Large-Scale Learning with Fourier Features and Tensor Decompositions

• URL: http://arxiv.org/abs/2109.01545v1
• Date: Fri, 3 Sep 2021 14:12:53 GMT
• Title: Large-Scale Learning with Fourier Features and Tensor Decompositions
• Authors: Frederiek Wesel, Kim Batselier
• Abstract summary: We exploit the tensor product structure of deterministic Fourier features, which enables us to represent the model parameters as a low-rank tensor decomposition. We demonstrate by means of numerical experiments how our low-rank tensor approach obtains the same performance of the corresponding nonparametric model.
• Score: 3.6930948691311007
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Random Fourier features provide a way to tackle large-scale machine learning problems with kernel methods. Their slow Monte Carlo convergence rate has motivated the research of deterministic Fourier features whose approximation error decreases exponentially with the number of frequencies. However, due to their tensor product structure these methods suffer heavily from the curse of dimensionality, limiting their applicability to two or three-dimensional scenarios. In our approach we overcome said curse of dimensionality by exploiting the tensor product structure of deterministic Fourier features, which enables us to represent the model parameters as a low-rank tensor decomposition. We derive a monotonically converging block coordinate descent algorithm with linear complexity in both the sample size and the dimensionality of the inputs for a regularized squared loss function, allowing to learn a parsimonious model in decomposed form using deterministic Fourier features. We demonstrate by means of numerical experiments how our low-rank tensor approach obtains the same performance of the corresponding nonparametric model, consistently outperforming random Fourier features.

Related papers

• Quantized Fourier and Polynomial Features for more Expressive Tensor Network Models [9.18287948559108]
  We exploit the tensor structure present in the features by constraining the model weights to be an underparametrized tensor network. We show that, for the same number of model parameters, the resulting quantized models have a higher bound on the VC-dimension as opposed to their non-quantized counterparts.
  arXiv  Detail & Related papers  (2023-09-11T13:18:19Z)
• Learning Unnormalized Statistical Models via Compositional Optimization [73.30514599338407]
  Noise-contrastive estimation(NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. In this paper, we study it a direct approach for optimizing the negative log-likelihood of unnormalized models.
  arXiv  Detail & Related papers  (2023-06-13T01:18:16Z)
• Theory on variational high-dimensional tensor networks [2.0307382542339485]
  We investigate the emergent statistical properties of random high-dimensional-network states and the trainability of tensoral networks. We prove that variational high-dimensional networks suffer from barren plateaus for global loss functions. Our results pave a way for their future theoretical studies and practical applications.
  arXiv  Detail & Related papers  (2023-03-30T15:26:30Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery [52.21846313876592]
  Low-rank tensor function representation (LRTFR) can continuously represent data beyond meshgrid with infinite resolution. We develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization. Our method substantiates the superiority and versatility of our method as compared with state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-12-01T04:00:38Z)
• Error Analysis of Tensor-Train Cross Approximation [88.83467216606778]
  We provide accuracy guarantees in terms of the entire tensor for both exact and noisy measurements. Results are verified by numerical experiments, and may have important implications for the usefulness of cross approximations for high-order tensors.
  arXiv  Detail & Related papers  (2022-07-09T19:33:59Z)
• Factorized Fourier Neural Operators [77.47313102926017]
  The Factorized Fourier Neural Operator (F-FNO) is a learning-based method for simulating partial differential equations. We show that our model maintains an error rate of 2% while still running an order of magnitude faster than a numerical solver.
  arXiv  Detail & Related papers  (2021-11-27T03:34:13Z)
• Function Approximation via Sparse Random Features [23.325877475827337]
  This paper introduces the sparse random feature method that learns parsimonious random feature models utilizing techniques from compressive sensing. We show that the sparse random feature method outperforms shallow networks for well-structured functions and applications to scientific machine learning tasks.
  arXiv  Detail & Related papers  (2021-03-04T17:53:54Z)
• Low-rank Characteristic Tensor Density Estimation Part I: Foundations [38.05393186002834]
  We propose a novel approach that builds upon tensor factorization tools. In order to circumvent the curse of dimensionality, we introduce a low-rank model of this characteristic tensor. We demonstrate the very promising performance of the proposed method using several measured datasets.
  arXiv  Detail & Related papers  (2020-08-27T18:06:19Z)
• A Solution for Large Scale Nonlinear Regression with High Rank and Degree at Constant Memory Complexity via Latent Tensor Reconstruction [0.0]
  This paper proposes a novel method for learning highly nonlinear, multivariate functions from examples. Our method takes advantage of the property that continuous functions can be approximated by bys, which in turn are representable by tensors. For learning the models, we present an efficient-based algorithm that can be implemented in linear time.
  arXiv  Detail & Related papers  (2020-05-04T14:49:14Z)
• Semiparametric Nonlinear Bipartite Graph Representation Learning with Provable Guarantees [106.91654068632882]
  We consider the bipartite graph and formalize its representation learning problem as a statistical estimation problem of parameters in a semiparametric exponential family distribution. We show that the proposed objective is strongly convex in a neighborhood around the ground truth, so that a gradient descent-based method achieves linear convergence rate. Our estimator is robust to any model misspecification within the exponential family, which is validated in extensive experiments.
  arXiv  Detail & Related papers  (2020-03-02T16:40:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.