A quadratic programming approach to find faces in robust nonnegative matrix factorization
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Date
2017-08-29
Authors
Ananthanarayanan, Sai Mali
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
Nonnegative matrix factorization (NMF) is a popular dimensionality reduction technique
because it is easily interpretable and can discern useful features. For a given matrix
M (dimension n x m) whose entries are nonnegative and an integer r smaller than both n and m,
NMF is the problem of finding nonnegative matrices A (dimension n x r) and W (dimension r x m) such that
M = AW. The matrix M could be noisy, in which case one seeks a robust algorithm
that solves M ≈ AW. The nonnegativity constraint in NMF has wide applications in
data science problems like document clustering, facial feature extraction,
hyperspectral unmixing etc.
Geometrically, the rows of M can be viewed as a set of points in m-dimensional space. If we think of
the rows of W as the vertices of an (unknown) W-simplex, then the data points lie in this
W-simplex. Therefore, NMF asks us to deduce the vertices of the simplex given the data
points.
NMF is a computationally hard problem though certain assumptions like separability
lead to polynomial time algorithms. This assumes that all the vertices of the
unknown simplex are already present as data points. In practice, this is not true in many
settings. Ge and Zou (2015) assumed subset separability which uses higher dimensional
structures and gave a polynomial time algorithm to find the NMF robustly. In this thesis,
we effectively replace one of their key algorithms that finds faces. We show a quadratic
programming based approach which is efficient and can be employed in practice. Under
bounded noise, our algorithm finds the faces of the simplex which contain enough data
points, thus helping in finding the NMF.
Description
Keywords
Continuous Optimization, Data Science