# 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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### Regularity results for ODEs weaker than Lipschitz in a single point

I have to solve the following minimal example for $t\geq 0$: $$\dot s(t)= \frac{1}{2s(t)}, \quad s(0)=0$$ This is a minimal example, I know I could solve it easily using the separation of variables....
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### Regularizing Jacobian by altering the differentiated function?

I have a function $f$ such that I want to solve for a linear system with its Jacobian: $$\left[\frac{\partial f}{\partial x}\right] a = b$$ I want to impose constraints to $x$ so that the space ...
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### Distribution (or Expectation) of Number of Non-Zero Coefficients in L1-Regularized Regression

Related to this question, suppose we're performing $L1$-regularized linear regression. For a given regularization coefficient $\lambda$, what is the distribution over the number of non-zero parameters?...
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### Is this function continuous? (Moreau-Yosida regularisation with additional function argument)

I'm considering a generalised form of Moreau-Yosida regularisation. Given a continuous function $f:X \times U \rightarrow \mathbb{R}$ where $f(x, \cdot)$ is convex for any $x \in X$ and $X$, $U$ ...
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### Continuity of pointwise infimum (of a special function); Moreau-Yosida regularisation

I'm seeking how to prove that Moreau-Yosida regularisation provides a continuous function: Given a convex function $f:X \rightarrow \mathbb{R}$ where $X$ is a compact subspace of an Euclidean space (...
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### Green’s function of poisson’s equation in general dimensions

Green’s function of the Helmholtz equation, in the general dimension D, can be calculated as follows: \begin{align} G(\mathbf{r})&=\int \frac{d^D{k}}{(2\pi)^D} \frac{e^{-i \mathbf{k}\mathbf{r}}}{\...
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### Relation between moments a measured PMF under matrix multiplication?

I'm working on a physics problem where we have a measured photon energy spectrum (I'm thinking of it as a probability mass function, PMF), which is created by an energy spectrum of electrons which ...
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### Heat Kernel on a compact manifold without boundary [closed]

I was wondering if the conservation of mass which is obvious for the heat equation in $\mathbb{R}^n$ holds also for the heat kernel in a general compact manifold without boundary. I mean I want to ...
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### Minimum of regularized cost function for a movie rating system

I'm currently reading Rafael Irizarry's Machine Learning notes and I was a little confused with the approach towards regularization. To analyse a movie recommendation system, the following equation is ...
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### Do these constants appear in other areas of mathematics?

I decided to consider divergent (to infinity) integrals as some new kind of number. Towards this end, I began by establishing certain rules defining equivalence of the integrals. It seems the usual ...
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### Dos the Euler-Mascheroni constant $\gamma$ correspond to infinite hyperbolic angle?

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 ...
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