Mathematics Stack Exchange
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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.

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When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty p_k = 4\pi^2$$ where $n\#$ is a primorial, and $p_k$…
Timmy Turner
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Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve an unconstrained minimization of the least-squares…
Tim
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Why do zeta regularization and path integrals agree on functional determinants?

When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions. The first definition is based on zeta function…
anon
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Is $1+2+3+4+\cdots=-\frac{1}{12}$ the unique ''value'' of this series?

I'm reading about zeta-function regularization in physics and I have some mathematical doubt. I understand that, since a sum of infinite terms is not well defined in a field, a series that is considered divergent in the usual meaning can have a…
Emilio Novati
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Can different choices of regulator assign different values to the same divergent series?

Physicists often assign a finite value to a divergent series $\sum_{n=0}^\infty a_n$ via the following regularization scheme: they find a sequence of analytic functions $f_n(z)$ such that $f_n(0) = a_n$ and $g(z) := \sum_{n=0}^\infty f_n(z)$…
tparker
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Regularization and $\int_{0}^{\infty}\sin x \;\mathrm{d}x$

In my grad quantum/E&M classes I had to do intuition-bending regularization of integrals that didn't seem mathematically justified (but got full credit and were repeated in the solutions) like the following: $$\int_{0}^{\infty}\sin x \;\mathrm{d}x =…
dr jimbob
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Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?

Regarding the proof by Tony Padilla and Ed Copeland that $1+2+3+\dotsb=-\frac{1}{12}$ popularized by a recent Numberphile video, most seem to agree that the proof is incorrect, or at least, is not sufficiently rigorous. Can the proof be repaired to…
ziggurism
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Values of the Riemann Zeta function and the Ramanujan Summation - How strong is the connection?

The Ramanujan Summation of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $$\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 (\mathfrak{R}) $$ (for non-negative integer $k$) and…
Max Muller
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Intuition of orthogonal polynomials?

Recently, while learning about the regularization methodologies(in machine learning), i came across orthogonal polynomials ( of Legendre's polynomials), I looked up on the Internet and there are formulas for the same, but i didn't get any intuition…
supermeat boy
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Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist? Why?

Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist, given that $\mathbf A$ is a square and positive (and possibly singular) matrix and $\epsilon$ is a small positive number? I want to use this to regularize a sample covariance matrix ($\mathbf…
ben
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Is $2 + 2 + 2 + 2 + ... = -\frac12$ or $-1$?

Using zeta function regularization, the divergent series $1+1+1+1+...$ can be evaluated to yield $$1+1+1+1+1+...=\sum_{n=1}^\infty\frac1{n^0}=\zeta(0)=-\frac12.$$ But what is $2+2+2+2+...$ then? On the one hand, it should be twice as much, but on…
Tobias Kienzler
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Is $\prod_{n=1}^\infty P_{2n-1}$ regularizable?

Assume that $P_n$ denotes the $n$'th prime for this entire question. Inspriation: I was dumbfounded by the fact that: $$\hat\prod_\limits{n=1}^\infty P_{n}=4\pi^2$$ After further investigation, I learned of many other properties of zeta-regulation,…
user285523
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Tikhonov regularization vs truncated SVD

To find $\mathbf{x}$ such that $$A\mathbf{x}=\mathbf{b}$$ we can use least squares when the problem is not well posed. Further, we can use Tikhonov regularization when $A$ is ill-conditioned. In Tikhonov regularization, we minimize…
fragapanagos
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Divergent Series

Thinking about divergent series and ways of "summing" them, they seem to fall into two categories (roughly): Series like $\sum_{k=1}^\infty \frac{1}{k}$, which defy all kinds of regularization or summing methods. Series which can be summed, in one…
Willem Noorduin
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Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also $\sqrt{2\pi}$? $$\infty!=\prod_{n=1}^\infty…
user76284
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