Online Learning Algorithms in Hilbert Spaces with $β-$ and $φ-$Mixing Sequences

In this paper, we study an online algorithm in a reproducing kernel Hilbert spaces (RKHS) based on a class of dependent processes, called the mixing process. For such a process, the degree of dependence is measured by various mixing coefficients. As a representative example, we analyze a strictly stationary Markov chain, where the dependence structure is characterized by the \(\beta-\) and \(\phi-\)mixing coefficients. For these dependent samples, we derive nearly optimal convergence rates. Our findings extend existing error bounds for i.i.d. observations, demonstrating that the i.i.d. case is a special instance of our framework. Moreover, we explicitly account for an additional factor introduced by the dependence structure in the Markov chain.

Upper Bounds for Learning in Reproducing Kernel Hilbert Spaces for Orbits of an Iterated Function System

One of the key problems in learning theory is to compute a function $f$ that closely approximates the relationship between some input $x$ and corresponding output $y$, such that $y\approx f(x)$. This approximation is based on sample points $(x_t,y_t)_{t=1}^{m}$, where the function $f$ can be approximated within reproducing kernel Hilbert spaces using various learning algorithms. In the context of learning theory, it is usually customary to assume that the sample points are drawn independently and identically distributed (i.i.d.) from an unknown underlying distribution. However, we relax this i.i.d. assumption by considering an input sequence $(x_t)_{t\in {\mathbb N}}$ as a trajectory generated by an iterated function system, which forms a particular Markov chain, with $(y_t)_{t\in {\mathbb N}}$ corresponding to an observation sequence when the model is in the corresponding state $x_t$. For such a process, we approximate the function $f$ using the Markov chain stochastic gradient algorithm and estimate the error by deriving upper bounds within reproducing kernel Hilbert spaces.

OCTID: Optical Coherence Tomography Image Database

Optical coherence tomography (OCT) is a non-invasive imaging modality which is widely used in clinical ophthalmology. OCT images are capable of visualizing deep retinal layers which is crucial for the early diagnosis of retinal diseases. In this paper, we describe a comprehensive open-access database containing more than 500 high-resolution images categorized into different pathological conditions. The image classes include Normal (NO), Macular Hole (MH), Age-related Macular Degeneration (AMD), Central Serous Retinopathy, and Diabetic Retinopathy (DR). The images were obtained from a raster scan protocol with a 2mm scan length and 512x1024 pixels resolution. We have also included 25 normal OCT images with their corresponding ground truth delineations which can be used for accurate evaluation of OCT image segmentation.