In Carnot groups, directional pliability allows curve extensions and approximations.
problem Existence of curve extensions and approximations in Carnot groups.
method Directional pliability in subsets of directions guarantees Whitney-type extensions and Lusin approximations.
result Every horizontal curve in the Engel group intersects a C1 curve in a set of positive measure. A Carnot group G admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve γ in G and ε>0, there is a C1 horizontal curve Γ such that Γ=γ and Γ′=γ′ outside a set of measure at most ε. We verify this property for free Carno…
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C1,1 functions. Strongly convex bodies can be approximated by smooth ones.
problem Approximating strongly convex bodies with smooth ones.
method Using C2 locally strongly convex bodies. result Smooth approximations of strongly convex bodies exist and can be controlled in terms of Hausdorff distance.
In this article we study the validity of the Whitney C1 extension property for horizontal curves in sub-Riemannian manifolds endowed with 1-jets that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a…
Develops calculus for tamed Dirichlet spaces using measure theory.
problem Defines calculus for measure spaces with Dirichlet forms.
method Introduces first and second order calculus on tamed Dirichlet spaces.
result Defines various geometric objects on tamed Dirichlet spaces.
The paper proves Rademacher's theorem for Heisenberg groups.
problem Proving Rademacher's theorem for Heisenberg groups.
method New definition of intrinsic Lipschitz graphs, extension and approximation theorems, use of Heisenberg currents and Rumin's complex.
result Rademacher's theorem for intrinsic Lipschitz graphs in Heisenberg groups.
Sharp lower bound found for integral varifolds' mean curvature.
problem Finding a sharp lower bound for the mean curvature integral of integral varifolds.
method Developed a new approach using integral varifolds and mean curvature.
result A sharp lower bound on the mean curvature integral with critical power for integral varifolds.
Convex functions and bodies can be approximated by smoother convex functions.
problem Approximating convex functions and bodies with smoother ones.
method Using properties of convex functions and bodies, constructing smoother approximations.
result Smooth approximations of convex functions and bodies exist for any given tolerance.
In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular …
The paper extends Frobenius' Theorem to non-involutive surfaces below C1,1 threshold.
problem Generalizing Frobenius' Theorem to surfaces with lower regularity.
method Combining fractional Sobolev estimates, Stokes-type theorem, and Lusin's Theorem.
result Sharp exponents for the tangency set's null measure.
The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.
problem Approximating Riemannian manifolds with polyhedral metrics.
method Conditions on curvature tensors for Lipschitz and local polyhedral approximations.
result Conditions are sufficient for local polyhedral approximations, conjectured to be sufficient for global approximations.
Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
Method approximates Riemannian barycenter on manifolds.
problem Computing the exact Riemannian barycenter is computationally expensive.
method Uses under- and over-approximations of Riemannian distance to compute an approximate barycenter.
result Approximation method is more efficient than exact methods and steepest descent.
We consider in this paper the optimal approximations of convex univariate functions with feed-forward Relu neural networks. We are interested in the following question: what is the minimal approximation error given the number of approximating linear pieces? We establish the necessary and sufficient conditions and uniqu…
Efficiently reduces tensor ranks using mean-field approximation.
problem Low-rank approximation of non-negative tensors.
method Mean-field approximation of tensor rank reduction.
result Our algorithm achieves faster and competitive tensor rank reduction.
Study approximates unknown function levels with queries.
problem Approximating unknown function levels through sequential queries.
method Introduce Bisect and Approximate algorithms to reduce to local function approximation.
result Rate-optimal sample complexity guarantees for H{ö}lder functions.
We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such …
Softmax attention approximates complex functions and subsumes many known universal approximators.
problem Universal approximation of continuous sequence-to-sequence functions.
method Interpolation-based analysis of attention's internal mechanism, showing its ability to approximate ReLU functions.
result Softmax attention is a universal approximator for continuous sequence-to-sequence functions.
Improved matrix approximation using randomized algorithms.
problem Finding better approximations of given matrices.
method Randomized algorithms to compute (HT) as an improved approximation. result Computed (HT) provides a better approximation than given F∗. Deviation inequalities for stochastic approximation methods.
problem Establishing bounds on the deviation of stochastic approximation methods.
method Martingale approximation method for separately Lipschitz functions.
result Established various deviation inequalities for stochastic approximation by averaging and minimization.
Neural approximate computing gains enormous energy-efficiency at the cost of tolerable quality-loss. A neural approximator can map the input data to output while a classifier determines whether the input data are safe to approximate with quality guarantee. However, existing works cannot maximize the invocation of the a…
Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmet…
We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matri…
Transformers use ReLUs to approximate softmax efficiently.
problem Analyzing resource usage in softmax transformer models.
method Translating ReLU approximation results to softmax attention mechanisms.
result Economic resource bounds for softmax attention mechanisms.
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…
Recently, variational approximations such as the mean field approximation have received much interest. We extend the standard mean field method by using an approximating distribution that factorises into cluster potentials. This includes undirected graphs, directed acyclic graphs and junction trees. We derive generaliz…
Adaptive approximations improve variational inference for complex models.
problem Efficiently approximate marginal distributions and partition functions in complex probabilistic models.
method Two classes of adaptive approximations that include Bethe, tree-reweighted, and convex free energies.
result Proposed approximations automatically adapt to a given model and outperform existing methods.
Non-negative L1-approximating polynomials for Gaussian distributions are proven for certain classes of sets.
problem Existence of non-negative L1-approximating polynomials for Gaussian distributions. method Proving the existence of degree-k non-negative polynomials that approximate indicator functions of sets with Gaussian surface area in L1-norm. result Proves the existence of non-negative L1-approximating polynomials for certain classes of sets with Gaussian surface area. Paper introduces new approximations for lognormal sums, matching comonotonicity and moments.
problem Approximating sums of lognormal random variables accurately.
method Introduces new approximations based on weighted distribution theory, emphasizing comonotonicity and moment matching.
result Approximations perform better than classical methods, especially in the right tail of the distribution.
Paper analyzes normal approximation for two-timescale stochastic algorithms, revealing interaction between fast and slow timescales.
problem Non-asymptotic bounds for accuracy of normal approximation in linear two-timescale stochastic approximation algorithms.
method Established bounds for normal approximation in terms of convex distance, focusing on last iterate and Polyak-Ruppert averaging.
result Normal approximation rate for the last iterate improves with increased timescale separation, while it decreases in the averaged setting.
One-pass algorithm finds small subset for ℓp subspace approximation with additive error.
problem Finding a small subset of data points for ℓp subspace approximation. method One-pass subset selection with additive approximation guarantee for p∈[1,∞). result First one-pass algorithm with additive error for ℓp subspace approximation. We are interested in approximation of a multivariate function f(x1,…,xd) by linear combinations of products u1(x1)⋯ud(xd) of univariate functions ui(xi), i=1,…,d. In the case d=2 it is a classical problem of bilinear approximation. In the case of approximation in the L2 space the bili…
A new method for efficient Gaussian process inference using sparse approximations.
problem Scalable and accurate inference for latent Gaussian processes.
method Variational approximation with sparse inverse Cholesky factors and double Kullback-Leibler minimization.
result The proposed method can achieve highly accurate approximations with polylogarithmic time complexity.
In this paper, we propose a low-rank approximation method based on discrete least-squares for the approximation of a multivariate function from random, noisy-free observations. Sparsity inducing regularization techniques are used within classical algorithms for low-rank approximation in order to exploit the possible sp…
Neural network based approximate computing is a universal architecture promising to gain tremendous energy-efficiency for many error resilient applications. To guarantee the approximation quality, existing works deploy two neural networks (NNs), e.g., an approximator and a predictor. The approximator provides the appro…
The paper approximates supply curves using a one-step basis method.
problem Computing supply curves accurately and efficiently.
method Derives L2 approximation expression and proposes node selection procedure.
result Illustrates the approach with European electricity market bid curves.
The paper defines a new concept of approximability for Lagrangian submanifolds.
problem Understanding the approximability of Lagrangian submanifolds.
method Introducing a new notion of categorical approximability for metric spaces, showing it applies to specific types of Lagrangian submanifolds.
result Examples of Lagrangian submanifolds are found that are approximable but not precompact.
Boosting Nyström improves accuracy of matrix approximations.
problem Generating low-rank approximations of large matrices efficiently.
method Iteratively generate multiple weak Nyström approximations, combine them to form a strong approximation.
result Boosting Nyström yields more efficient and accurate low-rank approximations.
High-probability bound for distributed stochastic approximation tracking error.
problem Analyzing the convergence of distributed stochastic approximation schemes.
method Analysis using ODE approach to stochastic approximation.
result High probability bound for tracking error between iterates and limiting differential equation.
Nyström KPCA balances computational efficiency and statistical accuracy.
problem Computational burden in large sample situations for kernel methods.
method Theoretical analysis of Nyström approximate kernel principal component analysis (KPCA).
result Nyström approximate KPCA matches statistical performance of non-approximate KPCA while being computationally beneficial.
Deep ReLU networks can approximate smooth functions nearly optimally.
problem Approximating smooth functions with deep neural networks.
method Using Taylor expansions and deep ReLU network approximations, the paper establishes optimal approximation error bounds.
result Deep ReLU networks of width and depth O(NlnN) and O(LlnL) can approximate f∈Cs([0,1]d) with an error O(∥f∥Cs([0,1]d)N−2s/dL−2s/d). Improves Laplace approximation for Bayesian inference on Riemannian manifolds.
problem Inaccurate Gaussian approximations for complex targets and finite-data posteriors.
method Develops alternative variants of the Laplace approximation using a Riemannian metric.
result Exact approximations at the limit of infinite data, improving practical performance.
We approximate derivatives of functions on manifolds by embedding them and applying vector-valued operators.
problem Derivatives of manifold-valued functions are harder to approximate than vector-valued functions.
method Embed the manifold into a higher space, approximate the derivative of the vector-valued function, and project back.
result We provide error bounds for the approximation of manifold-valued function derivatives.
We discuss Bayesian methods for learning Bayesian networks when data sets are incomplete. In particular, we examine asymptotic approximations for the marginal likelihood of incomplete data given a Bayesian network. We consider the Laplace approximation and the less accurate but more efficient BIC/MDL approximation. We …
Proposes efficient Gaussian approximations for non-Gaussian likelihoods.
problem Computational challenges in learning and inference with non-Gaussian likelihoods.
method Variational inference and moment matching in transformed bases.
result Good approximation quality for binary and multiclass classification.
Gradient descent trains shallow neural networks to approximate functions in 1D.
problem Approximating functions in 1D with shallow neural networks trained by gradient descent.
method Gradient descent optimization of non-convex weight space for finite width networks in 1D.
result Gradient descent can approximate functions in 1D with a minimal number of weights, balancing practical performance and theoretical capabilities.
Method uses DNNs to approximate functions with specific asymptotic behavior.
problem Approximating functions with given asymptotic behavior.
method Specifically constructed terms combined with unconstrained DNN.
result Enforcing asymptotic behavior leads to better approximation and faster convergence.