Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much …
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48 results for “set similarity”
Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. It is shown in [Dai, Rao and Zhang, Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule,https://doi.org/10.1088/1361-6544/ab1275] that for a connected self-similar set, space-filling curves can be c…
In this paper, we introduce the notion of asymptotic self-similar sets on general doubling metric spaces by extending the notion of self-similar sets, and determine their Hausdorff dimensions, which gives an extension of Balogh and Rohner 's result. This is carried out by introducing the notions of almost similarity ma…
Estimates set overlap and similarity using random samples.
problem Estimating set overlap and similarity with limited data.
method Binomial model for predicting set overlap, comparing to previous methods.
result Binomial model provides better estimates with small sample sizes.
New clustering method using point-set kernel measures similarity.
problem Measuring similarity between objects for clustering.
method Point-set kernel for similarity computation; clustering procedure uses this measure.
result Proposed method is more effective and faster than existing algorithms.
In [9] Kaimanovich introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown in [13] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in det…
Paper explains contrastive learning using cosine similarity and proposes mitigations for batch size effects.
problem Understanding and improving contrastive learning through batch size effects.
method Unified framework of cosine similarity, theoretical insights, and auxiliary loss.
result Performance improvement in small-batch settings through proposed auxiliary loss.
New stability measures for similar features improve feature selection accuracy.
problem Existing stability measures fail to distinguish similar features in highly correlated datasets.
method Introduce new adjusted stability measures that consider feature similarities.
result One new stability measure considers highly similar features as interchangeable.
Researchers create a Fredholm module on fractal shapes like the Cantor set.
problem Constructing Fredholm modules on complex fractal structures.
method Combining combinatorial techniques with higher-dimensional analogues.
result Calculated Dixmier trace of operators induced by the module.
Bayesian methods detect significant IIA violations in similarity choice data.
problem Detecting IIA violations in similarity choice data complicates classical models.
method Proposed two statistical methods: classical goodness-of-fit test and Bayesian PPC.
result Significant IIA violations confirmed in both datasets, driven by context effects.
Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similar…
The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on Rm. Atsushi Kameyama (Distances on Topological Self-Similar Sets, Proceedings of Symposia in Pure Mathematics, Volume 72.1, 2004) asked the following fundamental question: given a topological self-si…
We introduce GSimCNN (Graph Similarity Computation via Convolutional Neural Networks) for predicting the similarity score between two graphs. As the core operation of graph similarity search, pairwise graph similarity computation is a challenging problem due to the NP-hard nature of computing many graph distance/simila…
In this paper, we propose a family of graph partition similarity measures that take the topology of the graph into account. These graph-aware measures are alternatives to using set partition similarity measures that are not specifically designed for graph partitions. The two types of measures, graph-aware and set parti…
Proposes a new stability measure for model fitting on similar feature data sets.
problem Model fitting on data sets with similar features is challenging.
method Tuning hyperparameters in a multi-criteria fashion with predictive accuracy and feature selection stability.
result Our approach achieves similar or better predictive performance than single-criteria and stability selection approaches.
Learning-based Efficient Graph Similarity Computation via Multi-Scale Convolutional Set Matchingcs.LG
Graph similarity computation is one of the core operations in many graph-based applications, such as graph similarity search, graph database analysis, graph clustering, etc. Since computing the exact distance/similarity between two graphs is typically NP-hard, a series of approximate methods have been proposed with a t…
The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.
problem Understanding the structure and properties of homeomorphism groups of self-similar 2-manifolds.
method Survey of recent results, exposition of classical results, treatment of stable sets, and proof of new theorems.
result Characterization of homeomorphisms of perfectly self-similar 2-manifolds and extensions of existing results.
Develops a fast method to analyze image similarities without pre-trained models.
problem Lack of efficient methods to analyze fine-level similarities in large image datasets.
method Wavelet decomposition and numerical analysis tools.
result Identifies similar images in standard datasets (like CIFAR) in a few seconds.
In a context of document co-clustering, we define a new similarity measure which iteratively computes similarity while combining fuzzy sets in a three-partite graph. The fuzzy triadic similarity (FT-Sim) model can deal with uncertainty offers by the fuzzy sets. Moreover, with the development of the Web and the high ava…
ProtoBandit uses bandits to find prototypes efficiently.
problem Finding a compact set of prototypes from a large dataset.
method Stochastic greedy search and multi-armed bandits.
result ProtoBandit reduces similarity comparisons to , independent of target set size.
CatSIM measures image similarity robustly to small changes.
problem Measuring similarity between images, especially with small perturbations.
method Uses structural similarity image quality paradigm, robust to small location changes.
result Structural similarity between images rated higher when not entirely overlapping.
We consider the problem of predicting several response variables using the same set of explanatory variables. This setting naturally induces a group structure over the coefficient matrix, in which every explanatory variable corresponds to a set of related coefficients. Most of the existing methods that utilize this gro…
Automates similarity measure construction from data.
problem Defining similarity measures analytically is challenging.
method Applies machine learning to learn similarity measures from data.
result Data-driven similarity measure outperforms state of the art methods.
SOAK: Same/Other/All K-fold cross-validation for estimating similarity of patterns in data subsetsstat.ML
SOAK assesses data subset similarity for better model training.
problem Estimating similarity between data subsets for accurate predictions.
method Same/Other/All K-fold cross-validation method.
result SOAK estimates similarity of learnable/predictable patterns in data subsets.
Data Twinningstat.ML
Twinning splits data into fast, statistically similar sets.
problem Creating statistically similar data splits for Big Data.
method Twinning is a method based on SPlit for fast, model-independent dataset splitting.
result Twinning is orders of magnitude faster than SPlit.
Paper develops active learning for clustering unknown pairwise similarities.
problem Learning positive and negative pairwise similarities efficiently.
method Generic active learning framework for correlation clustering.
result Demonstrates effectiveness of query strategies in clustering.
Study properties of self-similar continua with finite intersection property.
problem Characterize self-similar continua with finite intersection property.
method Prove intersection graph criterion, finite order theorem, and parameter matching theorem.
result All Jordan arcs starting from a intersection point in such continuum on a plane should have the same slope parameter at that point.
Fractal Lipschitz-Killing curvature measures C^f_k(F,.), k = 0, ..., d, are determined for a large class of self-similar sets F in R^d. They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures C_k(F_r,.) from geometric measure theory of parallel sets F_r for small distances…
Hierarchical clustering based on pairwise similarities is a common tool used in a broad range of scientific applications. However, in many problems it may be expensive to obtain or compute similarities between the items to be clustered. This paper investigates the hierarchical clustering of N items based on a small sub…
Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show tha…
Study on self-similar sets on Riemannian manifolds with new separation conditions.
problem Analyzing self-similar sets on Riemannian manifolds with new separation conditions.
method Formulated weak separation and finite type conditions for conformal iterated function systems on Riemannian manifolds.
result Obtained formulas for Hausdorff dimensions of self-similar and graph self-similar sets.
Genus 2 Cantor setsmath.GT
First constructed genus 2 Cantor set in 3D space.
problem Constructing a geometrically self-similar Cantor set of genus 2.
method Geometrically self-similar construction in .
result First uniformly quasiregular mapping with a genus 2 Cantor set Julia set.
Excessive reuse of test data has become commonplace in today's machine learning workflows. Popular benchmarks, competitions, industrial scale tuning, among other applications, all involve test data reuse beyond guidance by statistical confidence bounds. Nonetheless, recent replication studies give evidence that popular…
Minwise hashing is the standard technique in the context of search and databases for efficiently estimating set (e.g., high-dimensional 0/1 vector) similarities. Recently, b-bit minwise hashing was proposed which significantly improves upon the original minwise hashing in practice by storing only the lowest b bits of e…
The problem of hierarchical clustering items from pairwise similarities is found across various scientific disciplines, from biology to networking. Often, applications of clustering techniques are limited by the cost of obtaining similarities between pairs of items. While prior work has been developed to reconstruct cl…
Study on self-similar solutions of supercritical Fujita equation, proving entropy and energy gap.
problem Characterization and stability of solutions to supercritical Fujita equation.
method Introduction of -functional, -stability, and entropy; use of mean curvature flows.
result Constant solution has lowest entropy among bounded positive self-similar solutions.
In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat -manifold , we show that the set of similarity classes of flat metrics on which occur as a cusp cross-section of a hyperbolic -orbifold is dense in the space of similarity classes of flat metri…
Towards Measuring Place Function Similarity at Fine Spatial Granularity with Trajectory Embeddingcs.AI
Study uses trajectory embedding to measure place function similarity at fine spatial granularity.
problem Measuring place function similarity at fine spatial granularity.
method Trajectory embedding to reduce dimensions and measure similarity of place functions.
result Embedding similarity can be a metric proxy for place functions at fine spatial granularity.
Algorithm aggregates rewards from multiple players to learn related tasks in online bandit learning.
problem Learning related but slightly different tasks in an online setting with heterogeneous feedback.
method RobustAgg algorithm that aggregates rewards from different players.
result Achieves instance-dependent regret guarantees and nearly matching lower bounds.
There are plenty of problems where the data available is scarce and expensive. We propose a generator of semi-artificial data with similar properties to the original data which enables development and testing of different data mining algorithms and optimization of their parameters. The generated data allow a large scal…
A Probabilistic Theory of Supervised Similarity Learning for Pointwise ROC Curve Optimizationstat.ML
The performance of many machine learning techniques depends on the choice of an appropriate similarity or distance measure on the input space. Similarity learning (or metric learning) aims at building such a measure from training data so that observations with the same (resp. different) label are as close (resp. far) a…
Multi-label classification is an important learning problem with many applications. In this work, we propose a principled similarity-based approach for multi-label learning called SML. We also introduce a similarity-based approach for predicting the label set size. The experimental results demonstrate the effectiveness…
Infinite fractal tree solves shortest connection problem.
problem Finding the shortest connection for a fractal set.
method Constructing an infinite planar self-similar binary tree.
result The tree is the unique solution to the Steiner problem.
Proposes a robust similarity measure for sparse time series data.
problem Sparse time course data in biological settings.
method Gaussian processes (GP) similarity measure based on log-likelihood ratio.
result Enhanced robustness to noise compared to Euclidean distance.
For many analytical problems the challenge is to handle huge amounts of available data. However, there are data science application areas where collecting information is difficult and costly, e.g., in the study of geological phenomena, rare diseases, faults in complex systems, insurance frauds, etc. In many such cases,…
BiLRP explains deep similarity models by decomposing scores into feature contributions.
problem Verifying meaningful patterns in complex similarity models.
method Augmenting similarity scores with feature explanations using LRP.
result BiLRP robustly explains deep neural network features and historical document similarities.
We found a new simple family of Cantor sets whose projections are one-dimensional.
problem Finding simple Cantor sets with specific projection properties.
method Developed a new series of self-similar Cantor sets in .
result All projections of these new Cantor sets are connected and one-dimensional.
CLS measures dataset similarity through decision rule performance.
problem Measuring dataset similarity in machine learning, especially for transfer learning and domain adaptation.
method Cross-Learning Score (CLS) measures similarity through bidirectional generalization performance of decision rules, linking to cosine similarity under canonical linear models.
result CLS effectively measures dataset similarity and transferability, validated on synthetic and real-world datasets.