Study slice-regular polynomial functions via twistor space group actions.
problem Characterize slice-regular functions and their polynomial subclasses.
method Employ the twistor construction and group actions of PGL(2,H). result Characterize slice-regular functions with planar twistor lifts and normal classes of polynomials.
Deep tensor factorization benefits from implicit regularization with polynomial growth.
problem Tensor factorization's implicit regularization effect in deep networks is not well understood.
method Investigated the implicit regularization in deep tensor factorization, showing polynomial growth.
result Implicit regularization in deep tensor factorization grows polynomially with depth, improving estimation accuracy and convergence.
Proves regularity of isomorphisms between hyperbolic 3-manifolds.
problem Regularity of isomorphisms between cusped hyperbolic 3-manifolds.
method Proves regularity of profinite completions of fundamental groups.
result Proves A-polynomial is a profinite invariant. Study introduces a new method for multiple parameter regularization in polynomial functional regression.
problem Handling varying regularization parameters in polynomial functional regression.
method Developed a theoretically grounded algorithm for multiple parameter regularization and model aggregation.
result Promising results from evaluations on synthetic and real-world data.
NO approximates non-Markovian BSDEs with polynomial scaling in 1/ε.
problem Complexity of NO approximations for structured families of BSDEs.
method Identifying structured families of non-Markovian BSDEs, informing NO's inductive bias.
result Polynomial scaling in 1/ε for NO approximations of BSDE solution operators.
The paper calculates Alexander polynomials for knots using finite group representations.
problem Calculating Alexander polynomials for knots using specific group representations.
method Defined twisted Alexander polynomials associated with regular representations of finite groups.
result Several formulas for the twisted Alexander polynomial are provided.
The study classifies polynomial relation tubular surfaces in 3-spaces.
problem Classifying tubular surfaces with polynomial curvature relations.
method Analyzing polynomial relations between Gaussian and mean curvatures in Euclidean, hyperbolic, and Lorentzian 3-spaces.
result Determination of sets of polynomial relations for tubular surfaces.
The paper investigates polynomial alternatives to softmax in transformer models.
problem The effectiveness of softmax attention in transformers is questioned.
method The authors explore polynomial activations as alternatives to softmax, focusing on their ability to regularize the attention matrix.
result Certain polynomials can serve as effective substitutes for softmax in transformer applications, achieving strong performance.
We accelerate CNF by reducing ODE truncation errors with polynomial regularization.
problem High computation cost of CNF due to large truncation errors in solving ODEs.
method Add polynomial regularization to approximate ODE trajectories with polynomial functions.
result 42.3% to 71.3% reduction of NFE on density estimation, 19.3% to 32.1% on variational auto-encoder.
Study shows convergence speed for Fekete points on specific sets.
problem Understanding convergence speed for Fekete points on certain sets.
method Demonstrates (Cα,Cα′)-regularity for uniformly polynomially cuspidal sets. result Established convergence speed for Fekete points on these sets.
No regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces are found.
problem Existence of regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces.
method Analyzing polynomials defining hypersurfaces of various degrees and shapes.
result Hyperspheres and round cylinders are the only such hypersurfaces defined by polynomials of degree ≤3.
We study incompressible surfaces constructed by Culler-Shalen theory in the context of twisted Alexander polynomials. For a 1st cohomology class of a 3-manifold the coefficients of twisted Alexander polynomials induce regular functions on the SL2(C)-character variety. We prove that if an ideal point giv…
Analytic networks with bounded coefficients can't outperform polynomial approximations.
problem Approximation limits of neural networks with analytic activation functions under coefficient constraints.
method Deterministic analysis using comparison argument and Bernstein-type estimates.
result Networks with analytic activation functions and controlled coefficients cannot outperform classical polynomial approximation rates on non-analytic targets.
Polynomial-time convex optimization for CNNs with ReLU activations.
problem Training Convolutional Neural Networks (CNNs) with ReLU activations.
method Developed a convex analytic framework using semi-infinite duality to formulate equivalent convex optimization problems for CNN architectures.
result Proved that two-layer CNNs can be globally optimized via an ℓ2 norm regularized convex program. New moving average adapts weight dynamically based on polynomial and wavefunction.
problem Lagging traditional moving averages in adjusting to changes in data.
method Develops a moving average with weight as a polynomial of a wavefunction from an eigenproblem.
result Immediate 'switch' without lag, adapting to changes in data.
Strongly polynomial algorithm for approximate Forster transforms and halfspace learning.
problem Computing approximate Forster transforms and halfspace learning.
method Strongly polynomial time algorithm for approximate Forster transforms and halfspace learning.
result First strongly polynomial time algorithm for distribution-free PAC learning of halfspaces.
Develops exact convex optimization formulations for neural networks.
problem Training two-layer neural networks with rectified linear units.
method Uses semi-infinite duality and minimum norm regularization to develop exact convex optimization formulations.
result Shows equivalence of ReLU networks trained with weight decay to block ℓ1 penalized convex models. We obtain polynomial Frobenius manifolds from classical W-algebras associated to regular nilpotent elements in simple Lie algebras using the related opposite Cartan subalgebras.
It is shown that for knots with a sufficiently regular character variety the Dubois' torsion detects the A-polynomial of the knot. A global formula for the integral of the Dubois torsion is given. The formula looks like the heat kernel regularization of the formula for the Witten-Reshetikhin-Turaev invariant of the dou…
For a complex polynomial in two variables we study the morphism induced in homology by the embedding of an irregular fiber in a regular neighborhood of it. We give necessary and sufficient conditions for this morphism to be injective, surjective. Particularly this morphism is an isomorphism if and only if the correspon…
In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a-priori bounds, the topolog…
We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show 'polynomially weighted' regularity in exterior domains as well.
3-manifolds with similar completions have matching slopes and polynomials.
problem Matching slopes and polynomials in cusped hyperbolic 3-manifolds.
method Prove similarity of profinite completions leads to matching A-polynomials and boundary slopes. result Strongly detected boundary slopes match between manifolds with similar completions.
Characterizes values at infinity for real polynomial maps with 2D fibers.
problem Understanding atypical values at infinity for real polynomial maps.
method Characterization using indices of gradient vector fields on spheres.
result Analogous to two-variable case, but for maps with 2D fibers.
Polynomial invariant derived from birack labelling of knots.
problem Developing a polynomial invariant for a broader class of knot theories.
method Generalizing biquandle colouring to birack labelling, reducing to biquandle invariant.
result Polynomial invariant for a class of knot theories.
The paper studies twisted Alexander polynomials for knot groups in various extensions.
problem Understanding twisted Alexander polynomials in knot groups for different extensions.
method Developed mod p formula for twisted Alexander polynomials and studied central extensions.
result Established formulas for twisted Alexander polynomials in knot groups for various extensions.
GD outperforms ridge regression and SGD in linear regression problems.
problem Comparing the risks of GD, ridge regression, and SGD in linear regression problems.
method Instance-wise finite-sample risk analysis of GD, ridge regression, and SGD.
result GD outperforms ridge regression and is incomparable with SGD in some cases.
In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in \cite{ge…
Proves curvature of conference graphs and finds local matchings.
problem Proving precise values of curvature in conference graphs.
method Combining parameter relations and combinatorial approach.
result Existence of local perfect matchings in broader classes of graphs.
Study on Monge-Ampère equations with polynomial growth rates.
problem Analyzing solutions to Monge-Ampère equations with polynomial right-hand sides.
method Utilizing polynomial growth analysis to study regularity and growth rates of solutions.
result Translators for sub-affine-critical curvature flows are smooth and convex with specific growth rates.
NGRC shows numerical instabilities with short lags and high-degree polynomials.
problem Numerical instabilities in NGRC feature matrix.
method Combining numerical linear algebra and dynamical systems theory, we study feature matrix conditioning. We evaluate different numerical algorithms for solving the regularized least-squares problem.
result SVD-based training achieves accurate forecasts without regularization, preferable for short lags and high-degree polynomials.
Path regularization reveals convex optimization in deep ReLU networks.
problem Understanding the optimization landscape of deep neural networks.
method Introducing path regularization to make the training problem convex and sparsity-inducing.
result Path regularized parallel ReLU networks are a parsimonious convex model in high dimensions.
Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.
problem Calculating measures on symplectic groupoid quotients.
method Using Hamiltonian groupoid actions and proper moment maps.
result Duistermaat-Heckman measure is polynomial.
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.
Gradient descent implicitly follows regularization for general losses.
problem The implicit bias of gradient descent methods in machine learning.
method Empirical risk minimization over linear predictors with arbitrary convex, strictly decreasing losses.
result Gradient descent and regularization paths converge to the same direction for non-attained risks.
It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A_2 invariant, for tangled trivalent graph diagrams. In this paper, a polynomial for a mar…
New polynomial-time solutions found for training ReLU networks, mirroring Max-Cut complexity.
problem Training two-layer ReLU neural networks with weight decay regularization.
method Developed a convex formulation and randomized algorithm to find approximate global optimizers.
result First polynomial-time approximation guarantees and hardness of approximation results for regularized ReLU networks.
The Vol-Det Conjecture relates the volume and the determinant of a hyperbolic alternating link in S3. We use exact computations of Mahler measures of two-variable polynomials to prove the Vol-Det Conjecture for many infinite families of alternating links. We conjecture a new lower bound for the Mahler measure of cer…
Analytic functions on specific domains are characterized by their smoothness and composites with polynomial curves.
problem Characterizing real analytic functions on closed subanalytic domains.
method Analyzing functions defined on closed uniformly polynomially cuspidal sets in Rn using composites with polynomial curves. result Conditions for a function to be real analytic are effectively related to the regularity of the boundary of the domain.
This paper improves neural network learning by escaping the NTK regime and efficiently learning sparse polynomials.
problem Learning sparse polynomials efficiently using neural networks.
method Spectral analysis of NTK, identifying 'good' directions, and constructing a regularizer.
result Gradient descent on a two-layer neural network can learn sparse polynomials efficiently, improving over the NTK and QuadNTK.
Polynomially inscribe 6 concyclic points into any smooth curve.
problem Inscribing points into smooth curves using polynomials.
method Combines Fukaya-Irie theorem and Floer homology computation.
result Supports conjecture about polynomial inscriptions.
It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describi…
The coefficients of twisted Alexander polynomials of a knot induce regular functions of the SL2(C)-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal genus Seifert surface by Culler-Shalen theory. It implies a partial affirmative an…
In this work, we examine the process of Tropical Polynomial Division, a geometric method which seeks to emulate the division of regular polynomials, when applied to those of the max-plus semiring. This is done via the approximation of the Newton Polytope of the dividend polynomial by that of the divisor. This process i…
Study supports recovery of PDEs from noisy data using a specific regularization method.
problem Support recovery of PDEs from a single noisy trajectory.
method Applying ℓ1-regularized Pseudo-Least Squares model to a given data set.
result Support of ℓ1-c coefficients asymptotically converges to the true signed-support of the PDE.
We consider regular surfaces M that are given as the zeros of a polynomial function p:R3→R, where the gradient of p vanishes nowhere. We assume that M has non-zero mean curvature and prove that there exist only two examples of such surfaces, namely the sphere and the circular cylinder.
New knot polynomials distinguish knot orientations without using knot groups.
problem Distinguishing knots based on their orientations without relying on knot groups.
method Constructing combinatorial 1-cocycles on moduli spaces of knots and cables, using Gauss diagram formulas and local parameterization.
result Polynomial invariants that can distinguish knot orientations.
Paper develops an online learning algorithm for functional data models.
problem Recovering slope functions or predictors in functional data models.
method Online regularized learning algorithm in reproducing kernel Hilbert spaces with polynomially decaying step-size.
result Established fast convergence rates for estimation error without capacity assumption.