Develops a novel classification boundary on Riemannian manifolds.
problem Classifying multivariate datasets on nonlinear Riemannian manifolds.
method Invents the principal boundary approach, defined as an optimal curve maximizing the margin between two classes.
result The principal boundary optimally separates classes and coincides locally with the SVM decision boundary.
New method visualizes decision boundaries of classification models.
problem Difficulty in understanding how classification models interpret data.
method Hybrid supervised-unsupervised technique for visualizing decision boundaries.
result Provides interpretable maps for qualitative and quantitative analysis.
Improved efficient learning of neighbor representations for large datasets.
problem Efficiently learn neighbor representations for large datasets.
method Differentiable Boundary Sets algorithm that overcomes computational issues and improves accuracy.
result Significant reduction in training time and improved classification accuracy.
New method classifies C-boundaries up to 6 crossings.
problem Classifying C-boundaries with up to 6 crossings. method Proposed a new construction method.
result Extended classification of C-boundaries up to 6 crossings. Study characterizes compact Einstein-type manifolds with boundary.
problem Characterize compact Einstein-type manifolds with nonempty boundary.
method Proved a sharp boundary estimate, obtained Hawking mass bounds, and provided a topological classification for the boundary.
result Obtained a gap result for compact Einstein-type manifolds with boundary.
The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.
problem Classifying solutions to a Liouville equation with a nonlinear Neumann boundary condition.
method Analyzing the n-Laplacian Liouville equation on the half-space R+n with positive nonlinear Neumann boundary condition. result The classification of solutions extends previous results for n=2 and p=n. The paper analyzes deep neural network classification regions and their decision boundaries.
problem Understanding the geometric properties of deep neural network classifiers.
method Empirical investigation of deep neural networks' classification regions and decision boundaries.
result Deep neural networks learn connected classification regions with flat decision boundaries.
Notes on linear classifiers and outliers, improving decision boundaries.
problem Negative impact of outliers on linear classifier decision boundaries.
method Use of least-squares cost function, scaling to reduce outlier significance.
result Improved decision boundaries through scaling outliers.
Classifies stability of flat-core p-elasticae pinned at boundaries.
problem Stability of flat-core p-elasticae under pinned boundary conditions. method Classification based on previous work for all p∈(1,∞) and d≥2. result Completes the classification of stable pinned p-elasticae in Rd. Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
New method reduces model bias and variance by adjusting training sample weights based on label uncertainty.
problem Tradeoff between model bias and variance in classification models.
method Estimate label uncertainty, adjust training sample weights, and fine-tune decision boundary.
result Improves model performance and reduces variance in physical activity recognition.
Paper shows similarity learning can lead to strong binary classification performance.
problem How similarity learning can lead to good classification performance.
method Product-type formulation of similarity learning is connected to binary classification through an excess risk bound.
result Similarity learning can directly elicit a decision boundary for binary classification.
The paper classifies boundaries of groups with isolated flats.
problem Classifying boundaries of groups with isolated flats.
method Classification theorem for 1-dimensional boundaries of groups with isolated flats.
result The boundary of a group with isolated flats is homeomorphic to a circle, Sierpinski carpet, or Menger curve.
Measures neural network decision boundary volume to predict model performance.
problem Understanding the geometry of deep learning models for better performance.
method Local surface volumes to measure decision boundary, applying Weyl's tube formula.
result Smaller surface volume correlates with higher classification accuracy.
Deep neural networks achieve optimal learning rates for high-dimensional classification.
problem Learning classification functions from noisy data with smooth boundaries.
method Empirical risk minimization over deep neural networks for locally Barron-regular decision boundaries.
result Optimal estimation rates are independent of dimension and can be achieved by deep neural networks.
The study classifies and constructs examples of surfaces with specific curvature and boundary conditions.
problem Classifying surfaces with parallel mean curvature and constant contact angle.
method Analytical and geometric methods, including classification and construction of examples.
result Sharp classification and examples of branched immersed disks and surfaces in space forms.
Classifies 4-manifolds with specific properties and fundamental group.
problem Classifying 4-manifolds with fundamental group Z and boundary. method Topological and smooth classification methods, including Hermitian forms and 2-handlebody constructions. result Every Hermitian form over Z[t±1] arises as the equivariant intersection form of exotic smooth 4-manifolds. Study mapping class groups of infinite type surfaces with noncompact boundaries.
problem Classify pure mapping class groups of infinite type surfaces.
method Developed a method to cut surfaces into simpler ones and combined recent results.
result Complete classification of perfect and uniformly perfect pure mapping class groups.
This paper classifies critical metrics on compact manifolds with boundary and harmonic Weyl tensor.
problem Classifying critical metrics on compact manifolds with boundary and harmonic Weyl tensor.
method Complete classification through geometric conditions and equivalence of assumptions.
result Critical metrics with harmonic Weyl tensor on simply connected compact manifolds with boundary are isometric to geodesic balls in simply connected space forms.
Classifies spaces with positive curvature and small boundary.
problem Understanding spaces with positive curvature and small boundary.
method Analyzes Alexandrov spaces with lower curvature bound 1 and small boundary.
result Classifies the total space X when the radius is π/2 and the boundary has diameter π/2.
Classifies ancient convex curves in convex domains.
problem Ancient convex curve shortening flows on convex domains.
method Classification of convex ancient solutions.
result Ancient convex curves in convex domains classified.
Ordinal data are often seen in real applications. Regular multicategory classification methods are not designed for this data type and a more proper treatment is needed. We consider a framework of ordinal classification which pools the results from binary classifiers together. An inherent difficulty of this framework i…
Automated vehicles need efficient testing, which this method addresses.
problem Current testing practices are insufficient for automated vehicles.
method Gaussian Process Classification to identify performance boundaries.
result It is feasible to test for challenging scenarios more efficiently.
Paper proposes a new classifier for hyperbolic spaces using horospherical boundaries.
problem Optimization of large margin classifiers in hyperbolic spaces.
method Horospherical decision boundaries for geodesically convex optimization.
result Geodesically convex optimization leads to globally optimal solutions.
In this paper we prove classification results to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball on Sn) of dimension n≥2 with prescribed constant mean curvat…
Deep rectifier networks can approximate high resolution boundaries with fewer parameters.
problem Classifying high-dimensional data with high resolution boundaries.
method Theoretical justification of deep rectifier networks' superior performance using PWL classifier boundaries.
result Deep rectifier networks can approximate high resolution boundaries with fewer parameters.
Unique ancient convex flow in a ball with free boundary found.
problem Classifying convex ancient free boundary mean curvature flows in the ball.
method Proof of existence and uniqueness in every dimension.
result A unique (modulo rotations and translations) convex ancient mean curvature flow found.
New method counts boundary pieces in ReLU classifiers for better complexity measure.
problem Current classification complexity measures are misleading and ineffective.
method Developed a novel method using tropical geometry to count exact boundary pieces.
result Boundary piece count is negatively correlated with robustness.
We give a classification of generic bifurcations of intersections of wavefronts generated by different points of a hypersurface with or without boundaries.
New adaptive strategy for active learning with smooth boundaries.
problem Adaptive active learning in multivariate classification with unknown distributional parameters.
method Combining insights from recent works, reduction to univariate-adaptive strategies.
result Near-optimal rates achieved without prior knowledge of distributional parameters.
Graphs with 4 colors classify 3-manifolds with toric boundaries.
problem Classifying 3-manifolds with toric boundaries.
method Using 4-colored graphs to represent and classify manifolds.
result Complete catalog of 3-manifolds up to 12 vertices.
Classifies convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
problem Classifying convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
method Contact isotopy classification, h-principle, fundamental groups, contact mapping class group.
result Establishes an h-principle for convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
This paper classifies discrete conformal structures on surfaces with boundary.
problem Classifying discrete conformal structures on surfaces with boundary.
method Axiomatic approach ensuring good geometric structure, classification based on triangulation and axioms.
result Unified and generalized existing discrete conformal structures on surfaces with boundary.
The study classifies quasi-Einstein manifolds with constant scalar curvature.
problem Characterizing quasi-Einstein manifolds with specific curvature properties.
method Classification and construction of examples of quasi-Einstein manifolds.
result Complete classification of quasi-Einstein manifolds with constant scalar curvature.
Homotopy theory for (2n+1)-dimensional manifold triads with fixed boundary.
problem Classifying stable moduli spaces of (2n+1)-dimensional manifold triads. method Homotopy-theoretic description of stable moduli spaces, stabilization by boundary connected sum with SnimesDn+1. result Established homology of stable moduli spaces for (2n+1)-dimensional manifold triads. Let X be a compact 2-manifold with nonempty boundary dX and let f: (X, dX) --> (X, dX) be a boundary-preserving map. Denote by MF_d[f] the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to f. The relative Nielsen number N_d(f) is the sum of the numbe…
Study shows how to approximate and estimate high-dimensional classification functions without the curse of dimensionality.
problem Approximating and estimating classification functions in high-dimensional spaces.
method Modified existing results to show that RBV2 functions can be approximated by neural networks with bounded weights. Proved the existence of a neural network with bounded weights approximating a classification function. Leveraged these bounds to quantify estimation rates. result Neural networks can approximate RBV2 functions without the curse of dimensionality, leading to efficient estimation rates. The study classifies geometrically finite polynomials on the boundary of Blaschke products.
problem Understanding the boundaries of hyperbolic components of Blaschke products.
method Combinatorial classification and construction of self-bumps.
result The closure of the main hyperbolic component is not a topological manifold with boundary for d≥4. Study rigidifies Einstein-type manifolds with boundary and constant curvature.
problem Classifying compact Einstein-type manifolds with boundary and constant scalar curvature.
method Applied recent results on gradient Einstein-type manifolds to prove rigidity.
result Rigidity results for compact Einstein-type manifolds with boundary and constant scalar curvature.
Study classifies Lie group representations with non-empty boundary orbit space.
problem Classifying representations of Lie groups with non-empty boundary orbit space.
method Detailed calculations based on previous work.
result Classification of Lie group representations with non-empty boundary orbit space.
FROB model improves robustness and reliable confidence for few-shot OoD detection.
problem Challenges in few-shot classification and OoD detection due to limited samples and adversarial attacks.
method FROB model combines support boundary generation and few-shot Outlier Exposure (OE) for improved robustness and reliable confidence.
result FROB achieves generalization to unseen OoD and maintains robustness independent of few-shot number.
The paper classifies algebraic curves in 4-balls and their boundaries.
problem Understanding algebraic curves in 4-dimensional balls and their boundaries.
method Analyzing algebraic curves in complex 2-space and their intersections with 4-balls.
result Classification of algebraic curves with up to 5 crossings.
For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the met…
The paper classifies all tight contact structures on a solid torus.
problem Classifying tight contact structures on a solid torus with specified dividing sets.
method Writing down a closed formula for the number of non-isotopic tight contact structures with any given dividing set.
result The complete classification of tight contact structures on a solid torus.
The paper classifies stable free boundary minimal hypersurfaces outside a ball.
problem Classifying stable free boundary minimal hypersurfaces outside a ball.
method Proved a Bôcher type result for positive Jacobi functions and used a symmetrization procedure.
result Stable free boundary minimal hypersurfaces outside a ball are catenoidal.
Classifies S1-invariant free boundary minimal annuli and Möbius bands in Bn.
problem Classifying S1-invariant free boundary minimal annuli and Möbius bands in Bn. method Analysis of the spectrum of the Dirichlet-to-Neumann map for S1-invariant metrics. result Existence and classification of S1-invariant free boundary minimal annuli and Möbius bands in Bn. RAPID efficiently samples SVDD subsets for better anomaly detection.
problem Efficiently sampling SVDD subsets for large datasets.
method Formulated as an optimization problem, RAPID selects samples that approximate the full SVDD decision boundary.
result RAPID outperforms competitors in classification accuracy, sample size, and runtime.
A new loss function improves classification accuracy in imbalanced datasets.
problem Suboptimal decision boundaries in classification with average, maximal, and average top-k losses. method Proposes a new classification objective called the close-k aggregate loss, which minimizes the loss for points close to the decision boundary. result Close-k aggregate loss achieves significant gains in 0-1 test accuracy compared to average, maximal, and average top-k losses.