# Tagged Questions

A high-dimensionality space is one that can only be spanned by a basis set with a large number of elements. High-dimensional problems often suffer from the *Curse of Dimensionality*, which is exponential growth in the problem size as a function of the number of dimensions.

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### PDEs in Many Dimensions

I know that most methods of finding approximate solutions to PDEs scale poorly with the number of dimensions, and that Monte Carlo is used for situations that call for ~100 dimensions. What are good ...
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### N-dimensional Delaunay Tesselation Software Libraries

I have a set of known points/nodes irregularly spaced in N-Dimensional space (N>=2), and I would like a way to generate the Delaunay triangulation of these points, and return the corresponding ...
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### Kolmogorovâ€“Smirnov test for multivariate data

I have a set of files consisting of randomly selected points from a dataset, each file belonging to a particular class. Each row in these files contains the coordinates in n-space of the point. I'd ...
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### Fastest PCA algorithm for high-dimensional data

I would like to perform a PCA on a dataset composed of approximately 40 000 samples, each sample displaying about 10 000 features. Using Matlab princomp function consistently takes over half an hour ...
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### Enumerating hexahedral cell vertices and faces in arbitrary dimension

I have a Cartesian mesh in $d$ dimensions, and I would like to enumerate all the subcells of a given hexahedral cell. If I am just enumerating the vertices of a cell (or cells that contain a vertex) I ...
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### Working with multi-dimensional functions

How would you represent functions of type $[-1, 1]^n \to \mathbb R \;$ for moderate $n$? How would you integrate them? For small $n$ (1-2) such functions can be represented as histograms, vectors in ...
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### Constructing the origin position by transforming distance information

Suppose a set of $n$ points, $n\in M$, is given in some $d-$dimensional space, $X\in\mathbb{R}^{n\times d}$. Among these $n$ points, some $k\in K$ are selected, so $k<n$, and the distances from ...
I have troubles imagining how to compute a volume and centroid of an n-dimesional convex polytope. For a polygon (especially for convex polygon) the area and centroid are described in (wiki) by $$... 2answers 129 views ### How to efficiently compute the total least squares with an inequality constraint I am looking for an efficient method to compute$$\sum_{i=1}^\left|B\right|\left|Ax_i-b_i\right|^2\rightarrow min$$under the condition$$\forall i, x_i\ge 0,$$where A is an n-by-m matrix and B ... 1answer 49 views ### Multivariate numerical integration with a non-uniform grid I want to approximate the integral:$$ I = \int f(\boldsymbol{x})d\boldsymbol{x} $$where \boldsymbol{x} is d-dimensional. I have a set of non-equally spaced points \boldsymbol{x}_1, \dots, ... 1answer 73 views ### High-dimensional representation of arbitrary input Given a symmetric matrix A\in\mathbb{R}^{n\times n} with positive entries and zero diagonal, is it always possible to construct a high-dimensional configuration in Euclidean space, such that these ... 1answer 171 views ### 3D to 2D projections, a generalization Given some data points in 3D, X\in\mathbb{R}^{n\times 3}, could one say that$$Y=XP,$$for some P\in\mathbb{R}^{3\times 2} actually corresponds to a particular viewpoint on a 3D data? Basically, ... 2answers 88 views ### Handling inconsistent solutions obtained by PCA In order to achieve a 2D representation X\in\mathbb{R}^{n\times 2} of some high-dimensional data residing in Y\in\mathbb{R}^{n\times k}, I use PCA:$$X=Y\cdot U,$$where U\in\mathbb{R}^{k\times 2} ... 2answers 85 views ### A sufficient number of distances to recover relative positions of n points On several places I found different claims on a sufficient number of distances to recover relative positions of n points in d-dimensional space. For instance, work from ... 1answer 89 views ### Recovering coordinates by eigendecomposition without double-centering Suppose an Euclidean distance D\in\mathbb{R}^{n\times n} matrix between a set of n objects is given. To obtain inner-products (which will be further be used to recover coordinates), entries of D ... 1answer 154 views ### Proof continuation for rigid transformation on PCA solution Suppose a matrix X\in\mathbb{R}^{n\times 3} is given as a Principal Component Analysis (PCA) projection from some high dimensional space. The 2D PCA solution on X, say Y\in\mathbb{R}^{n\times 2} ... 1answer 64 views ### Optimality criterion of PCA via recovered distances It is stated in http://users.eecs.northwestern.edu/~yingwu/teaching/EECS510/Reading/Williams_NIPS01.pdf that the PCA mapping from h-dimensional data to low k-dimensional space minimizes ... 1answer 143 views ### k-Nearest Neighbor Search using examples I want to perform k-Nearest Neighbor Search in multidimensional space, but not using for example L_2-distance. I want the user to specify some "similar"-pairs examples and then perform a search ... 1answer 77 views ### application of oscillatory high-dimensional functions Has anybody stumbled upon any kind of application of high-frequency high-dimensional problems (d\geq 4)? My interest comes from the following: there is quite a decent amount of papers where people ... 1answer 86 views ### Configuration shift to change the rank of a Gram matrix Suppose a matrix D\in\mathbb{R}^{n\times n} of Euclidean distances between n points is given. To obtain a Gram matrix (matrix of inner-products of points that give rise to distances in D), one ... 1answer 76 views ### What kinds of maths to learn for understanding dynamical systems in cognitive science? [closed] A current trend in cognitive science is to view the mind as a dynamical system (e.g., Continuity of Mind by Spivey, in which cognition is understood as a "continuous and often recurrent trajectory ... 1answer 61 views ### Normalizing axes prior to PCA For a given centered configuration of points X\in\mathbb{R}^{n\times 3}, the covariance matrix is denoted by S=\frac{1}{n}X^TX. Recall that the 2D PCA solution is obtained by Y=X\cdot U, where ... 0answers 21 views ### PCA performed on a configuration with scaled axes Suppose a configuration X\in\mathbb{R}^{n\times 2} is output of PCA on some high-dimensional data Y\in\mathbb{R}^{n\times h}. Note that this PCA is performed by$$X=Y\cdot U, where columns of ...
Given $d$-dimensional coordinates residing in a matrix $X\in\mathbb{R}^{n\times d}$, the Euclidean distance between items $i$ and $j$ is denoted as $g_{ij}$. Let $c\in\mathbb{R}^d$ denote the centroid ...