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Learning as Extraction of LowDimensional Representations
 Mechanisms of Perceptual Learning
, 1996
"... Psychophysical findings accumulated over the past several decades indicate that perceptual tasks such as similarity judgment tend to be performed on a lowdimensional representation of the sensory data. Low dimensionality is especially important for learning, as the number of examples required for a ..."
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Cited by 35 (8 self)
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Psychophysical findings accumulated over the past several decades indicate that perceptual tasks such as similarity judgment tend to be performed on a lowdimensional representation of the sensory data. Low dimensionality is especially important for learning, as the number of examples required for attaining a given level of performance grows exponentially with the dimensionality of the underlying representation space. In this chapter, we argue that, whereas many perceptual problems are tractable precisely because their intrinsic dimensionality is low, the raw dimensionality of the sensory data is normally high, and must be reduced by a nontrivial computational process, which, in itself, may involve learning. Following a survey of computational techniques for dimensionality reduction, we show that it is possible to learn a lowdimensional representation that captures the intrinsic lowdimensional nature of certain classes of visual objects, thereby facilitating further learning of tasks...
Machines, logic and quantum physics
 BULLETIN OF SYMBOLIC LOGIC
, 1999
"... Though the truths of logic and pure mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics. Recent progress in the quantum theory of computation has provided practical instances of th ..."
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Cited by 18 (0 self)
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Though the truths of logic and pure mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics. Recent progress in the quantum theory of computation has provided practical instances of this, and forces us to abandon the classical view that computation, and hence mathematical proof, are purely logical notions independent of that of computation as a physical process. Henceforward, a proof must be regarded not as an abstract object or process but as a physical process, a species of computation, whose scope and reliability depend on our knowledge of the physics of the computer concerned.
Ensembles and Experiments in Classical and Quantum Physics
 Int. J. Mod. Phys. B
, 2003
"... A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical real ..."
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Cited by 6 (3 self)
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A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization.
Noncommutative analysis and quantum physics  I. States and ensembles
"... In this sequence of papers, noncommutative analysis is used to give a consistent axiomatic approach to a unified conceptual foundation of classical and quantum physics. The present Part I defines the concepts of observables, states and ensembles, clarifies the logical relations and operations for th ..."
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Cited by 2 (2 self)
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In this sequence of papers, noncommutative analysis is used to give a consistent axiomatic approach to a unified conceptual foundation of classical and quantum physics. The present Part I defines the concepts of observables, states and ensembles, clarifies the logical relations and operations for them, and shows how they give rise to dynamics and probabilities. States are identified with maximal consistent sets of weak equations (instead of, as usual, with the rays in a Hilbert space). This leads to a concise foundation of quantum mechanics, free of unde ned terms, separating in a clear way the deterministic and the stochastic features of quantum physics. The traditional postulates of quantum mechanics are derived from wellmotivated axiomatic assumptions. No special quantum logic is needed to handle the peculiarities of quantum mechanics. Foundational problems associated with the measurement process, such as the reduction of the state vector, disappear. The new interpretation of quantum...
c World Scientic Publishing Company MATHEMATICAL TOPICS ON THE MODELLING COMPLEX MULTICELLULAR SYSTEMS AND TUMOR IMMUNE CELLS COMPETITION
, 2004
"... This paper deals with a critical analysis and some developments related to the mathematical literature on multiscale modelling of multicellular systems involving tumor immune cells competition at the cellular level. The analysis is focused on the development of mathematical methods of the classical ..."
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This paper deals with a critical analysis and some developments related to the mathematical literature on multiscale modelling of multicellular systems involving tumor immune cells competition at the cellular level. The analysis is focused on the development of mathematical methods of the classical kinetic theory to model the above physical system and to recover macroscopic equation from the microscopic description. Various hints are given toward research perspectives, with special attention on the modelling of the interplay of microscopic (at the cellular level) biological and mechanical variables on the overall evolution of the system. Indeed the nal aim of this paper consists of organizing the various contributions available in the literature into a mathematical framework suitable to generate a mathematical theory for complex biological systems.
Noncommutative analysis and quantum physics I. Quantities, ensembles and states
, 2000
"... Abstract. In this sequence of papers, noncommutative analysis is used to give a consistent axiomatic approach to a unified conceptual foundation of classical and quantum physics, free of undefined terms. The present Part I defines the concepts of quantities, ensembles, and states, clarifies the logi ..."
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Abstract. In this sequence of papers, noncommutative analysis is used to give a consistent axiomatic approach to a unified conceptual foundation of classical and quantum physics, free of undefined terms. The present Part I defines the concepts of quantities, ensembles, and states, clarifies the logical relations and operations for them, and shows how they give rise to probabilities and dynamics. The stochastic and the deterministic features of quantum physics are separated in a clear way by consistently distinguishing between ensembles (representing stochastic elements) and states (representing realistic elements). Ensembles are defined by extending the ‘probability via expectation ’ approach of Whittle to noncommuting quantities. This approach carries no connotations of unlimited repeatability; hence it can be applied to unique systems such as the universe. Precise concepts and traditional results about complementarity, uncertainty and nonlocality follow with a minimum of technicalities. Probabilities are introduced in a generality supporting socalled effects (i.e., fuzzy events). States are defined as partial mappings that provide reference values for certain quantities. An analysis of sharpness properties yields wellknown nogo theorems for hidden variables. By dropping the sharpness requirement, hidden variable theories such as Bohmian mechanics can be accommodated, but socalled ensemble states turn out to be a more natural realization of a realistic state concept. The weak law of large numbers explains the emergence of classical properties for macroscopic systems. Dynamics is introduced via a oneparameter group of automorphisms. A detailed conceptual analysis of the dynamics in terms of Poisson algebras will follow in the second part of this series. The paper realizes a strong formal implementation of Bohr’s correspondence principle. In all instances, classical and quantum concepts are fully parallel: a single common theory has a classical realization and a quantum realization.
The Sources of Certainty in Computation and Formal Systems
, 1999
"... In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathemati ..."
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In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...
The Sources of Certainty in Computation and Formal Systems
"... In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathemati ..."
Abstract
 Add to MetaCart
In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...