Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Estimating a vector from noisy quadratic observations is a task that arises naturally in many contexts, from dimensionality reduction, to synchronization and phase retrieval problems. It is often the case that additional information is available about the unknown vector (for instance, sparsity, sign or magnitude of its entries). Many authors propose non-convex quadratic optimization problems that aim at exploiting optimally this information. However, solving these problems is typically NP-hard. We consider a simple model for noisy quadratic observation of an unknown vector $\bvz$.

We study a deep learning inspired formulation for the blind demodulation problem, which is the task of recovering two unknown vectors from their entrywise multiplication.

Recovery of support of a sparse vector from simple measurements is a widely studied problem, considered under the frameworks of compressed sensing, 1-bit compressed sensing, and more general single index models. We consider generalizations of this problem: mixtures of linear regressions, and mixtures of linear classifiers, where the goal is to recover supports of multiple sparse vectors using only a small number of possibly noisy linear, and 1-bit measurements respectively. The key challenge is that the measurements from different vectors are randomly mixed. Both of these problems have also received attention recently. In mixtures of linear classifiers, an observation corresponds to the side of the queried hyperplane a random unknown vector lies in; whereas in mixtures of linear regressions we observe the projection of a random unknown vector on the queried hyperplane. The primary step in recovering the unknown vectors from the mixture is to first identify the support of all the individual component vectors. In this work, we study the number of measurements sufficient for recovering the supports of all the component vectors in a mixture in both these models. We provide algorithms that use a number of measurements polynomial in $k, \log n$ and quasi-polynomial in $\ell$, to recover the support of all the $\ell$ unknown vectors in the mixture with high probability when each individual component is a $k$-sparse $n$-dimensional vector.

Learning mixtures of linear regressions is a natural generalization of the basic linear regression problem.

Estimating a vector from noisy quadratic observations is a task that arises naturally in many contexts, from dimensionality reduction, to synchronization and phase retrieval problems. It is often the case that additional information is available about the unknown vector (for instance, sparsity, sign or magnitude of its entries). Many authors propose non-convex quadratic optimization problems that aim at exploiting optimally this information. However, solving these problems is typically NP-hard. We consider a simple model for noisy quadratic observation of an unknown vector $\bvz$.