Estimating a covariance matrix from incomplete realizations of arandom vector

It is desired to estimate the mean and the covariance matrix of a Gaussian random vector from a set of independent realizations, with the complication that not every component of each realization of the random vector is observed. Subject to some restrictions, the authors obtained an exact, noniterative solution for the maximum likelihood (ML) estimates of the mean and the covariance matrix. The ML estimate of the covariance matrix that is obtained from the set of incomplete realizations is guaranteed to be positive definite, in contrast to ad hoc approaches based on averaging products of components from the same realization. The key to obtaining the ML estimates is a tractable expression for the likelihood function in terms of the Cholesky factors of the inverse covariance matrix. With this formulation, the ML estimates are found by fitting regression operators to appropriate subsets of the data. The Cholesky formulation also leads to a simple calculation by Cramer-Rao bounds     

Computational concepts in classification: Neural networks,statistical pattern recognition,and model-based vision

A large number of algorithms have been developed for classification and recognition. These algorithms can be divided into three major paradigms: statistical pattern recognition, neural networks, and model-based vision. Neural networks embody an especially rich field of approaches based on a variety of architectures, learning mechanisms, biological and algorithmic motivations, and application areas. Mathematical analysis of these approaches and paradigms reveals that there are only a few computational concepts permeating all the diverse approaches and serving as a basis for all paradigms and algorithms for classification and recognition.These basic computational concepts are reviewed in this paper with the purposes of (i) providing a mathematical continuity to seemingly disparate techniques, (ii) establishing basic mathematical limitations on applicability of existing techniques, (iii) discerning fundamental questions facing the classification field, and (iv) searching for directions in which answers to these questions may be found.     

Evolution of Languages, Consciousness and Cultures

The knowledge instinct is a fundamental mechanism of the mind that drives evolution of higher cognitive functions. Neural modeling fields and dynamic logic describe it mathematically and relate to language, concepts, emotions, and behavior. Perception and cognition, consciousness and unconsciousness, are described, while overcoming past mathematical difficulties of modeling intelligence. The two main aspects of the knowledge instinct determining evolution are differentiation and synthesis. Differentiation proceeds from and unconscious states to more crisp and conscious, from less knowledge to more knowledge; it separates concepts from emotions, Its main mechanism is language. Synthesis strives to achieve unity and meaning of knowledge; it is necessary for resolving contradictions, concentrating will and for purposeful actions. Synthesis connects language and cognition. Its main mechanisms are emotionality of languages and the hierarchy of the mind. Differentiation and synthesis are in complex relationship of symbiosis and opposition. This Leads to complex dynamics of evolution of consciousness and languages. Its mathematical modeling predicts evolution of cultures. We discuss existing evidence and future research directions.     

Cognitive high level information fusion

Fusion of sensor and communication data currently can only be performed at a late processing stage after sensor and textual information are formulated as logical statements at appropriately high level of abstraction. Contrary to this it seems, the human mind integrates sensor and language signals seamlessly, before signals are understood, at pre-conceptual level. Learning of conceptual contents of the surrounding world depends on language and vice versa. The paper describes a mathematical technique for such integration. It combines fuzzy dynamic logic with dual cognitive-language models. The paper briefly discusses relationships between the proposed mathematical technique, working of the mind and applications to understanding-based search engines.     

Conundrum of combinatorial complexity

This paper examines fundamental problems underlying difficulties encountered by pattern recognition algorithms, neural networks, and rule systems. These problems are manifested as combinatorial complexity of algorithms, of their computational or training requirements. The paper relates particular types of complexity problems to the roles of a priori knowledge and adaptive learning. Paradigms based on adaptive learning lead to the complexity of training procedures, while nonadaptive rule-based paradigms lead to complexity of rule systems. Model-based approaches to combining adaptivity with a priori knowledge lead to computational complexity. Arguments are presented for the Aristotelian logic being culpable for the difficulty of combining adaptivity and a priority. The potential role of the fuzzy logic in overcoming current difficulties is discussed. Current mathematical difficulties are related to philosophical debates of the past     

Intersections of mathematical, cognitive, and aesthetic theories of mind.

New mathematical and cognitive theories of the mind are connected to psychological theories of aesthetics. I briefly summarize recent revolutionary advancements toward understanding the mind, due to new methods of neuroimaging studies of the brain and new mathematical theories modeling the brain–mind. These new theories describe abilities for concepts, emotions, instincts, imagination, adaptation, and learning. I consider the operation of these mechanisms in the mind hierarchy. I concentrate on the emotions of satisfaction or dissatisfaction related to understanding or misunderstanding of the surrounding world. These emotions are usually below the threshold of conscious registration at lower levels (of object perception). I discuss why, and in what sense, these emotions are aesthetic, I relate them to appraisal emotions, and I argue that at higher levels of abstract cognition these emotions are related to the perception of art. The contents of cognitive representations at the top of the mind hierarchy are analyzed, and aesthetic appraisal emotions at these highest levels are related to emotions of the beautiful. I emphasize that aesthetic emotions, so important in art, are not specific to art but to cognition at the highest levels of the mind hierarchy. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Evolving Compositionality in Evolutionary Language Games

Evolutionary language games have proved a useful tool to study the evolution of communication codes in communities of agents that interact among themselves by transmitting and interpreting a fixed repertoire of signals. Most studies have focused on the emergence of Saussurean codes (i.e., codes characterized by an arbitrary one-to-one correspondence between meanings and signals). In this contribution, we argue that the standard evolutionary language game framework cannot explain the emergence of compositional codes-communication codes that preserve neighborhood relationships by mapping similar signals into similar meanings-even though use of those codes would result in a much higher payoff in the case that signals are noisy. We introduce an alternative evolutionary setting in which the meanings are assimilated sequentially and show that the gradual building of the meaning-signal mapping leads to the emergence of mappings with the desired compositional property.     

Model-based neural network for target detection in SAR images

A controversial issue in the research of mathematics of intelligence has been that of the roles of a priori knowledge versus adaptive learning. After discussing mathematical difficulties of combining a priority with adaptivity encountered in the past, we introduce a concept of a model-based neural network, whose adaptive learning is based on a priori models. Applications to target detection in SAR images are discussed. We briefly overview the SAR principles, derive relatively simple physics-based models of SAR signals, and describe model-based neural networks that utilize these models. A number of real-world application examples are presented.