A new optimization scheme tackles convex objectives with non-separable penalties.
problem Optimizing convex objectives with non-separable penalties.
method Expectation-consistent approximation and vector approximate message-passing (VAMP) algorithm.
result Faster convergence compared to state-of-the-art approaches in tasks like classification and reconstruction.
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.
We study the problem of learning high dimensional regression models regularized by a structured-sparsity-inducing penalty that encodes prior structural information on either input or output sides. We consider two widely adopted types of such penalties as our motivating examples: 1) overlapping group lasso penalty, base…
A new method speeds up overlapping group lasso computations.
problem Time-consuming optimization of overlapping group lasso on large-scale problems.
method Non-overlapping statistical approximation to overlapping group lasso.
result The proposed penalty is statistically equivalent to overlapping group lasso.
New constructions from non-separating planar graphs improve understanding of graph linkability and knotability.
problem Understanding linkability and knotability of graph complements.
method Using maximal non-separating planar graphs to construct examples of maximal linkless and knotless graphs, and analyzing their Colin de Verdière invariant.
result The Colin de Verdière invariant of the complement of a maximal non-separating planar graph satisfies μ(cG) ≤ n-4, and equality holds.
Surgery on knots can produce non-separating spheres, using Heegaard Floer homology.
problem Conditions for surgery on knots to produce non-separating spheres.
method Heegaard Floer homology
result Sufficient conditions for a knot to be unknotted.
The complement of a non-separating planar graph contains a K_n minor.
problem Characterizing the structure of complements of planar graphs.
method Analyzing the structure of complements of non-separating planar graphs and using examples to illustrate hypotheses.
result The order 2n-3 is the lowest possible for a non-separating planar graph whose complement contains a K_n minor.
Finite rigid sets found in complex of curves for surfaces.
problem Finding finite rigid sets in curve complexes of surfaces.
method Exhaustion by finite rigid sets proved for surfaces of finite type and genus ≥3.
result Finite rigid sets exist in the non-separating curve complex of surfaces.
Neural networks can separate non-separable data using feature maps.
problem Non-separable data in neural networks.
method Characterization of feedforward neural networks and use of feature maps.
result ReLU neural networks can separate concentric data.
Paper examines Dehn twists on non-orientable surfaces and their limitations.
problem Limitations of generating Dehn twists on non-orientable surfaces.
method Analyzes the level 2 mapping class group of non-orientable surfaces and their subgroups.
result Dehn twist subgroup of M2(Ng) cannot be generated by squares of Dehn twists about non-separating curves. New convex relaxations solve sparse regression problems efficiently.
problem Sparse regression with ℓ0 constraint is NP-hard. method Rank-one convexification for semidefinite optimization.
result Stronger and more general convex relaxations for sparse regression.
We construct Bing houses in all dimensions n≥3, obtaining non-separating PL immersions of Sn→Rn+1.
A non-separating multicurve of a surface S of genus g with m punctures is a multicurve c so that S-c is connected. For k>0 define the graph of non-separting k-multicurves to be the graph whose vertices are non-separating multicurves with k components and where two such multicurves are connected by an edge if they can b…
New non-separable covariance kernels for spatiotemporal data derived from harmonic oscillator physics.
problem Capturing complex spatiotemporal dependencies in Gaussian processes.
method Hybrid spectral method based on the harmonic oscillator, deriving explicit covariance kernels.
result Explicit non-separable covariance kernels with space-time interactions.
If a 3--manifold Y contains a non-separating sphere, then some twisted Heegaard Floer homology of Y is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been proved for knots in L--spaces.
Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.
problem Computing shortest non-separating simple closed curves on non-orientable surfaces.
method Developed tools for computing shortest curves, proving NP-hardness and tractability.
result Proved NP-hardness and fixed-parameter tractability for computing shortest orienting curves, and polynomial-time algorithm for non-orienting curves.
Gradient descent converges to perfect classification in neural nets for non-separable data.
problem Classifying linearly non-separable data using neural networks.
method Analysis of gradient descent dynamics in neural networks with sufficient but not large number of neurons.
result Gradient descent converges to global minima with perfect classification in the landscape of minimization problems.
We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove t…
Tree-AMP simplifies inference in complex tree-structured models.
problem Inference in high-dimensional tree-structured models.
method Approximate Message Passing algorithms for various machine learning tasks.
result Theoretical performance predictions and automated entropy estimation.
This paper restricts efficient geodesics to non-separating curves.
problem Finding efficient geodesics in the complex of curves.
method Analysis of the dot graph and surgeries.
result Efficient geodesics can be restricted to the non-separating curve complex.
The paper analyzes condition numbers for logistic regression to understand first-order methods' performance.
problem Understanding the performance of first-order methods in logistic regression.
method Introducing condition numbers to measure non-separability and separability of data.
result Condition numbers inform the properties and convergence guarantees of first-order methods.
Minimal simplicial complexes in high dimensions always contain complex links.
problem Existence of complex links in high-dimensional embeddings.
method Demonstrated through minimal simplicial complexes in R2n. result Minimal simplicial n-complexes inevitably contain a nonsplittable two-component link. For n >2, we shall show that the group Aut(NS(M)) of simplicial automorphisms of the complex NS(M) of non-separating embedded spheres in the manifold M,connected sum of n copies of S^2 X S^1, isomorphic to the group Out(F_n) of outer automorphisms of the free group F_n, where Fn is identified with the fundamental gr…
New algorithms reduce slate bandit regret for large slates, outperforming existing methods.
problem Non-separable reward functions in slate bandits with many slates.
method Design of algorithms with sub-linear regret.
result Sub-linear regret with respect to the time horizon for large number of slates.
The paper proves local laws for non-separable sample covariance matrices.
problem Analyzing non-separable sample covariance matrices with dependent or nonlinearly transformed data.
method Tensor network framework for analyzing fluctuation averaging in the presence of higher-order cumulant structure.
result Optimal averaged local law and full anisotropic local law for non-separable sample covariance matrices.
We consider a class of sparse learning problems in high dimensional feature space regularized by a structured sparsity-inducing norm which incorporates prior knowledge of the group structure of the features. Such problems often pose a considerable challenge to optimization algorithms due to the non-smoothness and non-s…
We prove integral rigidity for Seiberg-Witten invariants of 4-manifolds with specific hypersurfaces.
problem Integral rigidity of Seiberg-Witten invariants in 4-manifolds with non-separating hypersurfaces.
method Floer theoretic conditions and interplay between irreducible and reducible solutions to Seiberg-Witten equations.
result Sum of Seiberg-Witten invariants is determined cohomologically for specific 4-manifolds.
Safe screening rule improves Group SLOPE efficiency.
problem Efficiently selecting groups of predictors in high-dimensional sparse learning.
method Safe screening rule for Group SLOPE, addressing block non-separable group effects.
result Significant computational efficiency gains without sacrificing accuracy.
Link framings can only change when a 3-manifold has a non-separating sphere.
problem Understanding how framings of links can change in 3-manifolds.
method Using McCullough's work on mapping class groups and the Dirac trick.
result Link framings can only change in specific 3-manifolds with a non-separating sphere.
In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance betwee…
Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.
problem Connectivity problem in reducing sphere complex for genus-4 Heegaard surface.
method Presented a sufficient condition for a non-separating weak reducing pair to be separated by a reducing sphere.
result Reduced connectivity problem to showing disjointness of representative reducing spheres from a fixed disk.
For an infinite cardinal κ let ℓ2(κ) be the linear hull of the standard othonormal base of the Hilbert space ℓ2(κ) of density κ. We prove that a non-separable convex subset X of density κ in a locally convex linear metric space if homeomorphic to the space (i) ℓ2f(κ) if and only if X can be…
We consider the problem of learning a high-dimensional multi-task regression model, under sparsity constraints induced by presence of grouping structures on the input covariates and on the output predictors. This problem is primarily motivated by expression quantitative trait locus (eQTL) mapping, of which the goal is …
We discuss the ζ−regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.
Safe screening rule reduces computational costs for Group OWL models.
problem High computational costs and memory usage in solving Group OWL models.
method Safe screening rule for Group OWL models that identifies and removes inactive features.
result Significant computational gain and memory savings achieved without loss of accuracy.
New graph kernels capture spatio-temporal interactions.
problem Lack of justified spatio-temporal graph kernels for graph problems.
method Derive graph kernels via SPDEs for spatio-temporal modelling.
result Non-separable spatio-temporal graph kernels outperform existing ones.
Unique 3-balls in S^4 can be non-isotopic.
problem Existence and uniqueness of spanning 3-balls in S^4.
method Introducing barbell diffeomorphisms, implantations, and twistings; 2-parameter calculus of embeddings; framed cobordism method.
result Non-uniqueness of spanning 3-balls in S^4.
To any compact Riemann surface of genus g one may assign a principally polarized abelian variety of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian is called a period Gram matrix. This paper provides upper and lower bounds for all the ent…
We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus g, the number N of non-separating vanishing cycles and the number D of singular fibers satisfy the inequality $N \…
We present a necessary and sufficient condition for existence of a contractible, non-separating and noncontractible separating Hamiltonian cycle in the edge graph of polyhedral maps on surfaces. In particular, we show the existence of contractible Hamiltonian cycle in equivelar triangulated maps. We also present an alg…
Identifies Heegaard Floer homology solid tori via Dehn fillings.
problem Characterizing Heegaard Floer homology solid tori.
method Using Dehn fillings to identify solid tori.
result Characterized Seifert fibered Heegaard Floer solid tori.
Modifying the method of [21], we compute the perturbed HF+ for some special classes of fibered three manifolds in the second highest spinc-structures Sg−2. The special classes considered in this paper include the mapping tori of Dehn twists along a single non-separating curve and along a transverse pair of c…
The paper explores nonconvex penalties for deep learning regularization.
problem Overfitting in deep learning neural networks.
method Examines and evaluates nonconvex penalties for DNN regularization.
result Nonconvex penalties, under certain conditions, can perform well in DNNs.
We show that a Hitchin representation is determined by the spectral radii of the images of simple, non-separating closed curves. As a consequence, we classify isometries of the intersection function on Hitchin components of dimension 3 and on the self-dual Hitchin components in all dimensions. As an important tool in t…
AMP algorithm for matrix tensor product model provides recovery conditions.
problem Generalization of standard spiked matrix models with multiple pairwise observations.
method Approximate message passing with optimal weighing and combining of estimates.
result Asymptotically exact performance description and necessary/sufficient recovery conditions.
New condition prevents hyperbolic spaces from matching curve complexes.
problem Identifying when hyperbolic spaces cannot match curve complexes.
method Analyzing specific hyperbolic complexes and identifying a condition.
result Identified a condition preventing quasi-isometry between hyperbolic spaces and curve complexes.
Experimental evidence for curve ratios on genus two surfaces.
problem Determining the ratio of topological curve types on surfaces.
method Experimental statistics applied to Mirzakhani's genus two surface results.
result Separating and non-separating curves occur in the ratio 1:48.
Gradient penalty improves GAN performance by inducing a large-margin classifier.
problem Improving GAN performance and addressing vanishing gradients.
method A unifying framework of expected margin maximization, showing gradient penalties induce large-margin classifiers.
result Gradient penalties reduce vanishing gradients and produce better generated outputs.