Browsing by Subject "Reproducing kernel Hilbert space"

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    Analytic properties of Besov spaces via Bergman projections
    (American Mathematical Society, 2008) Kaptanoğlu, H. T.; Üreyen, A. E.
    We consider two-parameter Besov spaces of holomorphic functions on the unit ball of CN: We obtain various exclusions between Besov spaces of di®erent parameters using gap series. We estimate the growth near the boundary and the growth of Taylor coe±cients of functions in these spaces. We ¯nd the unique function with maximum value at each point of the ball in each Besov space. We base our proofs on Bergman projections and imbeddings between Lebesgue classes and Besov spaces. Special cases apply to the Hardy space H2, the Arveson space, the Dirichlet space, and the Bloch space.
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    Aspects of multivariable operator theory on weighted symmetric Fock spaces
    (World Scientific Publishing, 2014) Kaptanoğlu, H. T.
    We obtain all Dirichlet spaces Fq, q ∈ ℝ, of holomorphic functions on the unit ball of ℂN as weighted symmetric Fock spaces over ℂN. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the Fq. We pick out the analytic Hilbert modules from among the Fq. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each Fq. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C∗-algebras generated by the shift operators on the Fq fit in exact sequences that are in the same Ext class. We identify the groups K0 and K1 of the Toeplitz algebras on the Fq arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces Fb that depend on a weight sequence b. © World Scientific Publishing Company.
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    Open-set object recognition
    (2022-07) Mohammad, Salman
    Despite significant advances in object recognition and classification over the past couple of decades, there are various situations where collecting representative training samples from all classes in real-world scenarios is quite expensive, or the system may be exposed to unpredictable novel samples at the test time. The pattern classification problem is commonly referred to as an open-set recognition task in such cases where limited and incomplete knowledge of the entire data distribution is provided to the model during the training time. During test phase, unknown classes can be faced which requires the classifier to accurately classify the previously seen classes while effectively rejecting unseen ones. Among others, one-class classification serves as a plausible solution to the open-set recognition problem. Nevertheless, current one-class classifiers have their limitations. Classical kernel-based approaches require carefully designed features to obtain reasonable performance but rest on a solid basis in statistical learning theory, providing good robustness against training set impurities. More recent deep learning-based methods, on the other hand, focus on learning relevant features directly from the data but typically rely on ad hoc one-class loss functions, which very often do not generalize well and are not robust against the omnipresent noise and contamination in the training set. In this thesis, we introduce a novel approach which leverages the advantages of both kernel-based and deep-learning approaches by bringing the two learning formalisms under a common umbrella. In particular, the proposed method learns deep convolutional features to optimize a kernel Fisher null-space loss subject to a Tikhonov regularisation on the discriminant in the Hilbert space. As such, it can be trained in a deep end-to-end fashion while being robust against training set contamination. Through extensive experiments conducted on different image datasets in various evaluation settings, the proposed approach is shown to be quite robust and more effective than the current state-of-the-art methods for anomaly detection in the scenario where the training set is corrupted and contains noisy samples. At the same time, the proposed approaches can be effectively utilized in an unsupervised scenario to rank the data points based on their conformity with the majority of samples.
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