A new method for faster spatial modeling on exascale computers.
problem Scalable, memory-efficient machine learning for spatially distributed data.
method Partitioned Sparse Variational Gaussian Process (PSVGP) with decentralized communication.
result Improved spatial predictions and better model fit with minimal overhead.
Clarifies connections between Nyström and SVGP methods for scalable GPs.
problem Lack of understanding between GP and kernel methods communities.
method Investigates Nyström and SVGP methods for scalable Gaussian processes.
result Establishes connections and equivalences between Nyström and SVGP methods.
Improves SVGP methods for faster and more accurate Gaussian process inference.
problem Efficient non-conjugate Gaussian process inference.
method Dual parameterization of SVGP methods using site parameters.
result Faster and more accurate inference with tighter evidence lower bound.
A new method for self-attention models that improves uncertainty estimation.
problem Overconfident predictions and lack of calibrated uncertainty in Transformers.
method Kernel-Eigen Pair Sparse Variational Gaussian Processes (KEP-SVGP) with Kernel SVD (KSVD) to handle asymmetry of attention kernels.
result Reduction in time complexity and improved performance on various benchmarks.
SVGP KAN integrates uncertainty quantification into Kolmogorov-Arnold networks.
problem Uncertainty quantification in scientific machine learning models.
method Sparse variational Gaussian process inference with Kolmogorov-Arnold topology.
result Demonstrated ability to distinguish aleatoric and epistemic uncertainty in various scientific applications.
A scalable online method for Gaussian processes that improves decision-making in various applications.
problem Scalability issues with Gaussian processes for online decision-making.
method Online variational conditioning (OVC) for SVGPs.
result OVC enables efficient online learning and decision-making with SVGPs.
Optimizes data acquisition in high-dimensional Bayesian optimization.
problem Suboptimal data acquisition in high-dimensional Bayesian optimization tasks.
method Utility-calibrated variational inference to align approximations with BO goals.
result Optimal data acquisition decisions under a limited computational budget.
SVGP KAN integrates sparse variational GP with KANs for scalable probabilistic inference.
problem Lack of probabilistic outputs in standard KANs and cubic scaling of Gaussian Process methods.
method Sparse Variational GP-KAN combines KAN topology with sparse variational inference and permutation-based importance analysis.
result Enables probabilistic KANs to handle larger datasets with linear computational complexity.
A scalable algorithm approximates Bayesian posteriors in RKHS with improved efficiency.
problem Scalable inference for Bayes posteriors in infinite-dimensional spaces.
method Approximate Langevin diffusion projection onto first M components, using law of total probability and sufficiency assumption.
result The method recovers SVGP as a special case and is provably close to optimal for convex and Lipschitz continuous likelihoods.
This paper proposes a method to approximate non-Gaussian likelihoods in Gaussian Processes.
problem Approximating non-Gaussian likelihoods in Gaussian Processes.
method Proposes a piece-wise constant approximation for the inverse-link function.
result Yields a closed form solution for the SVGP lower bound.
New method trains sparse Gaussian processes without matrix inversion.
problem Costly training of Gaussian processes at scale.
method Inverse-free approach using matmul-only natural-gradient updates.
result Significantly improved stability and convergence in training.
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 …
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.
Sparse Gaussian process hyperparameters optimized using MCMC.
problem Hyperparameter uncertainty leads to biased estimates and underestimation of predictive uncertainty.
method Proposes an MCMC algorithm to sample from the hyperparameter posterior in sparse Gaussian process regression.
result Significantly improves sampling efficiency in the Gaussian likelihood case.
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.
Graph partitioning is the problem of dividing the nodes of a graph into balanced partitions while minimizing the edge cut across the partitions. Due to its combinatorial nature, many approximate solutions have been developed, including variants of multi-level methods and spectral clustering. We propose GAP, a Generaliz…
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.
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.
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.
A novel online GP model captures long-term memory in sequential data.
problem Capturing long-term memory in sequential data online.
method Integrates HiPPO framework into interdomain GP, leveraging time-varying orthogonal projections as inducing variables.
result OHSVGP outperforms existing online GP methods in predictive performance, long-term memory preservation, and computational efficiency.
Space partitions of Rd underlie a vast and important class of fast nearest neighbor search (NNS) algorithms. Inspired by recent theoretical work on NNS for general metric spaces [Andoni, Naor, Nikolov, Razenshteyn, Waingarten STOC 2018, FOCS 2018], we develop a new framework for building space partitions re…
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.
This work improves Gaussian process model selection for large datasets.
problem Prohibitively high computational cost in Gaussian process model selection.
method Linear-time scaling and computational uncertainty tradeoff.
result Computation-aware Gaussian processes can be trained on large datasets efficiently.
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…
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.
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.
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.
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. 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…
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. 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…
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.
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.
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.
New online GP algorithm offers performance guarantees for streaming data.
problem Training and inference of GPs require all historic data, limiting online decision-making.
method Developed a new theoretical framework based on PAC-Bayes theory, optimizing empirical risk and parameter divergence.
result Offers both a guarantee of generalized performance and good accuracy.
Both supervised and unsupervised machine learning algorithms have been used to learn partition-based index structures for approximate nearest neighbor (ANN) search. Existing supervised algorithms formulate the learning task as finding a partition in which the nearest neighbors of a training set point belong to the same…
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.
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.
Stochastic partition models divide a multi-dimensional space into a number of rectangular regions, such that the data within each region exhibit certain types of homogeneity. Due to the nature of their partition strategy, existing partition models may create many unnecessary divisions in sparse regions when trying to d…