# Algebraic and Analytic Approaches for Parameter Learning in Mixture Models

arXiv: https://arxiv.org/abs/2001.06776 · HTML: https://mlsanity.com/paper/2001.06776/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jan 2020.

## Built on by (most cited first, 4 of 4)

- [Lower Bounds on the Total Variation Distance Between Mixtures of Two Gaussians](https://mlsanity.com/paper/2109.01064/) (2021)
- [Support Recovery in Mixture Models with Sparse Parameters](https://mlsanity.com/paper/2202.11940/) (2022)
- [Differentially Private Learning of Hawkes Processes](https://mlsanity.com/paper/2207.13741/) (2022)
- [Computationally efficient reductions between some statistical models](https://mlsanity.com/paper/2402.07717/) (2024)

## Abstract (arXiv)

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(N1/3)) 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 ε, O(k2(N/ε)^8/√ε) samples suffice to exactly recover the parameters. For some of these distributions, our results represent the first guarantees for parameter estimation.