1,878 research outputs found
On-stack replacement, distilled
On-stack replacement (OSR) is essential technology for adaptive optimization, allowing changes to code actively executing in a managed runtime. The engineering aspects of OSR are well-known among VM architects, with several implementations available to date. However, OSR is yet to be explored as a general means to transfer execution between related program versions, which can pave the road to unprecedented applications that stretch beyond VMs. We aim at filling this gap with a constructive and provably correct OSR framework, allowing a class of general-purpose transformation functions to yield a special-purpose replacement. We describe and evaluate an implementation of our technique in LLVM. As a novel application of OSR, we present a feasibility study on debugging of optimized code, showing how our techniques can be used to fix variables holding incorrect values at breakpoints due to optimizations
Computing a Nonnegative Matrix Factorization -- Provably
In the Nonnegative Matrix Factorization (NMF) problem we are given an nonnegative matrix and an integer . Our goal is to express
as where and are nonnegative matrices of size
and respectively. In some applications, it makes sense to ask
instead for the product to approximate -- i.e. (approximately)
minimize \norm{M - AW}_F where \norm{}_F denotes the Frobenius norm; we
refer to this as Approximate NMF. This problem has a rich history spanning
quantum mechanics, probability theory, data analysis, polyhedral combinatorics,
communication complexity, demography, chemometrics, etc. In the past decade NMF
has become enormously popular in machine learning, where and are
computed using a variety of local search heuristics. Vavasis proved that this
problem is NP-complete. We initiate a study of when this problem is solvable in
polynomial time:
1. We give a polynomial-time algorithm for exact and approximate NMF for
every constant . Indeed NMF is most interesting in applications precisely
when is small.
2. We complement this with a hardness result, that if exact NMF can be solved
in time , 3-SAT has a sub-exponential time algorithm. This rules
out substantial improvements to the above algorithm.
3. We give an algorithm that runs in time polynomial in , and
under the separablity condition identified by Donoho and Stodden in 2003. The
algorithm may be practical since it is simple and noise tolerant (under benign
assumptions). Separability is believed to hold in many practical settings.
To the best of our knowledge, this last result is the first example of a
polynomial-time algorithm that provably works under a non-trivial condition on
the input and we believe that this will be an interesting and important
direction for future work.Comment: 29 pages, 3 figure
Robustness Analysis of Hottopixx, a Linear Programming Model for Factoring Nonnegative Matrices
Although nonnegative matrix factorization (NMF) is NP-hard in general, it has
been shown very recently that it is tractable under the assumption that the
input nonnegative data matrix is close to being separable (separability
requires that all columns of the input matrix belongs to the cone spanned by a
small subset of these columns). Since then, several algorithms have been
designed to handle this subclass of NMF problems. In particular, Bittorf,
Recht, R\'e and Tropp (`Factoring nonnegative matrices with linear programs',
NIPS 2012) proposed a linear programming model, referred to as Hottopixx. In
this paper, we provide a new and more general robustness analysis of their
method. In particular, we design a provably more robust variant using a
post-processing strategy which allows us to deal with duplicates and near
duplicates in the dataset.Comment: 23 pages; new numerical results; Comparison with Arora et al.;
Accepted in SIAM J. Mat. Anal. App
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