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- Anima Anandkumar, Rong Ge, Daniel J. Hsu, Sham M. Kakade, Matus Telgarsky
- Journal of Machine Learning Research
- 2014

This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models—including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation—which exploits a certain tensor structure in their low-order observable moments (typically, of second-and third-order). Specifically,… (More)

- Matus Telgarsky
- COLT
- 2016

For any positive integer k, there exist neural networks with Θ(k 3) layers, Θ(1) nodes per layer, and Θ(1) distinct parameters which can not be approximated by networks with O(k) layers unless they are exponentially large — they must possess Ω(2 k) nodes. This result is proved here for a class of nodes termed semi-algebraic gates which includes the common… (More)

- Matus Telgarsky
- ArXiv
- 2015

This note provides a family of classification problems, indexed by a positive integer k, where all shallow networks with fewer than exponentially (in k) many nodes exhibit error at least 1/6, whereas a deep network with 2 nodes in each of 2k layers achieves zero error, as does a recurrent network with 3 distinct nodes iterated k times. The proof is… (More)

- Matus Telgarsky, Sanjoy Dasgupta
- ICML
- 2012

This manuscript develops the theory of ag-glomerative clustering with Bregman divergences. Geometric smoothing techniques are developed to deal with degenerate clusters. To allow for cluster models based on exponential families with overcomplete representations , Bregman divergences are developed for nondifferentiable convex functions.

- Matus Telgarsky
- Journal of Machine Learning Research
- 2012

Boosting combines weak learners into a predictor with low empirical risk. Its dual constructs a high entropy distribution upon which weak learners and training labels are uncorrelated. This manuscript studies this primal-dual relationship under a broad family of losses, including the exponential loss of AdaBoost and the logistic loss, revealing: • Weak… (More)

- Matus Telgarsky
- ICML
- 2013

This manuscript shows that AdaBoost and its immediate variants can produce approximate maximum margin classifiers simply by scaling step size choices with a fixed small constant. In this way, when the unscaled step size is an optimal choice, these results provide guarantees for Friedman's empirically successful " shrinkage " procedure for gradient boosting… (More)

- Matus Telgarsky, Sanjoy Dasgupta
- NIPS
- 2013

Suppose k centers are fit to m points by heuristically minimizing the k-means cost; what is the corresponding fit over the source distribution? This question is resolved here for distributions with p ≥ 4 bounded moments; in particular, the difference between the sample cost and distribution cost decays with m and p as m min{−1/4,−1/2+2/p}. The essential… (More)

- Maxim Raginsky, Alexander Rakhlin, Matus Telgarsky
- COLT
- 2017

Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of Stochastic Gradient Descent, where properly scaled isotropic Gaussian noise is added to an unbiased estimate of the gradient at each iteration. This modest change allows SGLD to escape local minima and suffices to guarantee asymptotic convergence to global minimizers for sufficiently… (More)

Can we effectively learn a nonlinear representation in time comparable to linear learning? We describe a new algorithm that explicitly and adaptively expands higher-order interaction features over base linear representations. The algorithm is designed for extreme computational efficiency, and an extensive experimental study shows that its… (More)