Norm-range partition improves MIPS search efficiency by reducing query complexity.
problem Efficiently searching for maximum inner product in large datasets.
method Norm-range partition technique that divides datasets into sub-datasets with similar norms and builds independent hash indexes.
result Significantly reduces the number of probed buckets for LSH-based MIPS algorithms.
Norm-ranging LSH improves MIPS performance by addressing 2-norm distribution issues.
problem Long tails in 2-norm distribution of real datasets affect Simple-LSH performance.
method Norm-ranging LSH partitions datasets into sub-datasets and builds independent hash indexes.
result Norm-ranging LSH achieves an order of magnitude speedup over Simple-LSH.
HBR improves normative modeling of neuroimaging data across multiple sites.
problem Dealing with nuisance variation in neuroimaging data across different sites.
method Hierarchical Bayesian regression (HBR) for multi-site normative modeling.
result HBR provides more accurate normative ranges compared to existing methods.
The study limits how many parts regular simplicial partitions can overlap.
problem Bounding the intersection number of regular simplicial partitions.
method Analyzing the properties of regular simplicial partitions.
result Established a maximum limit for the intersection number.
Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received …
New method improves nearest neighbor search using neural networks and graph partitioning.
problem Efficient nearest neighbor search in high-dimensional spaces.
method Developed a new framework for space partitioning using neural networks and graph partitioning.
result Neural LSH partitions outperform existing methods on standard benchmarks.
Rectangular Bounding Process (RBP) improves partitioning efficiency in multi-dimensional spaces.
problem Creating many unnecessary divisions in sparse regions when describing dense regions.
method Introduces Rectangular Bounding Process (RBP) to efficiently partition multi-dimensional spaces using a bounding strategy.
result The RBP is self-consistent and can be extended to infinite space, offering rich yet parsimonious expressiveness.
GAP uses deep learning to efficiently partition graphs.
problem Graph partitioning to minimize edge cut.
method Deep learning approach with a differentiable loss function.
result GAP achieves competitive partitions and generalizes to unseen graphs.
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…
The study examines the balancedness of random partition models and finds the rich-get-richer characteristic is a result of model assumptions.
problem The balancedness of random partition models is largely neglected in the literature.
method Formulated a framework to define and study the balancedness of exchangeable random partition models, analyzed using product-form exchangeability and projectivity assumptions.
result The 'rich-get-richer' characteristic is an inevitable consequence of the model assumptions.
The paper develops mixed-integer formulations for neural networks using partitioning.
problem Optimizing trained ReLU neural networks with balanced model size and tightness.
method Partitioning node inputs into groups, forming the convex hull via disjunctive programming.
result The proposed formulations outperform existing ones, especially with fewer partitions.
New partition designs reduce star discrepancy in high-dimensional sampling.
problem Improving the expected star discrepancy in high-dimensional sampling.
method Developed non-equal volume partitions to achieve lower expected star discrepancy.
result Explicit upper bounds for expected star discrepancy under non-equal volume partitions.
The paper constructs Markov partitions for geodesic flow on hyperbolic surfaces.
problem Understanding Markov partitions for general hyperbolic flows.
method Rigorous construction of Markov partitions for geodesic flow on Riemann surfaces of constant negative curvature.
result Explicit forms of rectangles and local cross sections provided for the geodesic flow.
New method unifies and formalizes data partitioning using a single vector.
problem Data partitioning and clustering methods.
method Rank-one matrix factorization and denoising of piecewise constant signals.
result Demonstrates robustness of denoising step in partitioning.
Survey of mass partition problems in geometry and topology.
problem Inducing partitions on measures or sets by dividing space.
method Recent progress in topology, discrete geometry, and computer science.
result Connections between different fields.
Locally isoperimetric partitions minimize perimeter in space.
problem Finding minimal perimeter partitions in space.
method Proving closure theorem to limit sequences of isoperimetric clusters.
result Examples of isoperimetric partitions in various dimensions.
Online BSP-Forest improves space partitioning for large-scale classification and regression.
problem Efficient space partitioning for large-scale classification and regression problems.
method Developed an online BSP-Forest framework that expands space coverage and refines partition structure in real-time.
result Guaranteed universal consistency for both classification and regression problems.
Study of Torelli groups of partitioned surfaces with bounds and asymptotic lengths.
problem Understanding Torelli groups of partitioned surfaces.
method Topological and dynamical analysis of Torelli groups of partitioned surfaces.
result Asymptotic translation lengths of Torelli groups of partitioned surfaces behave almost like the reciprocal of the Euler characteristic of the surface.
Efficiently calculates PL model likelihood for partitioned preference data.
problem Computational infeasibility of calculating PL model likelihood for partitioned preference data.
method Random utility model formulation and efficient numerical integration approach.
result Proposed method outperforms existing LTR baselines and scales to real-world tasks.
We argue that the standard graph Laplacian is preferable for spectral partitioning of signed graphs compared to the signed Laplacian. Simple examples demonstrate that partitioning based on signs of components of the leading eigenvectors of the signed Laplacian may be meaningless, in contrast to partitioning based on th…
Paper recovers lattice signal partitions efficiently.
problem Estimating lattice partition from noisy data.
method Uses dyadic CART for computationally-efficient partition recovery.
result Consistently estimates partition with optimal error rate.
New proof of a unique 3-part partition in 8D space.
problem Existence of a non-standard isoperimetric partition in high dimensions.
method Analytical proof showing existence of a specific partition.
result Existence of a non-standard isoperimetric partition in 8D space.
Hypergraph partitioning is an important problem in machine learning, computer vision and network analytics. A widely used method for hypergraph partitioning relies on minimizing a normalized sum of the costs of partitioning hyperedges across clusters. Algorithmic solutions based on this approach assume that different p…
We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.
Maps discrete manifolds to partitions to define new manifolds.
problem Creating manifolds from discrete structures.
method Mapping discrete d-manifolds onto (k+1)-partite complexes to define new manifolds.
result Defines a (d-k)-manifold from simplices in G mapped to P.
Proposes a new BSP-Tree process for flexible space partition modeling.
problem Limited modelling flexibility of axis-aligned partitions in Mondrian process.
method Introduces a self-consistent Binary Space Partitioning (BSP)-Tree process with oblique cuts.
result Clear inferential improvements over standard Mondrian process and related methods.
Standard bubbles and partitions are stable in various model spaces.
problem Stability of standard bubbles and partitions in different model spaces.
method New conjugated Brascamp-Lieb inequality and conformally flattening boundary potential.
result Stability of standard bubbles and partitions in Rn, Sn, and Hn. Algorithms learn and test variable partitions in various groups and error metrics.
problem Learning and testing variable partitions in different groups and error metrics.
method Algorithms for agnostically learning and testing k-partitionability over various groups and error metrics. result Learning algorithms for k-partitionability with polynomial time complexity and testing with adaptive queries. GRAF uses global partitioning to improve ensemble classifier performance.
problem Improving ensemble classifier performance.
method GRAF extends oblique decision trees to global partitioning.
result GRAF reduces generalization error and improves performance on benchmark datasets.
To devise efficient solutions for approximating a mean partition in consensus clustering, Dimitriadou et al. [3] presented a necessary condition of optimality for a consensus function based on least square distances. We show that their result is pivotal for deriving interesting properties of consensus clustering beyond…
A new multilabel classification framework improves ANN search performance.
problem Efficiently finding approximate nearest neighbors in large datasets.
method Formulated ANN search as a multilabel classification problem, using partitioning classifiers.
result Natural classifier leads to strictly improved performance in ANN search.
This paper presents Sparse Partitioning, a Bayesian method for identifying predictors that either individually or in combination with others affect a response variable. The method is designed for regression problems involving binary or tertiary predictors and allows the number of predictors to exceed the size of the sa…
The study explores continuous noncrossing partitions and their relation to weighted circular factorizations.
problem Understanding the structure of continuous noncrossing partitions on the unit circle.
method Analyzes degree-d continuous noncrossing partitions and their equivalence classes of weighted linear factorizations.
result Maximal elements in the poset of continuous noncrossing partitions form a subspace homeomorphic to the dual Garside classifying space for the d-strand braid group.
Proves an Euler-type formula for Möbius strip partitions.
problem No specific problem stated; focuses on a mathematical formula.
method Analyzes partitions of the Möbius strip.
result Proves an Euler-type formula for Möbius strip partitions.
Although consistency is a minimum requirement of any estimator, little is known about consistency of the mean partition approach in consensus clustering. This contribution studies the asymptotic behavior of mean partitions. We show that under normal assumptions, the mean partition approach is consistent and asymptotic …
Homology of partition algebras matches symmetric group homology under certain conditions.
problem Understanding homology of partition algebras and comparing it to symmetric groups.
method Inductive resolution and high acyclicity arguments, parallel to earlier work on Brauer algebras.
result Homology of partition algebras is isomorphic to symmetric group homology under specific conditions.
Topological recursion recovers a specific partition function for colored knots.
problem Recovering the extended Ooguri-Vafa partition function for colored HOMFLY-PT polynomials of torus knots.
method Applying topological recursion to the spectral curve of colored HOMFLY-PT polynomials of torus knots.
result Topological recursion reproduces the n-point functions of the extended Ooguri-Vafa partition function.
This thesis classifies pseudo-Anosov homeomorphisms using geometric Markov partitions.
problem Classifying pseudo-Anosov homeomorphisms up to topological conjugacy.
method Algorithmic approach using geometric Markov partitions.
result Geometric type is a complete invariant of conjugation.
New model of vague knowledge without strict partitions or transitivity.
problem Standard economic models of information fail to capture real-world vague knowledge.
method Relaxing assumptions of transitivity and partition structure to formalize vague knowledge.
result Vague knowledge can distinguish some states but not partition the state space.
POUnets combine partitions of unity and monomials for efficient deep learning.
problem Efficiently approximating functions with deep neural networks in high dimensions.
method Integrates partitions of unity and monomials into neural network architecture.
result POUnets achieve hp-convergence for smooth functions and outperform MLPs for discontinuous functions.
Enhances robustness of multi-view clustering via partition fusion.
problem Dealing with noises and inconsistency in multi-view data.
method Generates multiple partitions, integrates them, and co-evolves graph learning, partition generation, and view weight learning.
result Empirical results verify the effectiveness and robustness of the proposed approach.
Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of regions and region-specific models. This paper proposes novel convex region-specific…
Study on detecting hierarchical community structures in networks.
problem Detecting hierarchical community structures in networks.
method Analysis of planted hierarchies of partitions in networks, identifying additional detectability phases.
result There are additional phases in which the presence of multiple consistent partitions can either help or hinder detection of hierarchical structures.
Proposes SPFB method for optimizing partition functions in stochastic learning.
problem Optimizing partition functions in stochastic learning settings.
method Stochastic Gradient Bound (SPFB) method based on upper-bounding the partition function with a quadratic surrogate.
result Sub-linear convergence rate of SPFB method and efficient training of deep learning models.
Bayesian nonparametric method partitions shapes using curves.
problem Capturing complex shapes in multi-dimensional data.
method Proposes a novel spline partitioning approach using curves.
result Demonstrates improved shape modeling compared to existing methods.
Fitting statistical models is computationally challenging when the sample size or the dimension of the dataset is huge. An attractive approach for down-scaling the problem size is to first partition the dataset into subsets and then fit using distributed algorithms. The dataset can be partitioned either horizontally (i…
Proposes a novel multi-view clustering method by aligning partitions.
problem Challenges of integrating multi-view information and information loss.
method Aligns partitions through rotation matrices and assigns weights to views.
result Significant improvement over state-of-the-art methods on real datasets.
Paper proposes a method to predict optimal data partitioning based on query execution costs.
problem Finding optimal data partitioning for improved system performance and scalability.
method Formal model abstraction of workload queries, genetic algorithm for optimization, evaluation using PostgreSQL's query optimizer.
result The approach effectively reduces workload execution cost and improves system performance.