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 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%.
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.
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.
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…
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 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…
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. 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…
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. 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.
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. 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.
Improved neural network capacity analysis using simplified RDT.
problem Analyzing the memorization capabilities of sign perceptron neural networks.
method Developed a simplified, partially lifted Random Duality Theory (fl RDT) approach.
result Concrete capacity bounds universally improve over previous best known ones.
In this paper, we define a complete lift for semisprays. If S is a semispray on a manifold M, its complete lift is a new semispray Sc on TM. The motivation for this lift is two-fold: First, geodesics for Sc correspond to the Jacobi fields for S, and second, this complete lift generalizes and unifies previ…
Unified approach to constructing integrable systems using Stäckel lifts.
problem Constructing new integrable Hamiltonian systems.
method Generalized Stäckel geometry and Haantjes structure.
result Hamiltonian systems with momentum-dependent Stäckel matrices exhibit symplectic-Haantjes structures.
Geometric structures are lifted to higher tangent bundles preserving statistical properties.
problem Lifting statistical structures to higher tangent bundles while maintaining their properties.
method Natural lifts of geometric objects and potentials to higher tangent bundles, preserving statistical manifold structures.
result Lifted statistical structures on higher tangent bundles maintain pseudo-Riemannian metrics and are again statistical manifolds.
The study revisits Hopfield's associative memory model and calculates its capacity for two specific pattern basins.
problem Determining the capacity of a Hebbian-Hopfield network for storing binary patterns.
method Using fully lifted random duality theory and numerical analysis, the study calculates the capacity for two specific pattern basins.
result Explicit characterizations of the capacity for the AGS and NLT pattern basins, with remarkable fast lifting convergence.
The article classifies liftings of connections on differential manifolds for geodesic modeling.
problem Classifying liftings of connections on differential manifolds.
method Liftings of connections on frame bundles, induced and adjust liftings.
result Developed a method for geodesic modeling of differential equations.
Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
problem Embedding maps in higher dimensions without self-intersections.
method Lifting maps to embeddings in product spaces.
result Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
New framework analyzes regret in guided diffusion for optimizing structured inputs.
problem Understanding regret behavior in guided-diffusion black-box optimization for structured design problems.
method Developed a certificate-based expected simple-regret framework that avoids assumptions breaking down in modern diffusion BO pipelines.
result Explains how exponential and polynomial convergence can arise from mass lift in near-optimal designs.
In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold M to its Weil bundle TAM defined by a Frobenius Weil algebra A. For a Poisson manifold (M,w), we show that the complete lift wC and the vertical lift wV of the Poisson tensor w are Poisson tensors on $T^…
In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays. Firstly we consider basic examples. We give a projective line of diff(M)-equivariant pencil liftings for first order operators,…
Study covers of sphere with homeomorphisms lifting property.
problem Finite abelian covers of sphere with lifting homeomorphisms.
method Completely determined covers with specific lifting property.
result Properties of finite abelian covers with lifting homeomorphisms.
The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral ζ-invariants using lifted …
The paper constructs Sasakian lifts from Kähler manifolds and studies their properties.
problem Constructing Sasakian structures from Kähler manifolds and analyzing their geometric properties.
method Local construction of Sasakian manifolds from Kähler base using vector field operations.
result Existence and properties of α-Sasakian Ricci solitons in Sasakian lifts. New quantum code lacks sparse lift.
problem Existence of sparse lifts for quantum codes.
method Constructed a sparse Z2 chain complex without a sparse lift. result Found a quantum code without a sparse lift.
Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.
problem Lifting diffeomorphisms to vector bundles.
method Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
result Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
CLuP achieves near optimal ground state energies for positive and negative Hopfield models.
problem Finding near optimal ground state energies for positive and negative Hopfield models.
method Controlled Loosening-up (CLuP) algorithm with fully lifted random duality theory (fl RDT).
result Achieves ground state free energies of 1.77 and 0.33 for positive and negative Hopfield models respectively. Simple lifts of non-simple curves on surfaces.
problem Finding simple lifts of non-simple closed curves on surfaces.
method Constructing finite covers and curves on surfaces.
result Explicit non-simple closed curves with simple lifts on degree n covers.
This paper extends braid lifting to coloured braid groupoids for all simple disc covers.
problem Lifting braids to homeomorphisms on branched covers of the disc.
method Defines a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc.
result Characterizes the lift of every coloured braid, recovering classical lifting on liftable braids.
We construct some lift of an almost complex structure to the cotangent bundle, using a connection on the base manifold. This generalizes the complete lift defined by I.Sato and the horizontal lift introduced by K.Yano and S.Ishihara. We study some geometric properties of this lift and its compatibility with symplectic …
Proves harmonicity equivalence on manifold metrics.
problem Harmonicity of metrics on pseudo-Riemannian manifolds.
method Equivalence of harmonicity on manifold metrics and their lifts.
result Harmonicity of a metric is equivalent to its lifts' harmonicity.
In Finsler geometry the complete lift vector fields have distinguished geometric significance. For example a vector field on a Finsler manifold is said to be conformal if its complete lift is conformal in usual sense. In this work we define a new Riemannian or Pseudo-Riemannian metric on TM derived from a Finsler metri…
The Eisenhart lift connects Hamiltonian systems to geodesics in pp-wave spacetimes.
problem Studying the stability and dynamics of Hamiltonian systems.
method Eisenhart lift to pp-wave spacetimes and conformal classes of ODEs.
result Existence of a constant of the motion generalizing conservation of energy.
Survey of twistor lifts of surfaces in 4-dimensional spaces.
problem Understanding the properties of surfaces in 4-dimensional Euclidean space.
method Definitions of Gauss maps and twistor lifts using orthogonal complex structures.
result Holomorphicity and isotropicity of minimal surfaces in E4. Study of lifting maps in branched covers of 3-manifolds, showing non-injectivity.
problem Injectivity of lifting maps in branched covers of 3-manifolds.
method Analysis of branched covers and mapping class groups.
result Lifting map is generally not injective for most regular branched covers of 3-manifolds.
In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate …
Develops a new bidding system to maximize advertiser profit.
problem Inaccurate prediction of ad lift-effect due to biased log data.
method Unbiased Lift-based Bidding System that predicts lift-effect from biased log data.
result Demonstrates superior and practical high-performing lift-based bidding strategy.
We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection a…
We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.
Develops a new algebraic framework for differential geometry of infinite dimensional spaces.
problem Creating a differential geometry for infinite dimensional spaces without topology or local coordinates.
method Introduces a general algebraic framework for lifted geometry applicable to various infinite dimensional spaces.
result Stokes' Theorem appears as a form of differentiability in the lifted geometry of spaces of submanifolds.
Researchers lift knot coloring polynomial to Habiro ring.
problem Lifting colored Jones polynomial of knots to Habiro ring.
method Introduced new Habiro ring and map, used Alexander polynomial.
result Existence of loop expansion at roots of unity confirmed.