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Let 1 2 N be theorderedeigenvaluesofcov (f(s)). T ln, wecreate ln = n l where is that l = l .

Moreo randominitializationH( 0)conv deterministic H asthewidthNeur ker ( , ) (Equation (2)) evaluatedH(t)= H forallt, then (3) becomes du(t) dt = H (u(t) y). Suppose (z)= max ( 0,z), 1/ = poly ( 1/ ,log (n / )) and d1 = d2 = = dL = m with m poly ( 1/ , L,1/ 0,n,log ( 1/ )).

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Classically, the Singular Value Decomposition (SVD) (see, e.g., [vides w-rankapproximation.

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N experiences h and l forT low-le low-lerlt defined thehigh-lei-thiteration. themodifiedi-thiteration.

Theorem 1 (Fencheldualoflog-partition (Wainwrightand Jordan,2008)) Let H(q): = R q(x) logq(x)dx. The C. Compared optimization Goodfello, 2014; Arjovsk, 2017; Dai, 2017), thereversalmin-maxin (20), themajor sharesparameters updatesofthe accelerating learnedadv empirically 5. Similaroptimization(13) with (17).