New rules reduce SLOPE model fitting time by screening out irrelevant variables.
problem Expensive tuning of regularization parameter in penalized regression models.
method Strong screening rules for group-based SLOPE models.
result Significant acceleration of fitting process for Group SLOPE and sparse-group SLOPE.
A new screening rule improves SLOPE efficiency for high-dimensional data.
problem Efficiently selecting relevant predictors in high-dimensional data.
method Developed a screening rule for SLOPE based on its subdifferential.
result The screening rule improves SLOPE's performance significantly in high-dimensional settings.
Efficient packages solve SLOPE problem in multiple languages.
problem Efficiently solving SLOPE problem for various data types.
method Hybrid coordinate descent algorithm for GLMs with multiple loss functions.
result Packages outperform existing SLOPE implementations in speed.
A new fast algorithm solves SLOPE optimization problem.
problem Inefficient algorithms for SLOPE optimization in high dimensions.
method Combines proximal gradient descent and proximal coordinate descent steps.
result Our method outperforms competing algorithms in benchmarks.
Sparse multinomial logistic regression for multiclass classification with feature selection.
problem High-dimensional multiclass classification with a focus on sparse models.
method Penalized maximum likelihood with complexity penalty, feature selection using group Lasso and Slope classifiers.
result Achievement of minimax order in both small and large number of classes regimes.
Paper proposes efficient algorithms for designing SLOPE penalty sequences.
problem Designing SLOPE penalty sequences is computationally expensive.
method Developed two efficient algorithms: PGD and CD for Gaussian and general data matrices respectively.
result Demonstrated improved mean squared error performance of SLOPE with designed penalties.
2-level SLOPE improves high-dimensional inference with fewer hyperparameters.
problem High-dimensional linear regression with complex penalty sequences.
method 2-level SLOPE with three hyperparameters, reducing penalty space.
result Sharp trade-off between TPP and FDP, computationally efficient.
SLOPE optimizes portfolios by grouping similar assets and estimating weights with sorted ℓ1-Norm.
problem Optimizing financial portfolios with sparse selection.
method Sorted ℓ1-Norm penalization (SLOPE) and Alternating Direction Method of Multipliers (ADMM). result SLOPE optimizes portfolios from minimum variance to equally weighted, improving out-of-sample performance.
New method for robust regression and outlier detection in high dimensions.
problem Outliers detection and robust regression in high-dimensional settings.
method SLOPE penalization for simultaneous estimation of intercepts and coefficients.
result Guaranteed FDR and statistical power control for support selection of intercepts.
Unified analysis of multi-task functional linear regression with manifold and composite penalties.
problem Estimating slope functions from functional data with multi-task learning.
method Penalized splines with manifold constraint and composite quadratic penalty.
result Unified convergence upper bound and phase transition behaviors for estimators.
Unified framework for pattern recovery in penalized and thresholded estimation.
problem Pattern recovery in penalized and thresholded estimation methods.
method Defining a novel pattern notion based on subdifferentials, introducing accessibility and noiseless recovery conditions.
result Unified and extended conditions for pattern recovery in a broad class of penalized estimators.
Framework for designing nonlinearities in neural networks with slope constraints.
problem Designing nonlinearities with specific properties for signal processing.
method Variational framework with regularization for slope constraints and optimization of adaptive splines.
result Adaptive nonuniform linear splines achieve global optimum in constrained optimization.
Deep learning aids ADMM-based decoding for binary linear codes.
problem Improving decoding efficiency for binary linear codes.
method Designing a decoding network based on ADMM and deep learning.
result Numerical results show improved performance compared to original ADMM.
Proposes a model selection procedure for high-dimensional binary classification using sparse logistic regression.
problem High-dimensional binary classification with sparse logistic regression.
method Penalized maximum likelihood with complexity penalty on model size, Slope estimator for logistic regression.
result Proposed complexity penalty is rate-optimal in the minimax sense.
Paper extends chaining technique for empirical risk minimization bounds.
problem Empirical risk minimization with unbounded noise and estimates.
method Chaining technique applied to random design settings, proving excess risk bounds.
result Proves upper bounds for empirical risk minimization with sub-Gaussian or subexponential noise.
The Slope Conjecture is verified for a specific family of Montesinos knots.
problem Relating the degree of the colored Jones polynomial to boundary slopes of knots.
method Verification for a specific family of Montesinos knots with given conditions.
result The Slope Conjecture and Strong Slope Conjecture are confirmed for the specified knots.
The study confirms a conjecture for a specific type of knot.
problem Determining the topology of essential surfaces in knot theory.
method Analyzing twisted, generalized Whitehead doubles of knots.
result Twisted, generalized Whitehead doubles satisfy the Slope and Strong Slope Conjectures.
The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.
Study slopes in 3-manifolds, proving conjectures about knots.
problem Understanding slopes in 3-manifolds and their implications for knots.
method Upper bounds on distances between slopes, applications to knots and surgeries.
result Bounds on boundary and degeneracy slopes for knots in 3-manifolds.
Using the Hatcher-Oertel algorithm for finding boundary slopes of Montesinos knots, we prove the Slope Conjecture and the Strong Slope Conjecture for a family of 3-tangle pretzel knots. More precisely, we prove that the maximal degrees of the colored Jones polynomial of such knots determine a boundary slope as predicte…
The study identifies characterising slopes for hyperbolic knots and proves stronger results for L-space knots.
problem Characterizing slopes of hyperbolic knots and their properties.
method Combining Lackenby's proof and Heegaard Floer homology genus bounds.
result All but finitely many slopes are characterising for hyperbolic knots and L-space knots.
Bayesian SLOPE combines SLOPE with Bayesian methods for linear regression.
problem Estimating regression coefficients with variable selection and prediction properties.
method Introduces Bayesian SLOPE by treating the SLOPE estimate as the posterior mode and providing a Gibbs sampler for full Bayesian inference.
result Offers credible sets and standard error estimates for parameters, facilitating Bayesian prediction.
Constructs Lefschetz fibrations with slopes near 2.
problem Finding Lefschetz fibrations with slopes close to 2.
method Constructs minimal, simply connected genus-g Lefschetz fibrations over the sphere.
result The infimum and supremum of slopes are not achievable.
Study on geometry of mountain slopes as Finsler metrics.
problem Existence and behavior of slope metrics on surfaces of revolution.
method Analysis of Finsler metrics on surfaces of revolution and comparison with Riemannian areas.
result Existence of globally defined slope metrics on surfaces of revolution.
Paper analyzes origami slope gaps and their distribution, finding a unique pattern.
problem Analyzing slope gaps in origami surfaces.
method Derived slope gap distribution of a specific origami by considering return times under the horocycle flow.
result Found a unique distribution of origami slope gaps, not a sum of scaled Hall distributions.
Study provides concrete examples of knot slopes.
problem Finding explicit characterizing slopes for knots.
method Concrete examples for the (-2,3,7)-pretzel knot.
result Explicit characterizing slopes for the knot 12n242. The Strong Slope Conjecture is proven for specific knot types.
problem Proving the Strong Slope Conjecture for various knot types.
method Using connect sums and cabling, the conjecture is shown to be closed under these operations.
result The Strong Slope Conjecture is established for graph knots.
Concerning the set of exceptional surgery slopes for a hyperbolic knot, Lackenby and Meyerhoff proved that the maximal cardinality is 10 and the maximal diameter is 8. Their proof is computer-aided in part, and both bounds are achieved simultaneously. In this note, it is observed that the diameter bound 8 implies the m…
Rejoinder on slope heuristics for model selection in regression.
problem Model selection in least-squares fixed-design regression with biased models and general noise.
method Proves the slope heuristics works even with significant bias and computes expectations for Gaussian noise.
result The slope heuristics is valid even when models are biased and noise has a general dependence structure.
New slopes identified for torus knots, improving previous results.
problem Identifying characterizing slopes for torus knots.
method Applying recent work on L-space surgeries, improving previous bounds.
result Every nontrivial L-space slope of T5,2 is characterizing for T5,2. New research shows that many slopes are characterizing for satellite knots.
problem Characterizing slopes for satellite knots.
method Detailed examination of the JSJ decomposition of a surgery along a knot, combined with other authors' constraints on surgery slopes.
result Many non-integral slopes are characterizing for composite knots.
A non-trivial slope r on a knot K in S3 is called a characterizing slope if whenever the result of r-surgery on a knot K′ is orientation preservingly homeomorphic to the result of r-surgery on K, then K′ is isotopic to K. Ni and Zhang ask: for any hyperbolic knot K, is a slope r=p/q with $|p| +…
For torus knots, non-integer slopes are characterizing.
problem Characterizing slopes for torus knots in 3D space.
method Analysis of p/q-surgery on torus knots using Heegaard Floer homology. result The set of characterizing slopes for torus knots contains all but finitely many non-integer slopes.
New knots found with specific slope properties.
problem Finding knots with non-integral boundary slopes.
method Kabaya's method for boundary slopes and layered solid torus construction.
result Existence of knots with boundary slopes of any positive denominator.
Paper analyzes SLOPE via AMP, providing an asymptotically sharp analysis and algorithmic approach.
problem Analyzing SLOPE's solution under Gaussian random designs.
method Developed an asymptotically exact characterization using approximate message passing.
result AMP iterates converge to the SLOPE solution in an asymptotic sense.
Let X be a norm curve in the SL(2,C)-character variety of a knot exterior M. Let t = || b || / || a || be the ratio of the Culler-Shalen norms of two distinct non-zero classes a, b in H_1(\partial M, Z). We demonstrate that either X has exactly two associated strict boundary slopes \pm t, or else there are strict bound…
Large knots have very varied boundary slopes.
problem Understanding the variability of boundary slopes in knots.
method Analyzing alternating knots and their boundary slopes.
result The ratio of boundary slope diameter to crossing number can be arbitrarily large.
Study Mazur doubles of knots and their relation to the Slope Conjecture.
problem Proving the Strong Slope Conjecture for Mazur doubles of knots.
method Analyzing Mazur doubles under certain hypotheses and using the Slope Conjecture.
result Mazur doubles of knots satisfy the Strong Slope Conjecture under certain conditions.
Let r_m and r_M be the least and greatest finite boundary slopes of a hyperbolic knot K in S^3. We show that any cyclic surgery slopes of K must lie in the interval (r_m - 1/2, r_M + 1/2).
The study confirms conjectures about slopes of knots using knot Floer homology.
problem Verifying conjectures about non-integer characterizing slopes of knots.
method Using knot Floer homology, the study verifies conjectures for specific classes of knots.
result Almost all slopes are characterizing for many knots, and infinitely many for L-space knots. Proves existence of Lefschetz fibrations with arbitrary slopes.
problem Finding Lefschetz fibrations with specific slopes.
method Proves existence for rational numbers in (2,8) with large genus.
result Arbitrary slopes (r in (2,8)) are possible for genus-g Lefschetz fibrations.
This paper completes proofs for left orderable slopes of double twist knots.
problem Proving the left orderability of slopes for double twist knots.
method Continuous families of hyperbolic SL2(R) representations of knot groups.
result Any slope in (-4n, 4m) is a left orderable slope of C(2m+1, -2n).
The paper examines slopes and their norms in exceptional Dehn fillings.
problem Understanding slopes and norms in exceptional Dehn fillings.
method Investigates the relationship between Euclidean length and Culler-Shalen norm on horotori.
result Establishes two inequalities between Euclidean length and Culler-Shalen norm.
A slope qp is called a characterizing slope for a given knot K0 in S3 if whenever the qp-surgery on a knot K in S3 is homeomorphic to the qp-surgery on K0 via an orientation preserving homeomorphism, then K=K0. In this paper we try to find characterizing slopes for torus knots $…
New invariant slope for links helps complete signature formula.
problem Signature formula for splice of links.
method Developed slope invariant and computed it using various methods.
result Established close relation to Conway polynomials and Kojima-Yamasaki η-function.
No characterizing slopes for multi-component links.
problem No characterizing slopes for multi-component links.
method Demonstrated through infinitely many examples of non-homeomorphic complements.
result Infinitely many links with non-homeomorphic complements.
Formula found for energy slope in complex geometry.
problem Calculating the asymptotic slope of a K-energy.
method Established a formula for the asymptotic slope.
result Found a formula for the asymptotic slope of α-K-energy.
The paper studies slopes for knot fillings with left-orderable fundamental groups.
problem Understanding slopes for knot fillings with left-orderable fundamental groups.
method Using the Riley polynomial and root analysis, the paper computes and conjectures on slopes.
result The paper computes the range of rational slope r for left-orderable fillings of two-bridge knots.