# Questions tagged [geometry]

For on-topic questions involving geometry. For purely mathematical question about geometry it is better to ask on the math SE site

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### Understanding relation between axis of least and maximum second moment

I was going through computer vision lecture video. You can find the pdf of this lecture here. I was trying to understand how orientation of object corresponds to axis of least second moment aka ...
15 views

### Geometric interpretation of random effects design matrix

A fixed effects (FE) linear model can be understood in a precise way using concepts of geometry. Namely, the response vector in model space is projected onto the design subspace surface described by ...
40 views

### Describe a geometrical shape as a piecewise function

Consider a cube filled with random particles. Let's say the particles in cube are rotated around the z-axis through the center of the cube. Here, the rotation is proportional to the height of the cube,...
1 vote
24 views

### Is it possible to uniformly draw points over a $D-1$ dimensional disk, given that one has an algorithm to draw over the $D$ dimensional sphere?

Suppose I have the following scenario: And I am aware of an algorithm to draw uniformly from (in this case) the 2-sphere. Does this same algorithm readily extend to the situation where I randomly ...
59 views

### Geometric intuition for how ridge ($L_2$) regularization helps under multicollinearity

We have some nice posts (1, 2 and likely more) illustrating multicollinearity geometrically. Now, ridge regression ($L_2$ regularization) is known to be a remedy of multicollinearity. What is the ...
42 views

### Estimation of MLE - request for clarification

I am refering to an earlier post in Maximum Likelihood estimate of cell radius from observed cross section radii I am not sure if I understood this question. What exactly is the meaning of ...
1 vote
31 views

### Automatic selection of nuber or factors in PCA based on the Cattells scree plot in R

I would like to automatically select the number of factors after factor analysis (PCA). I mean the graphical selection method according to the Cattel criterion, which determines where the steepness of ...
14 views

### Penalize an attribute of points based on their distance from a line without use of any threshold

We have a set of points and a line. Each point in the set has a weight attribute that is an integer. How we could penalize this weight based on the distance of the point from the line without using ...
148 views

### Estimate weights given a weighted geometric median

Problem 1 Given a set of points and their weighted geometric median, find the weights associated with each point. For instance, given the points ...
33 views

### Using Machine Learning to Create Periodic Paths

Question: How can I use Machine Learning to predict the right initial conditions $(P_0,V_0)$, given the angles of the triangular table, that will result in periodic paths in the triangular billiard ...
1 vote
38 views

### Need help interpreting an answer about the Cauchy distribution and Huygens principle

I'm trying to understand the answer here, which provides a physical interpretation for why the Cauchy distributions mean doesn't exist makes the following statement: If a unit light source is located ...
74 views

### Does the existence of gradient in any function necessarily imply the existence of a subgradient at that point?

First , I apologize if the question is not supposed to be here, or if it is off topic for the subjects dealt with in here. I was reading on subgradients, with respect to convex functions in the ...
183 views

### What optimization algorithms are best at traversing complicated geometries, and what trade-offs exist between different algorithms?

Hi all :] so short version is just the title, and specifically as pertains to those algorithms included in R's optimx::optimx() under ...
85 views

### how to determine the equation of a hyperplane (W and b) using only support vectors points without a Lagrange multiplier?

In n-dimensional, assuming I have support vectors points (number of sv < n), can I find the hyperplan equation (w and b) that separates 2 sets of data using only support vectors points without a ...
87 views

### How is the set of probability distributions on $m$ values an $m-1$-dimensional simplex? [duplicate]

In Zuk et al. 2012, they claim: The set of probability distributions on $m$ values is the $m-1$-dimensional simplex denoted $S_m$. Probability distribution functions are familiar to me. I understand ...
140 views

### Calculating covariance matrix from given vertices of triangles

I am given 3D vertices of triangles and I want to build the covariance matrix from it. These vertices are pure geometric vertices no noise. I can easily get it from libraries of python but I wanted to ...
1 vote
72 views

1 vote
117 views

### PCA on polar coordinates (0°-360°)

I have a data set where one of the variables is in polar coordinates ($\varphi$ = 0° to 360° degrees) and PCA was applied in order to reduce dimensions. As far as I know, using PCA on polar ...
3k views

### What is the geometric relationship between the covariance matrix and the inverse of the covariance matrix?

The covariance matrix represents the dispersion of data points while the inverse of the covariance matrix represents the tightness of data points. How is the dispersion and tightness related ...
45 views

### graph convolution network

I am trying to understand papers and lectures on graph convolution networks but whenever I open some paper, I get lost on the very first page. I started with some videos like this and this and papers ...
137 views

### Approximate the mean area of 2D Voronoi cell

Consider a random uniform distribution of $N$ points in $2D$ space bounded by $[0, 1]$ in both dimensions. Example: If I want to estimate the mean area of their Voronoi cells, I have to obtain the ...
102 views

### Understanding sufficient statistics geometrically

Consider the distribution $\mathcal{P} = \mathcal{N}(\mu, 1)$, where the variance is known but the mean is unknown. Let $X_1,X_2\sim P$ i.i.d. In this case $T = X_1+X_2$ is a sufficient statistic. I ...
280 views

### Geometric interpretation of the difference between the means (ANOVA)

First of all, a disclosure: I'm a medical doctor trying to understand statistics for research. Coming from a non-mathematical background I can do many mistakes. I've read some traditional books of ...
269 views

1 vote
79 views

### D-optimal DOE suggest repeated samples

I tried to generate a D optimal design but the design output sounds very weird to me. I have a (real) process and I`d like to explore 3 factors, but the process have a lot of constraints so I provide ...
103 views

### Sampling from/near boundary of a region in R^n

Suppose $\Omega$ is a region in $\mathbb R^n$, and suppose we are given a function $\chi(x)$ with $\chi(x)=1$ if $x\in \Omega$ and $\chi(x)=0$ otherwise. If it helps we can assume $\Omega\subseteq B$ ...
82 views

### Boundary point errors in PCA projection using sklearn

I am preparing a small example of a projection using python, numpy and sklearn to perform <...
444 views

### Geometric interpretation of Cholesky Decomposition

I understand that a square matrix, say $A$, can be thought of as a linear transformation within the same space. I could be as simple as basis change or some other transformation. In this way of ...
1 vote
### Is there a geometric interpretation for the surprising result of $E[x(y-\mu_y)] = COV(x,y)$?
I was playing around with non-central second moments, and noticed that $E[x(y-\mu_y)] = E[(x-\mu_x + \mu_x)(y-\mu_y)] = COV[x,y] + E[\mu_x(y-\mu_y) = COV[x,y] + 0$. I find this very surprising. It ...