This paper generalizes the cut locus concept for almost distance functions.
problem Generalizing the cut locus concept for almost distance functions.
method Introducing 1-Lipschitz functions called almost distance functions and studying their singular loci.
result Obtained a structure theorem for the singular locus of an almost distance function on a 2D Finsler manifold.
New framework generalizes distance correlation for detecting dependencies.
problem Detecting general dependencies in complex data.
method Develops Multiscale Graph Correlation (MGC) using characteristic functions and nearest neighbor machinery.
result MGC is universally consistent for dependence testing against all joint distributions of finite moments.
The paper proves compactness theorems for specific types of tensors.
problem Proving compactness theorems for Riemannian manifolds with specific tensors.
method Using h-almost Ricci tensors and generalized quasi-Einstein tensors, the paper extends previous theorems. result Theorems are extended to cases where h has at most linear growth. Study convexity of Mabuchi functional in big cohomology classes.
problem Convexity of Mabuchi functional in big cohomology classes.
method Defined an invariant related to transcendental Fujita approximations and established convexity under vanishing of this invariant.
result Established almost convexity along weak geodesics in big cohomology classes.
Paper proves almost rigidity theorem and applies it to study RCD(0,N) spaces.
problem Understanding the structure of noncompact RCD(0,N) spaces with linear volume growth.
method Developed an almost rigidity theorem and applied it to study RCD(0, N) spaces.
result Obtained sublinear growth of diameter of geodesic spheres and non-existence of harmonic functions with polynomial growth.
The paper establishes Sobolev inequalities between Riemannian metrics and their distance functions.
problem Establishing a theory of Sobolev inequalities for Riemannian metrics and distance functions.
method Analyzing the sub-critical case $p < rac{m}{2}$, proving a Sobolev inequality linking $L^{ rac{p}{2}}$ bounds on metrics to Lq bounds on distance functions. result A Sobolev inequality exists between Riemannian metrics and their distance functions, leading to a convergence theorem.
The complex wave representation (CWR) converts unsigned 2D distance transforms into their corresponding wave functions. Here, the distance transform S(X) appears as the phase of the wave function φ(X)---specifically, φ(X)=exp(iS(X)/τwhere τis a free parameter. In this work, we prove a novel result using the higher-orde…
This work introduces a method for almost equivariance in neural networks using Lie algebra convolutions.
problem Real-world data often does not conform to strict group equivariances, leading to underperformance in models.
method Definition and practical implementation of almost equivariance through Lie algebra convolutions.
result Demonstrated the validity of the approach through benchmarking against fully equivariant settings.
Let M be a compact Riemannian manifold with boundary. We show that M is Gromov-Hausdorff close to a convex Euclidean region D of the same dimension if the boundary distance function of M is C1-close to that of D. More generally, we prove the same result under the assumptions that the boundary distance func…
Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtaine…
New GaussianSketch approximates kernel distances with almost relative error and small additive term.
problem Approximating kernel distances between point sets efficiently.
method Truncating Gaussian kernel expansions and using RecursiveTensorSketch.
result Approximates kernel distance with almost (1+ε)-relative error and small additive α term. Given a time function τ on a spacetime M, we define a `null distance function', d^τ, built from and closely related to the causal structure of M. In basic models with timelike ∇τ, we show that 1) d^τ is a definite distance function, which induces the manifold topology, 2) the causal struct…
The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
problem Investigating stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
method Assuming almost the same optimal constant, the paper shows that the cumulative distribution of almost extremal functions is close to that of an Aubin-Talenti bubble on the round sphere.
result Quantitative stability with sharp exponent for the Sobolev inequality in various curvature and dimension assumptions.
Study shows spherical hyperbolic manifolds almost rigidly converge to hyperbolic space.
problem Almost rigidity of positive mass theorem for spherical hyperbolic manifolds.
method Intrinsic flat distance to prove convergence.
result Spherically symmetric asymptotically hyperbolic manifolds converge to hyperbolic space if mass limit is zero.
We show that the distance of a link K with respect to a bridge surface of any genus determines a lower bound on the genus of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot. In particular, knots with high bridge distance do not admit non-trivial non-hy…
Study minimal hypersurfaces in manifolds with bounded Ricci curvature.
problem Angle estimate of distance functions from minimal hypersurfaces.
method Colding's method and Cheeger-Colding theory.
result Prove Frankel property for metric cones.
Optimal warping paths are unique for almost every pair of time series.
problem Adverse effects in learning due to non-unique optimal warping paths.
method Assumption of squared error local costs, measure-theoretic analysis.
result Optimal warping paths are unique almost everywhere.
Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.
problem Proving anisotropic extrinsic radius pinching inequality for hypersurfaces.
method Analyzes anisotropic mean curvatures and studies equality cases.
result Almost extremal hypersurfaces are close to Wulff shape.
We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a den…
Let (M,g) be a time oriented Lorentzian manifold and d the Lorentzian distance on M. The function τ(q):=supp<qd(p,q) is the cosmological time function of M, where as usual p<q means that p is in the causal past of q. This function is called regular iff τ(q)<∞ for all q and also $τ\to 0…
WDAIL uses Wasserstein distance for more effective reward shaping in IL.
problem Fixed reward functions in GAIL limit performance on complex tasks.
method Introduces Wasserstein distance and PPO for improved reward shaping and stability.
result Significant performance improvement in complex MuJoCo tasks.
New neural nets respect triangle inequality, improving graph and reinforcement learning performance.
problem Neural nets lack inductive bias for certain subadditive distances.
method Introduced novel architectures that universally approximate norm-induced metrics.
result Neural nets with triangle inequality inductive bias outperform existing approaches.
A new, efficient k-modes algorithm improves clustering of categorical data.
problem Clustering categorical data using existing methods like k-means is inefficient. method Developed a novel k-modes algorithm called OTQT, which improves on existing methods. result OTQT finds more accurate clusters per iteration and is faster overall.
New Einstein metrics found close to almost hyperbolic ones.
problem Finding Einstein metrics near almost hyperbolic ones.
method Extending Tian's work, using C2,α-topology. result Existence of Einstein metrics close to almost hyperbolic ones.
Under the definition of Ricci curvature bounded below for Alexandrov spaces introduced by Zhang-Zhu, we generalize a result by Colding that an n dimentional manifold with Ricci curvature greater or equal to n minus 1 and volume close to that of the unit n sphere is close (in the Gromov-Hausdorff distance) to the sphere…
Let F⊂Rn be a closed set and n=2 or n=3. S. Ferry (1975) proved that then, for almost all r>0, the level set (distance sphere, r-boundary) $S_r(F):= \{x \in \R^n: \dist(x,F) = r\}$ is a topological (n−1)-dimensional manifold. This result was improved by J.H.G. Fu (1985). We show that Ferry's resul…
New curvature concept preserves graph distances under operations.
problem Preserving graph distances under graph operations.
method Characterization of distance matrix and its null space.
result Linear system Dx=1 may not have a solution. Researchers prove a stability result for a 3-sphere inequality, extending previous work.
problem Quantitative stability of nonlinear Yamabe-type inequalities on the 3-sphere.
method Proved a two-term refinement of the Schur lemma inequality in the conformal class of the 3-sphere.
result Deduced quantitative stability of an entire family of nonlinear Yamabe-type inequalities.
The paper improves proximity estimates for hypersurfaces with almost constant curvature in space forms.
problem Proximity to a single sphere for hypersurfaces with curvature functions close to a constant.
method Unified approach using the method of moving planes.
result Sharp quantitative estimates of proximity to a single sphere.
The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capill…
Proposes Mutual Regression Distance for better distribution comparison.
problem Lack of effective distance measures for manifold data.
method Constrained mutual regression problem to exploit manifold properties.
result Mutual Regression Distance (MRD) is a pseudometric effective for manifold data.
The paper shows how almost isoperimetric domains are close to spheres.
problem Understanding the geometry of almost isoperimetric domains.
method Analyzing finite perimeter subsets with small isoperimetric deficit and applying integral curvature bounds.
result Finite perimeter subsets with small isoperimetric deficit are close to spheres up to a small measure.
Study shows surprising cobordism distances between certain torus knots.
problem Determining cobordism distances between thin and thick torus knots.
method Analyzes locally flat cobordisms between torus knots with small and large braid indices.
result Surprising fact about torus knots as cross-sections of almost minimal cobordisms.
This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, …
Optimal sample complexity for contrastive learning of distances.
problem Minimum labeled tuples needed for high accuracy in learning distances.
method Analyzes sample complexity in various distance settings, proving tight bounds.
result Almost optimal bound on sample complexity for learning ℓp distances. Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.
problem Uniform diameter and volume estimates for Chern-Ricci flow on Hermitian minimal models.
method Uniform diameter and volume estimates, local Kähler assumption, Perelman's reduced length, almost monotonicity formula for reduced volume.
result Gromov-Hausdorff convergence of the Chern-Ricci flow on Hermitian minimal models.
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher order rectifiability of the set of differentiabili…
Study cobordism distances between 3-braid links and trefoil knots.
problem Understanding the geometric relationship between 3-braid links and trefoil knots.
method Determined cobordism distances between 3-braid links and trefoil knots, and explored limits of Coxeter's braid group result.
result Found cobordism distances between 3-braid links and trefoil knots, up to a constant error.
Paper proves minimality of stable submanifolds in flat neighbourhoods.
problem Proving the uniqueness of minimizers of elliptic functionals in flat neighbourhoods.
method Introducing a penalized minimization problem to exploit almost minimizers' regularity.
result Quantitative estimates of minimizers in flat neighbourhoods.
For complex structures, there are strict limits on how close almost complex structures can be.
problem Limits on almost complex structures compatible with Kähler structures.
method Analyzing distances between almost complex structures and their holonomy.
result There are strict limits on how close almost complex structures can be.
3-manifolds can be embedded almost in the spine of a trisected 4-manifold.
problem Embedding 3-manifolds in trisected 4-manifolds.
method Defining and using the spine of a trisected 4-manifold, isotoping 3-manifolds to lie in the spine, and calculating bounds based on graph distances.
result Every 3-manifold can be embedded almost in the spine of a minimal genus trisection of connect sums of S2ildeimesS2. Study nondegenerate singularities in mean curvature flow.
problem Understanding the behavior of nondegenerate cylindrical singularities.
method New L2-distance monotonicity formula and discrete almost monotonicity. result Topology change agrees with level sets change near a critical point of a Morse function.
New algorithms improve reinforcement learning for complex tasks.
problem Improving reinforcement learning for complex tasks.
method Developed new algorithms using distributional reinforcement learning and Cram{é}r distance.
result Proved asymptotic almost-sure convergence of new algorithms for neural networks.
This paper refines bounds on random walk speed in Teichmüller space.
problem Understanding the speed of random walks on Teichmüller space.
method Analyzing Jenkins-Strebel directions and Lebesgue geodesics.
result The drift of random walks grows exponentially for typical geodesics and oscillates between linear and exponential for some geodesics.
Study soap films hanging from frames, proving surface limits and curvature conditions.
problem Understanding soap films with gravity and minimal surfaces.
method Compactness theorem for surfaces with vanishing mean curvature and fixed/converging boundaries.
result Minimal surfaces represent all possible limits of almost-minimal surfaces.
Proof that certain Anosov flows are almost equivalent.
problem Proving equivalence of suspension Anosov flows.
method Constructing a genus-one Birkhoff section and analyzing its first-return map.
result Explicit bounds on distances between suspension Anosov flows.
We announce the classification of complete, almost embedded surfaces of constant mean curvature, with three ends and genus zero: they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.
Sharp gradient estimate for Green functions on non-parabolic RCD(0,N) spaces.
problem Analyzing the gradient of Green functions on non-parabolic RCD(0,N) spaces.
method Defining a smoothed distance function and proving a sharp upper bound for its gradient.
result The gradient of the smoothed distance function has a sharp upper bound and is sharp at points where the space is isomorphic to an RCD(N-2, N-1) space.