Tight Bounds on Jensen's Gap: Novel Approach with Applications in Generative Modeling

• URL: http://arxiv.org/abs/2502.03988v1
• Date: Thu, 06 Feb 2025 11:44:31 GMT
• Title: Tight Bounds on Jensen's Gap: Novel Approach with Applications in Generative Modeling
• Authors: Marcin Mazur, Piotr Kościelniak, Łukasz Struski,
• Abstract summary: We provide a novel technique for finding lower and upper bounds on Jensen's gap. By studying in detail the case of the logarithmic function and the log-normal distribution, we explore a method for tightly estimating the log-likelihood of generative models trained on real-world datasets.
• Score: 0.5325390073522079
• License:
• Abstract: Among various mathematical tools of particular interest are those that provide a common basis for researchers in different scientific fields. One of them is Jensen's inequality, which states that the expectation of a convex function is greater than or equal to the function evaluated at the expectation. The resulting difference, known as Jensen's gap, became the subject of investigation by both the statistical and machine learning communities. Among many related topics, finding lower and upper bounds on Jensen's gap (under different assumptions on the underlying function and distribution) has recently become a problem of particular interest. In our paper, we take another step in this direction by providing a novel general and mathematically rigorous technique, motivated by the recent results of Struski et al. (2023). In addition, by studying in detail the case of the logarithmic function and the log-normal distribution, we explore a method for tightly estimating the log-likelihood of generative models trained on real-world datasets. Furthermore, we present both analytical and experimental arguments in support of the superiority of our approach in comparison to existing state-of-the-art solutions, contingent upon fulfillment of the criteria set forth by theoretical studies and corresponding experiments on synthetic data.

Related papers

• Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory [87.00653989457834]
  Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. Despite the empirical success, theory of conditional diffusion models is largely missing. This paper bridges the gap by presenting a sharp statistical theory of distribution estimation using conditional diffusion models.
  arXiv  Detail & Related papers  (2024-03-18T17:08:24Z)
• Seeing Unseen: Discover Novel Biomedical Concepts via Geometry-Constrained Probabilistic Modeling [53.7117640028211]
  We present a geometry-constrained probabilistic modeling treatment to resolve the identified issues. We incorporate a suite of critical geometric properties to impose proper constraints on the layout of constructed embedding space. A spectral graph-theoretic method is devised to estimate the number of potential novel classes.
  arXiv  Detail & Related papers  (2024-03-02T00:56:05Z)
• An analysis of Universal Differential Equations for data-driven discovery of Ordinary Differential Equations [7.48176340790825]
  We make a contribution by testing the UDE framework in the context of Ordinary Differential Equations (ODEs) discovery. We highlight some of the issues arising when combining data-driven approaches and numerical solvers. We believe that our analysis represents a significant contribution in investigating the capabilities and limitations of Physics-informed Machine Learning frameworks.
  arXiv  Detail & Related papers  (2023-06-17T12:26:50Z)
• On the Joint Interaction of Models, Data, and Features [82.60073661644435]
  We introduce a new tool, the interaction tensor, for empirically analyzing the interaction between data and model through features. Based on these observations, we propose a conceptual framework for feature learning. Under this framework, the expected accuracy for a single hypothesis and agreement for a pair of hypotheses can both be derived in closed-form.
  arXiv  Detail & Related papers  (2023-06-07T21:35:26Z)
• Demystifying Disagreement-on-the-Line in High Dimensions [34.103373453782744]
  We develop a theoretical foundation for analyzing disagreement in high-dimensional random features regression. Experiments on CIFAR-10-C, Tiny ImageNet-C, and Camelyon17 are consistent with our theory and support the universality of the theoretical findings.
  arXiv  Detail & Related papers  (2023-01-31T02:31:18Z)
• A Unifying Framework for Some Directed Distances in Statistics [0.0]
  Density-based directed distances -- particularly known as divergences -- are widely used in statistics. We provide a general framework which covers in particular both the density-based and distribution-function-based divergence approaches. We deduce new concepts of dependence between random variables, as alternatives to the celebrated mutual information.
  arXiv  Detail & Related papers  (2022-03-02T04:24:13Z)
• Loss function based second-order Jensen inequality and its application to particle variational inference [112.58907653042317]
  Particle variational inference (PVI) uses an ensemble of models as an empirical approximation for the posterior distribution. PVI iteratively updates each model with a repulsion force to ensure the diversity of the optimized models. We derive a novel generalization error bound and show that it can be reduced by enhancing the diversity of models.
  arXiv  Detail & Related papers  (2021-06-09T12:13:51Z)
• Marginal likelihood computation for model selection and hypothesis testing: an extensive review [66.37504201165159]
  This article provides a comprehensive study of the state-of-the-art of the topic. We highlight limitations, benefits, connections and differences among the different techniques. Problems and possible solutions with the use of improper priors are also described.
  arXiv  Detail & Related papers  (2020-05-17T18:31:58Z)
• A Survey on Causal Inference [64.45536158710014]
  Causal inference is a critical research topic across many domains, such as statistics, computer science, education, public policy and economics. Various causal effect estimation methods for observational data have sprung up.
  arXiv  Detail & Related papers  (2020-02-05T21:35:29Z)

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.