# A Spectral Algorithm for Learning Hidden Markov Models Source: [A Spectral Algorithm for Learning Hidden Markov Models]() Collection index: [llms.txt]() Data updated: 2025-10-28T13:56:28.148Z Authors: Daniel Hsu, Sham M. Kakade, Tong Zhang Published: 2008-11-26T20:22:51.000Z ## Abstract Hidden Markov Models (HMMs) are one of the most fundamental and widely used statistical tools for modeling discrete time series. In general, learning HMMs from data is computationally hard (under cryptographic assumptions), and practitioners typically resort to search heuristics which suffer from the usual local optima issues. We prove that under a natural separation condition (bounds on the smallest singular value of the HMM parameters), there is an efficient and provably correct algorithm for learning HMMs. The sample complexity of the algorithm does not explicitly depend on the number of distinct (discrete) observations---it implicitly depends on this quantity through spectral properties of the underlying HMM. This makes the algorithm particularly applicable to settings with a large number of observations, such as those in natural language processing where the space of observation is sometimes the words in a language. The algorithm is also simple, employing only a singular value decomposition and matrix multiplications. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Next-step prediction on Synthetic HMM Undercomplete case, N=4000](): Next step prediction; Synthetic HMM Undercomplete case, N=4000; Validation Loss: 1.624. ## Original sources and implementation [Original paper]() [PDF]()