Paper provides a performance guarantee for spectral clustering.
problem Finding the global solution to the minimum ratio cut problem.
method Two-step spectral clustering method with a rounding step, analyzed using two-to-infinity norm perturbation bounds.
result Spectral clustering is guaranteed to output the global solution under certain conditions.
A novel hypergraph partitioning method using tensor eigenvalue decomposition captures super-dyadic interactions.
problem Capturing super-dyadic interactions in k-uniform hypergraphs.
method Tensor-based representation and tensor eigenvalue decomposition for capturing interactions.
result Improved min-cut solution on 2-uniform hypergraphs (graphs) compared to standard spectral partitioning.
Spectral clustering is sensitive to how graphs are constructed from data particularly when proximal and imbalanced clusters are present. We show that Ratio-Cut (RCut) or normalized cut (NCut) objectives are not tailored to imbalanced data since they tend to emphasize cut sizes over cut values. We propose a graph partit…
Spectral clustering methods which are frequently used in clustering and community detection applications are sensitive to the specific graph constructions particularly when imbalanced clusters are present. We show that ratio cut (RCut) or normalized cut (NCut) objectives are not tailored to imbalanced cluster sizes sin…
Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced k-cut of the graph, are either based on greedy techniques or heuristics which have weak connection to th…
This paper establishes the consistency of a family of graph-cut-based algorithms for clustering of data clouds. We consider point clouds obtained as samples of a ground-truth measure. We investigate approaches to clustering based on minimizing objective functionals defined on proximity graphs of the given sample. Our f…
A recent theoretical analysis shows the equivalence between non-negative matrix factorization (NMF) and spectral clustering based approach to subspace clustering. As NMF and many of its variants are essentially linear, we introduce a nonlinear NMF with explicit orthogonality and derive general kernel-based orthogonal m…
Spectral clustering (SC) and graph-based semi-supervised learning (SSL) algorithms are sensitive to how graphs are constructed from data. In particular if the data has proximal and unbalanced clusters these algorithms can lead to poor performance on well-known graphs such as k-NN, full-RBF, ε-graphs. This is becaus…
We relax indicator matrices to form a manifold for faster optimization.
problem Optimizing indicator matrices is NP-hard.
method Developed a Riemannian manifold (RIM) and Riemannian optimization methods.
result RIM manifold optimization is significantly faster and yields better results.
Algorithms based on spectral graph cut objectives such as normalized cuts, ratio cuts and ratio association have become popular in recent years because they are widely applicable and simple to implement via standard eigenvector computations. Despite strong performance for a number of clustering tasks, spectral graph cu…
Spectral clustering has become one of the most widely used clustering techniques when the structure of the individual clusters is non-convex or highly anisotropic. Yet, despite its immense popularity, there exists fairly little theory about performance guarantees for spectral clustering. This issue is partly due to the…
Minimum Description Length prevents overfitting in noisy data.
problem Learning from noisy data with overfitting risk.
method Minimum Description Length learning rule with tempered guarantees.
result Tempered agnostic finite sample learning guarantees and asymptotic behavior characterization.
Minimum attention improves reinforcement learning performance in high-dimensional dynamics.
problem Improving reinforcement learning performance in high-dimensional nonlinear dynamics.
method Applying minimum attention as a regularization technique in reinforcement learning, including model-based and model-free approaches.
result Minimum attention outperforms state-of-the-art algorithms in few-shot adaptation and variance reduction.
The minimum number of colors is a challenging knot invariant since, by definition, its calculation requires taking the minimum over infinitely many minima. In this article we estimate and in some cases calculate the minimum number of colors for the Turk's head knots on three strands.
Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum braids to study graph trees, amphicheirality, unknotting numbers, and periodic table…
Study tightens bounds for interpolating noisy data using minimum l1-norm.
problem Predicting noisy data with minimum l1-norm interpolation.
method Provided matching upper and lower bounds for prediction error.
result Tight consistency up to negligible terms for d≫n. A new classification method based on Minimum Spanning Trees
problem Improving classification in supervised learning
method Proposing a classification algorithm based on Minimum Spanning Trees
result The proposed method is effective and computationally efficient
Minimum-norm solutions generalize well in over-parametrized neural networks.
problem Generalization error in over-parametrized neural networks.
method Analyzing three models: random feature model, two-layer neural network, and residual network.
result Generalization error for minimum-norm solutions is comparable to Monte Carlo rate, up to logarithmic terms.
The paper finds minimum Dehn colors for knots and defines useful graphs for coloring.
problem Finding the minimum number of colors for Dehn colorings of knots.
method Analyzes Dehn colorings for knots and defines R-palette graphs. result For Dehn p-colorable knots, the minimum number of colors is at least ⌊log2pfloor+2. The paper calculates genus bounds for multibranched surfaces.
problem Finding genus bounds for multibranched surfaces.
method Using the first Betti number and boundary genus, the paper provides lower bounds for maximum and minimum genus.
result The maximum and minimum genus of GimesS1 equals twice that of G. Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length $\mbox{Len}(K)$ of a knot K i…
Study shows how networks converge to minimum norm solutions with regularization.
problem Interpolating between known regions in shallow ReLU networks.
method Investigates empirical risk minimizers and weight decay regularizers.
result Empirical risk minimizers converge to minimum norm interpolants under specific conditions.
We find the minimum dilatation of pseudo-Anosov braids with many strands.
problem Finding the minimum dilatation of pseudo-Anosov braids with a large number of strands.
method Analyzing examples of Hironaka-Kin and Venzke to determine the minimum dilatation.
result The minimum dilatation is approximately 13.928 for large n. Study introduces AMVP and AMRR for dynamic portfolio optimization in volatile markets.
problem Optimizing portfolios in volatile and nonstationary financial markets.
method Adaptive Minimum-Variance Portfolio (AMVP) framework with ARFIMA-FIGARCH processes and non-Gaussian innovations.
result Demonstrated superior performance in risk reduction and portfolio stability during market breaks.
Minimum algebraic intersection found in hyperbolic surfaces, growing with genus.
problem Finding the minimum algebraic intersection form in hyperbolic surfaces.
method Analyzing algebraic intersection form in moduli space of hyperbolic surfaces.
result Minimum grows in the order of (logg)−2 with genus. Computed minimum crossing numbers for Turaev genus 2 links.
problem Verifying the Qazaqzeh-Chbili-Lowrance conjecture.
method Computed minimum crossing numbers for a specific family of links.
result Verified the Qazaqzeh-Chbili-Lowrance conjecture for the family.
This paper studies the geometry of minimum-volume confidence sets for multinomial parameters.
problem Determining if minimum-volume confidence sets for multinomial outcomes are disjoint.
method Enumerating and covering the continuous regions of the exact p-value function to study the geometry of minimum-volume confidence sets.
result The geometry of minimum-volume confidence sets for multinomial parameters is studied, providing insights into their structure and properties.
Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. Those are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have inde…
The paper calculates minimum Dehn colors for knots using symmetric local biquandle cocycles.
problem Determining the minimum number of Dehn colors for knots.
method Using symmetric local biquandle cocycle invariants to evaluate minimum Dehn colors.
result There exist knots distinguished by minimum numbers of Dehn colors.
Inference for normal and Monte Carlo distributions using minimum relative entropy.
problem Inference from partial information on expectations and covariances.
method Minimum relative entropy sub-manifolds, analytical formulas, Monte Carlo simulations.
result Improved numerical implementation for inference from partial information.
ML helps select variables for minimum-variance portfolios, reducing risk and improving performance.
problem Optimizing minimum-variance portfolios with relevant predictors.
method Parameterized minimum-variance portfolio weights using a large pool of firm-level characteristics and their transformations.
result ML-selected predictors lead to lower risk and better performance in minimum-variance portfolios.
We investigate the time series of the degree of minimum spanning trees obtained by using a correlation based clustering procedure which is starting from (i) asset return and (ii) volatility time series. The minimum spanning tree is obtained at different times by computing correlation among time series over a time windo…
Paper proves edge-connectivity equals minimum degree for graphs with non-negative curvature.
problem Edge-connectivity vs. minimum degree in graphs with non-negative curvature.
method Analyzes finite connected graphs with non-negative Lin-Lu-Yau curvature.
result Edge-connectivity equals minimum degree for graphs with non-negative curvature.
Simple graphs with 12 nodes and 6 neighbors always have a 6-node subgraph.
problem Finding a specific subgraph in simple graphs.
method Proving every graph of order 12 with minimum degree 6 contains a K_6 minor.
result Simple graphs of order 12 and minimum degree 6 contain K_6 minors.
New approach finds minimum width for deep, narrow MLPs.
problem Finding the minimum width for deep, narrow MLPs to approximate continuous functions.
method Proposes a framework to simplify finding minimum width into determining a geometrical function w(dx,dy) based on input and output dimensions. result Proves that w(dx,dy) equals the optimal minimum width for deep, narrow MLPs to achieve universality. We consider the relations between different measures of complexity for free homotopy classes of curves on a surface Σ, including the minimum number of self-intersections, the minimum length of the words representing them in a geometric presentation of π1(Σ), and the minimum degree of the coverings of Σ to which …
Unified framework for geometric computation of minimum-area homotopy.
problem Computing the minimum homotopy area of a closed curve.
method Unified combinatorial word approach combining geometric and algebraic methods.
result Unified geometric proof and constructive algorithm for minimum area homotopy.
The study tightens bounds on binomial probabilities and minimums using KL-divergence.
problem Tightening bounds on binomial probabilities and minimums of i.i.d. Binomials.
method Applied Sanov's theorem to derive upper and lower bounds on binomial tail probabilities and minimums, expressed in terms of KL-divergence.
result High probability upper and lower bounds on the minimum of i.i.d. Binomial random variables, finite sample, asymptotically tight.
In this paper we study the minimum dilatation pseudo-Anosov mapping classes coming from fibrations over the circle of a single 3-manifold, the mapping torus for the "simplest pseudo-Anosov braid". The dilatations that arise include the minimum dilatations for orientable mapping classes for genus g=2,3,4,5,8 as well as …
Originally, the SW-equations discovered by Seiberg-Witten are 1st-order PDE, which solutions (A,φ), with φ\ne 0, are known as SW-monopoles. It is known that the solutions of these 1st-order eq correspond to the minimum of SW-functional. However, it is not true, that for all spin^{c} class α, the minimum is always attai…
Investigates the long-only minimum variance portfolio in factor models.
problem Understanding the long-only minimum variance portfolio in factor models.
method Investigates the long-only global minimum variance portfolio in a factor model of returns, providing explicit and geometric descriptions for different factor models.
result Provides rigorous and explicit descriptions of the long-only solution in terms of covariance matrix parameters and geometric descriptions for multiple factors.
Paper shows minimum 10 vertices for hyperbolic origami 2-torus.
problem Finding minimum vertices for hyperbolic origami 2-torus.
method Geodesic triangulation and isometric polyhedral embedding.
result 10 vertices are the minimum required for a hyperbolic origami 2-torus.
Defines MER for Bayesian learning, a gap between achievable and optimal performance.
problem Analyzing the best performance of Bayesian learning under generative models.
method Two methods for deriving upper bounds for MER: conditional mutual information and minimum estimation error.
result Quantifies the rate at which MER decays to zero with more data and relates it to model richness.
Mathematical framework for minimum enclosing ball problem.
problem Determining the smallest sphere enclosing a set in d-dimensional space.
method Theoretical framework based on enclosing and partitioning theorems.
result Bounds and relations between circumradius, inradius, diameter, and width.
Consider the problem of estimating the minimum entropy of pseudo-Anosov maps on a surface of genus g with n punctures. We determine the behaviour of this minimum number for a certain large subset of the (g,n) plane, up to a multiplicative constant. In particular it has been shown that for fixed n, this minimum …
Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.
New framework for DNN training guarantees convergence to global minimum.
problem Training deep neural networks to converge to global minimum.
method Reformulated minimization problem with recursive algorithmic framework, using bounded style assumptions.
result Convergence to an ε-(global) minimum with O(1/ε^3) gradient computations.
Even knots with more than 30 crossings are not fertile.
problem Understanding fertility in knots with specific crossing numbers.
method Analyzing minimum crossing number diagrams and changing over-under information.
result Even knots with more than 30 crossings cannot be obtained from a minimum crossing number diagram by changing over-under information.