Study shows stability of travel time data reconstruction from closed subsets.
problem Reconstruction of length spaces from travel time data on a closed subset.
method Lipschitz stability proof for certain types of length spaces.
result Reconstruction of length spaces is Lipschitz stable from travel time data on a closed subset.
In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α:H→Rm is injective, with (α(x))k=∣<x,fk>∣2, where $…
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…
The paper stabilizes invertible neural networks by using Gaussian mixture models.
problem Invertible neural networks can have exploding Lipschitz constants, leading to numerical errors.
method The authors use Gaussian mixture models to stabilize the latent distribution of invertible neural networks.
result Numerical simulations confirm that this modification improves sampling quality in multimodal applications.
The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.
problem The expressivity of bi-Lipschitz Normalizing Flows in approximating specific target distributions.
method Characterization of expressivity through lower bounds on Total Variation distance and discussion of potential remedies.
result Several target distributions are difficult to approximate using bi-Lipschitz Normalizing Flows, and lower bounds on their approximation are provided.
New method controls posterior collapse in VAEs with theoretical guarantees.
problem Posterior collapse in VAEs where encoder ignores latent structure.
method Inverse Lipschitz constraint on decoder network.
result Controls degree of posterior collapse for various VAE models.
Study on estimating invertible functions with minimax analysis.
problem Minimizing risk of estimating invertible functions on a plane.
method Introduce two types of L2-risks, derive lower and upper rates for minimax values, develop an asymptotically almost everywhere invertible estimator. result Invertibility does not reduce the complexity of the estimation problem in terms of the rate.
This work tackles exploding inverses in INNs, revealing and mitigating their numerical non-invertibility.
problem Exploding inverses in INNs cause numerical non-invertibility, leading to failures in various tasks.
method Derived bi-Lipschitz properties of INN building blocks, proposed regularizers for local invertibility, and stable INN designs for global invertibility.
result Bi-Lipschitz properties and stable INN designs are crucial for addressing numerical non-invertibility.
Study tackles inverse problems on low-dimensional manifolds, proving stability and proposing a reconstruction algorithm.
problem Inverse problems in infinite-dimensional spaces with nonlinear and ill-posed nature.
method Assumption of low-dimensional manifold, proving stability, proposing Landweber-type algorithm.
result Global convergence of the proposed algorithm, Lipschitz stability for specific inverse problems.
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.
CGNNs use wavelets for continuous function generation in infinite-dimensional spaces.
problem Generating continuous functions in infinite-dimensional spaces for applications like inverse problems.
method Inspired by DCGAN, CGNNs use wavelet multiresolution analysis with convolutional and nonlinear layers.
result CGNNs can be injective under certain conditions on filters and nonlinearity, leading to Lipschitz stability estimates.
In this work we compute lower Lipschitz bounds of ℓp pooling operators for p=1,2,∞ as well as ℓp pooling operators preceded by half-rectification layers. These give sufficient conditions for the design of invertible neural network layers. Numerical experiments on MNIST and image patches confirm tha…
We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm that induces d. There is an Out(F_n)-invariant potential Ψon Outer space such that when the Lipschitz norm is corrected by the derivative of Ψ, the resulting norm is quasisymmetric. As an application, we give new …
This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each singular point. We prove the cone also evolves by the inverse mean curvature flow and e…
In this paper we prove that if we consider the standard real metric on simplicial rooted trees then the category Tower-Set of inverse sequences can be described by means of the bounded coarse geometry of the naturally associated trees. Using this we give a geometrical characterization of Mittag-Leffler property in inve…
Deep neural networks solve noisy, complex problems accurately.
problem Reconstructing solutions from noisy, high-dimensional, non-linear inverse problems.
method Restricting infinite-dimensional forward operators to finite-dimensional spaces, training neural networks to approximate these operators robustly to noise.
result Deep neural networks can accurately solve high-dimensional, noisy, non-linear inverse problems.
The subject of this article is the introduction of a new concept of well-posedness of Bayesian inverse problems. The conventional concept of (Lipschitz, Hellinger) well-posedness in [Stuart 2010, Acta Numerica 19, pp. 451-559] is difficult to verify in practice and may be inappropriate in some contexts. Our concept sim…
JacNet learns Jacobians to enforce structure on derivatives for invertibility and Lipschitz functions.
problem Enforcing structure on derivatives of neural network mappings.
method Proposes using a neural network to directly learn the Jacobian of the input-output function, allowing control over derivative structure.
result Demonstrates learning invertible approximations to simple and 1-Lipschitz functions.
Study examines stability of image-reconstruction algorithms using variational regularization.
problem Stability and robustness of image-reconstruction algorithms in medical imaging.
method Review and novel stability results for ℓp-regularized linear inverse problems, focusing on p∈(1,∞). result Guarantees Lipschitz continuity for small p and Hölder continuity for larger p in Lp(Ω) function spaces. New algorithms improve efficiency in learning from personalized rewards.
problem Learning from personalized rewards in recommendation systems.
method Developed provably efficient algorithms with sublinear regret for context-dependent feedback.
result Introduced a Lipschitz reward estimator that improves generalization performance.
A new machine learning method for Bayesian inverse problems in function spaces.
problem Bayesian inverse problems in function spaces with incompatibility of white noise sources.
method One-step generative transport with amortized neural operator and prior-aligned Gaussian random field.
result Generative operator trained on prior samples and noisy observations generates posterior samples efficiently.
Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map ever…
New algorithm tackles stochastic bilevel optimization under relaxed smoothness conditions.
problem Optimal algorithms for stochastic bilevel optimization under relaxed smoothness conditions.
method Introduces a novel fully single-loop and Hessian-inversion-free algorithmic framework for stochastic bilevel optimization.
result Demonstrates state-of-the-art oracle complexity results for multi-objective robust bilevel optimization.
This paper is an expository account of the development of soliton mathematics, from its inception in famous numerical experiments of Fermi-Pasta-Ulam and Zabusky-Kruskal to the recent synthesis of Terng-Uhlenbeck (dg-ga/9707004) that explains hidden symmetries of soliton equations in terms of loop-groups acting by dres…
Wide deep neural networks with Gaussian weights approximate Gaussian processes closely.
problem Understanding the approximation of deep neural networks with Gaussian weights to Gaussian processes.
method Established novel rates for the Gaussian approximation of random deep neural networks with Gaussian parameters and Lipschitz activation functions in the wide limit.
result The distance between the network output and the Gaussian approximation scales inversely with the width of the network.
We show that any smooth bi-Lipschitz h can be represented exactly as a composition hm∘...∘h1 of functions h1,...,hm that are close to the identity in the sense that each (hi−Id) is Lipschitz, and the Lipschitz constant decreases inversely with the number m of functions com…
A new method for target propagation using iterative approximations converges fast and is more biologically plausible.
problem Improving target propagation methods for neural networks.
method Iterative approximate inverses and local auto-encoders.
result The method converges exponentially fast under certain conditions.
This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric c…
LOT improves adversarial robustness by training 1-Lipschitz convolution layers.
problem Improving adversarial robustness of deep neural networks.
method LOT: Layer-wise Orthogonal Training for 1-Lipschitz convolution layers.
result LOT significantly enhances certified robustness of Lipschitz-bounded models.
The paper studies global invertibility of maps on Finsler manifolds.
problem Global invertibility of locally Lipschitz maps on Finsler manifolds.
method Introduces pseudo-Jacobian and studies its relations with local metric properties of the map.
result Conditions for a map to be globally invertible and covering.
As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold M into a Riemannian manifold N admits a smooth approximation via immersions if the map has no singular points on M in the sense of F.H. Clarke, where dimM≤dimN. As its corollary, we have that if a bi-Lipschitz homeomo…
Let (M,g) be a pseudo-Riemannian manifold of signature (p,q). We construct mutually quasi-inverse equivalences between the groupoid of bundles of weakly-faithful complex Clifford modules on (M,g) and the groupoid of reduced complex Lipschitz structures on (M,g). As an application, we show that (M,g) admits a …
Time-delayed embeddings avoid self-intersections for high enough delay.
problem Analyzing self-intersections in time-delayed embeddings.
method Study of time-delayed coordinate maps for diffeomorphisms on compact manifolds.
result For high enough delay, time-delayed embeddings avoid self-intersections almost everywhere.
Method learns conditional distributions using neural entropic optimal transport.
problem Challenges in learning multiple conditional distributions.
method Neural entropic optimal transport method with two networks and regularization.
result Effective learning of conditional distributions with limited samples.
Paper examines stability of Bayesian posterior measures using integral probability metrics.
problem Stability of Bayesian inference in large-scale inverse problems.
method New families of integral probability metrics for likelihood and prior perturbations.
result Constructs new stability results for Bayesian posterior measures.
Given a Moebius homeomorphism f:∂X→∂Y between boundaries of proper, geodesically complete CAT(-1) spaces X,Y, and a family of probability measures {μx}x∈X on ∂X, we describe a continuous family of extensions {f^p:X→Y}1≤p≤∞ of f, call…
A new method reduces computational cost for gene expression inference in large microarray data sets.
problem Efficiently predicting gene expression in large datasets with limited resources.
method Adaptive Lipschitz constant inspired learning rate, random sub-sampling, and A-ReLU activation function.
result Remarkable improvement in saving computational cost while maintaining prediction accuracy.
New proofs confirm travel time data determine simple metrics on a disc.
problem Determining a simple Riemannian metric from travel time data.
method Proofs based on Myers-Steenrod theorem, Lipschitz-type stability estimate.
result Travel time data determine a simple Riemannian metric on a disc up to natural gauge.
A deep learning approach solves probabilistic inverse problems with physical constraints.
problem Solving inverse problems with large inferred vectors and prior samples.
method Uses conditional Wasserstein generative adversarial networks (cWGAN) with full gradient penalty.
result Improves accuracy and robustness in sampling and solving inverse problems.
We prove that the Gromov boundary of every hyperbolic group is homeomorphic to some Markov compactum. Our reasoning is based on constructing a sequence of covers of ∂G, which is quasi-G-invariant wrt. the ball N-type (defined by Cannon) for N sufficiently large. We also ensure certain additional propert…
The paper accelerates ISTA and FISTA algorithms for composite optimization problems.
problem Improving convergence rates of ISTA and FISTA for composite optimization.
method Improved proximal subgradient norm minimization using Lyapunov function.
result Convergence rates of ISTA and FISTA are accelerated.
Novel Bayesian framework for Poisson inverse problems using Bregman geometry.
problem Solving Poisson inverse problems with non-Euclidean geometry and positivity constraints.
method Develops a Monte Carlo sampling algorithm that accounts for Bregman geometry, data augmentations, and conditional conjugacy properties.
result Efficient sampling via Gibbs steps and Hessian Riemannian Langevin Monte Carlo (HRLMC) for positivity constraints.
Generative Adversarial Nets (GANs) and Variational Auto-Encoders (VAEs) provide impressive image generations from Gaussian white noise, but the underlying mathematics are not well understood. We compute deep convolutional network generators by inverting a fixed embedding operator. Therefore, they do not require to be o…
Designs CNNs for better image reconstruction.
problem Image reconstruction from limited data.
method Parseval convolution operators and chaining of elementary modules.
result CNN-based algorithm yields better results than sparsity-based methods.
PGD algorithms solve nonlinear inverse problems with generative priors using noisy measurements.
problem Signal estimation from noisy nonlinear measurements with generative priors.
method Projected gradient descent algorithms for two cases: unknown and known nonlinearity.
result PGD algorithms converge linearly to optimal statistical rates using arbitrary initialization.
Normalizing flows optimize Jacobian determinant for unique likelihood objective.
problem Optimizing normalizing flows for unique likelihood.
method Showed Jacobian determinant is unique for given distributions, leading to a unique global optimum. Used eigenvalues of auto-correlation matrix for explicit likelihood expression.
result Explicit expression of likelihood for flows, independent of neural network parameterization, with theoretical optimal value.
Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.
problem Characterizing neural network approximations to Lipschitz functions.
method Analyzing L-Lipschitz neural networks and their density in L-Lipschitz functions. result One layer neural networks are dense in the set of all L-Lipschitz functions. Constructs a map with prescribed local Lipschitz constants on a subset of a manifold.
problem Creating a Lipschitz map with specific local Lipschitz constants on a subset of a manifold.
method Constructs a Lipschitz map that matches a given map on a subset and has a local Lipschitz constant defined by a continuous function.
result A Lipschitz map can be constructed with a local Lipschitz constant prescribed by a continuous function.