Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities

• URL: http://arxiv.org/abs/2406.13036v2
• Date: Fri, 21 Jun 2024 12:09:38 GMT
• Title: Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities
• Authors: Matthew T. C. Li, Tiangang Cui, Fengyi Li, Youssef Marzouk, Olivier Zahm,
• Abstract summary: We introduce a method for identifying and approximating a target measure $pi$ as a perturbation of a given reference measure $mu$. Our main contribution unveils a connection between the emphdimensional logarithmic Sobolev inequality (LSI) and approximations with this ansatz.
• Score: 0.5592394503914488
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Identifying low-dimensional structure in high-dimensional probability measures is an essential pre-processing step for efficient sampling. We introduce a method for identifying and approximating a target measure $\pi$ as a perturbation of a given reference measure $\mu$ along a few significant directions of $\mathbb{R}^{d}$. The reference measure can be a Gaussian or a nonlinear transformation of a Gaussian, as commonly arising in generative modeling. Our method extends prior work on minimizing majorizations of the Kullback--Leibler divergence to identify optimal approximations within this class of measures. Our main contribution unveils a connection between the \emph{dimensional} logarithmic Sobolev inequality (LSI) and approximations with this ansatz. Specifically, when the target and reference are both Gaussian, we show that minimizing the dimensional LSI is equivalent to minimizing the KL divergence restricted to this ansatz. For general non-Gaussian measures, the dimensional LSI produces majorants that uniformly improve on previous majorants for gradient-based dimension reduction. We further demonstrate the applicability of this analysis to the squared Hellinger distance, where analogous reasoning shows that the dimensional Poincar\'e inequality offers improved bounds.

Related papers

• A Bayesian Approach Toward Robust Multidimensional Ellipsoid-Specific Fitting [0.0]
  This work presents a novel and effective method for fitting multidimensional ellipsoids to scattered data in the contamination of noise and outliers. We incorporate a uniform prior distribution to constrain the search for primitive parameters within an ellipsoidal domain. We apply it to a wide range of practical applications such as microscopy cell counting, 3D reconstruction, geometric shape approximation, and magnetometer calibration tasks.
  arXiv  Detail & Related papers  (2024-07-27T14:31:51Z)
• Mean-field variational inference with the TAP free energy: Geometric and statistical properties in linear models [20.311583934266903]
  We show that the landscape of the TAP free energy is strongly convex in an extensive neighborhood of a local minimizer. We prove analogous properties for a local minimizer of the TAP free energy reachable by and show posterior inference based on this minimizer remains correctly.
  arXiv  Detail & Related papers  (2023-11-14T17:35:01Z)
• Stochastic Mirror Descent for Large-Scale Sparse Recovery [13.500750042707407]
  We discuss an application of quadratic Approximation to statistical estimation of high-dimensional sparse parameters. We show that the proposed algorithm attains the optimal convergence of the estimation error under weak assumptions on the regressor distribution.
  arXiv  Detail & Related papers  (2022-10-23T23:23:23Z)
• Adaptive Zeroth-Order Optimisation of Nonconvex Composite Objectives [1.7640556247739623]
  We analyze algorithms for zeroth-order entropy composite objectives, focusing on dependence on dimensionality. This is achieved by exploiting low dimensional structure of the decision set using the mirror descent method with an estimation alike function. To improve the gradient, we replace the classic sampling method based on Rademacher and show that the mini-batch method copes with non-Eucli geometry.
  arXiv  Detail & Related papers  (2022-08-09T07:36:25Z)
• A Dimensionality Reduction Method for Finding Least Favorable Priors with a Focus on Bregman Divergence [108.28566246421742]
  This paper develops a dimensionality reduction method that allows us to move the optimization to a finite-dimensional setting with an explicit bound on the dimension. In order to make progress on the problem, we restrict ourselves to Bayesian risks induced by a relatively large class of loss functions, namely Bregman divergences.
  arXiv  Detail & Related papers  (2022-02-23T16:22:28Z)
• Differentiable Annealed Importance Sampling and the Perils of Gradient Noise [68.44523807580438]
  Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective. We propose a differentiable algorithm by abandoning Metropolis-Hastings steps, which further unlocks mini-batch computation.
  arXiv  Detail & Related papers  (2021-07-21T17:10:14Z)
• 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)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• The Generalized Lasso with Nonlinear Observations and Generative Priors [63.541900026673055]
  We make the assumption of sub-Gaussian measurements, which is satisfied by a wide range of measurement models. We show that our result can be extended to the uniform recovery guarantee under the assumption of a so-called local embedding property.
  arXiv  Detail & Related papers  (2020-06-22T16:43:35Z)
• Effective Dimension Adaptive Sketching Methods for Faster Regularized Least-Squares Optimization [56.05635751529922]
  We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform (SRHT)
  arXiv  Detail & Related papers  (2020-06-10T15:00:09Z)

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.