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- W125514694 abstract "Suppose that an infinite sequence is produced by independent trials of a random variable with a fixed distribution. The Shannon-Weaver entropy of the sequence determines the minimum bit rate needed to transmit the values of the sequence. We show that if the source distribution is highly concentrated, as is commonly observed in practice, then its entropy is equal to the logarithm of the theoretical dimension of the sequence. We conclude that the best-basis algorithm, which minimizes this theoretical dimension over a library of transformations, both chooses the transformation that yields best compression and also gives an estimate of the compression rate. 1 Model We need to define some basic objects. First, suppose that p = p(t) is a probability density function, i.e., a real valued, nonnegative, integrable function defined on [0,1] which satisfies Jo p{t)dt = 1. For each measurable subset E C [0,1] we define the associated probability measure by P{E} ^ f p(t)dt. (1) •Research supported by NSF, AFOSR, and the Southwestern Bell Telephone Company Work partially supported by the Army Research Office through the Center for Nonlinear Analysis For technical reasons, we will assume that the density function p is continuous and strictly positive on (0,1). Secondly, fix 1 <& N < oo and define (uniform) quantization to N values by the formula" @default.
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- W125514694 date "1995-01-01" @default.
- W125514694 modified "2023-09-27" @default.
- W125514694 title "A relation between Shannon-Weaver entropy and theoretical dimension for classes of smooth functions" @default.
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