Related papers: Geometry and clustering with metrics derived from …

We present a generic dynamic programming method to compute the optimal clustering of $n$ scalar elements into $k$ pairwise disjoint intervals. This case includes 1D Euclidean $k$-means, $k$-medoids, $k$-medians, $k$-centers, etc. We extend…

Consensus algorithms are popular distributed algorithms for computing aggregate quantities, such as averages, in ad-hoc wireless networks. However, existing algorithms mostly address the case where the measurements lie in a Euclidean space.…

Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric…

We study expansion/contraction properties of some common classes of mappings of the Euclidean space ${\mathbb R}^n, n\ge 2\,,$ with respect to the distance ratio metric. The first main case is the behavior of M\"obius transformations of the…

Existing deep embedding methods in vision tasks are capable of learning a compact Euclidean space from images, where Euclidean distances correspond to a similarity metric. To make learning more effective and efficient, hard sample mining is…

This paper explores hierarchical clustering in the case where pairs of points have dissimilarity scores (e.g. distances) as a part of the input. The recently introduced objective for points with dissimilarity scores results in every tree…

We determine when a quasi-isometry between discrete spaces is at bounded distance from a bilipschitz map. From this we prove a geometric version of the Von Neumann conjecture on amenability. We also get some examples in geometric groups…

We study a family of parametric statistical models based on gamma distributions, which do give realistic descriptions for other stochastic porous media. Gamma distributions contain as a special case the exponential distributions, which…

This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at "quantum…

We consider a class of piecewise hyperbolic maps from the unit square to itself preserving a contracting foliation and inducing a piecewise expanding quotient map, with infinite derivative (like the first return maps of Lorenz like flows).…

Correlation matrices are fundamental summaries of functional brain networks, yet standard analyses often treat entries independently, ignoring the curved geometry of correlation space. Existing geometric methods frequently lack closed-form…

Tiling spaces are constructed using a metric in which two tilings of $\mathbb{R}^n$ are close if and only if, after a small translation, they agree on a large ball around the origin. We construct analogous spaces to study random…

Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as…

Gaussian mixture model is very useful in many practical problems. Nevertheless, it cannot be directly generalized to non Euclidean spaces. To overcome this problem we present a spherical Gaussian-based clustering approach for partitioning…

In this work, the possibility of clustering correlated random variables was examined, both because of their mutual similarity and because of their similarity to the principal components. The k-means algorithm and spectral algorithms were…

The contributions of the paper span theoretical and implementational results. First, we prove that Kd-trees can be extended to spaces in which the distance is measured with an arbitrary Bregman divergence. Perhaps surprisingly, this shows…

What is the shortest path between two data points lying in a high-dimensional space? While the answer is trivial in Euclidean geometry, it becomes significantly more complex when the data lies on a curved manifold -- requiring a Riemannian…

Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of…

Distance-based clustering and classification are widely used in various fields to group mixed numeric and categorical data. In many algorithms, a predefined distance measurement is used to cluster data points based on their dissimilarity.…

We introduce ordered and unordered configuration spaces of 'clusters' of points in an Euclidean space $\mathbb{R}^d$, where points in each cluster satisfy a 'verticality' condition, depending on a decomposition $d=p+q$. We compute the…