Fitting trees to $\ell_1$-hyperbolic distances

• URL: http://arxiv.org/abs/2409.01010v1
• Date: Mon, 2 Sep 2024 07:38:32 GMT
• Title: Fitting trees to $\ell_1$-hyperbolic distances
• Authors: Joon-Hyeok Yim, Anna C. Gilbert,
• Abstract summary: Building trees to represent or to fit distances is a critical component of phylogenetic analysis. We study the tree fitting problem as finding the relation between the hyperbolicity (ultrametricity) vector and the error of tree (ultrametric) embedding.
• Score: 4.220336689294244
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Building trees to represent or to fit distances is a critical component of phylogenetic analysis, metric embeddings, approximation algorithms, geometric graph neural nets, and the analysis of hierarchical data. Much of the previous algorithmic work, however, has focused on generic metric spaces (i.e., those with no a priori constraints). Leveraging several ideas from the mathematical analysis of hyperbolic geometry and geometric group theory, we study the tree fitting problem as finding the relation between the hyperbolicity (ultrametricity) vector and the error of tree (ultrametric) embedding. That is, we define a vector of hyperbolicity (ultrametric) values over all triples of points and compare the $\ell_p$ norms of this vector with the $\ell_q$ norm of the distortion of the best tree fit to the distances. This formulation allows us to define the average hyperbolicity (ultrametricity) in terms of a normalized $\ell_1$ norm of the hyperbolicity vector. Furthermore, we can interpret the classical tree fitting result of Gromov as a $p = q = \infty$ result. We present an algorithm HCCRootedTreeFit such that the $\ell_1$ error of the output embedding is analytically bounded in terms of the $\ell_1$ norm of the hyperbolicity vector (i.e., $p = q = 1$) and that this result is tight. Furthermore, this algorithm has significantly different theoretical and empirical performance as compared to Gromov's result and related algorithms. Finally, we show using HCCRootedTreeFit and related tree fitting algorithms, that supposedly standard data sets for hierarchical data analysis and geometric graph neural networks have radically different tree fits than those of synthetic, truly tree-like data sets, suggesting that a much more refined analysis of these standard data sets is called for.

Related papers

• Do you know what q-means? [50.045011844765185]
  Clustering is one of the most important tools for analysis of large datasets. We present an improved version of the "$q$-means" algorithm for clustering. We also present a "dequantized" algorithm for $varepsilon which runs in $Obig(frack2varepsilon2(sqrtkd + log(Nd))big.
  arXiv  Detail & Related papers  (2023-08-18T17:52:12Z)
• Tight and fast generalization error bound of graph embedding in metric space [54.279425319381374]
  We show that graph embedding in non-Euclidean metric space can outperform that in Euclidean space with much smaller training data than the existing bound has suggested. Our new upper bound is significantly tighter and faster than the existing one, which can be exponential to $R$ and $O(frac1S)$ at the fastest.
  arXiv  Detail & Related papers  (2023-05-13T17:29:18Z)
• Random graph matching at Otter's threshold via counting chandeliers [16.512416293014493]
  We propose an efficient algorithm for graph matching based on similarity scores constructed from counting a certain family of weighted trees rooted at each. This is the first graph matching algorithm that succeeds at an explicit constant correlation and applies to both sparse and dense graphs.
  arXiv  Detail & Related papers  (2022-09-25T20:00:28Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• HyperAid: Denoising in hyperbolic spaces for tree-fitting and hierarchical clustering [36.738414547278154]
  We propose a new approach to treemetric denoising (HyperAid) in hyperbolic spaces. It transforms original data into data that is more'' tree-like, when evaluated in terms of Gromov's $delta$ hyperbolicity. We integrate HyperAid with schemes for enforcing nonnegative edge-weights.
  arXiv  Detail & Related papers  (2022-05-19T17:33:16Z)
• Active-LATHE: An Active Learning Algorithm for Boosting the Error Exponent for Learning Homogeneous Ising Trees [75.93186954061943]
  We design and analyze an algorithm that boosts the error exponent by at least 40% when $rho$ is at least $0.8$. Our analysis hinges on judiciously exploiting the minute but detectable statistical variation of the samples to allocate more data to parts of the graph.
  arXiv  Detail & Related papers  (2021-10-27T10:45:21Z)
• Decentralized Learning of Tree-Structured Gaussian Graphical Models from Noisy Data [1.52292571922932]
  This paper studies the decentralized learning of tree-structured Gaussian graphical models (GGMs) from noisy data. GGMs are widely used to model complex networks such as gene regulatory networks and social networks. The proposed decentralized learning uses the Chow-Liu algorithm for estimating the tree-structured GGM.
  arXiv  Detail & Related papers  (2021-09-22T10:41:18Z)
• Unfolding Projection-free SDP Relaxation of Binary Graph Classifier via GDPA Linearization [59.87663954467815]
  Algorithm unfolding creates an interpretable and parsimonious neural network architecture by implementing each iteration of a model-based algorithm as a neural layer. In this paper, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we unroll a projection-free algorithm for semi-definite programming relaxation (SDR) of a binary graph. Experimental results show that our unrolled network outperformed pure model-based graph classifiers, and achieved comparable performance to pure data-driven networks but using far fewer parameters.
  arXiv  Detail & Related papers  (2021-09-10T07:01:15Z)
• SGA: A Robust Algorithm for Partial Recovery of Tree-Structured Graphical Models with Noisy Samples [75.32013242448151]
  We consider learning Ising tree models when the observations from the nodes are corrupted by independent but non-identically distributed noise. Katiyar et al. (2020) showed that although the exact tree structure cannot be recovered, one can recover a partial tree structure. We propose Symmetrized Geometric Averaging (SGA), a more statistically robust algorithm for partial tree recovery.
  arXiv  Detail & Related papers  (2021-01-22T01:57:35Z)
• Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding [7.381113319198104]
  We explore a new method for learning hyperbolic representations by taking a metric-first approach. Rather than determining the low-dimensional hyperbolic embedding directly, we learn a tree structure on the data. This tree structure can then be used directly to extract hierarchical information, embedded into a hyperbolic manifold.
  arXiv  Detail & Related papers  (2020-05-08T04:30:21Z)
• Explainable $k$-Means and $k$-Medians Clustering [25.513261099927163]
  We consider using a small decision tree to partition a data set into clusters, so that clusters can be characterized in a straightforward manner. We show that popular top-down decision tree algorithms may lead to clusterings with arbitrarily large cost. We design an efficient algorithm that produces explainable clusters using a tree with $k$ leaves.
  arXiv  Detail & Related papers  (2020-02-28T04:21:53Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.