Ionut ¸mironicȃ
email: imironica@alpha.imag.pub.ro

Bogdan Ionescu
email: bionescu@alpha.imag.pub.ro

Peter Knees
email: peter.knees@jku.at

Patrick Lambert
email: patrick.lambert@univ-savoie.fr

An In-Depth Evaluation of Multimodal Video Genre Categorization

In this paper we propose an in-depth evaluation of the performance of video descriptors to multimodal video genre categorization. We discuss the perspective of designing appropriate late fusion techniques that would enable to attain very high categorization accuracy, close to the one achieved with user-based text information. Evaluation is carried out in the context of the 2012 Video Genre Tagging Task of the MediaEval Benchmarking Initiative for Multimedia Evaluation, using a data set of up to 15.000 videos (3,200 hours of footage) and 26 video genre categories specific to web media. Results show that the proposed approach significantly improves genre categorization performance, outperforming other existing approaches. The main contribution of this paper is in the experimental part, several valuable interesting findings are reported that motivate further research on video genre classification.

I. INTRODUCTION

Searching for video content proves to be in many cases a tendinous task. This is mainly due to the lack of appropriate tools for handling, on one side the rich multimodal video contents, and on the other side the constantly growing large collections of video footage. To bridge this gap, existing media platforms (e.g., YouTube, Dailymotion, blip.tv) are still relying on user generated video categorization (e.g., based on user tags or metadata). The design of a system that would handle this task automatically is still an open challenge.

Machine learning techniques have been used extensively to solve various video categorization tasks as they can handle massive data, e.g., high descriptor dimensionality, tens of thousands of items. However, most of the approaches reported in the literature are limited to address very few categories, such as determining classic TV genres, e.g., action, comedy, horror, cartoon, sports. Best performance is reported with multimodal approaches that exploit the benefits of fusing various modalities: text, visual and audio.

From the modality point of view, the use of user generated textual information (e.g., synopsis, user tags, metadata) provides the most accurate results. The main drawback is that it cannot be generated automatically, which limits its applications. Text can also be obtained in an automated manner, either from scene text (e.g., graphic text, sub-titles), or from the transcripts of dialogs obtained with Automatic Speech Recognition (ASR) techniques [START_REF] Lamel | Speech Processing for Audio Indexing[END_REF]. Video footage however may contain different languages and also background noise that rends ASR highly inefficient. Video categorization using text is typically accomplished with classic Bag-of-Words model and Term Frequency-Inverse Document Frequency (TF-IDF) approaches.

Although reported as being less accurate than text, the use of audio-visual information is the most popular choice, mainly because it can be derived directly from the video footage itself. Audio-based information can be extracted from, both, time and frequency domains. Common time-domain approaches include the use of Root Mean Square of signal energy, subband information, Zero-Crossing Rate or silence ratio; while frequency-domain features include energy distribution, frequency centroid, bandwidth, pitch or Mel-Frequency Cepstral Coefficients -MFCC (see Yaafe audio features extraction toolbox http://yaafe.sourceforge.net/).

Visual-based information exploits both static and dynamic aspects either in the spatial domain using color, temporal structure, objects, feature points, motion, or in the compressed domain, e.g., using MPEG coefficients. Some of the most efficient approaches use feature points, e.g., Scale Invariant Feature Transform (SIFT) [START_REF] Uijlings | Real-Time Visual Concept Classification[END_REF], Space-Time Interest Points (STIP) [START_REF] Laptev | On space-time interest points[END_REF], Histogram of oriented Gradients (HoG), 3D-SIFT [START_REF] Reddy | Recognizing 50 Human Action Categories of Web Videos[END_REF], and Bag-of-Visual-Words representations [START_REF] Csurka | Visual Categorization with Bags of Keypoints[END_REF]. These methods are however known to be very computational expensive due to the computation of the visual word dictionaries.

In this paper we propose an in-depth evaluation of the performance of video descriptors to multimodal video genre categorization. We discuss the perspective of designing appropriate late fusion techniques that would enable to attain very high categorization accuracy, close to the one achieved with user-based text information.

The remainder of the paper is organized as follows. Section II discuses several relevant video genre categorization approaches and situates our work accordingly. The proposed multimodal descriptors and fusion strategies are presented in Section III and Section IV, respectively. Section V reports the experimental results. Finally, Section VI provides a brief summary and concludes the paper.

II. PREVIOUS WORK

Automatic video genre categorization has been studied extensively in the literature from now more than ten years [START_REF] Xu | Video classification using spatial-temporal features and PCA[END_REF]. Most of the work focuses on the categorization of movie genres, TV broadcasting [START_REF] Borth | Tubefiler: An automatic web video categorizer[END_REF] or online videos [START_REF] Yuan | Automatic video genre categorization using hierarchical SVM[END_REF]. Existing approaches range from exploiting single-modality to multimodal integration.

For instance, the approach in [START_REF] Song | Google challenge: Incremental-learning for web video categorization on robust semantic feature space[END_REF] uses only text-based information. It proposes an incremental Support Vector Machine (SVM) approach that makes use of online Wikipedia propagation to categorize large-scale web videos. It combines contextual and social information, such as metadata, user behavior, viewer's behavior and video relevance. A visual-based approach is the one in [START_REF] Wu | Efficient Genre-Specific Semantic Video Indexing[END_REF] that proposes a framework for distinguishing from two different level of concepts: a genre-specific concept classification layer and a frame-concept detection module. For genre classification, visual content is described using Bag-of-Visual-Words representation of Opponent SIFTs that are classified with a probabilistic model. [START_REF] Borth | Tubefiler: An automatic web video categorizer[END_REF] proposes the use of text and visual information for web video categorization. A genre-based categorization is first achieved using video tags and title, while sub-genres are further determined using visual features. Genre classification is addressed at different levels, according to a hierarchical ontology of genres.

A multimodal approach that considers also audio information is proposed in [START_REF] Xu | Video classification using spatial-temporal features and PCA[END_REF]. It combines synchronized audio MFCCs features and mean and standard deviation of motion vectors and MPEG-7 visual descriptors. Videos are classified with a Gaussian Mixture Models (GMM) based classifier. Another example is the approach in [START_REF] Montagnuolo | Parallel Neural Networks for Multimodal Video Genre Classification[END_REF] that extracts features from four information sources: visual-perceptual information (color, texture, and motion), structural information (shot length, shot distribution, shot rhythm, shot clusters duration, and saturation), cognitive information (e.g., numbers, positions, and dimensions of faces), and aural information (transcribed text, sound characteristics). These features are used to train a parallel Neural Network system and used to distinguish between seven TV genres. An interesting categorization approach is the cross-media retrieval proposed in [START_REF] Yang | Ranking with Local Regression and Global Alignment for Cross Media Retrieval[END_REF]. It is founded on the construction of a Multimedia Correlation Space (MMCS) which exploits semantic correlations between different modalities based on content description and cooccurrence information. The proposed video descriptors are related to color, texture and aural information.

Systems that combine multiple features using various fusion techniques have been proved to improve performance in various classification tasks [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF] [START_REF] Csurka | An empirical study of fusion operators for multimodal image retrieval[END_REF]. However, for video genre categorization most of the existing approaches are limited to make use of simple descriptor concatenation.

More recently the Genre Tagging Task of the MediaEval Benchmarking Initiative for Multimedia Evaluation [START_REF] Schmiedeke | Overview of the MediaEval 2012 Tagging Task[END_REF] set up a new perspective for benchmarking video genre categorization by addressing, both, large scale categorization (data set of up to 15,000 Internet videos) and multimodal approaches (textaudio-visual). It targets a real-world scenario, i.e., the video genre categorization employed by the blip.tv video platform.

In this paper we present an in-depth evaluation study of multimodal video genre categorization. We investigate the use of various state-of-the-art content descriptors extracted from different modalities and the efficiency of early and late fusion mechanisms to this task. With this work we attempt to respond to several research questions, such as to what extent aural and visual information can lead to similar performance or even surpass the highly semantic textual descriptors? How efficient would be an adequate combination of various modalities in achieving highly accurate classification? How really important is the contribution of video modalities in improving the accuracy of textual data. Experimentation is carried out in the context of the 2012 Genre Tagging Task of the MediaEval Benchmarking [START_REF] Schmiedeke | Overview of the MediaEval 2012 Tagging Task[END_REF].

Although the proposed methods have been more or less previously explored in the literature, the main contributions of this work are the following: (1) we provide an in-depth evaluation of truly multimodal video description (automated and user-generated text, audio and visual) in the context of a real-world genre-categorization scenario; (2) we demonstrate the potential of appropriate late fusion to genre categorization and achieve very high categorization performance; (3) we prove that notwithstanding the superiority of user-text based descriptors, late fusion can boost performance of automated content descriptors to achieve close performance; (4) we setup a new baseline for the Genre Tagging Task by outperforming the performance of the other participants; (5) evaluation is carried out on a standard data set [START_REF] Schmiedeke | Overview of the MediaEval 2012 Tagging Task[END_REF] making the results both relevant and reproducible.

III. CONTENT DESCRIPTION

It is well know that different modalities tend to account for different information providing complementary discriminative power. We experiment all the available sources of information, from the audio, visual, to highly semantic textual information obtained with Automatic Speech Recognition (ASR) as well as user generated data (e.g., metadata that typically accompanies video content on the Internet).

Aural information. Common video genres have very specific audio signatures, e.g., documentaries use a mixture of natural sounds and monologues, music clips contain different music genres, sports feature crowd noise and also monologues, in talk shows speech is predominant. To describe these aspects, we employ a set of standard audio features that provided good performance in audio genre categorization tasks [START_REF] Xu | Video classification using spatial-temporal features and PCA[END_REF]:

-standard audio features (196 values) [START_REF]Yaafe core features[END_REF] contains a common set of general-purpose audio descriptors, namely: Linear Predictive Coefficients, Line Spectral Pairs, MFCCs, Zero-Crossing Rate, spectral centroid, flux, rolloff and kurtosis, augmented with the variance of each feature over a certain window (a common setup for capturing enough local context is taking 1.28 s). Video temporal integration is achieved by taking the mean and standard deviation of these descriptors over all frames.

Visual information. From the visual point of view, the distribution of color and feature/object information highlight specific genre contents. For instance, music clips and movies tend to have darker colors, commercials use many visual effects, sports usually have a predominant hue, newscasts show a high frequency of faces, documentaries, sports and news have specific contour signatures, e.g., skyline contours, people silhouettes. To capture those particularities, we determine the following state-of-the-art descriptors that are classic approaches successfully employed in various image/video retrieval scenarios [START_REF] Over | TRECVID 2012 -An Overview of the Goals, Tasks, Data, Evaluation Mechanisms and Metrics[END_REF] [14] [START_REF] Nowak | New strategies for image annotation: Overview of the photo annotation task at ImageClef 2010[END_REF]:

-MPEG-7 related descriptors (1,009 values) [START_REF] Sikora | The MPEG-7 Visual Standard for Content Description -An Overview[END_REF] describe the global color and texture information over all the frames. We selected the following representative descriptors: Local Binary Pattern, autocorrelogram, Color Coherence Vector, Color Layout Pattern, Edge Histogram, Scalable Color Descriptor, classic color histogram and color moments. For each sequence, we aggregate the features by taking the mean, dispersion, skewness, kurtosis, median and root mean square statistics over all frames.

-structural descriptors (1,430 values) account for contour information and relation between contours. We use the approach in [START_REF] Rasche | An Approach to the Parameterization of Structure for Fast Categorization[END_REF] to parameterize the geometry and appearance of contours and regions of gray-scale images with the goal of object categorization. For each contour, a type of curvature space is created. This space is then abstracted into spectra-like functions, from which in turn a number of geometric attributes are derived, e.g., the degree of curvature, angularity, circularity, symmetry, "wiggliness".

-global Histograms of oriented Gradients (HoG -81 values) represent the average of the well known HoG features [START_REF] Ludwig | Trainable Classifier-Fusion Schemes: An Application To Pedestrian Detection[END_REF]. It exploits local object appearance and shape within an image via the distribution of edge orientations. This can be achieved by dividing the image into small connected regions (cells) and for each of them building a pixel-wise histogram of edge orientations. The combination of these histograms represent the descriptor which has the advantage of providing contextual information. For the entire sequence we compute the average histogram over all frames.

-Bag-of-VisualWords of SIFT descriptors (20,480 values), we compute a Bag-of-VisualWords (B-o-VW) model over a selection of key frames (uniformly sampled). For this task, we extract a visual vocabulary of 4,096 words. The keypoints are extracted with a dense sampling strategy and described using rgbSIFT features [START_REF] Uijlings | Real-Time Visual Concept Classification[END_REF]. Descriptors are extracted at two different spatial scales of a spatial pyramidal image representation (entire image and quadrants).

Textual information. Textual data is by far the most representative for providing content information. Specific keywords, e.g., "religion", "economy", "music", can reveal meaningful information about genres. For instance, metadata usually contains the video title, user tags, comments and content descriptions that are highly correlated to genre concepts. For text description we adapted a classic Term Frequency-Inverse Document Frequency (TF-IDF) approach. First, we filter the input text by removing the terms with a document frequency less than 5%-percentile of the frequency distribution. We reduce further the term space by keeping only those terms that discriminate best between genres according to the χ 2test. We generate a global list by retaining for each genre class, the m terms with the highest χ 2 values that occur more frequently than in complement classes. This results in a vector representation for each video sequence that is subsequently cosine normalized to remove the influence of the length of text data. We consider following TF-IDF descriptors:

-TF-IDF of ASR data [START_REF] Csurka | Visual Categorization with Bags of Keypoints[END_REF]466 values, m = 150) describes textual data obtained from Automatic Speech Recognition of the audio signal. For ASR we use the transcripts provided by [START_REF] Lamel | Speech Processing for Audio Indexing[END_REF] that proved highly efficient to genre classification [START_REF] Schmiedeke | Overview of the MediaEval 2012 Tagging Task[END_REF].

-TF-IDF of metadata (504 values, m = 20) describes textual data obtained from user metadata such as synopsis, user tags, video title, information that typically accompanies videos posted on the blip.tv platform.

IV. MULTIMODAL INTEGRATION

To combine various information sources one has to deploy efficient fusion strategies. In general, there are two approaches: early fusion and late fusion [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF]. These strategies are based on the hypothesis that an aggregated decision from multiple classifiers can be superior to a decision from a single one.

The early fusion combines the descriptors before the classification. The combination takes place in the feature space, namely the features are concatenated into one vector. The major drawback in this case is the dimensionality of the resulting feature space that is basically the sum of all the concatenated dimensions. High-dimensional spaces tend to scatter the homogeneous clusters of instances belonging to the same concepts reducing the performance.

In contrast, late fusion combines the confidence values from classifiers run on different descriptors. In our scenario, a classifier is supposed to provide some relevance scores indicating the probability of each sequence of belonging to some class (i.e., genre in our case). Naturally, each of the classifiers will tend to provide different scores for each class. Late fusion involves the design of an aggregated classifier combination function, f (x 1 , ..., x N ), with x i the relevance output of the classifier i, whose result is better than any of its individual classifiers and as good as possible. The aggregation is carried out for each individual class. To achieve the final categorization, videos are then sorted according to the new aggregated scoring formula results.

Late fusion focuses on the individual strength of modalities, whereas early fusion use the correlation of features in the mixed feature space. There are many strategies for late fusion, such as weighted majority voting, rank-level fusion methods or fusion techniques that uses second tier classifiers [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF] [30] [START_REF] Over | TRECVID 2012 -An Overview of the Goals, Tasks, Data, Evaluation Mechanisms and Metrics[END_REF]. However, there is no reported supremacy of one approach over the other, results show that typically the fusion mechanism is adapted to the specificity of each task [START_REF] Lan | Double Fusion for Multimedia Event Detection[END_REF].

For our study we have selected several popular approaches that are presented in the following. For each video document and feature pair, classification yields C confidence scores, one for each of the target genres. Simple linear combination represents a weighted average of the multimodal confidence values of each of the considered classifiers and is defined as:

CombM ean(d, q) = N i=1 α i • cv i (1)
where cv i is the confidence value of classifier i for class q (q ∈ {1, ..., C}), d is the current video, α i are some weights and N is the number of classifiers to be aggregated. In case of considering equal weights, α 1 =, ..., = α N , this is referred to as CombSum.

An extension of CombMean can be obtained by giving more importance to the documents that are more likely to be relevant for current concepts, which leads to:

CombM N Z(d, q) = F (d) γ • N i=1 α i • cv i (2)
where F (d) is the number of classifiers for which video d appears in the top K of the retrieved videos and γ ∈ [0, 1] is a parameter.

Finally, another useful perspective is to consider the rank of each confidence level. The score-based late fusion strategies require a normalization among all confidence values in order to balance the importance of each of them, which is not the case of the rank-based strategies. In our scenario, we use a common method for rank-based fusion, that is Borda Count. The document with the highest rank on each rank-list gets n votes, where n is the size of the dataset:

CombRank(d, q) = N i=1 α i • rank(cv i ) (3)
where rank() represents the rank of classifier i, α i are some weights and N is the number of classifiers to be aggregated.

V. EXPERIMENTAL RESULTS

Experimentation was conducted in the context of the Me-diaEval Benchmarking Initiative for Multimedia Evaluation, 2012 Genre Tagging Task [START_REF] Schmiedeke | Overview of the MediaEval 2012 Tagging Task[END_REF].

The data set consisted of up to 14,838 blip.tv videos that were divided into a training set of 5,288 videos (36%) and a test set of 9,550 movies (64%; we use the same scenario as for the official benchmark). Videos are labeled according to 26 video genre categories specific to the blip.tv media platform, namely (the numbers in brackets are the total number of videos): art (530), autos and vehicles [START_REF] Almeida | UNICAMP-UFMG at MediaEval 2012: Genre Tagging Task[END_REF], business (281), citizen journalism (401), comedy (515), conferences and other events (247), documentary (353), educational (957), food and drink (261), gaming (401), health (268), literature (222), movies and television (868), music and entertainment (1148), personal or auto-biographical (165), politics (1107), religion (868), school and education (171), sports (672), technology (1343), environment (188), mainstream media (324), travel (175), video blogging (887), web development an (116) and default category (2349, comprises movies that cannot be assigned to any of the previous categories). The main challenge of this scenario is in the high diversity of genres, as well as in the high variety of visual contents within each genre category (for more details see [12] [15]).

For classification, we have selected five of the most popular approaches that proved to provide high performance in various information retrieval tasks [START_REF] Schmiedeke | Overview of the MediaEval 2012 Tagging Task[END_REF] [13] [START_REF] Ionescu | An Audio-Visual Approach to Web Video Categorization[END_REF] [26] [START_REF] Marszaek | Learning representations for visual object class recognition[END_REF], namely Support Vector Machines (SVM, with various kernel functions: linear, Chi-square -CHI, Radial Basis Functions -RBF), k-Nearest Neighbor (k-NN), Random Trees (RT) and Extremely Random Forest (ERF).

To assess performance, we report the standard Mean Average Precision (MAP) that is computed as the average value of the Average Precision:

AP = 1 m n k=1 f c (v k ) k ( 4
)
where n is the number of videos, m is the number of videos of genre c, and v k is the k-th video in the ranked list {v 1 , ..., v n }.

Finally, f c () is a function which returns the number of videos of genre c in the first k videos if v k is of genre c and 0 otherwise (we used the trec eval scoring tool available at http: //trec.nist.gov/trec eval/).

A. Performance assessment of individual modalities

The first experiment consisted on assessing the discriminative power of each individual modality and group of descriptors. Table I summarizes some of the results (the best performance per modality is highlighted in bold).

The highest performance for visual information is achieved using MPEG-7 related descriptors and Extremely Random Forest (ERF) classifiers, MAP 26.17%, followed closely by HoG histograms on SVM and RBF kernel, MAP 25.63%. Surprisingly, Bag-of-Visual-Words representation of feature information (rgbSIFT) is not performing efficiently to this task, MAP is below 20%. The audio descriptors are able to provide a significantly higher discriminative power, the highest MAP of 42.33% being achieved with the ERF classifier.

In what concerns the text modality, the use of metadata and Random Forest classifiers led to the highest MAP of 58.66% which is an improvement of more than 16% over the audio. The use of ASR data alone is able only to provide a MAP up to 35.05% (with SVM and RBF kernel), which is less discriminative than using audio descriptors. Therefore, video descriptors can outperform at this point the automated text descriptors. This is mostly due to the fact that ASR data is extracted automatically, being inherently subject to errors (e.g., due to background noise).

From the classifier point of view, regardless the modality, the lowest performance tends to be obtained with SVM Linear and 5-NN classifiers. This proves that these video features have restraint linear separability and solving the genre classification problem will require more complex nonlinear classification schemes. In the following we investigate the advantage of combining different modalities as well as the impact of the fusion scheme. In all of the cases, late fusion tends to provide better performance than early fusion. Using only the visual descriptors the improvement is of more then 8% over simple descriptor concatenation (highest MAP is 38.21% using CombSUM). For audio descriptors, highest MAP of 44.5% is achieved with CombMNZ, that is an improvement of more than 2% over the simple use of all descriptors together. Audio still provides significant superior discriminative power than using only visual.

A significant improvement of performance is also achieved for textual descriptors. We obtain the highest MAP score with CombMean, namely 62.81%, which is an improvement of over 7% compared to early fusion. Although the simple concatenation of modalities manages to boost classification performance up to a MAP of 60.12%, late fusion is able to exploit better the complementarity between descriptors, achieving more than 5% of improvement. In what concerns the late fusion techniques, CombRank tends to provide the least accurate results in most of the cases, while the other approaches tend to provide more or less similar results. Therefore, in most of the cases late fusion proves to be a better choice for multimodal genre classification. Firstly, it provides significantly higher performance than early fusion. Secondly, late fusion is also less computational expensive than early fusion, because the descriptors used for each of the classifiers are shorter than using the concatenation of all features. Finally, late fusion systems scale up easier because no re-training is necessary if further streams or modalities are to be integrated.

C. Comparison to state-of-the-art

The final experiment consisted on comparing the late fusion strategies against other methods from the literature. As reference, we use the best team runs reported at MediaEval 2012 Video Genre Tagging Task [START_REF] Schmiedeke | Overview of the MediaEval 2012 Tagging Task[END_REF]. Results are presented in Table II by decreasing MAP values (one should note that the comparison with MediaEval results is indicative as the official runs were developed under time constraints and without a priori knowledge of the test set ground truth).

The most efficient modality remains the exploitation of textual information as it provides a higher semantic level of description than audio-visual information. In particular, the use of metadata proves to be the most efficient approach leading to the highest MAP at MediaEval 2012, 52.25% (see team TUB [START_REF] Schmiedeke | TUB @ MediaEval 2012 Tagging Task: Feature Selection Methods for Bag-of-(visual)-Words Approaches[END_REF]). In spite of this high classification rate, late fusion still allows for significant improvement, for instance CombMean on ASR and metadata achieves a MAP up to 62.81% -that is an improvement of more than 10% over the best run at MediaEval 2012 and of around 25% over using the same combination of textual descriptors (team ARF [START_REF] Ionescu | ARF @ MediaEval 2012: Multimodal Video Classification[END_REF]).

In what concerns the visual modality, best MAP at Medi-aEval 2012 is up to 35% (see team KIT [START_REF] Semela | KIT at MediaEval 2012 -Content-based Genre Classification with Visual Cues[END_REF]) and is obtained using a combination of classical color/texture descriptors (e.g., HSV color histogram, L*a*b* color moments, autocorrelogram, concurrence texture, wavelet texture grid and edge histograms) and B-o-VW of rgbSIFT descriptors. Results show that using only B-o-VW of feature descriptors (e.g., SIFT, SURF -Speeded-up Robust Features), in spite of their reported high performance in many retrieval tasks, is not that accurate, e.g., MAP 23.29% using SIFT, 23.01% with SURF-PCA (see detailed competition results at [START_REF] Schmiedeke | Overview of the MediaEval 2012 Tagging Task[END_REF] [16] [START_REF] Semela | KIT at MediaEval 2012 -Content-based Genre Classification with Visual Cues[END_REF]). The CombMean late fusion of visual descriptors provides an improvement over the best run of more than 3% (MAP 38.21%).

Using only audio information, best reported run at MediaEval 2012 achieves a MAP of 18.92% (see team ARF [START_REF] Ionescu | ARF @ MediaEval 2012: Multimodal Video Classification[END_REF]). In this case CombMean late fusion of audio descriptors provides an improvement of more than 25% (MAP 44.5%).

Combining all the descriptors with CombMNZ we achieve a very high classification accuracy as MAP is up to 65.82%, that is an improvement of more than 13% over the MediaEval 2012 best run. In spite of the high discriminative power of textual descriptors, the combination of all the modalities with late fusion is able to exploit data complementarity at some level as the improvement over using only textual information is of 3%. This is a significant achievement considering the scale of the data set.

From the modality point of view, metadata provides the highest discriminative power for genre categorization. However, one should note that this information is user generated (e.g., includes document title, tags and user comments and descriptions) and cannot be determined automatically from the video information, that limits its applicability in real-time categorization scenarios. Approaching the classification using only content information that can be computed automatically from video data (ASR and audio-visual descriptors), late fusion is still able to provide high classification performance leading to a MAP of 51.9%, surpassing even some metadata-based approaches, e.g., see team ARF [START_REF] Ionescu | ARF @ MediaEval 2012: Multimodal Video Classification[END_REF] and TUD-MM [START_REF] Shi | MediaEval 2012 Tagging Task: Prediction based on One Best List and Confusion Networks[END_REF].

VI. CONCLUSIONS

In this paper we addressed the problem of automatic video genre categorization. We studied the contribution of various modalities and the role of the fusion mechanisms in increasing the accuracy of the results. The study was carried out in a realworld scenario on 26 blip.tv web video categories and more than 3,200 hours of video footage. The design of appropriate descriptors and late fusion integration allows to achieve a MAP up to 65.8%, that is a significant improvement of more than 13% over the best approach reported at the 2012 MediaEval Genre Tagging Task. We prove that notwithstanding the superiority of employing user-generated textual information (e.g., user tags, metadata), the proposed multimodal integration allows to boost performance of automated content descriptors to achieve close performance. Future work will mainly consist in exploring spatio-temporal data representation in this context.

Table I .

 I CLASSIFICATION PERFORMANCE OF INDIVIDUAL MODALITIES (MAP).

	Descriptors	SVM Linear	SVM RBF	SVM CHI	5-NN	RF	ERF
	HoG	9.08 %	25.63%	22.44%	17.92%	16.62%	23.44%
	Bag-of-Visual-Words rgbSIFT	14.63 %	17.61%	19.96%	8.55%	14.89%	16.32%
	MPEG-7	6.12 %	4.26%	17.49%	9.61%	20.90%	26.17%
	Structural descriptors	7.55 %	17.17%	22.76%	8.65%	13.85%	14.85%
	Standard audio descriptors	20.68 %	24.52%	35.56%	18.31%	34.41%	42.33%
	TF-IDF of ASR	32.96 %	35.05%	28.85%	12.96%	30.56%	27.93%
	TF-IDF of metadata	56.33 %	58.14%	47.95%	57.19%	58.66%	57.52%

Table II .

 II COMPARISON WITH MEDIAEVAL 2012 VIDEO GENRE TAGGING TASK BEST RUNS[START_REF] Schmiedeke | Overview of the MediaEval 2012 Tagging Task[END_REF] (MAP) Fusion techniques tend to exploit complementarity among different information sources. In this experiment we assess the performance of various combination of modalities as well as of different fusion strategies, from late fusion schemes (see Section IV) to the simple concatenation of different descriptors (i.e., early fusion). For late fusion, weights (i.e., α k and F (d) values) are first estimated on the training set and tuned for best performance. To avoid overfitting, half of the training set is used for training and the other half for parameter evaluation. The actual classification is then carried out on the test set. MAP is reported in Table III (highest values per feature type are presented in bold).

	Team		Modality	Method		MAP
	proposed		all		Late Fusion CombMNZ with all descriptors	65.82%
	proposed		text		Late Fusion CombMean with TF-IDF of ASR and metadata	62.81%
	TUB [16]		text		Naive Bayes with Bag of Words on text (metadata)	52.25%
	proposed		all		Late Fusion CombMNZ with all descriptors except for metadata	51.9%
	proposed		audio		Late Fusion CombMean with standard audio descriptors	44.50%
	proposed		visual		Late Fusion CombMean with MPEG-7 related, structural, HoG and B-o-VW with rgbSIFT	38.21%
	ARF [17]		text		SVM linear on early fusion of TF-IDF of ASR and metadata	37.93%
	TUD [20]		visual & text	Late Fusion of SVM with B-o-W (visual word, ASR & metadata)	36.75%
	KIT [19]		visual		SVM with Visual descriptors (color, texture, B-o-VW with rgbSIFT)	35.81%
	TUD-MM [18]	text		Dynamic Bayesian networks on text (ASR & metadata)	25.00%
	UNICAMP -UFMG [21]	visual		Late fusion (KNN, Naive Bayes, SVM, Random Forests) with BOW (text ASR)	21.12%
	ARF [17]		audio		SVM linear with block-based audio features	18.92%
	B. Performance of multimodal integration
	Table III.	PERFORMANCE OF MULTIMODAL INTEGRATION (MAP).
	Descriptors	Comb	Comb	Comb	Comb	Early
		SUM	Mean	MNZ	Rank	Fusion
	all visual	35.82%	36.76%	38.21% 30.90%	30.11%
	all audio	43.86%	44.19%	44.50% 41.81%	42.33%
	all text	62.62%	62.81% 62.69%	50.60%	55.68%
	all	64.24%	65.61%	65.82% 53.84%	60.12%

VII. ACKNOWLEDGMENTS

Part of this work was carried out by Ionut ¸Mironicȃ during his research stage at the MHUG group at University of Trento, Italy. We would like to thank Prof. Nicu Sebe and Dr. Jasper Uijlings for the very constructive discussions. Also, Bogdan Ionescu's contribution was supported by the research grant EXCEL POSDRU/89/1.5/S/62557. Finally, we acknowledge the 2012 Genre Tagging Task of the MediaEval Multimedia Benchmark for the data set (http://www.multimediaeval.org/).


Martina Cardone
email: cardone@eurecom.fr

D A N I E L At U N I N E T T I *

Approximate Sum-Capacity of Full-and Half-Duplex Asymmetric Interference Channels with Unilateral Source Cooperation

This paper focuses on the causal cognitive radio channel.Thism o de lc o ns is tso fatw o -us e rG a us s ia ninte r fe r e nc e channel where a cooperative secondary/cognitive source can causally learn the message of a primary source through a noisy link. The generalized Degrees-of-Freedom (gDoF) and the sumcapacity to within a constant gap are characterized for channels where one destination does not experience interference, referred to as interference-asymmetric channels. Both cases where the cognitive source operates in full-and half-duplex mode are considered. The different models are compared among themselves and with the case of interference-symmetric channels with either bilateral or unilateral source cooperation. In particular, the parameter regimes where causal cognition, or unilateral cooperation, attains the same gDoF of classical channel models, such as the noncooperative interference channel and the noncausal cognitive channel, are identified in order to determine when causal cognitive radio offers benefits in practical systems.

I. INTRODUCTION

In this work we consider the Gaussian Interference Channel (IC) with unilateral source cooperation, which consists of two source-destination pairs that communicate over a shared wireless channel. In this model, each pair aims to reliably communicate at a certain rate and the two transmissions interfere with each other. Moreover, a primary user/user 1 is aided by a secondary user/user 2 to convey data to its intended destination. As opposed to the classical cognitive channel [START_REF] Devroye | Achievable rates inc o g n i t i v e radio channels[END_REF], where the cognitive source is assumed to have full non-causal aprioriknowledgeoftheprimary' smessage,hereforpractical reasons we impose that the secondary user learns the primary's message causally over a noisy link. This channel model is as p e c i a lc a s eo ft h eI Cw i t hg e n e r a l i z e df e e d b a c k [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF] , [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF] , [START_REF] Prabhakaran | Interference channelsw i t hs o u r c e cooperation[END_REF], or source cooperation, where only one source engages in cooperation. Understanding the fundamental performance limits of this channel is critical to the deployment of future wireless cognitive networks as it indicates whether substantial gains are possible over the noncooperative case and whether the performance of the idealistic non-causal cognitive channel can indeed be attained.

The interference-symmetric IC with source cooperation, wherein both destinations experience interference, has been widely studied in the literature [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF], [START_REF] Prabhakaran | Interference channelsw i t hs o u r c e cooperation[END_REF], [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF], [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF]. In this work we study the interference-asymmetric IC, where either the link from source 1 to destination 2 is nonexistent (Zchannel) or the link from source 2 to destination 1 is nonexistent (S-channel). Due to the asymmetry in the cooperation, the Z-channel and the S-channel have different capacities and we analyze them separately. In practice, interference-asymmetric models capture scenarios where one destination is in the coverage area of only one of the sources, see Fig. 1.

For the noncooperative Z-channel much more is known compared to the general interference-symmetric IC. For example, the sum-capacity of the Z-channel is known for all channel parameters [START_REF] Sason | On achievable rate regions for the gaussian interference channel[END_REF], while for the general interference-symmetric IC the sum-capacity is still an open problem in the weak interference regime [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]. Moreover, the sum-capacity achieving scheme for the Z-channel does not require rate splitting, as opposed to the general interference-symmetric IC. Finally,the sum-capacity of the Z-channel is larger than that of the general interference-symmetric IC because of the reduced interference at one destination. In this work we consider unilateral source cooperation and we aim to understand whether the absence of interference at one destination improves performance or simplifies the achievable schemes compared to the interferencesymmetric IC. Our result sheds light into the ultimate limits of practical cognitive radio channels as a function of the network topology and might guide the node placement in future systems.

A. Past Work

The presence of communication links between the sources enables cooperation. An IC with unilateral cooperation, or causal cognitive radio, is a special case of the general IC with bilateral source cooperation, or generalized feedback, where one cooperation link is nonexistent. The capacity of the IC with unilateral cooperation is lower bounded by that of the noncooperative IC [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] and upper bounded by that of the noncausal cognitive IC [START_REF] Rini | Inner and outer bounds for the gaussian cognitive interference channel and new capacity results[END_REF], both known to within 1 bit.

Two modes of operation for the cooperative cognitive source are considered: Full-Duplex (FD) and Half-Duplex (HD). A node operates in FD mode if it can receive and transmit over the same time-frequency resource, and in HD mode otherwise.

The Full-Duplex Interference-Symmetric Channel. Host-Madsen first studied outer and inner bounds for the Gaussian IC with either source or destination cooperation [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF]. A novel sum-rate outer bound for the "semi-deterministic injective" IC with source cooperation was developed in [4, eq.( 7)] as an extension of [10, eq.( 3)]. A novel sum-rate outer bound for the general memoryless IC with source cooperation appeared in [11, eqs.(4d)-(4e)] as an extension of [12, eq.( 19)]. The largest known achievable region for the general IC with source cooperation, to the best of our knowledge, is [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF]. With these bounds, the sum-capacity of the FD Gaussian IC with bilateral symmetric cooperation (i.e., by channel reciprocity the cooperation links between the sources are assumed to have the same strength) but with otherwise generic direct and cross/interference links, was characterized to within 20 bits in [START_REF] Prabhakaran | Interference channelsw i t hs o u r c e cooperation[END_REF]. The gap was reduced to 4 bits in [START_REF] Shuang | On interference channels with generalized feedback[END_REF] for the case of symmetric cooperation and symmetric interference links. Since causal cognition/unilateral FD cooperation assumes that one cooperation link is absent, capacity results to within ac o n s t a n tg a pf o ru n i l a t e r a lF Dc o o p e r a t i o na r en o ti m p l i e d by the gap result in [START_REF] Prabhakaran | Interference channelsw i t hs o u r c e cooperation[END_REF]. However, bounds valid for a general memoryless IC with bilateral FD source cooperation can be straightforwardly adapted to the case of unilateral FD cooperation. We followed this approach in [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF] and characterized the sum-capacity to within 7.3 bits for the interference-symmetric case (model depicted on the left in Fig. 1).

Cognitive channels were also studied in [START_REF] Mirmohseni | On the capacity of interference channel with causal and noncausal generalized feedback at the cognitive transmitter[END_REF]; here the authors analyzed the causal cognitive IC with delay. The zerodelay case coincides with the causal cognitive model studiedin this work; the main contribution is a capacity characterization for the degraded semi-deterministic channel in the strong interference regime.

The Half-Duplex Interference-Symmetric Channel. The case where nodes operate in HD mode can be analyzed within the framework of 'general memoryless IC with bilateral FD source cooperation' by introducing a state random variable that indicates whether a HD node is in receive-or in transmit-mode [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF]. In [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF] it was pointed out that higher rates can be achieved by considering random switch at the HD nodes compared to the case of deterministic switch.H e r e , deterministic switch refers to the case where the time and duration of the receive and transmit phases of each HD node is fixed a priori and known to all other nodes; when this is not the case, we say that a HD node uses a random switch to convey information to a destination (through the random times and durations of the receive and transmit phases). To the best of our knowledge, past work on the IC with HD cooperation has only considered the case of deterministic switch. A contributionof this work is to consider cooperative ICs with HD nodes that use random switch and quantify how much information can be carried by a random switch.

The case of HD source cooperation has received less attention compared to the FD case. In [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF], the authors studied the IC with HD bilateral and unilateral source cooperation and with deterministic switch. For HD bilateral cooperation, the sum-capacity of the Gaussian noise channel with symmetric cooperation, direct and cross links was characterized to within 17 + 3 = 20 bits [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.1]. For HD unilateral cooperation, the "cognitive capacity" (which is a sum-capacity achieving point that does not necessarily correspond to a corner point of the capacity region) with generic direct and cross links was characterized to within 23 + 2 × 7=37bits [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.2]. In both cases, the faction of time the cooperative source listens to the channel and the gDoF were not given in closed form but as a solution of a linear program. In this paper we will extend the work of [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF] by considering random switch and by analytically finding the gDoF in closed form.

Unilateral source cooperation was also considered in [START_REF] Wu | Partial decode-forward binning schemesf o rt h e causal cognitive relay channels[END_REF], [START_REF] Seyedmehdi | An improveda c h i e v a b l e rate region for causal cognitive radio[END_REF] where the authors studied FD and/or HD cooperation with deterministic switch by developing several achievable schemes and evaluating them in Gaussian noise. However, capacity guarantees in terms of constant gap results were not given.

The Full-Duplex Interference-Asymmetric Channel. In this work, we refer to an IC in which one destination does not experience interference as an interference-asymmetric channel. In the literature, this channel model is usually known as the Z-channel. For the Gaussian IC with symmetric bilateral FD cooperation (Fig. 1, left), the sum-capacity to within 20 bits follows from [START_REF] Prabhakaran | Interference channelsw i t hs o u r c e cooperation[END_REF], which considered symmetric cooperation links but generic direct and cross links. For unilateral cooperation, two interference-asymmetric scenarios are possible, as shown in Fig. 1. To distinguish among them, we refer to them as the Z-(Fig. 1, middle) and the S-channel (Fig. 1, left).

To the best of our knowledge the Gaussian FD Z-channel has not been considered in the literature. The Gaussian FD S-channel was studied in [START_REF] Cao | The cognitive z channel[END_REF] where a capacity corner point was determined for the case of weak cooperation and weak interference, but no 'capacity approximations' were given in other regimes. In this work we characterize the gDoF and the sum-capacity to within a constant gap for both the Z-and S-channel with unilateral FD cooperation.

The Half-Duplex Interference-Asymmetric Channel. For the Gaussian IC with unilateral HD cooperation with deterministic switch, the sum-capacity to within 20 bits follows from [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.2] for both the Z-and the S-channel, since [5, Thm.3.2] considered generic direct and cross links. However, [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.2] did not give closed-form expression for the gDoF and only considered deterministic switch. In this work we characterize the gDoF and the sum-capacity to within a constant gap for both the Z-and S-channel with unilateral HD cooperation and random switch by giving a closed-form expression for the gDoF and by substantially reducing the gap to no more than 5 bits.

B. Contributions and Paper Outline

In Section II the two-user Gaussian IC (G-IC) with source cooperation, where the cooperative source can operate either in FD or in HD mode, is introduced. The HD constraint is incorporated into the framework of memoryless FD framework by following [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF]. In Section III we summarize known results for the interference-symmetric G-IC with source cooperation. In particular, in Section III-B we extend [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.1] to the case of random switch and provide a closed form expression for the gDoF. In Section IV we derive in closed-form the gDoF and characterize the sum-capacity to within a constant gap for the interference-asymmetric G-IC with unilateral source cooperation. Both the Z-and the S-channel are considered, as well as, FD and HD operation modes for the cognitive source. For HD cooperation, we extend the work of [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.2] to random switch and we reduce the gap. We also highlight when the ultimate performance of the considered models in terms of gDoF is the same as that of the noncooperative IC, or the Relay Channel, or the non-causal cognitive IC, and systematically discuss the impact of the network topology (i.e., differences between the interference-symmetric andt h e interference-asymmetric scenarios) on the gDoF.

II. SYSTEM MODEL

A. The general memoryless interference channel with bilateral full-duplex source cooperation

An IC with source cooperation consists of two input alphabets (X 1 , X 2 ),f o u ro u t p u ta l p h a b e t s(Y 1 , Y 2 , Y 3 , Y 4 ) and a memoryless transition probability P Y1,Y2,Y3,Y4|X1,X2 . Source j/node j, j ∈ {1, 2},h a sam e s s a g eW j ∈ [1 : 2 NRj ] for destination j/node (j +2),w he reN denotes the codeword length and R j ≥ 0 the transmission rate measured in bits per channel use. The messages W 1 and W 2 are independent and uniformly distributed on their respective domains. At time t ∈ [1 : N ],s o u r c ej, j ∈ {1, 2},s e n d sX j,t (W j ,Y t-1 j ).A t time N ,d e s t i n a t i o nj, j ∈ {1, 2},m a k e sa ne s t i m a t eo fi t s intended message as W j (Y N j+2 ).T h ec a p a c i t yr e g i o ni st h e convex closure of all non-negative rate pairs (R 1 ,R 2 ) such that max j∈{1,2} P[ W j = W j ] → 0 as N → +∞.

B. The Gaussian IC with source cooperation

As i n g l e -a n t e n n aI Cw i t hb i l a t e r a ls o u r c ec o o p e r a t i o nh a s input/output relationship

    Y 1 Y 2 Y 3 Y 4     = H X 1 X 2 +     Z 1 Z 2 Z 3 Z 4     (1)
where the inputs are subject to a unitary average power constraint, the noises are independent proper-complex Gaussian random variables with zero mean and unit variance, and H is the constant complex-valued channel matrix that defines the connectivity of the network.

For FD source cooperation the channel matrix H is

H (FD) :=     ⋆ √ C 1 √ C 2 ⋆ √ S 3 √ I 3 e jθ3 √ I 4 e jθ4 √ S 4     , (2)
for non-negative (C 1 ,C 2 ,S 3 ,I 3 , θ 3 ,S 4 ,I 4 , θ 4 ) and where ⋆ indicates the channel gains that do not affect the capacity region (since a source can remove its transmit signal from its received signal). At each node, a channel gain can be taken to be real-valued and non-negative because a node can compensate for the phase of one of its channel gains.

For HD source cooperation the channel matrix H is

H (HD) :=     1 -S 1 00 0 01 -S 2 00 00 1 0 00 0 1     H (FD) S 1 0 0 S 2 (3)
for the same H (FD) in (2) since a HD channel is a special case of the memoryless FD framework [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF] where the channel input at source j, j ∈ {1, 2},i sn o wt h ep a i r(X j ,S j ),w h e r e X j ∈ X j as before and where S j ∈ {0, 1} is the state random variable that indicates whether the source is receiving (S j =0) or transmitting (S j =1).

An IC with unilateral cooperation is obtained with either C 1 =0or C 2 =0,aZ-channelwithI 4 =0and an S-channel with I 3 =0.

C. Performance Metrics: Generalized degrees-of-freedom and Sum-Capacity to within a Constant Gap

Let SNR > 0 and consider the parameterization

S 3 = S 4 := SNR, (4a)
I 4 := SNR β4 , β 4 ≥ 0, (4b)
I 3 := SNR β3 , β 3 ≥ 0, (4c)
C 2 := SNR β2 , β 2 ≥ 0, (4d)
C 1 := SNR β1 , β 1 ≥ 0, (4e)
where β 3 and β 4 measure the strength of the interference/cross links compared to the direct links (which are assumed to be equal), while β 1 and β 2 the strength of the cooperation links compared to the direct links. The gDoF is defined as [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] d := lim

SNR→+∞ max{R 1 + R 2 } 2log(1+SNR) (5)
where the maximization is intended over all possible achievable rate pairs (R 1 ,R 2 ).

The gDoF of the causal cognitive IC is lower bounded by the gDoF of the noncooperative IC and upper bounded by the gDoF of the non-causal cognitive IC. The gDoF of the noncooperative IC [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] (i.e., β 1 = β 2 =0 )w i t hβ 3 = β 4 ≥ 0 is the so-called W-curve [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF], and that of the noncooperative Z-IC (which coincides with the S-IC) with β 3 =0and β 4 ≥ 0 is the minimum between the so-called V-curve [START_REF] Suh | Feedback Capacity of the Gaussian Interference Channel to within 2 Bits[END_REF] and 1/2. The gDoF of the non-causal cognitive IC [START_REF] Rini | Inner and outer bounds for the gaussian cognitive interference channel and new capacity results[END_REF] (i.e., β 1 =0and β 2 =+∞)withβ 3 = β 4 ≥ 0 coincides with the V-curve, that of the Z-channel with β 3 ≥ 0 and β 4 =0is also the V-curve, and that of the S-channel with β 3 =0and β 4 ≥ 0 equals 1.

The gDoF region is an asymptotically exact characterization of capacity at infinite SNR.A tfi n i t eSNR the capacity is said to be known to within b bits if we can show an inner bound region I and an outer bound region

O such that (R 1 ,R 2 ) ∈ ConvexClosure[I]= ⇒ (R 1 + b,R 2 + b) ∈ O.
The capacity of the IC with unilateral cooperation is sandwiched between that of the noncooperative IC [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] (i.e., Y 1 = Y 2 = ∅)andthatofthenon-causalcogniti v eIC [START_REF] Rini | Inner and outer bounds for the gaussian cognitive interference channel and new capacity results[END_REF](i.e., Y 1 = ∅ and Y 2 is a noiseless bit-pipe of infinite capacity), which are both known to within 1 bit.

In this work we focus on the interference-asymmetric IC with unilateral source cooperation obtained as Z-channel:

β 1 =0, β 2 = β f , β 3 = β i , β 4 =0, S-channel: β 1 =0, β 2 = β f , β 3 =0, β 4 = β i
for some non-negative pair (β i , β f ),w h i c hw ew i l lc o m p a r e with the interference-symmetric IC, i.e., β 3 = β 4 = β i ,a n d the IC with symmetric bilateral source cooperation, i.e., β 1 = β 2 = β f .

III. GDOF AND CONSTANT GAP FOR INTERFERENCE-SYMMETRIC CHANNELS:S UMMARY OF

KNOWN RESULTS

A. FD Bilateral Cooperation

Thm. 1 reported next is from [START_REF] Shuang | On interference channels with generalized feedback[END_REF]. In [START_REF] Shuang | On interference channels with generalized feedback[END_REF] a smaller gap compared to [START_REF] Prabhakaran | Interference channelsw i t hs o u r c e cooperation[END_REF] was obtained. Note that [START_REF] Prabhakaran | Interference channelsw i t hs o u r c e cooperation[END_REF] considered symmetric cooperation links C 1 = C 2 but general (S 3 ,I 3 ,S 4 ,I 4 ), while [START_REF] Shuang | On interference channels with generalized feedback[END_REF] only considered the case C 1 = C 2 ,S 3 = S 4 ,I 3 = I 4 ,w h i c hj u s t i fi e st h es m a l l e rg a p .

Theorem 1 ([13]) The sum-capacity of the interferencesymmetric G-IC with bilateral FD source cooperation in known to within 4 bits.

Fig. 2(a) shows the gDoF and the gap for the interferencesymmetric G-IC with bilateral FD source cooperation. The whole set of parameters (β i , β f ) has been partitioned into multiple sub-regions depending upon different levels of cooperation (β f )a n di n t e r f e r e n c e( β i ). For each sub-region, the gDoF and constant gap results of [START_REF] Shuang | On interference channels with generalized feedback[END_REF] are reported.

By observing Fig. 2(a), we notice that cooperation might be worth implementing only in regions 5, 9 and 10, i.e., in the regimes where the gDoF is a function of β f .I nt hes t rong interference regime, i.e., 1 ≤ β i ≤ 2,s o u r c e2m a yu s et h e information learned through the noisy link to help source 1 to convey the message to receiver 1. In the weak interference regime, i.e., β i < 1,theacquiredinformationmaybeexploited to pre-code the message intended to receiver 2 by against the interference created by the learned message.

B. HD Bilateral Cooperation

Thm. 2 reported next is an extension to the case of random switch of the result first obtained in [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.1] for the case of deterministic switch. In addition, the gDoF in [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF] were given implicitly as the solution of a linear program, while here we give the gDoF in closed-form.

Theorem 2 (random switch extension of [5, Thm.3.1])

The sum-capacity of the interference-symmetric G-IC with bilateral HD source cooperation in known to within 26.011 bits.

The proof parallels that of [START_REF] Cardone | The symmetric sum-capacity of the gaussian half-duplex causal cognitive interference channel to within a constant gap[END_REF] (for the case of interferencesymmetric unilateral HD cooperation and random switch) and is not reported here for sake of space. In [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.1], the difference between upper and lower bounds gives a gap of 17+3 = 20 bits on the symmetric sum-capacity. With random switch at each of the two sources, the sum-rate upper bound is expected to be increased by 2 × log(2) = 2 bits (since each source can convey at most one bit of information through ab i n a r ys w i t c h ) ,t h e r e b ys u g g e s t i n gt h a tt h eg a ps h o u l db e of 22 bits. However, we used the bound from [4, page 179, bottom of 1st column]; this bound contains six entropy terms with positive sign, each of them contributes to one extra bit to the gap; for this reason our gap is about 6 bits larger than the one in [5, Thm. the system is the same as that of the noncooperative IC. Thus the same additive gap result found for the FD case holds in the case of HD cooperation. As an example consider the region 1 where β i ≥ 2 and β f ≤ 1,i . e . ,v e r ys t r o n gi n t e r f e r e n c ea n d weak cooperation; in this regime we have d =1,f o rb o t hF D and HD, thus the gap computed for the HD case (18.036 bits) can be reduced to the one obtained for the FD case (2 bits).

C. FD Unilateral Cooperation

Thm. 3 reported next is from [START_REF]On the interference channel with causal cognition[END_REF]. In [START_REF]On the interference channel with causal cognition[END_REF], we specialized the outer bounds of [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Prabhakaran | Interference channelsw i t hs o u r c e cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] to the case of unilateral cooperation (i.e., by setting C 1 =0)andde velopedachie v able schemes based on 'Block Markov superposition coding with joint decoding' and found that

Theorem 3 ([21]) The sum-capacity of the interferencesymmetric G-IC with unilateral FD source cooperation in known to within 7.322 bits.

Ad e t a i l e dp r o o fc a nb ef o u n di n [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF] .

Fig. 2(c) shows the gDoF and the gap for the symmetric G-IC with unilateral FD source cooperation. The whole set of parameters has been partitioned into multiple sub-regions depending upon different levels of cooperation (β f )a ndi nt e rference (β i )s t r e n g t h s . W en o t i c et h a t :

1) Unilateral source cooperation has the same gDoF of the classical noncooperative IC (given by min max{1 -

β i , β i }, max{1 -β i /2, β i /2}, 1 )
i nr e gi ons1,3,4a nd 5i nF i g .2 ( c ) .F o rt h i ss e to fp a r a m e t e r su n i l a t e r a l cooperation/causal cognitive radio might not be worth implementing in practical systems since the same gDoF is achieved without cooperation. 2) Unilateral source cooperation has the same gDoF of the classical relay channel (given by 1/2max{1, min{β i , β f }})f o rβ i ≥ 1 and β f ≥ β i , i.e., part of regions 2 and 3 in Fig. 2(c). For this set of parameters unilateral cooperation/causal cognitive radio might not be worth implementing in practical systems since zero rate is optimal for the cognitive user. 3) Unilateral cooperation behaves differently than other classical channel models in terms of gDoF only in part of region 2 (β f < β i )a n dr e g i o n6i nF i g .2 ( c ) . Region 6 requires more involved achievable schemes , i.e., Dirty-Paper-Coding (DPC),and points to a tradeoff between constant gap and complexity. In [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF] we showed that a DPC-based scheme achieves the sum-capacity to within 2 bits, rather than the 3 bits obtained by using superposition coding only. 4) Unilateral source cooperation has the same gDoF of the non-causal cognitive channel (given by min{1 -

β i /2, β i /2})inregions2(onlyforβ f ≥ β i -1
), 3, 4, 6a, and 6b in Fig. 2(c). Therefore, in these regimes, unilateral source cooperation attains the ultimate performance limits of the non-causal cognitive radio and is strictly better (in terms of gDoF) than the noncooperative case. 5) Unilateral source cooperation has the same gDoF of bilateral source cooperation (see Fig. 2(a)) when

β f ≤ 1 or β f ∈ [[β i -1] + , β i ]
except in the regimes 6c and 6d in Fig. 2(c). For this set of parameters unilateral cooperation attains the same gDoF of bilateral cooperation but with less resources and therefore represents a better trade-off in practical systems.

D. HD Unilateral Cooperation

Thm. 4 reported next is from [START_REF] Cardone | The symmetric sum-capacity of the gaussian half-duplex causal cognitive interference channel to within a constant gap[END_REF]. In [START_REF] Cardone | The symmetric sum-capacity of the gaussian half-duplex causal cognitive interference channel to within a constant gap[END_REF], we specialized the FD outer bounds of [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Prabhakaran | Interference channelsw i t hs o u r c e cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] to the case of unilateral HD cooperation by following the approach of [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF]. We also developed achievable schemes inspired by linear deterministic approximation of the Gaussian noise channel at high SNR [START_REF] Avestimehr | Wireless networki n f o r m a t i o n flow: A deterministic approach[END_REF] and found that

Theorem 4 ([20]) The sum-capacity of the interferencesymmetric G-IC with unilateral HD source cooperation in known to within 12.503 bits.

Fig. 2(d) shows the gDoF and the gap for the interferencesymmetric G-IC with unilateral HD source cooperation. The whole set of parameters has been partitioned into multiple subregions depending upon different levels of cooperation (β f ) and interference (β i )s t r e n g t h s . W en o t i c et h a t :

1) Everywhere, except in regions 3, 8 and 10 in Fig. 2(d), unilateral cooperation might not be worth implementing since the same gDoF is achieved without cooperation (classical IC). 2) Differently from the FD case, we notice that the gDoF of the IC with unilateral cooperation never equals the gDoF of the HD relay channel [START_REF] Cardone | Gaussian halfduplex relay channels: Generalized degrees of freedom and constant gap results[END_REF]. In other words, the gDoF of the HD relay channel is always a strict lower bound for the IC with unilateral cooperation.

3) The IC with unilateral HD source cooperation has the same gDoF of the non-causal cognitive channel in regions 4 and 5 in Fig. 2(d). Thus, in these two regions, the performance of the system in terms of gDoF is not worsened by allowing causal learning at one source. 4) In regions 1, 4, 5 and 6 of Fig. 2(d) the gDoF equals that of the equivalent FD channel. Thus, in these regimes, the same gap results found for the FD case hold in the case of unilateral HD source cooperation. 5) In regions 1, 2, 6, 7, 9, part of 5 (with

β f ≤ 3 -2β i )
and part of 4 (with β f ≤ 2β i -1)o fF i g .2 ( d ) ,t h e gDoF is the same as that computed in the bilateral case. Thus, for this set of parameters, unilateral cooperation attains the same gDoF of bilateral cooperation but with less resources and therefore represents a better trade-off in practical systems.

IV. GDOF AND CONSTANT GAP FOR INTERFERENCE-ASYMMETRIC CHANNELS:N EW RESULTS

A. Z-Channel: FD Unilateral Cooperation

For the Z-channel with unilateral cooperation, obtained from the general G-IC model by setting C 1 = I 4 =0,w eh a v e :

Theorem 5 The sum-capacity in known to within 1 bit for the Z-G-IC with unilateral FD source cooperation.

The proof can be found in [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF].

Fig. 3(a) shows the gDoF and the gap for the Z-G-IC with unilateral FD source cooperation. The whole set of parameters has been partitioned into multiple sub-regions depending upon different levels of cooperation (β f )a n di n t e r f e r e n c e( β i ) strengths. We notice that:

1) With no-cooperation we immediately have d ≥ min 1, max βi 2 , 1 -βi 2 from [START_REF] Sason | On achievable rate regions for the gaussian interference channel[END_REF]. Thus, for the Zchannel, cooperation improves the gDoF with respect to the noncooperative case in the regime β i ≥ 2 and β f ≥ 1 (regions 2 and 3 in Fig. 3(a)).

2) Unilateral cooperation, for the Z-channel, attains the same gDoF of the relay channel when 1 ≤ β i ≤ β f (part of regions 3 and 4 in Fig. 3(a)).

3) The Z-channel achieves the same gDoF of the noncausal cognitive channel, given by d =m a x {1 -

β i /2, β i /2},everywhereexceptinβ i > max{2, β f +1}
(regions 1 and 2 in Fig. 3(a)). 4) The gDoF of unilateral cooperation equals that of bilateral cooperation when β f ≤ max{1, β i } (regions 1 and 2, parts of regions 3, 4 and 5 in Fig. 3(a)). 5) By comparing Fig. 2(c) and Fig. 3(a), we observe that the gDoF of the Z-channel is always greater or equal than that of the symmetric G-IC with unilateral source cooperation. This is due to the fact that the source 1 does not cooperate in sending the signal of source 2.

Therefore by removing the link between source 1 and destination 2 we rid destination 2 of only interfering signal. The regimes where the Z-channel outperforms the symmetric channel are when 0 ≤ β i ≤ 2 3 and β f ≤ min{β i , 1β i } (part of region 5 in Fig. 3(a)).

B. Z-Channel: HD Unilateral Cooperation

Thm. 6 reported next is an extension to the case of random switch of [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.2] for the case of deterministic switch. In addition, the gDoF in [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF] were given implicitly as the solution of a linear program, while here we give the gDoF in closedform. Moreover, here we substantially reduce the gap from 20 bits to 4.507 bits: Theorem 6 The sum-capacity in known to within 4.507 bits for the Z-G-IC with unilateral HD source cooperation.

The proof parallels that of [START_REF] Cardone | The symmetric sum-capacity of the gaussian half-duplex causal cognitive interference channel to within a constant gap[END_REF] (for the case of interferencesymmetric unilateral HD cooperation and random switch).

Fig. 3(b) shows the gDoF and the gap for the Z-G-IC with unilateral HD source cooperation. The whole set of parameters has been partitioned into multiple sub-regions depending upon different levels of cooperation (β f )a n di n t e r f e r e n c e( β i ) strengths. The analysis is similar to that of the channel with symmetric interfering links in the same regimes. We notice: 1) By comparing Fig. 3(b) with Fig. 3(a), we notice that in regions 1, 4 and 5 of Fig. 3(b) the gDoF of the HD channel is as that of the FD. Moreover this happens when the performance of the system in terms of gDoF is not enhanced by allowing source cooperation. Thus the same gap result found for the FD case holds in the case of HD cooperation. Moreover, in region 2 in Fig. 3(b) the gDoF equals to that of the noncooperative IC. 2) Differently from the FD case, we notice that the gDoF of the Z-G-IC with unilateral source cooperation never equals the gDoF of the classical HD relay channel [START_REF] Cardone | Gaussian halfduplex relay channels: Generalized degrees of freedom and constant gap results[END_REF].

3) The Z-channel achieves the same gDoF of the noncausal cognitive channel everywhere except in β i > 2 (regions 1, 2 and 3 in Fig. 3(b)). 4) By comparing Fig. 3(b) and Fig. 2(d) we observe that the Z-channel outperforms the symmetric channel when 0 ≤ β i ≤ 2 3 (part of region 5 in Fig. 3(b)). The reason of this behavior is the same as the one stated for the FD case.

C. S-Channel: FD Unilateral Cooperation

For the S-channel with unilateral cooperation, obtained from the general G-IC model by setting C 1 = I 3 =0,w eh a v e : Theorem 7 The sum-capacity in known to within 4 bits for the S-G-IC with unilateral FD source cooperation.

The proof can can be found in [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF].

Fig. 3(c) shows the gDoF and the gap for the G-S channel with FD source cooperation. The whole set of parameters has been partitioned into multiple sub-regions depending upon different levels of cooperation (β f )a n d interference (β i )s t r e n g t h s .I n [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF] ,w ed e v e l o p e da c h i e v a b l e schemes. In contrast to the symmetric and Z channels, here we made use of DPC-based schemes in regions 1 and 2 of Fig. 3(c). We notice that:

1) Unilateral cooperation achieves the same gDoF of the noncooperative IC when β i ≥ 2 or β f ≤ max{1, β i } (regions 3 and 4 in Fig. 3(c)).

2) The gDoF of unilateral cooperation never equals the gDoF of the classical relay channel, which is equal to d = 1 2 .

3) The S-channel achieves the gDoF of the non-causal cognitive IC everywhere except in β i ≤ 2 and β f ≤ min{2, β i +1 } (regions 1, 3 and part of region 2 in Fig. 3(c)). Actually, when the link source 2→destination 1 is not present, the non-causal cognitive IC achieves d =1 ,i . e . ,t h ep e r f o r m a n c ei n terms of gDoF is the same as that of a two point-to-point communication channel. This is so because the cognitive source, by a-priori knowing the primary's message, can completely pre-cancel the interference experienced at its receiver. 4) The gDoF of unilateral and bilateral cooperation are equal when β i ≥ 2 and β f ≤ 1 or when β i ≤ 2 and β f ≤ min{2, β i +1 } (regions 1 and 3, and parts of regions 2 and 4 in Fig. 3(c)). 5) The S-channel outperforms the interference-symmetric IC with unilateral source cooperation when either 0 ≤ β i ≤ 2 3 and β f ≤ min{β i , 1β i } or when β i ≤ 2 and β f ≥ max{1, β i } (regions 1, 2 and part of region 3 in Fig. 3(c)). On the other hand, the interference-symmetric IC with unilateral source cooperation outperforms the Schannel when β i ≥ 2 and β f ≥ 1 (part of region 4 in Fig. 3(c)). This is so because, in the very strong interference and strong cooperation regime, the performance of the system is enhanced by allowing the cognitive source to help the primary user to convey the information, but this is not possible since I 3 =0. 6) When β i ≥ 2 (region 4 in Fig. 3(c)) we have an exact sum-capacity result, i.e., the gap between the sum-rate outer bound and inner bound is equal to zero.

D. S-Channel: HD Unilateral Cooperation

Thm. 8 reported next is an extension to the case of random switch of [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF]Thm.3.2] for the case of deterministic switch. In addition, the gDoF in [START_REF] Wu | Interference channels with half-duplex source cooperation[END_REF] were given implicitly as the solution of a linear program, while here we give the gDoF in closedform. Moreover, here we substantially reduce the gap from 20 bits to 5 bits: Theorem 8 The sum-capacity in known to within 5 bits for the S-G-IC with unilateral HD source cooperation.

The proof parallels that of [START_REF] Cardone | The symmetric sum-capacity of the gaussian half-duplex causal cognitive interference channel to within a constant gap[END_REF] (for the case of interferencesymmetric unilateral HD cooperation and random switch).

Fig. 3(d) shows the gDoF and the gap for the G-S channel with unilateral HD source cooperation. The whole set of parameters has been partitioned into multiple sub-regions depending upon different levels of cooperation (β f )a n di nterference (β i )s t r e n g t h s .W en o t i c et h a t :

1) By comparing Fig. 3(d) with Fig. 3(c), we notice that there are some regions (3 and 4 in Fig. 3(c)) in which the gDoF of the HD channel is as that of the FD channel. This happens when the gDoF of the system is the same as that of the pure IC. Thus, in these regions, the same additive gap results found for the FD case hold in the case of HD cooperation. Moreover, in region 3 in Fig. 3(d) the gDoF equals to that of the pure IC. 2) As obtained for the FD case, also the gDoF of the S-G-IC with unilateral HD source cooperation never equals that of the classical HD relay channel given by d = 1 2 .

3) The S-channel achieves the same gDoF of the non-causal cognitive channel, that is, d =1 ,f o rβ i ≥ 2 (region 1 in Fig. 3(d)). 4) The S-channel outperforms the interference-symmetric IC with unilateral source cooperation when either 0 ≤ β i ≤ 2 3 or when β i ≤ 2 and β f ≥ max{2β i , β i } (regions 4 and 5, and parts of regions 2 and 3 in Fig. 3(d)). On the other hand, the interference-symmetric IC with unilateral source cooperation outperforms the S-channel in very strong interference and very strong cooperation, i.e., β i ≥ 2 and β f ≥ 2 (region 2 in Fig. 3(d)). The reason of this behavior is the same as the one explained for the FD case. 5) When β i ≥ 2 (region 1 in Fig. 3(d)), i.e. very strong interference regime, we have an exact sum-capacity result, as in the FD case.

V. C ONCLUSIONS

In this work we considered the two-user Gaussian interference channel with unilateral source cooperation and with asymmetric interference. Our main contribution consisted in deriving closed-form expressions for the gDoF and characterizing the sum-capacity to within a constant gap for both cases where the cognitive source operates in full-duplex and half-duplex mode.

We also systematically compared the different channel models among themselves and with the case of: (i) bilateral symmetric cooperation and interference, (ii) unilateral cooperation and symmetric interference, (iii) non-causal cognitive interference channel, (iv) noncooperative interference channela n d (v) relay channel. In particular we highlighted regimes where cooperation might not be worth implementing in practical systems (because approximately equal to that of the relay channel or of the noncooperative channel), as well as regimes where causal cognition approximately equals the sum-capacity of the non-causal cognitive channel. Moreover, by comparing the symmetric-interference case with the asymmetric-interference case we discussed the impact on the sum-capacity of the network topology.

Fig. 1 :

 1 Fig. 1: The Gaussian Interference Channel with Unilateral Source Cooperation. Left: symmetric channel. Center: Z-channel.Right: S-channel.

  3.1]. Fig. 2(b) shows the gDoF and the gap for the interferencesymmetric G-IC with bilateral HD source cooperation. The whole set of parameters has been partitioned into multiple subregions depending upon different levels of cooperation (β f ) and interference (β i )strengths.F oreachsub-re gion,thegDoF and constant gap results are reported.By comparing Fig.2(b) with Fig.2(a), we see that there exist parameter regimes where the gDoF of the HD channel equals that of the FD channel. This happens when the gDoF of

  (a) Bilateral FD source cooperation. (b) Bilateral HD source cooperation. (c) Unilateral FD source cooperation. (d) Unilateral HD source cooperation.

Fig. 2 :

 2 Fig. 2: gDoF and constant gap results for interference-symmetric G-IC with source cooperation, either bilateral (cases( a )a n d (b)) or unilateral (cases (c) and (d)).

  (a) Z-IC with unilateral FD cooperation. (b) Z-IC with unilateral HD cooperation. (c) S-IC with unilateral FD cooperation. (d) S-IC with unilateral HD cooperation.

Fig. 3 :

 3 Fig. 3: gDoF and constant gap results for interference-asymmetric G-IC with unilateral source cooperation.

ACKNOWLEDGMENT

The work of Dr. D. Tuninetti was partially funded by NSF under award number 0643954; the contents of this article are solely the of the author and do not necessarily represent the official views of the NSF. The work of Dr. D. Tuninetti was possible thanks to the generous support of Telecom-ParisTech, Paris France, while the author was on a sabbatical leave at the same institution.

Eurecom's research is partially supported by its industrial partners: BMW Group Research & Technology, IABG, Monaco Telecom, Orange, SAP, SFR, ST Microelectronics, Swisscom and Symantec. The research carried out at Eurecom leading to these results has received funding from the EU Celtic+ Framework Program Project SPECTRA.

The research at IMC leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement iJOIN n 317941.