arXiv research

This paper connects ultrametric overlap gap properties to parametric RDT for symmetric binary perceptrons.

It is well-known that quasi-isometries between R-trees induce power quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper investigates power quasi-symmetric homeomorphisms between bounded, complete, uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising up to similarity …

We announce ultrametric analogues of the results of Kleinbock-Margulis for shrinking target properties of semisimple group actions on symmetric spaces. The main applications are S-arithmetic Diophantine approximation results and logarithm laws for buildings, generalizing the work of Hersonsky-Paulin on trees.

Following a review of metric, ultrametric and generalized ultrametric, we review their application in data analysis. We show how they allow us to explore both geometry and topology of information, starting with measured data. Some themes are then developed based on the use of metric, ultrametric and generalized ultrame…

Finite approximations help reconstruct countable metric and ultrametric spaces.

Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is gen…

The increasing needs of clustering massive datasets and the high cost of running clustering algorithms poses difficult problems for users. In this context it is important to determine if a data set is clusterable, that is, it may be partitioned efficiently into well-differentiated groups containing similar objects. We …

We model anomaly and change in data by embedding the data in an ultrametric space. Taking our initial data as cross-tabulation counts (or other input data formats), Correspondence Analysis allows us to endow the information space with a Euclidean metric. We then model anomaly or change by an induced ultrametric. The in…

Tessellations cover planes without gaps or overlaps.

We study the problem of fitting an ultrametric distance to a dissimilarity graph in the context of hierarchical cluster analysis. Standard hierarchical clustering methods are specified procedurally, rather than in terms of the cost function to be optimized. We aim to overcome this limitation by presenting a general opt…

New model addresses instability in hierarchical clustering of social networks.

Develops a variational method for ultrametric phylogenetic trees.

Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. "Structure" can be understood as symmetry and a range of symmetries are expressed by hierarchy. Such symmetries directly point to invariants, that pinpoint intrinsic properties of the data and of the …

We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (all scale uniform equivalences). We prove that these equivalences can be characterized with parallel constructions using a combinatoric tool called common zi…

Decomposes ultrametric spaces into scaled simplices.

This note explores norms beyond ultrametric inequalities in non-Archimedean analysis.

The Baire metric induces an ultrametric on a dataset and is of linear computational complexity, contrasted with the standard quadratic time agglomerative hierarchical clustering algorithm. In this work we evaluate empirically this new approach to hierarchical clustering. We compare hierarchical clustering based on the …

We consider a sparse high dimensional regression model where the goal is to recover a $k$-sparse unknown vector $β^*$ from $n$ noisy linear observations of the form $Y=Xβ^*+W \in \mathbb{R}^n$ where $X \in \mathbb{R}^{n \times p}$ has iid $N(0,1)$ entries and $W \in \mathbb{R}^n$ has iid $N(0,σ^2)$ entries. Under certa…

It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete \br-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that, when the ultrametric spac…

This paper introduces hierarchical quasi-clustering methods, a generalization of hierarchical clustering for asymmetric networks where the output structure preserves the asymmetry of the input data. We show that this output structure is equivalent to a finite quasi-ultrametric space and study admissibility with respect…

The paper derives upper bounds on eigenvalues of Laplace-Beltrami operator on hyperbolic surfaces.

New method for estimating mean in SS inference with selection bias and decaying overlap.

Method preserves order in hierarchical clustering of ordered data.

This paper uses two hierarchical techniques, a minimal spanning tree and an ultrametric hierarchical tree, to extract a topological influence map for major currencies from the ultrametric distance matrix for 1996-2001. We find that these two techniques generate a defined and robust scale free network with meaningful ta…

New model for detecting communities in weighted bipartite networks.

The high-frequency cross-correlation existing between pairs of stocks traded in a financial market are investigated in a set of 100 stocks traded in US equity markets. A hierarchical organization of the investigated stocks is obtained by determining a metric distance between stocks and by investigating the properties o…

Consider observation data, comprised of n observation vectors with values on a set of attributes. This gives us n points in attribute space. Having data structured as a tree, implied by having our observations embedded in an ultrametric topology, offers great advantage for proximity searching. If we have preprocessed d…

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale …

The study simplifies assessing overlap in logistic regression models using empirical likelihood.

Failing to distinguish between a sheepdog and a skyscraper should be worse and penalized more than failing to distinguish between a sheepdog and a poodle; after all, sheepdogs and poodles are both breeds of dogs. However, existing metrics of failure (so-called "loss" or "win") used in textual or visual classification/r…

New findings show that common optimization algorithms struggle with random problems.

This paper presents a novel spectral algorithm with additive clustering designed to identify overlapping communities in networks. The algorithm is based on geometric properties of the spectrum of the expected adjacency matrix in a random graph model that we call stochastic blockmodel with overlap (SBMO). An adaptive ve…

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let $X$ be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension $d$. Informally, the theorem states that if $X$ has sufficiently strong higher-dimensional expansion properties (which generali…

Study potential computational gaps in symmetric binary perceptrons using fl-RDT.

CausalMix generates synthetic data with causal controls for mixed-type tables.

New method uses cohomology to quantify molecular similarity.

New GLPs split Lévy bridges into non-overlapping subprocesses.

New method detects overlapping communities in weighted graphs without pure nodes assumption.

AdaDEM decouples EM into two parts to improve class overlap and uncertainty.

New LT-O-learners improve HLTE estimation with low overlap.

Deconfounding scores improve causal effect estimation with weak overlap.

FCPCA fuzzy clusters high-dimensional time series data efficiently.

There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even c…

The availability of large microarray data has led to a growing interest in biclustering methods in the past decade. Several algorithms have been proposed to identify subsets of genes and conditions according to different similarity measures and under varying constraints. In this paper we focus on the exclusive row bicl…

New metrics assess class overlap and imbalance in datasets.

Proposes a meta-algorithm for classification with overlapping classes in high-energy physics.

C-Learner improves stability of plug-in estimators for causal inference.

We prove that if X is a complete geodesic metric space with uniformly generated first homology group and $f: X\to R$ is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and adapting the definition of hyperbolic approximation we obtain an intrinsic sufficent …