• Publications
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On the mathematical foundations of learning
TLDR
We try to emphasize relations of the theory of learning to the mainstream of mathematics. Expand
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Emergent Behavior in Flocks
  • F. Cucker, S. Smale
  • Mathematics, Computer Science
  • IEEE Transactions on Automatic Control
  • 15 May 2007
TLDR
We provide a model (for both continuous and discrete time) describing the evolution of a flock. Expand
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On a theory of computation and complexity over the real numbers: $NP$- completeness, recursive functions and universal machines
We present a model for computation over the reals or an arbitrary (ordered) ring R. In this general setting, we obtain universal machines, partial recursive functions, as well as JVP-completeExpand
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Learning Theory Estimates via Integral Operators and Their Approximations
The regression problem in learning theory is investigated with least square Tikhonov regularization schemes in reproducing kernel Hilbert spaces (RKHS). We follow our previous work and apply theExpand
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Mathematical problems for the next century
V. I. Arnold, on behalf of the International Mathematical Union has written to a number of mathematicians with a suggestion that they describe some great problems for the next century. This report isExpand
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On the mathematics of emergence
Abstract.We describe a setting where convergence to consensus in a population of autonomous agents can be formally addressed and prove some general results establishing conditions under which suchExpand
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Differentiable dynamical systems
This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is aExpand
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The Mathematics of Learning: Dealing with Data
  • T. Poggio, S. Smale
  • Computer Science
  • International Conference on Neural Networks and…
  • 13 October 2005
TLDR
We outline the mathematical foundations of learning theory and describe a key algorithm of it. Expand
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Complexity of Bezout’s Theorem II Volumes and Probabilities
In this paper we study volume estimates in the space of systems of n homegeneous polynomial equations of fixed degrees d i with respect to a natural Hermitian structure on the space of such systemsExpand
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