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Provable Algorithms for Machine Learning Problems

Report ID:
October 2013
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Modern machine learning algorithms can extract useful information from text, images and
videos. All these applications involve solving NP-hard problems in average case using heuristics.
What properties of the input allow it to be solved efficiently? Theoretically analyzing
the heuristics is 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
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 sample
complexity analysis for tensor power method, which reduces the number of sample required
for learning 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.

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