Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L 2 L^2 L 2 for metrics with finitely differentiable tensor. Smoothness analysis of adversarial training reveals L ∞ L_\infty L ∞ constraints cause more non-smoothness.
problem Non-smoothness of adversarial training loss function.
method Analyzed the smoothness of adversarial training loss function using optimal attacks for model parameters.
result The L ∞ L_\infty L ∞ constraint causes more non-smoothness than L 2 L_2 L 2 constraint. Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain smoothed analysis estimates for elements in this class depending only on geometric inva…
Study on optimal rates for sequential probability assignment using smoothed analysis.
problem Optimal rates for sequential probability assignment under smoothed adversaries.
method General-purpose reduction from minimax rates to transductive learning, development of an efficient algorithm using MLE oracle.
result Optimal (logarithmic) fast rates for parametric and finite VC dimension classes, sublinear regret for general classes.
Regularized MFPCA smooths multivariate functional data for clearer patterns.
problem Challenges in controlling roughness of multivariate functional PCs.
method ReMFPCA incorporates a roughness penalty in a penalized framework to smooth PCs.
result Smoothed multivariate functional PCs reveal clearer patterns.
Generalizes smoothness conditions for optimization methods.
problem Optimization under non-uniform smoothness conditions.
method Develops a new analysis technique for bounding gradients.
result Obtains convergence rates for gradient descent and Nesterov's method.
Improved learning algorithms with privacy using smoothed analysis.
problem Designing robust and private learning algorithms.
method Smoothed analysis of adversarial and differentially private learning.
result Stronger regret and privacy error guarantees with smoothed adversaries.
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
Smoothed analysis is a powerful paradigm in overcoming worst-case intractability in unsupervised learning and high-dimensional data analysis. While polynomial time smoothed analysis guarantees have been obtained for worst-case intractable problems like tensor decompositions and learning mixtures of Gaussians, such guar…
Quantifies polynomial approximation rates for smooth functions under various distributions.
problem Approximating smooth functions with polynomials under different distributional constraints.
method Develops a quantitative analogue of Carleman's theorem using complex analysis.
result Establishes superexponential rates of approximation for certain function classes over general distributions.
Unified approach to discrete and smooth isoperimetric inequalities of arbitrary order.
problem Finding higher order isoperimetric inequalities for both discrete and smooth curves.
method Unified approach via Fourier analysis of linear operators.
result Unified upper and lower bounds for isoperimetric deficit in smooth curves.
Unified analysis of stochastic gradient methods for convex and smooth optimization.
problem Minimizing composite convex and smooth functions.
method Unified convergence analysis of various stochastic gradient methods.
result Unified convergence rates for a variety of methods including proximal SGD, variance reduced methods, quantization, and coordinate descent.
Proposes a neural network autoencoder for smoothing and representation learning of functional data.
problem Lack of sufficient nonlinear representations in existing methods for functional data analysis.
method Develops a neural network autoencoder architecture to process functional data directly, learning both smoothing and representation.
result Outperforms traditional methods in prediction, classification, and computational efficiency.
Study shows solutions to certain equations form smooth manifolds.
problem Understanding moduli spaces of solutions to non-linear elliptic equations.
method Analyzes moduli spaces as derived log smooth manifolds.
result Moduli spaces of solutions are derived log smooth manifolds.
The paper improves smoothed analysis for online problems with adaptive adversaries.
problem Online prediction, discrepancy minimization, and online optimization with adaptive adversaries.
method General technique to prove smoothed guarantees against adaptive adversaries, reducing to simpler oblivious adversaries.
result Strong smoothed guarantees for three online problems, matching or improving previous results.
SFM generates smooth functional data without exposing real data.
problem Challenges in statistical analysis of functional data.
method Copula framework and smooth flow construction.
result SFM produces high-quality synthetic functional data.
Geodesic X-ray transform proves injective for smooth one-forms on gas giant manifolds.
problem Injectivity of geodesic X-ray transform for one-forms on specific manifolds.
method Pestov identity and asymptotic analysis of short geodesics.
result Geodesic X-ray transform is solenoidally injective for smooth one-forms on gas giant manifolds.
Improved dynamic regret analysis for strongly convex and smooth functions.
problem Analyzing dynamic regret for online learning algorithms.
method Improved analysis of the Online Multiple Gradient Descent (OMGD) algorithm.
result Achieved a best-of-three-worlds guarantee for dynamic regret.
New method for faster convergence in non-convex optimization with unbounded smoothness.
problem Finding first-order stationary points of non-convex functions with unbounded smoothness.
method Developed a stopped analysis technique to prove convergence rates for ( L 0 , L 1 ) (L_0,L_1) ( L 0 , L 1 ) -smooth functions. result Achieved O ( p o l y log ( T ) T ) \mathcal{O}(\frac{\mathrm{poly}\log(T)}{\sqrt{T}}) O ( T poly l o g ( T ) ) convergence rates without uniform noise bounds. New analysis improves denoising of modulo signals on graphs.
problem Robustly unwrapping noisy modulo samples of smooth functions.
method Analyzing sphere-relaxation and unconstrained relaxation of a QCQP on a graph.
result Proves denoising of modulo observations w.r.t the ℓ 2 \ell_2 ℓ 2 norm in Gaussian noise. Decision tree learning heuristics fail even in smoothed analysis for complex targets.
problem Greedy decision tree learning heuristics fail for complex target functions in the smoothed analysis model.
method Construct counterexamples and analyze the behavior of heuristics in the smoothed setting and agnostic setting.
result Greedy decision tree learning heuristics can build trees of exponential depth before achieving high accuracy for certain complex target functions.
This paper improves coreset size via smoothed analysis.
problem Efficiently computing small subsets that approximate query errors.
method Smoothed analysis for approximate average error over queries.
result Deterministic and randomized algorithms for smaller coresets.
DNAMite creates interpretable, calibrated survival analysis models.
problem Limited interpretability in survival analysis models, especially for healthcare applications.
method Feature discretization and kernel smoothing in embedding module for flexible shape functions.
result DNAMite produces calibrated shape functions interpretable as contributions to cumulative incidence function.
Surrogate-based analysis of interactions via local effect smooths
problem Detecting and characterizing feature interactions in machine learning models
method Surrogate-based analysis using generalized additive models
result Empirical validation of effectiveness for pairwise interactions
We propose a new indicator for technical analysis. The indicator emphasizes maximums and minimums in price series with inherent smoothing and has a potential to be useful in both mechanical trading rules and chart pattern analysis.
The paper proposes a method to compute higher infinitesimals in numerical and symbolic analysis.
problem Computing higher-order derivatives with higher infinitesimals.
method Automatic differentiation in terms of C-infinity rings and Weil algebras.
result A unifying theoretical framework for multivariate higher-order derivatives.
Regularized variants of Principal Components Analysis, especially Sparse PCA and Functional PCA, are among the most useful tools for the analysis of complex high-dimensional data. Many examples of massive data, have both sparse and functional (smooth) aspects and may benefit from a regularization scheme that can captur…
We consider the problem of finding local minimizers in non-convex and non-smooth optimization. Under the assumption of strict saddle points, positive results have been derived for first-order methods. We present the first known results for the non-smooth case, which requires different analysis and a different algorithm…
Time series analysis is used to understand and predict dynamic processes, including evolving demands in business, weather, markets, and biological rhythms. Exponential smoothing is used in all these domains to obtain simple interpretable models of time series and to forecast future values. Despite its popularity, expon…
New framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.
problem Comparative analysis of regularization norms in ill-posed problems.
method Small noise analysis framework for Tikhonov and RKHS regularizations.
result Optimal convergence rates achieved with adaptive fractional RKHS, but hyper-parameters decay too fast.
The paper analyzes MACD using operator theory.
problem Understanding the mathematical foundation of MACD.
method Developed a functional-analytic framework interpreting MACD as a phase-corrected, smoothed derivative operator.
result MACD is structurally equivalent to a band-pass filter and can be expressed as a finite difference of delayed and doubly averaged signals.
New method generates private synthetic data with optimal utility for smooth queries.
problem Achieving strong utility guarantees for meaningful downstream analysis of sensitive datasets.
method Proposes a polynomial-time algorithm for generating ( ε , δ ) (\varepsilon,δ) ( ε , δ ) -differentially private synthetic data with minimax optimal error rates for smooth queries. result Achieves a minimax error rate of O k , d ( n − min { 1 , k d } ) O_{k,d}(n^{-\min \{1, \frac{k}{d}\}}) O k , d ( n − m i n { 1 , d k } ) for k k k -smooth queries, up to a log ( n ) \log(n) log ( n ) factor. FedAc accelerates Federated Averaging for distributed optimization.
problem Efficiently optimizing distributed machine learning models.
method Federated Accelerated Stochastic Gradient Descent (FedAc) using a potential-based perturbed iterate analysis.
result FedAc achieves faster convergence and lower communication costs than previous methods.
Smoothed analysis shows that many classes become learnable from positive-only samples.
problem Learning from positive-only samples is challenging due to negative results in worst-case settings.
method Smoothed analysis of positive-only learning, assuming samples from a reference distribution smooth with respect to the true distribution.
result All VC classes become learnable in the smoothed model with O ( V C / ε 2 ) O(VC/ε^2) O ( V C / ε 2 ) positive samples for ε ε ε classification error. New findings show learning deeper neural networks is hard even with Gaussian inputs and non-degenerate weights.
problem The computational complexity of learning neural networks, especially deeper ones.
method Smoothed analysis framework and local pseudorandom generators.
result Learning depth-3 ReLU networks under Gaussian input distribution is hard even if weight matrices are non-degenerate.
Develops PRPCA for smooth image recovery combining low-rank and smoothness.
problem Image matrix recovery under low-rank and smoothness assumptions.
method Projected Robust PCA (PRPCA) framework combining low-rank and smoothness.
result Explicit statistical guarantees for PRPCA, reducing matrix dimensionality.
Robust learning method combines kernel smoothing and robust optimization.
problem Certifying robustness against distribution shifts in machine learning models.
method Adapting integral operator using supremal convolution for robustness, leveraging optimal transport.
result The method provides theoretical guarantees for certified robustness and competitive performance.
Improved regret bounds for structured linear contextual bandits with Gaussian noise.
problem Optimizing bandit learning algorithms for structured contexts with Gaussian perturbations.
method Proposed simple greedy algorithms for structured linear contextual bandits with Gaussian noise.
result Unified regret analysis for structured parameters with geometric quantities as bounds.
Oracle-efficient algorithms for online learning with smoothed and hint-adversaries.
problem Online learning with beyond worst-case adversaries.
method Oracle-efficient algorithms for two settings: smoothed analysis and K K K -hint transductive learning. result Oracle-efficient regret bounds for learning real-valued and binary-valued functions.
Improved analysis and new algorithm for gradient-free optimization of smooth functions.
problem Minimization of highly smooth functions with noisy oracle information.
method Two zero-order projected gradient descent algorithms based on randomization over the ℓ 2 \ell_2 ℓ 2 and ℓ 1 \ell_1 ℓ 1 spheres, with improved analysis and theoretical guarantees. result Improved convergence rates and theoretical guarantees for various function classes.
New algorithm learns halfspaces over hypercube with random bit flips.
problem Agnostic learning of Boolean halfspaces over discrete domains is computationally hard.
method Smoothed analysis with random bit flips for discrete inputs.
result First efficient algorithm for smoothed agnostic learning of halfspaces over Boolean hypercube.
Theoretical analysis of cross-entropy loss functions and their robustness.
problem Guarantees for using cross-entropy as a surrogate loss function.
method Theoretical analysis of a broad family of loss functions, including cross-entropy.
result First H H H -consistency bounds for comp-sum losses and smooth adversarial comp-sum losses. New Morse theory for shapes at distances.
problem Understanding shapes at distances from a reference point.
method Defining Morse functions and using non-smooth analysis, geometric measure theory.
result Homotopy type changes at critical values, with one cell added per critical point.
In high dimensional sparse regression, pivotal estimators are estimators for which the optimal regularization parameter is independent of the noise level. The canonical pivotal estimator is the square-root Lasso, formulated along with its derivatives as a "non-smooth + non-smooth" optimization problem. Modern technique…
Unified framework for ensemble transport-based smoothing of non-Gaussian time series.
problem Bayesian time series re-analysis with non-Gaussian distributions.
method Measure transport approach to derive consistent prior-to-posterior transformations.
result General ensemble framework for transport-based smoothing of state-space models.
Estimates time-varying network connections using multi-stage smoothing.
problem Estimating edge probabilities of time-varying networks.
method Multi-stage smoothing: temporal local smoothing followed by node-domain smoothing.
result Captures both smooth temporal evolution and structural patterns in connectivity.
In this article, we mathematically study several GAN related topics, including Inception score, label smoothing, gradient vanishing and the -log(D(x)) alternative. --- An advanced version is included in arXiv:1703.02000 "Activation Maximization Generative Adversarial Nets". Please refer Section 6 in 1703.02000 for deta…
This paper analyzes the convergence of Federated Average under relaxed assumptions.
problem Lack of theoretical analysis for Federated Average under assumptions beyond smoothness.
method Relaxing assumptions of strong smoothness to semi-smoothness and semi-Lipschitz properties, and introducing a bound on the gradient.
result Provides a theoretical convergence study on Federated Learning under new assumptions.