The simplicial condition and other stronger conditions that imply it have recently played a central role in developing polynomial time algorithms with provable asymptotic consistency and sample complexity guarantees for topic estimation in separable topic models. Of these algorithms, those that rely solely on the simplicial condition are impractical while the practical ones need stronger conditions. In this paper, we demonstrate, for the first time, that the simplicial condition is a fundamental, algorithm-independent, information-theoretic necessary condition for consistent separable topic estimation. Furthermore, under solely the simplicial condition, we present a practical quadratic-complexity algorithm based on random projections which consistently detects all novel words of all topics using only up to second-order empirical word moments. This algorithm is amenable to distributed implementation making it attractive for big-data scenarios involving a network of large distributed databases.
