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Joaquim P. Marques de Sá, Luís M.A. Silva, Jorge M.F.'s Minimum Error Entropy Classification PDF

By Joaquim P. Marques de Sá, Luís M.A. Silva, Jorge M.F. Santos, Luís A. Alexandre

ISBN-10: 3642290280

ISBN-13: 9783642290282

ISBN-10: 3642290299

ISBN-13: 9783642290299

This booklet explains the minimal errors entropy (MEE) proposal utilized to facts class machines. Theoretical effects at the internal workings of the MEE thought, in its software to fixing quite a few type difficulties, are awarded within the wider realm of danger functionals.

Researchers and practitioners additionally locate within the publication an in depth presentation of sensible information classifiers utilizing MEE. those comprise multi‐layer perceptrons, recurrent neural networks, complexvalued neural networks, modular neural networks, and determination bushes. A clustering set of rules utilizing a MEE‐like idea is usually awarded. Examples, checks, overview experiments and comparability with comparable machines utilizing vintage methods, supplement the descriptions.

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Briefly, the tik do not form a valid probability distribution. They should be interpreted as mere switches: when a particular tik is equal to 1 (meaning that xi belongs to class ωk ), yik should be maximum and we then just minimize − ln(yik ), since all the remaining til , with l = k, are zero. 16) as cross-entropy is incorrect, we will keep it given its wide acceptance. ˆ CE risk, one should note that whenever the When applying the empirical R ˆ CE is also continuous classifier outputs are continuous and differentiable, R and differentiable.

Let us now consider two-class classifiers such that for the {−1, 1} target coding, the classifier output, Y , is restricted to the [−1, 1] interval. g. by sigmoid functions at neural network outputs) turns out to be advantageous. In this case E = T − Y takes value in [−2, 2]. If the class-conditional PDFs of Y have no discontinuities at support ends an important constraint on fE is imposed: fE (0) = 0 since lim →0 fY |−1 (−1 + ) = lim →0 fY |1 (1 + ) = 0, as illustrated in Fig. 3. Also note that e ∈ [0, 2] for fY |1 and e ∈ [−2, 0] for fY |−1 .

One can also, of course, consider −VR2 (E) as the risk functional expressed in terms of a loss −f (e). As an initial motivation to use entropic risk functionals, let us recall that entropy provides a measure of how concentrated a distribution is. For discrete distributions its minimum value (zero) corresponds to a discrete Dirac-δ PMF. For continuous distributions the minimum value (minus infinite for HS and zero for HR2 ) corresponds to a PDF represented by a sequence of continuous Dirac-δ functions, a Dirac-δ comb.

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Minimum Error Entropy Classification by Joaquim P. Marques de Sá, Luís M.A. Silva, Jorge M.F. Santos, Luís A. Alexandre

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