Study large deviation in stationarized fully lifted blirp interpolation.
problem Understanding atypical solutions in random optimization problems.
method Large deviation theory applied to fully lifted blirp interpolation.
result Elegant relations uncovered for fundamental interpolating parameters.
New method improves statistical interpolation for analyzing complex random structures.
problem Analyzing atypical random structures in statistical models.
method Introduces a large deviation upgrade to fully lifted interpolation.
result Allows for easier analysis of atypical random structures.
New method lowers spherical perceptron capacity using fully lifted random duality theory.
problem Tackles the negative spherical perceptron capacity, a long-standing open problem.
method Develops fully lifted random duality theory (fl RDT) to characterize capacity.
result Shows remarkable closed-form analytical relations for practical capacity values.
The study calculates the injectivity capacity of ReLU networks using a novel mathematical approach.
problem Determining the injectivity capacity of ReLU networks layers.
method Employing fully lifted random duality theory (fl RDT) to handle the ℓ0 spherical perceptron and implicitly the ReLU layers injectivity. result The lifting mechanism converges remarkably fast with relative corrections not exceeding 0.1%.
Continuous-time interpolation of volatility surfaces preserving mixtures and arbitrage-free.
problem Interpolation of volatility surfaces
method Constructing a mixture-preserving, arbitrage-free interpolation
result Lifts Brigo-Mercurio to time-varying weights with additive cost
New approach uses interpolation models and error bounds for verifiable scientific machine learning.
problem Challenges in verifying and validating modern scientific machine learning workflows.
method Combines multiple standard interpolation techniques with error bounds for efficient computation and comparative performance analysis.
result Error bounds for interpolation techniques can be computed or estimated efficiently, aiding in validation goals.
Study binary perceptrons' capacity using random duality theory.
problem Characterize the capacity of binary perceptrons with general thresholds.
method Utilized fully lifted random duality theory (fl RDT) to characterize the capacity.
result Characterizations match replica symmetry breaking predictions and uncover the capacity for zero-threshold scenario.
Characterizes kernel interpolation in large dimensions, revealing optimal and sub-optimal regions.
problem Understanding the phase diagram of kernel interpolation in large dimensions.
method Characterization of variance and bias under various source conditions.
result Determined the (s,γ)-phase diagram of large-dimensional kernel interpolation. Despite the recent successes of deep neural networks, the corresponding training problem remains highly non-convex and difficult to optimize. Classes of models have been proposed that introduce greater structure to the objective function at the cost of lifting the dimension of the problem. However, these lifted methods…
This paper develops optimal transport methods on the roto-translation group SE2.
problem Optimal transport on the roto-translation group SE2 for image analysis.
method Develops a computational framework for optimal transportation over Lie groups, focusing on SE2. Uses Sinkhorn-like algorithm with efficient distance approximations.
result Advances in image barycentric interpolation, orientation field interpolation, and Wasserstein flows on SE2.
The paper improves convergence rates of curvature approximations using Regge elements.
problem Improving convergence rates of curvature approximations using Regge elements.
method Investigates the interplay between polynomial degree of curvature lifting and metric tensor degree in Regge finite element space.
result Higher convergence rates are achieved by reducing the polynomial degree of curvature lifting and using linear Regge elements.
Wide hidden layer TCM nets capacity analyzed using RDT and fl RDT.
problem Capacity analysis of wide hidden layer TCM nets.
method Employed Fully Lifted Random Duality Theory (fl RDT) for capacity characterization.
result Explicit, closed form capacity characterizations for a generic class of hidden layer activations.
Automorphisms of Lie algebras and their root systems are fully lifted.
problem Understanding automorphisms of real semisimple Lie algebras and their root systems.
method Proving every automorphism of the restricted root system can be lifted to a Lie algebra automorphism.
result Automorphisms of restricted root systems can be fully lifted to Lie algebras.
This article concerns cotangent-lifted Lie group actions; our goal is to find local and ``semi-global'' normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies to a…
The study examines deep convolutional neural networks and their learning ability.
problem Understanding the learning ability of deep convolutional neural networks (DCNNs).
method Examines DCNNs under both underparameterized and overparameterized settings, using a novel network deepening scheme.
result Establishes the first learning rates of underparameterized DCNNs and shows how adding layers can create interpolating DCNNs with good learning rates.
The paper provides a geometric framework for understanding non-equilibrium thermodynamics.
problem Unclear geometric structure of GENERIC in non-equilibrium thermodynamics.
method Cotangent lifts of dynamics, splitting into holonomic and vertical representatives, and formulation within contact geometry.
result Physical meaning and explicit formulation of the second law of thermodynamics within evolution equations.
Statistical relational models provide compact encodings of probabilistic dependencies in relational domains, but result in highly intractable graphical models. The goal of lifted inference is to carry out probabilistic inference without needing to reason about each individual separately, by instead treating exchangeabl…
New bounds for linear interpolators show how they generalize under covariate shifts.
problem Understanding how linear interpolators generalize under covariate shifts.
method Proved non-asymptotic excess risk bounds for benignly-overfit linear interpolators in transfer learning.
result Identified beneficial and malignant covariate shifts based on overparameterization degree.
Study potential computational gaps in symmetric binary perceptrons using fl-RDT.
problem Potential statistical-computational gaps in symmetric binary perceptrons.
method Parametric utilization of fully lifted random duality theory (fl-RDT).
result Observation of a computational gap SCG=αc−αa in SBP. Study reveals phase transition in neural networks near interpolation.
problem Understanding generalization and learning transitions in neural networks.
method Effective theory for approximating Bayes-optimal generalisation error.
result Unveils a discontinuous phase transition between universal and specialisation phases.
The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.
problem Boundary estimates for fully nonlinear Yamabe equations on Riemannian manifolds.
method Deriving a priori second derivative estimates for subsolutions.
result Existence of smooth solutions with uniform estimates.
New RL framework simulates financial market dynamics.
problem Complex financial market dynamics under various scenarios.
method Two RL families learn simultaneously, using Deep RL and parametrized reward.
result Agents learn a shared policy for diverse behaviors.
Study geometric structures in transfer learning to avoid negative transfer.
problem Understanding information-theoretic limits of transfer learning without exploiting domain geometry.
method Integrates geometric structure into linear regression models, using Gram matrices of source and target domains.
result Proposes an interpolation estimator that matches minimax lower bound and outperforms existing methods.
GenSDR tackles SDR by leveraging generative models to fully recover lower-dimensional structures.
problem Challenges in identifying low-dimensional sufficient structures in nonlinear SDR.
method Proposes GenSDR, a method that uses modern generative models to fully recover information in the central σ-field.
result Establishes consistency of GenSDR estimator for sample-level data and extends its applicability to non-Euclidean responses.
New geometry for optimal transport cost based on Bregman divergences.
problem Optimal transport cost calculation with Bregman divergences.
method Established properties, defined interpolations, constructed dualistic geometry.
result Derived generalized Pythagorean inequality and Bregman-Wasserstein barycenters.
New method finds rare dense clusters in asymmetric binary perceptrons, resolving algorithmic hardness.
problem Resolving algorithmic hardness in asymmetric binary perceptrons.
method Fully lifted random duality theory (fl RDT) and large deviation upgrade (sfl LD RDT).
result Local entropy breaks down for constraint densities in (0.77, 0.78) interval, matching current solver limits.
Lifted Relational Neural Networks (LRNNs) describe relational domains using weighted first-order rules which act as templates for constructing feed-forward neural networks. While previous work has shown that using LRNNs can lead to state-of-the-art results in various ILP tasks, these results depended on hand-crafted ru…
We prove overfitting in minimal and random NNs, tempering the effect.
problem Overfitting in minimal and random neural networks.
method Analyzing binary weight fitting to noisy data, proving overfitting is tempered.
result The overfitting of minimal and random neural networks is tempered.
Variable Annuity (VA) products expose insurance companies to considerable risk because of the guarantees they provide to buyers of these products. Managing and hedging these risks requires insurers to find the value of key risk metrics for a large portfolio of VA products. In practice, many companies rely on nested Mon…
Study on theoretical limits of ℓ0 sparse-regression algorithms using Fl RDT.
problem Understanding the performance limits of ℓ0 norm based optimization algorithms in compressed sensing and sparse regression. method Utilized Fully lifted random duality theory (Fl RDT) to analyze the maximum-likelihood (ML) decoding performance.
result Uncovered phase-transition (PT) and descending ℓ0 (dℓ0) curves that separate successful and unsuccessful algorithm performance. HFNO enhances interpretability of turbulent flows through parallel wavenumber bin processing.
problem Opaque inner workings of Fourier Neural Operators (FNOs) hinder physical interpretability.
method Introduces HFNO, a novel FNO-based architecture that processes wavenumber bins in parallel, enhancing interpretability.
result HFNO decomposes turbulent flows across various scales, enabling increased interpretability and multiscale modeling.
We present a discriminative clustering approach in which the feature representation can be learned from data and moreover leverage labeled data. Representation learning can give a similarity-based clustering method the ability to automatically adapt to an underlying, yet hidden, geometric structure of the data. The pro…
Study resolves conjecture on overparameterized linear models' generalization.
problem Asymptotic generalization of multiclass classification with overparameterized models.
method Gaussian covariates bi-level model, Hanson-Wright inequality variant.
result Min-norm interpolating classifier can be suboptimal compared to noninterpolating classifiers.
The annihilating filter-based low-rank Hankel matrix approach (ALOHA) is one of the state-of-the-art compressed sensing approaches that directly interpolates the missing k-space data using low-rank Hankel matrix completion. The success of ALOHA is due to the concise signal representation in the k-space domain thanks to…
Convexity preserved in curved surfaces moving at concave speeds.
problem Deforming convex surfaces with concave speeds.
method Nonlinear geometric flows with concave speed functions.
result High curvature regions remain approximately convex.
Calibrates historical and implied correlations in energy markets.
problem Challenges in aligning historical correlations of futures contracts with implied volatility smiles.
method Multiplicative multi-factor Heath-Jarrow-Morton model combined with stochastic volatility from lifted Heston model, using Kemna-Vorst approximation and Fourier-based techniques.
result Remarkable joint historical and implied calibration fits on the German power market.
New method uses neural networks to interpolate stellar atmospheres with high precision.
problem Recover precise stellar model atmospheres from grids of models.
method Deep neural network with 1D convolutional auto-encoder for feature extraction.
result Higher precision compared to traditional methods.
Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove bot…
DFFL tackles federated learning with heterogeneous objectives and constraints.
problem Federated learning with clients having different objectives and feasible regions.
method Derived heterogeneity bounds for cost-vector distances and support-function/shape-distance terms. Lifted pointwise bounds to local-versus-federated excess-risk comparison.
result Federation is beneficial when the statistical advantage of pooling exceeds a client-specific heterogeneity penalty.
Existence and rigidity results for lifts in Carnot groups.
problem Existence and properties of lifts for maps between Carnot groups.
method Use central extensions to define lifts and prove existence and rigidity results for Lipschitz, Sobolev, and quasiconformal maps.
result Quasiconformal maps admit contact lifts that are bi-Lipschitz.
Mixup improves feature learning by mixing common and rare features.
problem Improving generalization in deep learning models.
method Mixup, a data augmentation technique, is applied to feature learning. Theoretical and experimental studies are conducted to understand its benefits.
result Mixup effectively learns rare features from common ones, leading to better generalization.
Study lift metrics and connections on tangent bundles of Riemannian manifolds.
problem Investigate geometric properties of tangent bundles and their lifts.
method Analyze lift metrics and connections on TM of (M,g), and study statistical and Codazzi couples. result Prove a result on 1-Stein and Osserman structures on TM. Generative Distribution Embeddings learn multiscale representations of distributions.
problem Learning representations of entire distributions for multiscale reasoning.
method Introducing GDE framework that lifts autoencoders to the space of distributions, using conditional generative models and distributional invariance.
result GDEs learn predictive sufficient statistics embedded in Wasserstein space, recovering distances and trajectories for Gaussian and Gaussian mixture distributions.
Study shows convergence rates for Cheeger cuts on data clouds.
problem Optimizing graph cuts for clustering data sampled from a manifold.
method Analyzes statistical properties of Cheeger cuts on proximity graphs built from data.
result Obtains high probability convergence rates for Cheeger constant and cuts.
Paper develops exact convex optimization for neural networks with polynomial activations.
problem Training two-layer neural networks with nonlinear polynomial activations.
method Exact convex optimization using semidefinite programming.
result Global optimization of neural networks is polynomial-time computable.
The paper proposes using function approximations to reduce the computational burden in measuring counterparty credit exposure.
problem The need for regular exposure calculations in finance, balancing between computational cost and risk simplification.
method Replacing derivative pricers with function approximations, proving error bounds, and using Chebyshev interpolation for convergence.
result Derives probabilistic and finite sample error bounds, showing significant run-time reductions and asymptotic efficiency gains.
Improved online convex optimization bounds between stochastic and adversarial settings.
problem Understanding optimization tasks that are neither i.i.d. nor fully adversarial.
method Establishing novel regret bounds exploiting smoothness of expected losses.
result Regret bounds improve on previous results by reducing dependence on maximum gradient length to variance of gradients.
Study precise estimators for correlated data using RDT.
problem Analyzing estimators in correlated linear regression models.
method Utilized Random Duality Theory to characterize prediction risk.
result Precise closed form characterizations of estimators' risk.