Algebraic and Analytic Approaches for Parameter Learning in Mixture Models

• URL: http://arxiv.org/abs/2001.06776v1
• Date: Sun, 19 Jan 2020 05:10:56 GMT
• Title: Algebraic and Analytic Approaches for Parameter Learning in Mixture Models
• Authors: Akshay Krishnamurthy, Arya Mazumdar, Andrew McGregor, Soumyabrata Pal
• Abstract summary: We present two different approaches for parameter learning in several mixture models in one dimension. For some of these distributions, our results represent the first guarantees for parameter estimation.
• Score: 66.96778152993858
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present two different approaches for parameter learning in several mixture models in one dimension. Our first approach uses complex-analytic methods and applies to Gaussian mixtures with shared variance, binomial mixtures with shared success probability, and Poisson mixtures, among others. An example result is that $\exp(O(N^{1/3}))$ samples suffice to exactly learn a mixture of $k<N$ Poisson distributions, each with integral rate parameters bounded by $N$. Our second approach uses algebraic and combinatorial tools and applies to binomial mixtures with shared trial parameter $N$ and differing success parameters, as well as to mixtures of geometric distributions. Again, as an example, for binomial mixtures with $k$ components and success parameters discretized to resolution $\epsilon$, $O(k^2(N/\epsilon)^{8/\sqrt{\epsilon}})$ samples suffice to exactly recover the parameters. For some of these distributions, our results represent the first guarantees for parameter estimation.

Related papers

• Learning general Gaussian mixtures with efficient score matching [16.06356123715737]
  We study the problem of learning mixtures of $k$ Gaussians in $d$ dimensions. We make no separation assumptions on the underlying mixture components. We give an algorithm that draws $dmathrmpoly(k/varepsilon)$ samples from the target mixture, runs in sample-polynomial time, and constructs a sampler.
  arXiv  Detail & Related papers  (2024-04-29T17:30:36Z)
• Stochastic Approximation Approaches to Group Distributionally Robust Optimization [96.26317627118912]
  Group distributionally robust optimization (GDRO) Online learning techniques to reduce the number of samples required in each round from $m$ to $1$, keeping the same sample. A novel formulation of weighted GDRO, which allows us to derive distribution-dependent convergence rates.
  arXiv  Detail & Related papers  (2023-02-18T09:24:15Z)
• On-Demand Sampling: Learning Optimally from Multiple Distributions [63.20009081099896]
  Social and real-world considerations have given rise to multi-distribution learning paradigms. We establish the optimal sample complexity of these learning paradigms and give algorithms that meet this sample complexity. Our algorithm design and analysis are enabled by our extensions of online learning techniques for solving zero-sum games.
  arXiv  Detail & Related papers  (2022-10-22T19:07:26Z)
• Joint Probability Estimation Using Tensor Decomposition and Dictionaries [3.4720326275851994]
  We study non-parametric estimation of joint probabilities of a given set of discrete and continuous random variables from their (empirically estimated) 2D marginals. We create a dictionary of various families of distributions by inspecting the data, and use it to approximate each decomposed factor of the product in the mixture.
  arXiv  Detail & Related papers  (2022-03-03T11:55:51Z)
• Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions [79.35722941720734]
  Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks. We prove exacts characterising the estimator in high-dimensions via empirical risk minimisation. We discuss how our theory can be applied beyond the scope of synthetic data.
  arXiv  Detail & Related papers  (2021-06-07T16:53:56Z)
• Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models [39.27013036481509]
  We introduce the nonparametric maximum likelihood (NPMLE) model for general Gaussian mixtures. We show that with high probability the NPMLE based on a sample size has $O(log n)$ atoms (mass points) Notably, any mixture is statistically in from a finite one with $Olog selection.
  arXiv  Detail & Related papers  (2020-08-19T03:39:13Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)
• Robustly Learning any Clusterable Mixture of Gaussians [55.41573600814391]
  We study the efficient learnability of high-dimensional Gaussian mixtures in the adversarial-robust setting. We provide an algorithm that learns the components of an $epsilon$-corrupted $k$-mixture within information theoretically near-optimal error proofs of $tildeO(epsilon)$. Our main technical contribution is a new robust identifiability proof clusters from a Gaussian mixture, which can be captured by the constant-degree Sum of Squares proof system.
  arXiv  Detail & Related papers  (2020-05-13T16:44:12Z)
• Outlier-Robust Clustering of Non-Spherical Mixtures [5.863264019032882]
  We give the first outlier-robust efficient algorithm for clustering a mixture of $k$ statistically separated d-dimensional Gaussians (k-GMMs) Our results extend to clustering mixtures of arbitrary affine transforms of the uniform distribution on the $d$-dimensional unit sphere.
  arXiv  Detail & Related papers  (2020-05-06T17:24:27Z)
• The EM Algorithm gives Sample-Optimality for Learning Mixtures of Well-Separated Gaussians [21.780375994511324]
  We prove a new result for the ExpectationMaximization (EM): we show that EM converges locally, under separation $Omega(sqrtlog k)$. Our results do not assume or use prior knowledge of the (potentially different) Gaussian components.
  arXiv  Detail & Related papers  (2020-02-02T05:09:26Z)

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.