Differential Priors for Elastic Nets

The elastic net and related algorithms, such as generative topographic mapping, are key methods for discretized dimension-reduction problems. At their heart are priors that specify the expected topological and geometric properties of the maps. However, up to now, only a very small subset of possible priors has been considered. Here we study a much more general family originating from discrete, high-order derivative operators. We show theoretically that the form of the discrete approximation to the derivative used has a crucial influence on the resulting map. Using a new and more powerful iterative elastic net algorithm, we confirm these results empirically, and illustrate how different priors affect the form of simulated ocular dominance columns.

The real-time refinement calculus: A foundation for machine-independent real-time programming

The real-time refinement calculus is an extension of the standard refinement calculus in which programs are developed from a precondition plus post-condition style of specification. In addition to adapting standard refinement rules to be valid in the real-time context, specific rules are required for the timing constructs such as delays and deadlines. Because many real-time programs may be nonterminating, a further extension is to allow nonterminating repetitions. A real-time specification constrains not only what values should be output, but when they should be output. Hence for a program to implement such a specification, it must guarantee to output values by the specified times. With standard programming languages such guarantees cannot be made without taking into account the timing characteristics of the implementation of the program on a particular machine. To avoid having to consider such details during the refinement process, we have extended our real-time programming language with a deadline command. The deadline command takes no time to execute and always guarantees to meet the specified time; if the deadline has already passed the deadline command is infeasible (miraculous in Dijkstra's terminology). When such a realtime program is compiled for a particular machine, one needs to ensure that all execution paths leading to a deadline are guaranteed to reach it by the specified time. We consider this checking as part of an extended compilation phase. The addition of the deadline command restores for the real-time language the advantage of machine independence enjoyed by non-real-time programming languages.