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Questions tagged [regularization]

Regularization, in mathematics and statistics and particularly in the fields of machine learning and inverse problems, refers to a process of introducing additional information in order to solve an ill-posed problem or to prevent overfitting. (Def: http://en.wikipedia.org/wiki/Regularization_(mathematics))

Regularization, in mathematics and statistics and particularly in the fields of machine learning and inverse problems, refers to a process of introducing additional information in order to solve an ill-posed problem or to prevent overfitting. Reference: Wikipedia.

This information is usually of the form of a penalty for complexity, such as restrictions for smoothness or bounds on the vector space norm.

362 questions
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Regularized sum

How could I prove that as $ \epsilon \to 0 $ the regularized series goes as $$ \sum _{n=1}^{\infty} \frac{\exp(-n\epsilon)}{n}=-\log(1-e^{-\epsilon}) $$ and how could I prove that the finite part $$ \text{F.P}\sum _{n=1}^{\infty}…
Jose Garcia
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Regulators and uniqueness

Does the regularization of a divergent infinite sum yield a unique value? I.e. do different regularization schemes acting on the same infinite sum produce the same exact value independent of the regulator? What, exactly, do these values mean? Or…
user122066
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Abel regularization formula

given the sum $$ \sum_{n=0}^{\infty}n^k \exp(-n\epsilon), $$ for given $ k >0 $ how can it be Abel regularizable? According to this paper the regularized value agrees up to some pole term to zeta function regularization that is $$…
Jose Garcia
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proving that the series $ 1+2^{s}+3^{s}++ $ is divergent but borel summable

suppose that in the sense of distribution $ \int_{0}^{\infty}dxx^{n}T(x,s) =n^{s} $ for some distribution $ T(x) $ i do not know :( so if we apply borel generalized resummation $$ 1+2^{s}+3^{s}+......= \int_{0}^{\infty}(…
Jose Garcia
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determinant property of an operator

for a matrix we know that $$\det(aA)=a^n \det(A) $$ but what happens for an INFINITE dimensional operator ?? should we have $$\det(aA)=a^{Z(0)}\det(A) $$ $$ Z(s)= \sum_{n=0}^\infty \lambda_n^{-s} $$ since $ \sum_{n=0}^\infty 1 =Z(0) $ and the zeta…
Jose Garcia
  • 8,506
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abel summation and Harmonic series

Is it possible to prove that for the regularized Harmonic series $$ \tag 1\sum_{n=1}^{\infty} \frac{e^{-n\epsilon}}{n}=\gamma + 1/\epsilon $$ if epsilon is very small $ \epsilon \to 0 $ i can use $ e^{-n\epsilon}=1-n\epsilon $ , to the Harmonic…
Jose Garcia
  • 8,506
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Frequency dependent regularization of an ill-conditioned matrix

A x = b This is a frequency dependent problem, where the interested frequency range is 1-300 Hz. A is a square matrix of 246 by 246, and it consists elements that are dependent on $\omega^2$, and the rest of the elements are either zero or have…
ivanfield
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Regularization of underdetermined system to favour low frequency solutions?

Consider the ill-posed system $$ \mathbf A \mathbf x= \mathbf b.$$ One method to regularize the solution is the Tikhonov method which effectively minimizes $ ||\mathbf A \mathbf x - \mathbf b ||^2 + || \mathbf \Gamma \mathbf x||^2$. Letting the…
xyz
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The “units” of the terms of Tikhonov’s first order regularization functional

There’s one point about the order of regularization that I can’t understand. Let’s suppose that we have an ill-posed problem $Af=g$, where $g$ is the observation (with noise), $A$ is a linear operator, $f$ is the input we need to find. Let’s assume…
John Doe
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zeta regularization separation of series

in the sense of infinite series and for an integer 'a' is then correct that $$ \sum_{n=1}^{\infty}n^{k} = \sum_{n=1}^{a}n^{k}+ \sum_{n=a+1}^{\infty}n^{k} $$ opther that works only when ·$ re(k) > 1 $ ?? i have tried only the case $ a =2 $ and $ k=1$…
Jose Garcia
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