Addressing RL's agent-environment boundary issues, a novel analysis ensures optimal value functions are invariant.
problem Fundamental RL concepts like value functions are not uniquely defined due to the agent-environment boundary.
method A boundary-invariant analysis of Fitted Q-Iteration, ensuring optimality guarantees are independent of the boundary choice.
result Theoretical analyses of RL algorithms, including Fitted Q-Iteration, are made invariant to the boundary choice.
We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on manifolds with boundary. Given a metric on the boundary, this is the issue of whether any Killing field of the boundary metric extends to a Killing field of any bulk or filling Einstein metric inducing the given data on …
BOCK optimizes Bayesian Optimization by transforming the search space to reduce boundary evaluations.
problem Bayesian Optimization struggles with boundary issues, wasting evaluations near the search space boundary.
method BOCK uses a cylindrical transformation to redirect Gaussian Process efforts away from the boundary and towards the center of the search space.
result BOCK achieves better accuracy and efficiency, scaling to high-dimensional problems and optimizing neural network layers and hyperparameters.
The paper studies the properties of maps with free boundaries, focusing on the obstacle case.
problem Properties of the projected image and its regularity in maps with free boundaries.
method Dividing the map into distance and projected image parts; applying classical obstacle problem methods and proving higher regularity for the projected image.
result The projected image is at most of class C2,1 and globally of class W3,BMO, locally of C2,1 around the regular part of the free boundary. A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehor…
We analyse the issue of uniqueness of solutions of the static vacuum Einstein equations with prescribed geometric or Bartnik boundary data. Large classes of examples are constructed where uniqueness fails. We then discuss the implications of this behavior for the Bartnik quasi-local mass. A variational characterization…
A new classifier improves one-class predictions on unevenly sampled data.
problem Non-uniformly sampled data affects one-class classifier performance.
method Dynamic decision boundary based on minimum spanning tree.
result Proves effectiveness and robustness compared to state-of-the-art classifiers.
Neural network solves BVPs with unstructured data.
problem Solving Boundary Value Problems (BVPs) with numerical methods.
method Neural Network based numerical method for solving BVPs.
result Validated the method for Laplace and Poisson equations.
Study Zoll manifolds with boundary, showing unique geodesic properties.
problem Characterize Zoll manifolds with boundary.
method Analyzing geodesics, boundary conditions, and manifold structure.
result All free boundary geodesics have the same length and Morse index.
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.
Paper improves stochastic collocation for local volatility models.
problem Improving local volatility models for assets with boundaries.
method Applied stochastic collocation to lognormal distributions, derived analytical local volatility.
result Simple analytical Dupire local volatility derived from option prices.
New connected sum method for zero scalar curvature with constant mean curvature boundary.
problem Prescribing zero scalar curvature with constant mean curvature boundary on connected sums of manifolds.
method Boundary connected sum construction, exploiting nonlocal aspects and recent tools.
result Construction of a connected sum with zero scalar curvature and constant mean curvature boundary.
Paper proposes methods to estimate minimal adversarial perturbations for deep neural networks.
problem Quantifying robustness of deep neural networks against adversarial attacks.
method Proposes two lightweight strategies to find minimal adversarial perturbation.
result Approximates theoretical distance for samples close to classification boundary, providing robustness guarantees.
A new numerical framework simplifies elastic surface matching and comparison.
problem Challenging problem in surface comparison and matching in computer vision.
method Relaxing the geodesic boundary constraint using a varifold fidelity metric.
result Flexibility to deal with arbitrary topologies and sampling patterns, scalability to large meshes.
Study on heat flow across two half-lines with special boundary conditions.
problem Low energy mode of heat flow transmission across a Grushin-type cylinder.
method Analysis of heat equation with inverse-square potential and bridging boundary conditions.
result First insight into qualitative features of the heat flow solution at later times.
The paper identifies potential adversarial samples near decision boundaries of neural networks.
problem Vulnerability of deep neural networks to small perturbations of inputs.
method Developed a method to explore near decision boundaries of trained classifiers to identify potential adversarial samples.
result Potential adversarial samples represent only 61% of the test data but cover more than 82% of adversarial samples produced by iFGSM and 92% of those by DeepFool on CIFAR10.
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…
GCAO improves clustering of high-dimensional data by grouping low-density boundary points.
problem Stability and accuracy of clustering in high-dimensional, non-uniform data.
method Group-level optimization with gravitational attraction and optimization.
result GCAO outperforms 11 clustering methods on multiple datasets.
Proposes a novel method for generating hard negatives near time series data boundaries.
problem Challenges in generating effective negative samples for time series anomaly detection.
method Reconstruction-driven boundary negative generation framework using reinforcement learning.
result Improves anomaly representation learning and achieves competitive detection performance.
Let σ be the scattering relation on a compact Riemannian manifold M with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let ℓ be the length of that geodesic ray. We study the que…
We prove that the flat product metric on Dn×S1 is scattering rigid where Dn is the unit ball in Rn and n≥2. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map S:U+∂M→U−∂M from unit vectors V at the boundary that point inward to unit vecto…
Proves well-posedness for Einstein equations with totally geodesic timelike boundary condition.
problem Initial boundary value problem for Einstein equations with specific geometric boundary condition.
method ADM system, parallelly propagated orthonormal frame, modified evolution equations, hyperbolic systems, constraints propagation.
result First well-posedness result for Einstein equations with totally geodesic timelike boundary condition.
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.
I'll describe a general geometric setup allowing for a generalization of Rehren duality to asymptotically anti-de Sitter spacetimes whose classical matter distribution is sufficiently well-behaved as to prevent the occurence of singularities in the sense of null geodesic incompleteness. I'll also comment on the issues …
This paper addresses the importance of defining locality for accurate surrogate explanations.
problem Accurate approximation of local black-box decision boundaries for generating explanations.
method Proposes a novel approach to generate surrogate-based explanations centered on relevant places of the decision boundary, rather than on predictions.
result The proposed approach outperforms state-of-the-art methods and a straightforward improvement thereof on UCI datasets.
Flow maps into minimal surfaces with free boundary.
problem Mapping surfaces into minimal submanifolds with free boundary.
method Combining Plateau-flow and Teichmüller harmonic flow to achieve half-harmonic maps.
result Flow produces a branched minimal immersion as time tends to infinity.
New methods control false discoveries near the boundary in conformal novelty detection.
problem Over-optimistic assessments near the rejection threshold in conformal novelty detection.
method Support line (SL) correction and alternative procedures to control boundary false discovery rate (bFDR).
result New procedures control the boundary false discovery rate (bFDR) in the conformal setting.
A new PU classifier PUAL tackles trifurcate data issues.
problem Training classifiers on trifurcate data containing only labeled-positive instances and unlabeled instances.
method PUAL classifier with asymmetric loss and kernel-based algorithm.
result PUAL achieves satisfactory classification on trifurcate data.
Hidden cost: Smoothing shrinks decision boundaries, affecting class-wise accuracy.
problem The fragility of machine learning models and the need for robustness verification.
method Randomized smoothing approach to achieve statistical robustness.
result Smoothed classifiers' decision boundaries shrink, leading to class-wise accuracy disparity.
Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at confor…
We address the issue of strong cosmic censorship for T^2-symmetric spacetimes with positive cosmological constant. In the case of collisionless matter, we complete the proof of the C^2 formulation of the conjecture for this class of spacetimes. In the vacuum case, we prove that the conjecture holds for the special case…
Novel approach analyzes ReLU networks' training dynamics and proposes GmP for improved optimization.
problem Stochastic optimization instability in ReLU networks impedes convergence and generalization.
method Characteristic activation boundaries analysis and Geometric Parameterization (GmP) technique.
result GmP resolves instability, leading to better optimization, convergence, and generalization.
Survival trees exhibit end-cut preference, leading to biased splits.
problem End-cut preference in survival trees causes biased splits and poor interpretability.
method Proposed a smooth sigmoid surrogate (SSS) approach to replace hard-threshold indicator function.
result Smooth sigmoid surrogate (SSS) effectively mitigates end-cut preference in survival trees.
Paper solves boundary rigidity and lens rigidity problems for Riemannian manifolds.
problem Boundary and lens rigidity problems for Riemannian manifolds.
method Analysis of geodesic X-ray transform in normal coordinates.
result Metric can be determined from boundary distance function and lens relation.
Determining Finsler manifold structure from boundary distance map and elastic wave measurements.
problem Determine the structure of a compact Finsler manifold from its boundary distance map.
method Construct optimal fiberwise open subset of tangent bundle, use inverse problem in elasticity to measure travel times of waves.
result Finsler function can be uniquely determined from boundary distance map and travel times of waves.
Proves path connectedness of asymptotically flat metrics with boundary.
problem Proving path connectedness of asymptotically flat metrics with boundary.
method Generalization of Marques' result to compact manifolds with boundary, differential topology, and a new proof.
result Space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on R^3\B^3 is path connected.
Develops moduli theory for Calabi-Yau pairs, constructing a projective space.
problem Constructing a moduli space for Calabi-Yau pairs at the Calabi-Yau wall.
method Develops moduli theory, proving S-completeness and Θ-reductivity, constructing projective moduli space. result Constructs a projective moduli space for degenerate P2 pairs. In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky's conjecture was soon proven by the second aut…
The paper solves Monge-Ampère equations on reflexive polytopes, linking solvability to SYZ conjecture and tropical geometry.
problem Solvability of Monge-Ampère equations on reflexive polytopes.
method Analyzes reflexive polytopes with height functions, proving conditions for Monge-Ampère solvability and linking to SYZ conjecture.
result Conditions for Monge-Ampère solvability are necessary and sufficient, and solvability implies the SYZ conjecture for Calabi-Yau hypersurfaces.
Maxout networks show similar complexity issues as ReLU networks.
problem Understanding the complexity of maxout networks and decision boundaries.
method Analyzing the parameter space and decision boundaries, obtaining lower bounds, and investigating initialization procedures.
result Maxout networks exhibit a wide range of complexity, similar to ReLU networks.
mfEGRA uses active learning to efficiently locate failure boundaries in reliability analysis.
problem Prohibitive cost of reliability analysis using Monte Carlo sampling for high-fidelity models.
method Develops a multifidelity active learning method using data-driven adaptively refined surrogates.
result Significant computational savings (46-48%) compared to single-fidelity EGRA.
A new gradient estimator reduces variance near boundaries for binary latent variables.
problem Explosive gradient variance near boundaries in binary latent variable models.
method Introduces a new gradient estimator (bitflip-1) and an aggregated estimator (UGC) that uses either bitflip-1 or DisARM for each coordinate.
result UGC has uniformly lower variance than DisARM and achieves optimal optimization objectives.
Soft-Radial Projection solves gradient saturation in constrained deep learning.
problem Gradient saturation in deep learning models when integrating hard constraints.
method Introduces Soft-Radial Projection, a differentiable layer that maps predictions onto constraint boundaries without rank-deficient Jacobians.
result Improves convergence and solution quality over state-of-the-art methods.
Local decision boundary approximation improves model explanations for complex models.
problem Challenges in explaining complex, opaque machine learning models.
method Train a variational autoencoder to learn a latent space and map it to meaningful attributes. Use these attributes to approximate the local decision boundary and explain model predictions.
result Can recover latent attributes that determine class decisions in a new benchmark data set.
Researchers identify surfaces with special fluid flow fields.
problem Understanding fluid flows on curved surfaces.
method Defined and analyzed hydrodynamic Killing vector fields (HKVF) on surfaces.
result Any connected, orientable surface with HKVF is conformally equivalent to one of 14 canonical Riemann surfaces.
BeBold improves exploration in sparse-reward tasks by regulating visitation counts.
problem Efficient exploration in deep reinforcement learning under sparse rewards.
method Regulated difference of inverse visitation counts.
result BeBold solves 12 challenging tasks in MiniGrid with fewer steps than previous state-of-the-art.
This research deals with the mathematical modeling of the physical capital diffusion through the borders of the countries. The physical capital is considered an important variable for the economic growth of a country. Here we use an extension of the economic Solow model to describe how the smuggling affects the economi…
New insights into X-ray transform on hyperbolic disk, with functional relations and range characterizations.
problem Understanding the X-ray transform on hyperbolic geometry.
method Derived new singular value decompositions, range characterizations, and intertwining relations with wedge-type differential operators.
result Sharp understanding of boundary behavior and invertibility settings for the X-ray transform.