On the Rademacher maximal function

Tools known as maximal functions are frequently used in harmonic analysis when studying local behaviour of functions. Typically they measure the suprema of local averages of non-negative functions. It is essential that the size (more precisely, the L^p-norm) of the maximal function is comparable to the size of the original function. When dealing with families of operators between Banach spaces we are often forced to replace the uniform bound with the larger R-bound. Hence such a replacement is also needed in the maximal function for functions taking values in spaces of operators. More specifically, the suprema of norms of local averages (i.e. their uniform bound in the operator norm) has to be replaced by their R-bound. This procedure gives us the Rademacher maximal function, which was introduced by Hytönen, McIntosh and Portal in order to prove a certain vector-valued Carleson's embedding theorem. They noticed that the sizes of an operator-valued function and its Rademacher maximal function are comparable for many common range spaces, but not for all. Certain requirements on the type and cotype of the spaces involved are necessary for this comparability, henceforth referred to as the “RMF-property”. It was shown, that other objects and parameters appearing in the definition, such as the domain of functions and the exponent p of the norm, make no difference to this. After a short introduction to randomized norms and geometry in Banach spaces we study the Rademacher maximal function on Euclidean spaces. The requirements on the type and cotype are considered, providing examples of spaces without RMF. L^p-spaces are shown to have RMF not only for p greater or equal to 2 (when it is trivial) but also for 1 < p < 2. A dyadic version of Carleson's embedding theorem is proven for scalar- and operator-valued functions. As the analysis with dyadic cubes can be generalized to filtrations on sigma-finite measure spaces, we consider the Rademacher maximal function in this case as well. It turns out that the RMF-property is independent of the filtration and the underlying measure space and that it is enough to consider very simple ones known as Haar filtrations. Scalar- and operator-valued analogues of Carleson's embedding theorem are also provided. With the RMF-property proven independent of the underlying measure space, we can use probabilistic notions and formulate it for martingales. Following a similar result for UMD-spaces, a weak type inequality is shown to be (necessary and) sufficient for the RMF-property. The RMF-property is also studied using concave functions giving yet another proof of its independence from various parameters.

Computing the Stochastic Complexity of Simple Probabilistic Graphical Models

Minimum Description Length (MDL) is an information-theoretic principle that can be used for model selection and other statistical inference tasks. There are various ways to use the principle in practice. One theoretically valid way is to use the normalized maximum likelihood (NML) criterion. Due to computational difficulties, this approach has not been used very often. This thesis presents efficient floating-point algorithms that make it possible to compute the NML for multinomial, Naive Bayes and Bayesian forest models. None of the presented algorithms rely on asymptotic analysis and with the first two model classes we also discuss how to compute exact rational number solutions.

Matrix Decomposition Methods for Data Mining : Computational Complexity and Algorithms

Matrix decompositions, where a given matrix is represented as a product of two other matrices, are regularly used in data mining. Most matrix decompositions have their roots in linear algebra, but the needs of data mining are not always those of linear algebra. In data mining one needs to have results that are interpretable -- and what is considered interpretable in data mining can be very different to what is considered interpretable in linear algebra. --- The purpose of this thesis is to study matrix decompositions that directly address the issue of interpretability. An example is a decomposition of binary matrices where the factor matrices are assumed to be binary and the matrix multiplication is Boolean. The restriction to binary factor matrices increases interpretability -- factor matrices are of the same type as the original matrix -- and allows the use of Boolean matrix multiplication, which is often more intuitive than normal matrix multiplication with binary matrices. Also several other decomposition methods are described, and the computational complexity of computing them is studied together with the hardness of approximating the related optimization problems. Based on these studies, algorithms for constructing the decompositions are proposed. Constructing the decompositions turns out to be computationally hard, and the proposed algorithms are mostly based on various heuristics. Nevertheless, the algorithms are shown to be capable of finding good results in empirical experiments conducted with both synthetic and real-world data.

Linguistic grammars with very low complexity

We have presented an overview of the FSIG approach and related FSIG gram- mars to issues of very low complexity and parsing strategy. We ended up with serious optimism according to which most FSIG grammars could be decom- posed in a reasonable way and then processed efficiently.

Language universals and linguistic complexity : Three case studies in core argument marking

In this dissertation I study language complexity from a typological perspective. Since the structuralist era, it has been assumed that local complexity differences in languages are balanced out in cross-linguistic comparisons and that complexity is not affected by the geopolitical or sociocultural aspects of the speech community. However, these assumptions have seldom been studied systematically from a typological point of view. My objective is to define complexity so that it is possible to compare it across languages and to approach its variation with the methods of quantitative typology. My main empirical research questions are: i) does language complexity vary in any systematic way in local domains, and ii) can language complexity be affected by the geographical or social environment? These questions are studied in three articles, whose findings are summarized in the introduction to the dissertation. In order to enable cross-language comparison, I measure complexity as the description length of the regularities in an entity; I separate it from difficulty, focus on local instead of global complexity, and break it up into different types. This approach helps avoid the problems that plagued earlier metrics of language complexity. My approach to grammar is functional-typological in nature, and the theoretical framework is basic linguistic theory. I delimit the empirical research functionally to the marking of core arguments (the basic participants in the sentence). I assess the distributions of complexity in this domain with multifactorial statistical methods and use different sampling strategies, implementing, for instance, the Greenbergian view of universals as diachronic laws of type preference. My data come from large and balanced samples (up to approximately 850 languages), drawn mainly from reference grammars. The results suggest that various significant trends occur in the marking of core arguments in regard to complexity and that complexity in this domain correlates with population size. These results provide evidence that linguistic patterns interact among themselves in terms of complexity, that language structure adapts to the social environment, and that there may be cognitive mechanisms that limit complexity locally. My approach to complexity and language universals can therefore be successfully applied to empirical data and may serve as a model for further research in these areas.