# 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))

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### Penalty function $f(a, b)$ for $a, b \in \mathbb{R}^+$ which adds a high positive penalty only when both $a, b$ are very small

I'm doing Bayesian recruitment curve fitting where my curve has two parameters $a$ and $b$. Both $a, b \in \mathbb{R}^+$. I have put a Truncated Normal prior on $a$ and a Half Normal prior on $b$. I'm ...
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### regularization and punishment method in least square fitting

I have a least square problem: $$\min \sum_j^p| f(x_j)-y_j |^2 \\ \text{where } f(x) = a x - \sum_{i}^{n} b_i J_1(c_i x)$$ where $J_1$ is the first order Bessel function. I have to find a set ...
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### Sobolev regularity required for $|\nabla u|^2 \in H^1(\Omega)$

We have $n=2$ and $\Omega=\{x \in \mathbb{R}^2\| ||x||_2<1 \}$. I need $|\nabla u|^2 \in H^1(\Omega)$. So i would choose $u \in W^{2,4}(\Omega)$. But since we have $n=2$ I thought that a lower ...
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### Morozov Discrepancy for PDE-Constrained Optimization With Bound Constraints

$\textbf{Background and context}$ Let $\beta:\Omega\rightarrow\mathbb{R}$, be an unknown spatially distributed parameter (over the spatial domain $\Omega$). Let $\mathcal{F}$ represent a parameter-to-...
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### Weighted sum of N images, which minimizes their TV norm

I have $K$ images $I_i, i\in{1 \ldots K}$ of the size $M \times N$. I wish to find weights $w_i$, s.t. $w_i \in [0,1]$ and $\sum_1^K w_i = 1$ so that $$|\sum_{i=1}^{K} w_i I_i|_{TV}$$ is minimal. I ...
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### Constant terms of asymptotic expansions of smoothed sums of all prime numbers

Consider the series $$\sum_{n=1}^\infty n e^{-n \varepsilon}$$ For $\varepsilon \leq 0$, it diverges. For $\varepsilon > 0$, it converges and equals $$\frac{e^\varepsilon}{(e^\varepsilon - 1)^2}$$ ...
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### Convergence of Tikohnov regularization for convex objective

I have the following question: Assume we want to minimize some convex (not strictly convex) and coercive ($f(x) \to \infty$ as $||x|| \to \infty$) function $f \in C^2(R^n,R)$, which has possibly ...
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### Indefinite generalized Tikhonov regularization

Suppose we want to choose $x$ to minimise the following (generalized) Tikhonov regularized least squares objective: $$(Ax-b)^\top (Ax-b) + \lambda [(x-c)^\top W (x-c)],$$ where $W$ is symmetric, but ...
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### Principal value integral with cosine

We are doing a practice set for an exam where we have to find the principal value of the integral $$I = P.V. \int^\infty_{-\infty} \frac{\cos(x)}{(x-1)(x^2+1)} dx$$ Firstly, I am unsure if this ...
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### Optimal Transport and Entropic Regularization

We are working with discrete optimal transport. Let $P$ be a matrix and let $H(P) =- \sum_{i,j} P_{i,j} (\log(P_{i,j})-1)$. Let $C$ be the cost matrix. And $\langle C,P\rangle$ the Frobenius inner ...
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### Best optimization technique for solving overdetermined systems with a constraint

I am trying to make a prediction model based on a system of linear equations: $A\vec{x}=\vec{b}$, where $\vec{x}$ ($m\times1$) is my learning parameters, $A (m\times n)$ and $\vec{b}$ $(m\times1)$ are ...
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### For $u \in L^2([0,T])$ there exists $u_m \in C^\infty$ such that $u_m \to u$ in $L^2_{loc}(]0,T[)$.

I am reading Navier-Stokes Equations by Roger Temam and there is a point in a proof I do not understand. Let me explain: We have a function $u: [0,T] \to H \in L^2([0,T]; H)$, for $H$ some Hilbert ...
In A Generalization of the Cauchy Principal Value, the author presents a way to assign values for hypersingular integrals of the form $$I=\int_a^b\frac{f(x)\,\mathrm dx}{(x-u)^n},\quad u\in(a,b)$$ ...