Let K be a knot of genus g. If K is fibered, then it is well known that the knot group pi(K) splits only over a free group of rank 2g. We show that if K is not fibered, then pi(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then pi(K) splits over every free group of rank at …
Proximal splitting methods solve rank-constrained convex problems locally.
problem Solving optimization problems with rank constraints.
method Proximal splitting algorithms with conditions on rank constraint convex envelopes.
result Proximal splitting methods converge locally to solutions under convex relaxation conditions.
Study calculates the maximal dimension of totally geodesic submanifolds in symmetric spaces.
problem Understanding the maximal dimension of totally geodesic submanifolds in symmetric spaces.
method Defined and computed the splitting rank for symmetric spaces, and analyzed bounded cohomology.
result Explicitly computed the splitting rank for each irreducible symmetric space and showed the surjectivity of a comparison map.
Random survival forests (RSF) are a powerful method for risk prediction of right-censored outcomes in biomedical research. RSF use the log-rank split criterion to form an ensemble of survival trees. The most common approach to evaluate the prediction accuracy of a RSF model is Harrell's concordance index for survival d…
A new method selects variables for random survival forests using maximally selected rank statistics.
problem Random survival forests can be biased in selecting variables, especially for non-linear effects.
method Use maximally selected rank statistics for variable selection in random survival forests, comparing on p-value scale.
result The new method outperforms other approaches in prediction performance and computational speed.
We prove that any isometry of the graph of cyclic splittings of a finitely generated free group FN of rank N≥3 is induced by an outer automorphism of FN. The same statement also applies to the graphs of maximally-cyclic splittings, and of very small splittings.
We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.
iSplit LBI predicts individualized partial rankings from ties, outperforming state-of-the-art methods.
problem Predicting partial rankings from pairwise comparisons with ties, considering individual preferences.
method Variable splitting-based algorithm (iSplit LBI) that generates a sequence of estimations with a regularization path, decomposing parameters into abnormal signals, personalized signals, and random noise.
result iSplit LBI significantly outperforms state-of-the-art alternatives in predicting individualized partial rankings.
Global rigidity theorem for certain lattice actions on manifolds.
problem Volume-preserving actions of higher rank lattices on manifolds with dominated splitting.
method Proves standard conjugacy of actions with dominated splitting.
result Actions must be standard, manifold is flat torus with affine action.
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
Study new invariant to measure flat directions on complex manifolds.
problem Investigate numerical rank invariant on projective Kähler manifolds with semi-negative holomorphic sectional curvature.
method Introduce and analyze a new differential geometric numerical rank invariant.
result Bounding the new invariant by nef dimension and numerical Kodaira dimension.
This work simplifies proximal mapping for low-rank norms.
problem Efficient computation of proximal mappings for low-rank inducing norms.
method Reduces proximal mapping to nested binary search, solving simpler problems analytically.
result Simplified computation of proximal mappings for various norms.
Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.
problem CAT(0) spaces of higher rank with geometric group actions.
method Proving rigidity for spaces containing periodic flats and geodesics in flats.
result CAT(0) spaces of higher rank n≥2 are rigid if they contain a periodic n-flat. SBSS uses similarity to split data for better classifier training.
problem Training better classifiers with realistic performance estimation.
method SBSS uses both input and output space information to split data using similarity functions.
result SBSS outperformed ordinary stratified 10-fold cross-validation in 75% of scenarios.
This work introduces a transformation-based learner model for classification forests. The weak learner at each split node plays a crucial role in a classification tree. We propose to optimize the splitting objective by learning a linear transformation on subspaces using nuclear norm as the optimization criteria. The le…
A new method identifies class-specific covariates in multi-class prediction tasks.
problem Identifying covariates specifically associated with one or more outcome classes in multi-class prediction tasks.
method Introducing multi forests (MuFs) with multi-way and binary splits to measure class-associated discriminatory ability.
result The multi-class VIM specifically ranks class-associated covariates highly, unlike conventional VIMs.
In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating combinatorial descriptions and geometric properties of hyperbolic 3-manifolds.
A new Boosting algorithm with differential inclusion approach for sparse parameter control.
problem Sparse parameter control in machine learning models.
method Iterative regularization path with differential inclusions and variable splitting.
result Split LBI outperforms generalized Lasso in theory and experiments.
Cao's splitting theorem says that for any complete Kähler-Ricci flow (M,g(t)) with t∈[0,T), M simply connected and nonnegative bounded holomorphic bisectional curvature, (M,g(t)) is holomorphically isometric to $\C^k\times (N,h(t))$ where (N,h(t)) is a Kahler-Ricci flow with positive Ricci curvature for $t…
We prove that given a Hitchin representation in a real split rank 2 group G0, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermitian bundle over Teichmüller space. The proof goes through intr…
Let G be the real points of a semisimple algebraic Q-group, let H be an arithmetic subgroup of G and let T be the real points of an R-split torus in G. We prove that if there is a divergent T-orbit in G/H, and Q-rank(G) > 1, then the dimension of T is not larger than Q-rank(G). This provides a partial answer to a quest…
Paper proposes a novel SVM method for creating survival trees.
problem Creating non-linear survival trees for right-censored data.
method L2-regularized dipole splitting criteria with kernel methods.
result Non-linear splits using polynomial and Gaussian kernels show similar predictive power but often smaller tree sizes.
New G2 symmetry found in 8D distribution with 6D square.
problem Discovering new G2 symmetry in geometric distributions. method Analyzing rank 3 distribution on 8D manifold with growth vector (3,6,8).
result Maximally symmetric rank 3 distribution with 6D square.
Graph neural networks struggle with fair evaluation.
problem Evaluation of graph neural networks is flawed.
method Thorough empirical evaluation of four GNN models.
result Different splits of data lead to different model rankings.
We show that if M is a fibered, orientable 3-manifold, and if π1M has 1-relator presentation, then the presentation is induced by a Heegaard splitting of M. A corollary is that, for these manifolds, the rank of π1M is equal to the "restricted" Heegaard genus of M. We also explore the analogy between 1-rel…
Study shows infinite volumes of moduli spaces for certain groups.
problem Infinite volumes of Hitchin-Riemann moduli spaces for specific groups.
method Employed Goldman flows to find infinite disjoint subsets of identical volume.
result Proved infinite Atiyah-Bott-Goldman covolume for mapping class group actions.
CIT and CIF improve feature selection for downstream prediction.
problem Feature selection bias in machine learning models.
method Conditional inference trees and forests with Bonferroni correction.
result CIF ranks top 3 among 18 regression methods and top 4 among 17 classification methods.
The study proves conditions for CAT(0) spaces with higher rank rigidity.
problem Conditions for rigidity in CAT(0) spaces with higher rank.
method Geometric group action and analysis of geodesics and flats.
result CAT(0) spaces with specific properties are classified.
Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthog…
PSI-LinUCB improves scalability for large recommender systems.
problem Efficiently training and inferring for large action spaces in recommender systems.
method Represent inverse design matrix as diagonal + low-rank correction, derive stable rank-1 and batched updates, use projector-splitting integrator.
result Demonstrated effectiveness on recommender system datasets, achieving scalable training and inference.
Survival trees exhibit end-cut preference, leading to biased splits.
problem End-cut preference in survival trees causes biased splits and poor interpretability.
method Proposed a smooth sigmoid surrogate (SSS) approach to replace hard-threshold indicator function.
result Smooth sigmoid surrogate (SSS) effectively mitigates end-cut preference in survival trees.
An almost complex structure J on a 4-manifold X may be described in terms of a rank 2 vector bundle E. A splitting of J consists of a pair of line bundles spanning E. A hypersurface M in X satisfying a nondegeneracy condition inherits a CR-structure from J and a path geometry from the splitting. Using the Cartan-Kähler…
New invariant real rank identifies constant real Lie algebroids.
problem Characterizing complex Lie algebroids with constant real rank.
method Introducing real rank and minimal complex subalgebroid.
result Local splitting and characterization of complex Lie algebroids.
Study of rank 2 Kleinian groups with specific parabolic elements.
problem Characterizing discrete representations of rank 2 free groups into PSL(2,C). method Analyzing representations sending three disjoint curves to parabolics.
result Only maximal cusp groups of infinite covolume, and new finite covolume groups.
Unique submaximal symmetry found for certain parabolic geometries.
problem Determining the next realizable symmetry dimension in parabolic geometries.
method Analyzing submaximally symmetric structures of type (G,P) for specific Lie groups. result Local uniqueness of submaximally symmetric structures established.
Rank inequality proven for annular Khovanov homology of 2-periodic links.
problem Proving a rank inequality for annular Khovanov homology of 2-periodic links.
method Spectral sequence analysis with quantum and sl2 weight spaces.
result Proven rank inequality rank AKhj,k(L)≤rank AKh2j−k,k(ildeL). Complete classification of sub-Riemannian spaces with step and rank three.
problem Classifying sub-Riemannian model spaces with specific step and rank.
method Complete classification based on nilpotentization and algebra realization.
result No nontrivial sub-Riemannian model spaces of step and rank three with free nilpotentization.
Tests factor models by decomposing market into body and tail legs, revealing inconsistent results.
problem Inconsistency between factor models and market behavior.
method Decomposes market into body and tail legs, testing factor models at daily and monthly frequencies.
result q5 model shows inconsistent results, with negative body and positive tail alphas at all split ratios.
Groups acting on CAT(0) spaces without 3-flats have rigid properties.
problem Understanding group actions on CAT(0) spaces without 3-flats.
method Analyzing the structure of CAT(0) spaces and group actions.
result Groups acting geometrically on such spaces have rigid properties.
Free group automorphisms group rigidity proven.
problem Proving rigidity of Out(F_N).
method Measure equivalence rigidity, new canonical splittings.
result Superrigidity of Out(F_N).
Computes the decomposition of rank-three bundles over the projective line with three marked points.
problem Decomposing rank-three bundles over the projective line with three marked points.
method Using the monodromy derivative to compute the roots of the bundles.
result Computes the exact decomposition of rank-three bundles for m=3. We characterize complete nonnegatively curved steady gradient soliton with curvature in L^1. We show that there are isometric to a product (R^2,g_{cigar}) times(R^{n-2}, eucl))/Gamma where Gamma is a Bieberbach group of rank n-2. We prove also a similar local splitting result under weaker curvature assumptions.
Gradient descent recovers low-rank matrices from random rank-one measurements.
problem Recovering low-rank matrices from random rank-one measurements.
method Directly estimate the low-rank factor by minimizing a nonconvex quadratic loss function via vanilla gradient descent with tailored spectral initialization.
result The algorithm converges to the ground truth with near-optimal sample and computational complexity when the true rank is small.
Researchers compute the rank and trace of Kauffman bracket skein algebra over its center.
problem Computing the rank and trace of Kauffman bracket skein algebra.
method Using finite type surfaces and complex roots of unity, they compute the rank and trace of Kζ(F) over its center. result They extend a theorem about the skein algebra having a splitting coming from two pants decompositions of F. Homotopy equivalent to spheres, free factor complex of a free group.
problem Understanding the homotopy type of free factor complexes.
method Proving homotopy equivalence and connectivity of relevant complexes.
result Homotopy equivalence of free factor complexes to spheres and other complexes.
This work analyzes tree-based methods from a ranking perspective, providing insights and new statistics.
problem Understanding the effectiveness of tree-based methods in finite-sample settings, especially symbolic feature selection.
method Local ranking perspective, finite-sample analysis, oracle bounds, posterior contraction results, concordant divergence statistics.
result New insights and statistics for evaluating symbolic feature mappings.
Upper bounds on topological dimensions of certain graph boundaries.
problem Calculating topological dimensions of specific graph boundaries.
method Linear upper bounds in terms of rank.
result Linear bounds on topological dimensions, dependent on rank.
Many 2D Artin groups are residually finite.
problem Residual finiteness of 2D Artin groups.
method Showed these groups split as free products with amalgamation or HNN extensions of finite rank free groups.
result Many 2D Artin groups are residually finite.