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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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 ...
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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 ...
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138 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 ...
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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 ...
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82 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 ...
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83 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 $D$ ...
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281 views
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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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 ...
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128 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 $B$ ...
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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 $U\in\mathbb{R}^{k\times 2}$ ...
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59 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 ...
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Relation to all-pairs Euclidean distances
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 ...
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Convex polytope volume and centroid calculation
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
$$
...
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81 views
A sufficient number of distances to recover relative positions of n points
On several places I found different claims on a sufficient number of distances to recover relative positions of $n$ points in $d$-dimensional space.
For instance, work from
...
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77 views
application of oscillatory high-dimensional functions
Has anybody stumbled upon any kind of application of high-frequency high-dimensional problems ($d\geq 4$)?
My interest comes from the following: there is quite a decent amount of papers where people ...
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171 views
3D to 2D projections, a generalization
Given some data points in 3D, $X\in\mathbb{R}^{n\times 3}$, could one say that
$$Y=XP,$$ for some $P\in\mathbb{R}^{3\times 2}$ actually corresponds to a particular viewpoint on a 3D data? Basically, ...
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106 views
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 ...
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152 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 $Y\in\mathbb{R}^{n\times 2}$ ...
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163 views
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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211 views
Kolmogorov–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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430 views
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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162 views
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 ...
