The paper studies geometric properties of group equivariant operators and their Riemannian structure.
problem Understanding the geometric structure of group equivariant operators.
method Endowing the space of group equivariant non-expansive operators with a Riemannian manifold structure and using gradient descent methods.
result Gradient descent methods can be applied to minimize cost functions on the space of group equivariant non-expansive operators.
The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group-equivariant non-expansive operators (GENEOs), which are maps between function spaces…
Paper defines mathematical framework for neural network explainability.
problem Neural network explainability and equivariant operators.
method Mathematical framework based on Group Equivariant Non-Expansive Operators (GENEOs) and complexity measures.
result Formal properties and interpretability of Group Equivariant Operators (GEOs) defined.
In this paper known results of symmetric orthogonality, as introduced by G. Birkhoff, and non-expansive nearest point projections are extended from the linear to the metric setting. If the space has non-positive curvature in the sense Busemann then it is shown that those concepts are actually equivalent. In the end it …
New analysis of stochastic approximation with non-expansive mappings.
problem Finite-time analysis of two-time-scale stochastic approximation with non-expansive mappings.
method Studied two-time-scale stochastic approximation algorithms with non-expansive mappings and projection steps.
result Last-iterate mean square residual error decays at a rate O(1/k1/4−ε). Maximal concentration bounds for stochastic approximation with heavy-tailed noise.
problem Analyzing the convergence of stochastic approximation algorithms under heavy-tailed Markovian noise.
method Novel Lyapunov function and black-box truncation argument.
result Tail behavior of the error can be sub-Gaussian, sub-Weibull, or lighter than any Pareto but heavier than any Weibull.
A softmax operator applied to a set of values acts somewhat like the maximization function and somewhat like an average. In sequential decision making, softmax is often used in settings where it is necessary to maximize utility but also to hedge against problems that arise from putting all of one's weight behind a sing…
Value function estimation is an important task in reinforcement learning, i.e., prediction. The Boltzmann softmax operator is a natural value estimator and can provide several benefits. However, it does not satisfy the non-expansion property, and its direct use may fail to converge even in value iteration. In this pape…
New method recovers signals from compressed measurements using generative networks with contractive layers.
problem Signal recovery from compressed measurements with generative network priors.
method Developed a new matrix concentration inequality (R2WDC) to relax expansivity conditions for generative networks.
result Signals in the range of a Gaussian generative network can be recovered from few linear measurements with contractive layers.
Improved stochastic approximation method reduces residual error.
problem Reducing residual error in stochastic approximation algorithms.
method Fixed-schedule one-quarter barrier and bias-corrected acceleration.
result Achieves T−1/2+o(1) residual reduction with O(1) primitive samples. A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, t…
Proposes a topological model for partial equivariance in neural networks.
problem Capturing partial equivariance in neural networks for data analysis.
method Introduces P-GENEOs and studies spaces of measurements and P-GENEOs between them.
result Spaces of measurements and P-GENEOs have convenient approximation and convexity properties.
SCENE-Net improves 3D point cloud segmentation with low resource usage and transparency.
problem Lack of resources and transparency in 3D semantic segmentation models.
method SCENE-Net uses signature shapes identified via GENEOs to achieve semantic segmentation with minimal resources.
result SCENE-Net achieves comparable IoU to state-of-the-art methods with less data and computational resources.
We examine overlapping clustering schemes with functorial constraints, in the spirit of Carlsson--Memoli. This avoids issues arising from the chaining required by partition-based methods. Our principal result shows that any clustering functor is naturally constrained to refine single-linkage clusters and be refined by …
Convergence of the Kalman filter is best analyzed by studying the contraction of the Riccati map in the space of positive definite (covariance) matrices. In this paper, we explore how this contraction property relates to a more fundamental non-expansiveness property of filtering maps in the space of probability distrib…
This work extends GNNs to handle multiple graphs with non-commuting operators, proving transferability.
problem Handling multiple graphs with non-commuting operators in graph neural networks.
method Developed a mathematical theory for graph-tuple neural networks (GtNNs) with non-commuting non-expansive operators.
result Proved universal transferability of GtNNs, ensuring no non-transferable energy under convergence.
Abstracts a theorem for non-smooth maps in infinite dimensions.
problem Generalizing inverse mapping theorem for non-smooth maps.
method Introduces property A and applies it to non-smooth maps.
result Generalized inverse mapping theorems for non-smooth maps.
Neural network quantization is becoming an industry standard to efficiently deploy deep learning models on hardware platforms, such as CPU, GPU, TPU, and FPGAs. However, we observe that the conventional quantization approaches are vulnerable to adversarial attacks. This paper aims to raise people's awareness about the …
Novel algorithm accelerates PnP methods for image deblurring and super-resolution.
problem Efficiently solving inverse problems and imaging with provable convergence guarantees.
method Incorporates quasi-Newton steps into provable PnP framework based on proximal denoisers.
result 2--8x faster convergence compared to other provable PnP methods with similar quality.
Deep networks are commonly used to model dynamical systems, predicting how the state of a system will evolve over time (either autonomously or in response to control inputs). Despite the predictive power of these systems, it has been difficult to make formal claims about the basic properties of the learned systems. In …
This work uses sampling theory to analyze smoothness and error bounds of finite neural networks.
problem Analyzing the function space of finite neural networks and providing error bounds.
method Applying sampling theory to finite neural networks with non-expansive activation functions, considering both deterministic and random sampling.
result Novel error bounds for univariate neural networks under band-limited input assumption, highlighting the advantage of deterministic uniform sampling.
Framework for designing nonlinearities in neural networks with slope constraints.
problem Designing nonlinearities with specific properties for signal processing.
method Variational framework with regularization for slope constraints and optimization of adaptive splines.
result Adaptive nonuniform linear splines achieve global optimum in constrained optimization.
CCDF reduces diffusion sampling steps for inverse problems.
problem Slow sampling from diffusion models in inverse problems.
method Starting from a single forward diffusion step with better initialization, followed by stochastic contraction.
result Significantly reduced sampling steps for state-of-the-art reconstruction.
NAST generalizes scattering transform for non-stationary time series analysis.
problem Analyzing non-stationary time series data.
method Neural activation of scattering transform with various activation functions and high pass filters.
result Central and non-central limit theorems for NAST of Gaussian processes.
We develop algorithms to learn non-linear dynamical systems without mixing assumptions.
problem Learning non-linear dynamical systems from dependent data.
method We introduce an offline algorithm and a one-pass streaming method with SGD-RER.
result Our methods achieve optimal or near-optimal performance for learning non-linear systems.
This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for …
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
Introduces a new elliptic operator with positive eigenvalue.
problem None explicitly stated in the abstract.
method Introduces a new elliptic operator called the two-radical Laplace operator.
result The eigenvalue of the new operator is the positive square root of the Laplace operator's eigenvalue.
Proves Kato inequalities for various conformal operators.
problem Proving inequalities for differential operators.
method Analyzes a class of first order differential operators, including Dirac and Penrose twistor operators.
result Derives Kato inequalities that interpolate between classical and refined versions.
We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…
Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.
problem Characterizing biharmonic hypersurfaces with recurrent operators.
method Analysis of various recurrent operators and their impact on biharmonic hypersurfaces.
result Some well-known recurrent operators play a significant role in making biharmonic hypersurfaces minimal.
Local index theorem for chiral geometric operators proved using heat kernel.
problem Proving a local index theorem for geometric first-order differential operators.
method Using Gilkey's invariance theory and heat kernel techniques.
result Supertrace of heat kernel converges to Chern-Weil form.
Study estimates eigenvalues for concave Hessian operators on convex domains.
problem Estimating eigenvalues for concave elliptic Hessian operators.
method Investigates Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators.
result Existence and properties of the first nonzero eigenvalue and eigenfunction.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
GJMS operators connect geometry, analysis, and physics.
problem None explicitly stated; focus on operators and their impact.
method Construction of conformally invariant differential operators.
result GJMS operators have significant impact in geometry, analysis, and physics.
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
The paper proves new theorems about specific types of operator perturbations.
problem Analyzing conformal perturbations of Dirac and signature operators.
method Developed Kastler-Kalau-Walze type theorems for specific operator types.
result Established new theorems for six-dimensional manifolds with boundary.
Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
Study essential spectrum of differential operators on geometrically finite orbifolds.
problem Analyzing the essential spectrum of differential operators over specific geometric structures.
method Investigates first order and Laplace type elliptic differential operators on Riemannian vector bundles over geometrically finite orbifolds.
result Discovers properties of essential spectra for these operators.
The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
Study on opers over complex manifolds of dimension one.
problem Investigating opers over complex manifolds of dimension one.
method Introducing relative opers and differential operators, analyzing their equivalence.
result Bijective correspondence between relative opers and differential operators.
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.
Formula for Hadamard coefficients from Green's operators on spacetimes.
problem Calculating Hadamard coefficients from Green's operators on spacetimes.
method Developed formulas for diagonal values and integrals over the diagonal of Hadamard coefficients.
result Formulated analogues of Hadamard expansions and resolvents for Green's operators.
Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.
problem Understanding perturbations of Dirac operators in various dimensions.
method Analyzes canonical perturbations of Dirac operators on Hermitian Clifford modules.
result Characterizes the low-energy spectrum of these operators on complete surfaces.
Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.
problem Determining when the Calabi operator can identify the range of the Killing operator in Lorentzian locally symmetric spaces.
method Developed criteria for a connection to be in the range of a connection, applied to the Killing connection.
result For indecomposable spaces, the Calabi operator suffices to identify the range of the Killing operator; for products, it fails.
Researchers create a family of conformally covariant operators.
problem Developing a comprehensive set of conformally covariant operators.
method Constructing a family of conformally covariant tridifferential operators as tangential operators in the Fefferman--Graham ambient space.
result Symmetrization of ambient operators is formally self-adjoint.