All Questions
Tagged with approximation-theory st.statistics
12 questions
1
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0
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85
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Approximation of continuous function by multilayer Relu neural network
For continuous/holder function $f$ defined on a compact set K， a fix $L$ and $m_1,m_2,\dots,m_L$, can we find a multilayer Relu fully connected network g with depth $L$ and each $i$-th layer has width ...
- 781
1
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1
answer
202
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Approximating a smooth function under some restrictions
Let $C^{m,\alpha}_M([0,1])$ be a Holder ball consisting of real-valued functions $g$ on $[0,1]$ such that
$$ \|g\|_{C^{m,\alpha}} := \max_{0\leq j \leq m } \sup_{x\in [0,1]} |g^{(j)}(x)| + \sup_{x,y\...
- 93
1
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1
answer
223
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Bound error in approximating $E_x [H(f(x))]$ with random $(1/n) \sum_{i=1}^n \Phi(f(x_i)/h)$ where $H$ is Heaviside function and $\Phi$ is normal CDF
Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently smooth" function. For simplicity, we may consider $f$ to be an affine function, i.e $f(x) \equiv b-x^\top w$, for some $(w,b) \in \mathbb ...
- 6,853
3
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0
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113
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Image restoration quality general lower bounds
A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe
$$
\tilde f = f \star h + n
$$
where $\star$ is convolution, $h$ is the point spread function (caused,...
- 1,069
5
votes
1
answer
330
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Constructive approximation of Hölder functions using kernel functions
Suppose I have a function $f \in \mathcal C^{\alpha, L}([0,1])$, where
$\mathcal C^{\alpha, L}([0,1])$ is the space of $\alpha$-smooth Hölder
functions with norm $L$. I am interested in efficiently ...
- 155
11
votes
1
answer
1k
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What are some of the surprising results of finite sample statistical estimation?
I'm trying to familiarize myself with the latest results in finite sample statistics. It seems to me that these results can be classified into two categories:
Unsurprising results confirm that the ...
- 395
3
votes
2
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2k
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Empirical estimator for total variation distance between two product distributions
Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...
- 35
3
votes
1
answer
244
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Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p?
Let $\mathbf{p}=(p_1,\dots,p_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X_1,\dots,X_m)$ be a vector of independently-distributed binomial random variables such that $X_i\sim \text{Binom}(n,p_i)$...
- 103
0
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0
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160
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Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
- 103
6
votes
5
answers
3k
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Approximation to the ratio of a Gaussian CDF to PDF
Johnstone and Silverman (2005) claimed that for large x
$\frac{1-\Phi(x)}{\phi(x)} \approx \frac{1}{x}$
where $\Phi(x)$ and $\phi(x)$ are the CDF and PDF for a normal random variable.
I was able ...
- 203
1
vote
1
answer
502
views
nonnegative series expansion of nonnegative functions
The title says it all! When using orthogonal series expansions like the Gram-Charlier expansion to approximate probability density function, a big problem (making this approach less usefull and less ...
- 2,600
10
votes
3
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22k
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Approximation of a normal distribution function
I am reviewing and documenting a software application (part of a supply chain system) which implements an approximation of a normal distribution function; the original documentation mentions the same/...
- 211