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On the mathematical foundations of learning
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
We provide a model (for both continuous and discrete time) describing the evolution of a flock. Expand
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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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Learning Theory: An Approximation Theory Viewpoint: Index
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SIAM Journal on Optimization
The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. Expand
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Best Choices for Regularization Parameters in Learning Theory: On the Bias—Variance Problem
We find a unique minimizer of E(γ ) which is found by an easy algorithm to yield the “best” regularization parameter γ = γ ∗. Expand
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Learning Theory: An Approximation Theory Viewpoint
Preface Foreword 1. The framework of learning 2. Basic hypothesis spaces 3. Estimating the sample error 4. Polynomial decay approximation error 5. Estimating covering numbers 6. Logarithmic decayExpand
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A new condition number for linear programming
Abstract.In this paper we define a new condition number ?(A) for the following problem: given a m by n matrix A, find x∈ℝn, s.t. Ax<0. We characterize this condition number in terms of distance toExpand
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Avoiding Collisions in Flocks
  • F. Cucker, J. Dong
  • Mathematics, Computer Science
  • IEEE Transactions on Automatic Control
  • 5 February 2010
We show that, for this modified model, convergence to flocking is established along the same lines while, in addition, avoidance of collisions (i.e., the respect of a minimal distance between particles) is ensured. Expand
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