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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
The main result shows that when beta<1/2 convergence of the flock to a common velocity is guaranteed, while for betages1/ 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only.
Finding the Homology of Submanifolds with High Confidence from Random Samples
This work considers the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space and shows how to “learn” the homology of the sub manifold with high confidence.
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
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
The regression problem in learning theory is investigated with least square Tikhonov regularization schemes in reproducing kernel Hilbert spaces (RKHS) and the sampling operator is applied to the error analysis in both the RKHS norm and the L2 norm.
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
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