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In coding and information theory, it is desirable to construct maximal codes that can be either variable length codes or error control codes of fixed length. However deciding code maximality boils down to deciding whether a given NFA is…
We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We analyze numerically various ensembles of linear programming problems and obtain, for each of these…
The concept of randomized measurements on individual particles has proven to be useful for analyzing quantum systems and is central for methods like shadow tomography of quantum states. We introduce $\textit{collective}$ randomized…
Complexity is an interdisciplinary concept which, first of all, addresses the question of how order emerges out of randomness. For many reasons matrices provide a very practical and powerful tool in approaching and quantifying the related…
A good process model is expected not only to reflect the behavior of the process, but also to be as easy to read and understand as possible. Because preferences vary across different applications, numerous measures provide ways to reflect…
Today's probabilistic language generators fall short when it comes to producing coherent and fluent text despite the fact that the underlying models perform well under standard metrics, e.g., perplexity. This discrepancy has puzzled the…
The classical binary hypothesis testing problem is revisited. We notice that when one of the hypotheses is composite, there is an inherent difficulty in defining an optimality criterion that is both informative and well-justified. For…
Though it has been recognized that recommending serendipitous (i.e., surprising and relevant) items can be helpful for increasing users' satisfaction and behavioral intention, how to measure serendipity in the offline environment is still…
We extend previously proposed measures of complexity, emergence, and self-organization to continuous distributions using differential entropy. This allows us to calculate the complexity of phenomena for which distributions are known. We…
Random projections offer an appealing and flexible approach to a wide range of large-scale statistical problems. They are particularly useful in high-dimensional settings, where we have many covariates recorded for each observation. In…
The problem of defining and studying complexity of a time series has interested people for years. In the context of dynamical systems, Grassberger has suggested that a slow approach of the entropy to its extensive asymptotic limit is a sign…
Algorithmic efficiency is essential to reducing energy and time usage for computational problems. Optimizing efficiency is important for tasks involving multiple resources, for example in stochastic calculations where the size of the random…
The linear complexity of a sequence $s$ is one of the measures of its predictability. It represents the smallest degree of a linear recursion which the sequence satisfies. There are several algorithms to find the linear complexity of a…
A two-parameter family of complexity measures $\tilde{C}^{(\alpha,\beta)}$ based on the R\'enyi entropies is introduced and characterized by a detailed study of its mathematical properties. This family is the generalization of a continuous…
In an attempt to better understand generalization in deep learning, we study several possible explanations. We show that implicit regularization induced by the optimization method is playing a key role in generalization and success of deep…
Given the increasing popularity of algorithms for overlapping clustering, in particular in social network analysis, quantitative measures are needed to measure the accuracy of a method. Given a set of true clusters, and the set of clusters…
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and…
The wealth of data being gathered about humans and their surroundings drives new machine learning applications in various fields. Consequently, more and more often, classifiers are trained using not only numerical data but also complex data…
This work starts from definition of randomness, the results of algorithmic randomness are analyzed from the perspective of application. Then, the source and nature of randomness is explored, and the relationship between infinity and…
In this paper, we develop efficient randomized algorithms for estimating probabilistic robustness margin and constructing robustness degradation curve for uncertain dynamic systems. One remarkable feature of these algorithms is their…