanced music retrieval interfaces[END_REF], the authors combine users' listening behavior with social taggings to derive social audio features of the songs. Probabilistic Latent Semantic Analysis (PLSA) is consequently applied for dimensionality reduction. In the context of Neighborhood-based CF, classical Multi Dimensional Scaling (MDS) [START_REF] Cox | Multidimensional Scaling, Second Edition[END_REF][START_REF] Borg | Modern Multidimensional Scaling: Theory and Applications[END_REF] is directly applicable. MDS has been widely used as a visualization technique in many domains including data mining and machine learning. Namely, a symmetric matrix of item-item dissimilarities is computed. A user-specific list of top-k relevant items is then visualized through MDS. This method has the drawback of not mapping a user together with its recommended items, as similarity is only defined between items but not between a user and an item.

Another possible solution to CF visualization is to model data as a bipartite graph with vertices representing users and items, and edges representing the interactions (e.x. [START_REF] Xiong | Visualizations of collaborative information for end-users[END_REF]). The graph is then visualized by graph drawing algorithms like Spring Embedder [START_REF] Eades | A heuristic for graph drawing[END_REF] or Spring electrical model [START_REF] Fruchterman | Graph drawing by forcedirected placement[END_REF]. Graph drawing algorithms are aimed at drawing readable layouts according to a number of predetermined aesthetic criteria. Consequently, although they lead to visually pleasing results, they do not have convincing predicting ability. In particular, they offer no prediction function to estimate missing ratings.

This work restates the MF approach in the goal of targeting visualizability of the results, while preserving prediction precision. Relating the relevance of users and items to their distance paves the way for usage of dimensionality reduction techniques. Unlike Neighborhood-based visualization through MDS, our approach can project users and items together on the recommendation map. It is also worth noting the conceptual difference between Neighborhood-based and MF approaches. A study of the literature shows the two approaches treat data in different scales [START_REF] Koren | Factorization meets the neighborhood: a multifaceted collaborative filtering model[END_REF]. Specifically, MF processes the whole data in a global scale. On the contrary, Neighborhoodbased approach concentrates on local similarities inside data. This implies that MF maps and Neighborhood-based maps visualize the correlations inside the data in different scales.

Another asset of this work is aligning the properties of the projection approach with those of the predictor function. Namely, the more items are relevant, the hyperbolically closer to the user they are placed by the predictor function. CCA then captures this property of the predictor by preferring local topology over global topology. As far as we know, there is no prior work adopting such a strategy to CF visualization. Our approach is also a solution to alleviate the expainability issues of the MF approach. It is important for a recommender system to be able to explain the reason behind recommendations. Item-based kNN approach has the advantage of good explainability as it recommends based on the items already known to users. On the contrary, explaining the results of MF approach is more challenging, although this latter slightly outperforms its Neighborhood-based counterpart. Visual representation of the MF data provides users with the possibility of choosing an item close to their past favorite items.

Matrix Factorization for Visual Prediction

Describing data with predefined features needs content information which in turn requires complicated gathering and cleaning tasks. One way to overcome this problem is to use dimensionality reduction techniques to describe the data by latent features. In machine learning, it is known as Latent Semantic Analysis (LSA). Singular Value Decomposition (SVD) is the most common MF approach to CF. The rating matrix is represented by R, where rows represent users and columns represent items. Each entry r ij is the rating of user i for item j. SVD factors R in the form of U ΣV T , where U m×m and V n×n are orthonormal matrices, and Σ m×n is a diagonal matrix whose entries are the singular values of R. A low rank approximation matrix R is made by picking the largest singular values and corresponding left and right singular vectors. It is proved that SVD provides the best low rank estimation of R according to the Frobenious norm R -R F . In CF, it is convenient to reformulate the factorization in the form of R = P Q T , where P = U Σ -1/2 and Q T = Σ -1/2 V T . In this case the low rank estimation of P and Q denoted by P and Q contain user and item latent features.

Original SVD-like Approach

SVD is computable if all matrix entries are known. In CF however, the majority of ratings are missing. One way to deal with this problem is to fill in the missing entries by imputation. Nevertheless, inexact imputation can decrease the precision of recommendations. On the other hand, it significantly increases the complexity of the computations. A more successful alternative, known as the SVD-like approach, is to estimate the latent features through minimizing a regularized cost function.

min p * ,q * ,r * ∈R (r ui -p u • q i ) 2 + λ( p u 2 + q i 2 ),
where

p u = (x u (1), • • • , x u (k)) and q i = (y i (1), • • • , y i (k))
are the latent features of user u and item i in an embedding space of k dimensions. The first term strives to minimize the prediction error over the observed ratings. Since the goal is to predict future ratings, the second terms is added to avoid overfitting the observed ratings. λ is called the regularization factor. This function is minimized using the stochastic gradient descent method. Users and items are moved iteratively by a step length γ in the opposite direction of the gradient until the movement of users and items is negligible. At each iteration, coordinates are updated as:

p u ← p u + γ(e ui p u -λq i ) q i ← q i + γ(e ui q i -λp u )
,

where e ui = rui -r ui is the prediction error of item i for user u. A local minimum is usually found after few iterations (less than 100). λ and γ are optimized by cross validation on a smaller subset of the ratings.

Basic Visual Predictor

The original SVD-like approach is not suitable for visualization purposes.

The reason is that the inner product of vectors is not natural to be understood by the end user. To bypass this problem, we express our predictor as a function of the Euclidean distance. Let rui = f ( p u -q i ) be the predicted rating of user u for item i. The cost function has then the following form:

min p * ,q * ,r * (r ui -rui ) 2 + λ( p u 2 + q i 2 ).
Taking the partial derivatives, the coordinate update equations are obtained:

x u (j) ← x u (j) -γ(e ui xu(j)-y i (j) pu-q i ∂ rui ∂x j + λ xu(j) pu ) y i (j) ← y i (j) -γ(e ui y i (j)-xu(j) pu-q i ∂ r ui ∂y j + λ y i (j) q i )
.

We set an inverse relation between the distance of an item from a user and the corresponding rating:

p u -q i = α( 1 r ui -β).
The hyperbolic shape of the distance function is particularly useful in CF as the more irrelevant items become w.r.t a user, the faster they get further from her. This property helps us concentrate on visualizing local topology in PIMs while relaxing long distances. Taking the inverse function of the above equation, the objective estimation function is derived:

rui = 1 1 α p u -q i + β . (1)
In this case, partial derivatives are:

∂ rui ∂x i = -1 α p u -q i ( 1 α p u -q i + β) 2 (x i -y i ), ∂ rui ∂y i = 1 α p u -q i ( 1 α p u -q i + β) 2 (x i -y i ). If we define = 1 α p u -q i ( 1 α p u -q i + β) 2 = r2 ui α p u -q i ,
coordinates are updated at each iteration as:

p u ← p u -γ(e ui (q i -p u ) + λp u ) q i ← q i -γ(e ui (p u -q i ) + λq i ) , = r2 ui α p u -q i .
α and β are found by cross validation. In theory, other functions varying inversely with the distance may also be used as predictor. We tried hyperbola square and tangent inverse functions. The precision was however worse than the presented predictor. This approach is very similar in the spirit to MF approaches as it embeds users and items in a feature space. Despite, since it is not formulated in form a factorization like SVD, we refer to it as an MF-like approach.

We compare different predictors on the MovieLens and Netflix datasets. To our best knowledge, these are the biggest datasets publically available. MovieLens dataset is provided by GroupLens team in three versions [START_REF]MovieLens Datasets[END_REF]. Each user has rated at least 20 movies. The algorithm is trained on 95% of the data. Predictions are made on the remaining 5%. To form the training and test sets, we have split each user profile into 20 uniform regular slices. 19 slices are chosen as the training profile, while the remaining one serves as the test profile. This way, we assure there is at least one rating in each user's test profile. The Netflix data set was made public for the Netflix Prize competition [START_REF] Bennett | The netflix prize[END_REF]. The dataset includes 100, 480, 507 ratings. 1, 480, 300 of the same ratings were provided as the probe set for test purposes. We trained our algorithm on all ratings, after having discarded those included in the probe set. The predictions are made on the probe set. Both MovieLens and Netlix are based on a 5-star rating scheme, that is, users rate movies between 1 and 5 stars. While the real ratings are integers, the predictions need not be. In our algorithm, predicted ratings are real numbers. Table 1 summarizes some properties of the four datasets. The quality of the results is compared by the Root Mean Squared Error (RMSE) defined as:

RMSE = r ui ∈R T (r ui -r ui ) 2 |R T | ,
where R T is the set of ratings in the test profiles of users. Lower values of RMSE indicate higher prediction precision. For both MoviesLens and Netflix datasets α = 2.5 and β = 0.2 led to good results. For all datasets λ = 0.01. In the Netflix dataset, γ = 0.005 for the SVD-like approach and 0.00125 for the visual predictor. In the MovieLens datasets γ = 0.002 for the SVD-like approach, and 0.0005 for the visual predictor. Precision of the basic visual predictor is shown in Figure 2. It is seen that the precision improves with the number of dimensions up to some threshold. After that, the error increments because the data is not enough to train extra dimensions. In other words, the larger the dataset, the more dimensions can be trained. Figure 1 shows the distribution of MovieLens users and movies in the 2-dimensional space.

It is obvious how much the distribution differs from one approach to the other.

Enhancing the Basic Visual Predictor

Users have in general different rating habits to express their opinions about items. For example, they may rate movies with different personal scales, that is, one rates between 1 and 3, while the other rates between 3 and 5. In this case, the same score 3 shows like for the former but dislike for the latter. Consequently, the two users should not be considered similar even if they put the same notes for a number of common items. While a recommender system must be able to handle personal rating habits, the basic visual predictor cannot model it adequately. The reason is that it uses the same model parameters for all users. Figure 1b shows the distribution of users and items in a 2-dimensional embedding space using the basic visual predictor. It is seen that items are spread around a straight line with the end of the line slightly curved to the left. It seems that items can not be moved freely according to the needs of users. As a result, they are put along a line; Users then take some position around the area where the collection of their favorite items is denser. This issue decreases considerably the precision of recommendations. In order to inject more flexibility into the model, we set α u as a user-specific variable of the model learned automatically from the data. The prediction function is then: rui = 1

1 αu p u -q i + β .
The regularized cost function is:

min p * ,q * ,r * ,α * (r ui -rui ) 2 + λ( p u 2 + q i 2 + α 2 u ).
The coordinates are updated as:

       p u ← p u -γ(e ui (q i -p u ) + λp u ) q i ← q i -γ(e ui (p u -q i ) + λq i ) , = r2 ui αu pu-q i . α u ← α u + γ α (e ui /α u -λα u )
The values of λ, γ and β are the same as before. We observed that α must be moved with a bigger step length to improve the results. A proper value for all datasets is γ α = 9γ. The distribution of users and items is seen in Figure 1c. The pattern of items is more twisted this time (see Figure 1b) compared to the basic visual predictor, indicating they have been placed with more freedom. It is seen in Figure 2 that for all datasets precision is greatly improved compared to the basic visual predictor. For the small dataset of MovieLens100K, the visual predictor with user specific α slightly outperforms the SVD-like approach.

Improved Visual Predictor

The precision of the visual predictor with user-specific α is still worse than the SVD-like approach for larger datasets. To improve further the results, we also set β as a user-specific variable. This helps modeling user behaviors in finer levels, resulting in better positioning of users and items in the embedding space. The prediction function is: rui = 1

1 αu p u -q i + β u .
The regularized cost function is:

min p * ,q * ,r * ,α * ,β * (r ui -rui ) 2 + λ( p u 2 + q i 2 + α 2 u + β 2 u ).
The coordinates are updated using the following equations:

           p u ← p u -γ(e ui (q i -p u ) + λp u ) q i ← q i -γ(e ui (p u -q i ) + λq i ) , = r2 ui αu pu-q i . α u ← α u + γ α (e ui /α u -λα u ) β u ← max{β u -γ β (e ui r2 ui + λβ u ), β min u }
The values of λ, γ, γ α are the same as before, and γ β = γ/29. We observed during the experiments that for a few users β u becomes negative. In order to prevent it, we set a minimum value β min u > 0 for β u . This is a necessary condition for unification of the scale of PIMs in Section 4.2. We set β min u = 0.05. The distribution of uses and items is shown in Figure 1d. The item pattern takes the shape of a horseshoe. It is wider and more symmetric than in Figure 1c. It is seen in Figure 2 that precision of the improved visual predictor is considerably better than the visual predictor with user specific α. For the small dataset of MovieLens100K, precision gets worse because the data is not enough to train the new variables. However, interestingly, it is almost as precise as the SVD-like approach for larger datasets. The difference in precision is always less than 0.01, being 0.0096, 0.0032 and 0.0057 for MovieLens1M, MovieLens and Netflix, respectively. This is a good sign as it allows for visual representaion of the results without considerable loss in precision. Note it is still possible to improve slightly the precision adding item-specific parameters as in [START_REF] Koren | Collaborative filtering with temporal dynamics[END_REF] or [START_REF] Khoshneshin | Collaborative filtering via euclidean embedding[END_REF]. However, we avoid doing so because the predictions would not only depend on the active user and her distance from the corresponding item, but also on the properties of the item. Consequently, it is no more certain that items closer to a user have necessarily higher predicted ratings than those further from her.

Projection to Lower Dimensions

We embedded users and items into a high-dimensional space because the prediction error is less in higher dimensions. where the relevance between two items or that between a user and a movie can be conjectured visually through their Euclidean distance. Hence, we need proper projection methods to come back to 2 dimensions. We suggest two types of maps: Global Item Map (GIM) and Personalized Item Map (PIM). GIM projects the whole set of items indicating the semantic correlation between them. PIM is aimed at representing a small set of items to a particular user such that their semantic correlation and predicted ratings conform to the Euclidean distance between them. In addition, predicted ratings depend inversely on the distance of the corresponding item from the active user.

Global Item Map

A Global Item Map (GIM) is very helpful to have an estimation of how items are similar or dissimilar according to consumers' point of view. It provides a way for clustering items without having any knowledge about their content.

In the high-dimensional embedding space, each dimension is related to some latent feature of the data. The distance of users or items on each dimension is a measure of how similar they are in that feature. One way to visualize the data is to show only two features of the data [START_REF] Koren | Matrix factorization techniques for recommender systems[END_REF]. However, the information of other features is missed. In addition, features are correlated and may contain redundant information. We use Principal Component Analysis (PCA) to generate a GIM. Data projection using PCA has backgrounds in Multi Dimensional Scaling (MDS) and graph drawing [START_REF] Harel | Graph drawing by high-dimensional embedding[END_REF]. PCA finds the linear combination of all dimensions capturing as much variance of the data as possible. It omits linear dependencies between dimensions so that axes are uncorrelated in the output space. In other words, PCA represents the data from the most informative point of view. In CF, the two axes resulted from PCA are indeed two uncorrelated super features, giving the best explanation about the semantic correlation between items than any other linear combination of the original features.

Algorithm 1 Power iteration algorithm ← 0.0000001 % is some tolerance.

for i = 1 to k do ûi ← random ûi ← ûi ûi u i ← 1 while u T i ûi < 1 -do u i ← ûi %orthogonalize against previous eigenvectors. for j = 1 to i-1 do u i ← u i -(u T i u j )u j end for ûi ← Su i ûi ← û û
end while % ûi are the estimated eigenvectors. end for Item latent features are kept in Q. PCA uses the first eigenvectors of the Covariance matrix (those corresponding to the highest eigenvectors) as the principal components. To compute the principal components, Q is first centered:

Q =    q1 (1) • • • q1 (k) . . . . . . . . . qn (1) • • • qn (k)    ,
where qi (j) = qi (j) -m j , and m j is the mean of column j. Each column qi , representing one data point in the input space, is projected along the principal components:

y i = qT i u.
Writing it in the matricial form, the projection is defined as Y = QT u. Since Q is centered, Y is also centered. We need to maximize the Covariance of the projection:

max (Y T Y ) = max (u T Q QT u) = max (u T Su),
where S = Q QT is the Covariance matrix. This is a standard linear algebra maximization problem whose answer is the eigenvector of S corresponding to the highest eigenvalue. In the same way, second eigenvector may be Figure 3 shows the GIM of Netflix movies projected from a 20-dimensional space. It is seen that the distribution of movies is more uniform than in Fig- ure 1d. This is an advantage of visualization as the drawing surface is used more optimally. Every subset of movies can consequently be picked up for the means of comparison. For example Figure 4 shows 47 movies with more than 140, 000 ratings in the Netflix dataset. Some names are omitted for clarity. With a little knowledge about the movies we see that the two very similar artsy movies Lost in Translation and Royal Tenenbaums are close to each other while they are far from Pearl Harbor, being a patriotic movie with elements of romance and drama. Also the two episodes of Lord of The Rings are close to each other. The day after tomorrow and Twister are both about natural disasters and are on top right of the map. Some continous consistency of movie genre is seen on the map. Specifically, movies on the top left of the map are mostly artsy movies, while those on the top right are actions and advantures. These two groups smoothly join each other in the center just before meeting drama, romance and comedy movies on the bottom.

Personalized Item Map

Our second objective is to build Personalized Item Maps (PIMs). One possible solution is to take a desired set of items with a user and apply classical MDS (ex. [START_REF] Khoshneshin | Collaborative filtering via euclidean embedding[END_REF]). If the data to be projected contains distances, which is the case in our application, classical MDS is equivalent to PCA. In CF, when all presented points are of the same type (for example movies), PCA positions pretty well similar points close to each other. This is intuitionally understandable as each new feature recapitulates a combination of the semantics of the input features. Despite, when a set of items is to be represented to a user on a PIM, PCA disturbs so much the original distances that they no more conform to the predicted ratings. Indeed, PCA is a linear mapping preserving best the input distances, but giving the same weight to all distances. In order to increase the quality of PIMs to an acceptable level, we use nonlinear mapping techniques to prioritize the preservation of shorter distances over longer ones. This idea is also in line with the hyperbolic relation we set in Section 3 between the predicted ratings and the Euclidean distance (between the corresponding user and item), as such relation emphasizes the closeness of relevant items.

Sammon mapping was the first variant of nonlinear maps addressing topology preservation [START_REF] Sammon | A nonlinear mapping for data structure analysis[END_REF]. However, it cannot reproduce all distances, because the output space has in general lower dimensions. Curvilinear Component Analysis (CCA) [START_REF] Demartines | Curvilinear component analysis: a selforganizing neural network for nonlinear mapping of data sets[END_REF] was suggested as an alternative to Sammon mapping. The idea is to give priority to the preservation of local topology. This is done by starting from projecting long distances, and overriding them by shorter ones whenever compromises must be made.

We have slightly altered the original CCA approach to obtain better results in CF. The details are as follows. Suppose we want to show a subset of items together with an active user on a PIM such that first, the semantic correlation between items, represented by their distance, is inherited from the high dimensional embedding space. In the same way, items take distance from the active user based on their irrelevance. Second, preservation of shorter distances is preferred over longer ones. Finally, preserving distances w.r.t the active user is more important as this latter is the central identity to whom the map is presented. We require the distances in the output space to conform with the original spaces as much as possible. The cost function to be minimized is then:

E = 1 2 i j =i (X ij -Y ij ) 2 F (Y ij , λ y ),
where

F (Y ij , λ y ) = 1 if Y ij ≤ λ y 0 if Y ij > λ y .
X ij and Y ij are the distances in the original and the output space, respectively. i and j may refer either to an item or to the active user. We use the principal components as the initial distribution of the points. F (Y ij , λ y ) is the step function discarding all distances larger than λ y in the output space. λy is decreased per cycle in order to prioritize shorter distances in the output space over longer ones.

The cost function can be minimized using the usual stochastic gradient descent approach. However, the complexity of each iteration cycle would be as O(n 2 ). To decrease the complexity, a different procedure was proposed in [START_REF] Demartines | Curvilinear component analysis: a selforganizing neural network for nonlinear mapping of data sets[END_REF]. Instead of moving every point according to the influence of all other points, one random point is fixed. Other points are moved with respect to the fixed point, but without consideration of the mutual influence between them. This approach reduces the complexity of each cycle to O(n). The cost may occasionally increase in each cycle, but it decreases in average. If F (Y ij , λ y ) is the step function, the movement of each point j w.r.t the fixed point i is:

∆y j (i) = α(t)F (Y ij , λ y )(X ij -Y ij ) y j -y i Y ij ∀i = j.
λ y is decreased per cycle, starting with λ 0 and ending with λ tmax : For each user we set λ tmax = du , where du = α u ( 1 ru -β u ) is the target distance corresponding to the average rating of user u. An item is relevant if its distance from the active user is less than du . Setting λ tmax = du ensures that items within some relevant distance from the user on the output space are not discarded until the end of the execution. In our experiments, we set λ 0 = 2.0. The algorithm is run a number of cycles on the active user and the selected items, denoted by Q t . In order to give more weight to the preservation of the distances w.r.t the active user, whenever the fixed point is an item, the algorithm is run one extra time on the active user with probability 0.2. This strategy can be validated from a game theoretical point of view. Namely, each point of the data moves the others in the goal of aligning their position in the output space with their original distances from it. Since the algorithm is run in average 0.2 |Q t | times more on the active user, compromises are made to her benefit.

λ y (t) = λ 0 ( λ tmax λ 0 ) t/tmax .
Projection error of CCA is seen in Figure 6 for an anonymous MovieLens user. Each point shows the distance between two points in the input space dx versus their distance in the output space dy. Closeness to the 45 • line indicates better consistency between the original and the output distances. Notice that inconsistencies are less for shorter distances.

Application of user specific parameters to the visual predictor leads to different interpretation of distance in PIMs. It would be nice that all PIMs are represented in the same scale for users can compare their PIMs with each other. To unify the scale of PIMs, we first set the origin to the position of the active user (p u = 0), and reassign all the item coordinates in the translated coordinate system. We look for a transformation such that the predictions remain in the form of Equation (3.2) with the same α and β for all users:

rscaled ui = 1 1 α q scaled i + β .
Setting runscaled ui = rscaled ui , we obtain:

q scaled i = α α u q unscaled i + α(β u -β).
Since the Euclidean distance is non-negative, q scaled i ≥ 0. This condition is satisfied for all users if β ≤ β min u . The above equation is then held if:

q scaled i = α α u q unscaled i + q unscaled i q unscaled i α(β u -β).
In our experiments we set α = 2.5 and β = β min u = 0.05. Figure 5 is the PIM of the same user. Shown in the figure, are the movies in her test profile together with a number of movies previously rated by the user. Scores of the known movies are the real ratings of the user. Scores of the test items are the predicted scores, computed in the high dimensional feature space. Note, the predicted rating is a decreasing function of the distance in the embedding space. Hence, the consistency between the predicted ratings and the distance of the user from the corresponding item on PIM is a measure of how much the original distances have been preserved. It is seen that, in general, more relevant items are closer to the user than less relevant ones. The semantic correlation between items is preserved in a very good level. The movies Friday and Clerks on the right of the user are both comedies. In the same way, Sixteen Candles, St. Elmo's Fire and ThreeSome are movies targeting teenagers with a background of romance and comedy. An American in Paris and Willy Wonka & the Chocolate Factory are both musicals. This map can help the user choose a movie similar to one of her previous favorites based on her mood in the time of selection. PIMs are also useful in offering serendipitous recommendations to a user. Serendipity addresses the diversity of recommended items. Serendipitous recommendations entice users to go through new experiences in their consuming behavior. If the user views a cluster of recommended items on her PIM, none of which is known to her, she is encouraged to go further and try them.

Discussion

We studied visual representation of CF data. Our approach establishes a link between semantic correlations within data points and their distance in the Euclidean space. Usually, visualization techniques are likely to sacrifice precision for readability. This work presents a unified approach capitalizing on high precision of the MF approach with user-friendly representation of the results. Relating the correlation of data points to their distance in a highdimensional Euclidean space provides a framework for using the existing projection techniques. We defined our predictor with a hyperbolic function of distance to insist on similarities. Experiments on the three version of MovieLens dataset and the Netflix dataset shows that the visual predictor has almost the same precision as the inner product predictor. Interestingly enough, for the small MovieLens100K ratings dataset, the visual predictor even outperforms the SVD-like algorithm.

A Global Item Map (GIM) was generated using Principal Component Analysis (PCA). It can project the whole set of items at the same time. GIM has a satisfactory performance in regrouping similar items. An altered version of the Curvilinear Component Analysis (CCA) was used to generate Personalized Item Maps (PIMs). PIMs show a small set of items together with the active user. The main difference between PIM and GIM is that local topology is prioritized over global topology in PIMs. We also gave more weight to the preservation of distances w.r.t to the active user. All PIMs were reexpressed with the same prediction parameters for intra-user comparison purposes.

The running time of the MF approach is about half an hour on the Netflix dataset. Fortunately, the algorithm is run offline. It is recomputed with some frequency (for example daily or weekly). On the contrary, running times of projection algorithms is very short. Consequently, they both can be run on demand. However, there is no advantage in generating a GIM more than once after each execution of the MF algorithm as it is a global map.

The magic of this approach is its capability in communicating latent information which are hard to understand even if one is disposed of the content information. Indeed, presented maps are the result of the collective wisdom of a huge number of users, sometimes being more promising than sniffing into tones of expert generated information. The application of the proposed techniques is not limited to recommendations. For example GIM can be used to monitor public opinion about products. Likewise, PIMs can be used to represent a visual comparison between the items of a digital catalogue. It is also worth mentioning that although we discussed only item maps, the very same approach can be applied to users. For example a general user map can be generated to detect social groups of consumers showing the same habits in their consumption behavior.

  (a) Original SVD-like (b) basic visual predictor (c) visual Predictor with αu (d) improved visual predictor

Figure 1 :

 1 Figure 1: User and item distribution of different predictors in 2-dimensional projection space.

Figure 2 :

 2 Figure 2: RMSE of various approaches on different datasets.

Figure 3 :

 3 Figure 3: Netflix movies projected from 20d space by PCA

Figure 4 :

 4 Figure 4: GIM of Netflix movies projected from 20-dimensional space.

Figure 5 :

 5 Figure 5: PIM of an anonymous MovieLens user projected from 10dimensional space.

Figure 6 :

 6 Figure 6: d x -d y of the PIM of an anonymous MovieLens user.

Table 1 :

 1 Properties of the datasets is only half of the problem. Remember the goal was to provide item maps

	Nevertheless, the embedding


B A Van Tiggelen

S Kawka

G L J A Rikken

QED Corrections to the Electromagnetic Abraham Force Casimir Momentum of the Hydrogen atom?

We develop a QED approach to find the contribution of the quantum vacuum to the electromagnetic Abraham force. Semi-classical theories predict diverging contributions from the quantum vacuum.

We show that the divergencies disappear by Kramers-Bethe mass-renormalization. The finite remainder is compared to the relativistic corrections to the Abraham force. This work generalizes an earlier paper [1], dedicated to the harmonic oscillator, to the hydrogen atom and corrects two subtle errors.

Introduction

The Abraham force refers to the small force exerted by time-dependent electromagnetic fields on neutral matter. Despite our complete knowledge on classical electromagnetism, a longstanding controversy exists about the precise expression of this force, given macroscopic, phenomenological constants such as dielectric constant and magnetic permeability. This so-called "Abraham-Minkowski" controversy stems mainly from the fact that the macroscopic Maxwell's equations seem to favor a radiative momentum density D × B [START_REF] Jackson | Classical Electrodynamics[END_REF] (the Minkowski version), whereas the Send offprint requests to: microscopic version leads to ε 0 E × B (the Nelson version [START_REF] Nelson | [END_REF]). In addition, neither one of them is in general equal to ε 0 µ 0 E × H, that is radiative momentum does not always seem to be just equal to energy flow S = c -1 0 E × H, divided by c 0 (the Abraham version, also advocated by Jackson [START_REF] Jackson | Classical Electrodynamics[END_REF]). The different versions lead to different expressions for the Abraham force [4].

Following Nelson [START_REF] Nelson | [END_REF], it is instructive to calculate directly the force density f = ∂ t (ρ m v) exerted on the matter, given by the Lorentz force f = ρ q E + J q × B with ρ q = ∇ • P q and J q = ∇ × M q -∂ t P q the bound charge density and bound charge current density. For nonmagnetic materials without free charges this leads to

∂ t (ρ m v -P q × B) = ∇ • U - 1 2 (∇ε r ) ε 0 E 2 (1)
with U a second rank stress tensor specified elsewhere [5].

Since the space integral of ∇ • U vanishes, and so does the last term in a sufficiently homogeneous field, the force on the atom is equal to the classical "Abraham force",

F A = ∂ t dr(P q × B) = ε 0 (ε r -1)V ∂ t (E × B) (2)
Only a few observational claims exist, such as the work by

Walker [6], and our recent work on gases [7].

Since a few years several authors have discussed the possibility of a contribution to the Abraham force stemming from the electromagnetic quantum vacuum. In the remainder we will refer to the momentum associated with this QED contribution to the Abraham force as "Casimir momentum". Its existence seems even more controversial.

The first approaches are based on a semi-classical analysis, i.e. classical for the matter, and quantum for the radiation. Following the early work in Ref. [8], Refs. [12] and [START_REF] Birkeland | [END_REF] demonstrate that Casimir momentum can exist in socalled bi-anisotropic media. These are media where electric fields induce magnetic moments, and by symmetry, magnetic fields induce electric moments. All media can be made bi-anisotropic, for instance by moving them or by exposing them to crossed electric and magnetic fields. However, two major problems show up for Casimir momentum in such media. First, the end result diverges significantly in the UV and needs to be regularized, and the question is whether the effect survives the regularization. Secondly, the question about Lorentz-invariance arises, and more generally, the need to include relativistic effects. In a recent Reply [5] a fully Lorentz-invariant model was discussed in which Casimir momentum exists, although still with a diverging value.

In Ref. [8] the divergent frequency integral was solved by a simple cut-off frequency in the X-ray regime, as had previously been proposed by Schwinger [START_REF] Schwinger | Proc. Natl. Acad. Sci[END_REF] to calculate the Casimir energy of an oscillating bubble. Dimensional regularization was proposed by Brevik etal [11]. When applied to Casimir momentum [12], one finds values inversely proportional to the sample size but still resulting in extremely small values for micro-sized media. In a recent work [1] we considered the quantum-mechanical harmonic oscillator exposed to quasi-static fields as well as to the quantum vacuum. We showed that Kramer's mass renormalization [13,[START_REF] Milonni | The Quantum vacuum[END_REF] eliminates the divergencies. Unfortunately, this work suffers from two errors that will be repaired in this work. In addition we shall consider the hydrogen atom with Coulomb interaction. This will allow us to conclude that relativistic effects are (here) much smaller, typically by a factor m e /M .

UV catastrophe in bi-anisotropic media

The problem that occurs in bianisotropic media is most easily illustrated by considering an infinite, dispersionless dielectric medium moving at speed v with respect to the observer. Such medium is subject to the Fizeau effect and has a bi-anisotropic coupling tensor equal to as P A = ρ C v that in Eq. ( 1) adds up to the momentum density ρ m v of "matter" , gives

g ij = ε 0 (1 -ε r )ǫ ikj v k , i
ρ C = 2 3 h π 3 c 5 0 ∞ 0 dω(ε r -1)ω 3 (3)
This integral diverges. Usually the divergence of Casimir energy poses no problem as long since the observable forces are finite. The Casimir energy between two ideal mirrors is known to have a finite inertial mass [START_REF] Jaeckel | [END_REF]. Being an observable quantity itself, the diverging inertial mass density ρ C does pose a problem. We can identify E • P q = dω(ε r -1)hω 3 /π 3 c 3 0 as the potential energy density of the quantum vacuum. Indeed, what we might expect to find for ρ C is (the inertial mass due to) the binding energy associated with Coulomb, Van-der-Waals and Casimir Polder forces [16] that can be viewed as longitudinal and transverse degrees of freedom of the vacuum electromagnetic field, and which are also known to find their way to the dielectric constant [START_REF] Lagendijk | [END_REF]. Apart from the divergence, the front factor 2/3 in Eq.( 3) is also strange, and reminiscent of the problem of electromagnetic self-mass of the electron [START_REF] Jackson | Classical Electrodynamics[END_REF].

If we assume free electron optical dispersion (ε r = 1 -n e e 2 /ε 0 m e ω 2 ) at high frequencies [START_REF] Jackson | Classical Electrodynamics[END_REF], the divergence of Casimir momentum will still persist as dωω. If we as-sume that frequencies larger than π/r e do not contribute by some unknown physical principle (valid at length scales smaller than the free electron radius) the mass density

(3) would typically be equal to n e m e /α, i.e. a factor 1/α larger than the actual mass density of the electrons. It seems likely that this mass is already counted in the physical values attributed to the masses of particles, as was already suggested in literature [START_REF] Milonni | The Quantum vacuum[END_REF]. In this paper we shall validate this argument explicitly for the hydrogen atom, both for the Fizeau effect and for the magneto-electric effect. For the first this is quite naturally suggested by Eq.

(3). For magneto-electric materials the quantum vacuum gives a similar UV divergence [8] for which mass renormalization is much less obvious.

In the following we consider the quantum mechanics of a hydrogen atom exposed to crossed, quasi-static electromagnetic fields. We calculate the total (pseudo-) momentum of the atoms and identify diverging terms when coupled to the quantum vacuum. We regularize the infinities, and finally find the finite contribution of the quantum vacuum to the momentum, into lowest order of the fine structure constant α.

A moving hydrogen atom coupled to EM quantum vacuum and external fields

We consider a hydrogen atom moving at non-relativistic speed, in crossed electric and magnetic fields E 0 and B 0 respectively. Particle 1 is the proton, particle 2 the electron. We shall use the reduced coordinates R = (m 1 r 1 + m 2 r 2 )/M and r = r 1r 2 for the center of mass position and the interparticle distance, with conjugate momenta

P = p 1 + p 2 and p = µ(p 1 /m 1 -p 2 /m 2 )
. The presence of an external stationary magnetic field B 0 will be treated in the Coulomb gauge and it is instructive to make the

unitary transformation U = exp[ie(B 0 × R) • r/h]
H 0 = 1 2µ p + ∆m M e 2 B 0 × r 2 + 1 2M (P -eB 0 × r) 2 - e 2 4πε 0 r -eE 0 • r (4)
with ∆m = m 1 -m 2 > 0 [1]
. This transformed Hamiltonian commutes clearly and conveniently with the total conjugated momentum P, even when E 0 depends on time, meaning that (when transformed back) the pseudo mo-

mentum Q = U * PU = P + e 2 B 0 × r = P kin + eB 0 × r is
a conserved quantity when time-evolution is governed by H 0 , as in the classical theory. The relativistic correction H rel will be added later. The transverse degrees of freedom of the electromagnetic quantum vacuum are described by the Hamiltonian

H F = kǫ hω k a † kǫ a kǫ + 1 2 (5)
The interaction with the quantum vacuum, when transformed reads [1],

W = - e m 1 p + m 1 M P - e 2 B 0 × r • A R + m 2 M r - e m 2 p - m 2 M P + e 2 B 0 × r • A R - m 1 M r + e 2 2m 1 A 2 R + m 2 M r + e 2 2m 2 A 2 R - m 1 M r (6)
where the vector potential of the electromagnetic field is given by

A(x) = kǫ A k ǫ a kǫ exp(ik • x) + a † kǫ exp(-ik • x) (7)
with A k = h/2ε 0 c 0 kV in a quantization volume V . In the presence of the quantum vacuum the pseudo momentum that commutes with H is

K = P + e∆A + kǫ hka † kǫ a kǫ ( 8
)
Because K is a conserved quantity, the time variation of P kin , and thus the total exerted force, equals minus the time-derivative of all other contributions to K. If |Ψ 0 is the ground state of the joint atom and photon field, with | Ψ0 = U * |Ψ 0 the transformed ground state, the pseudomomentum follows from,

K = Ψ0 | K| Ψ0 = Ψ0 | Pkin | Ψ0 + e 2 B 0 × Ψ0 |r| Ψ0 +e Ψ0 |∆A| Ψ0 + Ψ0 | kǫ hk a † kǫ a kǫ | Ψ0 ≡ P kin + P A + δ P pola + P long + P trans (9)
The last two terms are the "longitudinal" and "transverse" contributions to the momentum stemming from the electromagnetic quantum vacuum [START_REF] Cohen-Tannoudji | Photons et Atomes[END_REF]. The classical Abraham momentum P A is found from the second term in on the righthand side when we ignore the coupling with the quantum vacuum. In this case -eE 0 • r becomes the only perturbing potential and we find,

| Ψ0 (W = 0) =   |0 + j =0 |j j|eE 0 • r|0 E j -E 0   ⊗ |{0} ≡ |0, E 0 ⊗ |{0} (10)
where |{0} denotes the quantum vacuum, empty for all plane waves, and |j represent the unperturbed eigen states of the atom. We will denote by |0, E 0 the atomic ground state with explicit reference to the the external electric field, and given by the first factor in the above expression.

Insertion directly generates P A = ε 0 α(0)B 0 × E 0 with the well-known expression for the static polarizability,

α(0) = 2 3 e 2 ε 0 j =0 | j|r|0 | 2 E j -E 0 (11)
which for the hydrogen atom equals α(0) = 18πa 3 0 [START_REF]In that case is the induced atomic polarization pq = ε0αE, with α the dimension of a volume, and the induced local polarization density Pq = ε0N αE loc with N[END_REF].

To find the contribution of the quantum vacuum to K, we need to know how the ground state |i = 0, Q 0 , {n = 0} of an atom with pseudo-momentum Q 0 in an empty vacuum is perturbed by the coupling W . Let |jQ{n} be an eigen state of H0 , P and HF , the second with eigenvalue Q. We will restrict ourself to processes involving absorption and creation of one photon, i.e. involving the products of only two A(r) photon fields. The ground state is, up to second order in the coupling, equal to (the accent ′ is to avoid summing over ground state itself),

| Ψ0 =   1 - 1 2 ′ iQn |W iQn,0Q00 | 2 (E 0Q00 -E iQn ) 2   |0Q 0 {0} + ′ iQn W iQn,0Q00 E 0Q00 -E iQn |iQn + ′ iQn ′ i ′ Q ′ n ′ W iQn,i ′ Q ′ n ′ W i ′ Q ′ n ′ ,0Q00 (E 0Q00 -E iQn )(E 0Q00 -E i ′ Q ′ n ′ ) |iQn -W 0Q00,0Q00 ′ iQn W iQn,0Q00 (E 0Q00 -E iQn ) 2 |iQn (12)
Note that the A 2 (r) terms contained in W do not contribute in this expression since i = 0|A 2 (r)|0 = 0 for any atomic state other than the ground state. Note also that the last term in the perturbed wave function does not contain one-photon processes since {0}|A(r)|{0} = 0. Several terms yield diverging contributions to the momentum K and will below be identified. The factor in the first term proportional to the unperturbed ground state is imposed by normalization. In Appendix A we estimate it to be of order α 3 , which makes it beyond the scope of this work.

δ P pola

The momentum δ P pola can be regarded as a quantum vacuum contribution to the static polarizability at B 0 = 0 that finds its way to the Abraham momentum via the classical expression. As such it is arguably not a true "Casimir" momentum. It takes the form,

δ pola = eB 0 × 0, Q 0 , E 0 {0}|r ′ iQn ′ i ′ Q ′ n ′ W iQn,i ′ Q ′ n ′ W i ′ Q ′ n ′ ,0Q00 (E 0Q00 -E iQn )(E 0Q00 -E i ′ Q ′ n ′ ) |iQn + c.c. ( 13
)
The matrix elements of W contain exponentials of the form exp(±ik • r i ) stemming from A(r). Those terms for which products occur of exponentials with opposing phase, diverge. They can be re-arranged to give,

δ P pola div = 2 Re e 2 m 2 1 kǫ A 2 k 0, E 0 |e (B 0 × r) 1 ′ H0 (p, Q 0 ) -E 0Q00 p + m 1 M Q 0 • ǫ 1 H0 (p + m2 M hk, Q 0 -hk) -E 0Q00 + hω k p + m 1 M Q 0 • ǫ|0, E 0
δ P pola div = 1 2 δm 1 m 2 1 0, E 0 |e (B 0 × r) 1 ′ H0 -E 0Q00 p + m 1 M Q 0 2 |0, E 0 + 1 2 δm 2 m 2 2 0|e (B 0 × r) 1 ′ H0 -E 0Q00 p - m 2 M Q 0 2 |0, E 0 +c.c
where we have introduced the masses,

δm i = 4 3π αh 2 ∞ 0 dkk h2 k 2 /2m i + hω k (14)
The diverging mass is recognized as the nonrelativistic QED contribution to the bare mass [START_REF] Milonni | The Quantum vacuum[END_REF], and is supposed to be absorbed by the value attributed to the observed mass. This "Kramer-Bethe" mass renormalization method seems reasonable provided that all masses acquire the same diverging contribution. The polarizability is in principle a complicated function of the reduced mass. It is easy

to check that δ(1/µ) = -δm 1 /m 2 1 -δm 2 /m 2 2 .
As a result, the p 2 term in the expression for δ P pola div contributes

δ P pola div = eB 0 × 0, E 0 |r -1 ′ H0 -E 0 δ(1/2µ)p 2 |0, E 0 + c.c = eB 0 × ( 0, E 0 | r|δ0, E 0 + c.c.) = ε 0 δα(0)B 0 × E 0 where |δ0, E 0 = -( H0 -E 0 ) ′-1 δ Ṽ |0, E 0 is the first-order
modification of the atomic ground state due to the per-turbation δ Ṽ = δ(1/2µ)p 2 in the kinetic energy. We thus conclude that this divergence disappears when all masses in α(0) absorb consistently the same diverging QED contribution [START_REF] Milonni | The Quantum vacuum[END_REF].

The terms proportional to Q 2 0 and p•Q 0 in δ P pola div can be seen to vanish. The other terms in δ P pola are finite. A straightforward analysis comparable to the one for δ P long below shows it be a factor α 2 m e /M smaller than P A . In this work we shall restrict ourselves to orders α 2 . We conclude that no quantum vacuum contribution of the order α 2 exists to the static polarizability.

P long

The quantum expectation of the momentum of the longitudinal vacuum field reads,

P long = 0, E 0 , Q 0 , {0}| (e∆A(r 1 ) -e∆A(r 2 )) × ′ iQn A 2 k W iQn,0Q00 (E 0Q00 -E iQn ) |iQn + c.c. (15)
If we acknowledge the subsequent annihilation and creation of one virtual photon, we find that

P long = 2e 2 Re kǫ A 2 k 0, E 0 | 1 -e -ik•r × 1 H0 (p -m2 M hk, Q 0 -hk) -E 0Q00 + hω k ǫ • p m 1 + Q 0 M - e 2m 1 B 0 × r |0, E 0 + 0, E 0 | e ik•r -1 × 1 H0 (p + m1 M hk, Q 0 -hk) -E 0Q00 + hω k ǫ • p m 2 - Q 0 M + e 2m 2 B 0 × r |0, E 0
It is clear that terms involving exponents exp(±ik • r) ensure finite k-integrals. We will first focuss on the diverging terms. We can extract the divergence by keeping only recoil plus photon energy in the denominator, to get

P long div = δm 1 0, E 0 | p m 1 + Q 0 M - e 2m 1 B 0 × r |0, E 0 + δm 2 0, E 0 | - p m 2 + Q 0 M - e 2m 2 B 0 × r |0, E 0
with the masses δm i defined earlier in Eq. ( 14). From the relation [r, H0 ] = ih∂ p H0 it is easy to show that 0, E 0 |p+ (∆m/M )(e/2)B 0 × r|0, E 0 = 0. This leads to

P long div = Q 0 - e 2 0, E 0 |B 0 × r|0, E 0 δm 1 + δm 2 M = 0, E 0 | Pkin |0, E 0 δM M = P kin δM M (16)
Hence, the diverging contribution disappears entirely into the inertial mass featuring in the kinetic momentum P kin = M v of the atom.

We proceed with the converging terms. When we remove the divergencies from the expression even the part with compensating phase factors will yield a finite contribution. This can be shown to be of order α 3 P A and will be ignored here. The terms with exponentials typically have k ≈ 1/a 0 since the ground state has r ≈ a 0 .

In that case hω k ≫ e 2 /4πε 0 a 0 and the atomic energy H 0 can be neglected. Similarly the recoil energy h2 k 2 /2m i is much smaller than hω k . This simplification leads to the following expression,

P long = 2e 2 k A 2 k hω k 0, E 0 |∆ k • p 1 m 2 e ik•r - 1 m 1 e -ik•r -∆ k • Q 0 1 M e ik•r + e -ik•r + e 2 ∆ k • (B 0 × r) 1 m 2 e ik•r + 1 m 1 e -ik•r |0, E 0 (17)
with ∆ k = ǫ k ǫ k ǫ k the transverse projection matrix.

The integral over k can be carried out by using that ∞ 0 dk dΩ exp(ik•r)∆ k = π 2 (1 + rr)/r and we get,

P long = e 2 8πε 0 c 2 0 0, E 0 | 1 + rr r • p 1 m 2 - 1 m 1 -2 Q 0 M + e 2µ B 0 × r |0, E 0 (18) Since 0, Q 0 , E 0 | -e 2 /4πε 0 r|0, Q 0 , E 0 = 2E 0 + O(E 2 0 ),
the middle term generates a contribution,

P long 2 = 8 3 E 0 M c 2 0 Q 0 ( 19
)
This can, in fact, be recognized as the contribution to the ground state energy of the modified Coulomb field seen by the moving charges, as expressed by the Darwin interaction [22], and thus, via the relativistic equivalence, to the inertial mass. The first term in the expression for P long is more elaborate. We can insert the expression [START_REF] Birkeland | [END_REF] for the polarized ground state |0, E 0 , and expand H0 linearly into the external magnetic field. Some algebra leads to the expression

P long 1 = - e 4 16πε 0 c 2 0 (m 2 -m 1 ) 2 m 2 1 m 2 2 ǫ nml B n E k 0|p • 1 + rr r 1 ′ ( H0 -E 0 ) 2 r m p l r k |0 + c.c (20)
This can be further simplified for the hydrogen isotropic 1S ground state for which p i |0 = (ih/a 0 )r i |0 , and using that m 1 ≫ m 2 = m e so that

P long 1 = -B 0 ×E 0 e 4 h2 4πε 0 c 2 0 m 2 e a 0 1 3 0| 1 r rm 1 ′ ( H0 -E 0 ) 2 r m |0 (21)
If we write α(0) = 18πa 3 0 , this becomes

P long 1 = -κ 1 α 2 P A ( 22
)
with P A = ε 0 α(0)B 0 × E 0 and the dimensionless number

κ 1 = 2 27 e 2 4πε 0 a 0 2 n =0ℓm 0|r -1 r|nℓm • nℓm|r|0 (E n -E 0 ) 2 (23)
Since κ 1 > 0, this contribution lowers the classical Abraham momentum. The sum over all excited states (with ℓ = 1 imposed by selection rule, the sum over m equals 1)

involves both the bound states and the continuous spectrum. The sum over discrete states can easily be done numerically using a recursion formula for the hypergeometric functions [23]. With E n = -e 2 /8πε 0 a 0 n 2 the numerical factor equals κ 2 (D) = 0.21. The part associated with continuous spectrum is much harder and we shall here assume that the continuous spectrum consists of plane waves exp(iq • r)/ √ V with energy E q = h2 q 2 /2m e , which is a usual approximation in treatises of the photo-electric effect [24] and valid in principle only for q > 1/a 0 . This leads to (y = qa 0 ),

κ 1 (C) = 8 27 q 0|r -1 r|q • q|r|0 (a 2 0 q 2 + 1) 2 = 8 27 16 π ∞ ymin dy y 3 (y 2 + 1) 3 arctan y y 2 - 1 y y 2 + 1 ( 24
)
Choosing y min = 0 or y min = 1 as lower limits gives 9.3 •

10 -3 < κ 1 (C) < 1.4 • 10 -2
This is small compared to κ 1 (D). We shall adopt κ 1 = 0.22 [25].

We finally evaluate the last term in Eq. [START_REF] Cohen-Tannoudji | Photons et Atomes[END_REF]. Upon inserting the ground state [START_REF] Birkeland | [END_REF] perturbed by the electric field. this leads to

P long 2 = e 2 4πε 0 e 2 6m e c 2 0 B 0 × E 0 ′ j 0|r|j • j|r|0 E j -E 0 ( 25
)
We can write this as

P long 2 = +κ 2 α 2 P A (26)
with the dimensionless number

κ 2 = 1 27 e 2 4πε 0 a 2 0 n =0ℓ=1m 0|r|nℓm • nℓm|r|0 E n -E 0 (27)
The same method as above yields κ 2 (D) = 0.0796 for the discrete spectrum. For the continuous spectrum we find, again assuming perfect plane waves,

κ 2 (C) = 256 27π ∞ ymin dy y 4 (y 2 + 1) 6 ≈ 0.018 (28)
assuming a lower bound at y = qa 0 = 1. Thus κ 2 ≈ 0.1.

We see that P long 1 en P long 2 have opposite sign.

We conclude that P long ≈ -0.12α 2 P A

P trans

The leading contribution to the quantum expectation value of the transverse momentum of electromagnetic field is given by,

P trans = e 2 kǫ A 2 k hk × 0, E 0 |ǫ • p m e - Q 0 M + e 2m e B 0 × r 1 H0 (p + hk, Q 0 -hk) -E 0Q00 + hω k 2 ǫ • p m e - Q 0 M + e 2m e B 0 × r |0, E 0
and a similar term for the proton (particle 1), which in fact can be neglected for m 2 = m e ≪ m 1 . In leading order, atomic energies can be neglected compared to typical photon energies hω k . We need to expand the denominator into k in order to find a non-vanishing contribution. This leads to,

P trans = e 2 h 2ε 0 c 0 d 3 k (2π) 3 2hk k(h 2 k 2 /2m e + hω k ) 3 0, E 0 | p m e - Q 0 M + e 2m e B 0 × r • ∆ k 1 m e hk • e 2 (B 0 × r) + 1 m e hk • p + 1 M hk • Q 0 • p m e - Q 0 M + e 2m e B 0 × r |0, E 0 Since ∞ 0 dkk 3 /(h 2 k 2 /2m e + hω k ) 3 = m e /h 4 c 2 0
this can be evaluated. For instance, the typical contribution linear to Q 0 will be,

P trans ≃ e 2 4πε 0 1 πhc 3 0 m e 0|p 2 |0 Q 0 M ≃ e 2 4πε 0 1 πhc 3 0 m e m e E 0 Q 0 M ≃ α E 0 M c 2 0 Q 0
Similarly we can estimate that terms proportional to B 0 × r generate a momentum of order α 3 P A . Thus, P trans is typically a factor α smaller than the longitudinal momentum P long found in Eqs. ( 19) and ( 22). It will be neglected in this work.

Relativistic corrections to the Pseudo-momentum

A particle with mass m and kinetic momentum p achieves a relativistic correction -p 4 /8c 2 0 m 3 to its kinetic energy.

When exposed to external electromagnetic fields , the particles Hamiltonian achieves a term

H rel = - 1 8c 2 0 m 3 1 p + m 1 M P -eA 1 4 - 1 8c 2 0 m 3 2 -p + m 2 M P + eA 2 4 (29)
It is easily shown that Hrel = U * H rel U commutes with the canonical total momentum P. The kinetic momentum related to the conjugate total momentum by the relation P kin = M Ṙ = M ∂ P H. Equation ( 8) for the pseudomomentum is still valid but Eq. ( 9) achieves an extra term

P rel = -M ∂ P H rel . Straightforward algebra leads to Prel = 1 2m 2 1 c 2 0 p + m 1 M P - e 2 B 0 × r 3 + 1 2m 2 2 c 2 0 -p + m 2 M P - e 2 B 0 × r 3 ( 30
)
with notation w 3 := (w • w)w. In the following we collect different contributions to 0, E 0 | Prel |0, E 0 , that are either

linear in Q 0 or in B 0 × E 0 .
The contributions linear to P add up to

P rel 1 = 0, E 0 | 1 2M c 2 0 1 µ p 2 P + 2(p • P)p |0, E 0 = 1 M c 2 0 5 3 Q 0 0, E 0 | p 2 2µ |0, E 0 = - 5 3
E 0 M c 2 0 Q 0 (31)
If we next split off Prel,1 from the equation and assume that m 1 ≫ m 2 = m e , we obtain,

Prel = Prel,1 - 1 + O(m 2 e /m 2 p ) 2m 2 e c 2 0 p + e 2 B 0 × r 3 (32)
In Appendix B we show that 0,

E 0 | p + e 2 ∆m M B 0 × r 3 |0, E 0 = 0
Using this identity and a little algebra leads to,

P rel = P rel 1 - m e M 1 2m 2 e c 2 0 × e 2 0|(E 0 • r) (B 0 • L) ′ H0 -E 0 p|0 + e 2 0, E 0 |p 2 B 0 × r|0, E 0
with L = r × p. This expression can be evaluated for the hydrogen atom, but we will here restrict to an order of magnitude. It follows straightforwardly that (• • •)/m 2 e c 2 0 ∼ α 2 P A . We thus conclude that

P rel = - 5 3
E 0 M c 2 0 Q 0 + O α 2 m e M ε 0 α(0)B 0 × E 0 (33)
Both terms are of order α 2 m e /M and together with the QED contribution [START_REF]In that case is the induced atomic polarization pq = ε0αE, with α the dimension of a volume, and the induced local polarization density Pq = ε0N αE loc with N[END_REF] the first adds up to the value E 0 /M c 2 0 that we could have anticipated from the relativistic equivalence principle that includes the bounding energy E 0 /c 2 0 into inertial mass. This finding fits into the general notion that Casimir energy has inertial mass that respects the equivalence principle [START_REF] Jaeckel | [END_REF]. The relativistic correction to the Abraham momentum is a factor m e /M smaller than the contribution from the quantum vacuum, found in Eq. ( 22).

There is also a modification of the static polarizability imposed by special relativity. For the hydrogen atom the relative correction was calculated by Bartlett and Power [21] to be -28 27 α 2 . This will give a similar contribution to the Abraham momentum as the one stemming from the quantum vacuum. We will here take the point of view that the atom's relativistic correction to the Abraham momentum entering via the static polarizability is not counted as Casimir momentum, and adopt P A with either the observed or the exactly calculated α(0) as the true Abraham momentum. This is also our experimental procedure [7].

Conclusion

In this work we have calculated the contribution of the quantum vacuum to the Abraham force. Our approach treats the kinetic momentum M v and the magneto-electric Abraham contribution on equal footing. The conserved pseudo-momentum is the sum of kinetic and Abraham momentum, K = K kin + P A , so that the force is

∂ t K kin = M R = -∂ t P A .
As expected from the equivalence principle, our approach shows that the inertial mass M is affected by the binding energy of the atom, with a significant contribution that can be viewed as stemming from the longitudinal electromagnetic field of the quantum vacuum, but which in electrodynamics is better known as the Darwin interaction associated with moving charges The QED approach in this work, very likely to be valid albeit more complex for more complex atoms and molecules, solves the UV catastrophe encountered in a semi-classical approach [8]. 25. The same assumption of plane waves in the continuous spectrum can be adopted for the static polarizability α(0) = 4πνa 3 0 , for which it is known that ν = 9/2. The discrete spectrum contributes νD = 3.66. A lower limit q = 0 would yield νC = 2.32 clearly much too large. The lower limit q = 1/a0 yields νC = 0.48, too small. A reasonable choice is somewhat lower than q = 1/a0.

A Appendix A

To leading order, the normalization factor in Eq. ( 6 The k-integral has an effective lower limit at k = (E j -E 0 )/hc 0 below which k-dependence disappears and we estimate it as,

E j -E 0 hc 0 0 dk k (E j -E 0 ) 2 + ∞ E j -E 0 hc 0 dk k (h 2 k 2 /2m e + hkc 0 ) 2 ≈ 1 (hc 0 ) 2 -log E j -E 0 hc 0 - 1 2
so that,

N ≈ α 3 1 π ∞ n=2 -log E n -E 0 hc 0 - 1 2 | 0|r|n, ℓ = 1 | 2 (34)
Using a similar method to evaluate Eq.( 23), the sum over Rydberg states at constant logarithm can be evaluated to be 0.336. The typical value for the logarithm used in calculations of the Lamb shift is -8.35 [START_REF] Milonni | The Quantum vacuum[END_REF]. We conclude that N ≈ 0.84α 3 . The continuous spectrum will have a small additional contribution.

B Appendix B

Let w ≡ p+e(∆m/M )B 0 ×r/2. We prove that 0, E 0 |w 3 |0, E 0 = 0, with notation w 3 := (w • w)w. To this end we notice that (w 2 ) 2 , r = ih∂ p (w 2 ) 2 = 4ihw 3 and that H0 = w 2 /2µ + F (Q 0 , r) in the subspace of the eigenvalue Q 0 of P. We see that

w 2 2µ 2 = ( H0 ) 2 -F (Q 0 , r) 2 - w 2 2µ F (Q 0 , r) -F (Q 0 , r) w 2 2µ = ( H0 ) 2 -H0 F (Q 0 , r) -F (Q 0 , r) H0 + F (Q 0 , r) 2
Since [F (Q 0 , r), r] = 0 and H0 |0, E 0 = E 0 |0, E 0 it follows that 0, E 0 |w 3 |0, E 0 = 1 4ih 0, E 0 | (w 2 ) 2 , r |0, E 0 = 0

  .e the magnetic field induces a polarization density equal to ε 0 (1 -ε r )v × B. It is straightforward to see that the quantum expectation value of the Abraham momentum density P A = -P q × B is given by (2v/3c 2 0 )(ε r -1) B 2 /µ 0 and is directed along the direction v. Defining the "Casimir inertial mass density" ρ C

plus a similar term for m 2 .

 2 The extra momentum and energy in denominator of the second line stem from the absorption of a virtual photon in the intermediate photon state {n}. This changes the total state and thus removes the accent in the sum over the remaining atomic states.It generates a term h2 k 2 /2m 1(2) + hω k and leads to a k-integral that diverges in the UV. Upon extracting the divergency leads to,

[

  22]. As for contribution of the quantum vacuum to the Abraham momentum, our main result is that, at least for the simple case of the hydrogen atom, divergencies can be uniquely renormalized into the masses of electron and proton. The finite small remainder is of relative order -0.12α 2 ± O(α 3 ) ∼ 6 • 10 -6 , thus reducing the classical value, and stemming from the gauge potential eA of the quantum vacuum. The transverse virtual photons with momentum hk contribute only to order α 3 . The static polarizability α(0) itself has a well-known relativistic correction of order -α 2 , whereas the contribution from the quantum vacuum to α(0) is much smaller, only of order α 2 m e /M . We have corrected an error in a previous publication[1] where a modification of order α was predicted the relative change in the Abraham momentum. In the present work, we have also considered relativistic corrections to the Abraham force and have concluded them to be factor m e /M = 5 • 10 -4 smaller than the contribution of the quantum vacuum.

1 ′|

 1 ) will affect the expectation values of P or B 0 × r. It is therefore sufficient to evaluate it for Q 0 = 0, E 0 = 0 and B 0 = 0 .If we also assume that m 1 ≫ m 2 = m e we can writeH0 (p, Q, r) + H F -E 0 2 W |00, , Q = k, r) + hω khk, Q = k, r) + hω k -E 0 2 ǫ • p|0If we neglect the p • hk/m e term in the denominator, and use p|0 = i(h/a 0 )r|0 , one obtains 0|r|j | 2 (E j -E 0 + h2 k 2 /2m e + hω k ) 2

  . L.D. Landau and E.M. Lifschitz, The Classical Theory of Fields (Pergamon, 1975), section 65. 23. A. Matsumoto, Physica Scripta 44, 154 (1991). 24. R. Loudon, The Quantum Theory of Light second edition (Oxford University Press, 1984).

	Casimir momentum exists, is
	finite, but is of relative order α 2 . How such "Casimir mo-
	mentum" reveals itself in more complex quantum systems
	is an important though difficult many-body problem. One
	could speculate it to scale like (Zα) 2 , with Z the atomic
	number, as is known to be true for relativistic corrections
	to the static polarizability of hydrogen-like atoms [21]. If
	this is true the "Casimir momentum" could become within
	reach of experimental observation.

22

This work was supported by the ANR contract PHO-TONIMPULS ANR-09-BLAN-0088-01. We would like to thank Denis Basko and Thierry Champel for useful help.