5 research outputs found
The cutting plane method is polynomial for perfect matchings
The cutting plane approach to finding minimum-cost perfect matchings has been discussed by several authors over past decades. Its convergence has been an open question. We develop a cutting plane algorithm that converges in polynomial-time using only Edmonds’ blossom inequalities, and which maintains half-integral intermediate LP solutions supported by a disjoint union of odd cycles and edges. Our main insight is a method to retain only a subset of the previously added cutting planes based on their dual values. This allows us to quickly find violated blossom inequalities and argue convergence by tracking the number of odd cycles in the support of intermediate solution
The Cutting Plane Method is Polynomial for Perfect Matchings
The cutting plane approach to optimal matchings has been discussed by several
authors over the past decades (e.g., Padberg and Rao '82, Grotschel and Holland
'85, Lovasz and Plummer '86, Trick '87, Fischetti and Lodi '07) and its
convergence has been an open question. We give a cutting plane algorithm that
converges in polynomial-time using only Edmonds' blossom inequalities; it
maintains half-integral intermediate LP solutions supported by a disjoint union
of odd cycles and edges. Our main insight is a method to retain only a subset
of the previously added cutting planes based on their dual values. This allows
us to quickly find violated blossom inequalities and argue convergence by
tracking the number of odd cycles in the support of intermediate solutions
Belief Propagation for Linear Programming
Belief Propagation (BP) is a popular, distributed heuristic for performing
MAP computations in Graphical Models. BP can be interpreted, from a variational
perspective, as minimizing the Bethe Free Energy (BFE). BP can also be used to
solve a special class of Linear Programming (LP) problems. For this class of
problems, MAP inference can be stated as an integer LP with an LP relaxation
that coincides with minimization of the BFE at ``zero temperature". We
generalize these prior results and establish a tight characterization of the LP
problems that can be formulated as an equivalent LP relaxation of MAP
inference. Moreover, we suggest an efficient, iterative annealing BP algorithm
for solving this broader class of LP problems. We demonstrate the algorithm's
performance on a set of weighted matching problems by using it as a cutting
plane method to solve a sequence of LPs tightened by adding ``blossom''
inequalities.Comment: To appear in ISIT 201
Metric and Representation Learning
All data has some inherent mathematical structure. I am interested in understanding the intrinsic geometric and probabilistic structure of data to design effective algorithms and tools that can be applied to machine learning and across all branches of science.
The focus of this thesis is to increase the effectiveness of machine learning techniques by developing a mathematical and algorithmic framework using which, given any type of data, we can learn an optimal representation. Representation learning is done for many reasons. It could be done to fix the corruption given corrupted data or to learn a low dimensional or simpler representation, given high dimensional data or a very complex representation of the data. It could also be that the current representation of the data does not capture the important geometric features of the data.
One of the many challenges in representation learning is determining ways to judge the quality of the representation learned. In many cases, the consensus is that if d is the natural metric on the representation, then this metric should provide meaningful information about the data. Many examples of this can be seen in areas such as metric learning, manifold learning, and graph embedding. However, most algorithms that solve these problems learn a representation in a metric space first and then extract a metric.
A large part of my research is exploring what happens if the order is switched, that is, learn the appropriate metric first and the embedding later. The philosophy behind this approach is that understanding the inherent geometry of the data is the most crucial part of representation learning. Often, studying the properties of the appropriate metric on the input data sets indicates the type of space, we should be seeking for the representation. Hence giving us more robust representations. Optimizing for the appropriate metric can also help overcome issues such as missing and noisy data. My projects fall into three different areas of representation learning.
1) Geometric and probabilistic analysis of representation learning methods.
2) Developing methods to learn optimal metrics on large datasets.
3) Applications.
For the category of geometric and probabilistic analysis of representation learning methods, we have three projects. First, designing optimal training data for denoising autoencoders. Second, formulating a new optimal transport problem and understanding the geometric structure. Third, analyzing the robustness to perturbations of the solutions obtained from the classical multidimensional scaling algorithm versus that of the true solutions to the multidimensional scaling problem.
For learning optimal metric, we are given a dissimilarity matrix , some function and some a subset of the space of all metrics and we want to find that minimizes . In this thesis, we consider the version of the problem when is the space of metrics defined on a fixed graph. That is, given a graph , we let , be the space of all metrics defined via . For this , we consider the sparse objective function as well as convex objective functions. We also looked at the problem where we want to learn a tree. We also show how the ideas behind learning the optimal metric can be applied to dimensionality reduction in the presence of missing data.
Finally, we look at an application to real world data. Specifically trying to reconstruct ancient Greek text.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169738/1/rsonthal_1.pd