Abstract operator calculus solves fermionic quantum harmonic oscillator problems.
problem Eigenvalue problems of fermionic quantum harmonic oscillators.
method Abstract operator calculus using homotopy operator.
result Formulated eigenvalue problem resembling fermionic quantum harmonic oscillator.
Abstract cone operators prove scalar curvature comparisons on singular manifolds.
problem Proving scalar curvature inequalities on manifolds with cone singularities.
method Using index theory for twisted Dirac operators on Lipschitz bundles.
result Lipschitz rigidity for scalar curvature on odd-dimensional manifolds.
Studied stable solutions for symmetric systems with hypoelliptic operators.
problem Analyzing stable solutions for symmetric systems with hypoelliptic operators.
method Examined stable solutions for symmetric systems with hypoelliptic operators.
result Identified stable solutions for symmetric systems with hypoelliptic operators.
The paper calculates indices for families of Fredholm operators and their extensions.
problem Calculating indices for families of Fredholm operators and their extensions.
method Passing from a Fredholm operator to its graph, deforming the horizontal subspace.
result Index formulas for families of Fredholm realizations and self-adjoint extensions.
Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.
problem Understanding scalar curvature on manifolds with cone-like singularities.
method Analysis of abstract cone operators, spectral flow argument, and twisted Dirac operators.
result Lipschitz rigidity for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions.
Abstract: A new approach to technical indicators without lag.
problem Defining classical technical indicators as bounded operators for lag-free trading.
method Using linear algebra to redefine technical indicators as bounded operators in l∞(N) space. result Demonstrated the no-lag versions of technical indicators are simpler and more effective.
In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use t…
Study investigates how simple speech sounds can form abstract categories.
problem How do abstract categories like phonemes emerge from speech exposure?
method Used modeling techniques to test Memory-Based Learning and Error-Correction Learning.
result Error-Correction Learning models can learn abstractions, identifying phone inventory and grouping.
Abstract: A possibilistic portfolio choice problem using expected utility operators.
problem A possibilistic portfolio choice problem in the framework of expected utility operators.
method Using expected utility operators, the paper formulates a possibilistic choice problem and derives two approximate calculation formulas for optimization.
result Two approximate calculation formulas for optimization of possibilistic portfolio choice problem.
Minimal learning agents can infer unobserved variables in complex environments.
problem How to infer unobserved variables in complex environments using minimal learning agents.
method Concrete operational definition of abstract concepts, minimal architecture supporting abstraction, reinforcement learning.
result Minimal learning agents can infer the existence of unobserved variables.
We prove the rigidity of presymplectic actions of a compact semisimple Lie algebra on a presymplectic manifold of constant rank in the local and global case. The proof uses an abstract normal form theorem we had stated in a previous work, based on an iterative process of Nash-Moser type. In order to use correctly this …
The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincaré dual to the continuous trace algebra. Index formulas for pseudo-differential opera…
NHC learns scalable algorithmic solutions from diverse tasks.
problem Neural networks struggle to learn algorithmic strategies.
method Memory-augmented network with abstraction mechanism and evolutionary training.
result Reliable learning of robust and scalable algorithmic solutions.
Automatically verifies robustness of SVMs against adversarial examples.
problem Ensuring SVMs are robust to adversarial attacks.
method Abstract verification using interval domain and reduced affine forms.
result Encouragingly high percentages of provable robustness on MNIST test set.
Abstract: Study Kähler identities on almost complex manifolds.
problem Generalize Kähler identities to almost complex manifolds.
method Study via Clifford bundle and Dirac operator, then translate results to exterior bundle.
result Obtain generalization of Kähler identities for compact almost complex manifolds.
Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.
problem Generalize Picard iteration for nonlinear parabolic PDEs.
method Formulate Picard iteration as abstract state-transition model, derive generalization error bounds.
result Picard depth reduction reduces Picard truncation error without increasing estimation error.
We introduce a fourth order CR invariant operator on pluriharmonic functions on a three-dimensional CR manifold, generalizing to the abstract setting the operator discovered by Branson, Fontana and Morpurgo. For a distinguished class of contact forms, all of which have vanishing Hirachi-Q curvature, these operators d…
Abstract Szegedy walks on simplicial complexes are studied, revealing connections to combinatorial and geometric properties.
problem Investigating spectral structures of abstract Szegedy walks on simplicial complexes.
method Introduced modified Grover walks on simplicial complexes, focusing on orientations of simplices.
result Strong relationships between the spectrum of discriminants and combinatorial/geometry/topology properties of simplicial complexes.
Abstract reviews geometric wave and Dirac equations on manifolds.
problem Analyzing geometric equations on manifolds.
method Well-posedness and stability for initial value problems, structure of equations on black-hole spacetimes, index theorem for hyperbolic Dirac operators, properties of Green-hyperbolic operators.
result Results on the structure of wave and Dirac equations on black-hole spacetimes, including the Kerr solution.
Maps sets to probability distributions to minimize information loss.
problem Learning to map sets to probability distributions to preserve information.
method Relates set operations to probability distribution interpolations and demonstrates a preliminary solution.
result Experimental results show the effectiveness of the set embedding approach.
Abstract: Determinants and formulas for operators on various spaces.
problem Determinants and formulas for operators on different algebras and spaces.
method Use of Poincaré type determinants, invariant operators, and full matrix-symbols.
result Explicit formulas for determinants of elliptic operators and periodic pseudo-differential operators.
Differentiates conic optimization problems with unique solutions.
problem Conic optimization problems with unique solutions.
method Differentiability of the solution map under regular conditions.
result Derivative of the solution map is an abstract linear operator.
Study solves optimal portfolio selection using HJB equation.
problem Optimal portfolio selection problem.
method Maximal monotone operator method, Banach fixed-point theorem, Fourier transform, monotone operators technique.
result Existence and uniqueness of solution to HJB equation.
Abstract machinery finds obstructions to uniform positive scalar curvature.
problem Finding obstructions to uniform positive scalar curvature.
method Coarse index theory and embedding submanifolds.
result Abstract machinery constructs wrong way maps on K-theory. The paper proves a new semiclassical limit result for abstract Schrödinger operators.
problem Abstract Schrödinger operators on locally compact spaces.
method Probabilistic proof using the principle of not feeling the boundary.
result A new semiclassical limit result for partition functions.
The analysis of classical consensus algorithms relies on contraction properties of adjoints of Markov operators, with respect to Hilbert's projective metric or to a related family of seminorms (Hopf's oscillation or Hilbert's seminorm). We generalize these properties to abstract consensus operators over normal cones, w…
Interprets APS index theorem using coarse homotopy theory.
problem Motivic boundary value problem and its index theorem.
method Coarse homotopy theory interpretation of APS index theorem.
result Abstract version related to classical case for Dirac operators.
Abstract: Studies differential systems on compact Lie groups, extending Greenfield and Wallach's methods.
problem Global properties of left-invariant differential systems on compact Lie groups.
method Abstract: Extends Greenfield and Wallach's methods to systems, obtaining characterizations for regularity, range closeness, and cohomology spaces.
result Abstract: Derives generalizations of results and global versions of Caetano and Cordaro's result.
We survey some Lp-vanishing results for solutions of Bochner or Simons type equations with refined Kato inequalities, under spectral assumptions on the relevant Schrödinger operators. New aspects are included in the picture. In particular, an abstract version of a structure theorem for stable minimal hypersurfaces…
Mid-training improves RL by identifying compact action abstractions.
problem Unlocking full potential of RL with large language models.
method RA3 algorithm that optimizes sequential variational lower bound and discovers latent structures via RL.
result Improves performance by 8 points on HumanEval and 4 points on MBPP.
We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of the Cauchy-Schwarz inequality--is used to produce ``parameter-free'', ``projection-free'' vers…
The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have pos…
In this note we determine the first two derivatives of the classical Boltzmann-Shannon entropy of the conjugate heat equation on general evolving manifolds. Based on the second derivative of the Boltzmann-Shannon entropy, we construct Perelman's F and W entropy in abstract geometric flows. Monotonicity of the entropies…
ART trains neural nets to be both accurate and safe.
problem Ensuring neural nets are both accurate and safe during training.
method Integrates an optimization-based abstraction refinement loop into the learning process.
result Enables training provably correct networks with respect to safety properties.
Develops SCMs for latent selection to simplify causal analysis.
problem Latent selection complicates causal analysis.
method Introduces a conditioning operation for SCMs to encode latent selection.
result Conditioning operation preserves simplicity, acyclicity, and linearity of SCMs.
Paper uses weakly-supervised clustering to automatically create network protocol abstractions.
problem Manual definition of abstraction by domain experts is time-consuming.
method Weakly supervised clustering algorithm for automatic abstraction.
result The method successfully matches the reference abstraction with minimal labeled examples.
In a series of papers, we will develop systematically the basic spectral theory of (self-adjoint) boundary value problems for operators of Dirac type. We begin in this paper with the characterization of (self-adjoint) boundary conditions with optimal regularity, for which we will derive the heat asymptotics and index t…
Abstracts index for ML4H workshop at NeurIPS 2019.
problem No specific problem stated; index of accepted abstracts.
method Not specified; index of accepted abstracts.
result No specific result stated.
The study classifies cellular pseudomanifolds and their properties.
problem Understanding the structure of cellular pseudomanifolds.
method Analyzing the combinatorial and geometric properties of cellular pseudomanifolds.
result Complete classification of cellular pseudomanifolds with excess < 2, and progress towards excess 2.
We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family $X_\eps$ of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances o…
The paper aims to mathematically define and learn abstractions from data.
problem Defining and learning abstractions from data.
method Characterize abstractions as summaries for answering queries, define leakiness as a loss function, and generalize classical statistics.
result A mathematical theory of abstraction can be learned from data.
Unified framework for causal models at different levels of abstraction.
problem Relating causal models at varying levels of abstraction.
method Categorical framework using natural transformations between Markov functors.
result Generalized and unified causal abstractions with categorical proofs.
Abstract MDPs enable strategic exploration and fast reward transfer in complex environments.
problem Challenging to learn accurate MDPs for high-dimensional states.
method Learn an abstract MDP over low-dimensional coarse states, using an abstraction function.
result Achieves superhuman performance on Pitfall! and higher reward with fewer samples.
Discrete exterior calculus shows natural properties of wedge product and averaging.
problem Naturalness of discrete exterior calculus operations.
method Showed naturalness of discrete wedge product and averaging interpretation.
result Discrete wedge product is natural and equals Wilson's cochain product.
Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold without boundary. We examine the geometric content of such an operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients…
The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.
problem Characterizing and studying deformations and cohomologies of relative Rota-Baxter operators.
method Constructing graded Lie algebras and studying their Maurer-Cartan elements, cohomology, and deformations.
result Homomorphisms between cohomology groups of relative Rota-Baxter operators and deformation cohomology groups of left-symmetric algebroids.
Abstract: Shows nonexistence of Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.
problem Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.
method Constructs a metric on a compact manifold to demonstrate the nonexistence of Courant-type bounds.
result Provides a negative answer to the existence of Courant-type nodal domain bounds.
Large dataset for medical abstracts classified by sentence role.
problem Efficiently classify long medical abstracts for researchers.
method Labeled dataset of 200k abstracts, each with 2.3M sentences.
result Improved sentence classification for medical literature.