Model complexities of shallow networks representing highly varying functions

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• 作者：Věra Kůrková ; ^a ; ^{vera@cs.cas.cz" class="auth_mail" title="E-mail the corresponding author}Author Vitae ; Marcello Sanguineti^b ; ^{marcello.sanguineti@unige.it" class="auth_mail" title="E-mail the corresponding author}Author Vitae
• 关键词：Shallow networks ; Model complexity ; Highly varying functions ; Chernoff Bound ; Perceptrons ; Gaussian kernel units
• 刊名：Neurocomputing
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：171
• 期：Complete
• 页码：598-604
• 全文大小：384 K

文摘

Model complexities of shallow (i.e., one-hidden-layer) networks representing highly varying multivariable class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0925231215009893&_mathId=si0001.gif&_user=111111111&_pii=S0925231215009893&_rdoc=1&_issn=09252312&md5=4d8d62288befbb140652af041aeb98ed" title="Click to view the MathML source">{−1,1}class="mathContainer hidden">class="mathCode">$\{-1,1\}$-valued functions are studied in terms of variational norms tailored to dictionaries of network units. It is shown that bounds on these norms define classes of functions computable by networks with constrained numbers of hidden units and sizes of output weights. Estimates of probabilistic distributions of values of variational norms with respect to typical computational units, such as perceptrons and Gaussian kernel units, are derived via geometric characterization of variational norms combined with the probabilistic Chernoff Bound. It is shown that almost any randomly chosen class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0925231215009893&_mathId=si0001.gif&_user=111111111&_pii=S0925231215009893&_rdoc=1&_issn=09252312&md5=4d8d62288befbb140652af041aeb98ed" title="Click to view the MathML source">{−1,1}class="mathContainer hidden">class="mathCode">$\{-1,1\}$-valued function on a sufficiently large d-dimensional domain has variation with respect to perceptrons depending on d exponentially.