*Speaker*:**Baver Okutmuştur**
(Department of Mathematics, METU)

*Date / Time*:**Thursday, April 25, 2024 / 14:00**

*Place*:**Online Participation** (please self-register first)

**Abstract:** In this talk a short introduction to reproducing kernel and reproducing kernel Hilbert spaces (RKHS) is provided. We first memorize a brief overview of the Hilbert space with its fundamental characteristics. Here the basic facts on the inner product spaces including the notion of norms, pre-Hilbert spaces, and hence Hilbert spaces are briefly presented (for further details see [3]). The main part is then devoted to the definitions and fundamental properties of reproducing kernels and RKHSs. The proofs of most of the theorems are adressed to the lecture notes [1] of T. Ando, the paper [2] of N. Aronszajn and particularly the recent book [5] of S. Saitoh and Y. Sawano.

**References:**

- T. Ando, Reproducing Kernel Spaces and Quadratic Inequalities, Lecture Notes, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo, Japan, 1987.
- N. Aronszajn, Theory of reproducing kernels, TAMS Vol. 68, No.3, 1950, pp. 337-404.
- J.B. Conway, A Course in Functional Analysis, Springer Verlag, Berlin - Heidelberg - New York, 1989.
- P.L. Duren, A. Schuster, Bergman Spaces, Amer. Math. Soc., Providence R.I. 2004.
- S. Saitoh, Y. Sawano, Theory of Reproducing Kernels and Applications Springer, 2016.

*Speaker*:**Ömür Uğur**
(Institute of Applied Mathematics, METU)

*Date / Time*:**Tuesday, March 26, 2024 / 15:40**

*Place*:**S-212 @ IAM** /
**Online Participation** (please self-register first)

**Abstract:** In this talk, we will be reviewing the vectors and matrices to prepare ourselves for the inner products as well as the maps that are essential for RKHSs. Hopefully, many examples will be provided to illustrate and possibly to verify the underlying idea and the key points in and usage of the finite-dimensional RKHSs in applications. The talk will be somewhat a modified version of the Second Chapter of the book “A Primer on Reproducing Kernel Hilbert Spaces” by J. H. Manton and P.-O. Amblard (Manton & Amblard, 2015); and hopefully, it will bridge the gap between applications and theoretical considerations of RKHS that will follow within these seminar series.

- Manton, J. H., & Amblard, P.-O. (2015).
*A Primer on Reproducing Kernel Hilbert Spaces*. https://doi.org/10.1561/2000000050

*Speaker*:**Aydın Aytuna**
(Emeritus, Affiliated Faculty at the Institute of Applied Mathematics, METU)

*Date / Time*:**Thursday, March 21, 2024 / 15:40**

*Place*:**S-212 @ IAM** /
**Online Participation** (please self-register first)

**Abstract:** These notes are written for the ‘‘RKHS Learning Seminars’’ at the Institute of Applied Mathematics, METU. It aims to introduce the audience to the fundamental ideas of elementary Hilbert space theory. We assume the participants have good knowledge of linear algebra and advanced calculus. The material covered is relatively standard and contains no new mathematics. The book “A Primer on Reproducing Kernel Hilbert Spaces” by J. H. Manton and P. Amblard (Manton & Amblard, 2015) was chosen as the principal reference for this seminar series. Hence, it shaped the structure of these notes. Additional references used in these notes, plus references that can be used for further studies, are listed below.

*Additional Material*: Lecture Notes of Aydın Aytuna

- Manton, J. H., & Amblard, P.-O. (2015).
*A Primer on Reproducing Kernel Hilbert Spaces*. https://doi.org/10.1561/2000000050

- H. Aronszajn: Theory of Reproducing Kernels, TAMS Vol. 68, No.3, 1950, pp. 337-404
- E. Kreyszig: Introductory functional analysis with applications, John Wiley & Sons, 1978
- R. Meise, D. Vogt: Introduction to Functional Analysis, Oxford Graduate Texts in Mathematics, Band 2, 1997
- B. Okutmuştur: “A Survey on Hilbert Spaces and Reproducing Kernels” in Functional Calculus, London: Intech Open, 2020, pp 61-77
- V. Paulsen, M. Raghupathi: An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge University Press 2016
- W. Rudin: Functional Analysis, 2. Edition Mc Graw-Hill

*Speaker*:**Ömür Uğur**
(Institute of Applied Mathematics, METU)

*Date / Time*:**Thursday, March 14, 2024 / 15:40**

*Place*:**S-212 @ IAM** /
**Online Participation** (please self-register first)

**Abstract:** In this talk, we will be reviewing the vectors and matrices to prepare ourselves for the inner products as well as the maps that are essential for RKHSs. Hopefully, many examples will be provided to illustrate and possibly to verify the underlying idea and the key points in and usage of the finite-dimensional RKHSs in applications. The talk will be somewhat a modified version of the Second Chapter of the book “A Primer on Reproducing Kernel Hilbert Spaces” by J. H. Manton and P.-O. Amblard (Manton & Amblard, 2015); and hopefully, it will bridge the gap between applications and theoretical considerations of RKHS that will follow within these seminar series.

- Manton, J. H., & Amblard, P.-O. (2015).
*A Primer on Reproducing Kernel Hilbert Spaces*. https://doi.org/10.1561/2000000050

*Speaker*:**Aydın Aytuna**
(Emeritus, Affiliated Faculty at the Institute of Applied Mathematics, METU)

*Date / Time*:**Tuesday, March 19, 2024 / 15:40**

*Place*:**S-212 @ IAM** /
**Online Participation** (please self-register first)

**Abstract:** These notes are written for the ‘‘RKHS Learning Seminars’’ at the Institute of Applied Mathematics, METU. It aims to introduce the audience to the fundamental ideas of elementary Hilbert space theory. We assume the participants have good knowledge of linear algebra and advanced calculus. The material covered is relatively standard and contains no new mathematics. The book “A Primer on Reproducing Kernel Hilbert Spaces” by J. H. Manton and P. Amblard (Manton & Amblard, 2015) was chosen as the principal reference for this seminar series. Hence, it shaped the structure of these notes. Additional references used in these notes, plus references that can be used for further studies, are listed below.

*Additional Material*: Lecture Notes of Aydın Aytuna

*A Primer on Reproducing Kernel Hilbert Spaces*. https://doi.org/10.1561/2000000050

- H. Aronszajn: Theory of Reproducing Kernels, TAMS Vol. 68, No.3, 1950, pp. 337-404
- E. Kreyszig: Introductory functional analysis with applications, John Wiley & Sons, 1978
- R. Meise, D. Vogt: Introduction to Functional Analysis, Oxford Graduate Texts in Mathematics, Band 2, 1997
- B. Okutmuştur: “A Survey on Hilbert Spaces and Reproducing Kernels” in Functional Calculus, London: Intech Open, 2020, pp 61-77
- V. Paulsen, M. Raghupathi: An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge University Press 2016
- W. Rudin: Functional Analysis, 2. Edition Mc Graw-Hill

*Speaker*:**Ömür Uğur**
(Institute of Applied Mathematics, METU)

*Date / Time*:**Tuesday, March 12, 2024 / 15:40**

*Place*:**S-212 @ IAM** /
**Online Participation** (please self-register first)

**Abstract:** In this talk, we will be reviewing the vectors and matrices to prepare ourselves for the inner products as well as the maps that are essential for RKHSs. Hopefully, many examples will be provided to illustrate and possibly to verify the underlying idea and the key points in and usage of the finite-dimensional RKHSs in applications. The talk will be somewhat a modified version of the Second Chapter of the book “A Primer on Reproducing Kernel Hilbert Spaces” by J. H. Manton and P.-O. Amblard (Manton & Amblard, 2015); and hopefully, it will bridge the gap between applications and theoretical considerations of RKHS that will follow within these seminar series.

*A Primer on Reproducing Kernel Hilbert Spaces*. https://doi.org/10.1561/2000000050

*Speaker*:**Bülent Karasözen**
(Emeritus, Affiliated Faculty at the Institute of Applied Mathematics, METU)

*Date / Time*:**Tuesday, March 5, 2024 / 15:40**

*Place*:**Online Participation** (please self-register first)

**Abstract:** In this talk, application kernels in machine learning are
presented such as separating and detecting similarity between the objects.
Construction and operation with kernels,
such as the kernel trick and different types of kernels are introduced.
A non-technical introduction to reproducing kernel Hilbert spaces (RKHSs) are given,
and some basic algorithms in RHKS are described.
Ridge regression as an application of the kernels is discussed.

The *tentative* outline of the (*learning part* of the) seminars
is as follows.

- Introduction and Motivation: kernels, kernel principle component analysis, kernel ridge regression (See RKHS in Machine Learning and (Ghojogh et al., 2023))
- Finite Dimensional RKHSs (See (Manton & Amblard, 2015))
- Prerequisites: metric, normed, and unitary spaces; Cauchy sequences and Completion, Banach and Hilbert spaces, bounded linear operators and the Riesz Theorem (See Lecture Notes of Aydın Aytuna)
- Finite and Infinite Dimensional RKHSs (See (Manton & Amblard, 2015) and (Okutmuştur, 2020))
- Applications to Stochastic Processes (See (Manton & Amblard, 2015))

- Ghojogh, B., Crowley, M., Karray, F., & Ghodsi, A. (2023).
*Elements of Dimensionality Reduction and Manifold Learning*. Springer International Publishing. https://doi.org/10.1007/978-3-031-10602-6 *A Primer on Reproducing Kernel Hilbert Spaces*. https://doi.org/10.1561/2000000050- Okutmuştur, B. (2020). A Survey on Hilbert Spaces and Reproducing Kernels. In K. Shah & B. Okutmuştur (Eds.),
*Functional Calculus*. IntechOpen. https://doi.org/10.5772/intechopen.91479