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.
Optimizes minimum-volume prediction sets for multivariate regression.
problem Lack of efficient methods for multivariate conformal prediction.
method Optimization-driven framework for minimum-volume covering sets.
result Efficient and informative prediction sets with tight coverage.
Paper certifies intersection of minimum-volume confidence sets for multinomial outcomes.
problem Certifying intersection of minimum-volume confidence sets for multinomial outcomes.
method Exploits likelihood ordering to induce halfspace constraints, enabling adaptive geometric partitioning and computable bounds on p-values.
result Efficient and provably sound algorithm for certifying intersection, disjointness, or indeterminate result.
SLS optimizes minimum-volume regions for conditional quantiles, bypassing density estimation.
problem Constructing minimum-volume prediction regions that satisfy conditional coverage.
method Super-level-set regression (SLS) directly optimizes geometric boundaries of conditional level sets.
result SLS optimizes regions directly, capturing complex conditional structures end-to-end.
A new topic modeling method that minimizes topic simplex volume.
problem Topic modeling efficiency and accuracy.
method Reformulates LDA as minimizing topic simplex volume, uses convex relaxation and ADMM.
result Relaxed problem has same global minimum as original under assumptions.
Anomaly detection algorithm using nearest neighbor graphs and max-margin learning.
problem Anomaly detection in high-dimensional data.
method Rank nearest neighbor scores, train max-margin models to imitate, declare anomalies based on percentile.
result Asymptotically optimal decision region converges to minimum volume level set.
Extends conformal prediction to contrastive learning for better coverage of positive samples.
problem Lack of principled guarantees on coverage in contrastive learning.
method Introduces minimum-volume covering sets with learnable constraints.
result Improves inclusion-exclusion trade-offs in positive and negative samples.
Paper finds minimum volume for specific anti-de Sitter 3-manifolds.
problem Finding the minimum volume of globally hyperbolic anti-de Sitter 3-manifolds.
method Analyzes maximal globally hyperbolic Cauchy-compact anti-de Sitter 3-manifolds.
result Minimum volume is π²|χ(M)|, attained for Fuchsian manifolds.
We enumerate the small-volume manifolds that can be obtained by Dehn filling on Mom-2 and Mom-3 manifolds as defined by Gabai, Meyerhoff, and the author. In so doing we complete the proof that the Weeks manifold is the minimum-volume compact hyperbolic 3-manifold, as well as enumerating the 10 smallest one-cusped hyper…
Paper compares two entropy concepts for finite presentation groups.
problem Comparing two entropy concepts for groups of finite presentation.
method Analyzes and contrasts minimum volume entropy for geometrically finite groups.
result Two entropy concepts coincide in dimension 1 but differ in others.
In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the…
The Hessian of the renormalized volume of geometrically finite hyperbolic 3-manifolds without rank-1 cusps, computed at the hyperbolic metric g with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichmüller space. The metric g is known fro…
New method for NMF without tuning parameter.
problem Finding latent structures in noisy data matrices.
method Inspired by square-root lasso, proposes a tuning-free minimum-volume NMF.
result Optimal tuning parameter value is noise level-independent.
Minimal Gaussian surfaces partitioning space with minimum area.
problem Partitioning space with minimal Gaussian surface area.
method Second variation argument using infinitesimal translations, combined with Colding-Minicozzi theory and Euclidean Double Bubble Conjecture arguments.
result Triple and Quadruple Bubble Conjectures for Gaussian measure.
Four hyperbolic 24-cell 4-manifolds with one cusp are identified.
problem Identifying hyperbolic 24-cell 4-manifolds with one cusp.
method Analyzing hyperbolic geometry and isometry.
result Found four one-cusped hyperbolic 4-manifolds of minimum volume.
Minimal Gaussian surface area is achieved by cones over a regular simplex for m>3 sets partitioning Rn.
problem Finding the minimal Gaussian surface area of m sets partitioning Rn. method Volume-preserving variations of the sets, avoiding matrix-valued partial differential inequalities.
result Strengthened Milman-Neeman Gaussian multi bubble theorem and first known dimension-independent bounds for the Plurality is Stablest Conjecture.
We develop an efficient algorithm to find confidence ellipsoids with volume guarantees in high dimensions.
problem Finding robust confidence ellipsoids in high-dimensional data.
method Polynomial time algorithm using primal-dual structure and geometric Brascamp-Lieb inequality.
result Algorithm finds ellipsoids within a O(β)γd volume factor of best β-conditioned ellipsoid. A new method for SSMF improves upon existing algorithms.
problem Identify identifiable solutions in simplex-structured matrix factorization.
method Dual simplex volume maximization approach.
result The proposed method outperforms state-of-the-art SSMF algorithms.
Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space CV(Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding. We also prove that for every $…
Paper proves min-vol NMF robust to noise under expanded condition.
problem Robustness of min-vol NMF to noise.
method Proved robustness under expanded sufficiently scattered condition.
result Proves min-vol NMF identifies groundtruth factors in noise.
We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points.
Gradient flow converges to a minimal convex structure.
problem Finding the minimal convex structure in hyperbolic manifolds.
method Weil-Petersson gradient vector field of renormalized volume.
result The flow converges to the structure with minimum convex core volume.
This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic 3-manifold with volume <= 2.848 can be obtained by a Dehn filling on one of 21 cusped hyperbolic 3-manifolds. We also sho…
Study shows continuity of renormalized volume for geometrically convergent hyperbolic structures.
problem Continuity of renormalized volume under geometric limits.
method Extended renormalized volume concept for geometrically finite hyperbolic 3-manifolds and showed continuity for geometrically convergent sequences.
result Renormalized volume attains its minimum at the geodesic class.
Optimizes test set size for accurate diagnosis using machine learning.
problem Determining the minimum test set size for accurate diagnosis.
method Proposes machine learning methods (LASSO and SVM) to predict optimal test set size.
result SVM achieves 90.4% accuracy with a reduced test set by 35.24%.
Researchers found the minimum volume of a 3-cusped hyperbolic 3-manifold.
problem Finding the minimum volume of a 3-cusped orientable hyperbolic 3-manifold.
method Using guts in sutured and pared manifolds.
result The volume of a 3-cusped orientable hyperbolic 3-manifold is at least 5.49... = 6 × Catalan's constant.
In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the systolic volume depen…
Study finds a minimum volume for vector fields on a punctured sphere.
problem Finding the minimum volume of unit vector fields on a punctured sphere.
method Analyzes the volume of vector fields tangent to an antipodally punctured unit 2-sphere.
result Provides a lower bound for the volume of unit vector fields.
New measure of knot complexity tied to twist number.
problem Understanding knot complexity.
method Defining alternating volume and showing coarsely equivalent to twist number.
result Alternating volume of a knot is coarsely equivalent to its twist number.
The Ekeland variational principle implies what can be regarded as a strong version, in the C1 category, of the Yau minimum principle: under the appropriate hypotheses {\it every} minimizing sequence admits a {\it good shadow}, a second minimizing sequence that has good properties and is asymptotic to the original on…
Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.
problem Developing an ABP approach to Sobolev and Michael-Simon inequalities under volume noncollapsing assumptions.
method Using a refinement of Brendle's contact-set argument to derive lower bounds for the volumes of geodesic balls.
result A Michael-Simon type inequality for immersed submanifolds with nonnegative sectional curvature and volume noncollapsing.
The study finds upper bounds and computes volumes of ideal right-angled polyhedra in Lobachevsky space.
problem Finding upper bounds and computing volumes of ideal right-angled polyhedra in Lobachevsky space.
method Analyzing a class of right-angled polyhedra with vertices on the absolute, obtaining upper bounds on volumes, computing volumes for polyhedra with up to 23 faces, and introducing the class of polyhedra with isolated triangles.
result Minimum volumes are realized on antiprisms and twisted antiprisms, and the first 248 values of volumes are presented.
The paper provides bounds for the empirical angular measure and applies them to improve statistical learning in extreme regions.
problem Estimating the angular measure in high-dimensional data with different distributions.
method Established bounds for the maximal deviations of the empirical angular measure from the true measure, using rank transformation and analyzing the most extreme observations.
result The bounds provide performance guarantees for statistical learning procedures in extreme regions, such as binary classification and anomaly detection.
This is an expository paper on Mom-technology, describing the recent work of the authors in this area (found in arXiv:math/0606072, arXiv:0705.4325, and arXiv:0809.0346) concerning the use of Mom-technology to find the minimum-volume compact hyperbolic 3-manifold and the 10 smallest cusped hyperbolic 3-manifolds. In ad…
Study finds knots with ideal length need not have smallest volume.
problem Tackles the conjecture that ideal knot length equals smallest volume.
method Measures convex hull volume of knots during length annealing.
result Identifies knots with non-ideal global minimum volume.
We prove that among all constant width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant width body obtained by rotation of the Reuleaux triangle about an axis of symmetry.
Proves a conjecture about the maximum tet-volume of triangulations of a 2-sphere.
problem Proving the conjectured maximum tet-volume for all triangulations of a 2-sphere.
method Simplified version of Mathieu and Thurston's combinatorial proof using more general volume notions.
result Proves the full conjecture about the maximum tet-volume for all triangulations of a 2-sphere.
The paper develops optimal confidence regions for categorical data.
problem Constructing tight confidence regions for categorical data.
method Develops new theory for minimum average volume confidence regions.
result Shows optimality of the regions for categorical data and its implications for machine learning.
The volume distance from a point p to a convex hypersurface M of the (N+1)-dimensional space is defined as the minimum (N+1)-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane H(p) we ob…
In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of…
MMCGAN uses explicit manifold learning to improve GAN performance.
problem GAN mode collapse and unstable training.
method Introduces Minimum Manifold Coding (MMC) as a prior to guide GAN training.
result MMCGAN effectively alleviates mode collapse and stabilizes GAN training.
A Seifert surface F for a knot K is free if the complement of F is a handlebody (i.e., has free fundamental group). The free genus of K is the minimum genus among all free Seifert surfaces for K. In this paper we show that there exist families of hyperbolic knots with arbitrarily large volume, which each have free genu…
Study reveals how model volume affects learning curves in machine learning.
problem Understanding the double descent risk phenomenon in machine learning.
method Investigates the role of model volume using MDL, Occam's Razor, and information geometry.
result Model volume can explain the double descent risk, suggesting better generalization with increased dimensionality.
Study σ2-curvature and volume of compact manifolds, proving conditions for Einstein metrics and geodesic balls.
problem Understanding σ2-curvature and volume in compact manifolds. method Critical point analysis, volume comparison, variational properties, geodesic balls.
result Sufficient and necessary condition for a critical metric to be Einstein, volume comparison results.
The paper introduces a new method to rank anomalies in multivariate data.
problem Ranking multivariate unlabeled observations based on their degree of abnormality.
method Formulates the problem as a M-estimation problem using the Mass Volume curve (MV curve).
result Established generalization bounds for the MV curve estimation.
This paper improves volume bounds for orbifolds of symmetric spaces.
problem Finding minimum volumes for orbifolds modeled on symmetric spaces.
method Combining H. C. Wang's radius estimate with Gunther's volume comparison theorem.
result Explicit uniform lower volume bounds for arbitrary orbifold quotients of irreducible symmetric spaces.
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. MaxVol NMF maximizes the volume of H in NMF for better sparse and interpretable solutions.
problem Finding interpretable and unique NMF solutions.
method Dual approach to MinVol NMF, maximizing the volume of H. result MaxVol NMF solutions correspond to clustering columns in disjoint clusters.