Graph continuous operators become Riesz continuous after multiplication by unitary operators.
problem Characterizing Riesz continuity of graph continuous operators.
method Multiplication by unitary operators to transform graph continuity to Riesz continuity.
result The index of graph continuous families of Fredholm operators coincides with N. Ivanov's index.
Notes on continuity of discrete-spectrum Fredholm operators.
problem Continuity properties of discrete-spectrum families of Fredholm operators.
method Relates recent work on discrete-spectrum families to classical continuity properties.
result Establishes connections between new and classical concepts.
Study Cowen-Douglas operators from analytic function spaces.
problem Analytic continuation and spectrum of Cowen-Douglas operators.
method Investigate Banach spaces of analytic functions and their operators.
result Analytic continuations of functions relate to the spectrum of Cowen-Douglas operators.
We prove that a topological manifold (possibly with boundary) admitting a continuous cancellative binary operation is orientable. This implies that the Möbius band admits no cancellative continuous binary operation. This answers a question posed by the second author in 2010.
Formulas derived for operators on forms in anti-de Sitter spaces.
problem Operators on forms in anti-de Sitter spaces.
method Explicit formulas for codifferential and Laplace-de Rham operators.
result Formulas for restriction and continuation between spaces.
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…
Continuous family of elliptic operators' projections maintain Cauchy data spaces.
problem Maintaining Cauchy data spaces for a continuous family of elliptic operators.
method Elementary tools and classical results applied to operator graphs, Sobolev spaces, and Green's formula.
result Orthogonalized Calderón projections form a continuous family of projections.
A simple continuous relaxation for argsort improves performance and is easy to implement.
problem Discrete nature of argsort operator makes it unsuitable for gradient-based learning.
method Proposed a continuous relaxation for argsort operator that is simple, fast, and achieves state-of-the-art performance.
result The proposed relaxation achieves state-of-the-art performance and is faster than competing approaches.
We first analyze the integrated density of states (IDS) of periodic Schrödinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with both continuous and discontinuous IDS'. Subsequently, alloy-type perturbations of th…
Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our ma…
We model how Lipschitz continuity changes during neural network training.
problem Understanding how Lipschitz continuity evolves during training.
method We use a system of stochastic differential equations to capture the dynamics of Lipschitz continuity under SGD.
result We identify three factors driving the evolution of Lipschitz continuity: gradient flow projection, gradient noise, and Hessian projection.
This paper describes recent development and test implementation of a continuous time recurrent neural network that has been configured to predict rates of change in securities. It presents outcomes in the context of popular technical analysis indicators and highlights the potential impact of continuous predictive capab…
New methods solve MI problems with locally Lipschitz operators, improving solution efficiency.
problem Solving monotone inclusions with locally Lipschitz continuous operators.
method Primal-dual extrapolation methods using backtracking line search.
result Improved operation complexity for solving MI problems.
Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.
problem Quantifying unique continuation for Schrödinger operators on manifolds with specific curvature conditions.
method Proving quantitative unique continuation estimates for Schrödinger operators on manifolds with Ricci curvature bounded below.
result Upper bound for energy range and constant in terms of Ricci curvature and parameters of relatively dense set.
Graph Laplace operators uniquely identify metrics and densities on manifolds.
problem Identifying Riemannian metrics and sampling densities from graph Laplace operators.
method Analyzing intrinsic and extrinsic graph Laplace operators on compact Riemannian manifolds.
result Graph Laplace operators uniquely determine metrics and densities under certain conditions.
The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this p…
Geometrically studies Moore-Penrose inverse and polar decomposition continuity.
problem Continuity of Moore-Penrose inverse for perturbations by operator ideals.
method Geometric construction using essential codimension and Banach-Lie group action.
result Moore-Penrose inverse is a real analytic map between manifolds.
The paper quantifies the regularity of attention operations.
problem Quantifying the regularity of attention operations.
method Proposes a new mathematical framework using measure theory and integral operators.
result Proves attention operation is Lipschitz continuous on compact domains and provides an estimate of its Lipschitz constant.
Sharp estimates derived for quasilinear equations on metric measure spaces.
problem Estimating solutions and eigenvalues of quasilinear equations on smooth metric measure spaces.
method Sharp estimates derived using comparisons with one-dimensional equations.
result Optimal lower bounds for the first Dirichlet eigenvalue of quasilinear operators.
Unified framework for analyzing graph neural operators converging to graph limits.
problem Analyzing convergence of graph neural operators to graph limits.
method Develops a unified spectral framework for graph neural operators under various graphon assumptions.
result Unified framework enables direct comparison of convergence rates and tradeoffs.
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…
The paper explains how continuous language models can produce discrete, interpretable meanings.
problem Semantic collapse in continuous systems of large language models.
method Formalizing large language models as Continuous State Machines (CSMs) and analyzing the associated transfer operator.
result The leading eigenfunctions of the transfer operator induce a finite number of invariant meaning basins, explaining how continuous computation can produce discrete, interpretable semantics.
New method learns policies from offline data using operator models.
problem Limited understanding of approximation errors in offline reinforcement learning.
method Linking reinforcement learning to Hamilton-Jacobi-Bellman equation, proposing operator-theoretic algorithm.
result Global convergence of the value function and finite-sample guarantees derived.
Automatic continuity of polynomial maps and cocycles proved.
problem Proving continuity of polynomial maps and cocycles.
method The approach involves proving continuity of polynomial maps and cocycles.
result Automatic continuity of polynomial maps and cocycles.
In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in algebraic topology. Generic maps which switch between the continuous differential for…
We give partial answers to the following question: if F is an m by m matrix on Rn satisfying a second order linear elliptic equation, does detF satisfy the strong unique continuation property? We give counterexamples in the case when the operator is a general non-diagonal operator and also for som…
Researchers prove an equivariant index theorem on Euclidean space.
problem Calculating the equivariant index of the Bott-Dirac operator on R2n. method Continuous field of C∗-algebras and equivariant index theorem. result Explicit calculation of the equivariant index of the Bott-Dirac operator on R2n. Researchers compute Wodzicki residue for pseudo-differential operators on compact Lie groups.
problem Computing the Wodzicki residue for pseudo-differential operators on compact Lie groups.
method Analytic continuation of traces and matrix-valued symbols.
result Main theorem complementary to [2], removing ellipticity hypothesis.
Neural networks learn discrete tasks on continuous data via emergent geometry.
problem Understanding how neural networks perform discrete computations on continuous data.
method Analysis of Riemannian pullback metric across neural network layers.
result Neural networks learn to discretize continuous inputs and perform logical operations on these discretized variables.
Novel framework proves fast RL convergence in continuous spaces.
problem Analyzing stability in continuous state-action RL.
method Introduces a novel framework to analyze stability properties of RL.
result Highlights two key stability properties and demonstrates their satisfaction in RL.
Mixtures of neural operators reduce active complexity in operator learning.
problem Reduction of active complexity in operator learning models.
method Constructive comparison between routed mixtures of neural operators (MoNOs) and a fixed single-neural-operator construction.
result Every scalar uniformly continuous nonlinear operator can be approximated by a MoNO whose active expert has smaller depth, width, and rank scaling.
This paper introduces a neural operator for probabilistic conditioning.
problem Probabilistic conditioning of random variables X given Y. method Develops a single operator that maps any joint density to its conditional, approximated by neural operators.
result Neural operators can approximate the conditioning operator to arbitrary accuracy.
We provide two examples of spectral analysis techniques of Schroedinger operators applied to geometric Laplacians. In particular we show how to adapt the method of analytic dilation to Laplacians on complete manifolds with corners of codimension 2 finding the absence of singular continuous spectrum for these operators,…
Proposes a new convolutional neural network for non-grid data.
problem Limited applicability of standard CNNs to non-grid structured data.
method Introduces Parametric Continuous Convolution (PCC) with learnable kernel functions.
result Significant improvement in point cloud segmentation and lidar motion estimation.
Continuous analysis techniques for deforming domains in manifolds.
problem Analyzing continuity of analysis objects in deforming domains.
method Introducing quasi-Lipschitz domains and studying monotone C0-deformations. result Global Morse index theorem holds for arbitrary Lipschitz domains.
Improved genetic programming by optimizing mutation operators for continuous program search.
problem Small syntactic mutations in genetic programming can lead to unpredictable behavioral shifts.
method Learned a compact trading-strategy DSL, created a block-factorized embedding, and designed geometry-compiled mutation operators.
result Geometry-compiled mutation operators discover strong strategies using fewer evaluations and achieve higher Sharpe ratios.
Stochastic representation for determinants derived from Brownian loop soups.
problem Determinants of Laplace-type operators on manifolds.
method Stochastic representation using Brownian loop soups.
result Derives continuity of determinants and conformal invariance on surfaces.
New methods for calculating curvature in graph theory.
problem Calculating curvature in graphs and random walks.
method Analyzing continuous and discrete-time Ollivier-Ricci curvatures of weighted graphs.
result Generalized existence and properties of Ollivier-Ricci curvature for various random walks.
We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a …
Deep RBVFs improve continuous control in RL.
problem Challenges in finding optimal actions for continuous actions in RL.
method Introduced deep radial-basis value functions (RBVFs) for continuous control.
result RBF-DQN significantly outperforms value-function-only baselines and is competitive with actor-critic algorithms.
Let M be a manifold, possibly with boundary. We show that the deRham differential from k-forms to exact (k+1)-forms has a continuous right inverse when both spaces are given the weak Whitney topology. This antidifferential operator is given a fairly explicit formula depending on the choice of a suitable good cover of M…
New method calculates eta invariant without analytic continuation.
problem Spectral asymmetry of non-semibounded systems.
method Direct pseudodifferential technique for curl operator.
result Eta invariant can be traced as spectral projection difference.
Model proposes neural network for continuous time dynamics with inductive biases.
problem Training neural networks for small datasets with nonlinear dynamics.
method Inductive biases on decay rates and frequencies using Koopman operator theory.
result Higher forecasting performance with single short training sequence.
This paper tackles continuous domain generalization, improving model performance across unseen domains.
problem Existing domain generalization approaches fail to capture the complex, multidimensional nature of real-world variation.
method Introduces Continuous Domain Generalization (CDG), a principled framework grounded in geometric and algebraic theories. Proposes a Neural Lie Transport Operator (NeuralLio) for structure-preserving parameter transitions and a gating mechanism for robust generalization.
result Demonstrates significant improvement in generalization accuracy and robustness across various datasets.
We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are i…
We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous fa…
Study shows how information loss and operation loss are related in feature representations.
problem Understanding the relationship between information loss and operation loss in feature representations.
method Analyzes the interplay between weak information loss and operation loss in continuous representations.
result Specific forms of vanishing information loss imply vanishing MPE loss in classification.