We show that recent results in [3] on risk bounds for regularized least-squares on reproducing kernel Hilbert spaces can be straightforwardly extended to the vector-valued regression setting. We first briefly introduce central concepts on operator-valued kernels. Then we show how risk bounds can be expressed in terms of a generalization of effective dimension.

This paper presents an approach to model selection for regularized least-squares on reproducing kernel Hilbert spaces in the semi-supervised setting. The role of effective dimension was recently shown to be crucial in the definition of a rule for the choice of the regularization parameter, attaining asymptotic optimal performances in a minimax sense. The main goal of the present paper is showing how the effective dimension can be replaced by an empirical counterpart while conserving optimality. The empirical effective dimension can be computed from independent unlabelled samples. This makes the approach particularly appealing in the semi-supervised setting.

Quasi-Monte Carlo (QMC) methods are playing an increasingly important role in computational finance. This is attributed to the increased complexity of the derivative securities and the sophistication of the financial models. Simple closed-form solutions for the finance applications typically do not exist and hence numerical methods need to be used to approximate their solutions. QMC method has been proposed as an alternative method to Monte Carlo (MC) method to accomplish this objective. Unlike MC methods, the efficiency of QMC-based methods is highly dependent on the dimensionality of the problems. In particular, numerous researches have documented, under the Black-Scholes models, the critical role of the generating matrix for simulating the Brownian paths. Numerical results support the notion that generating matrix that reduces the effective dimension of the underlying problems is able to increase the efficiency of QMC. Consequently, dimension reduction methods such as principal component analysis, Brownian bridge, Linear Transformation and Orthogonal Transformation have been proposed to further enhance QMC. Motivated by these results, we first propose a new measure to quantify the effective dimension. We then propose a new dimension reduction method which we refer as the directional method (DC). The proposed DC method has the advantage that it depends explicitly on the given function of interest. Furthermore, by assigning appropriately the direction of importance of the given function, the proposed method optimally determines the generating matrix used to simulate the Brownian paths. Because of the flexibility of our proposed method, it can be shown that many of the existing dimension reduction methods are special cases of our proposed DC methods. Finally, many numerical examples are provided to support the competitive efficiency of the proposed method.

Quasi-Monte Carlo (QMC) methods are playing an increasingly important role in computational finance. This is attributed to the increased complexity of the derivative securities and the sophistication of the financial models. Simple closed-form solutions for the finance applications typically do not exist and hence numerical methods need to be used to approximate their solutions. QMC method has been proposed as an alternative method to Monte Carlo (MC) method to accomplish this objective. Unlike MC methods, the efficiency of QMC-based methods is highly dependent on the dimensionality of the problems. In particular, numerous researches have documented, under the Black-Scholes models, the critical role of the generating matrix for simulating the Brownian paths. Numerical results support the notion that generating matrix that reduces the effective dimension of the underlying problems is able to increase the efficiency of QMC. Consequently, dimension reduction methods such as principal component analysis, Brownian bridge, Linear Transformation and Orthogonal Transformation have been proposed to further enhance QMC. Motivated by these results, we first propose a new measure to quantify the effective dimension. We then propose a new dimension reduction method which we refer as the directional method (DC). The proposed DC method has the advantage that it depends explicitly on the given function of interest. Furthermore, by assigning appropriately the direction of importance of the given function, the proposed method optimally determines the generating matrix used to simulate the Brownian paths. Because of the flexibility of our proposed method, it can be shown that many of the existing dimension reduction methods are special cases of our proposed DC methods. Finally, many numerical examples are provided to support the competitive efficiency of the proposed method.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22975号 / 理博第4652号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 山口 孝男, 教授 藤原 耕二, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM