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). 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…
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.
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 …
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.
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. 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…
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 …
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.
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.
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.
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.
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…
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.
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…
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.
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.
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.
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.
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.
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.
The article explores the mapping class group using unicellular maps and provides filtrations.
problem Understanding the structure of the mapping class group.
method Using unicellular maps and surgeries, the article describes the mapping class group.
result Provides filtrations of the mapping class group.
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
Maps are an important medium that enable people to comprehensively understand the configuration of cultural activities and natural elements over different times and places. Although massive maps are available in the digital era, how to effectively and accurately access the required map remains a challenge today. Previo…
The paper constructs biharmonic maps between spheres using polynomial maps.
problem Creating biharmonic maps between spheres.
method Using harmonic homogeneous polynomial maps of different degrees to generate proper biharmonic maps.
result Established a method for constructing proper biharmonic product maps.
Both bi-harmonic map and f-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study f-bi-harmonic maps as the critical points of the f-bi-energy functional 21∫Mf∣τ(φ)∣2dvg. This class of maps generalizes both …
Research explores real algebraic realization of round fold maps of codimension -1.
problem Real algebraic realization of round fold maps of codimension -1.
method Generalizes canonical projections of unit spheres to round fold maps and discusses their real algebraic realization.
result Developed new studies in real algebraic geometry focusing on round fold maps of codimension -1.
The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.
problem Deriving Liouville theorems for generalized maps on Riemannian manifolds.
method Using conservation laws and monotonicity formulas, the paper derives Liouville theorems for different types of maps under various conditions.
result The paper establishes Liouville theorems for several types of generalized maps, including φ-F harmonic maps, φ-F symphonic maps, and φ-F-V-harmonic maps. This paper shows semi-equivelar toroidal maps are vertex-transitive covers.
problem Understanding the relationship between semi-equivelar and vertex-transitive toroidal maps.
method Proving semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps.
result Each semi-equivelar toroidal map has a finite vertex-transitive cover.
Paper defines and studies Clairaut warped product Riemannian maps.
problem Understanding the geometry of specific Riemannian maps.
method Identify geodesic conditions, derive conditions for Clairaut maps, and calculate curvature.
result Found conditions for a warped product Riemannian map to be Clairaut.
The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.
problem Investigating unique continuation principles for polyharmonic maps.
method Analyzing critical points of higher order functionals to prove extensions of known results in harmonic and biharmonic cases.
result Proving extensions of unique continuation principles for k-harmonic maps.
The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping…
This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable…
The paper constructs gluing maps for harmonic maps between Riemannian manifolds.
problem Constructing harmonic maps between Riemannian manifolds.
method Gluing construction of extended harmonic maps.
result Construction of gluing maps for harmonic maps under specific conditions.
Characterizes a general range decreasing group homomorphism.
problem Understanding range decreasing group homomorphisms in the entire mapping group.
method Characterization of a general range decreasing group homomorphism.
result Computes a particular class of homomorphisms and identifies all range decreasing group homomorphisms on specific mapping groups.
The paper examines HM-tensional and HS-tensional maps between Riemannian manifolds.
problem Analyzing tension fields of maps between Riemannian manifolds.
method Investigating harmonic maps and harmonic sections as tension fields.
result Characterization and properties of HM-tensional and HS-tensional maps. The paper proves a Liouville theorem for specific harmonic maps with free boundary.
problem Analyzing harmonic maps with free boundary conditions.
method Developed Liouville theorem for φ-F-symphonic, φ-F-harmonic, and φ-ΦS,p,ε harmonic maps. result Established Liouville theorem for the specified harmonic maps with free boundary.
Dirac-harmonic maps are uncoupled under certain conditions.
problem Understanding the uncoupling of Dirac-harmonic maps.
method Critical points of a super-symmetric energy functional, with focus on harmonic maps.
result Dirac-harmonic maps are uncoupled under minimality assumption.
We introduce slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of slant immersions, invariant Riemannian maps and anti-invariant Riemannian maps. We give examples, obtain characterizations and investigate the harmonicity of such maps. We also obtain necessary and sufficie…
Analyzes harmonic and biharmonic maps from gradient Ricci solitons.
problem Characterizing maps from gradient Ricci solitons.
method Derives conditions for maps to be constant or harmonic.
result Biharmonic maps of finite energy from the two-dimensional cigar soliton are harmonic.
The paper studies maps from pseudo-Hermitian to Kähler manifolds, proving harmonic map properties.
problem Analyzing maps between pseudo-Hermitian and Kähler manifolds.
method Investigates partial energy functionals and critical maps, proving foliated results for ∂b- and ∂b-harmonic maps. result Generalizes Siu's holomorphicity result to ∂b- and ∂b-harmonic maps. In this paper, we show that one can interrelate pluriharmonic maps with para-pluriharmonic maps by means of the loop group method. As an appendix, we give examples for the interrelation between pluriharmonic maps and para-pluriharmonic maps. Moreover, we investigate the relation among CMC-surfaces by use of such maps.
The article explores constructing biharmonic and conformal biharmonic maps to spheres.
problem Constructing biharmonic and conformal biharmonic maps to spheres.
method Geometric algorithm to render harmonic maps biharmonic or conformally biharmonic.
result Explicit critical points for conformal-biharmonic maps between spheres are found.