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.
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.
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.
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 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.
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 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. 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.
New insights into binary perceptron reveal phase transitions and algorithmic thresholds.
problem Understanding the statistical-computational gap in binary perceptron models.
method Application of fully lifted random duality theory (fl RDT) to uncover structural changes.
result Numerical estimates of constraint density thresholds align with theoretical predictions.
This paper connects ultrametric overlap gap properties to parametric RDT for symmetric binary perceptrons.
problem Characterizing statistical computational gaps in symmetric binary perceptrons.
method Developed an analytical union-bounding program to rigorously upper-bound constraint densities of ultrametric overlap gap properties.
result Obtained tightest bounds at the first two levels of ultrametric overlap gap properties, closely approaching parametric RDT estimates.
New analysis shows capacity of treelike neural networks with various activations.
problem Analyzing the capacity of treelike neural networks with diverse activations.
method Utilized Random Duality Theory and its partially lifted version to handle various activations.
result The capacity of treelike neural networks decreases for large network width but converges to a constant value.
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. 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.
New algorithms handle phase retrieval with rank d measurements, revealing phase transitions.
problem Phase retrieval with rank d measurements.
method Random duality theory (RDT) and descending phase retrieval algorithms (dPR).
result Minimal sample complexity ratio for dPR's success exhibits phase transitions.
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.
Study precise sample covariance error for Gaussian centered data.
problem Precise characterization of sample covariance error for Gaussian data.
method Developed a Random Duality Theory (RDT) framework to determine upper and lower bounds.
result Upper and lower bounds match in large-dimensional contexts, matching the spectral norm's limiting value.
Descending phase retrieval algorithms show a phase transition with increasing sample complexity.
problem Theoretical limits of descending phase retrieval algorithms.
method Utilizing Random duality theory (RDT), the study develops a generic program to characterize algorithm performance.
result As sample complexity increases, the parametric manifold transitions from multi to single funneling points, leading to a phase transition in algorithm success.
This work adapts RDT for mental program construction, showing benefits and costs.
problem Applying RDT to mental programs with trade-offs between description length, error, and computational costs.
method Proposed a three-way trade-off and used simulations and partial information decomposition.
result Constructing a shared program library provides global benefits but is sensitive to curricula.
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 algorithm nearly achieves ground state free energy of SK model.
problem Determining the ground state free energy of the SK model.
method Controlled Loosening-up (CLuP) algorithm applied to SK models.
result Achieves ground state free energy of ~0.76 for n in the thousands.
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…
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…
New bounds on neural network capacity for treelike sign perceptrons using RDT.
problem Determining the capacity of treelike sign perceptrons neural networks.
method Random Duality Theory (RDT) to establish upper bounds.
result Mathematically rigorous bounds on network capacity for any number of neurons.
We connect Causal inference and low-rank recovery via RDT and free probability theory.
problem Determining the applicability of causal inference via low-rank recovery.
method Random Duality Theory, free probability theory, and mathematical rigor.
result Exact closed-form worst case phase transitions for causal inference.
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…
We analyze ridge interpolators in correlated factor regression models using RDT.
problem Performance analysis of ridge interpolators in correlated factor regression models.
method Utilizing Random Duality Theory (RDT), we obtain precise closed form characterizations of optimization problems.
result Ridge interpolators can smooth out the excess prediction risk and exhibit double-descent behavior.
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…
Deep ReLU networks show that 4 layers suffice for unique input recovery.
problem Injectivity capacity of deep ReLU networks.
method Developed a program connecting deep ReLU injectivity to an l-extension of the ℓ0 spherical perceptrons, using random duality theory. result Only 4 layers are needed for unique input recovery, showing expansion saturation effect.
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. Optimal spectral initializers impact phase retrieval phase transitions.
problem Understanding the limits of phase retrieval algorithms.
method Developed Random duality theory (RDT) to characterize optimal spectral initializers.
result Optimal spectral initializers can fall into flat regions of the phase retrieval manifold, making phase retrieval difficult.
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.
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 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.