Related papers: Technical report: Two observations on probability …
Random projections (RP) are a popular tool for reducing dimensionality while preserving local geometry. In many applications the data set to be projected is given to us in advance, yet the current RP techniques do not make use of…
This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in R^p as the number of points n -> infinity, while the dimension p is either fixed or growing with n. For both…
We construct symmetric representations of distributions over two-dimensional plane with given mean values as convex combinations of distributions with supports containing not more than three points and with the same mean values.
This paper studies the problem of identifying directions of axial symmetry in multivariate distributions. Theoretical results are derived on how the measure or cardinality of the set of symmetry directions relates to spherical symmetry. The…
Posterior distribution over a countable set M of continuous data-sampling distributions piles up at L-projection of the true distribution r on M, provided that the L-projection is unique. If there are several L-projections of r on M, then…
One way of defining probability distributions for circular variables (directions in two dimensions) is to radially project probability distributions, originally defined on $\mathbb{R}^2$, to the unit circle. Projected distributions have…
Random projection is often used to project higher-dimensional vectors onto a lower-dimensional space, while approximately preserving their pairwise distances. It has emerged as a powerful tool in various data processing tasks and has…
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying…
Random Projection is a foundational research topic that connects a bunch of machine learning algorithms under a similar mathematical basis. It is used to reduce the dimensionality of the dataset by projecting the data points efficiently to…
A theoretical framework is developed to describe the transformation that distributes probability density functions uniformly over space. In one dimension, the cumulative distribution can be used, but does not generalize to higher…
A random vector whose norm and overlap (inner product with an independent copy) concentrates is shown to have random low-dimensional projections that are approximately random Gaussians. Conversely, asymptotically random Gaussian projections…
The manifold of empirical mean values of statistical data ad infinitum has a geometric shape that depends on the probability measure that governs the generating model. Large deviation theory produces entropy functions that depend on both…
We consider products of uniform random variables from the Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^n$, $k \le n$, and random vectors from the $n$-dimensional $\ell_p^n$-ball $\mathbb{B}_p^n$ with certain $p$-radial…
Computing the similarity between two probability distributions is a recurring theme across control. We introduce a unified family of distances between the probability distributions of two random variables that is based on the discrepancy…
Recent theoretical work has identified random projection as a promising dimensionality reduction technique for learning mixtures of Gausians. Here we summarize these results and illustrate them by a wide variety of experiments on synthetic…
In ref [math.ST/0411462] the notion of statistically dual distributions is introduced. The reconstruction of confidence density [AIP Conference Proceedings 803 (2005) 398] for the location parameter for several pairs of statistically dual…
A general piecewise (including pointwise) probability distribution with space-saving notation and its hierarchical particular cases are considered. The explicit closed-form normalization, expectation, and variance formulas along with the…
Information theoretical measures, such as entropy, mutual information, and various divergences, exhibit robust characteristics in image registration applications. However, the estimation of these quantities is computationally intensive in…
We investigate $(2+1)$-dimensional discretized directed polymers in Gaussian random media. By numerically calculating the probability distribution function of overlap between two independent and identical systems on a common random…
Random projections offer an appealing and flexible approach to a wide range of large-scale statistical problems. They are particularly useful in high-dimensional settings, where we have many covariates recorded for each observation. In…