Locally adaptive activation functions boost deep and physics-informed neural networks.
problem Improving the performance and training speed of deep and physics-informed neural networks.
method Layer-wise and neuron-wise locally adaptive activation functions with a slope recovery term.
result The proposed methods accelerate convergence and reduce training cost.
The paper improves precision matrix estimation by SLOPE, especially in high-dimensional settings.
problem Estimating precision matrices with structured edge patterns.
method Graphical SLOPE, focusing on sparsity and cluster recovery.
result The method converges to the optimal solution and accurately identifies cluster structures.
We show that the martingale component in the long-term factorization of the stochastic discount factor due to Alvarez and Jermann (2005) and Hansen and Scheinkman (2009) is highly volatile, produces a downward-sloping term structure of bond Sharpe ratios, and implies that the long bond is far from growth optimality. In…
The paper analyzes the mathematics of the relationship between the default risk and yield-to-maturity of a coupon bond. It is shown that the yield-to-maturity is driven not only by the default probability and recovery rate of the bond but also by other contractual characteristics of the bond that are not commonly assoc…
A regularized risk minimization procedure for regression function estimation is introduced that achieves near optimal accuracy and confidence under general conditions, including heavy-tailed predictor and response variables. The procedure is based on median-of-means tournaments, introduced by the authors in [8]. It is …
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.
The study tests a functional-form restriction on risk exposure dynamics using margin debt data.
problem Understanding risk exposure dynamics under capital constraints and slack.
method Testing a regime-conditional functional-form restriction on aggregate risk-exposure dynamics implied by VaR-constrained intermediary models.
result The contraction and growth of exposures under capital constraints and slack are observed and tested.
StrTransformer recovers sources without labels by optimizing latent matrices and enforcing structural constraints.
problem Unsupervised blind source recovery in signal processing.
method Source-wise structured Transformer framework with latent source matrix optimization, structural regularization, and branch-specific weights.
result StrTransformer learns distinct temporal-scale structures and recovers source-aligned latent trajectories.
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…
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.
The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies th…
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. 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.
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…
A slope p/q is a characterising slope for a knot K in S3 if the oriented homeomorphism type of p/q-surgery on K determines K uniquely. We show that when K is a hyperbolic knot its set of characterising slopes contains all but finitely many slopes p/q with q≥3. We prove stronger results for hyper…
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 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.
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. The Slope Conjecture relates the degree of the colored Jones polynomial to the boundary slopes of a knot. We verify the Slope Conjecture and the Strong Slope Conjecture for Montesinos knots M(r1,s−u11,t1) with r,u,t odd, s even and u≤−1, r<−1<1<s,t.
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| +…
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.
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.
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.
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…
New method identifies physical constants from video data alone.
problem Identifying physical constants from video data.
method Proves level-set slope-coverage condition ensures local affine mapping to true physical state, enabling exact parameter recovery.
result Underdamped systems identifiable from a single video clip, other regimes require three diverse trajectories.
The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots satisfying 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).
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.
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.
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.
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).
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 $…
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.
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.
SLOPE is a relatively new convex optimization procedure for high-dimensional linear regression via the sorted l1 penalty: the larger the rank of the fitted coefficient, the larger the penalty. This non-separable penalty renders many existing techniques invalid or inconclusive in analyzing the SLOPE solution. In this pa…
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. We describe an algorithm for computing boundary slopes of 2-bridge links. As an example, we work out the slopes of the links obtained by 1/k surgery on one component of the Borromean rings. A table of all boundary slopes of all 2-bridge links with 10 or less crossings is also included.
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.
The slope invariant is shown to be unchanged by concordance of colored links.
problem Describing the concordance of colored links with a distinguished component.
method Proved the slope invariant is invariant under concordance using C-complexes and generalized Seifert forms.
result The slope invariant is invariant under concordance of links.
The geometry on a slope of a mountain is the geometry of a Finsler metric, called here the {\it slope metric}. We study the existence of globally defined slope metrics on surfaces of revolution as well as the geodesic's behavior. A comparison between Finslerian and Riemannian areas of a bounded region is also studied.
Certain torus knots have infinitely many slopes that do not uniquely identify them.
problem Identifying non-characterizing slopes for certain torus knots.
method Applying a condition from Baker and Motegi to show infinitely many non-characterizing slopes.
result The knots T2,2n+3#T−2,2n+1 have infinitely many non-characterizing slopes. SLOPE outperforms LASSO in low noise scenarios but is suboptimal in large noise scenarios.
problem Sparse linear regression with high-dimensional data.
method Characterized SLOPE's estimation error under specific conditions and compared it with LASSO and bridge regression.
result SLOPE is optimal for low noise scenarios but suboptimal in large noise scenarios.