Related papers: Lowness, randomness, and computable analysis
The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed…
The purpose of this article is to introduce a new analytical framework dedicated to measuring performance of recommender systems. The standard approach is to assess the quality of a system by means of accuracy related statistics. However,…
Several concepts on the measure of observability, reachability, and robustness are defined and illustrated for both linear and nonlinear control systems. Defined by using computational dynamic optimization, these concepts are applicable to…
Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in…
The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure.
Here we briefly discuss how negative numbers, or "negative probabilities", can naturally arise in probabilistic expressions and be given an operational interpretation. Like the use of negative numbers in arithmetical expressions, the use of…
Algorithmic robustness refers to the sustained performance of a computational system in the face of change in the nature of the environment in which that system operates or in the task that the system is meant to perform. Below, we motivate…
We survey the diverse approaches to the notion of information content: from Shannon entropy to Kolmogorov complexity. The main applications of Kolmogorov complexity are presented namely, the mathematical notion of randomness (which goes…
The aim of this note (as well as of the course itself) is to give a largely self-contained proof of two of the main results in the field of low-rank matrix recovery. This field aims for identification of low-rank matrices from only limited…
Exponentiation makes the difference between the bit-size of this line and the number (<< 2^{300}) of particles in the known Universe. The expulsion of exponential time algorithms from Computer Theory in the 60's broke its umbilical cord…
Recent controversy regarding the meaning and usefulness of weak values is reviewed. It is argued that in spite of recent statistical arguments by Ferrie and Combes, experiments with anomalous weak values provide a useful amplification…
Herein we consider various concepts of entropy as measures of the complexity of phenomena and in so doing encounter a fundamental problem in physics that affects how we understand the nature of reality. In essence the difficulty has to do…
The rationalizability concept was introduced in \cite{Ber84} and \cite{Pea84} to assess what can be inferred by rational players in a non-cooperative game in the presence of common knowledge. However, this notion can be defined in a number…
A method for testing nonlinearity in time series is described based on information-theoretic functionals -- redundancies, linear and nonlinear forms of which allow either qualitative, or, after incorporating the surrogate data technique,…
What is a question? According to Cox a question can be identified with the set of assertions that constitute possible answers. In this paper we propose a different approach that combines the notion that questions are requests for…
In-degree, PageRank, number of visits and other measures of Web page popularity significantly influence the ranking of search results by modern search engines. The assumption is that popularity is closely correlated with quality, a more…
Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…
The Boolean lattice of logical statements induces the free distributive lattice of questions. Inclusion on this lattice is based on whether one question answers another. Generalizing the zeta function of the question lattice leads to a…
The use of terms from natural and social scientific titles and abstracts is studied from the perspective of sublanguages and their specialized dictionaries. Different notions of sublanguage distinctiveness are explored. Objective methods…
We introduce probability estimation, a broadly applicable framework to certify randomness in a finite sequence of measurement results without assuming that these results are independent and identically distributed. Probability estimation…