Questions tagged [regularization]
The regularization tag has no usage guidance.
51
questions
0
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0answers
35 views
What intuitive meaning “determinant” of a divergency (divergent integral, series, germ, pole or a singularity) can have?
I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs.
So, I decided to construct something similar to the modulus or determinant of a matrix of these ...
2
votes
0answers
138 views
Did anyone ever propose the distinction between “divergent to infinity” as opposed to “divergent but with finite average”?
There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences.
Still, in my view there is fundamental difference between divergent ...
-1
votes
1answer
187 views
A question on assigning finite values to divergent sums involving expression of primes
We know the following:
$$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$
This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.
...
2
votes
0answers
69 views
Regularization of the area under hyperbola
So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
0
votes
0answers
114 views
What is the regularized numerocity of prime numbers?
Basically, I want to find this sum:
$$\lim_{s\to0}\sum_{n=1}^\infty \frac{p(n)}{n^s}$$
where $p(n)$ is the membership function of the set of primes. That is, $p(n)=1$ if $n$ is prime and $0$ otherwise....
0
votes
0answers
21 views
Assigning negative integer moments to random variables with Hadamard regularization
Let $X\sim F_X$ denote a continuous random variable that admits a density $f_X$ with support $\mathcal S=\operatorname{supp}(X)\ni 0$ and assume $f_X(0)>0$. I am interested in defining a ...
2
votes
1answer
223 views
A set of divergent integrals that I think, equal to $-\gamma$
So, we take $\frac{\text{sgn}(x-1)}{x}$ and apply $\mathcal{L}_t[t f(t)](x)$ four times. The transform is known to keep area under the curve. These integrals, I think, are equal to minus Euler-...
1
vote
2answers
187 views
Can we meaningfully ascribe values to these divergent integrals?
My gut feeling is that
$\int_0^\infty (1-\frac1{x^2})dx=0$
$\int_0^\infty (x-\frac2{x^3})dx=0$
$\int_0^\infty (x^2-\frac6{x^4})dx=0,$
etc, and in general,
$\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$
...
8
votes
2answers
863 views
On modified Euler product
I know this forum is for professional Mathematicians. I didn't know if the following question is fit or not fit for the site. If not, feel free to delete it.
Consider the modified Euler product as ...
19
votes
4answers
963 views
Is the pseudoinverse the same as least squares with regularization?
Given a linear system $Ax=b$, the pseudoinverse of $A$ is found as the matrix $A^+$ such that $x=A^+ b$ where $x$ solves the least squares problem $\min \| Ax - b \|^2 $ and $x \perp \mathcal{N}(A)$. ...
2
votes
1answer
161 views
The zeta regularization of $\prod_{m=-\infty}^\infty (km+u)$
Background: I'm facing the computation of the zeta regularization of the infinite product given by
$$\prod_{m=-\infty}^\infty (km+u)$$
for a real positive $k$ and $\Im(u)\neq 0$. From J. R. Quine, S. ...
2
votes
0answers
65 views
Sparse signal recovery (nonlinear case)
Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
5
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0answers
196 views
More or less universal formula for regularization of divergent integrals?
Is there a simple formula that would produce the regularized value for the most common divergent integrals?
I know, there is a formula for Cesaro integration, but it is applicable only to Cesaro-...
2
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0answers
87 views
Theta-function in the lower half-plane
Standard theta function
$$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$
has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...
6
votes
3answers
304 views
Is regularization of infinite sums by analytic continuation unique?
There are ill-posed summations that we can assign values to, take for concreteness,
$$ S = \sum_{k=0}^\infty k $$
to which we can assign $-1/12$ by several methods. Is there a fundamental and rigorous ...
6
votes
2answers
501 views
Less fundamental applications of Zeta regularization:
As we all know, zeta regularization is used in Quantum field theory and calculations regarding the Casimir effect.
Are there less fundamental applications of zeta function regularization? By "less ...
1
vote
0answers
124 views
Are the shapes of the $\mathbb{R}^2$ plane and a disk of infinite radius different? Or otherwise, why their areas differ by $\frac\pi{12}$?
The calculation of the area of the $\mathbb{R}^2$ plane depends on filtering used. I think, the most natural filtering is along the radius in polar coordinates:
$$S_{\mathbb{R}^2}=\int_0^\infty 2\pi ...
0
votes
1answer
183 views
Generalised limits via derivatives of integrals?
Assuming that $f$ is a continuous function, we have that
$$f(x) = \frac{d}{dx}\int f(t)\,dt.$$
Assuming instead that $f$ has a removable singularity at $x=a$, and is otherwise continuous, we have ...
8
votes
2answers
596 views
Regularizing the sum of all primes
In the spirit of a similar question for the harmonic series, is there a way to regularize the (divergent) sum of all primes?
$$ \sum_{p \text{ prime}} p $$
Neither of these questions obtained a ...
4
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2answers
689 views
Is there a sensible way to regularize $\int_0^\infty \tan x\, dx$?
I wonder if there is any sensible generalization of regularization which would be able to ascribe finite values to $\int_0^\infty \tan x \,dx$ and $\int_{-\infty}^0 \psi(x)dx$?
Perticularly, since $\...
4
votes
1answer
150 views
Interesting questions for inverse parabolic problems
I'm looking for some interesting questions and maybe open problems in inverse problems theory, especially in the framework of parabolic PDEs (basically the heat equation). As key words here we can ...
4
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1answer
78 views
Proximal Operator image of convex functionals
Let $\Gamma_0$ denote the set of lower-semi-continuous convex functionals on a Hilbert space $H$. What exactly is the image of $\Gamma_0$ under the proximal operator
$$
\begin{aligned}
&\Gamma_0\...
0
votes
1answer
90 views
How to derive the solution of Tikhonov Regularization via SVD [closed]
The solution to Tikhonov Regularization is
$$x=(A^HA+\sigma^2_{min}I)^{-1}A^Hb$$
where $\sigma^2_{min}$ is the minimum of the singular values of $A$.
Then we apply $SVD$ to $A$ such that,
$$A=U\Sigma ...
3
votes
0answers
336 views
New/useful method for summation of divergent series?
Questions
$$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$
Also obeys (see background for argument):
$$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...
1
vote
1answer
174 views
Why we cannot speak about the main or natural regularization?
Often when asking about a regularized value of an integral or series, I encounter a negative reaction of the sorts that "regularization is what you define it".
But in practice if we consider some ...
8
votes
3answers
257 views
Regularized linear vs. RKHS-regression
I'm studying the difference between regularization in RKHS regression and linear regression, but I have a hard time grasping the crucial difference between the two.
Given input-output pairs $(x_i,y_i)...
2
votes
0answers
99 views
What is the generalized sum of the following series? $\sum _{x=1}^{\infty } \sqrt{s^2 x^2-1}$ [closed]
I tried Mathematica, various regularization methods, including Borel, with no result.
On Math.SE the question was attacked with claims that divergent series cannot have a sum, so I decided to ask at ...
1
vote
0answers
172 views
Regularization on divergent series [closed]
I want to compute $$ \sum ^{\infty }_{n=1}\dfrac {\left( 2n+k-2\right) !\zeta \left( 2n\right) \left( -1\right) ^{n-1}}{\left( 2\pi \right) ^{2n}}. $$
This series is surely not convergent for any ...
3
votes
1answer
85 views
Why to multiply the penalty by $n$ in the penalized least squares and likelihood?
In the SCAD paper by Fan and Li (2001), there exist two forms of penalized least squares as follows:
$$\frac{1}{2}\left \| y-X\beta \right \|^2+\lambda \sum_{j=1}^{d}p_j (\left | \beta _j \right |),$$
...
2
votes
0answers
280 views
When will the upper regularization of a bounded function not defined?
Suppose $E$ is a compact metric space.
A function $f :E \rightarrow \mathbb{R}$ is upper semicontinous if for all $c \in \mathbb{R}$, $f^{-1}(-\infty, c)$ is open in $E.$
For any real-valued ...
1
vote
1answer
109 views
Existence of analysis regularization solution
I am interested in the optimization problem known as "analysis regularization":
$$ {\rm argmin}_{x \in \mathbb{R}^{p}}\frac{1}{2}\|y - Ax\|_2^2 + \lambda \|D^T x\|_1,$$
where $y \in \mathbb{R}^n$, $...
1
vote
0answers
44 views
Is there optimal sparse and minimum energy solution?
I am interested to know if anyone worked on the the inverse solution of the problem defined below, which means I want to find $\tilde{y}$.
$M_{\delta}^{\alpha}[\tilde{y}]=\|A\tilde{y}-\tilde{b}\|_{...
3
votes
2answers
371 views
Regularisation of $\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
As from the title, I am currently dealing with this sum
$\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
in particular with $p=1/2,3/2,...$ (but once solved for $p=1/2$ one can derive wrt $a$ and find the ...
2
votes
0answers
506 views
Difference between Sobolev norm and L2 norm for regularization?
I am interested to find a inverse solution of the problem defined below, which means I want to find $\tilde{y}$.
$M_{\delta}^{\alpha}[\tilde{y}]=\|A\tilde{y}-\tilde{b}\|_{L_2}^2 + \alpha\|\tilde{y}\|...
1
vote
0answers
202 views
Significance of Tikhonov matrix
I am looking for a tutorial on Tikhonov matrix, in the sense what it can do or it cannot do. The definition of the matrix can be obtained in the wikipedia link. https://en.wikipedia.org/wiki/...
16
votes
2answers
652 views
Comparing sizes of sets of natural numbers
It seems natural to consider $\lim_{q \rightarrow 1^-} \sum_{n \in S} q^n - \sum_{n \in T} q^n$, when it exists, as a way of comparing the sizes of two sets $S,T \subseteq {\bf N}$ that have the same ...
5
votes
1answer
241 views
Regularization by mean curvature flow
I have a $C^{1,\alpha}$ surface defined as the graph of some function $\varphi : B \to \Bbb{R}_+$ ($B$ is a ball). This surface has positive and bounded mean curvature in the weak sense (since the ...
5
votes
1answer
193 views
Multidimensional integrals that diverge by oscillation
It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq 0$...
2
votes
1answer
208 views
A question about some notation involving the exclamation mark [closed]
What does the symbol ‘!’ signify? Is it $ \text{argmin} $? For example, $ \| A x - y \| = \min! $.
3
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0answers
361 views
On increasing the penalty term in convex optimization with regularization
Given the two strictly convex (unique solution) optimization problems as:
$$Problem\:1:\min_{X} f(X)+\|X\|_{F}^2 \hspace{2cm}Problem \:2:\min_{X}f(X)+n\|X\|_2^2$$
where $X\in\mathbf{S}_{++}^{n}$ (...
2
votes
2answers
493 views
double integral and Hadamard finite part
Given the divergent integral
$$ \int _{0}^{\infty}dx \int_{0}^{\infty}dy \frac{x^{2}y+1}{1+x+y} $$
how can I apply Hadamard's finite part to give a finite meaning to it ?
It is just made by ...
3
votes
1answer
630 views
General method for under and over determined systems?
Suppose I have a system:
$$
Ax = b
$$
where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.
I'd like a combination of a ...
0
votes
0answers
791 views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
8
votes
1answer
862 views
What are some geometric / physical / probabilistic interpretations of the Riemann zeta function at integer arguments n ≤ 1?
Introduction: This is slightly edited and generalised version of a question I asked on the Physics Stack Exchange website. This question has a twin brother asked here on MO, only now we consider ...
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2answers
3k views
Choosing the order of Tikhonov regularization of an inverse problem
This question is migrated from math.stackexchange.
Let me first describe the problem I am trying to solve and then the question I have. I greatly appreciate anyone who can shine some light on it.
...
23
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3answers
3k views
Understanding zeta function regularization
I attended a talk this morning on Ray-Singer torsion, in which Rafael Siejakowski introduced zeta function regularization in a compelling way. The goal is to define the determinant of a positive self-...
2
votes
1answer
419 views
Regularization of Zygmund functions
Dear community.
I would like to derive a "good" estimate on $\frac{d}{dt}f_\epsilon(t)$, where $f_\epsilon$ is a regularization of a Zygmund-continuous function $f$, i.e.
$|f(x-\tau)+f(x+\tau)-2f(x)|...
3
votes
3answers
1k views
Nonlinear circle fit with known radius
I have data points from a half circle and I already know the approximate radius. I want to find the circle which best fits the points using a fixed radius. How can I do this? If I solve the problem ...
2
votes
1answer
402 views
What is the regularity of the argument of a complex function?
Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
6
votes
2answers
1k views
What's the correct notion of determinant of a bilinear pairing?
By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...
