Sub-Gaussian Matrices on Sets: Optimal Tail Dependence and Applications

• URL: http://arxiv.org/abs/2001.10631v2
• Date: Wed, 20 Jan 2021 21:30:18 GMT
• Title: Sub-Gaussian Matrices on Sets: Optimal Tail Dependence and Applications
• Authors: Halyun Jeong, Xiaowei Li, Yaniv Plan, \"Ozg\"ur Y{\i}lmaz
• Abstract summary: We study when a sub-Gaussian matrix can become a near isometry on a set. We show that previous best known dependence on the sub-Gaussian norm was sub-optimal. We also develop a new Bernstein type inequality for sub-exponential random variables, and a new Hanson-Wright inequality for quadratic forms of sub-Gaussian random variables.
• Score: 6.034622792544481
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Random linear mappings are widely used in modern signal processing, compressed sensing and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful information. This is done by approximately preserving the distances between data points, which are assumed to belong to $\mathbb{R}^n$. Thus, the performance of these mappings is usually captured by how close they are to an isometry on the data. Gaussian linear mappings have been the object of much study, while the sub-Gaussian settings is not yet fully understood. In the latter case, the performance depends on the sub-Gaussian norm of the rows. In many applications, e.g., compressed sensing, this norm may be large, or even growing with dimension, and thus it is important to characterize this dependence. We study when a sub-Gaussian matrix can become a near isometry on a set, show that previous best known dependence on the sub-Gaussian norm was sub-optimal, and present the optimal dependence. Our result not only answers a remaining question posed by Liaw, Mehrabian, Plan and Vershynin in 2017, but also generalizes their work. We also develop a new Bernstein type inequality for sub-exponential random variables, and a new Hanson-Wright inequality for quadratic forms of sub-Gaussian random variables, in both cases improving the bounds in the sub-Gaussian regime under moment constraints. Finally, we illustrate popular applications such as Johnson-Lindenstrauss embeddings, null space property for 0-1 matrices, randomized sketches and blind demodulation, whose theoretical guarantees can be improved by our results (in the sub-Gaussian case).

Related papers

• Highly Adaptive Ridge [84.38107748875144]
  We propose a regression method that achieves a $n-2/3$ dimension-free L2 convergence rate in the class of right-continuous functions with square-integrable sectional derivatives. Har is exactly kernel ridge regression with a specific data-adaptive kernel based on a saturated zero-order tensor-product spline basis expansion. We demonstrate empirical performance better than state-of-the-art algorithms for small datasets in particular.
  arXiv  Detail & Related papers  (2024-10-03T17:06:06Z)
• Learning Sparse High-Dimensional Matrix-Valued Graphical Models From Dependent Data [12.94486861344922]
  We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional, stationary matrix- Gaussian time series. We consider a sparse-based formulation of the problem with a Kronecker-decomposable power spectral density (PSD) We illustrate our approach using numerical examples utilizing both synthetic and real data.
  arXiv  Detail & Related papers  (2024-04-29T19:32:50Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Implicit Manifold Gaussian Process Regression [49.0787777751317]
  Gaussian process regression is widely used to provide well-calibrated uncertainty estimates. It struggles with high-dimensional data because of the implicit low-dimensional manifold upon which the data actually lies. In this paper we propose a technique capable of inferring implicit structure directly from data (labeled and unlabeled) in a fully differentiable way.
  arXiv  Detail & Related papers  (2023-10-30T09:52:48Z)
• Random Smoothing Regularization in Kernel Gradient Descent Learning [24.383121157277007]
  We present a framework for random smoothing regularization that can adaptively learn a wide range of ground truth functions belonging to the classical Sobolev spaces. Our estimator can adapt to the structural assumptions of the underlying data and avoid the curse of dimensionality.
  arXiv  Detail & Related papers  (2023-05-05T13:37:34Z)
• Gaussian Universality of Linear Classifiers with Random Labels in High-Dimension [24.503842578208268]
  We prove that data coming from a range of generative models in high-dimensions have the same minimum training loss as Gaussian data with corresponding data covariance. In particular, our theorem covers data created by an arbitrary mixture of homogeneous Gaussian clouds, as well as multi-modal generative neural networks.
  arXiv  Detail & Related papers  (2022-05-26T12:25:24Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• Sparse Gaussian Processes with Spherical Harmonic Features [14.72311048788194]
  We introduce a new class of inter-domain variational Gaussian processes (GP) Our inference scheme is comparable to variational Fourier features, but it does not suffer from the curse of dimensionality. Our experiments show that our model is able to fit a regression model for a dataset with 6 million entries two orders of magnitude faster.
  arXiv  Detail & Related papers  (2020-06-30T10:19:32Z)
• Robust Compressed Sensing using Generative Models [98.64228459705859]
  In this paper we propose an algorithm inspired by the Median-of-Means (MOM) Our algorithm guarantees recovery for heavy-tailed data, even in the presence of outliers.
  arXiv  Detail & Related papers  (2020-06-16T19:07:41Z)
• Optimal Iterative Sketching with the Subsampled Randomized Hadamard Transform [64.90148466525754]
  We study the performance of iterative sketching for least-squares problems. We show that the convergence rate for Haar and randomized Hadamard matrices are identical, andally improve upon random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.
  arXiv  Detail & Related papers  (2020-02-03T16:17:50Z)

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.