This work deals with the problem of estimating the parameters (i.e., pole multiplicity structure, pole locations, and complex amplitudes) of exponential polynomials for a given model order. We start from an exponential polynomial signal model and algebraically demonstrate the reasoning behind and the manner in which this model generalizes to the broader class of signals containing multiple poles. The poles and complex amplitudes are estimated by a subspace-based method. With the given model order as prior, a heuristic algorithm is developed to estimate the multiplicity structure of poles. Numerical simulations are conducted to study the performance of the subspace-based method and the sensitivity of poles and multiplicity estimation. Our simulations show that our pole estimation method is accurate for signals with poles of multiplicity greater than one, particularly for signals with lower orders. This allows for successful signal regeneration, producing estimates that closely match the original data. The estimation is sensitive, particularly in the presence of noise, and when the poles are complex and have a higher order of multiplicity.

Discovering the latent structure from many observed variables is an important yet challenging learning task. Existing approaches for discovering latent structures often require the unknown number of hidden states as an input. In this paper, we propose a quartet based approach which is agnostic to this number. The key contribution is a novel rank characterization of the tensor associated with the marginal distribution of a quartet. This characterization allows us to design a nuclear norm based test for resolving quartet relations. We then use the quartet test as a subroutine in a divide-and-conquer algorithm for recovering the latent tree structure. Under mild conditions, the algorithm is consistent and its error probability decays exponentially with increasing sample size. We demonstrate that the proposed approach compares favorably to alternatives. In a real world stock dataset, it also discovers meaningful groupings of variables, and produces a model that fits the data better.

We approach the problem of estimating the parameters of a latent tree graphical model from a hierarchical tensor decomposition point of view. In this new view, the marginal probability table of the observed variables in a latent tree is treated as a tensor, and we show that: (i) the latent variables induce low rank structures in various matricizations of the tensor; (ii) this collection of low rank matricizations induce a hierarchical low rank decomposition of the tensor. Exploiting these properties, we derive an optimization problem for estimating the parameters of a latent tree graphical model, i.e., hierarchical decomposion of a tensor which minimizes the Frobenius norm of the difference between the original tensor and its decomposition. When the latent tree graphical models are correctly specified, we show that a global optimum of the optimization problem can be obtained via a recursive decomposition algorithm. This algorithm recovers previous spectral algorithms for hidden Markov models (Hsu et al., 2009; Foster et al., 2012) and latent tree graphical models (Parikh et al., 2011; Song et al., 2011) as special cases, elucidating the global objective these algorithms are optimizing for. When the latent tree graphical models are misspecified, we derive a better decomposition based on our framework, and provide approximation guarantee for this new estimator. In both synthetic and real world data, this new estimator significantly improves over the-state-of-the-art.

Matrix lower rank approximations such as non-negative matrix factorization (NMF) have been successfully used to solve many data mining tasks. In this paper, we propose a new matrix lower rank approximation called Bounded Matrix Low Rank Approximation (BMA) which imposes a lower and an upper bound on every element of a lower rank matrix that best approximates a given matrix with missing elements. This new approximation models many real world problems, such as recommender systems, and performs better than other methods, such as singular value decompositions (SVD) or NMF. We present an efficient algorithm to solve BMA based on coordinate descent method. BMA is different from NMF as it imposes bounds on the approximation itself rather than on each of the low rank factors. We show that our algorithm is scalable for large matrices with missing elements on multi core systems with low memory. We present substantial experimental results illustrating that the proposed method outperforms the state of the art algorithms for recommender systems such as Stochastic Gradient Descent, Alternating Least Squares with regularization, SVD++, Bias-SVD on real world data sets such as Jester, Movie lens, Book crossing, Online dating and Netflix.

Newton's method for solving the matrix equation F(X) identical to AX-XX(T) AX = 0 runs up against the fact that its zeros are not isolated. This is due to a symmetry of F by the action of the orthogonal group. We show how differential-geometric techniques can be exploited to remove this symmetry and obtain a "geometric" Newton algorithm that finds the zeros of F. The geometric Newton method does not suffer from the degeneracy issue that stands in the way of the original Newton method.