Algebraic and Analytic Approaches for Parameter Learning in Mixture Models

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.Comment: 22 pages, Accepted at Algorithmic Learning Theory (ALT) 202

Recovery of Sparse Signals from a Mixture of Linear Samples

Mixture of linear regressions is a popular learning theoretic model that is used widely to represent heterogeneous data. In the simplest form, this model assumes that the labels are generated from either of two different linear models and mixed together. Recent works of Yin et al. and Krishnamurthy et al., 2019, focus on an experimental design setting of model recovery for this problem. It is assumed that the features can be designed and queried with to obtain their label. When queried, an oracle randomly selects one of the two different sparse linear models and generates a label accordingly. How many such oracle queries are needed to recover both of the models simultaneously? This question can also be thought of as a generalization of the well-known compressed sensing problem (Cand\`es and Tao, 2005, Donoho, 2006). In this work, we address this query complexity problem and provide efficient algorithms that improves on the previously best known results.Comment: International Conference on Machine Learning (ICML), 2020. (26 pages, 3 figures