Scaling-up Empirical Risk Minimization: Optimization of Incomplete U-statistics

In a wide range of statistical learning problems such as ranking, clustering or metric learning among others, the risk is accurately estimated by $U$-statistics of degree $d\geq 1$, i.e. functionals of the training data with low variance that take the form of averages over $k$-tuples. From a computational perspective, the calculation of such statistics is highly expensive even for a moderate sample size $n$, as it requires averaging $O(n^d)$ terms. This makes learning procedures relying on the optimization of such data functionals hardly feasible in practice. It is the major goal of this paper to show that, strikingly, such empirical risks can be replaced by drastically computationally simpler Monte-Carlo estimates based on $O(n)$ terms only, usually referred to as incomplete $U$-statistics, without damaging the $O_{\mathbb{P}}(1/\sqrt{n})$ learning rate of Empirical Risk Minimization (ERM) procedures. For this purpose, we establish uniform deviation results describing the error made when approximating a $U$-process by its incomplete version under appropriate complexity assumptions. Extensions to model selection, fast rate situations and various sampling techniques are also considered, as well as an application to stochastic gradient descent for ERM. Finally, numerical examples are displayed in order to provide strong empirical evidence that the approach we promote largely surpasses more naive subsampling techniques.Comment: To appear in Journal of Machine Learning Research. 34 pages. v2: minor correction to Theorem 4 and its proof, added 1 reference. v3: typo corrected in Proposition 3. v4: improved presentation, added experiments on model selection for clustering, fixed minor typo

Comment on "Support Vector Machines with Applications"

Comment on ``Support Vector Machines with Applications'' [math.ST/0612817]Comment: Published at http://dx.doi.org/10.1214/088342306000000484 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Positive Definite Kernels in Machine Learning

This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as reproducing kernel Hibert spaces, the natural extension of the set of functions $\{k(x,\cdot),x\in\mathcal{X}\}$ associated with a kernel $k$ defined on a space $\mathcal{X}$. We discuss at length the construction of kernel functions that take advantage of well-known statistical models. We provide an overview of numerous data-analysis methods which take advantage of reproducing kernel Hilbert spaces and discuss the idea of combining several kernels to improve the performance on certain tasks. We also provide a short cookbook of different kernels which are particularly useful for certain data-types such as images, graphs or speech segments.Comment: draft. corrected a typo in figure

A survey on perceived speaker traits: personality, likability, pathology, and the first challenge

The INTERSPEECH 2012 Speaker Trait Challenge aimed at a unified test-bed for perceived speaker traits – the first challenge of this kind: personality in the five OCEAN personality dimensions, likability of speakers, and intelligibility of pathologic speakers. In the present article, we give a brief overview of the state-of-the-art in these three fields of research and describe the three sub-challenges in terms of the challenge conditions, the baseline results provided by the organisers, and a new openSMILE feature set, which has been used for computing the baselines and which has been provided to the participants. Furthermore, we summarise the approaches and the results presented by the participants to show the various techniques that are currently applied to solve these classification tasks

Force field feature extraction for ear biometrics

The overall objective in defining feature space is to reduce the dimensionality of the original pattern space, whilst maintaining discriminatory power for classification. To meet this objective in the context of ear biometrics a new force field transformation treats the image as an array of mutually attracting particles that act as the source of a Gaussian force field. Underlying the force field there is a scalar potential energy field, which in the case of an ear takes the form of a smooth surface that resembles a small mountain with a number of peaks joined by ridges. The peaks correspond to potential energy wells and to extend the analogy the ridges correspond to potential energy channels. Since the transform also turns out to be invertible, and since the surface is otherwise smooth, information theory suggests that much of the information is transferred to these features, thus confirming their efficacy. We previously described how field line feature extraction, using an algorithm similar to gradient descent, exploits the directional properties of the force field to automatically locate these channels and wells, which then form the basis of characteristic ear features. We now show how an analysis of the mechanism of this algorithmic approach leads to a closed analytical description based on the divergence of force direction, which reveals that channels and wells are really manifestations of the same phenomenon. We further show that this new operator, with its own distinct advantages, has a striking similarity to the Marr-Hildreth operator, but with the important difference that it is non-linear. As well as addressing faster implementation, invertibility, and brightness sensitivity, the technique is also validated by performing recognition on a database of ears selected from the XM2VTS face database, and by comparing the results with the more established technique of Principal Components Analysis. This confirms not only that ears do indeed appear to have potential as a biometric, but also that the new approach is well suited to their description, being robust especially in the presence of noise, and having the advantage that the ear does not need to be explicitly extracted from the background

Relational visual cluster validity

The assessment of cluster validity plays a very important role in cluster analysis. Most commonly used cluster validity methods are based on statistical hypothesis testing or finding the best clustering scheme by computing a number of different cluster validity indices. A number of visual methods of cluster validity have been produced to display directly the validity of clusters by mapping data into two- or three-dimensional space. However, these methods may lose too much information to correctly estimate the results of clustering algorithms. Although the visual cluster validity (VCV) method of Hathaway and Bezdek can successfully solve this problem, it can only be applied for object data, i.e. feature measurements. There are very few validity methods that can be used to analyze the validity of data where only a similarity or dissimilarity relation exists – relational data. To tackle this problem, this paper presents a relational visual cluster validity (RVCV) method to assess the validity of clustering relational data. This is done by combining the results of the non-Euclidean relational fuzzy c-means (NERFCM) algorithm with a modification of the VCV method to produce a visual representation of cluster validity. RVCV can cluster complete and incomplete relational data and adds to the visual cluster validity theory. Numeric examples using synthetic and real data are presente

Facial expression recognition in the wild : from individual to group

The progress in computing technology has increased the demand for smart systems capable of understanding human affect and emotional manifestations. One of the crucial factors in designing systems equipped with such intelligence is to have accurate automatic Facial Expression Recognition (FER) methods. In computer vision, automatic facial expression analysis is an active field of research for over two decades now. However, there are still a lot of questions unanswered. The research presented in this thesis attempts to address some of the key issues of FER in challenging conditions mentioned as follows: 1) creating a facial expressions database representing real-world conditions; 2) devising Head Pose Normalisation (HPN) methods which are independent of facial parts location; 3) creating automatic methods for the analysis of mood of group of people. The central hypothesis of the thesis is that extracting close to real-world data from movies and performing facial expression analysis on movies is a stepping stone in the direction of moving the analysis of faces towards real-world, unconstrained condition. A temporal facial expressions database, Acted Facial Expressions in the Wild (AFEW) is proposed. The database is constructed and labelled using a semi-automatic process based on closed caption subtitle based keyword search. Currently, AFEW is the largest facial expressions database representing challenging conditions available to the research community. For providing a common platform to researchers in order to evaluate and extend their state-of-the-art FER methods, the first Emotion Recognition in the Wild (EmotiW) challenge based on AFEW is proposed. An image-only based facial expressions database Static Facial Expressions In The Wild (SFEW) extracted from AFEW is proposed. Furthermore, the thesis focuses on HPN for real-world images. Earlier methods were based on fiducial points. However, as fiducial points detection is an open problem for real-world images, HPN can be error-prone. A HPN method based on response maps generated from part-detectors is proposed. The proposed shape-constrained method does not require fiducial points and head pose information, which makes it suitable for real-world images. Data from movies and the internet, representing real-world conditions poses another major challenge of the presence of multiple subjects to the research community. This defines another focus of this thesis where a novel approach for modeling the perception of mood of a group of people in an image is presented. A new database is constructed from Flickr based on keywords related to social events. Three models are proposed: averaging based Group Expression Model (GEM), Weighted Group Expression Model (GEM_w) and Augmented Group Expression Model (GEM_LDA). GEM_w is based on social contextual attributes, which are used as weights on each person's contribution towards the overall group's mood. Further, GEM_LDA is based on topic model and feature augmentation. The proposed framework is applied to applications of group candid shot selection and event summarisation. The application of Structural SIMilarity (SSIM) index metric is explored for finding similar facial expressions. The proposed framework is applied to the problem of creating image albums based on facial expressions, finding corresponding expressions for training facial performance transfer algorithms