Vedran Vukotić

Guillaume Gravier

A Crossmodal Approach to Multimodal

Fusion in Video Hyperlinking

Multimodal representations are typically obtained with autoencoders by focusing on reconstruction of multimodal data. We propose an alternative possible approach that focuses on crossmodal translations between initially disjoint modalities to perform multimodal fusion in a new, common, representation space and evaluate it in multimodal retrieval and video hyperlinking tasks.

With the recent resurgence of neural networks and the proliferation of massive amounts of multimodal unlabeled data, recommendation systems and multimodal retrieval systems based on continuous representation spaces and deep learning methods are becoming of great interest. In this work, we present a method to perform high-level multimodal fusion by focusing on crossmodal translation by means of symmetrical encoders cast into a bidirectional deep neural network (BiDNN).

We analyze different continuous single-modal representations and evaluate BiDNNs in a multimodal retrieval setup. Using the notions learnt from multimodal retrieval we craft a system based on BiDNNs to perform video hyperlinking and recommend interesting video segments to a viewer. Results established within the TRECVID's 2016 video hyperlinking benchmarking initiative show that our method obtained the best score, thus defining the state of the art.

INTRODUCTION

Dealing with data originating from multiple modalities often requires to either combine them, fuse them into a joint multimodal representation or translate from one modality into another. While combining initially disjoint modalities, e.g., late score fusion by combining the scores obtained with each modality or early feature fusion by concatenating features from the different modalities, is the simplest approach, results obtained in such a manner are usually underperforming. State-of-the-art results are today typically obtained by defining a new representation space that fuses the initially disjoint modalities, incorporating a multimodal autoencoder in the pipeline. [START_REF] Ngiam | Multimodal deep learning[END_REF] In this setting, multimodal autoencoders focus on reconstructing the initially disjoint modalities through a common representation space of lower dimension. In addition, they typically increase robustness by adding noise to their inputs and by learning to reconstruct the various modalities even one is zeroed out. Instead of focusing on reconstruction of initially disjoint modalities, it is possible to focus on the task of crossmodal translation. [START_REF] Feng | Cross-modal retrieval with correspondence autoencoder[END_REF] In this case, one learns mappings from one initial modality to another and vice versa. The importance of focusing on crossmodal translation rather than modality reconstruction, appears to be the key to the improvement in multimodal fusion. By creating translations between modalities, it becomes easier to tweak the system in such a manner that a common multimodal representation space is created where all the initially disjoint modalities are projected and later fused.

When dealing with a video collection, there are two main possible tasks: retrieval and hyperlinking. The task of video retrieval aims at retrieving similar videos or video segments given a query, the latter also being a video or video segment. In this scenario, all the videos or video segments are predefined and stored in a database. Contrary to retrieval, video hyperlinking does not start with a collection of video segments stored in a database but rather with full-length videos in which segments of interest must be found. The task thus requires a segmentation step to create video segments that differ from their neighbouring segments in either one of the modalities before casting hyperlinking into a retrieval task. While video retrieval can easily be assessed by evaluating retrieval results against groundtruth relevance annotation, video hyperlinking evaluations are typically done by human evaluators who assess the importance of a proposed hyperlink and state how likely they would be to follow such a hyperlink if they were watching the source video of the hyperlink.

CONTENT REPRESENTATION

Multimodal feature fusion necessarily builds in singlemodal representations that are to be combined. We thus start by analyzing different single-modal representations for both the visual and speech modalities. After choosing the best-performing ones, we progress to methods for either combining them or fusing them into a joint multimodal representation. We do so by firstly introducing multimodal autoencoders that focus on multimodal reconstruction and that we use as a baseline, and secondly by introducing our proposed BiDNN architecture that focuses on crossmodal translation

Initial Single-modal Representations

In this work, two modalities are used: i) automatic transcripts of the speech contained in the audio track and ii) video keyframes. We do not utilize any human-generated information available in the datasets (e.g., metadata, subtitles, etc.) in order to keep our systems fully autonomous and without a human-in-the-loop element in the pipeline. Recent results show that metadata information is beneficial, however, used as a filtering step to filter out non-relevant matches. [START_REF] Awad | Trecvid 2017: Evaluating ad-hoc and instance video search, events detection, video captioning and hyperlinking[END_REF][START_REF] Demirdelen | IRISA at TRECVid 2017: Beyond crossmodal and multimodal models for video hyperlinking[END_REF] Regarding speech transcripts, we evaluate two different representations of texts in a continuous representation space: paragraph vectors and Word2Vec. Paragraph vectors provide directly a representation of textual segments. Contrary to paragraph vectors, Word2Vec is used to embed single words and thus it is necessary to aggregate the vectors of each word within the speech segment into one representation. We classically perform aggregation by taking the average of the vectors over each words. [START_REF] Campr | Comparing semantic models for evaluating automatic document summarization[END_REF] For the visual modality, we rely on keyframes representation, considering two approaches. A fist possibility consists in describing a keyframe with visual concepts which are further embedded and aggregated into a continuous representation space. The alternative is to directly embed the image into a continuous representation space. We used the ImageNet concepts to describe keyframes. When directly embedding video keyframes into a continuous representation space, we again use deep convolutional neural networks that have been shown to perform well in a multitude of computer vision tasks. [START_REF] Sharif Razavian | CNN features off-theshelf: an astounding baseline for recognition[END_REF] More precisely, we use a less deep convolutional neural network, namely AlexNet and two very deep convolutional deep neural networks, namely VGG-16 and VGG-19. [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF][START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] Aggregation, both for visual concepts and multiple keyframes, is performed by averaging.

Multimodal and Crossmodal Approaches

Multiple modalities can be used without actually fusing them. Two very common methods of using multiple modalities are concatenation, of the representations and score fusion. [START_REF] Chen | Audio-visual integration in multimodal communication[END_REF][START_REF] Guinaudeau | HITS and IRISA at MediaEval 2013: Search and hyperlinking task[END_REF] When performing score fusion, each modality is processed separately, yielding a classification or decision score for each, and the final score is computed by weighting the scores obtained with each modality.

Multimodal Autoencoders

Autoencoders are now widely used for multimodal fusion, with approaches that can be broken down into two main families: i) concatenating the different modalities and utilizing a standard autoencoder or ii) keeping the modalities separated and presenting them to a multimodal autoencoder with a modified architecture that contains separate input and output branches for each modality, as illustrated in Figure 1 in the left part. This last architecture also allows the autoencoder to provide a better crossmodal translation if necessary. [START_REF] Ngiam | Multimodal deep learning[END_REF] In both cases, the central layer, typically of low dimension, is used to obtain a multimodal representation of the input modalities. To ensure robustness, noise is often added to the inputs and one modality can be sporadically zeroed while expecting a complete reconstruction of both modalities. We, however, believe autoencoders to have a few downsides:

• Whatever the family of approaches considered (concatenated or separate input), all modalities are mixed and must be present: they all influence the central layer, even when zeroed out. • Autoencoders have to produce the same output when both modalities are presented to their input and when one modality is zeroed out. These two tasks might not necessarily point towards the same local optimum and might be detrimental for training. • Autoencoders can perform crossmodal translation by taking one modality and a zero vector at their inputs and reconstructing both modalities, including the missing one, at their output. This is however not as optimal as a direct crossmodal translation.

These potential downsides were tackled by introducing bidirectional (symmetrical) deep neural networks, [START_REF] Vukotić | Bidirectional joint representation learning with symmetrical deep neural networks for multimodal and crossmodal applications[END_REF] which we discuss next. This allows for one modality of a datapoint to be compared against both modalities of another datapoint.

Formally, let ℎ ! (!) denote a hidden layer at depth 𝑗 in network 𝑖 (𝑖 = 1,2; one for each modality), 𝑥 ! the feature vector for modality 𝑖 and 𝑜 ! the output of the network for modality 𝑖. In turn, for each network, 𝑾 ! (!) i denotes the weight matrix and 𝒃 ! (!) the bias vector of layer 𝑗. Finally, we assume that each layer admits 𝑓as an activation function. The architecture is then defined by:

𝒉 ! ! = 𝑓 𝑾 ! ! ×𝒙 ! + 𝒃 ! ! 𝒉 ! ! = 𝑓 𝑾 ! ! ×𝒙 ! + 𝒃 ! ! 𝒉 ! ! = 𝑓 𝑾 ! ×𝒉 ! ! + 𝒃 ! ! 𝒉 ! ! = 𝑓 𝑾 (!) ! ×𝒉 ! (!) + 𝒃 ! ! 𝒉 ! ! = 𝑓 𝑾 ! ×𝒉 ! ! + 𝒃 ! ! 𝒉 ! ! = 𝑓 𝑾 (!) ! ×𝒉 ! (!) + 𝒃 ! ! 𝒐 ! = 𝑓 𝑾 ! (!) ×𝒉 ! ! + 𝒃 ! ! 𝒐 ! = 𝑓 𝑾 ! (!) ×𝒉 ! (!) + 𝒃 ! !
In the above equations, the weight matrices 𝑾 (!) and 𝑾 (!) are used twice due to weight tying, for computing 𝒉 ! (!) , 𝒉 ! (!) and 𝒉 ! (!) , 𝒉 ! (!) respectively. Training is performed to minimize the mean squared error of 𝒐 ! , 𝒙 ! and 𝒐 ! , 𝒙 ! thus effectively minimizing the reconstruction error in both directions and creating a joint representation in the middle, where both representations can be projected.

Crossmodal translation is performed by presenting a single modality to its respective input xi and generating the output 𝒐 ! of the appropriate network, that represents the projection of the given modality into the representation space of the other modality. Multimodal fusion is performed by presenting one or both modalities, (𝒙 ! and/or 𝒙 ! ), to their respective inputs and by taking the activation outputs of the central layers 𝒉 ! (!) and/or 𝒉 ! (!) . If the two modalities are available, they are both presented to their respective inputs and the activations of the two central layers are taken and concatenated to form a fused multimodal representation. In the case where only one modality is available, it is presented to its respective input and the activation of the corresponding central layer is taken and replicated to replace the missing central layer activations. This is made possible by the symmetrical nature of the BiDNN architecture and allows to have representations of the same dimension when all modalities are present or when one is missing.

Given the multimodal embedding defined by the BiDNN architecture, the similarity of two video segments is obtained using a cosine similarity on the embedded representation of each segment as defined by the respective activations at the central layer, as illustrated in Figure 2.

EXPERIMENTS

In this section, we start by describing the datasets used for evaluation. We then proceed to evaluate different single-modal representations, as described theoretically in the previous section.

After choosing the best performing single-modal evaluations, we evaluate methods for performing fusion, first preliminarily in multimodal retrieval and then we use the learned notions and evaluate the best methods in video hyperlinking.

Datasets Video Retrieval

The methods discussed in this work were first evaluated in a multimodal video retrieval setup, based on the groundtruth that was formed after the MediaEval 2014 video hyperlinking task by collecting all the video segments proposed by the participants and their judged relevance. [START_REF] Eskevich | The search and hyperlinking task at MediaEval 2014[END_REF] The dataset exploits broadcast videos provided by the BBC and consists of 30 anchors (acting as queries) and 10,809 targets. There are about 34.3 keyframes on each video segment on average. The groundtruth consists of 12,340 anchor-target pairs with the corresponding relevance judgment, i.e., either related or not. Among these video segments, not all of them contain the two modalities.

Although multiple modalities available, each with possible variations, we used two main ones: for speech, we use automatic transcripts embedded in different ways, while for video keyframes, we either use embedded visual concepts (ImageNet classes) provided by KU Leuven, [START_REF] Chatfield | Visor: Towards on-the-fly largescale object category retrieval[END_REF] or CNN features we computer ourselves.

Video Hyperlinking

Video hyperlinking or, more specifically, the generation of hyperlinks within a collection of video segments is evaluated within the "Search and Hyperlinking" benchmarking initiative, first at MediaEval, [START_REF] Eskevich | The search and hyperlinking task at MediaEval 2014[END_REF] and more recently at TRECVid. [START_REF] Over | Trecvid 2015 an overview of the goals, tasks, data, evaluation mechanisms, and metrics[END_REF] Our proposed methods were evaluated in a video hyperlinking setup by participating to TRECVID's 2016 video hyperlinking benchmarking initiative. Contrary to the previous dataset that contained videos provided by BBC, this dataset is formed with videos from the BlipTV video sharing platform and contains different usersubmitted videos with different topics, styles, and languages. The dataset contained 14,838 videos an average duration of 13 minutes. We used automatic speech transcripts provided by LIMSI, [START_REF] Lamel | Speech processing for audio indexing[END_REF] and video keyframes provided with the dataset.

In the case of video hyperlinking, we perform segmentation and we recommend relevant video segments out of the multitude of obtained segments, where the big majority of them is irrelevant. Video segmentation must result in a set of 10 to 120 seconds long potential targets, given the limitations imposed by the benchmark organizers. We chose to segment videos by taking only 30 seconds of continuous speech and cut at the following speech pause, as detected by the speech transcription system. Additionally, we run this segmentation another time, using an offset of one speech segment at the second pass, in order to obtain an overlapping segmentation. This results in 307,403 video segments with an average duration of 45 seconds.

Table 1: Results of the evaluated methods over five runs with their respective precision at 10 (%) and standard deviation.

Modalities

Initial Representations

Speech is represented either with paragraph vectors or averaged Word2Vec 5 -in both cases with a representation of size 100. For Word2Vec we used a skip-gram model with hierarchical sampling, as this work best. Visual concepts were treated in the same manner, except for being sorted before being processed with Word2Vec. All the speech models were trained in an unsupervised manner on the dataset.

For representing video keyframes, we used the output of the last fully connected layer of a convolutional neural network, thus yielding a representation of size 4096. We analyze 3 different CNN architectures, namely AlexNet, VGG-16 and VGG-19, in an implementation called KerasConvnets -a CNN framework based on Keras, offering models already pretrained on ImageNet. Within video segments, representations were obtained for each keyframe and were then averaged. [START_REF] Vukotić | Multimodal and Crossmodal Representation Learning from Textual and Visual Features with Bidirectional Deep Neural Networks for Video Hyperlinking[END_REF] Averaged VGG-16 provide the best visual embedding, yielding a result of 70.67% in precision at 10. The performance of the different methods is shown in Table 1.

Using Multiple Modalities

Multiple modalities can be used either separately, in their original representation spaces or by fusing them into a new representation. We start by evaluating simple concatenation and score fusion. After that, we progress to multimodal fusion were we first evaluate state-of-the-art multimodal autoencoders, that we use as our baseline and our proposed BiDNN architecture. To have comparable experiments, we use a representation size of 100 for speech and a representation size of 4096 for video keyframes in all experiments. Both multimodal autoencoders and BiDNN were designed to yield a multimodal representation of size 1000. In our experiments and for the given datasets, bigger multimodal representation spaces did not improve the results.

All the weights in the different neural architectures were randomly initialized with an appropriate uniform distribution. [START_REF] Glorot | Understanding the difficulty of training deep feedforward neural networks[END_REF] Training was performed by stochastic gradient descent (SGD) with Nesterov momentum in mini-batches of 100 samples. For regularization, dropout of 20% was applied. Training was performed for 1000 epochs to ensure convergence was achieved for all systems. No system started to diverge after converging. Every setup was run 5 times, the results were averaged and the respective standard deviation computed, as shown in Table 1.

Simple Multimodal Approaches

It is possible to utilize multimodal data without fusing the initially disjoint modalities. Two typical methods to do so are concatenation and score fusion. [START_REF] Guinaudeau | HITS and IRISA at MediaEval 2013: Search and hyperlinking task[END_REF] Although these methods are typically underperforming, we still include them for the sake of completeness and to obtain a better overall picture of the improvement brought by multimodal autoencoders and BiDNNs. Although there is no significant improvement when concatenating speech transcripts and visual concepts, combining VGG-16 embeddings with embedded transcripts yields 75.33% (precision at 10) over the initial performance of 70.67% and 58.67% respectively.

Multimodal Embedding with Autoencoders

Multimedia autoencoders are a state-of-the-art method for performing multimodal fusion. Although a simple autoencoder can be used for multimodal fusion by concatenating the different modalities, we implemented a multimodal autoencoder with separate branches for each modality, as illustrated in Figure 1 in the left part. Simple autoencoders performed worse than multimodal autoencoders and were thus not used as a baseline. [START_REF] Vukotić | Bidirectional joint representation learning with symmetrical deep neural networks for multimodal and crossmodal applications[END_REF] The results are illustrated in Table 1. Multimodal embedding clearly performs better than every single modality separately. Combining embedded transcripts and VGG-19 features yield 74.73%, compared to 58.67% and 68.07% respectively. In some cases, multimodal embedding did not improve the results significantly. We believe this to be the case for already good initial single-modal representations given the fact that autoencoders have to train to represent the correct output with both modalities being present at their input and with one modality. In cases where the initial embeddings perform less good (e.g., embedded visual concepts combined with embedded transcripts), autoencoders seem to improve in a more significant way.

BiDNN Multimodal Embedding

To have comparable results, we performed all the BiDNN experiments with an architecture of comparable dimensionality and a number of layers as the architectures of multimodal autoencoders previously evaluated: the input branches were either of 100 or 4096 dimensional and the central representation layer was of size 1000. Training was done for 1000 epochs though convergence was again achieved earlier. Just like before, each model was run for 5 times and the averages with their respective standard deviations were computed and reported in Table 1. When comparing different models, we use this information to determine whether there is a significant improvement by performing a single-tailed t-test. Multimodal fusion created with BiDNNs has the benefit of a common joint representation space where both modalities are projected from their initial representation spaces. This provides multimodal embedding superior to those obtained with multimodal autoencoders and brings significant improvement. For instance, combining embedded transcripts with VGG-19 embeddings yields a precision at 10 of 80.00%, compared to 58.67% and 68.67% respectively. All the other tested combinations also yielded better results and high-quality multimodal embeddings.

BiDNN Single Modality Embedding

When performing multimodal fusion, both modalities are projected into the new representation space created in the network's central layers. However, it is possible to only project one modality into the new representation space and use it as an improved multimodal representation. The network is still trained in a crossmodal manner but the only a single modality is projected into the newly learned representation space. When doing so, automatic speech transcripts improve from 58.67% to 66.78%, visual concepts from 50.00% to 54.92% and VGG-19 embeddings from 68.67% to 70.81%. Although using only a single modality cannot achieve the same results as multimodal fusion, these experiments show that the newly learned representation space offers better performance and can even improve single-modal representations after they are projected into it.

BiDNN Crossmodal Query Expansion

BiDNNs are crafted for crossmodal translation in mind, so when a single modality of a datapoint is missing it is possible to easily reconstruct it from the other. By doing so, we expand the queries in such a way that all datapoints have both their modalities present and are compared to both modalities. It is important to note that in case of crossmodal query expansion, we remain into the original representation spaces of each modality and we only fill in the very few missing modalities. When combining transcripts and visual concepts (originally 58.00%), we obtain, e.g., 62.35%. No significant improvement is obtained for pairs computed with highperforming deep convolutional neural networks and automatic transcripts. We believe this to be caused by the fact that only a few datapoints are missing one modality, so by filling in the missing modalities and staying into the same representation spaces, we do not see an improvement if the original representation spaces are already well performing

Video Hyperlinking Evaluation at TRECVID 2016

To further analyze BiDNNs and evaluate whether they also work in a video hyperlinking setup, we decided to evaluate them through the TRECVID's 2016 video hyperlinking benchmark. [START_REF] Awad | Trecvid 2016: Evaluating video search, video event detection, localization, and hyperlinking[END_REF] In this case, as explained before, we also perform segmentation and we recommend video segments to a human evaluator out of the multitude of mostly irrelevant video segments. To see not only how well the proposed system performs in regards to the systems proposed by other participants but also how much does it improve over the initial, disjoint single-modal representations, we decided to evaluate the best performing single-modal representations, according to our previous evaluation in multimodal retrieval, together with the two modalities fused in a crossmodal fashion by BiDNNs. Each submission was made by finding the top 10 most similar video segments, for each given anchor, out of the 307,403 overlapping video segments. In case two proposed segments overlapped, only the most similar one was kept and the other was removed from the stack.

The results of the submitted runs, as well as the statistics of all the team that participated, are given in Table 2. The official scores at TRECID's 2016 video hyperlinking task were given in precision at 5. It is important to emphasize that, contrary to some participants, we did not use any additional information available with the dataset and we utilized solely the video keyframes and automatic speech transcripts. Also, many participants did not use deep learning methods. The video hyperlinking evaluation confirmed what we have previously shown in multimodal retrieval evaluations: focusing on crossmodal translations with added restrictions (namely enforced symmetry) yields improved multimodal embeddings. Not only did BiDNN improve the two initially disjoint modalities (40% and 45% into a representation that yields 52% in terms of precision at 5) but it also proved that obtaining multimodal fusion in a crossmodal fashion outperforms multimodal fusion methods. This shows how a simple, unsupervised learning system, with no additional information and no handcrafted features, can compete with more complex multimodal systems. It also proves that crossmodal translations with enforced symmetry improve over classic ways of multimodal fusion and define the new state of the art. Post TRECVID 2016 Evaluation

While we have seen that BiDNNs define the new state of the art in video hyperlinking, it is still interesting to further evaluate the overall system and see how would the different parts impact the evaluation if altered. After the TRECVID 2016 video hyperlinking evaluation was performed, a new groundtruth was again collected by the organizers, containing the targets proposed by all their participants and their relevance, as judged by AMT human evaluators. These targets are not consistent with our evaluated video segments as every participant had their own segmentation. We thus use our proposed system (multimodal representations of each video segment, fused with BiDNNs) to analyze the similarity of the targets proposed by all the participants for each anchor and the rank (determined by similarity) of the targets we proposed. On average, there are 79.19 proposed targets for each anchor.

Figure 3 illustrates the distribution of our proposed targets between all the targets that were proposed for a specific anchor by all the participants, according to their similarity given by multimodal embeddings obtained with a BiDNN model. While most of the targets are within the top 10 by similarity to the given anchor, we proposed many targets that are quite low ranking according to our own system. The average rank for our proposed targets is 27.98 and furthermore, 61.17% of our proposed targets are not within the top 10 video segments when evaluating all the video segments proposed by all participating teams. According to BiDNN similarity, within the top 10 best targets for any given anchor, there are on average only 3.02 targets that we proposed. This indicates that, had we evaluated the same video segments as some other participants, the system could further be improved, at least in regards to similarity between targets and anchors given by BiDNN based representations.

Additionally, by performing a multimodal retrieval analysis on all the proposed targets (without retraining the model) and reranking the targets proposed by all the participants, we obtain a result of 49.56% in precision at 5, compared to 30.8% for the video hyperlinking evaluation, which strongly indicates that although multimodal retrieval evaluations are acceptable for choosing the best performing method, they largely differ from video hyperlinking evaluations in terms of performance.

CONCLUSIONS

We started by analyzing different methods for obtaining single-modal continuous representations and methods to fuse them into a single multimodal representation. After evaluating them in a multimodal retrieval setup, we use the learned notions and the best performing methods in video hyperlinking, where we perform a more exhaustive evaluation. Expectedly, visual embeddings obtained with deep convolutional neural networks outperformed embedded visual concepts and proved to be more relevant than automatic transcripts. Very deep VGG convolutional neural network architectures significantly outperformed the less deep AlexNet architecture. VGG-16 performed best and produced a single-modal visual embedding that yields 70.68% in precision at 10. We have shown that the few downsides of autoencoders can affect their results and that BiDNNs successfully tackle these problems and clearly outperform multimodal autoencoders by a significant margin. Although VGG-16 performed better than VGG-19 in a single-modal setup, the best performance was obtained by multimodal fusion of embedded automatic transcripts and embedded VGG-19 features, yielding a precision at 10 of 80.00%.

The video hyperlinking evaluation confirmed that focusing on crossmodal translations with added restrictions, namely in the form of symmetry, is a feasible method to perform multimodal fusion and BiDNNs defined the new state of the art by achieving the best performance at the challenge. After the post-evaluation groundtruth was formed, we used it to further analyze our model and found that, although our submissions were most often ranked between the top 10 ones, multimodal representations obtained with BiDNNs often favored other targets proposed by other participants. This indicates that further improvements can be made solely by changing the initial segmentation. BiDNNs have already been shown to successfully tackle the downsides of multimodal autoencoders and to provide superior multimodal embeddings. In this work, we extensively tested BiDNNs first in a preliminary evaluation in multimodal retrieval and then more extensively in video hyperlinking. This reinforces the points already made in preliminary research, [START_REF] Vukotić | Bidirectional joint representation learning with symmetrical deep neural networks for multimodal and crossmodal applications[END_REF] and shows that BiDNNs defined the new state of the art at the last TRECVID's 2016 video hyperlinking benchmarking initiative.

Figure 1 .

 1 Figure 1. Illustration of the architectures: left -classical multimodal autoencoder; right -our proposed bidirectional symmetrical deep neural network.

Figure 2 .

 2 Figure 2. With BiDNNs, the similarity of two video segments is obtained by presenting the respective modalities of each of the two samples to a BiDNN and taking the activations of the respective central layers after propagating the information. Afterword, the obtained representations are used to compute a cosine similarity measure.

Figure 3 .

 3 Figure 3. Histogram illustrating the distribution of ranks of our proposed targets in the collection of targets proposed by all the participants, ordered by similarity according to BiDNN fused representations. The blue histogram represents all our proposed targets while the one in red solely those that we proposed and were judged relevant.

Table 2 :

 2 Results of our submitted runs containing solely the initial modalities, fused modalities with BiDNN and overall statistics for all the participants of TRECVID's 2016 video hyperlinking benchmark.

	Evaluation	P@5 (%)
	Speech transcripts only	40
	VGG-19 features only	45
	BiDNN -transcripts and VGG-19	52
	Min (all teams)	24
	1	st quartile (all teams)	32
	Mean (all teams)	35
	3	rd quartile (all teams)	41
	Max (all teams)	52

Guillaume Gravier is senior research scientist at CNRS, France. He leads the Linkmedia research group focusing on content-based media analysis, indexing and linking with the ultimate goal of enabling better multimedia applications and new innovative services. His current research interests are in media analytics, multimedia collection modeling, deep learning and multimodality, graph-based methods for multimedia content representation.


Maiga Adama
email: mahamadoumaiga@yahoo.fr

Mahamadou

A property of the space-time: space-time occupied by a particle is an inverse function of its mass

The two great theories of physics, the theory of relativity and quantum theory, contradict each other in the space-time description. However, these two great theories and some theories that are trying to merge them are concordant that space-time no longer constitutes a simple inert screen on which natural phenomena are projected. This unanimity should encourage to explore more advantage the fundamental properties of space-time. Also, on the basis on the observations that quantum mechanics shows that the electron seems to follow several trajectories simultaneously and the theory of relativity which shows that the trajectory of a planet is very well defined, we postulate the following principle: In a space-time, the smaller the mass of a particle is, the bigger the spacetime occupied by the particle is.

For example, if we consider three particles without internal structure of sizes of an electron, a tennis ball and the planet earth, with this principle, it can be stated that the electron particle occupies more space-time than the tennis ball, which itself occupies more space-time than the planet earth particle. This principle of space-time occupied raises a number of questions. An initial analysis shows that the principle is compatible with the inertial mass, the gravitational mass and de Broglie's hypothesis. The agreement between the principle and de Broglie's hypothesis indicates that the principle can be extended to particles of zero theoretical mass. The question of the influence of interactions (weak, strong, electromagnetism and gravitational) on this principle remains central. If these influences are proven and are of the same order of magnitude, then they can constitute a fundamental basis for a merger of these interactions.

2

A first idea for a possible application of the principle may be to analyze the evolution of stars especially that of the Sun or a black hole devouring a star. A preliminary analysis to the evolution of the Sun provides some possible explanations why the Sun's corona temperature is much higher than that of its surface layer (see annex).

Introduction

Space-time is the fundamental basis of Science. That they were the humanities and social sciences (history, sociology, psychology ...) or the exact sciences (mathematics, physics, chemistry ...). The phenomena or laws that locate or predict these sciences are projected in space-time. The latter is no doubt fundamental in physics that studies the external world and the laws of its variation and evolution in space and time. Specifically, there are two great theories in physics: quantum theory and the theory of relativity.

Quantum theory studies the behavior of elementary particles and atoms, i.e. very small scale phenomena [START_REF] Hobson | Relativité générale[END_REF] (Mehra & Rechenberg, 2002). The theory of relativity studies large scale phenomena (planets, stars, galaxies...) [START_REF] Mehra | The Historical Development of Quantum Theory[END_REF] [START_REF] Hobson | Relativité générale[END_REF].

Although these two great theories have each been confirmed by experimental studies, nevertheless, they are not compatible. They contradict each other in the space-time description. The theory of relativity considers a space-time continuum, i.e. continuous, smooth but deformable while quantum theory predicts a space-time foam. This glaringly contradiction or incompatibility may legitimately lead to think that these two theories cannot both be just. However, great theories as relativity and quantum mechanics that on the one hand have repeatedly been verified by many measurements and observations: [START_REF] Bohr | On the Constitution of Atoms and Molecules, Part I[END_REF][START_REF] Bohr | On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus[END_REF][START_REF] Compton | A Quantum Theory of the Scattering of X-rays by Light Elements[END_REF][START_REF] Aad | Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC[END_REF][START_REF] Chatrchyan | Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC[END_REF][START_REF] Bevan | BaBar and Belle Collaborations)[END_REF] Resonance Imaging, integrated circuit, laser… and for theory of relativity: Global Positioning System, CRT screen, electricity generators…). Therefore, these theories can by no means be false, at worst, they can only be insufficient as Rutherford-Bohr model, Galileo-Newton's mechanics and special relativity have been before.

Nevertheless, the incompatibility between the theory of relativity and quantum mechanics must lead to question on the conception of space-time of these two theories. Also, our understanding of spacetime deserves to be better investigated. Thus, this is logically that some theories, Loop Quantum Gravity and String Theory that are trying to merge quantum mechanics and the theory of relativity, explore the conception of space-time. These theories explore particularly the structure of spacetime: with Loop Quantum Gravity who tries to get rid of the time and, most importantly, that space presents a discrete structure [START_REF] Rovelli | Loop Quantum Gravity[END_REF][START_REF] Rovelli | Par delà le visible: La réalité du monde physique et la gravité quantique[END_REF] [START_REF] Rovelli | Loop Quantum Gravity[END_REF][START_REF] Rovelli | Par delà le visible: La réalité du monde physique et la gravité quantique[END_REF]. String Theory assumes the existence of the multiple universes and included especially more than 4 dimensions [START_REF] Tong | String Theory[END_REF][START_REF] Greene | L'Univers élégant: Une révolution scientifique de l'infiniment grand à l'infiniment petit, l'unification de toutes les théories de la physique[END_REF] Almost all the theories (theory of relativity, quantum mechanics, loop quantum gravity, string theory) are concordant that space-time no longer constitutes a simple inert screen on which natural phenomena are projected. The unanimity shown by these models, on this property of space-time, is experimentally highlighted within the scope of validation of the general relativity [START_REF] Dyson | A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29[END_REF][START_REF] Abbott | Observation of Gravitational Waves from a Binary Black Hole Merger[END_REF] [START_REF] Dyson | A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29[END_REF][START_REF] Abbott | Observation of Gravitational Waves from a Binary Black Hole Merger[END_REF]. This evidence should encourage to explore more advantage the fundamental properties of space-time.

One of the primary questions, raised by the absorption and emission of light by matter, is to understand the process that gives a color to objects. It is a fact that if an object appears of green color, it is because the whole spectrum of visible light is absorbed by the matter except for the green part which is reflected. So the color perception of objects is strictly virtual. And if the space perception occupied by objects is also strictly virtual: i.e. between three particles without internal structure of sizes of an electron, a tennis ball and the planet earth, which actually occupies more space?

The main objective of this present paper is to postulate a principle which defines the space-time occupied by a particle as a function to its mass. This principle may appear as a property of spacetime.

Principle of space-time occupied

On the basis on the observations that quantum mechanics shows that the electron seems to follow several trajectories simultaneously and the theory of relativity which shows that the trajectory of a planet is very well defined, we postulate the following principle: « In a space-time, the smaller the mass of a particle is, the bigger the space-time occupied by the particle is » Example if we consider three particles without internal structure of sizes of an electron, a tennis ball and the planet earth, respectively. With this principle of space-time occupied, it can be stated that the electron particle occupies more space-time than the tennis ball, which itself occupies more space-time than the planet earth particle.

This principle raises a number of questions:

 The inverse function is of what type: proportional, polynomial, exponential…?  How is the structure of this space-time occupied, is it a space-time with more than 4 dimensions, a space without time, a time without space, a zone without space-time...?

 By which process the mass occupies the space-time, is it by the support of a field...?

 Does this principle impose the conservation of a quantity in the inertial reference systems?

 What about particles with zero theoretical mass?

 Is it the inertial mass, gravitational mass or another form of mass?  This principle, is it compatible with the inertial mass and gravitational mass?  How do interactions (weak, strong, electromagnetism and gravitational) influence this principle?

The question of the influence of interactions (weak, strong, electromagnetism and gravitational) on this principle remains central. If these influences are proven and are of the same order of magnitude, then they can constitute a fundamental basis for a merger of these interactions. But the compatibility between this principle and the inertial and gravitational masses remains the primary question to be explored.

The principle of space-time occupied affirms that the smaller the mass of a particle, the bigger the space-time occupied by the particle is. As a result, the particles of very small mass, especially at the atomic and subatomic scale (elementary particles, atoms...), occupy more space-time than particles comparable to a material point (cartridge, tennis ball, star...).

It can be stated that, the space-time in which the atomic and subatomic particles manifests itself, is bigger than that in which the particles described by a material point manifests itself. So, it takes more space-time to properly identify the atomic and subatomic particles that it is necessary to properly identify the particles described by a material point. After all, as the space-time occupied by a particle is an inverse function to its mass, on the one hand, it is possible for a particle, with a very small mass, to occupy sufficient space-time for an investigation on the particle depends, in whole or in part, on the space-time occupied. On the other hand, a particle can have enough mass to occupy a sufficiently small space-time and be very well localized. Thus, it can be assumed that an investigation on the particle is independent, in whole or in part, of the space-time occupied.

Compatibility with the inertial mass

The inertial mass is a property of matter which manifests itself by the bodies's inertia. In an inertial movement of a particle case, if an investigation on the particle depends, in whole or in part, on the space-time occupied, only a wave approach will allow to properly identify the space-time occupied by the particle. This conclusion that some particles behave as a wave, is in perfect agreement with the experimental observations and the theoretical models which show that, in an experiment where the energy of the particles is conserved, the atomic and subatomic particles (atoms, elementary particles ...) behave like waves.

In the case where the particle can have enough mass to occupy a space-time sufficiently small and to be very well localized. Thus, at any moment, the position of the particle can be determined in an inertial movement case. Also, this conclusion is in perfect agreement with the experimental observations and the theoretical models which show that particles described by a material point have a trajectory well determined.

Compatibility with the gravitational mass

The gravitational mass is a property of matter which manifests itself by the universal attraction of bodies.

General relativity states that the gravitation is not a force but the manifestation of the curvature of space-time. This curvature is due to the distribution of energy, in the form of mass or kinetic energy. The principle of space-time occupied states that the space-time occupied by a particle is an inverse function to its mass. As a result, the atomic and subatomic particles manifests itself in a large space-time and an investigation on these particles depends on this space-time. As a consequence, the distribution of the atomic and subatomic particles energy is more spatial, i.e. a very low energy density, and therefore a lower curvature. However, the material point particles manifest itself in a very small space-time and are well localized. So, the distribution of the material point particles energy is more concentric, i.e. a very high energy density, and therefore a higher curvature.

If we consider two particles of mass m 1 and m 2 , with m 1 greater than m 2 (m 1 > m 2 ), the principle states that the mass particle m 2 occupies more space-time than m 1 . Assuming an interaction between m 1 and m 2 leading m 1 to acquire an energy from the m 2 mass. And that the interaction process occurs through an exchange of a mediator or interaction vector. This exchange of energy results in a variation of masses m 1 and m 2 . And appear as an interaction between two new particles of masses m 1 + dm 2 and m 2 -dm 2 . With the principle, the particle of mass m 1 + dm 2 will occupy a space-time smaller than that of m 1 . And the particle of mass m 2 -dm 2 occupies a spacetime greater than that of m 2 . Also, the curvature of the space-time due to the mass m 1 + dm 2 is stronger than that due to m 1 . This variation of the curvature, i.e. an increase in curvature around m 1 + dm 2 and a decrease in curvature around m 2 -dm 2 , can accelerate the interaction between m 1 + dm 2 and m 2 -dm 2 . And if the interaction between the particles continues at the end there will remain only a particle of mass m 1 + m 2 , which will occupy a space-time smaller than occupied both masses at the beginning. In addition, we have a much stronger space-time curvature around m 1 + m 2 .

Compatibility with the de Broglie hypothesis

By identifying the velocity of the particle at the group velocity of the wave, as well as their energy, de Broglie proposes to associate to material particles a wavelength. This hypothesis leads to a relation of proportionality between the momentum and the wave vector for a particle. The equation shows that the wavelength will be greater if the mass is smaller. That is the smaller the mass of the particle is, the larger the space in which the associated wave manifests itself is. Thus, the principle of space-time occupied, which predicts that: the smaller is the mass of a particle, the bigger is the space-time occupied by the particle, is in perfect agreement with de Broglie's hypothesis. This agreement indicates that the principle of space-time occupied can be extended to particles of zero theoretical mass which one can be associated a wavelength.

Possible application of the principle of space-time occupied

The application of the principle of space-time occupied is probably possible in cases where the mass of the particle varies over time and this variation is sufficient to influence the properties of the space-time.

A first idea to explore the principle may be to analyze the evolution of stars especially that of the Sun or a black hole devouring a star. To this end, we consider the Sun, black hole and star as particles without internal structure.

The emission of light by the Sun is the result of a very complex process and, above all, leads to a decrease in its mass. As the mass of the Sun decreases over time, the principle states that the spacetime occupied by the Sun increases over time: i.e. the space-time occupied by the Sun is expanding.

It will be interesting to explore if this expansion of the space-time occupied have an influence on the evolution of the Sun and particularly on the temperature of its corona that exceeds one million Kelvins, much hotter than that of its surface layer (the photosphere) which is of the order of 6000 K [START_REF] Mcintosh | Alfvenic waves with sufficient energy to power the quiet solar corona and fast solar wind[END_REF][START_REF] Amari | Small-scale dynamo magnetism as the driver for heating the solar atmosphere[END_REF] [START_REF] Mcintosh | Alfvenic waves with sufficient energy to power the quiet solar corona and fast solar wind[END_REF][START_REF] Amari | Small-scale dynamo magnetism as the driver for heating the solar atmosphere[END_REF]. This phenomenon is contrary to the second law of thermodynamics.

In the case of the black hole devouring a star, we have the mass of the black hole increases and that of the star decreases. Thus, considering alone black hole, principle states that its space-time occupied decreases over time. However, it states that the space-time occupied by the star increases over time. This expansion is much greater than the case where the star is not devoured and loses mass than by emission of light.

Conclusion

A principle which defines the space-time occupied by a particle as an inverse function to its mass is postulated.

The predictions derived from this principle concerning the behavior of inertial and gravitational masses are in agreement with the experimental observations and the theoretical models. They show above all that the principle is compatible with the inertial mass, gravitational mass and de Broglie's hypothesis.

The agreement between the principle and de Broglie's hypothesis indicates that the principle can be extended to particles of zero theoretical mass.

A possible application of the principle may be to analyze the evolution of stars especially that of the Sun or a black hole devouring a star.

It can be concluded that, the principle that the space-time occupied by a particle is an inverse function to its mass is an intrinsic properties of the space-time.

Annex A possible explanation of the Sun's corona temperature

The experimental measurements show that the Sun's corona temperature exceeds one million Kelvins, much hotter than that of its surface layer (the photosphere) which is of the order of 6000 K [START_REF] Mcintosh | Alfvenic waves with sufficient energy to power the quiet solar corona and fast solar wind[END_REF][START_REF] Amari | Small-scale dynamo magnetism as the driver for heating the solar atmosphere[END_REF] [START_REF] Mcintosh | Alfvenic waves with sufficient energy to power the quiet solar corona and fast solar wind[END_REF][START_REF] Amari | Small-scale dynamo magnetism as the driver for heating the solar atmosphere[END_REF]. This phenomenon is contrary to the second law of thermodynamics.

The emission of light by the Sun is the result of a very complex process and, above all, leads to a decrease in its mass. The principle of space-time occupied states that, in a space-time, the smaller the mass of a particle is, the bigger the space-time occupied by the particle is. If it is assumed the Sun as particles without internal structure, as the mass of the Sun decreases over time, the principle states that the space-time occupied by the Sun increases over time: i.e. the space-time occupied by the Sun is expanding. It is assumed that, the space-time occupied gain for the Sun comes from, in whole or in part, directly of its surrounding space-time (the photosphere and the corona). Also but above all, the loss in mass of the Sun supplies energy and matter to the photosphere and the corona. Thus, one can consider that the space-time of the photosphere and the corona is in contraction in time.

The principle predicts that, on the one hand, the space-time occupied by the Sun is in expansion. This expansion phase consumes energy, i.e. of the temperature. Some or all of this temperature is withdrawn directly in the space-time of the external layer to that of the Sun, i.e. the photosphere layer. On the other hand, the space-time occupied by photosphere and the corona is in contraction.

In this contraction phase, the interaction between the space-time parts is very strong and led to a noticeable increase in the temperature. However, the temperature of the photosphere layer, which is withdrawn by the space-time occupied by the Sun, is lower than that of corona.

The application of the principle of space-time occupied to analyze the evolution of the Sun of its surrounding space-time (the photosphere and the corona) is a first idea to explore the applicability of this principle. This application provides some possible explanations why the Sun's corona temperature can exceed one million Kelvins, much hotter than that of its surface layer (the photosphere) which is of the order of 6000 K [START_REF] Mcintosh | Alfvenic waves with sufficient energy to power the quiet solar corona and fast solar wind[END_REF][START_REF] Amari | Small-scale dynamo magnetism as the driver for heating the solar atmosphere[END_REF] Thus, this application deserves to be thorough and if it is possible modeled.

  (Bohr 1913a; Bohr 1913b;[START_REF] Compton | A Quantum Theory of the Scattering of X-rays by Light Elements[END_REF]; Aad et al. 2012; Chatrchyan et al. 2012; Bevan et al. 2014) for quantum theory and [9-12] (Dyson et al. 1920; Pound & Rebka 1959; Hulse & Taylor 1974; Abbott et al. 2016) for theory of relativity. On the other hand, these theories are at the base of the significant progress of science (for quantum theory: nuclear energy, Magnetic

  photosphere) which is of the order of 6000 K[START_REF] Mcintosh | Alfvenic waves with sufficient energy to power the quiet solar corona and fast solar wind[END_REF][START_REF] Amari | Small-scale dynamo magnetism as the driver for heating the solar atmosphere[END_REF] (Mcintosh et al. 2011; Amari et al. 2015).


José Saavedra

Christopher Flores

Anthony Busson

Pierre Duhamel

Coverage estimation of multicast OFDM systems based on stochastic geometry

Keywords: Wireless Multicast, OFDM, Coverage Capacity

Quality of service has become crucial in wireless communications, due to the high data rates needed to provide multimedia services with sufficient quality. In this sense, the integration of OFDM in wireless technologies has made it possible to grant reliable communications, reducing the effects of attenuation and multipath propagation that degrades the services in environments where the devices are located. This problem may become even more important if, as it is expected, the classical broadcast TV is left to the wireless network operators, in order to recover the frequencies that are currently used for this service. This paper aims to provide efficient tools to estimate the coverage of an OFDM transmission system using stochastic geometry and information theory to model the capacity of a cell when users are known to the transmitter (multicast situation). We consider that users are distributed uniformly over a fixed radius ball from the base stations. We derive the equations and develop simulations to model the distribution of channel capacity to ensure a certain quality of service. Our results indicate that for a user, on average, wireless multicast can be improved by 20% compared to classical broadcast.

I. INTRODUCTION

I N the last decade wireless technologies have grown quickly.

The devices that deliver telephony, data and multiple services, are widely deployed. Between the standards, Orthogonal Frequency Division Multiplexing (OFDM) are more and more present. By the side of wireless packet data transmission we have 802.11a/g/n/ac/ad [START_REF] Perahia | Next Generation Wireless LANs: Throughput, Robustness, and Reliability in 802[END_REF], popularly known as Wireless Fidelity (WI-FI) and by the side of mobile communications we have the consortium of the 3rd Generation Partnership Project Long Term Evolution (3GPP-LTE) with 5G [START_REF] Wong | Key Technologies for 5G Wireless Systems, ser. Key Technologies for 5G Wireless Systems[END_REF] [START_REF] Wang | Cellular architecture and key technologies for 5g wireless communication networks[END_REF], both have integrated OFDM given its capacity to overcome impairments due to scattering environments where the devices are deployed. The main trick of OFDM is to take a channel bandwidth and split them in many orthogonal sub-channels with low data rate, granting reliable communication.

Usually, in wireless networks, when the same data stream is provided to a group of users, Time Division Multiplexing is used to allocate the same data to different block time resources, for each user, with power allocation and the modulation and coding scheme negotiated between the base station (BS) and the user device. This is very costly in terms of resource (one resource allocated to each user). In classical broadcasting, in contrast, the same information is transmitted via the same resource to all users, which are a priori unknown to the transmitter. This corresponds to a large saving in terms of resource but no power allocation can be performed by the transmitter, which corresponds to a waste of energy. This strategy is unavoidable when the number of users is very large. However, there is a possibility that, in the near future, pure broadcast (for terrestrial TV) will disappear, and the wireless networks will be in charge of performing this service [START_REF] Nakahara | Digital transmission scheme for isdb-t and reception characteristics of digital terrestrial television broadcasting system in japan[END_REF] [START_REF] Institute | Digital Video Broadcasting (DVB): Implementation guide for DVB terrestrial services; transmission aspects[END_REF]. If this happens, the number of users to be served simultaneously by the same transmitter (base station) will be reasonable, and one can envision to use a wireless multicast strategy, where the users are known to the transmitter, which can therefore perform some optimization. The frontier between these two cases is however not obvious, and this paper tries to clarify this trade-off.

Our work focuses on find an objective criterion to decide if the optimization is still useful, or if the uniform power allocation is the best way to transmit data to a group of users. We name wireless multicast when a group of users subscribe to the same service and the CSI are available on BS. When the uniform power allocation is used to transmit the same data, we name it broadcast, like the old-fashioned broadcast-TV.

In such a situation the transmitter has no hope that the transmitted signal will be received by all users, because the randomness of channels makes some of them really intractable. In such case, one way to take these events into account is via outage capacity: a broadcast channel is always designed in such a way that x % of the users are certain to obtain a given rate [START_REF] Ozarow | Information theoretic considerations for cellular mobile radio[END_REF] [START_REF] Goldsmith | Wireless Communications[END_REF].

In practical situations, the coverage of an access point or base station can be made using long simulations using a merge between analytic and empirical methods [START_REF] Rappaport | Wireless Communications: Principles and Practice[END_REF]. However, in the last decade, techniques based on stochastic geometry [START_REF] Stoyan | Introduction to Stochastic Geometry[END_REF] [10] were proposed, which allow to solve analytically such problems related to wireless communications.

Stochastic geometry (SG) is a discipline of applied mathematics that studies spatial random patterns with emphasis on the theory of Point Processes (PP) [START_REF] Baccelli | Stochastic geometry and wireless networks: Volume i: Theory[END_REF]. The classic SG models are mainly based on the study of Poisson Point Processes (PPP) [START_REF] Calka | Stochastic geometry: Boolean model and random geometric graphs[END_REF]. The simplest model is the Boolean (BM), which is composed of random balls centered at each point of the process [START_REF] Baccelli | Stochastic geometry and wireless networks: Volume i: Theory[END_REF]. Another model commonly used in SG are the Voronoi Tessellations (VT), which are composed of finite polyhedrons (cells) containing all the points that are closest to the center of the cell [START_REF] Baccelli | Stochastic geometry and wireless networks: Volume i: Theory[END_REF] [START_REF] Yang | Sir coverage analysis in cellular networks with temporal traffic: A stochastic geometry approach[END_REF]. The SG has diverse applications in biology, astronomy, wireless communications, among others.

For instance, in the context of wireless communications, the location of base stations (BSs) can be modeled using a PPP, while coverage areas can be defined using a Voronoi cell [START_REF] Klatt | Cell shape analysis of random tessellations based on minkowski tensors[END_REF].

We intend to extend the stochastic geometry techniques to OFDM channels systems, and it will be shown that analytical solutions are not fully feasible, and that only mixed analytical/numerical solutions are obtained. This process will anyway be faster than the many simulations usually required.

It's known that the percentage of users meeting the given bitrate constraint depend on their location. Therefore, we assume that the users are distributed according to a certain spatial point process. It allows us to consider the spatial distribution of the users around the base station,from which we deduce the distribution of the capacity. Then, for a typical user, we compute a coverage area where a minimum/fixed capacity is guaranteed with a certain probability. This corresponds to an outage probability on the capacity. This outage probability may also be interpreted as the proportion of users for which this fixed capacity will be offered, which is precisely the desired design criterion for old-fashioned broadcast transmission.

Our solution involves some approximations to obtain semianalytical results, therefore, a first set of numerical results checks that the different approximations of the distributions (channels, capacity of a given channel, etc.) involved in the computation of the final capacity are accurate. Then, the extrapolated capacity is compared to simulations, in order to check if our optimization is correct over this scenario, meaning that the required percentage of users indeed meets the desired bit rate.

II. SYSTEM MODEL

Our model consider an ideal OFDM system were the channels are perfectly orthogonal. The user is one user among L-total users. The received signal of the user is:

y = H x + z (1)
where x = (x (1) , .., x (N ) ) ∼ CN (0, S x ) is the input vector of N OFDM symbols and S x is the covariance matrix of the vector x. Since the modulated symbols are independent and orthogonal, the covariance matrix S x is a diagonal matrix such that S x = diag{p n }, n ∈ {1, ..., N }, where p n is the power allocated to the n-th subcarrier. The total power available to optimize is P , such that N n=1 p n ≤ P . The Matrix H is the a diagonal channel gain matrix which dimension is N × N , the elements are the channel impulse response of channel on each orthogonal sub-carrier frequency, H = diag{h ( ,n) }) and is the Channel Sate Information (CSI) present at the transmitter. The vector z is the AWGN at the receiver z ∼ CN (0, S z ). The problem of broadcast the same information for L-users are faced in [START_REF] Saavedra | Multicast or broadcast in multiuser ofdm systems[END_REF]. Where we develop an optimal algorithm to allocate efficiently the power among the subcarriers. From that we recall the capacity of one user (L = 1), since we use it to perform a coverage statistic over a spatial distribution. And we can compare with previous results.

We recall the capacity formula for the ideal OFDM scheme, as parallel independent channel capacity [START_REF] Saavedra | Multicast or broadcast in multiuser ofdm systems[END_REF] [START_REF] Tse | Fundamentals of Wireless Communications[END_REF].

C = 1 N N n=1 log 1 + |h (n) | 2 p n σ 2 z [bits/s/Hz] (2)
The complex subcarrier gain h (n) is expressed in the frequency domain.

III. OFDM REALISTIC CHANNEL AND COVERAGE

MODELING

The goal is to find a closed-form expression to calculate the coverage, and have an impartial instrument to say if the multicast optimization is still useful under a spatial distribution of the users, with an outage criterion in mind.

Let P (C ≥ C out ) denote the outage probability. It is the probability that the capacity of a typical user, denoted C, is greater than a predetermined value C out [START_REF] Goldsmith | Wireless Communications[END_REF], [START_REF] Tse | Fundamentals of Wireless Communications[END_REF]. Given a capacity requirement C out and an outage probability, we can calculate the maximal coverage, i.e. the maximal distance where users can be distributed while statistically meeting these two constraints.

A. Realistic Channel

The first step towards characterizing properly the capacity is to first evaluate the probability density function of the channel h (n) . We know that [START_REF] Rappaport | Wireless Communications: Principles and Practice[END_REF] [START_REF] Perahia | Next Generation Wireless LANs: Throughput, Robustness, and Reliability in 802[END_REF] it is modelled by ( 3), evaluated at frequency f n :

h (n) = I i=1 β i e -j(2πfnτi+θi) (3)
Where β i is the product of the path loss gain 1 L(D) and the complex gain of the path g i :

β i = 1 L(d) • g i .
We can express it as:

h (n) = 1 L(D) I i=1 g i e -j(2πfnτi+θi) (4)
The multipath part of this expression can be written as:

g (n) = I i=1 g i e -j(2πfnτi+θi) (5)
Given that the path gains {g i } are Rayleigh distributed with parameter σ i and that the phases {θ i } are uniformly distributed on the interval [0, 2π], we define ḡi = g i e -jθi as a single random variable with a complex gaussian distribution with zero mean and variance σ 2 i . The random variables { ḡi } i=1,..,I form a sequence of independent complex Gaussian random variables with zero mean and different variances. The real and imaginary parts are assumed to be independent. The multipath part thus reduces to:

g (n) = I i=1 ḡi e -j2πfnτi (6)
Finally the complex channel gain h (n) can be viewed as:

h (n) = 1 L(D) g (n) (7)
But we can see that g (n) is a combination of different random variables, all of them with zero mean, but of various variances. Under this situation the central limit theorem does not apply, because the involved random variables have different variances. However, in order to simplify our next computations, we will make a Gaussian approximation of g (n) . In fact, for the considered model, the model E described in [START_REF] Medbo | Channel models for HIPERLAN/2 in different indoor scenarios[END_REF], we can empirically show by simulation that the real and the imaginary part of g (n) are very close to Gaussian distributions. We illustrate in Fig, 1 the distribution of imaginary and the real part of g (n) for this model. In this figure we can appreciate the actual model in red line and the approximated by the complex gaussian function in cross blue line. The curves are almost superimposed, and the complex normal distribution seems to be a good approximation for our purpose. Therefore, in the rest of this work, we assume that the random variables (g (n) ) n follow complex normal distributions.

Hence, the probability distribution function of g (n) is approximated by:

f g (n) (z) = 1 π I i=1 σ 2 i exp {- zz I i=1 σ 2 i } (8)
which will be useful for a proper characterization of the capacity distribution.

B. Outage probability

Unfortunately, analytical computation of the capacity pdf is not tractable for our model. Instead, we approximate it with a known distribution. Parameters of this distribution will be deduced from mean and variance of the capacity. Therefore, in this Section, we compute mean and variance of the capacity for our model.

If we retake the formula (2) and put inside the channel expression [START_REF] Goldsmith | Wireless Communications[END_REF] we obtain:

C = 1 N N n=1 log 1 + 1 L(D) |g (n) | 2 p n σ 2 z [bits/s/Hz] (9)
The random variables that compose the equation of the capacity ( 9) are the distance D and the complex channel gains g (n) .

The distance D is the distance between a typical user and the base station. We assume that users are distributed in an observation window. In our numerical evaluation, it consists in a ball with radius R. Within this ball, users location can follow any distribution. Here we consider that users are independently and uniformly distributed in this window. Since the base station is at the center of this ball, the pdf of the distance from a user to the BS is given by:

f D (d) = 2d R 2 (10)
where d belongs to [0, R].

The multipath complex channel gain g (n) has f g (n) (z) as probability density function [START_REF] Rappaport | Wireless Communications: Principles and Practice[END_REF]. In order to characterize the capacity random variable, we start with an expression of the expectation considering the cited distributions and we follow with the variance.

The computation of the mean capacity is straightforward. It is given by:

E[C] = 1 N N n=1 E log 2 1 + 1 L(D) |g (n) | 2 pn σ 2 z = 1 N N n=1 R 0 C log 2 1 + 1 L(d) |z| 2 pn σ 2 z ... ...f g (n) (z)f D (d)dzdd (11)
For the variance, we get:

V ar(C) = N n=1 V ar(C n ) + 2 N -1,N n=1,i>n Cov(C i , C n ) (12)
where C n is the capacity over one subcarrier.

Considering that the channels can have a correlation between the subcarriers, the covariances are not nil. This correlation is due to the random variables ḡi (see formula 6) which are the same for all the channels. The covariance Cov(C i , C n ) has the form:

Cov(C i , C n ) = R 0 C C c n (z n , d)c i (z i , d)... ...f g (n) ,g (i) (z (n) , z (i) )f D (d)dz (n) dz (i) dd (13)
where c n (z n , d) is the capacity of the subcarrier n for a given sample of g n (equals to z n ) and a sample of the distance D (d). f (g (n) ,g (i) ) (z (n) , z (i) ) is the joint probability density function of the random variables (g n , g i ).

If we assume that the vector (g (n) , g (i) ) is still normal complex, the joint distribution is:

f (g (n) ,g (i) ) (z (n) , z (i) ) = 1 π 2 det(Γ) ... exp I i=1 σ 2 i det(Γ) z (n) z (n) + z (i) z (i) -2Re(γ n,i z (i) z (n) ) ( 14
)
with det(Γ)

= I i=1 σ 2 i 2 -|γ n,i | 2 and γ n,i = l u=1 σ 2 u exp {-j2π(f n -f i )τ u } ( 15
)
The joint distribution above is obtained under the assumption that (g (n) , g (i) ) is normal complex, and is function of variance, covariance of these two random variables [START_REF] Van Den Bos | The multivariate complex normal distribution-a generalization[END_REF], [START_REF] Picinbono | Second-order complex random vectors and normal distributions[END_REF].

It can be seen that the mean and the variance do not have an easy form. These integrals must be calculated numerically.

C. Modelling the capacity pdf

P (C ≥ C out ) = 1 -P (C ≤ C out ) = 1 - Cout 0 f (E[C], V ar[C], x)dx (16)
Once the mean and variance are obtained, we will compare the shape of the actual pdf obtained by simulation with a well known function, in order to find the best fit. Good candidates belong to the log-normal probability distributions.

For instance f (E[C], V ar[C], x) can be the log-normal pdf. It can be used to approach the empirical pdf of the capacity under a realistic environment. This function has the following form:

f (E[C], V ar[C], x) = 1 xσ V exp -(ln(x)-µ E ) 2 2σ V (17
)

Whereµ E = ln E[C] 2 (V ar[C] + E[C] 2 ) σ V = ln V ar[C] E[C] 2 + 1
where the parameters µ E and σ V are deduced from mean and variance of the capacity C. Once we have [START_REF] Van Den Bos | The multivariate complex normal distribution-a generalization[END_REF] parameterized, we can calculate numerically C out from ( 16). The mean E[C] and the V ar[C] was calculated using numerical integration. We took the probabilistic distribution [START_REF] Baccelli | Stochastic geometry and wireless networks: Volume i: Theory[END_REF] with R = 200. We use the parameters of the model E [START_REF] Medbo | Channel models for HIPERLAN/2 in different indoor scenarios[END_REF] to model the pdf of the fading channel, and parameterize [START_REF] Rappaport | Wireless Communications: Principles and Practice[END_REF]. For the path loss, we use the equations from [START_REF] Saavedra | Multicast or broadcast in multiuser ofdm systems[END_REF], taking in account the distance breakpoint. Using this parameters, the calculated mean has value: E[C] = 0.4529 and the variance: V ar[C] = 0.8827. On the other hand, in order to have values to compare, we perform an empirical test generating samples, the empirical mean are 0.4525 and the variance of the samples are 0.9996. The difference are essentially due to the iterations needed to perform the integrations. With that the log-normal distribution are parameterized and the pdf is shown in the Fig. 2. The curve in blue is the pdf of the empirical random variable, made with many samples. The curve in magenta, is the log-normal pdf but parameterized with the empirical values of the mean and variances of the samples. And finally, the black curve shown is parameterized with the computed values using the theoretical pdf. It appears that the curves are quite different on the peak of the distributions, but the distribution tails are very close to each other.

Fig. 2: PDF of the capacity of a given scenario

The formula ( 16) can be used in different ways. It gives the distribution of the capacity offers to broadcast communications when the parameters are given. It can also be used to parameterize the system in order to achieve a certain quality of service: fix the maximum radio range R of the BS to ensure a capacity C out with a certain probability, etc. In the next section we show numerical results that can be found performing many realizations.

IV. NUMERICAL RESULTS

The simulations are performed under the assumption that the users are uniformly distributed over a ball of radius R = 200[m]. To aim a SNR close to 10dB we set the noise with a power of 1.0 × 10 -12 [W ]. And to aim a SNR close to 15dB we set the noise with a power of 1.0 × 10 -13 [W ]. We perform the simulations using our optimization algorithm for both cases and compare with the non optimized case. We set the P (C ≥ C out ) = 0.95 and we calculate C out . The target here is to check how the optimization criterion (sum capacity) matches the broadcast optimization criterion (outage capacity) . Also we see if the extrapolation by a known pdf are useful in order to establish a design criteria for the broadcast channel situation.

A. Numerical results at 10 dB Fig. 3 shows the results and present the followed concepts: At first glance we can see that the capacity gain in mean is weak. From the Fig. 3, we can see 20 % of increase in one user and 13 % when we have 2 users. The mean of the sub-set that is over 95 % of probability is equal with the one with all the samples. The mean of the optimized capacity shows weak improvement between one and eight users. The outage capacity of the samples is near 0.03 [bits/s/Hz]. We try to approximate the actual pdf by a log normal pdf. When we parameterize the log-normal pdf with the mean and the variance of the samples, the log-normal based empirical outage capacity is 0.0103[bits/s/Hz]. And when we parameterize with the mean and the variance obtained computing E[C] and V ar[C], the log-normal based computed capacity is 0.0127[bits/s/Hz].

On the other hand, we consider the percentage increase histograms (Fig. 4). It is computed for each sample as:

( Optimized Capacity Broadcast Capacity -1) • 100 (18)
Fig. 4 shows the distribution of this percentage increase. A percentage increase of 0% means that the optimized and broadcast capacity are the same (the ratio is 1). We can observe that for approximately 30% of the samples, there is a percentage increase superior to 100%, meaning that the optimized capacity is at least two times greater than the broadcast one. Also, it is worth noting that for one user the mean percentage increase is 72 %. The difference with the results shown in Fig. 3 may be explained by the fact that these improvements mainly concerns users with small capacity keeping mean capacity quite constant. The later can be appreciated comparing the histograms in Fig. 5, for one user. It clearly appears that the distribution of the capacity is different for the first part (approximately for capacity less than 0.5 bit/s/Hz), and similar for higher capacities. More precisely, for one user the optimized mean capacity is 0.55 bits/s/Hz, the broadcast mean capacity is 0.46 bits/s/Hz. If we compare the means we have 20 % of increase between the means. For 2, 3 , 4 and 5 users this comparison gives 13 %, 8.7 %, 8.8 % and 6.66 % respectively.

V. CONCLUSION

In this work we explore the capacity of the OFDM with new elements. We consider that the user are distributed uniformly over a ball with a fixed radius from the base station. Following the goal of characterizing the outage probability, we derive expressions for the mean and the variance of the capacity under the studied scenario. These two quantities could be used to extrapolate the capacity distribution. The derived expression are difficult to handle and finally we perform numerical simulations. From this distribution, the broadcast can be designed in order to ensure a certain quality of service. For example, the system can be tuned in order to ensure to a certain proportion of users and in a certain radio range a given capacity. Comparisons between the optimized and the classical broadcasts have shown that there is not a significant improvement in average. The mean capacity increases of 20 % for one user and decreases slowly with the number of users. But, it concerns users with small capacity that explains the poor improvement in average. However, this is an issue of practical importance, since the largest improvement is provided to the users with the smallest quality of service. A topic that could extend this work would consist in comparing the capacity improvement with different spatial distributions. Indeed, it appeared clearly that the distance between the users and the BS impacts the capacity gain of the optimized case. Spatial distributions where the users are far from the BS should greatly increase the benefit of an optimized allocation power, whereas distributions where the users are gathered near the BS should lead to cases where this optimization is almost useless. It could be particularly interesting to collect real users spatial distributions and real scenarios, to have better insight of the situations where multicast is really worth the additional complexity needed for the optimization.

  Fig. 1: Multipath Channel Distribution

  Let f (E[C], V ar[C], x), the pdf parameterized with the mean E[C] and the variance V ar[C]. To calculate the outage probability under the given scenario, we use the following formula:

•Fig. 3 :

 3 Fig. 3: Mean capacity, Outage capacity for 95 % and Mean over 95 % for SNR 10 dB

Fig. 4 :

 4 Fig. 4: Percentage increase at 10 dB

Fig. 5 :

 5 Fig. 5: Optimized and Broadcast Capacity, one user at 10 dB

ACKNOWLEDGMENT

The authors would like to thank to Dirección de Investigación e Innovación from Universidad Católica de la Santísima Concepción (INDIN Grant # 0317) for their support to this research.