This work investigates variants of Lloyd's heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and proposes and justifies a clusterability criterion for data sets.Expand

A deterministic method for simulating noiseless-channel protocols on noisy channels, with only a constant slowdown is described, an analog for general, interactive protocols of Shannon's coding theorem, which deals only with data transmission, i.e., one-way protocols.Expand

A fairly general method for finding deterministic constructions obeying k-restrictions, which yields structures of size not much larger than the probabilistic bound and imply the very efficient derandomization of algorithms in learning, of fixed-subgraph finding algorithms, and of near optimal /spl Sigma/II/Spl Sigma/ threshold formulae.Expand

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The author relates the noisy channel and the standard (noise less channel) complexities of a communication problem by establishing a 'two-way' or interactive analogue of Shanon's coding theorem, which involves simulating the original protocol while implementing a hierarchical system of progress checks which ensure that errors of any magnitude in the simulation are, with high probability, rapidly eliminated.Expand

It is shown that any distributed protocol which runs on a noiseless network in time T, can be simulated on an identical noisy network with a slow-down factor proportional to log(d + 1), where d is the maximum degree in the network, and with exponentially small probability of error.Expand

A deterministic method for simulating noiseless-channel protocols on noisy channels, with only a constant slow-down is described, an analog for general interactive protocols of Shannon's coding theorem, which dealt only with data transmission, i.e. one-way protocols.Expand

We provide positive and negative results concerning the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer.

Let <i>X</i> be a space and <i>F</i> a family of 0, 1-valued functions on <i>X</i>. Vapnik and Chervonenkis showed that if <i>F</i> is "simple" (finite VC dimension), then for every probability… Expand

The information carried by a signal decays when the signal is corrupted by random noise is studied and a tight bound is obtained on the rate at which information decreases as a signal crosses a noisy channel to obtain depth lower bounds in the noisy circuit model of computation defined by von Neumann.Expand