Optimality of Universal Bayesian Sequence Prediction for General Loss and Alphabet

Various optimality properties of universal sequence predictors based on Bayes-mixtures in general, and Solomonoff's prediction scheme in particular, will be studied. The probability of observing $x_t$ at time $t$, given past observations $x_1...x_{t-1}$ can be computed with the chain rule if the true generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If $\mu$ is unknown, but known to belong to a countable or continuous class \M one can base ones prediction on the Bayes-mixture $\xi$ defined as a $w_\nu$-weighted sum or integral of distributions \nu\in\M. The cumulative expected loss of the Bayes-optimal universal prediction scheme based on $\xi$ is shown to be close to the loss of the Bayes-optimal, but infeasible prediction scheme based on $\mu$. We show that the bounds are tight and that no other predictor can lead to significantly smaller bounds. Furthermore, for various performance measures, we show Pareto-optimality of $\xi$ and give an Occam's razor argument that the choice $w_\nu\sim 2^{-K(\nu)}$ for the weights is optimal, where $K(\nu)$ is the length of the shortest program describing $\nu$. The results are applied to games of chance, defined as a sequence of bets, observations, and rewards. The prediction schemes (and bounds) are compared to the popular predictors based on expert advice. Extensions to infinite alphabets, partial, delayed and probabilistic prediction, classification, and more active systems are briefly discussed.Comment: 34 page

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE

Avtonomna segmentacija slik z Markovim slučajnim poljem

Segmentacija slik je zelo raziskovano področje, za katero so na voljo številni algoritmi. Naš cilj je segmentacija slike s pomočjo superpikslov na več skladnih delov in na nenadzorovan način. Da bi to dosegli, predlagamo iterativni segmentacijski algoritem. Algoritem predstavlja sliko kot slučajno polje Markova (MRF), katerega vozlišča so superpiksli, ki imajo barvne in teksturne atribute. Superpikslom dodelimo oznake na podlagi njihovih atributov s pomočjo metode podpornih vektorjev (SVM) in že omenjenega MRF in iterativno zmanjšujemo število segmentov. Negotovo segmentacijo po vsaki iteraciji se izboljšuje in rezultat je segmentacija slike na več semantično smiselnih delov, brez pomoči uporabnika. Algoritem je bil testiran na segmentacijsko podatkovno bazo in F ocene so podobne najsodobnejšim algoritmom. Glede fragmentacije slike naš pristop bistveno prekosi stanje tehnike z zmanjšanjem števila segmentov, iz katerih je sestavljen predmet zanimanja.Image segmentation is a widely-researched topic with many algorithms available. Our goal is to segment an image, in an unsupervised way, into several coherent parts with the help of superpixels. To achieve that, we propose an iterative segmentation algorithm. The algorithm models the image by a Markov random field, whose nodes are the superpixels, and each node has both color and texture features. The superpixels are assigned labels according to their features with the help of support vector machines and the aforementioned MRF and the number of segments is iteratively reduced. The result is a segmentation of an image into several regions with requiring any user input. The segmentation algorithm was tested on a standard evaluation database, and performs on par with state-of-the-art segmentation algorithms in F-measures. In terms of oversegmentation, our approach significantly outperforms the state of the art by greatly reducing the oversegmentation of the object of interest