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Questions tagged [geometry]

For on-topic questions involving geometry. For purely mathematical questiona about geometry it is better to ask on math SE https://math.stackexchange.com/

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Is there work for statistical properties analysis of road curvatures?

I looking for an analysis about the most suitable stochastic process to model distribution of road curvatures and its properties. I have been googling for several days and could not find it yet. For ...
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0answers
52 views

What is the geometric meaning of correlation matrix

I recently read this article explaining the geometric meaning of covariance matrix. http://www.visiondummy.com/2014/04/geometric-interpretation-covariance-matrix/ My question is : is there an ...
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How is the frenet frame along an asymptotic curve related to the geometry of the surface? [migrated]

I'm reading Differential Geometry: A first course in curves and surfaces by Theodore Shifrin and here is one of the questions from the exercise. I just can't seem to make the connection between the ...
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12 views

Does skipgram(word2vec) decreases the euclidean distance or increases cosine similarity between similar words?

The skip-gram model tends to predict the surrounding words or in other words, it tries to maximize the co-occurrence in the output of the Network. According to my knowledge, this makes a similar word ...
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Expected number of quadrilaterals

There are N points on the plane and the probability that the probability that two points are connected is p. What is the expected number of Quadrilaterals you can find? Assume that there is no three ...
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39 views

Is the Franke-Wolfe algorithm the same as Manifold optimization?

The Frank-Wolfe optimization algorithm describes optimization over a constrained domain. In the Manifold Optimization literature (e.g. [1]) a Gradient-Step is done using an exponential map. This maps ...
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44 views

Limitations to using the principal of least squares to fit an ellipse to experimental data?

I have used the method described below many times without issue to determine the major and minor axes of the "data ellipse". However, for this set of data (Fig. A) the method described below is ...
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18 views

Probability that some coefficients will be zero with lasso regression

Maybe it is duplicate, but I don't know how to ask a right question. Suppose I fixed some lambda, for instance lambda = 1. I draw the usual "square" |x| + |y| = 1. Now I want to find probability, ...
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How would one carry out principal component analysis in the Beltrami-Klein model of hyperbolic space?

This is a follow up to a comment by @whuber in response to this question: Is there an extension of PCA for data embedded in hyperbolic spaces? Sorry, I would have posted it there as a comment but it ...
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1answer
202 views

Drawing 95% ellipse over scatter plot

The context is regression analysis using Eviews, but first I wanted to create a few scatter plots and overlay error ellipses on them. Eviews doesn't support that kind of graph ornamentation so I am ...
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1answer
21 views

Sampling from the surface of a sphere in n dimensions with specific centre

I have a point $p$ on the surface of a unit sphere. I want to sample points from the surface of the sphere, such that the probability density of a point $q$ on the surface is given by \begin{equation} ...
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1answer
21 views

When is the result of multidimensional-scaling unique up to isometry?

What conditions on the ambient space and/or the given matrix of dissimilarities guarantee that all point configurations that minimise the error function of multidimensional scaling (MDS) are congruent,...
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44 views

Expected value as an orthogonal projection

I'm reading a paper in which the expected value of a random variable, $\mathbb{E}[X]$, is characterized as an orthogonal projection. This is on page 10. I've seen the geometric interpretation of ...
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1answer
16 views

Location of “x standard deviation(s)” for multivariate normals

In 1D with scalar parameters $(\mu,\sigma^2)$, it is common to represent normally distributed data with error bars spanning $[\mu-\sigma, \mu+\sigma]$. In n-D with parameters $(\mu, \Sigma)$, where $\...
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1answer
62 views

Is an geometric approach of support vector machine part of supervised learning?

The support vector machine can be approximated geometrically by enclosing the data of the classes by convex hulls. Then you can e.g. using the Rotating Calipers, you can search the widest band/border ...
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2answers
35 views

Can I create a test dataset with known errors to validate accuracy assessment? [closed]

I developed a procedure to measure the geometric accuracy of 3D building models based on the similarity to a 3D point cloud. Therefore I created mainly two quality criteria. The result of my automatic ...
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125 views

How are functional margin and geometric margin used in SVM?

I believe, geometric margin is euclidean distance between the point and hyperplane, whereas the functional margin just gives the confidence. At which stage is geometric margin and functional margin ...
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0answers
19 views

Mean distance from the centre to any point in a sphere and a cylinder [closed]

What is the mean distance from the centre to any point within a sphere of radius r? What is the mean distance from the centre to any point within a cylinder of radius r and length l?
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1answer
110 views

Geometric interpretation of mathematical expectation of a random variable

Is there any nice geometric interpretation of the mathematical expectation of a random variable (preferably based on density or cumulative density plot)? (For example, median has a nice geometric ...
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116 views

Vapnik-Chervonenkis Dimension

My question has to do with the VC dimension of the class of convex polygons with $m$ vertices. A solution to this problem is given in the following: Advanced Algorithms. Call the class of all ...
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0answers
72 views

Lower bounds on covering numbers for sparse vectors

Consider the set $S_k$, which is defined as the subset of $k$-sparse vectors in the unit sphere in $d$ dimensions: $$ S_k \triangleq \left\{ x \in \mathbb{R}^d : \| x \|_2 = 1, \, \left|\operatorname{...
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107 views

ML techniques to classify simple polygons

I'm curious about suitability of Machine Learning techniques to classify grayscale images of polygons into categories defined by number of their sides, which will be kept small. Images will be "...
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1answer
104 views

Covering the unit sphere with sparse vectors

I'm looking for references for covering the $d$-dimensional unit sphere $$ \mathbb{S}^{d - 1} = \left\{ x \in \mathbb{R}^d : \| x \| = 1 \right\} $$ I'm trying to cover $\mathbb{S}^{d-1}$ with ...
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616 views

Why word embeddings learned from word2vec are linearly correlated

I was playing with CBOW from the word2vec program downloaded from https://code.google.com/archive/p/word2vec/ with some sequence data (peptides in this case). I was trying to get embeddings for amino ...
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2answers
741 views

What type of data are dates?

According to Yale: Categorical variables represent types of data which may be divided into groups (Lacey M, 1997) To me, dates do not fit this definition. They are ordinal, as one date is ...
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Is there any geometric intuition on least absolute deviation regression?

There are a lot of geometric intuitions for regression with least square, e.g., projection, orthogonal, etc. (This and this answers are good examples.) Is there similar geometric intuition for least ...
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Fitting data (and getting normal vectors) to planes that can be vertical [duplicate]

I have many individual datasets of $(x,y,z)$ coordinates. I'd like to see if the coordinates form a plane (and if they do, I'd then like to compute a unit normal to the plane). My original method was ...
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1answer
48 views

SVM in a 2D plane

I was just watching this tutorial about Support Vector Machines, and I came to a halt because of the following problem. Given that $\vec{w}$ is a vector perpendicular to a hyperplane separating two ...
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1answer
119 views

What is the physical intuition behind the equality $\sum_i (x_i - \bar x)^2 = \sum_i (x_i - \bar x) x_i$?

Suppose that $x_1,\ldots,x_n$ are real numbers, and let $\bar x$ denote the average $\frac{\sum_i x_i}n.$ I know how to prove on paper that the equality $$\sum_i (x_i - \bar x)^2 = \sum_i (x_i - \bar ...
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2answers
379 views

Average absolute value of a coordinate of a random unit vector?

Let $\vec x$ be a random unit vector (that is, a random vector on the unit sphere). Let $x_i$ be the $i$'th coordinate (if it is easier, you can assume we are in 3-dimensional space). What is the ...
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1answer
59 views

Can CCA model any linear transformation?

I have recently been looking into canonical correlation analysis (CCA) as a way to map between different spaces. As I understand it, CCA maps data from both distinct spaces to a common (possibly ...
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0answers
57 views

Metrics to identify the presence of more than one circle in a set of (x,y) coordinates

I have a set of $(x,y)$ coordinates that represent any number of circles in a plane. What I am trying to do is determine whether there is 1 or more circles present in the data set. This would normally ...
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2answers
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How does a quadratic kernel look like?

Based on visualized decision boundaries, we have to decide what kind of classifier has generated it. One example is shown in the image below - this is from a quadratic kernel Support Vector Machine (...
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0answers
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Algorithm to find smallest ellipse containing p% of points in 2D plane?

This question is the same one I want answered: https://stackoverflow.com/questions/26810092/how-to-find-the-smallest-ellipse-covering-a-given-fraction-of-a-set-of-points-in Except I need to either ...
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0answers
24 views

Pairwise distances between 3 random points in a ball

Say I have a 3-D ball with radius $R$ and I randomly pick points from the volume of the ball. If I randomly pick two points, the probability density function of inter-point distances $P(r_{12})$ is ...
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1answer
852 views

Ellipse formula from points

Question: Best way to derive an ellipse formula when given bunch of points (say 60 points which when connected draws an ellipse)? Background: Using R Momocs library function conf_ell which returns ...
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1answer
312 views

Is the expectation of the sufficient statistics $S(X)$ transverse the whole space in an exponential family?

An exponential family is defined using two ingredients: - a base density $q_0(x)$ - a number of sufficient statistics $S_i(x)$ The family is all probability densities which can be written as: $$ q(x| ...
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2answers
395 views

How is the spherical elevation angle distributed when $(x,y,z)$ are uniformly and normally chosen?

As a follow up to How the polar coordinate, $\theta$, is distributed when $(x,y) \sim U(-1,1) \times U(-1,1)$ and if $(x,y) \sim N(0,1)\times N(0,1)$? Assume $(x,y,z) \sim U(-10,10) \times U(-10,10) \...
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31 views

mean and least error on the line

Suppose I have a line on which i have points in non-decreasing order. My intuition tells me that if I want to minimize the squared mean error on some subset of 3 (or other number) points, I would like ...
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4answers
357 views

How can I solve this joint probability problem?

Two points are dropped at random onto the unit interval, creating three sub intervals this way. Find the probability that the central sub interval will be two times shorter than the right ...
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3answers
791 views

Distribution of quadratic equation roots where coefficients are generated uniformly

We draw two points $p$ and $q$ at random from the interval $[−1, 1]$. Let $x_1$ and $x_2$ denote the roots of the equation $x^2 + px + q = 0$. Find the probability that, $x_1, x_2 \in \...
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1answer
166 views

Temporal Multi Dimensional Scaling

Let's say I apply a multidimensional scaling(MDS) to a dynamic dataset of $n$ points (eg, time series). At each step I will obtain a projection (in 2/3D) of the $n$ points. If nothing meaningful ...
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1answer
84 views

What is Geometrical Probability? [closed]

Is it a technique where Geometry is used to solve probabilistic problems? Is it a kind of probability which grows Geometrically when we conduct experiments? Is it a kind of distribution? I am ...
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0answers
73 views

Finding the non-linear curve that minimize the (sum of squared) distance to a set of point

I'm looking for the best approach to find the non-linear curve that minimize the mean square of perpendicular distance from a set of points. In essence, I would like to do non-linear major axis ...
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1answer
67 views

Geometry of Rejection regions for tests (NP lemma, Karl-Rubin, UMPU, LMP, etc)?

The Neyman-Pearson lemma, the Karlin-Rubin theorem, and the other for UMPU tests for the exponential family, etc. They all define a most powerful Rejection Region (RR), in a certain class of tests. I ...
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1answer
166 views

Centroid of nearest-neighbours on a hypersphere as a method for applying crossover in genetic algorithms

I am currently building a genetic algorithm to tune n parameters where n will probably be in the range of ...
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0answers
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Learning areas of intersections

I will try my best to make this question clear. I have many circles intersected randomly ( wireless communication application WSN) , i want to let each circle knows the intersection area with its ...
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0answers
470 views

How to train neural net to output a 2D polygon?

Lets consider a net, that should map image to an arbitrary four-sided 2D polygon (all vertices are scaled to (0,0) - (1,1)). We need a loss function with a gradient to train it. If we try to use some ...
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1answer
4k views

Data space, variable space, observation space, model space (e.g. in linear regression)

Suppose we have the data matrix $\mathbf{X}$, which is $n$-by-$p$, and the label vector $Y$, which is $n$-by-one. Here, each row of the matrix is an observation, and each column corresponds to a ...
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0answers
43 views

Intuitive relation between metrics and measures

I understand that a measure is essentially a volume element in a given geometry. That seems to suggest that a metric is the fundamental object and from it you can derive the measure of a space. Is ...