# Questions tagged [geometry]

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### Can a regularizer be reverse engineered to induce precise modifications to the associated unregularized regression problem's solution?

The following is a picture of regularization in a regression problem... The blue line is unregularized or less regularized, and the green line is more regularized. Problems of this sort are often ...
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### Finding the corners of noisy polygons

I have some polygons that look for example like this: If I zoom in very close on one side, you can see the noise. The data is a list of x coordinates and a corresponding list of y coordinates. I ...
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Edited to focus on the math (thanks whuber): Given a sequence $z_1,…,z_n$ of complex numbers and a fixed real number $σ$, find a sequence $x_1,…,x_n$ from the set $\{0,\pm1,\pm2\}$ minimizing \sum_{... • 121 9 votes 2 answers 184 views ### Origin of the term "spherical" in relation to covariance matrices? I understand that a covariance matrix with all diagonal elements equal, and all off-diagonal elements also equal (but different to the diagonal elements) is called "spherical". I am curious ... • 63.4k 4 votes 1 answer 315 views ### Regression coefficient on a triangle using geometry I am encountering a question as follows: Let X, Y be two independent uniform random variable on (0,1). We consider the regression model Y = \beta_1 X + \beta_0, given the restriction that X + Y ... • 143 0 votes 0 answers 19 views ### For KNN: How to justify that the probability of points to fall into a sphere of volume V is p(X)V https://www.cs.cmu.edu/~lwehbe/10701_S19/files/Lecture_3.pdf At the end of these notes, there is a short paragraph. Let x be a test point. Let x_1, \ldots, x_K be its K nearest neighbors. Let ... 3 votes 1 answer 235 views ### Bivariate normal covering circles and ellipses I am looking at covering circles for cartesian coordinates given by independent bivariate random variables X, Y \sim N(0, \sigma). The radius of a circle that will cover proportion p of these ... • 1,152 3 votes 0 answers 102 views ### Uniformly sampling surface of an ellipsoid using multivariate Gaussian Sampling uniformly from the surface of an ellipsoid (in the sense of \mu(dA) = \frac{1}{A}) seems very nontrivial: How to sample uniformly from the surface of a hyper-ellipsoid (constant ... 0 votes 1 answer 46 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,... • 113 4 votes 3 answers 110 views ### Is it possible to uniformly draw points over a D-2 sphere, given that one has an algorithm to draw over the D-1 sphere in D-dimensional space? 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 ... • 545 3 votes 0 answers 88 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 ... • 68.5k 2 votes 1 answer 63 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 ... • 293 0 votes 0 answers 227 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 ... 2 votes 0 answers 40 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 ... • 141 1 vote 1 answer 66 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 ... • 2,600 0 votes 0 answers 100 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 ... • 313 2 votes 1 answer 275 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 ... 2 votes 0 answers 93 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 ... • 21 3 votes 1 answer 121 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 ... • 9,452 0 votes 0 answers 189 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 1 answer 75 views ### Why does my simulation of nearest neighbors as circle origins provide different non-contact probability compared to theoretical? I am simulating spatially distributed points in \mathbb{R}^2 with intensity \lambda (units 1/area), which act as circle origins with radii being a random variable R_k. Given the distance to the ... 77 votes 1 answer 4k views ### Impractical question: is it possible to find the regression line using a ruler and compass? The ancient greeks famously sought to construct geometrical relationships using only a ruler and a compass. Given a set of points in a two dimensional plane, is it possible to find the OLS line using ... • 818 1 vote 0 answers 160 views ### How to find the average distance between randomly distributed points in a rectangle? Assume there are n points randomly distributed in a rectangle (x being the height y being the width) shown below in the figure. I would like to calculate the average distance between 2 random red ... 1 vote 0 answers 18 views ### Geometric properties of independence between random variables [closed] Suppose Y and X are two independent random variables. Are there any geometrical properties (of PDF/CDF/PMF/etc.) that capture this independence? I know that the Support has to be rectangular and ... • 3,540 6 votes 2 answers 100 views ### Does the distribution f(x) \propto (1-x^2)^{n/2} have a name? The distribution f(x) \propto (1-x^2)^{n/2} for -1 \leq x \leq 1 It occurs in a problem like Law of the norm of the empirical mean of uniforms on the sphere? It relates to intersections of high ... • 82.6k 1 vote 0 answers 32 views ### Calculating the mean or median "whereabout" of a joystick I have recently collected data using the NASA TLX tool, which includes a tracking task, where subjects have to keep the joystick deadcenter while having to react to multiple tasks. The raw data looks ... • 11 1 vote 0 answers 125 views ### Finding a Projection Plane in Dimensionality Reduction (e.g., Multidimensional Scaling) I have a set of data points in high-dimensional space that I wish to map onto a lower dimension (3D or 2D). Question : How do I obtain the Projection (Hyper)Plane (e.g., its normal vector or its set ... 2 votes 1 answer 1k views ### How to find the weighted midpoint between n-points? I was hoping if someone could guide me to the right algorithm to find the weighted midpoint of n-points. The photo attached below perfectly describes my problem. Let's just say we have three points ... 2 votes 1 answer 164 views ### Non-Euclidean analogue to MSE loss The most basic machine learning model called OLS uses the RSS (squared loss) or its average, mean squared error (MSE), for its loss function, which is aligned with Euclidean geometry. What is the ... • 4,005 7 votes 1 answer 347 views ### Why do we need to triangulate a convex polygon in order to sample uniformly from it? Suppose I want to uniformly sample points inside a convex polygon. One of the most common approaches described here and on the internet in general consists in triangulation of the polygon and generate ... • 71 1 vote 0 answers 261 views ### What are the principal components of a set of points lying on a circle? [closed] So if there are a set of points lying on a circle (2 dimensional), what will be the principal components given that the variance of points vary equally along any 2 perpendicular directions • 85 4 votes 0 answers 38 views ### How to do statistics on different geometries? The original motivation: Let's say, we have two particles, doing random one motion on an infinite 1-D lattice \mathbb{Z}, and we are interested in the autocorrelation of the trajectory of each ... • 237 5 votes 1 answer 66 views ### Expectation of differences between arcs on a circle Consider a circle with a circumference of n. On this circle, I define two arcs of length k<n, A_1 and A_2. The centres of the two arcs are uniformly distributed on the circle. Let \Omega_{... 1 vote 0 answers 138 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 ... • 111 12 votes 2 answers 4k 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 ... • 242 0 votes 1 answer 48 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 ... 2 votes 0 answers 164 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 ... • 4,332 4 votes 2 answers 116 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 ... • 855 4 votes 0 answers 310 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 ... 0 votes 0 answers 296 views ### Bivariate normal probability of being inside ellipse Assume that \mathbf{X} is a bivariate normal random variable\mathbf{\mu} = E\mathbf{X} = \begin{bmatrix} 0 \\ 2 \end{bmatrix} \ \text{and} \ \Sigma = Cov \ \mathbf{X} = \begin{bmatrix} 3 & 1 ...
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Let $S^2:=\{x\in\mathbb R^3:|x|=1\}$ denote the unit 2-sphere, $$\omega_{x\to y}:=\frac{y-x}{|y-x|}\;\;\;\text{for }x,y\in\mathbb R^3\text{ with }x\ne y,$$ $M\subseteq\mathbb{R}^3$ be the disjoint ...