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On the mathematical foundations of learning
(1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the…
Emergent Behavior in Flocks
TLDR
Finding the Homology of Submanifolds with High Confidence from Random Samples
- P. Niyogi, S. Smale, S. Weinberger
- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 4 March 2008
TLDR
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-complete…
Newton’s Method Estimates from Data at One Point
- S. Smale
- Mathematics
- 1986
Newton’s method and its modifications have long played a central role in finding solutions of non-linear equations and systems. The work of Kantorovich has been seminal in extending and codifying…
Learning Theory Estimates via Integral Operators and Their Approximations
- S. Smale, Ding-Xuan Zhou
- Mathematics, Computer Science
- 15 March 2007
TLDR
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 such…
Mathematical problems for the next century
- S. Smale
- Art
- 1 March 1998
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 is…
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