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41
Efficient Distributed Topic Modeling with Provable Guarantees
"... Topic modeling for largescale distributed webcollections requires distributed techniques that account for both computational and communication costs. We consider topic modeling under the separability assumption and develop novel computationally efficient methods that provably achieve the statisti ..."
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Topic modeling for largescale distributed webcollections requires distributed techniques that account for both computational and communication costs. We consider topic modeling under the separability assumption and develop novel computationally efficient methods that provably achieve the statistical performance of the stateoftheart centralized approaches while requiring insignificant communication between the distributed document collections. We achieve tradeoffs between communication and computation without actually transmitting the documents. Our scheme is based on exploiting the geometry of normalized wordword cooccurrence matrix and viewing each row of this matrix as a vector in a highdimensional space. We relate the solid angle subtended by extreme points of the convex hull of these vectors to topic identities and construct distributed schemes to identify topics. 1
Necessary and sufficient conditions for novel word detection in separable topic models
 In Advances in on Neural Information Processing Systems (NIPS), Workshop on Topic Models: Computation, Application, Lake Tahoe
, 2013
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A Vavasis, “Semidefinite programming based preconditioning for more robust nearseparable nonnegative matrix factorization,” arXiv preprint arXiv:1310.2273
, 2013
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Ellipsoidal Rounding for Nonnegative Matrix Factorization Under Noisy Separability
, 2013
"... We present a numerical algorithm for nonnegative matrix factorization (NMF) problems under noisy separability. An NMF problem under separability can be stated as one of finding all vertices of the convex hull of data points. The research interest of this paper is to find the vectors as close to the ..."
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We present a numerical algorithm for nonnegative matrix factorization (NMF) problems under noisy separability. An NMF problem under separability can be stated as one of finding all vertices of the convex hull of data points. The research interest of this paper is to find the vectors as close to the vertices as possible in a situation in which noise is added to the data points. Our algorithm is designed to capture the shape of the convex hull of data points by using its enclosing ellipsoid. We show that the algorithm has correctness and robustness properties from theoretical and practical perspectives; correctness here means that if the data points do not contain any noise, the algorithm can find the vertices of their convex hull; robustness means that if the data points contain noise, the algorithm can find the nearvertices. Finally, we apply the algorithm to document clustering, and report the experimental results.
Random projections for nonnegative matrix factorization. arXiv preprint arXiv:1405.4275
, 2014
"... Nonnegative matrix factorization (NMF) is a widely used tool for exploratory data analysis in many disciplines. In this paper, we describe an approach to NMF based on random projections and give a geometric analysis of a prototypical algorithm. Our main result shows the protoalgorithm requires κ̄k ..."
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Nonnegative matrix factorization (NMF) is a widely used tool for exploratory data analysis in many disciplines. In this paper, we describe an approach to NMF based on random projections and give a geometric analysis of a prototypical algorithm. Our main result shows the protoalgorithm requires κ̄k log k optimizations to find all the extreme columns of the matrix, where k is the number of extreme columns, and κ ̄ is a geometric condition number. We show empirically that the protoalgorithm is robust to noise and wellsuited to modern distributed computing architectures.
Noisy Matrix Completion under Sparse Factor Models
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2014
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Provable Algorithms for Machine Learning Problems
, 2013
"... Modern machine learning algorithms can extract useful information from text, images and videos. All these applications involve solving NPhard problems in average case using heuristics. What properties of the input allow it to be solved efficiently? Theoretically analyzing the heuristics is often v ..."
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Modern machine learning algorithms can extract useful information from text, images and videos. All these applications involve solving NPhard problems in average case using heuristics. What properties of the input allow it to be solved efficiently? Theoretically analyzing the heuristics is often very challenging. Few results were known. This thesis takes a different approach: we identify natural properties of the input, then design new algorithms that provably works assuming the input has these properties. We are able to give new, provable and sometimes practical algorithms for learning tasks related to text corpus, images and social networks. The first part of the thesis presents new algorithms for learning thematic structure in documents. We show under a reasonable assumption, it is possible to provably learn many topic models, including the famous Latent Dirichlet Allocation. Our algorithm is the first provable algorithms for topic modeling. An implementation runs 50 times faster than latest MCMC implementation and produces comparable results. The second part of the thesis provides ideas for provably learning deep, sparse representations. We start with sparse linear representations, and give the first algorithm for dictionary learning problem with provable guarantees. Then we apply similar ideas to deep learning: under reasonable assumptions our algorithms can learn a deep network built by denoising autoencoders. The final part of the thesis develops a framework for learning latent variable models. We demonstrate how various latent variable models can be reduced to orthogonal tensor decomposition, and then be solved using tensor power method. We give a tight perturbation analysis for tensor power method, which reduces the number of samples required to learn many latent variable models. In theory, the assumptions in this thesis help us understand why intractable problems in machine learning can often be solved; in practice, the results suggest inherently new approaches for machine learning. We hope the assumptions and algorithms inspire new research problems and learning algorithms. iii
Scalable Methods for Nonnegative Matrix Factorizations of Nearseparable Tallandskinny Matrices
"... Numerous algorithms are used for nonnegative matrix factorization under the assumption that the matrix is nearly separable. In this paper, we show how to make these algorithms scalable for data matrices that have many more rows than columns, socalled “tallandskinny matrices. ” One key component ..."
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Numerous algorithms are used for nonnegative matrix factorization under the assumption that the matrix is nearly separable. In this paper, we show how to make these algorithms scalable for data matrices that have many more rows than columns, socalled “tallandskinny matrices. ” One key component to these improved methods is an orthogonal matrix transformation that preserves the separability of the NMF problem. Our final methods need to read the data matrix only once and are suitable for streaming, multicore, and MapReduce architectures. We demonstrate the efficacy of these algorithms on terabytesized matrices from scientific computing and bioinformatics. 1 Nonnegative matrix factorizations at scale A nonnegative matrix factorization (NMF) for an m × n matrix X with realvalued, nonnegative entries is X = WH (1) where W is m × r, H is r × n, r < min(m, n), and both factors have nonnegative entries. While
Nonnegative matrix factorization under heavy noise.
 In Proceedings of the 33nd International Conference on Machine Learning,
, 2016
"... Abstract The Noisy Nonnegative Matrix factorization (NMF) is: given a data matrix A (d × n), find nonnegative matrices B, C (d × k, k × n respy.) so that A = BC + N , where N is a noise matrix. Existing polynomial time algorithms with proven error guarantees require each column N ·,j to have l 1 ..."
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Abstract The Noisy Nonnegative Matrix factorization (NMF) is: given a data matrix A (d × n), find nonnegative matrices B, C (d × k, k × n respy.) so that A = BC + N , where N is a noise matrix. Existing polynomial time algorithms with proven error guarantees require each column N ·,j to have l 1 norm much smaller than (BC) ·,j  1 , which could be very restrictive. In important applications of NMF such as Topic Modeling as well as theoretical noise models (eg. Gaussian with high σ), almost every column of N ·j violates this condition. We introduce the heavy noise model which only requires the average noise over large subsets of columns to be small. We initiate a study of Noisy NMF under the heavy noise model. We show that our noise model subsumes noise models of theoretical and practical interest (for eg. Gaussian noise of maximum possible σ). We then devise an algorithm TSVDNMF which under certain assumptions on B, C, solves the problem under heavy noise. Our error guarantees match those of previous algorithms. Our running time of O((n + d) 2 k) is substantially better than the O(n 3 d) for the previous best. Our assumption on B is weaker than the "Separability" assumption made by all previous results. We provide empirical justification for our assumptions on C. We also provide the first proof of identifiability (uniqueness of B) for noisy NMF which is not based on separability and does not use hard to check geometric conditions. Our algorithm outperforms earlier polynomial time algorithms both in time and error, particularly in the presence of high noise.