Scalable kernel methods for large datasets using Fourier representations and NUFFT.
problem Cubic complexity in kernel methods limits their use on large-scale datasets.
method Fourier representation of kernels combined with NUFFT for O(n log n) complexity.
result Achieves minimax convergence rates and processes up to tens of billions of samples.
Enhances Fourier estimator performance for asynchronous event-data.
problem Improving correlation and covariance estimation on event-data.
method Implement and test NUFFT methods with different averaging kernels.
result Demonstrates improved performance and relationship between averaging scales.
A new FFT method for Heston model option pricing with explicit error bounds.
problem Efficiently pricing European options in the Heston model with high accuracy.
method Convolution-FFT method leveraging a continuously differentiable joint characteristic function.
result Explicit error bounds for FFT-based convolution method in Heston model.
Deep learning has delivered its powerfulness in many application domains, especially in image and speech recognition. As the backbone of deep learning, deep neural networks (DNNs) consist of multiple layers of various types with hundreds to thousands of neurons. Embedded platforms are now becoming essential for deep le…
Paper extends FFT-based differential privacy method to heterogeneous compositions.
problem Computing accurate differential privacy guarantees for mixed mechanisms.
method Uses Fast Fourier Transform (FFT) for error analysis and parameter selection.
result Provides tighter bounds for heterogeneous compositions compared to homogeneous cases.
A new algorithm computes Fourier coefficients for a specified range efficiently.
problem Inefficiency in FFT due to fixed output size for all applications.
method Fast Partial Fourier Transform (PFT) that allows specifying the range of Fourier coefficients to compute.
result PFT achieves significant speedup over state-of-the-art FFT algorithms for small output sizes.
In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stol…
We discuss various analytic and numerical methods that have been used to get option prices within a framework of the VG model. We show that some popular methods, for instance, Carr-Madan's FFT method could blow up for certain values of the model parameters even for an European vanilla option. Alternative methods - one …
The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.
problem Finding thin Hitchin representations in non-uniform lattices of Lie groups.
method Arithmetic methods to construct thin Hitchin representations.
result Infinitely many orbits of thin Hitchin representations in non-uniform lattices.
New approach to adversarial robustness with non-uniform perturbations.
problem Real-world adversaries craft adversarial examples with non-uniform perturbations.
method Proposes non-uniform perturbations based on feature dependencies and data distribution.
result Shows improved robustness to real-world attacks compared to uniform perturbations.
This work studies the robustness certification problem of neural network models, which aims to find certified adversary-free regions as large as possible around data points. In contrast to the existing approaches that seek regions bounded uniformly along all input features, we consider non-uniform bounds and use it to …
New method upsamples sparse, non-uniform point clouds more accurately.
problem Suboptimal results from existing point cloud upsampling methods.
method Imposes manifold distribution constraints using Gaussian functions.
result Generates higher-quality, more uniformly distributed dense point clouds.
DAFNO learns surrogates for complex systems on irregular geometries.
problem Learning accurate surrogates for complex physical systems on irregular geometries.
method DAFNO incorporates a smoothed characteristic function in the integral layer architecture of FNOs, leveraging FFT for rapid computations.
result DAFNO achieves state-of-the-art accuracy on material modeling and airfoil simulation datasets.
New approach finds minima of geodesic lengths for non-uniform fillings.
problem Finding minima of geodesic length functions for non-uniform fillings.
method Elementary optimization for 4-regular topological fillings, analysis of fat graphs and optimization techniques.
result Minima of geodesic length functions are found to be at triangle surfaces in both analyzed classes of non-uniform fillings.
New method uses graphene transistors for efficient non-uniform random number generation.
problem Generating non-uniform random variates efficiently.
method GFET-based hardware non-uniform random number generator.
result Demonstrated speedup of Monte Carlo integration by up to 2x.
Efficiently accelerates attention calculation for Transformers with relative positional encoding.
problem Quadratic complexity of attention in long sequences.
method Kernelized attention with Fast Fourier Transform (FFT) for RPE.
result Achieves O(n log n) time complexity, mitigates training instability, and outperforms other models.
Study shows how non-uniform scaling affects persistence diagrams.
problem Stability of persistence diagrams under non-uniform scaling.
method Explicit bounds on bottleneck distance derived for Euclidean scaling.
result Explicit bounds on the stability of persistence diagrams under non-uniform scaling.
Unified framework for non-uniform materials evolving over time.
problem Dealing with non-uniform materials evolving over time.
method Constructing a material groupoid and material distribution.
result Unified framework for general non-uniform evolution materials.
An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional Value at Risk (CVaR) of compound distributions using characteristic functions. A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation …
New rigidity theorem for product of lattices.
problem Understanding quasi-isometry of product lattices.
method Demonstrated rigidity for product of non-uniform rank one lattice and nilpotent lattice.
result Any quasi-isometric group is an extension of a non-uniform rank one lattice by a nilpotent lattice.
Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
problem Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
method Proving non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
result Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
We apply stochastic average gradient (SAG) algorithms for training conditional random fields (CRFs). We describe a practical implementation that uses structure in the CRF gradient to reduce the memory requirement of this linearly-convergent stochastic gradient method, propose a non-uniform sampling scheme that substant…
We compare the CPU effort and pricing biases of seven Fourier-based implementations. Our analyses show that truncation and discretization errors significantly increase as we move away from the Black-Scholes-Merton framework. We rank the speed and accuracy of the competing choices, showing which methods require smaller …
A new FFT-based method for fast rigid alignment of 2D closed curves.
problem Rigid alignment of 2D closed curves with application to shape analysis.
method FFT-based algorithm for optimal rigid alignment of closed curves with O(N log N) complexity.
result Order of magnitude speed-up in curve alignment compared to previous methods.
We study primal-dual type stochastic optimization algorithms with non-uniform sampling. Our main theoretical contribution in this paper is to present a convergence analysis of Stochastic Primal Dual Coordinate (SPDC) Method with arbitrary sampling. Based on this theoretical framework, we propose Optimality Violation-ba…
The paper prices energy spread options using a complex stochastic model.
problem Pricing energy spread options with specific stochastic dynamics.
method Uses an exponential Ornstein-Uhlenbeck process driven by variance gamma processes, applying the Esscher transform and FFT method.
result Derives an analytical formula for pricing forwards and spread options.
New method detects communities in complex hypergraphs, matching theoretical limits.
problem Detecting communities in non-uniform hypergraphs with varying hyperedge sizes.
method Developed a spectral theory for weighted non-backtracking operators on non-uniform hypergraphs.
result Achieved the Kesten-Stigum bound for weak recovery in a general class of non-uniform HSBMs.
We present a novel method for neural network quantization that emulates a non-uniform k-quantile quantizer, which adapts to the distribution of the quantized parameters. Our approach provides a novel alternative to the existing uniform quantization techniques for neural networks. We suggest to compare the results as …
Sharp threshold for exact recovery in non-uniform hypergraph stochastic block model.
problem Community detection in random hypergraphs with non-uniform hyperedge probabilities.
method Sharp threshold established; two efficient algorithms for exact recovery.
result Sharp threshold for exact recovery; information-theoretic lower bound on misclassification.
We prove that if G is a non-uniform lattice in a rank-one semi-simple Lie group $\ne Isom(\H^2_\R)$ then G is quasi-isometrically co-Hopf. This means that every quasi-isometric embedding G→G is coarsely onto and thus is a quasi-isometry.
Improved sampling accuracy in SG-MCMC methods via non-uniform gradient subsampling.
problem Computational inefficiency and sampling error in stochastic gradient MCMC methods.
method Proposes a non-uniform subsampling scheme to reduce sampling error in EWSG, a variant of SG-MCMC.
result EWSG reduces sampling error compared to uniform subsampling, improving accuracy without sacrificing convergence speed.
A groupoid Ω(B) called material groupoid is naturally associated to any simple body B. The material distribution is introduced due to the (possible) lack of differentiability of the material groupoid. Thus, the inclusion of these new objects in the theory of material bodies opens th…
In this note, we study deformations of a non-uniform real hyperbolic lattice in quaternionic hyperbolic spaces. Specially we show that the representations of the fundamental group of the figure eight knot complement into PU(2,1) cannot be deformed in PSp(2,1) out of PU(2,1) up to conjugacy.
We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are di…
This work introduces 'Artificial Entanglement' to understand LLMs' fine-tuning effectiveness.
problem Understanding the effectiveness of parameter-efficient fine-tuning methods for large language models.
method Adopting a quantum-information-inspired perspective, the study measures 'Artificial Entanglement' in neural networks.
result LoRA and FFT induce distinct internal entanglement signatures but not external ones, suggesting a 'no-hair' property.
A new FFT-based method simplifies causal structure recovery for linear dynamical systems.
problem Efficiently identifying dynamic causal effects from time-series data.
method FFT-based approach to reduce computational complexity to O(Tn3logN). result Significant computational advantage for graph reconstruction.
WideBNet learns inverse scattering from wide-band data efficiently and stably.
problem Learning the inverse scattering map from wide-band scattering data.
method Combines butterfly factorization, FFT, and deep learning.
result WideBNet requires fewer training points and has stable training dynamics.
The paper improves privacy accounting for discrete-valued mechanisms and the subsampled Gaussian mechanism.
problem Improving the accuracy and efficiency of differential privacy accounting for discrete outputs.
method Uses fast Fourier transform (FFT) for rigorous error analysis and accounting of privacy loss.
result Provides strict lower and upper bounds for (ε,δ)-values, demonstrating up to 75% reduction in noise variance. We study the effectiveness of non-uniform randomized feature selection in decision tree classification. We experimentally evaluate two feature selection methodologies, based on information extracted from the provided dataset: (i) \emph{leverage scores-based} and (ii) \emph{norm-based} feature selection. Experimenta…
Improved matrix completion for non-uniformly sampled data.
problem Estimating unobserved entries in a matrix with varying sampling probabilities.
method Developed entry-specific bounds for low-rank matrix completion under structured non-uniform sampling.
result Error bounds for each entry match minimax lower bounds under certain conditions.
New loss function equivalence reveals PER's uniform sampling can be improved.
problem Improving Prioritized Experience Replay (PER) for better learning efficiency.
method Transforming non-uniformly sampled data loss functions into uniformly sampled ones.
result Some environments can replace PER with a new loss function without performance loss.
Convolutional Neural Networks (CNN) has become more popular choice for various tasks such as computer vision, speech recognition and natural language processing. Thanks to their large computational capability and throughput, GPUs ,which are not power efficient and therefore does not suit low power systems such as mobil…
Proof shows volumes of certain geometric representations are always integers.
problem Integrality of volumes of specific geometric representations.
method Elementary, combinatorial-geometrical proof.
result Volumes of representations are integers when n≥2. Two algorithms converge to dictionary learning with geometric rate for non-uniform data.
problem Dictionary learning for non-uniform data models.
method Derivation of convergence conditions for MOD and ODL.
result Both algorithms converge to the generating dictionary with geometric rate under certain conditions.
Spectral algorithm recovers community structure in sparse hypergraphs.
problem Community detection in sparse random hypergraphs with community structure and higher-order interactions.
method Spectral algorithm with three steps: hyperedge selection, spectral partition, and correction/merging.
result Weak consistency achieved for weak signal-to-noise ratio.
The paper proves Zimmer's conjecture for non-uniform lattices by controlling mass escape and Lyapunov exponents.
problem Proving Zimmer's conjecture for non-uniform lattices in higher-rank semisimple Lie groups.
method Establishes finiteness of low-dimensional actions, introduces novel techniques to control mass escape and Lyapunov exponents.
result Proves Zimmer's conjecture for many non-uniform lattices, improving previous results.
New simulation technique speeds up Lévy-driven OU process pricing.
problem Inefficient Monte Carlo simulations of Lévy-driven OU processes.
method Numerical inversion of characteristic function combined with FFT for fast and accurate simulations.
result The proposed technique is at least one order of magnitude faster than existing methods.
The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.
problem The existence of thin surface subgroups in non-uniform arithmetic lattices.
method Analyzes arithmetic hyperbolic manifolds and their fundamental groups.
result Fundamental groups of non-compact arithmetic hyperbolic manifolds contain thin surface subgroups.