Sharp bounds found for energy in projective space mappings.
problem Finding bounds for energy in mappings of real projective spaces.
method Sharp lower and upper bounds for energy in homotopy classes of mappings from real projective space to Riemannian manifolds.
result Characterization of maps that achieve the lower bound for energy and determination of the infimum of energy in a homotopy class.
Paper bounds subspace estimator error from noisy projections.
problem Estimating subspaces from noisy data.
method Derives perturbation bound on optimal subspace estimator.
result Fundamental result with implications in matrix completion and clustering.
Study examines Lp-boundedness of Hodge projection on manifolds with ends.
problem Understanding Lp-boundedness of Hodge projection on manifolds with ends. method Investigates the relationship between Hodge projection, Riesz transform, and bounded harmonic functions.
result Connects Lp-boundedness of Hodge projection to the structure of L2 harmonic one-forms and bounded harmonic functions. Study energy-minimizing maps in projective spaces, proving sharp bounds.
problem Finding optimal mappings in projective spaces.
method Proving lower bounds and characterizing energy-minimizing maps.
result Sharp lower bounds and characterization of energy-minimizing maps.
The known upper bounds for the multiplicities of the Laplace-Beltrami operator eigenvalues on the real projective plane are improved for the eigenvalues with even indexes. Upper bounds for Dirichlet, Neumann and Steklov eigenvalues on the real projective plane with holes are also provided.
New algorithm reduces adaptive regret without projections.
problem Computational expense of projections in online convex optimization.
method Lazy gradient-based algorithm with set-membership computations.
result Near-optimal adaptive regret bounds for general convex functions.
The study finds lower bounds for the warping degree of a knot projection.
problem Determining the warping degree of a knot projection.
method Examining the maximal number of regions sharing no crossings for a fixed crossing in a knot projection.
result Lower bounds for the warping degree of a knot projection are provided.
Estimates spectral projections restricted to uniformly embedded submanifolds.
problem Estimating spectral projections on submanifolds of manifolds with nonpositive curvature.
method Estimates the L2(M)oLq(Σ) norm of spectral projection operators. result Sharp spectral projection estimates for small spectral windows.
Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.
problem Understanding geometric structures of singular Kahler spaces.
method Proving RCD spaces homeomorphic to 3D projective varieties with bounded Nash entropy and Ricci curvature.
result Compact RCD spaces are equivalent to underlying projective varieties.
Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on how many projections are needed to accurately preserve the geometry of these man…
Bounds projective structure norms by bending lamination lengths.
problem Bounding the L2-norm of projective structures. method Using the Thurston parameterization and Krasnov-Schlenker's W-volume theory. result Upper bounds on L2-norm of holomorphic quadratic differential by the length of bending lamination. Bounding geodesic length variation for surface projective structures.
problem Understanding how geodesic lengths change under projective structure variations.
method Bounding the derivative of complex length in terms of the Schwarzian norm.
result Application to cone-manifold deformations of hyperbolic 3-manifolds.
Uniform bounds prove connection between Kähler metrics and RCD spaces.
problem Bounding Nash entropy and Calabi energy for Kähler metrics.
method Proving uniform Sobolev bounds for Kähler manifolds.
result Establishes connection to RCD spaces and provides examples.
We extend some results of [BF12] on subfactor projections to show that the projection of a free factor B to the free factor complex of the free factor A is well-defined with uniformly bound diameter, unless either A is contained in B or A and B are vertex stabilizers of a single splitting of F_n, i.e. they are disjoint…
New number bounds knot complexity, including unknotting and crosscap numbers.
problem Bounding knot complexity and understanding knot types.
method Introducing an unknotting-type number to estimate crosscap number.
result Determines set of knots with crosscap number at most two.
New extension theorem for projective manifolds.
problem Extension of twisted canonical forms on hypersurfaces.
method Established a new extension result for canonical forms on hypersurfaces with simple normal crossings.
result Obtained sharp bounds for the extension.
Formula for projecting geodesics in hyperbolic 3-manifolds, relating lengths to subsurface projections.
problem Relating lengths of geodesics to projections in hyperbolic 3-manifolds.
method Formula with explicit constants relating subsurface projections to geodesic lengths.
result Effective and computable large projections versus short curves relation.
Investigates projections onto explicit subspaces and their variance effects.
problem Understanding the variance preservation in explicit subspace projections.
method Investigates projections onto explicit subspaces of varying dimensionality and analyzes the variance effects.
result Developed new bounds for Euclidean distances and inner products.
Optimizes energy of mappings from complex projective spaces.
problem Finding energy-minimizing mappings between complex projective spaces and Riemannian manifolds.
method Establishes optimal lower bounds for energy functionals and characterizes optimal mappings.
result Optimal lower bounds for energy functionals are characterized for mappings from real and complex projective spaces.
We prove that the largest first eigenvalue of the Dirac operator among all hermitian metrics on the complex projective space of odd dimension m, larger than the Fubini-Study metric is bounded by (2m(m+1))1/2.
Projections to a graph have bounded diameter for certain group structures.
problem Analyzing the diameter of projections to a specific graph for group structures.
method Analysis of the boundary of hyperbolic groups relative to subgroups and cyclic subgroups.
result The diameter of projections is bounded and depends only on the length bound.
Study examines Hilbert area of inscribed polygons in projective geometry.
problem Understanding Hilbert area of inscribed polygons in projective geometry.
method Examined correspondence between Fock-Goncharov and Cartesian coordinates, analyzed degeneration and Hilbert area of inscribed quadrilaterals, developed microlocal condition.
result Sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.
We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for this functional and construct some minima for it via the twistors spaces of comple…
This paper focuses on projection-free methods for solving smooth Online Convex Optimization (OCO) problems. Existing projection-free methods either achieve suboptimal regret bounds or have high per-iteration computational costs. To fill this gap, two efficient projection-free online methods called ORGFW and MORGFW are …
We prove a continuity property for ending invariants of convergent sequences of Kleinian surface groups. We also analyze the bounded curve sets of such groups and show that their projections to non-annular subsurfaces lie a bounded Hausdorff distance from geodesics joining the projections of the ending invariants.
The study examines the distribution of projections of Gaussian data points and its implications for learning models.
problem Understanding the distribution of projections of Gaussian data points in high dimensions.
method Analyzes the asymptotic behavior of projections of i.i.d. standard Gaussian vectors in Rd onto m-dimensional subspaces. result Establishes bounds on the Wasserstein radius of the set of probability distributions arising from these projections.
The paper extends inequalities for projection bodies to arbitrary measures.
problem Sharp bounds for volume ratios of convex bodies and their projection bodies.
method Generalizations of Zhang's inequality to arbitrary measures and extensions of the projection body operator.
result New Zhang-type inequalities for arbitrary measures and functions.
Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.
problem Optimal transport in unbounded settings with heavy-tailed distributions.
method Hilbert's projective metric for integrable functions of bounded growth, kernel integral operators as contractions.
result Exponential convergence of Sinkhorn's algorithm for light-tailed marginal distributions.
Random projections have been applied in many machine learning algorithms. However, whether margin is preserved after random projection is non-trivial and not well studied. In this paper we analyse margin distortion after random projection, and give the conditions of margin preservation for binary classification problem…
In many online learning problems the computational bottleneck for gradient-based methods is the projection operation. For this reason, in many problems the most efficient algorithms are based on the Frank-Wolfe method, which replaces projections by linear optimization. In the general case, however, online projection-fr…
We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds o…
The paper explores properties of projections and gradient methods in hyperbolic space forms.
problem Optimization problems in hyperbolic space forms.
method Intrinsic κ-projection and gradient projection methods.
result Every accumulation point of the sequence generated by the gradient projection method is a stationary point.
Study bounds on cusp volumes of alternating knots on surfaces.
problem Bounding cusp volumes of knots on surfaces.
method Analyzing hyperbolic knots with alternating projections on embedded surfaces.
result Two-sided bounds on cusp area in terms of twist number and surface genus.
We revisit the challenge of designing online algorithms for the bandit convex optimization problem (BCO) which are also scalable to high dimensional problems. Hence, we consider algorithms that are \textit{projection-free}, i.e., based on the conditional gradient method whose only access to the feasible decision set, i…
Generalizing previous constructions, we present a dual pair of decompositions of the complement of a link L into bipyramids, given any multi-crossing projection of L. When L is hyperbolic, this gives new upper bounds on the volume of L given its multi-crossing projection. These bounds are realized by three closely rela…
Study bounds changes in hyperbolic 3-manifold structures after drilling short geodesics.
problem Bounding changes in complex projective structures after drilling short geodesics.
method Analyzes L2-bounds on changes in conformally compact hyperbolic 3-manifolds. result Change is bounded by a universal constant times the square root of the length of the drilled geodesics.
We establish a correspondence on a Riemann surface between hyperbolic metrics with isolated singularities and bounded projective functions whose Schwarzian derivatives have at most double poles and whose monodromies lie in PSU(1,1). As an application, we construct explicitly a new class of hyperbolic metrics …
A dissertation on scalable projection-free optimization methods.
problem Efficient optimization algorithms for large-scale machine learning problems.
method Study of Frank-Wolfe variants and their extensions to distributed and derivative-free settings.
result Development of 1-SFW and QFW, achieving state-of-the-art complexity and efficiency.
We introduce a method to produce bounds for the non secant defectivity of an arbitrary irreducible projective variety, once we know how its osculating spaces behave in families and when the linear projections from them are generically finite. Then we analyze the relative dimension of osculating projections of Grassmann…
Sharp inequality for complex projective space's systoles under scalar curvature bounds.
problem Proving a sharp stable 2-systolic inequality for complex projective space.
method Uses Spin^c Dirac operators, comass estimate, and stable norm-comass duality.
result Equality holds only for the Fubini-Study metric, up to biholomorphism.
It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed positive Ricci metrics on …
Optimizes bounds for multiple T-singularities on surfaces.
problem Bounding T-singularities on non-rational projective surfaces with many singularities.
method Analyzes combinatorial configurations and classifies them to find optimal bounds.
result Classifies all combinatorial configurations leading to high bounds, proving their non-existence gives optimal bounds.
For any sequence of properly convex domains in the real projective plane such that the zeros of Pick differentials have bounded multiplicity and get further and further apart, we determined all Hausdorff limit domains that one can obtain after normalizing each member of the sequence by a projective transformation. We t…
We develop a probabilistic framework for sequential random projection.
problem Challenges of sequential decision-making under uncertainty.
method Novel construction of a stopped process and method of mixtures.
result Achieved a non-asymptotic probability bound for random projection.
Study shows bounds on volumes of weakly generalised alternating knots.
problem Volume bounds for weakly generalised alternating knots.
method Analysis of weakly generalised alternating knots in 3-manifolds.
result Upper volume bound does not hold for weakly generalised alternating knots.
The study connects projective codes to the distribution of zeros of odd maps.
problem Understanding the distribution of zeros of odd maps from spheres to Euclidean space.
method Using the topology of the space of probability measures on the sphere.
result Generalization of the Borsuk-Ulam theorem and its four consequences.
The paper develops strong lower bounds for projective Stiefel manifolds, resolving special cases with the Browder-Dupont invariant.
problem Developing strong lower bounds for the span of projective Stiefel manifolds.
method Elementary stability properties of vector bundles, with special attention to the Browder-Dupont invariant for odd dimensions.
result Characterization of n for which the Browder-Dupont invariant is well-defined and use of this invariant to obtain lower bounds for the span. Study on Kähler manifolds shows rigidity of eigenvalues with positive Ricci bound.
problem Optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
method Established optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
result Complex projective space is the only Kähler manifold with the largest multiplicity of the first eigenvalue.