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-5
votes
0answers
82 views

Is $\infty=-\frac12$? [duplicate]

edit I'd like to dispute the dupe-closing: The other question ($1+1+1+\cdots=−\frac12$) asks for a proof of that formula, while this questions asks whether its implication is valid in a more general ...
0
votes
0answers
17 views

sum of divisors function $\sum \tau(n) = \frac{1}{4}$

These notes on multiplicative number theory mention the convolution $ 1 \ast 1 = \tau$ (where $\tau$ is the divisor function not Ramanujan tau function. Therefore $$ \bigg(\sum \frac{1}{n^s} ...
1
vote
1answer
61 views

Zeta and Gamma function regularization with $\omega=1/0$

I have recently read about zeta function regularization, a way of ascribing values to functions having simple poles in a point and to divergent series. The values obtained are the same as those ...
3
votes
1answer
79 views

Is there a metric in which $1+2+3+4+\cdot$ converges to $-\frac1{12}$?

It is well known that the sum $1+2+3+4+\ldots$, which tends to infinity in the regular sense, can be assigned the value $-\frac{1}{12}$ by different means, e.g., zeta regularization or different ...
0
votes
1answer
42 views

Optimization Problem (Linear Algebra)

I am not trying to cheat or anything, so any reference to online literature or MOOCs, that teach this stuff, will be highly appreciated. The problem is to prove that the following optimization ...
2
votes
0answers
15 views

Can this divergent integral transform be regularized?

The integral $$\int_0^{\infty} e^u \ K_{i t}(u) du$$ is the adjoint Kontorovich-Lebedev transform of the increasing exponential function, but unfortunately this integral is divergent because $$e^u \ ...
0
votes
0answers
15 views

How is this the variance and bias of the least square estimator calculated?

How is the variance and bias of beta calculated? For variance, what does it have to do with eigenvalues? The lambdas in the picture are eigenvalues of X'X
0
votes
1answer
11 views

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) > ...
0
votes
0answers
19 views

Regularization of equivalent optimization problems

In portfolio optimization three equivalent optimization problems exist. I am wondering if they are still equivalent when regularized, e.g by ridge regularization. E.g. are the following equivalent ...
2
votes
0answers
32 views

Fourier transform of an exponentially singular radial function

I am trying to compute the 3D Fourier transform of a spherically symmetric function of the form $$f(r) = e^{\frac{1}{r} e^{-r}} - 1\, ,$$ which entails the integral $$\begin{aligned}F(k) =& \int ...
0
votes
1answer
11 views

Support of vector $w$ in graph sparsity

I'm reading about graph sparsity and I have one problem in a paper I'm reading I don't understand, maybe someone can clarify: Graph Sparsity: In graph sparsity, we have a directed acyclic graph ...
0
votes
1answer
17 views

Differentiable L-1 Regularization

In machine learning we are often faced with optimization problems where we want to minimize some energy function using L1 regularization over some of the parameters, e.g.: $$ E(a,w) = [\text{sum of ...
0
votes
0answers
21 views

Differentiability of Moreau-Yosida Regularization? [duplicate]

I'm looking for a proof of the differentiability of the Moreau-Yosida regularization of a proper closed convex function $f(y)$ defined on an n-dimensional Banach space $Y$. namely the function is ...
0
votes
1answer
23 views

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 + ...
0
votes
0answers
48 views

Function smoothing using convolution

I have a function $\hat f$ which is an estimator of an unknown function $f$. The estimator $\hat f$ looks pretty irregular (see the red line). I would like to smooth it with some kernel function ...
1
vote
1answer
27 views

Integral in regularization involving exp of a parameter in the denominator

Can someone please point me a way to compute $$\int_0^\infty\frac1{s+t}\exp\left(-\alpha t+\frac{t^2\beta}{s+t}\right)dt$$ ? How about the following one? $$\int_0^\infty ds\int_0^\infty ...
4
votes
1answer
77 views

Is it possible to sum the divergent series with prime coefficients?

This is a follow-up of this question. It is known that the divergent series $$ P := \sum_{n=1}^\infty p_n \qquad \text{where } p_n \text{ is the $n$th prime} $$ cannot be summed by means of (prime) ...
0
votes
1answer
76 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
171 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
34 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
51 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 ...
2
votes
1answer
109 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
80 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
55 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
51 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
44 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
20 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
102 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
45 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 ...
2
votes
0answers
27 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
193 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
21 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
27 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
220 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
74 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
25 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 ...
1
vote
0answers
31 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
15 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
25 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 ...
3
votes
1answer
90 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
33 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} ...
1
vote
0answers
110 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, ...
0
votes
1answer
140 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 ...
1
vote
1answer
176 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
133 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
48 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
160 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
286 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 ...