1,978 research outputs found
Algorithms and Hardness for Robust Subspace Recovery
We consider a fundamental problem in unsupervised learning called
\emph{subspace recovery}: given a collection of points in ,
if many but not necessarily all of these points are contained in a
-dimensional subspace can we find it? The points contained in are
called {\em inliers} and the remaining points are {\em outliers}. This problem
has received considerable attention in computer science and in statistics. Yet
efficient algorithms from computer science are not robust to {\em adversarial}
outliers, and the estimators from robust statistics are hard to compute in high
dimensions.
Are there algorithms for subspace recovery that are both robust to outliers
and efficient? We give an algorithm that finds when it contains more than a
fraction of the points. Hence, for say this estimator
is both easy to compute and well-behaved when there are a constant fraction of
outliers. We prove that it is Small Set Expansion hard to find when the
fraction of errors is any larger, thus giving evidence that our estimator is an
{\em optimal} compromise between efficiency and robustness.
As it turns out, this basic problem has a surprising number of connections to
other areas including small set expansion, matroid theory and functional
analysis that we make use of here.Comment: Appeared in Proceedings of COLT 201
A Polynomial Time Algorithm for Lossy Population Recovery
We give a polynomial time algorithm for the lossy population recovery
problem. In this problem, the goal is to approximately learn an unknown
distribution on binary strings of length from lossy samples: for some
parameter each coordinate of the sample is preserved with probability
and otherwise is replaced by a `?'. The running time and number of
samples needed for our algorithm is polynomial in and for
each fixed . This improves on algorithm of Wigderson and Yehudayoff that
runs in quasi-polynomial time for any and the polynomial time
algorithm of Dvir et al which was shown to work for by
Batman et al. In fact, our algorithm also works in the more general framework
of Batman et al. in which there is no a priori bound on the size of the support
of the distribution. The algorithm we analyze is implicit in previous work; our
main contribution is to analyze the algorithm by showing (via linear
programming duality and connections to complex analysis) that a certain matrix
associated with the problem has a robust local inverse even though its
condition number is exponentially small. A corollary of our result is the first
polynomial time algorithm for learning DNFs in the restriction access model of
Dvir et al
Learning Topic Models - Going beyond SVD
Topic Modeling is an approach used for automatic comprehension and
classification of data in a variety of settings, and perhaps the canonical
application is in uncovering thematic structure in a corpus of documents. A
number of foundational works both in machine learning and in theory have
suggested a probabilistic model for documents, whereby documents arise as a
convex combination of (i.e. distribution on) a small number of topic vectors,
each topic vector being a distribution on words (i.e. a vector of
word-frequencies). Similar models have since been used in a variety of
application areas; the Latent Dirichlet Allocation or LDA model of Blei et al.
is especially popular.
Theoretical studies of topic modeling focus on learning the model's
parameters assuming the data is actually generated from it. Existing approaches
for the most part rely on Singular Value Decomposition(SVD), and consequently
have one of two limitations: these works need to either assume that each
document contains only one topic, or else can only recover the span of the
topic vectors instead of the topic vectors themselves.
This paper formally justifies Nonnegative Matrix Factorization(NMF) as a main
tool in this context, which is an analog of SVD where all vectors are
nonnegative. Using this tool we give the first polynomial-time algorithm for
learning topic models without the above two limitations. The algorithm uses a
fairly mild assumption about the underlying topic matrix called separability,
which is usually found to hold in real-life data. A compelling feature of our
algorithm is that it generalizes to models that incorporate topic-topic
correlations, such as the Correlated Topic Model and the Pachinko Allocation
Model.
We hope that this paper will motivate further theoretical results that use
NMF as a replacement for SVD - just as NMF has come to replace SVD in many
applications
A performance index approach to aerodynamic design with the use of analysis codes only
A method is described for designing an aerodynamic configuration for a specified performance vector, based on results from several similar, but not identical, trial configurations, each defined by a geometry parameter vector. The theory shows the method effective provided that: (1) the results for the trial configuration provide sufficient variation so that a linear combination of them approximates the specified performance; and (2) the difference between the performance vectors (including the specifed performance) are sufficiently small that the linearity assumption of sensitivity analysis applies to the differences. A computed example describes the design of a high supersonic Mach number missile wing body configuration based on results from a set of four trial configurations
On minimizing the number of calculations in design-by-analysis codes
A method is presented for aerodynamic design for a specified pressure distribution, using analysis codes only. The method requires a very conservative number of analysis runs, and therefore is appropriate when the analysis code is a large code in terms of storage and/or running time. Three model problems illustrate some capabilities and limitations of the method
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