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0
votes
1answer
56 views

Why use regularization to reduce over-fitting

I'm having trouble understanding why should we use regularization for over-fitting when we can simply reduce the number of order to our polynomial function? Is it because it saves us time from having ...
6
votes
0answers
143 views

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 ...
1
vote
0answers
23 views

When does l1 regularisation give a sparse solution?

I was maximising a likelihood function, which is convex. I know that the system has a K-sparse solution. I wanted to know the conditions (or some sufficient conditions) on the likelihood function ...
1
vote
0answers
28 views

Regularity of semilinear heat equation

I'm facing following regularity issue and i wonder if anyone of you guys is able to help me. I'd like to show that the solution of a semilinear heat equation is classical, i.e. $C^2$ in space and ...
1
vote
1answer
104 views

Playing fast and loose with divergent series [closed]

I have been playing around recently with the regularization of infinite divergent sums and products, e.g. $$1+1+1+1+1+\ldots=\zeta(0)=-\frac{1}{2}$$ $$1+2+3+4+5+\ldots=\zeta(-1)=-\frac{1}{12}$$ ...
1
vote
1answer
65 views

Assigning values to divergent integrals

I'm interested in the (obviously divergent) integral $$ \int_{-\infty}^\infty dx e^{-x f}\ ,$$ where $f$ is real. Is there any way to meaningfully assign a value to this integral? I was thinking of ...
0
votes
0answers
20 views

Implementation of Total Variation Regularization Algorithm (Lagged Diffusivity Algorithm)

I am trying to compute the derivative of an experimentally-measured quantity as a function of time. The data are fairly noisy, which causes problems. For instance, using finite differences (central ...
0
votes
0answers
35 views

Laplacian Regularization with Sparse group Lasso

I have an optimization problem that is of the form: $\{\textbf{A}\} = argmin \{tr(\textbf{A}^\top L \textbf{A}) + \lambda_1||\textbf{A}||_1 + \lambda_2||\textbf{A}||_{2,1}\}$ where $\textbf{A}$ is a ...
-3
votes
1answer
37 views

Eigenvalues and Eigenvectors of a singular Covariance matrix

I am working on a research in which my data matrix $\bf X$ has dimension of $N\times P$ where $P>>>>N$.ie. its a small sample size problem. I need to compute the covariance of $\bf X$, ...
1
vote
0answers
17 views

integration of a multiple Laurent series

let be an multiple integral given by $$ \int_{0}^{\infty}dx_{1}\dots\int_{0}^{\infty}dx_{n}F(x_{1} ,x_{2},...x_{n}) $$ i have a question can i solve this integral equation by exapnding the integrand ...
4
votes
1answer
97 views

a doubt with the series $ \sum_{n=0}^{\infty}e^{-nx} $

I know that the series is equal to $$ \sum_{n=0}^{\infty}e^{-nx}= \frac{1}{1-e^{-x}}$$ However, if I expand each exponential term into a Taylor series I get $$ \sum_{n=0}^{\infty}e^{-nx}= ...
0
votes
0answers
39 views

regu tools l_curve regularization stanford ee 263

I am trying to solve one of the famous stanford EE263 problems, which gives me matrix A representing blurring of an image and y, representing the blurred image. For that I have been trying to use ...
1
vote
0answers
25 views

Regularized least squares

In Image Restoration, a true image f (in vector form)can be related to degraded data y through a linear model of the form $$y = Hf + n$$ where H is 2d blurring matrix and n denotes noise vector and ...
0
votes
0answers
18 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$ [duplicate]

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
5
votes
4answers
186 views

Find the regularized sum $\frac{1}{\ln(2)}+\frac{1}{\ln(3)}+\frac{1}{\ln(4)}+…$

By considering the integral Zeta function $$F(s)=s+\frac{1}{2^s\ln(2)}+\frac{1}{3^s\ln(3)}+\frac{1}{4^s\ln(4)}+...$$ Evaluate $$\frac{1}{\ln(2)}+\frac{1}{\ln(3)}+\frac{1}{\ln(4)}+...$$ EDIT: ...
0
votes
0answers
20 views

Regularization for unordered vectors

Let suppose we have two vectors u, v $\in \mathbb{R}^n$ and we want a function that returns $0$ if the ordering of the elements of both vectors are the same or a positive number otherwise, where the ...
0
votes
0answers
26 views

Why do the following regularisations of a function in Sobolev space exist?

Suppose $v_1, v_2$ satisfy $\mu \leq v_1(x,t), v_2(x,t) \leq M$ a.e. in $Q:=\Omega\times(0,T)$ and $$(v_1, \eta_1) \quad\text{and} \quad (v_2, \eta_2) \in L^2(0,T;H^1(\Omega)) \cap L^2(Q).$$ Define ...
0
votes
0answers
12 views

regularization parameters

Could someone out here kindly help me to state whether or not I am wrong on how to compute the tuning parameter for penalized neural network training problem... Given the objective function, I ...
2
votes
1answer
102 views

Understanding Regularization parameters in Machine Learning/Statistics

Suppose I have the following $k$ degree polynomial regression model with a data set of size $n$ which includes a $k$-dimensional feature vector $x$ and an outcome denoted $t_i$ for each vector in the ...
0
votes
2answers
58 views

Regex for strings with at least three unique characters

I'm trying to represent some string conditions in terms of regex. One of those conditions I find hard to transform is that the string must have at least three different characters. So is there any ...
0
votes
0answers
24 views

Regularising a function that is constant on an interval (related to Heaviside)

Define the function $f:\mathbb R \to \mathbb R$ by $$f(x) = \begin{cases} x &\text{for $x < 0$}\\ 0 &\text{for $x \in [0,1]$}\\ x-1 &\text{for $x > 1$}& \end{cases} $$ Note that ...
0
votes
0answers
53 views

Exponential integral in the limit $ \epsilon \to 0 $

what would be the asymptotic expansion near $ \epsilon =0 $ of the integral $$ \int_{1}^{\infty} x^{a}e^{-\epsilon x}\,dx= I(\epsilon) $$ in the limit $ \epsilon \to 0 $ my guess $ \Gamma(a+1) ...
1
vote
0answers
29 views

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 $$ ...
0
votes
0answers
14 views

what happens with zeta regularized determinant if the spectrum is not discrete ??

given the operator with eigenvalues $ \lambda _{n} $ the zeta-regularized determinant is $$ detA= e^{-\zeta ' _{A} (0)} $$ with $ \zeta _{A} (s)= \sum_{n=0}^{\infty} \lambda _{n} ^{-s} $ but my ...
0
votes
0answers
17 views

in the sense of Zeta regularization $ \zeta (0)=-1/2 $ should we then have $ \int_{0}^{\infty}dx =1/2 $

from the Euler Maclaurin sum formula $$ \int_{0}^{\infty}dx = 1+ \sum_{n=1}^{\infty}1 $$ since for $ f(x)=1 $ we have $ f(0)=f(\infty) =1 $ , therefore if $ \sum_{n=1}^{\infty }1=-1/2 $ therefore ...
0
votes
0answers
22 views

analytic continuation of an integral and Euler-Maclaurin formula

let be the Euler Maclaruin formula $$ \int_{0}^{a} dx x^{m}= \frac{m}{2}\int_{0}^{a}dx x^{m-1}+ \sum_{i=1}^{a-1}i^{m}- \sum_{r=1}^{\infty} \frac{B_{2r}}{(2r)!}\frac{ \Gamma ...
3
votes
1answer
73 views

Euler-Maclarurin summation formula and regularization

let be $ f(x)= x^{a} $ with $ -1<a<0 $ if i use Euler Maclaurin summation formula $$ \int_{1}^{\infty}dxx^{a}= \frac{1}{2}+ ...
1
vote
1answer
32 views

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} ...
0
votes
0answers
75 views

Second-Order Tikhonov Regularization

In the second-order Tikhonov regularization approach $\min\left\|Gm - d \right\|_2^2 + \alpha\ ^2 \left\|\Gamma x\right\|_2^2$ (1) given that $\Gamma\ $ contains second order derivatives, ...
2
votes
0answers
54 views

Divergent product, regularization [closed]

It is no secret that the Zeta function can do magic things. What we can do with this expression $$\prod_{k,m\in \mathbb Z}(k^2+m^2)$$ that it will be not infinity?
0
votes
1answer
112 views

The gradient of the standard mollifier

Please check my proof for the following result: I want to prove a result for $D\eta_{\epsilon}$ the gradient of the standard mollifier $\eta$. The function $\eta$ is defined as follows: Let ...
0
votes
1answer
111 views

Uniform convergence with Lp functions

I have a convergence question: Say we have a sequence of functions $\{u_{m}\}_{m=1}^{\infty}$ where $\{u_{m}\}_{1}^{\infty} \subset L^{p}(U)$ and where $U$ is bounded. Consider $u^{\epsilon} := ...
2
votes
1answer
112 views

Approximation in Sobolev Spaces

Consider the following proof in Lawrence Evans book 'Partial Differential Equations': How does it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in ...
1
vote
0answers
45 views

Finite part integral for terms involving absolute value

Can the simple "recipe" for Cauchy principle value and Hadamard Finite Part be extended to find the finite part of integrals of the form $$\int_a^b f\big(\left|x-c\right|\big) \ dx$$ with $c \in ...
1
vote
1answer
150 views

Support of Convolution and Smoothing

I just want to know how it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in C^{\infty}(V)$ by using the translations, but I'm having difficulty seeing how it ...
4
votes
2answers
226 views

Contour integration for functions with residues that form an infinite oscillating sequence

I would like to evaluate some complicated integrals involving the hyperbolic secant, but the extension of the usual contour integration evaluation using the residue theorem isn't clear to me. I've ...
2
votes
1answer
140 views

Is the regularization of an otherwise diverging two-sided sum always equal to zero?

As a first example, take the divergent series of all powers of two $1+2+4+8+...=\sum\limits_{k=0}^\infty 2^k$ which can be regularized by using the analytical continuation of the geometric series ...
9
votes
1answer
200 views

Is $2 + 2 + 2 + 2 + … = -\frac12$ or $-1$?

Using zeta function regularization, the divergent series $1+1+1+1+...$ can be evaluated to yield $$1+1+1+1+1+...=\sum_{n=0}^\infty\frac1{n^0}=\zeta(0)=-\frac12.$$ But what is $2+2+2+2+...$ then? On ...
0
votes
2answers
85 views

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 $ ...
2
votes
0answers
60 views

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 ...
3
votes
1answer
56 views

$f\in L^2(\mathbb{R}^3)$ implies $v(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}dy\in W^{2,2}$?

Let $f\in L^2(\mathbb{R}^3)$ be a function with compact support and define $v:\mathbb{R}^3\to\mathbb{R}$ by $$v(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}dy$$ Is true that $v\in W^{2,2}(\mathbb{R}^3)$ ...
2
votes
1answer
75 views

Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?

I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution $$v=\Gamma\star f$$ They claim that $v\in ...
1
vote
0answers
40 views

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 ...
1
vote
0answers
24 views

Shothostsky's formula regularization

let be the integra $$ I(a)=\int_{0}^{b} dx \frac{f(x)}{(x+ie-a)} $$ with 'e' an small quantity going to 0 from Shothotsky's formula $$ (x+ie-a)^{-1}= P(1/(x-a))-i\pi \delta (x-a) $$ so $$ ...
1
vote
1answer
36 views

Regularity method: size of $\epsilon$-regular graph parts

I am applying the Regularity method, as described for example here. I do not understand why the size $l$ of the $k$ resulting sets $V_i$, ie: $l = |V_1| = |V_2| = ... = |V_k|$ has the following ...
1
vote
1answer
54 views

Regularizing a $L^p(B)$ function in $B$ and in $\partial B$.

Let $B_1=\{x\in\mathbb{R}^N:\ |x|<1\}$. Let $u\in L^p(B_1)$ with $p\in (1,\infty)$ and suppose that $u$ is also defined in the boundary of $B_1$ and satisfies $u_{|\partial B_1}\in L^p(\partial ...
2
votes
0answers
38 views

Is there a regularized Ricci flow which will not generate singularity?

Define Hamilton's Ricci flow as $$\frac{\partial}{\partial t} g_{ij}(t) = -2 R_{ij}(t)$$ Q: Does there exist a regularized Ricci flow like $$\frac{\partial}{\partial t} g_{ij} = -2 F(g_{ij}, ...
2
votes
0answers
40 views

Hadamard finite part in 2 dimensions

given the divergent integral $$ \int_{0}^{\infty}dx \int_{0}^{\infty}dy \frac{xy+x^{3}-y^{2}}{1+xy+x+y} $$ how could i get a finite value in the sense of Hadamard finite part integral ?? thanks, ...
1
vote
1answer
82 views

Convergence rate when solving L1 regularized optimization via coordinate descent with tiny step?

Wondering if there is an established result for the convergence rate when solving L1 regularized optimization via coordinate descent with tiny step? By "tiny step" I mean the step is always set to a ...
1
vote
0answers
57 views

find example from some regularities. euclids elements

I heared that there are 6 basic math abilities. and one of them is to find an example from regularities. I want to develop this ability so I tried to find an example from real world. but I am not ...