Mismatch in the Classification of Linear Subspaces: Sufficient Conditions for Reliable Classification

This paper considers the classification of linear subspaces with mismatched classifiers. In particular, we assume a model where one observes signals in the presence of isotropic Gaussian noise and the distribution of the signals conditioned on a given class is Gaussian with a zero mean and a low-rank covariance matrix. We also assume that the classifier knows only a mismatched version of the parameters of input distribution in lieu of the true parameters. By constructing an asymptotic low-noise expansion of an upper bound to the error probability of such a mismatched classifier, we provide sufficient conditions for reliable classification in the low-noise regime that are able to sharply predict the absence of a classification error floor. Such conditions are a function of the geometry of the true signal distribution, the geometry of the mismatched signal distributions as well as the interplay between such geometries, namely, the principal angles and the overlap between the true and the mismatched signal subspaces. Numerical results demonstrate that our conditions for reliable classification can sharply predict the behavior of a mismatched classifier both with synthetic data and in a motion segmentation and a hand-written digit classification applications.Comment: 17 pages, 7 figures, submitted to IEEE Transactions on Signal Processin

Assessment of Immature Platelet Fraction in the Diagnosis of Wiskott-Aldrich Syndrome.

Children with Wiskott-Aldrich syndrome (WAS) are often first diagnosed with immune thrombocytopenia (ITP), potentially leading to both inappropriate treatment and the delay of life-saving definitive therapy. WAS is traditionally differentiated from ITP based on the small size of WAS platelets. In practice, microthrombocytopenia is often not present or not appreciated in children with WAS. To develop an alternative method of differentiating WAS from ITP, we retrospectively reviewed all complete blood counts and measurements of immature platelet fraction (IPF) in 18 subjects with WAS and 38 subjects with a diagnosis of ITP treated at our hospital. Examination of peripheral blood smears revealed a wide range of platelet sizes in subjects with WAS. Mean platelet volume (MPV) was not reported in 26% of subjects, and subjects in whom MPV was not reported had lower platelet counts than did subjects in whom MPV was reported. Subjects with WAS had a lower IPF than would be expected for their level of thrombocytopenia, and the IPF in subjects with WAS was significantly lower than in subjects with a diagnosis of ITP. Using logistic regression, we developed and validated a rule based on platelet count and IPF that was more sensitive for the diagnosis of WAS than was the MPV, and was applicable regardless of the level of platelets or the availability of the MPV. Our observations demonstrate that MPV is often not available in severely thrombocytopenic subjects, which may hinder the diagnosis of WAS. In addition, subjects with WAS have a low IPF, which is consistent with the notion that a platelet production defect contributes to the thrombocytopenia of WAS. Knowledge of this detail of WAS pathophysiology allows to differentiate WAS from ITP with increased sensitivity, thereby allowing a physician to spare children with WAS from inappropriate treatment, and make definitive therapy available in a timely manner

Robust Large Margin Deep Neural Networks

The generalization error of deep neural networks via their classification margin is studied in this paper. Our approach is based on the Jacobian matrix of a deep neural network and can be applied to networks with arbitrary nonlinearities and pooling layers, and to networks with different architectures such as feed forward networks and residual networks. Our analysis leads to the conclusion that a bounded spectral norm of the network's Jacobian matrix in the neighbourhood of the training samples is crucial for a deep neural network of arbitrary depth and width to generalize well. This is a significant improvement over the current bounds in the literature, which imply that the generalization error grows with either the width or the depth of the network. Moreover, it shows that the recently proposed batch normalization and weight normalization reparametrizations enjoy good generalization properties, and leads to a novel network regularizer based on the network's Jacobian matrix. The analysis is supported with experimental results on the MNIST, CIFAR-10, LaRED, and ImageNet datasets

Lessons from the Rademacher complexity for deep learning

Understanding the generalization properties of deep learning models is critical for successful applications, especially in the regimes where the number of training samples is limited. We study the generalization properties of deep neural networks via the empirical Rademacher complexity and show that it is easier to control the complexity of convolutional networks compared to general fully connected networks. In particular, we justify the usage of small convolutional kernels in deep networks as they lead to a better generalization error. Moreover, we propose a representation based regularization method that allows to decrease the generalization error by controlling the coherence of the representation. Experiments on the MNIST dataset support these foundations

Generalization Error in Deep Learning

Deep learning models have lately shown great performance in various fields such as computer vision, speech recognition, speech translation, and natural language processing. However, alongside their state-of-the-art performance, it is still generally unclear what is the source of their generalization ability. Thus, an important question is what makes deep neural networks able to generalize well from the training set to new data. In this article, we provide an overview of the existing theory and bounds for the characterization of the generalization error of deep neural networks, combining both classical and more recent theoretical and empirical results

The clinical utility of molecular diagnostic testing for primary immune deficiency disorders: a case based review

Primary immune deficiency disorders (PIDs) are a group of diseases associated with a genetic predisposition to recurrent infections, malignancy, autoimmunity and allergy. The molecular basis of many of these disorders has been identified in the last two decades. Most are inherited as single gene defects. Identifying the underlying genetic defect plays a critical role in patient management including diagnosis, family studies, prognostic information, prenatal diagnosis and is useful in defining new diseases. In this review we outline the clinical utility of molecular testing for these disorders using clinical cases referred to Auckland Hospital. It is written from the perspective of a laboratory offering a wide range of tests for a small developed country

Modulation of the endocannabinoids N-Arachidonoylethanolamine (AEA) and 2-Arachidonoylglycerol (2-AG) on Executive Functions in Humans

Animal studies point to an implication of the endocannabinoid system on executive functions. In humans, several studies have suggested an association between acute or chronic use of exogenous cannabinoids (Δ9-tetrahydrocannabinol) and executive impairments. However, to date, no published reports establish the relationship between endocannabinoids, as biomarkers of the cannabinoid neurotransmission system, and executive functioning in humans. The aim of the present study was to explore the association between circulating levels of plasma endocannabinoids N-arachidonoylethanolamine (AEA) and 2-Arachidonoylglycerol (2-AG) and executive functions (decision making, response inhibition and cognitive flexibility) in healthy subjects. One hundred and fifty seven subjects were included and assessed with the Wisconsin Card Sorting Test; Stroop Color and Word Test; and Iowa Gambling Task. All participants were female, aged between 18 and 60 years and spoke Spanish as their first language. Results showed a negative correlation between 2-AG and cognitive flexibility performance (r = −.37; p<.05). A positive correlation was found between AEA concentrations and both cognitive flexibility (r = .59; p<.05) and decision making performance (r = .23; P<.05). There was no significant correlation between either 2-AG (r = −.17) or AEA (r = −.08) concentrations and inhibition response. These results show, in humans, a relevant modulation of the endocannabinoid system on prefrontal-dependent cognitive functioning. The present study might have significant implications for the underlying executive alterations described in some psychiatric disorders currently associated with endocannabinoids deregulation (namely drug abuse/dependence, depression, obesity and eating disorders). Understanding the neurobiology of their dysexecutive profile might certainly contribute to the development of new treatments and pharmacological approaches