Refined Arithmetic, Geometric and Harmonic Mean Inequalities

Mercer, Peter R.

doi:10.1216/rmjm/1181075474

Cited by 19 publications

(13 citation statements)

References 4 publications

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“…We note here that (4.5) also improves Mercer's proposition [20,Proposition 4], as one checks directly. To end this section, we give a refinement of (1.2) based on the results in [20].…”

Section: Some Equivalent Inequalitiessupporting

confidence: 54%

A new approach to Ky Fan‐type inequalities

Gao

2005

International Journal of Mathematics and Mathematical Sciences

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“…We note here that (4.5) also improves Mercer's proposition [20,Proposition 4], as one checks directly. To end this section, we give a refinement of (1.2) based on the results in [20].…”

Section: Some Equivalent Inequalitiessupporting

confidence: 54%

A new approach to Ky Fan‐type inequalities

Gao

2005

International Journal of Mathematics and Mathematical Sciences

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“…This inequality was originally proved by Cartwright and Field [20]. Several refinements, such as

\frac{\max_{i} O_{i} - G}{2 \max_{i} O_{i}} V a r (O) \leq E - G \leq \frac{\min_{i} O_{i} - G}{2 \min_{i} O_{i} (\min_{i} O_{i} - E)} V a r (O)

\frac{1}{2 \max_{i} O_{i}} \sum_{i} p_{i} {(O_{i} - G)}^{2} \leq E - G \leq \frac{1}{2 \min_{i} O_{i}} \sum_{i} p_{i} {(O_{i} - G)}^{2}

as well as other interesting bounds can be found in [4, 5, 31, 32, 1, 2]. …”

Section: Weighted Arithmetic Geometric and Normalized Geometric Mmentioning

confidence: 99%

The dropout learning algorithm

Baldi

Sadowski

2014

Artificial Intelligence

324

227

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Dropout is a recently introduced algorithm for training neural network by randomly dropping units during training to prevent their co-adaptation. A mathematical analysis of some of the static and dynamic properties of dropout is provided using Bernoulli gating variables, general enough to accommodate dropout on units or connections, and with variable rates. The framework allows a complete analysis of the ensemble averaging properties of dropout in linear networks, which is useful to understand the non-linear case. The ensemble averaging properties of dropout in non-linear logistic networks result from three fundamental equations: (1) the approximation of the expectations of logistic functions by normalized geometric means, for which bounds and estimates are derived; (2) the algebraic equality between normalized geometric means of logistic functions with the logistic of the means, which mathematically characterizes logistic functions; and (3) the linearity of the means with respect to sums, as well as products of independent variables. The results are also extended to other classes of transfer functions, including rectified linear functions. Approximation errors tend to cancel each other and do not accumulate. Dropout can also be connected to stochastic neurons and used to predict firing rates, and to backpropagation by viewing the backward propagation as ensemble averaging in a dropout linear network. Moreover, the convergence properties of dropout can be understood in terms of stochastic gradient descent. Finally, for the regularization properties of dropout, the expectation of the dropout gradient is the gradient of the corresponding approximation ensemble, regularized by an adaptive weight decay term with a propensity for self-consistent variance minimization and sparse representations.

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“…This is motivated by a recent result of Mercer [12], which refines (1.1) for the case r = 1, s = 0 by using Hadamard's inequality. A version of his result is given by Theorem 4.3 of [8] (note there is a typo in the original statement though):…”

Section: More Ky Fan-type Inequalitiesmentioning

confidence: 92%

“…Cartwright and Field [5] first proved the validity of (1.1) for r = 1, s = 0. For other extensions and refinements of (1.1), see [3,7,[10][11][12] and [8]. Inequality (1.2) is commonly referred as the additive Ky Fan's inequality.…”

Section: Introductionmentioning

confidence: 99%