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We address the issue of binary classification in Banach spaces in presence of uncertainty. We show that a number of results from classical support vector machines theory can be appropriately generalised to their robust counterpart in Banach spaces. These include the Representer Theorem, strong duality for the associated Optimization problem as well as their geometric interpretation. Furthermore, we propose a game theoretic interpretation by expressing a Nash equilibrium problem formulation for the more general problem of finding the closest points in two closed convex sets when the underlying space is reflexive and smooth.

There remains an open question about the usefulness and the interpretation of Machine learning (MLE) approaches for discrimination of spatial patterns of brain images between samples or activation states. In the last few decades, these approaches have limited their operation to feature extraction and linear classification tasks for between-group inference. In this context, statistical inference is assessed by randomly permuting image labels or by the use of random effect models that consider between-subject variability. These multivariate MLE-based statistical pipelines, whilst potentially more effective for detecting activations than hypotheses-driven methods, have lost their mathematical elegance, ease of interpretation, and spatial localization of the ubiquitous General linear Model (GLM). Recently, the estimation of the conventional GLM has been demonstrated to be connected to an univariate classification task when the design matrix is expressed as a binary indicator matrix. In this paper we explore the complete connection between the univariate GLM and MLE \emph{regressions}. To this purpose we derive a refined statistical test with the GLM based on the parameters obtained by a linear Support Vector Regression (SVR) in the \emph{inverse} problem (SVR-iGLM). Subsequently, random field theory (RFT) is employed for assessing statistical significance following a conventional GLM benchmark. Experimental results demonstrate how parameter estimations derived from each model (mainly GLM and SVR) result in different experimental design estimates that are significantly related to the predefined functional task. Moreover, using real data from a multisite initiative the proposed MLE-based inference demonstrates statistical power and the control of false positives, outperforming the regular GLM.

Understanding implicit bias of gradient descent has been an important goal in machine learning research. Unfortunately, even for a single-neuron ReLU network, it recently proved impossible to characterize the implicit regularization with the square loss by an explicit function of the norm of model parameters. In order to close the gap between the existing theory and the intriguing empirical behavior of ReLU networks, here we examine the gradient flow dynamics in the parameter space when training single-neuron ReLU networks. Specifically, we discover implicit bias in terms of support vectors in ReLU networks, which play a key role in why and how ReLU networks generalize well. Moreover, we analyze gradient flows with respect to the magnitude of the norm of initialization, and show the impact of the norm in gradient dynamics. Lastly, under some conditions, we prove that the norm of the learned weight strictly increases on the gradient flow.

Support vector machines (SVMs) play a very dominant role in data classification due to their good generalization performance. However, they suffer from the high computational complexity in the classification phase when there are a considerable number of support vectors (SVs). Then it is desirable to design efficient algorithms in the classification phase to deal with the datasets of real-time pattern recognition systems. To this end, we propose a novel classifier called HMLSVMs (Hierarchical Mixing Linear Support Vector Machines) in this paper, which has a hierarchical structure with a mixing linear SVMs classifier at each node and predicts the label of a sample using only a few hyperplanes. We also give a generalization error bound for the class of locally linear SVMs (LLSVMs) based on the Rademacher theory, which ensures that overfitting can be effectively avoided. Experimental evaluations shows, while maintaining a comparable classification performance to kernel SVMs (KSVMs), the proposed classifier achieves the high efficiency in the classification stage.

Microarray experiments are capable of measuring the expression level of thousands of genes simultaneously. Dealing with this enormous amount of information requires complex computation. Support Vector Machines (SVM) have been widely used with great efficiency to solve classification problems that have high dimension. In this sense, it is plausible to develop new feature selection strategies for microarray data that are associated with this type of classifier. Therefore, we propose, in this paper, a new method for feature selection based on an ordered search process to explore the space of possible subsets.

This paper studies an interesting problem: is it possible to predict the crowd opinion about a movie before the movie is actually released? The crowd opinion is here expressed by the distribution of ratings given by a sufficient amount of people. Consequently, the pre-release crowd opinion prediction can be regarded as a Label Distribution Learning (LDL) problem. In order to solve this problem, a Label Distribution Support Vector Regressor (LDSVR) is proposed in this paper. The basic idea of LDSVR is to fit a sigmoid function to each component of the label distribution simultaneously by a multi-output support vector machine. Experimental results show that LDSVR can accurately predict peoples's rating distribution about a movie just based on the pre-release metadata of the movie.

Part deformation has been a longstanding challenge for object parsing, of which the primary difficulty lies in modeling the highly diverse object structures. To this end, we propose a novel structure parsing model to capture deformable object structures. The proposed model consists of two de-formable layers: the top layer is an undirected graph that incorporates inter-part deformations to infer object structures; the base layer is consisted of various independent nodes to characterize local intra-part deformations. To learn this two-layer model, we design a layer-wise learning algorithm,which employs matching pursuit and belief propagation for a low computational complexity inference. Specifically, active basis sparse coding is leveraged to build the nodes at the base layer, while the edge weights are estimated by a structural support vector machine. Experimental results on two benchmark datasets (i.e., faces and horses) demonstrate that the proposed model yields superior parsing performance over state-of-the-art models.

In this paper, we propose a new local spatio-temporal descriptor for videos and we propose a new approach for action recognition in videos based on the introduced descriptor. The new descriptor is called the Video Covariance Matrix Logarithm (VCML). The VCML descriptor is based on a covariance matrix representation, and it models relationships between different low-level features, such as intensity and gradient. We apply the VCML descriptor to encode appearance information of local spatio-temporal video volumes, which are extracted by the Dense Trajectories. Then, we present an extensive evaluation of the proposed VCML descriptor with the Fisher vector encoding and the Support Vector Machines on four challenging action recognition datasets. We show that the VCML descriptor achieves better results than the state-of-the-art appearance descriptors. Moreover, we present that the VCML descriptor carries complementary information to the HOG descriptor and their fusion gives a significant improvement in action recognition accuracy. Finally, we show that the VCML descriptor improves action recognition accuracy in comparison to the state-of-the-art Dense Trajectories, and that the proposed approach achieves superior performance to the state-of-the-art methods.