We construct a finitely presented group with non-quadratic Dehn function majorizable by a quadratic function on arbitrary long intervals.
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48 results for “quadratic functional”
Ricci solitons as critical points of quadratic curvature functionals
problem Einstein metrics and Ricci solitons as critical points of quadratic Riemannian functionals
method Study of Ricci solitons as critical points of a special quadratic curvature functional
result Ricci solitons are non-Einstein critical points of these functionals
Researchers found all special metrics in 4D for certain curvature functionals.
problem Identifying special metrics in 4D for quadratic curvature functionals.
method Determined all homogeneous metrics that are critical for quadratic curvature functionals.
result All homogeneous metrics in 4D for some quadratic curvature functionals have been identified.
Recently, deep learning has achieved huge successes in many important applications. In our previous studies, we proposed quadratic/second-order neurons and deep quadratic neural networks. In a quadratic neuron, the inner product of a vector of data and the corresponding weights in a conventional neuron is replaced with…
New rigidity results for critical metrics of a quadratic curvature functional.
problem Proving uniqueness of critical metrics for a specific curvature functional.
method Analyzing complete, possibly non-compact, critical metrics of the quadratic curvature functional.
result Critical metrics with finite energy are scalar flat (global minima) for dimensions n≥10.
We investigate classification results for general quadratic functions on torsion abelian groups. Unlike the previously studied situations, general quadratic functions are allowed to be inhomogeneous or degenerate. We study the discriminant construction which assigns, to an integral lattice with a distinguished characte…
Improved SVRG for quadratic functions achieves better performance and running times.
problem Minimizing quadratic functions with a specific type of Hessian matrix.
method Variant of SVRG algorithm for quadratic functions with improved analysis.
result Improved performance and running times for quadratic functions compared to state-of-the-art methods.
The paper studies special surfaces with a new type of support function.
problem Characterizing surfaces with a specific quadratic support function.
method Developed a Weierstrass type representation involving holomorphic functions.
result Classified surfaces of rotation with this new type of support function.
Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.
problem Characterizing the behavior of volume functions associated with quadratic differentials.
method Analyzing the Thurston volume of unit balls in measured lamination spaces.
result The volume function is not proper and is -integrable for any .
Paper optimizes estimation of quadratic functionals in nonparametric IV models.
problem Optimal estimation of a nonlinear functional in ill-posed inverse regression.
method Adaptive, minimax estimation using leave-one-out, sieve NPIV estimator with data-driven sieve dimension selection.
result Adaptive estimator achieves minimax optimal rate in various ill-posed cases.
Study on quadratic L-functions using hyperelliptic curves and homology.
problem Understanding moments of families of quadratic L-functions.
method Homological stability theorem and computations of homology.
result Confirmations of Conrey-Farmer-Keating-Rubinstein-Snaith predictions for large prime powers.
We consider a proximal operator given by a quadratic function subject to bound constraints and give an optimization algorithm using the alternating direction method of multipliers (ADMM). The algorithm is particularly efficient to solve a collection of proximal operators that share the same quadratic form, or if the qu…
In this paper we construct a distinguished Riemannian geometrization on the dual 1-jet space J^{1*}(T,M) for the multi-time quadratic Hamiltonian functions. Our geometrization includes a nonlinear connection N, a generalized Cartan canonical N-linear connection (together with its local d-torsions and d-curvatures), nat…
In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are always critical points. The purpose of this article is to show that, under some c…
The paper extends a variance gamma model to quadratic functions, reducing arbitrage and computational costs.
problem Creating an arbitrage-free interpolation for option pricing models.
method Generalizing the local variance gamma model to a piecewise quadratic local variance function.
result The quadratic model results in an arbitrage-free interpolation of class C3, reducing knots and computational cost.
We show how the space of complex spin structures of a closed oriented three-manifold embeds naturally into a space of quadratic functions associated to its linking pairing. Besides, we extend the Goussarov-Habiro theory of finite type invariants to the realm of compact oriented three-manifolds equipped with a complex s…
In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold of dimension on the space of Riemannian metrics on with unit volume. We study the stability of these functionals at the metric with constant sectional curvature as its critical point.
We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to simila…
This paper concerns a method of selecting a subset of features for a sequential logit model. Tanaka and Nakagawa (2014) proposed a mixed integer quadratic optimization formulation for solving the problem based on a quadratic approximation of the logistic loss function. However, since there is a significant gap between …
A new QHR model extends HR model with a quadratic variance function.
problem Modeling volatility with greater flexibility and stationarity.
method Introducing a quadratic variance function to the HR model, maintaining Markovian property.
result Stationary distribution of the QHR model is Pearson type IV.
The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that th…
Paper constructs multivalued harmonic functions on R^3 using twistor methods.
problem Constructing multivalued harmonic functions on R^3.
method Twistor methods to construct multivalued harmonic functions.
result Found a family of multivalued harmonic functions with branching sets as ellipses and quadratic growth at infinity.
We consider a class of continuous functions on that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in admits a linear pathwise quadratic variatio…
A Piecewise Lyapunov Analysis of Sub-quadratic SGD: Applications to Robust and Quantile Regressionstat.ML
New analysis improves SGD for robust and quantile regression with sub-quadratic convergence.
problem Improving SGD for robust and quantile regression with sub-quadratic convergence.
method Piecewise Lyapunov function for first-order differentiable functions.
result First geometrical convergence result for sub-quadratic SGD.
Paper classifies conic submanifolds in control systems.
problem Characterizing and classifying conic submanifolds in control systems.
method Feedback equivalence of control-affine and fully nonlinear systems.
result Complete description of non-degenerate conic submanifolds.
Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.
problem Solving special Lagrangian equations near infinity with specific conditions.
method Modified Kelvin transforms to characterize remainders in asymptotic expansions.
result Remainders in asymptotic expansions are characterized by a single smooth function in even dimensions and in odd dimensions.
We address the problem of which functions can arise as Dehn functions of Kähler groups. We explain why there are examples of Kähler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a Kähler group which has Dehn function bounded below by a cubic function a…
Given an oriented rational homology 3-sphere M, it is known how to associate to any Spin^c-structure σon M two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo 1 of the Reidemeister-Turaev torsion of (M,σ), while the other one can be defined using the intersectio…
Paper connects MoE and self-attention, proposing active-attention.
problem Improving efficiency and performance of self-attention mechanisms.
method Established connection between MoE and self-attention, analyzed quadratic gating functions, proposed active-attention mechanism.
result Active-attention outperforms standard self-attention in various tasks.
Quadratic differentials on Riemann surfaces uniquely determine foliations.
problem Understanding the relationship between quadratic differentials and foliations on Riemann surfaces.
method Extending prior results to arbitrary Fuchsian groups, analyzing measured foliations and their Dirichlet integrals.
result A finite-area holomorphic quadratic differential uniquely determines a horizontal foliation on a Riemann surface.
The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.
problem Approximating functionals of non-geometric rough paths.
method Extending rough paths with time and quadratic variation terms, proving uniform approximation.
result Linear functionals of extended signatures uniformly approximate continuous functionals.
A new method for exponentially weighted moving models using approximations.
problem Efficiently updating moving averages for time series data.
method Approximates EWMM using a fixed window and quadratic term, solving non-growing problems.
result Approximation produces estimates similar to exact EWMM.
Precise computations of Dehn functions for subgroups of free group products.
problem Computing precise Dehn functions for subgroups of direct products of free groups.
method Analyzing specific subgroups and using algebraic methods to compute Dehn functions.
result Quartic and quadratic Dehn functions for specific subgroups of free group products.
Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of càdlàg functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation …
Most of machine learning approaches have stemmed from the application of minimizing the mean squared distance principle, based on the computationally efficient quadratic optimization methods. However, when faced with high-dimensional and noisy data, the quadratic error functionals demonstrated many weaknesses including…
Geodesic flows with diagonalisable integrals are orthogonal.
problem Understanding geodesic flows with specific integrals.
method Analyzing quadratic integrals for geodesic flows.
result Diagonalisable integrals imply orthogonal separation of variables.
QENDy learns quadratic dynamics from nonlinear systems data.
problem Identifying governing equations of highly nonlinear dynamical systems.
method QENDy embeds nonlinear dynamics into a quadratic feature space, requiring trajectory data and preselected basis functions.
result QENDy accurately identifies quadratic dynamics and outperforms SINDy and deep learning methods.
In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the t…
In this paper, we prove some rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds, characterized by some point-wise inequalities. Moreover, we also provide a few rigidity results that involve the Weyl curvature, the trace-less Ricci cu…
In this paper, we study Riemannian functionals defined by -norms of Ricci curvature, scalar curvature, Weyl curvature, and Riemannian curvature. We try to understand stability of their critical points that are products of Einstein metrics. In particular, we prove that the product of a spherical space form and a co…
Given a smooth function f on R^n and a submanifold M, we prove that the set of diagonal quadratic forms q such that the restriction of f+q to M is Morse is a dense set (in the n-dimensional space of diagonal quadratic forms). The standard transversality argument seems not to work and we need a more refined approach.
This paper develops a new method for eliciting more flexible metrics, improving fairness and applicability.
problem Limited flexibility in existing metric elicitation strategies for reflecting user preferences.
method Develops a strategy for eliciting quadratic metrics based on predictive rates, requiring only relative preference feedback.
result Achieves near-optimal query complexity and broadens the use cases for metric elicitation.
The paper debiases mini-batch approximations in deep learning for more accurate optimization and uncertainty quantification.
problem Bias in mini-batch approximations distorts the shape of quadratic approximations used in deep learning.
method Developed and evaluated debiasing strategies for mini-batch approximations.
result Debiasing strategies improve the accuracy of second-order optimization and uncertainty quantification in deep learning.
We study closed -dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with unit volume. Under some additional integral conditions, we classify such manifol…
We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvat…
New algorithm achieves logarithmic regret for adversarial online control.
problem Online linear-quadratic control in systems with adversarial disturbances.
method Characterization of optimal offline control law, reduced to online learning with approximate advantage functions.
result First algorithm with logarithmic regret for arbitrary adversarial disturbance sequences.
Optimizes quadratic bandits with tight Hessian-dependent sample complexity bounds.
problem Understanding optimal sample complexity for quadratic functions.
method Introduces energy allocation and optimal energy spectrum to prove tight lower bounds. Solves for Hessian-independent optimal algorithm.
result Proves optimal Hessian-dependent sample complexities and existence of a universally optimal algorithm.
Automatic, Dynamic, and Nearly Optimal Learning Rate Specification by Local Quadratic Approximationstat.ML
A new method automatically and dynamically sets learning rates in deep learning.
problem Determining the appropriate learning rate in deep learning tasks is challenging and often subjective.
method Local Quadratic Approximation (LQA) to automatically and dynamically set learning rates.
result The proposed method leads to nearly optimal learning rates in a computationally efficient way.