All Questions
Tagged with nonlinear-optimization st.statistics
19 questions
2
votes
1
answer
170
views
Equivalence of minimizing trace and determinant over matrix quadratic form in multivariate regression
Consider the multivariate regression model
$$Y = XB + E$$
where $Y$ is $n \times p$ and corresponds to the dependent variables, $X$ is $n \times k$ and corresponds to the independent variables, $B$ is ...
2
votes
0
answers
87
views
A complex problem involving densities (likelihood functions) and optimization
Consider the following autoregressive process with normal errors:
\begin{equation}\label{7YlUV4i8nuO}\tag{I}
y_t = \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim} N(0,\sigma^2)
\end{equation}
We ...
- 13
4
votes
1
answer
287
views
Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
- 1,465
1
vote
1
answer
132
views
Local maxima of the sum of Gaussian functions in *one dimension* are always strict local maxima - proof?
Motivated by this question asked earlier, I was wondering whether one can prove easily that the local maxima of the sum of Gaussians:
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, \quad x_1 < x_2 < \...
- 1,465
1
vote
0
answers
41
views
Fitting a non-periodic sum of periodic time series
The problems is as follows: you have $n$ points $(x_1,y_1),\dots,(x_n,y_n)$ and you want to fit the following equation to the data points:
$$y=\theta_1\cos(\theta_2 x+\theta_3) + \theta_4\cos(\theta_5 ...
- 3,098
1
vote
0
answers
78
views
Minima of a cdf of multivariate normal distribution with respect to a parameter
Let $\mathrm{X}\sim\mathcal{N}_{3}(\boldsymbol{\mu},\mathrm{\Sigma})$ where
\begin{equation}
\boldsymbol{\mu} = n[(\mu_1-\mu_2)\sqrt{\xi_1\xi_2/(\xi_1+\xi_2)}, (\mu_1-\mu_3)\sqrt{\xi_1\xi_3/(\xi_1+\...
- 11
4
votes
5
answers
2k
views
Reference request: importance of Lipschitz continuity
I see that Lipschitz continuity is a common assumption used in optimisation, statistics, machine learning, etc.
Could you point me in the direction of some literature that discusses why Lipschitz ...
- 161
1
vote
0
answers
73
views
Reduce the asymptotic variance for a class of Metropolis-Hasting estimates
I'm running the Metropolis-Hastings algorithm with state space $E$, target distribution $\mu=p\lambda$ and proposal kernel $Q$ to estimate $\mu(hf)$ for a fixed function $f:E\to[0,\infty)^3$ and a ...
- 167
1
vote
1
answer
232
views
Maximize a Lebesgue integral subject to an equality constraint
I want to maximize $$\Phi_g(w):=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)\int\lambda({\rm d}y)\left(w_i(x)p(x)q_j(y)\wedge w_j(y)p(y)q_i(x)\right)\sigma_{ij}(x,y)|g(x)-g(y)|^2$$ over all choices ...
- 167
3
votes
0
answers
202
views
Maximize an $L^p$-functional subject to a set of constraints
Let
$(E,\mathcal E,\lambda)$ and $(E',\mathcal E',\lambda')$ be measure spaces
$f\in L^2(\lambda)$
$I$ be a finite nonempty set
$\varphi_i:E'\to E$ be bijective $(\mathcal E',\mathcal E)$-measurable ...
- 167
0
votes
1
answer
98
views
1D functional equation: solve for function with given expected value w.r.t normal density
Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation
$$
\begin{split}
\mathbb ...
- 6,853
3
votes
0
answers
121
views
Finding a mixture of 1st and 0'th order Markov models that is closest to an empirical distribution
I am interested in finding the distribution "$p^*$" closest to an empirical distribution $\hat{p}$ where $p^*$ is a mixture of first and zeroth order Markov models. That is, I want to find $$
p^* = \...
- 283
2
votes
0
answers
380
views
Matrix optimization of a random quadratic form
I am interested in maximizing a quadratic form which looks like
$$f(\Sigma) = E(\operatorname{trace}(SJ)) = E(1^{\top} S 1)$$
where $J$ is a matrix of $1$'s, $S= \Sigma_{mm} - \Sigma_{mo} \Sigma_{oo}...
- 21
5
votes
1
answer
581
views
Strong duality for a particular moment problem
Reading the paper in this Link (see pag 13) with the objective of understanding a topic related to stochastic optimization I came across a problem in demonstrating one of the theorems. The situation ...
- 159
2
votes
0
answers
101
views
Best describing a stochastic process in terms of others
Intuitive Question
Suppose I'm given a set of $k$ time-series $\{X_t^1,\dots X_t^k\}$. Is there a way to determine how much of each series is dependent on the others.
Formal Question
More ...
- 5,405
1
vote
0
answers
64
views
Posterior consistency of non linear model
This is possibly a reference request. Let $G$ : $\mathbb{R}^p \to \mathbb{R}^q$ be a continuous injective/bijective function. Let $\mu$(we may also assume this to be a non degenerate Gaussian) be ...
- 157
0
votes
0
answers
444
views
How to decide a value of learning rate for Stochastic Gradient Descent?
I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla E_n(...
4
votes
0
answers
213
views
Optimization problem involving Multivariate Normal
I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
$$h(\mu_{1},\ldots,\...
- 41
3
votes
1
answer
2k
views
The average number of people that can sit on a bench of a given length.
Let me explain what I mean:
The width of the average person varies, perhaps with a normal distribution.
Given a specific variance, how many people (on average) can sit side-by-side on a bench of a ...
- 43