A quasi-geodesic is Morse if and only if it is strongly contracting in injective spaces.
problem Characterizing Morse quasi-geodesics in injective spaces.
method Proving equivalence between Morse and strongly contracting quasi-geodesics.
result Injective metric spaces have the Morse local-to-global property and acylindrically hyperbolic groups with Morse elements.
Injectivity of ReLU networks is characterized for generative models and inverse problems.
problem Injectivity in ReLU networks for generative models and inverse problems.
method Layerwise analysis, worst-case Lipschitz constants, differential topology, random projections.
result Global injectivity of ReLU networks requires expansivity between 3.4 and 10.5 for Gaussian matrices.
Maximal spacetimes have unique past/future sets.
problem Characterizing maximal spacetimes.
method Developing map analysis and diamond properties.
result Maximal spacetimes have unique past/future sets.
Proposes a differentially private bandit algorithm reducing noise over time.
problem Privacy concerns in interactive recommendation systems.
method Tree-based mechanism to add Laplace or Gaussian noise to model parameters, focusing on dynamic global sensitivity.
result Demonstrates (ε,δ)-differential privacy with reduced noise and improved regret. The function on the Teichmueller space of complete, orientable, finite-area hyperbolic surfaces of a fixed topological type that assigns to a hyperbolic surface its maximal injectivity radius has no local maxima that are not global maxima.
We give sharp conditions on a local biholomorphism F:X→Cn which ensure global injectivity. For n≥2, such a map is injective if for each complex line l⊂Cn, the pre-image F−1(l) embeds holomorphically as a connected domain into CP1, the embedding bei…
Paper discusses conditions for global injectivity of semi-algebraic local diffeomorphisms.
problem Conditions for global injectivity of semi-algebraic local diffeomorphisms in higher dimensions.
method Analyzes foliations and simply connectedness of leaves, relates to fibrations and Jacobian conjecture.
result Relates simply connectedness of foliation leaves to locally trivial fibrations and provides computable regularity conditions.
We consider the problem of finding sufficient conditions for a locally Lipschitz mapping between Finsler manifolds to be a global homeomorphism. For this purpose, we develop the notion of Clarke generalized differential in this context and, using this, we obtain a version of the Hadamard integral condition for invertib…
Injective and surjective neural operators for function spaces.
problem Tackles injective and surjective neural operators in function spaces.
method Combines prior work in ReLU and operator learning, uses Fredholm theory and Leray-Schauder degree theory.
result Injective and surjective neural operators are universal approximators and maintain their properties in finite-rank implementations.
We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates.…
Noise in RNNs promotes flatter minima and more stable dynamics.
problem Understanding and optimizing the training of RNNs with noise.
method Formalizing RNNs as stochastic differential equations and analyzing the effect of noise in the hidden states.
result Noise injection in RNNs leads to flatter minima, more stable dynamics, and improved robustness.
Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar func…
Topological methods improve neuron analysis and tracer injection summary.
problem Traditional methods fail to capture the tree-like structure of neurons.
method Discrete Morse (DM) Theory for neuron skeletonization and consensus tree summarization.
result Significant performance improvements over non-topological methods.
Paper proposes methods to reduce financial contagion by targeted cash injections.
problem Financial contagion through interconnected networks.
method Dynamic model of payments with external control term for corrective cash injections.
result Targeted cash injections can significantly reduce default propagation.
Compact hyperkaehler manifolds are higher-dimensional generalizations of K3 surfaces. The classical Global Torelli theorem for K3 surfaces, however, does not hold in higher dimensions. More precisely, a compact hyperkaehler manifold is in general not determined by its natural weight-two Hodge structure. The text gives …
Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces…
Proves surjectivity of certain smooth maps with non-properness sets.
problem Surjectivity of linear operators and global diffeomorphisms of semialgebraic maps.
method Analytic proof involving C∞ semialgebraic local diffeomorphisms and linear partial differential operators. result A new analytic conjecture for polynomial local diffeomorphisms of Rn implies known results. We study global injectivity of proper branched coverings defined on the Euclidean n-ball in the case when the branch set is compact. In particular we show that such mappings are homeomorphisms when n=3 or when the branch set is empty. This proves the corresponding cases of a question of Vuorinen from [Vuo79].
Uniform bounds for eigenvalues of Hodge Laplacian on manifolds with lower Ricci curvature.
problem Establishing bounds for eigenvalues of Hodge Laplacian under lower Ricci curvature.
method Using geometric assumptions including lower Ricci curvature, injectivity radius, and diameter bounds.
result Uniform eigenvalue bounds for the Hodge Laplacian and connection Laplacian.
As global greenhouse gas emissions continue to rise, the use of stratospheric aerosol injection (SAI), a form of solar geoengineering, is increasingly considered in order to artificially mitigate climate change effects. However, initial research in simulation suggests that naive SAI can have catastrophic regional conse…
The paper proves global invertibility for certain local diffeomorphisms and biholomorphisms in higher dimensions.
problem Global invertibility of local diffeomorphisms and biholomorphisms in higher dimensions.
method The approach uses conformal geometry, complex analysis, elliptic PDEs, and topology.
result The main theorem guarantees global invertibility for specific local diffeomorphisms and biholomorphisms in higher dimensions.
Electronic power inverters are capable of quickly delivering reactive power to maintain customer voltages within operating tolerances and to reduce system losses in distribution grids. This paper proposes a systematic and data-driven approach to determine reactive power inverter output as a function of local measuremen…
Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) is a momentum version of stochastic gradient descent with properly injected Gaussian noise to find a global minimum. In this paper, non-asymptotic convergence analysis of SGHMC is given in the context of non-convex optimization, where subsampling techniques are used o…
In our previous paper, we discussed the hyperbolization of the configuration space of n(> 4) marked points with weights in the projective line up to projective transformations. A variation of the weights induces a deformation. It was shown that this correspondence of the set of the weights to the Teichmüller space when…
Replica exchange Langevin diffusion accelerates nonconvex optimization.
problem Nonconvex optimization challenges in machine learning.
method Replica exchange Langevin diffusion, discretization analysis.
result Replica exchange accelerates convergence to global minima.
This paper analyzes deep and wide transformer training dynamics.
problem Understanding the training dynamics of infinitely deep and wide transformers.
method Develops a mean-field framework for gradient-based training of transformers, controlling a neural PDE.
result Establishes a rigorous foundation for gradient-based transformer training, proving convergence to global minima.
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
Geometric analysis improves noise injection in GANs.
problem Unclear mechanism of noise injection in GANs.
method Geometric framework based on Riemannian geometry.
result A new strategy for noise injection is devised.
Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.
problem Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.
method A perturbed form of gradient descent with arbitrary initialization.
result Gradient descent with noise converges to a unique optimum.
The paper analyzes optimal dividend and capital injection strategies under time-inconsistent preferences.
problem Optimal dividend and capital injection strategies under time-inconsistent preferences.
method Diffusion risk model with general discount functions, weak equilibrium definition, HJB equation system.
result Explicit solutions and threshold types of optimal strategies derived under different discount functions.
This paper connects noise injection to Bayesian inference for neural networks, improving model uncertainty.
problem Improving the reliability and confidence of neural network predictions through uncertainty quantification.
method Introducing noise into neural network parameters during training and inference to estimate prediction uncertainty.
result The MCNI method outperforms baseline models in regression and classification tasks.
Compact theorem for minimal surfaces with lower injectivity radius.
problem Proving compactness of minimal surfaces with lower injectivity radius.
method Variant of Choi--Schoen compactness theorem, focusing on injectivity radius.
result Proved compactness theorem for minimal surfaces.
Lecture notes on group actions on injective spaces and Helly graphs.
problem Understanding group actions on specific metric spaces.
method Review of injective metric spaces and Helly graphs, elementary properties, constructions, and exercises.
result Presentation of various constructions of injective metric spaces and Helly graphs with interesting group actions.
Proves polynomial injectivity of Fubini-Study map for ample line bundles.
problem Injectivity of Fubini-Study map for ample line bundles.
method Polynomial injectivity proof with polynomial dependence on ample line bundle exponent.
result Quantitative version of injectivity proved, polynomial in ample line bundle exponent.
Lower bound on boundary injectivity radius for specific tubes.
problem Estimating the boundary injectivity radius of Margulis tubes.
method Using curvature bounds to derive a lower bound.
result A lower bound on the boundary injectivity radius is provided.
Reservoir subspace injection improves online ICA by preserving injected features.
problem Discarding injected features in top-n whitening can degrade performance. method Formalized reservoir subspace injection (RSI) and developed diagnostics (IER, SSO, ρ_x) to identify and mitigate the failure mode.
result RSI controller preserves passthrough retention, improving performance by up to 2.2 dB.
Proposes a method to enhance graph models by injecting unseen connections.
problem Enhancing graph models to utilize unseen connections.
method Parametric link injection layer to find and inject weak connections.
result Improves performance on node classification and link prediction tasks.
Let M be a Riemannian globally symmetric space of compact type, M′ its set of maximal flat totally geodesic tori, and ad(M) its adjoint space. We show that the kernel of the maximal flat Radon transform τ:L2(M)→L2(M′) is precisely the orthogonal complement of the image of the pullback map…
The paper proves that certain spaces are injective and Helly graphs.
problem Understanding the structure of certain geometric and algebraic spaces.
method Building Helly graphs and injective metric spaces from lattices.
result The natural piecewise ℓ∞ metric on Euclidean buildings and Deligne complexes is injective. Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
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…
In this paper, we obtain two-sided bounds for the volumes of the Aloff-Wallach spaces W(p,q), compute maximal and minimal sectional curvature for the spaces W(n,n+1), and use this information to estimate the injectivity radii: We derive an upper bound for the injectivity radii of W(p,q) and a lower bound for the …
Injectivity of geodesic ray transform on specific Finsler manifolds proven.
problem Injectivity of geodesic ray transform on spherically symmetric reversible Finsler manifolds.
method Reduction to invertibility of generalized Abel transforms using angular Fourier series and Taylor expansions of geodesics.
result Injectivity of geodesic ray transform proven on specified Finsler manifolds.
Noise Injection probes deep learning dynamics during training phases.
problem Understanding the learning mechanism of deep neural networks.
method Noise Injection Nodes (NINs) are used to perturb DNN architectures without altering the optimization algorithm.
result Distinct training phases are observed based on the scale of injected noise.
Generalizes a soul-bound for noncompact Alexandrov spaces.
problem Finding a lower bound for injectivity radius in Alexandrov spaces.
method Introduces the soul of Alexandrov spaces and applies a generalized bound.
result Injectivity radius is at least πK⁻¹/² if not equal to the soul's.
Optimal control problem for firm cash flow with dividend and capital injection strategies.
problem Maximizing dividends while managing capital injections in a firm's cash flow.
method Proved two optimal strategies: mean-reverting dividends with capital injections or no injections until ruin.
result Optimal strategies are dichotomous: either mean-reverting dividends with injections or no injections.
A labeled oriented tree is called injective if each generator occurs at most once as an edge label. We show that injective labeled oriented trees are aspherical. The proof relies on a new relative asphericity test based on a lemma of Stallings.